Divisão de Serviços Técnicos

Transcription

UNIVERSIDADE FEDERAL DO RIO GRANDE DO NORTE
CENTRO DE CIÊNCIAS EXATAS E DA TERRA
INSTITUTO DE QUÍMICA
PROGRAMA DE PÓS-GRADUAÇÃO EM QUÍMICA
Novos protocolos teóricos utilizados nos estudos das ligações
hidrogênio-hidrogênio, na estabilidade do tetraedrano, seus derivados
e das reações de diels-alder e na quantificação da afinidade de fármacos
Norberto de Kássio Vieira Monteiro
____________________________________________
Tese de Doutorado
Natal/RN, outubro de 2015
Norberto de Kássio Vieira Monteiro
NOVOS PROTOCOLOS TEÓRICOS UTILIZADOS NOS ESTUDOS DAS
LIGAÇÕES HIDROGÊNIO-HIDROGÊNIO, NA ESTABILIDADE DO
TETRAEDRANO, SEUS DERIVADOS E DAS REAÇÕES DE DIELS-ALDER E NA
QUANTIFICAÇÃO DA AFINIDADE DE FÁRMACOS.
Orientador: Caio Lima Firme
NATAL/RN
2015
Divisão de Serviços Técnicos
Catalogação da Publicação na Fonte. UFRN
Biblioteca Setorial do Instituto de Química
Monteiro, Norberto de Kássio Vieira.
Novos protocolos teóricos utilizados nos estudos das ligações hidrogênio-hidrogênio,
na estabilidade do tetraedrano, seus derivados e das reações de diels-alder e na
quantificação da afinidade de fármacos / Norberto de Kássio Vieira Monteiro. – Natal,
RN, 2015.
167 f. : il.
Orientador: Caio Lima Firme.
Tese (Doutorado em Química) - Universidade Federal do Rio Grande do Norte. Centro
de Ciências Exatas e da Terra. Programa de Pós-Graduação em Química.
1. Diels-Alder – Tese. 2. Ligação H-H – Tese. 3. Tetraedrano – Tese. 4. CYP17 – Tese
I. Firme, Caio Lima. II. Universidade Federal do Rio Grande do Norte. III. Título.
RN/UFRN/BSE- Instituto de Química
CDU 544.142.4:615 (043.3)
AGRADECIMENTOS
Às minhas três mães: Marilú, Vitória e Dona Francisca. Sem elas, nem sei o que seria
de mim.
Ao meu amor, Richele, a qual tenho o privilégio de chamar de esposa, pela ajuda e por
me aguentar nos momentos de ansiedade e estresse nos anos que me dediquei ao
doutorado.
Ao meu gordinho! Filhão, você foi a melhor coisa que já aconteceu na minha vida!
Papai te ama!
À minha querida afilhada Shara Maria, a qual considero como uma filha.
Aos meus padrinhos que sempre estiveram comigo durante toda a minha vida.
Aos alunos de iniciação científica: Cidinha, Luís, Franklin, Fabrício, Bruno, Elizeu e
Valéria.
Aos alunos e amigos da pós-graduação: Gutto, Caio Sena, João Batista, Acássio,
Roberta, William, Tamires e Francinaldo.
Aos meus amigos: Elton, Isabelle, Amisson, Clesiane, Aninha e, em especial, Sérgio,
que além de amigo também é meu padrinho de casamento.
Aos meus sogros, seu Laércio e Dona Rita, que me tratam como um filho.
À minha cunhada Raynara por me dar caronas à UFRN quando eu estava sem moto.
Muito especialmente, agradeço ao meu orientador Prof. Caio de Lima Firme, pela
disponibilidade, paciência, dedicação e profissionalismo e, acima de tudo, por ter
acreditado em mim desde o início.
Aos professores Jones de Andrade, Gorette Cavalcante e Luiz Seixas pelas longas e
produtivas conversas.
Aos professores Carlos Alberto Martinez e Sibele Berenice pela competência em dirigir
o Programa de pós-graduação em Química.
Aos professores componentes da banca de qualificação, Dr. Euzébio Guimarães, Dr.
Davi Serradella e Dr. Caio de Lima Firme, pelas correções e aperfeiçoamento desta tese.
Aos professores componentes da banca de defesa de tese, Dr. Euzébio Guimarães, Dr.
Davi Serradella, Dr. Cláudio Costa dos Santos, Dra. Thereza A. Soares e Dr. Caio de
Lima Firme.
“... com o passar dos tempos, você mudou. Deixou de ser você.
Deixou as pessoas apontarem pra sua cara e dizerem que você não
é bom. E quando as coisas ficaram difíceis, você começou a
procurar alguém para culpar... como uma grande sombra. O mundo
não é um mar de rosas. É um lugar ruim e asqueroso... e não
importa o quão durão você é... ele te deixará de joelhos e te
manterá assim se permitir. Nem você, nem eu, nem ninguém baterá
tão forte quanto à vida. Mas isso não se trata de quão forte
você pode bater. Se trata de quão forte pode ser atingido e
continuar
seguindo
em
continuar
seguindo
em
frente.
frente.
Quanto
É
você
assim
pode
que
a
apanhar
e
vitória
é
conquistada!”
Rocky Balboa
“Empty your mind, be formless, shapeless —
like water. If you put water in a cup, it
becomes the cup; You put water into a bottle
it becomes the bottle; You put it in a teapot
it becomes the teapot. Water can flow or it
can crash. Be water, my friend”.
Lee Jun Fan (Bruce Lee).
RESUMO
Esta tese foi realizada em quatro capítulos, todos a nível teórico, enfocados
principalmente na densidade eletrônica. No primeiro capítulo, nos descrevemos a
aplicação de um minicurso para estudo das reações de Diels-Alder na Universidade
Federal do Rio Grande do Norte. Utilizando ferramentas de química computacional, os
estudantes puderam construir um determinado conhecimento e puderam associar
importantes aspectos da físico-química e da química orgânica. No segundo capítulo,
estudamos um novo tipo de interação química envolvendo átomos de hidrogênio de
carga neutra, denominada de ligação hidrogênio-hidrogênio (H-H). Nesse estudo
realizado com alcanos complexados, fornece novas e importantes informações acerca
das suas estabilidades envolvendo esse tipo de interação. Mostramos que a ligação H-H
desempenha um papel secundário na estabilidade de alcanos ramificados em
comparação aos seus isômeros menos ramificados ou lineares. No terceiro capítulo,
estudamos a estrutura eletrônica e a estabilidade do tetraedrano, de tetraedranos
substituídos e seus parentes de silício e germânio. Avaliamos o efeito do substituinte na
gaiola de carbono dos derivados do tetraedrano e os resultados indicam que fortes
grupos retiradores de elétrons (GRE) causam pequena instabilidade na gaiola carbônica,
em contrapartida, fracos GRE resulta em grande instabilidade. Mostramos que na
aromaticidade-σ, GRE e grupos doadores de elétrons (GDE) resultam em um
decréscimo e em aumento, respectivamente, dos índices de aromaticidade NICS e
D3BIA. Em adição, outro fator pode ser utilizado para explicar a estabilidade do tetratert-butiltetraedrano, assim como a ligação H-H. GVB e ADMP também foram
utilizados para avaliar o efeito de substituintes na gaiola do tetraedrano. No quarto
capítulo, nós realizamos uma investigação teórica do efeito inibitório do fármaco
abiraterona (ABE), utilizado no tratamento do câncer de próstata, frente à enzima
CYP17, comparando as energias de interação e densidade eletrônica desse fármaco com
o substrato natural, a pregnenolona (PREG). Dinâmica molecular e docking foram
utilizados para obter os complexos CYP17-ABE e CYP17-PREG. Á partir da dinâmica
molecular, foi obtido que a ABE possui uma maior tendência de difusão água  sítio de
interação da enzima CYP17, quando com a mesma tendência da PREG. Com o método
ONIOM (B3LYP:AMBER), encontramos que a energia de interação da ABE é 21,38
kcal mol-1 mais estável em comparação à PREG. Os resultados obtidos através da
QTAIM indicam que essa estabilidade se dá devido a uma maior densidade eletrônica
das interações entre a ABE e a CYP17.
Palavras-chave: Diels-Alder; Ligação H-H; Tetraedrano; CYP17.
ABSTRACT
This thesis was performed in four chapters, at the theoretical level, focused mainly on
electronic density. In the first chapter, we have applied an undergraduate minicourse of
Diels-Alder reaction in Federal University of Rio Grande do Norte.
By using
computational chemistry tools students could build the knowledge by themselves and
they could associate important aspects of physical-chemistry with Organic Chemistry.
In the second chapter, we studied a new type of chemical bond between a pair of
identical or similar hydrogen atoms that are close to electrical neutrality, known as
hydrogen-hydrogen (H-H) bond. In this study performed with complexed alkanes,
provides new and important information about their stability involving this type of
interaction. We show that the H-H bond playing a secondary role in the stability of
branched alkanes in comparison with linear or less branched isomers. In the third
chapter, we study the electronic structure and the stability of tetrahedrane, substituted
tetrahedranes and silicon and germanium parents, it was evaluated the substituent effect
on the carbon cage in the tetrahedrane derivatives and the results indicate that stronger
electron withdrawing groups (EWG) makes the tetrahedrane cage slightly unstable
while slight EWG causes a greater instability in the tetrahedrane cage. We showed that
the sigma aromaticity EWG and electron donating groups (EDG) results in decrease and
increase, respectively, of NICS and D3BIA aromaticity indices. In addition, another
factor can be utilized to explain the stability of tetra-tert-butyltetrahedrane as well as HH bond. GVB and ADMP were also used to explain the stability effect of the
substituents bonded to the carbon of the tetrahedrane cage. In the fourth chapter, we
performed a theoretical investigation of the inhibitory effect of the drug abiraterone
(ABE), used in the prostate cancer treatment as CYP17 inhibitor, comparing the
interaction energies and electron density of the ABE with the natural substrate,
pregnenolone (PREG). Molecular dynamics and docking were used to obtain the
CYP1ABE and CYP17-PREG complexes. From molecular dynamics was obtained that
the ABE has higher diffusion trend water  CYP17 binding site compared to the
PREG. With the ONIOM (B3LYP:AMBER) method, we find that the interaction
electronic energy of ABE is 21.38 kcal mol-1 more stable than PREG. The results
obtained by QTAIM indicate that such stability is due a higher electronic density of
interactions between ABE and CYP17.
Keywords: Diels-Alder; H-H bond; Tetrahedrane; CYP17.
LISTA DE FIGURAS – REFERÊNCIAL TEÓRICO
Figura 1. Algoritmo mostrado o procedimento autoconsistente para resolução das
equações Kohn-Sham......................................................................................................30
Figura 2. Gráfico molecular do Ferroceno.....................................................................32
Figura 3. Mapa de contorno da distribuição de carga do cloreto de sódio e o gradiente
do campo vetorial ∇ρ.......................................................................................................33
Figura 4. Orbital Hartree-Fock da molécula H2 para (A) R = 0,741 Å e (B) R = 6,35
Å......................................................................................................................................37
Figura 5. Energias da molécula H2 obtidas à partir dos métodos Hartree-Fock, MO, VB
e GVB, comparadas com a energia exata não-relativística.............................................37
Figura 6. Representação do método ONIOM de um sistema proteico dividido em duas
camadas. Na camada baixa (low layer), os átomos são representados por estruturas do
tipo arame (wireframe) e na camada alta (high layer) os átomos são representados por
estruturas do tipo bola e bastão (ball-and-stick)..............................................................38
LISTA DE FIGURAS
Artigo 1 - Teaching thermodynamic, geometric and electronic aspects of Diels-Alder
cycloadditions by using computational chemistry – an undergraduate experiment.
Figure 1. (A) Plot of enthalpy change, ∆H, versus Gibbs free energy change, ∆G, of
Diels-Alder reactions (1)-(7). (B) Plot of ln K versus ∆H of Diels-Alder reactions (1)(7)………………………………………………………………………………………48
Figure 2. HOMO and LUMO of buta-1,3-diene and ethene (A); buta-1,3-dien-2-amine
and nitroethene (B); and 2-nitrobuta-1,3-diene and ethenamine (C)………………...…50
Figure 3. Optimized geometries of corresponding reactants and products from DielsAlder reactions 4 (A) and 7 (B). Selected bond lengths, in Angstroms, are
depicted………………………………………………………………………….…...…51
LISTA DE FIGURAS
Artigo 2 - Hydrogen-hydrogen bonds in highly branched alkanes and in alkane
complexes: a DFT, ab initio, QTAIM and ELF study
Figure 1. Virial graphs of C8H18 isomers. (A) Octane, (B) 3-methylheptane, (C) 3,3
dimethylhexane, (D) 2,2,3-trimethylpentane and (E) 2,2,3,3-tetramethylbutane. The
geometry optimization of the molecules were performed at ωB97xd/6-311++G(d,p)
level of theory…………………………………………………………………………..76
Figure 2. Randomized, non-optimized and optimized geometries of methane-methane,
ethane-ethane, propone-propane and butane-butane complexes depicting their
corresponding selected CH--HC dihedral angles, in degrees. The geometry optimization
of the molecules were performed at G4/CCSD(T)/6-311++G(d,p) level of theory. The
dashed lines represent the selected dihedral angles………………………………….…78
Figure 3. Randomized non-optimized initial geometry and optimized geometry of
methane-methane, ethane-ethane, propone-propane, butane-butane, pentane-pentane and
hexane-hexane complexes depicting their corresponding selected H-H interatomic
distances, in angstroms. The geometry optimization of the molecules were performed at
ωB97XD/6-311++G(d,p) level of theory. The dashed lines represent the selected
interatomic distances………………………………………………………………...…80
Figure 4. Virial graphs of Alkane complexes optimized at G4/CCSD(T)/6-311++G(d,p)
level of theory. Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane
(Pro-Pro) and butane-butane (But-But) along with the charge density, ρ(r), and
Laplacian of the electron density, ∇2 ρ(r) at H-H bond critical, in au. The others
complexes were not optimized due to lack of computational resources……………….81
Figure 5. Virial graphs of Alkane complexes optimized at ωB97XD/6-311++G(d,p)
(Pro-Pro), butane-butane (But-But), pentane-pentane (Pen-Pen) and hexane-hexane
(Hex-Hex) complexes along with the charge density, ρ(r), and Laplacian of the electron
density, ∇2 ρ(r) at H-H bond critical, in au…………………………………………..…82
Figure 6. Plot of number of H-H bond paths versus ΔG (complex-2*isolated), in
kcal/mol, of selected alkanes at ωB97XD/6-311++G(d,p) level of theory…………….83
Figure 7. Plot of number of H-H bond paths, from ωB97XD/6-311++G(d,p) wave
function, versus boiling point, in °C, of selected alkanes…………………………...….84
Figure 8. Plot of energy, in hartrees, of a hydrogen atom belonging to H-H bond (EH(HH))
in ethane-ethane complex versus the energy (E) in ethane-ethane complex of selected
steps in the optimization steps……………………………………………………...…..85
Figure 9. Plot of energy, in hartrees of a hydrogen atom not belonging to H-H bond
(EH) in ethane-ethane complex versus the energy (E) in ethane-ethane complex of
selected steps in the optimization steps………………………………………………...86
Figure S1. Optimized geometries of alkane molecules. The alkane molecules were
optimized at (a) B3LYP/6-311++G(d,p), (b) M062X/6-311++G(d,p), (c) G4 method, (d)
G4/CCSD/6-311++G(d,p) (T) and (e) wB97XD/6-311++G(d,p). (1) Methane, (2)
ethane, (3) propane, (4) butane, (5) pentane and (6) hexane………………………..….95
Figure S2. Virial graphs of Alkane complexes optimized at B3LYP/6-311G++(d,p)
(Hex-Hex) complexes………………………………………………………………..…96
Figure S3. Virial graphs of Alkane complexes optimized at M062X/6-311G++(d,p)
(Hex-Hex) complexes………………………………………………………………..…97
Figure S4. Virial graphs of Alkane complexes optimized at G4 method. Methanemethane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro), butane-butane
(But-But), pentane-pentane (Pen-Pen) and hexane-hexane (Hex-Hex) complexes……98
Figure S5. Plot of energy (E), in Hartree, of ethane-ethane complex versus of
optimization steps………………………………………………………………………99
LISTA DE FIGURAS
Artigo 3 - Stability and Electronic Structure of Substituted Tetrahedranes, Silicon and
Germanium Parents – a DFT, ADMP, QTAIM and GVB study.
Figure 1: Optimized geometries of the substituted tetrahedranes (A-F), silicon and
germanium cages (G-I) with corresponding C 1-C4 , C1-C5 , C1-Si, C1 -N, Si-Si, Si-H,
Ge-Ge, Ge-H and Ge-C bond lengths (in Angstroms); values of selected angles (in
degree)………………………………………………………………………………107
Figure 2: Virial graphs of substituted tetrahedranes (A-F), silicon and germanium cages
(G-I) with the electronic density, ρ(r), Laplacian of the electron density, ∇2 ρ(r), local
energy density, H(r) and relative kinetic energy density, G(r)/ρ at cage-substituent bond
critical, in au. Values of electronic density, ρ(r), Laplacian of the electron density, ∇2
ρ(r), local energy density, H(r), relative kinetic energy density, G(r)/ρ, delocalization
index, DI, bond order, n, and ellipticity, ε, from carbon, silicon or germanium atoms in
the tetrahedrane (or parent) cage for 3a – 3f, 4a, 5a and 5b………………………….108
Figure 3: Gibbs free energy difference, in kcal mol-1, from diazocompound (as a
reference) to cyclobutadiene and tetrahedrane structures, along with nitrogen, with
respect to the syntheses of 3a (A), 3b (B), 3f (C) and 3e (D)…………...……………109
Figure 4: Doubly occupied generalized valence bond (GVB) orbitals of lone pairs of
oxygen of nitro (-NO2) functional group substituent (LP1, LP1’, LP2 and LP2’) (A and
B) and singly occupied GVB orbitals of C-C sigma bond [VB(C-C)1, VB(C-C)2 and
VB(C-C)’] (B)………………………………………………………………………...113
Figure 5: (A) Plot of NICS versus D3BIA for 3a-3e, 4a, 4b and 5a; (B) Plot of NICS
versus D3BIA for 3a-3f, 4a, 4b and 5a……………………………………………….115
Figure 6: Singly occupied generalized valence bond (GVB) orbitals of C-C sigma bond
[VB(C-C)1, VB(C-C)2 and VB(C-C)’] of 3a (A), 3b (B), 3f (C) and cubane (D)…...117
Figure 7: Plot of total energy, in Hartree, versus the time of trajectory, in femtoseconds,
of tetrahedrane (3a) using ADMP method. Some structures are depicted at the
corresponding selected points of the trajectory…………………………………….…118
Figure 8: Plot of total energy, in Hartree, versus the time of trajectory, in femtoseconds,
of 3c using ADMP method. The structure at the end of trajectory is also depicted…..119
Figure 9. Plot of total energy, in Hartree, versus the time of trajectory, in femtoseconds,
of 3e using ADMP method. A couple of structures are depicted at the selected
corresponding points of the trajectory………………………………………………...119
LISTA DE FIGURAS
Artigo 4 - Docking, Molecular Dynamics, ONIOM, QTAIM: theoretical investigation
of inhibitory effect of Abiraterone on CYP17.
Figure 1. Inside the cell, testosterone is converted to DHT and binds to AR. AR-DHT
complex undergoes phosphorylation and is translocated to the nucleus in the form of
dimer. Inside the nucleus, AR-dimer binds to DNA coregulators leading to transcription
activation……………………………………………………………………………...125
Figure 2. Chemical structure of ABE……………………………..………………….126
Figure 3. The protonation states of histidine at pH 7.0. HID (A) HIE (B) and HIP
(C)……………………………………………………………………………………..128
Figure 4. Schematic representations of pockets of the CYP17-ABE (A) and CYP17PREG (B) complexes …………………………………………………………………133
Figure 5. Plots of RMSD (in nm) versus time of simulation (in ns) for CYP17-ABE
complex of each protonation state of His 373. HISD (A), HISE (B) and HISH
(C)………………………………………………………………………………..……135
Figure 6. Plots of RMSD (A) and χ (B) of the His373 of each protonation state. The
data represent the average of five 10-ns snapshots for each protonation state expressed
as mean standard deviation. It was considered a p value less than 0.05………..…….135
Figure 7. The binding poses of PREG in the active site of the enzyme CYP17….….136
Figure 8. Plot of RMSD (in nm) versus time of simulation (in ns) for CYP17-PREG
complex……………………………………………………………………………….136
Figure 9. ABE, heme group and CYS442 of CYP17-ABE complex (A) and PREG,
heme group and CYS442 of CYP17-PREG complex (B)…………………………….137
Figure 10. Optimized structures of the QM (highlighted) and MM (gray lines) layers of
the CYP17-ABE (A) and CYP17-PREG (B) complexes and CYP17 enzyme (C) at the
ONIOM (B3LYP:AMBER) level……………………………………………………..139
Figure 11. Virial graphs of binding pockets of CYP17-ABE (A) and CYP17-PREG (B)
complexes…………………………………………………………………………….139
Figure S1. Optimized geometries of Fe(NH3)2(NH2)2 at B3LYP/6-311G(d,p) level in
the triplet and quintet states …………………………………………………………157
Figure S2. Fitting structures of CYP17-ABE (A) and CYP17-PREG (B) complexes
before (yellow)
and after (red) molecular dynamics simulation. The picture was
rendered using VMD……………………………………………………….…………158
Figure S3. B-factors for the CYP17-ABE (A) and CYP17-PREG (B) complexes. The
picture was rendered using VMD, “blue” indicates high, “white” indicates intermediate
and “red” low values. High temperature factors imply exibility and low ones
rigidity……………………………………………………………………………...…158
LISTA DE TABELAS – REFERÊNCIAL TEÓRICO
Tabela 1. Acrônimos, sinais dos autovalores e denominações dos pontos críticos........32
LISTA DE TABELAS
Artigo 1 - Teaching thermodynamic, geometric and electronic aspects of Diels-Alder
Table 1. Values of enthalpy, H, Gibbs free energy, G, in kcal mol-1, and entropy, S, in
cal mol-1 K-1 of all studied molecules…………………………………………………..48
Table 2. Gibbs free energy change, ∆G, Enthalpy change, ∆H, entropy change and
temperature product, in kcal mol-1 K-1, T∆S, entropy change, in kcal mol-1, ∆S, and
equilibrium constant logarithm, ln K, of Diels-Alder reactions (1)–(7)………………..49
Table 3. Energy values of HOMO (EHOMO) and LUMO (ELUMO) and energy gap
between HOMO and LUMO orbitals, |EHOMO-ELUMO|, in a.u., of the reagents of the
Diels-Alder reactions (1) – (7)……………………………………………….…………50
Table 4. Bond lengths of reagents (r1, r2, r3 and r4) and products (r’1, r’2, r’3 and r’4), in
Angstroms, of the studied Diels-Alder reactions…………………………….…………52
LISTA DE TABELAS
Artigo 2 - Hydrogen-hydrogen bonds in highly branched alkanes and in alkane
complexes: a DFT, ab initio, QTAIM and ELF study
Table 1. Amount of branching, amount of H-H bonds and heats of combustion (ΔHcomb)
in branched alkanes…………………………………………………………………….75
Table 2. Number of H-H bonds, boiling point, in °C, Gibbs energy of alkane complexes
(Gcomplex), two-fold Gibbs energy of the corresponding isolated alkane (2*Gisolated), and
complex stability (G(complex-2*isolated)) of the alkane complexes………………………..82
Table 3. ELF value, η(r), electron density, ρ(r), and Laplacian of the electron density,
∇2 ρ(r) at bond critical point of a selected H-H bond in alkane complexes…………....87
LISTA DE TABELAS
Artigo 3 - Stability and Electronic Structure of Substituted Tetrahedranes, Silicon and
Germanium Parents – a DFT, ADMP, QTAIM and GVB study
Table 1. Values of Gibbs free energy difference and enthalpy difference for the
corresponding tetrahedranes 3a – 3f, 4a, 5a and 5b from diazocompound to
cyclobutadiene (and derivatives), ΔG1 and ΔH1, from diazocompound to tetrahedrane
(and derivatives), ΔG2 and ΔH2, and from cyclobutadiene (and derivatives) to
tetrahedrane (and derivatives), ΔGf and ΔHf, in kcal mol-1, respectively. Values of
delocalization index, DI, bond order, n, and ellipticity, ε, from carbon, silicon or
germanium atoms in the tetrahedrane (or parent) cage, bond path angle, in degree, from
bonds in the tetrahedrane (or parent) cage for 3a – 3f, 4a, 5a and 5b……………...…110
Table 2. Values of AIM atomic charge, q(Ω), atomic energy, E(Ω), atomic dipole
moment, M(Ω), atomic volume, V(Ω) and localization index, LI of carbon, silicon and
germanium atoms in tetrahedrane, silicon and germanium derivatives, respectively...111
Table 4. Overlap matrix of selected GVB orbitals of molecule 3f of two oxygen lone
pairs (LP1, LP2, LP1’ and LP2’) and of two C-C sigma bonds [VB(C-C)1, VB(C-C)’
and VB(C-C)2]………………………………………………………………………...113
Table 3. Average DI, DI, of the carbon or silicon or germanium atoms in the
tetrahedrane (or parent) cage, the corresponding DIUs, the degree of degeneracy (δ)
involving the atoms of the tetrahedrane (or parent) cage, the charge density of the cage
critical points [ρ(3,+3)], D3BIA and NICS values for 3a, 3b, 3c, 3d, 3e, 3f, 4a, 5a and
5b……………………………………………………………………………………...115
Table 5. Overlap matrix of selected GVB orbitals, VB(C-C)’, VB (C-C)1 and VB (CC)2, of 3a, 3b, 3f and cubane. ……………………………………………………...….117
LISTA DE TABELAS
Artigo 4 - Docking, Molecular Dynamics, ONIOM, QTAIM: theoretical investigation
of inhibitory effect of Abiraterone on CYP17.
Table 1. Binding energies of CYP17 with PREG…………….…………...……….…135
Table 2. Values of Lennard-Jones potential (VLJ) and Coulomb potential (VC), in kcal
mol-1, between CYP17 (p)/water (w) and ligands ABE and PREG…………………..138
Table 3. Values of electronic energy of CYP17 enzyme, ABE, PREG, CYP17-ABE and
CYP17-PREG complexes……………………………………………………………..138
Table 4. Electronic density (ρ), in au., of the critical points between abiraterone (or
pregnenolone) and CYP17 active site……………………………………………..….139
TABLE S1. Electronic energy, in a.u., bond length Fe-NH2 and bond length Fe-NH3, in
angstroms, of triplet and quintet states of the Fe(NH3)2(NH2)2 with different functionals
using 6-311G(d,p) basis set……………………………………………………..…….157
LISTA DE ABREVIATURAS, ACRÔNIMOS E SIGLAS EMPREGADAS.
ρ
Densidade eletrônica
𝛁𝟐𝝆
Laplaciano da densidade eletrônica
ΔG
Variação da energia livre de Gibbs
AMBER
Assisted Model Building with Energy Refinement (Campo de força ou
pacote de programas de simulação )
BCP
Bond Critical Point (Ponto Crítico de Ligação)
DFT
Density Functional Theory (Teoria Funcional da Densidade)
GVB
Generalized Valence Bond (Ligação de Valência Generalizada)
HF
Hartree-Fock
Hb
Total Energy Density (Densidade de Energia Total)
QM/MM
Mecânica Quântica/Mecânica Molecular
QTAIM
Quantum Theory of Atoms in Molecules (Teoria Quântica de
Átomos em Moléculas)
RESP
Restrained Electrostatic Potential
ABE
Abiraterona
PREG
Pregnenolona
SUMÁRIO
1.
INTRODUÇÃO GERAL ................................................................................... 24
2.
REFERENCIAL TEÓRICO ............................................................................. 25
2.1.
TEORIA FUNCIONAL DA DENSIDADE (Density functional theory – DFT)
...........................................................................................................................25
2.2. TEORIA QUÂNTICA DE ÁTOMOS EM MOLÉCULAS (Quantum Theory of
Atoms in Molecules - QTAIM) ................................................................................ 30
2.3. FUNÇÃO DE LOCALIZAÇÃO ELETRÔNICA (Electron Localization
Function - ELF)....................................................................................................... 34
2.4. MÉTODO HARTREE-FOCK (HF) E TEORIA DA LIGAÇÃO DE
VALÊNCIA GENERALIZADA (Generalized Valence Bond – GVB)..................... 36
3.
2.5.
CÁLCULOS DE QM/MM ............................................................................ 38
2.6.
SIMULAÇÃO MOLECULAR ..................................................................... 39
2.7.
DOCAGEM MOLECULAR ......................................................................... 40
CAPÍTULO 1 – Artigo publicado no periódico World Journal of Chemical
Education.................................................................................................................... 41
4.
CAPÍTULO 2 – Artigo publicado no periódico The Journal of Physical Chemistry
A. ..................................................................................................................................70
5.
CAPÍTULO 3 – Artigo publicado no periódico New Journal of Chemistry. ...... 100
6.
CAPÍTULO 4 – Formatação de artigo a ser submetido para o periódico The
Journal of Physical Chemistry B. .............................................................................. 124
7.
REFERÊNCIAS GERAIS ............................................................................... 159
24
1. INTRODUÇÃO GERAL
A química é a ciência que lida com a construção, transformação e propriedades das moléculas. A
química teórica é uma subárea da química onde os métodos matemáticos são combinados com as leis
fundamentais da física com o objetivo de se estudar processos de relevância química. (Clark, 1985)
A química computacional é uma disciplina que se estende muito além dos limites tradicionais que
separam a química, a física, a bioquímica e a ciência da computação, permitindo o estudo aprofundado de
átomos, moléculas e macromoléculas através da ferramenta computacional. A química computacional pode
atuar como ferramenta de apoio na análise e interpretação de dados experimentais, através de informações
que muitas vezes não são possíveis de serem obtidas diretamente de experimentos (Morgon et al., 1995;
Morgon et al., 1997; Morgon e Riveros, 1998; Morgon et al., 2000), ou na previsão de propriedades
diversas. Além disso, a química computacional pode ser aplicada nas mais diversas áreas da química, assim
como: Físico-química, química orgânica, química inorgânica, química analítica, bioquímica, além de ser
utilizada como ferramenta de ensino.
Na química orgânica, por exemplo, existem muitas explicações consolidadas acerca da estabilidade de
alcanos e sua relação o ponto de fusão e ebulição. A química computacional pode nos fornecer novas e
importantes informações acerca do modo de interação dessas moléculas e relacioná-las com as suas
propriedades macroscópicas.
A química computacional também pode ser aplicada no desenvolvimento de moléculas de alta energia de
interesse militar. Moléculas orgânicas de alta tensão em formato de gaiola podem ser usadas como
potenciais explosivos em resposta a estímulos externos. (Katz e Acton, 1973; Zhou et al., 2004)
O estudo teórico na área de bioquímica possui uma gama de aplicações como a análise conformacional
de grandes sistemas moleculares de importância biológica; o estudo da interação enzima-substrato e
principalmente no planejamento racional de novos fármacos. Ainda existe um novo cenário surgindo com o
aparecimento de uma recente estratégia de cálculos híbridos que envolvem a mecânica quântica e a
mecânica molecular, conhecida como QM/MM (Quantum Mechanics/Molecular Mechanics). Assim, por
exemplo, existe a possibilidade de descrição da quebra e formação de ligações químicas em reações que
envolvam grandes sistemas através da mecânica quântica, onde é sabido que a mecânica molecular é falha
na descrição das propriedades eletrônicas.
Portanto, para essa tese, foram realizados quatro estudos teóricos das áreas descritas acima, utilizando
métodos quânticos, de mecânica molecular e híbridos objetivando um estudo mais aprofundado, assim como
fornecer novas perspectivas em cada tema abordado.
25
2. REFERENCIAL TEÓRICO
2.1. TEORIA FUNCIONAL DA DENSIDADE (Density functional theory – DFT)
A teoria funcional da densidade (DFT) é um dos métodos quanto-mecânicos mais populares e bemsucedidos da atualidade. (Capelle, 2006) Essa metodologia é rotineiramente aplicada para cálculos de
energia de ligação de moléculas na química e para cálculos de estrutura de bandas de sólidos em física. Por
outro lado, as primeiras aplicações em campos considerados mais distantes da mecânica quântica, assim
como a biologia (Devipriya e Kumaradhas, 2012; Sicolo et al., 2014; Bigler et al., 2015; Jitonnom et al.,
2015) e a mineralogia (Deng et al.; Liu et al., 2013; Mikhlin et al., 2014; Kahlenberg et al., 2015) estão
começando a aparecer.
Para se ter uma ideia do que é a teoria funcional da densidade, se faz necessário algum conhecimento
prévio da mecânica quântica. Na mecânica quântica, toda e qualquer informação relevante a um sistema está
contida na sua função de onda, Ψ. Os graus de liberdade nucleares aparecem somente na forma de energia
potencial, ν(r), atuando nos elétrons devido à aproximação de Born-Oppenheimer.(Woolley e Sutcliffe,
1977) Devido a isso, a função de onda depende apenas das coordenadas eletrônicas. Essa função de onda
não-relativística é calculada a partir da equação de Schrödinger para um elétron se movendo em um
potencial ν(r):
[−
ħ 2 ∇2
+ 𝜈 (𝒓)] 𝛹 (𝒓) = 𝜀𝛹 (𝒓)
2𝑚
(1)
onde ∇2 é o operador Laplaciano definido como a soma dos operadores diferenciais em coordenadas
cartesianas:
𝜕2
𝜕2
𝜕2
∇ = 2+ 2+ 2
𝜕𝑥
𝜕𝑦
𝜕𝑧
2
(2)
para um sistema multieletrônico, a equação de Schrödinger se torna:
𝑁
[∑ (−
𝑖
ħ 2 ∇2
+ 𝜈 (𝒓)) + ∑ 𝑈(𝒓𝑖 , 𝒓𝑗 )] 𝛹 (𝒓1 , 𝒓2 , … , 𝒓𝑁 ) = 𝐸𝛹 (𝒓1 , 𝒓2 , … , 𝒓𝑁 ) (3)
2𝑚
𝑖<𝑗
onde N é o número de elétrons e U(ri,rj) é a interação elétron-elétron. Para um sistema Coulômbico, temos:
̂ = ∑ 𝑈(𝒓𝑖 , 𝒓𝑗 ) = ∑
𝑈
𝑖<𝑗
e o operador energia cinética é dado por:
𝑖<𝑗
𝑞2
|𝒓𝑖 − 𝒓𝑗 |
(4)
26
𝑇̂ = −
ħ2
∑ ∇2𝑖
2𝑚
(5)
𝑖
e o potencial de interação elétron-núcleo é dado por:
𝑉̂ = ∑ 𝜈 (𝒓𝑖 ) = ∑
𝑖
𝑖
𝑄𝑞
| 𝒓 𝑖 − 𝑹|
(6)
Onde Q é a carga nuclear (Pople, 1999) e R é a posição do núcleo.
A função de onda Ψ em si, não é uma observável, pois não é mensurável. Entretanto, o quadrado do
valor absoluto da função de onda possui uma interpretação física. (Koch e Holthausen, 2001) Essa
interpretação só é possível porque o produto de um número complexo pelo seu complexo conjugado é real e
não negativo, onde:
2
|𝛹(𝑥⃗1, 𝑥⃗2, … 𝑥⃗𝑁, )| 𝑑𝑥⃗1 𝑑𝑥⃗2 … 𝑑𝑥⃗𝑁 (7)
representa a probabilidade que os elétrons 1, 2, ..., N são encontrados simultaneamente em volumes
𝑑𝑥⃗1 𝑑𝑥⃗2 … 𝑑𝑥⃗𝑁 . Desde que esses elétrons sejam indistinguíveis, não deve haver troca de coordenadas de
quaisquer dois elétrons (i e j) se os mesmos são comutados, ou seja:
2
|𝛹(𝑥⃗1, 𝑥⃗2 , … , 𝑥⃗𝑖 , 𝑥⃗𝑗 , … , 𝑥⃗𝑁, )| = |𝛹(𝑥⃗1, 𝑥⃗2 , … , 𝑥⃗𝑗 , 𝑥⃗𝑖 , … , 𝑥⃗𝑁, )|
2
(8)
Isso garante que as funções de onda possuam sinais contrários resultando em funções antissimétricas
aplicadas à férmions com spins semi-inteiros. Elétrons são férmions com spin = ½ e Ψ deve ser, portanto,
antissimétrica com respeito a permutação das coordenadas espaciais e spin de quaisquer dois elétrons:
𝛹(𝑥⃗1, 𝑥⃗2 , … , 𝑥⃗𝑖 , 𝑥⃗𝑗 , … , 𝑥⃗𝑁, ) = −𝛹(𝑥⃗1, 𝑥⃗2 , … , 𝑥⃗𝑖 , 𝑥⃗𝑗 , … , 𝑥⃗𝑁, )
(9)
esse princípio da antissimetria, representa uma generalização quanto-mecânica denominada de princípio da
exclusão de Pauli (onde dois elétrons não podem ocupar o mesmo estado). Uma consequência lógica da
interpretação probabilística da função de onda, é que integrando a eq. (7) em todo o intervalo sobre todas as
variáveis, o resultado é igual a um. Em outras palavras, a probabilidade de encontrar os N elétrons em
qualquer lugar do espaço corresponde a uma unidade,
2
∫ … ∫|𝛹(𝑥⃗1, 𝑥⃗2, … 𝑥⃗𝑁, )| 𝑑𝑥⃗1 𝑑𝑥⃗2 … 𝑑𝑥⃗𝑁 = 1
(10)
a função de onda que satisfaz a eq. (10) é dita ser normalizada.
A interpretação probabilística da eq. (10) da função de onda leva diretamente à densidade eletrônica,
ρ(r), que é definida como a integral múltipla sobre todas as coordenadas spin e sobre todas as variáveis
espaciais de todos os elétrons:
27
2
⃗⃗) = 𝑁 ∫ … ∫|𝛹(𝑥⃗1, 𝑥⃗2, … 𝑥⃗𝑁, )| 𝑑𝑥⃗1 𝑑𝑥⃗2 … 𝑑𝑥⃗𝑁
𝜌 (𝒓
(11)
⃗⃗) determina a probabilidade de encontrar qualquer um dos N elétrons dentro de um volume 𝑑(⃗⃗⃗⃗⃗).
⃗⃗)
𝜌 (𝒓
𝒓𝟏 𝜌 ( 𝒓
é a densidade de probabilidade, comumente denominada de densidade eletrônica.
Diferentemente de uma função de onda, a densidade eletrônica é uma observável e pode ser
mensurada experimentalmente, como por exemplo, através da difração de raios X. (Averkiev et al., 2014) A
teoria funcional da densidade é um método baseado na análise do sistema molecular através da densidade
eletrônica. O primeiro trabalho com a densidade eletrônica foi realizado no início do século XX com o
objetivo de explicar as propriedades térmicas e elétricas dos metais. (Chudnovsky, 2007) Mas foi somente
em 1964, através do trabalho de Hohenberg e Kohn (Hohenberg e Kohn, 1964) que mostra que a densidade
eletrônica, de fato, determina o operador Hamiltoniano e, assim, todas as propriedades do sistema. Ainda
assim, o método precisava buscar os funcionais adequados, os chamados funcionais exatos, necessários para
serem incorporados na expressão da energia total eletrônica. Somente com os trabalhos de Kohn-Sham
(Kohn e Sham, 1965) melhorando o tratamento matemático das equações de densidade e do funcional, que a
teoria funcional da densidade avançou como ferramenta de cálculo teórico para estudo das propriedades
moleculares com aplicações computacionais. (Kohn et al., 1996)
O primeiro teorema de Hohenberg-Kohn afirma que a densidade eletrônica determina o potencial
externo [ν(r)] devido aos núcleos, assim como também determina o número total de elétrons, N, via sua
normalização ∫ 𝜌(𝒓)𝑑𝑟 = 𝑁. N e ν(r) determinam o operador Hamiltoniano molecular (Hop) [vide eq. (3)]
que, por sua vez, determina a energia do sistema através da equação de Schrödinger 𝐻𝑜𝑝 𝛹 = 𝐸𝛹. A
densidade eletrônica determina a energia do sistema e todas as outras propriedades eletrônicas do sistema no
estado fundamental. (Geerlings et al., 2003) O esquema 1 mostra a relação existente entre as variáveis
descritas acima.
Esquema 1:
ν
Hop
N
consequentemente, E é um funcional de ρ:
E
28
𝐸 = 𝐸 [𝜌 ]
(12)
O segundo teorema de HK estabelece que o funcional FHK é universal, ou seja, o mesmo é usado para
todos os problemas de estrutura eletrônica (Burke, 2007), que é dado por:
𝐹𝐻𝐾 = 𝑇[𝜌] + 𝑉𝑒𝑒 [𝜌] (13)
onde T[ρ] é o funcional de energia de energia cinética eletrônica e Vee[ρ] é o funcional de interação elétronelétron. Esse funcional é um componente da expressão do funcional da energia (Geerlings et al., 2003):
𝐸 [𝜌] = ∫ 𝜌(𝒓)𝜈(𝒓)𝑑𝒓 + 𝐹𝐻𝐾 (14)
dessa forma, o segundo teorema de HK estabelece um princípio variacional onde 𝐸 [𝜌] ≥ 𝐸0 .
Para contornar o problema dos teoremas de HK, Kohn e Sham propuseram uma abordagem para a
aproximação dos funcionais de energia cinética e potencial elétron-elétron. (Kohn e Sham, 1965) Eles
introduziram um sistema fictício de N elétrons não-interagentes para serem descritos através de um único
determinante em N orbitais ϕi.
Kohn e Sham reescreveram a equação (14) e explicitaram a repulsão eletrônica de Coulomb e
definiram uma nova função universal G[ρ]:
1
𝜌(𝑟1 )𝜌(𝑟2 )
𝐸𝜈 [𝜌] = 𝐺 [𝜌] + ∫ ∫
𝑑𝑟1 𝑑𝑟2 + ∫ 𝜌(𝑟)𝜈 (𝑟)𝑑𝑟
|𝑟1 − 𝑟2 |
2
(15)
em que
𝐺 [𝜌] = 𝑇𝑠 [𝜌] + 𝐸𝑥𝑐 [𝜌]
(16)
onde 𝑇𝑠 [𝜌] é o funcional energia cinética de um sistema de elétrons que não interagem e 𝐸𝑥𝑐 [𝜌] inclui o
termo de interação elétron-elétron não-clássica (troca e correlação) e a parte residual da energia cinética,
𝑇[𝜌] − 𝑇𝑠 [𝜌], em que 𝑇[𝜌] é a energia cinética exata para o sistema de elétrons que interagem.
O Hamiltoniano dos elétrons que não interagem com o potencial efetivo, 𝜈𝑒𝑓 , é dado por:
1
𝐻𝐾𝑆 = − ∇2 + 𝜈𝑒𝑓
2
(17)
A obtenção da função de onda Kohn-Sham, ΨKS, do estado fundamental do sistema de elétrons que
não interagem, descrito pelo Hamiltoniano da equação (17), é dada pelo produto anti-simetrizado de N
orbitais de um elétron, ϕi(r), através do determinante de Slater (Morgon e Coutinho, 2007):
29
𝛹 𝐾𝑆
𝜙1𝐾𝑆 (𝑟1 ) 𝜙2𝐾𝑆 (𝑟1 ) … 𝜙𝑁𝐾𝑆 (𝑟1 )
1 𝜙1𝐾𝑆 (𝑟2 ) 𝜙2𝐾𝑆 (𝑟2 ) … 𝜙𝑁𝐾𝑆 (𝑟2 )
|
|
=
⋮
⋮
⋱
⋮
√𝑁!
𝜙1𝐾𝑆 (𝑟𝑁 ) 𝜙2𝐾𝑆 (𝑟𝑁 ) … 𝜙𝑁𝐾𝑆 (𝑟𝑁 )
(18)
Por sua vez, os orbitais Kohn-Sham, 𝜙𝑖𝐾𝑆 , são obtidos a partir da equação de Schrödinger para um elétron:
1
(− ∇2 + 𝜈𝑒𝑓 ) 𝜙𝑖𝐾𝑆 = 𝜀𝑖 𝜙𝑖𝐾𝑆
2
(19)
A relação existente entre o sistema real e o sistema que não interage é dada por:
𝑁
2
𝜌𝑠 (𝑟) = ∑ 2|𝜙𝑖𝐾𝑆 (𝑟)| = 𝜌0 (𝑟)
(20)
𝑖
onde ρ0(r) é a densidade eletrônica no estado fundamental. A energia cinética é dada por:
𝑁
1
𝑇𝑠 [𝜌] = − ∑〈𝜙𝑖𝐾𝑆 |∇2 |𝜙𝑖𝐾𝑆 〉
2
(21)
𝑖
e o potencial efetivo é dado por:
𝜈𝑒𝑓 = 𝜈 (𝑟) +
𝜌(𝑟1 )
𝑑𝑟 + 𝜈𝑥𝑐 (𝑟)
|𝑟 − 𝑟1 | 1
(22)
em que
𝜈𝑥𝑐 (𝑟) =
𝛿𝐸𝑥𝑐 [𝜌]
𝛿𝜌(𝑟)
(23)
A equação (23) é o funcional proveniente da energia de correlação e troca [equação (15)] em relação à
densidade.
Das equações acima, percebe-se que o potencial efetivo é dependente da densidade, ρ(r), a qual
depende dos orbitais Kohn-Sham, 𝜙𝑖𝐾𝑆 , que deverão ser encontrados. Para isso, utiliza-se um procedimento
autoconsistente que é mostrado na Figura 1.
Primeiramente é construída a densidade inicial, ρ0, para se calcular o potencial efetivo, 𝜈𝑒𝑓 , para se
obter o Hamiltoniano. Assim, resolve-se a equação de Kohn-Sham para se obter um conjunto de
autofunções, 𝜙𝑖𝐾𝑆 , que irão compor o determinante de Slater. Em seguida é obtida a função de onda KohnSham, ΨKS, do qual a nova densidade eletrônica, ρ1, pode ser calculada. A densidade recém-calculada, ρn é
comparada com a anterior ρn-1. Se ρn ≠ ρn-1 o ciclo é reiniciado até que a densidade não se altere de um ciclo
30
para outro, ou seja, ρn = ρn-1. Quando essa condição for atingida, significa dizer que essa densidade de carga
minimiza a energia e a convergência foi alcançada.
Cálculo do potencial efetivo
ρ0
Resolução da equação de KS
Determinante de Slater
Obtenção de ΨKS
Cálculo da nova densidade
Não
Sim
ρn é a densidade procurada
Figura 1. Algoritmo mostrado o procedimento autoconsistente para resolução das equações Kohn-Sham.
Fonte: Autor, 2015.
2.2.TEORIA QUÂNTICA DE ÁTOMOS EM MOLÉCULAS (Quantum Theory of Atoms in
Molecules - QTAIM)
A teoria quântica de átomos em moléculas (QTAIM) é um modelo quântico considerado eficiente no
estudo da ligação química (Cortés-Guzmán e Bader, 2005) e caracterização das interações intra e
intermoleculares. (Grabowski et al., 2004; Oliveira et al., 2009; Monteiro e Firme, 2014).
A teoria quântica de átomos em moléculas (Bader, 1990), desenvolvida pelo Professor Richard F. W.
Bader e seus colaboradores, as propriedades observáveis dos átomos constituintes de um sistema molecular
estão contidas na sua densidade eletrônica, ρ. As trajetórias da densidade eletrônica são obtidas à partir do
vetor gradiente da densidade eletrônica (∇ρ) (Popelier, 2000), que é dada pela primeira derivada da
31
densidade eletrônica sobre todas as coordenadas (Equação 24) e o conjunto de trajetória desse gradiente
formam as bacias atômicas (Ω).
∇𝜌 = 𝒊
𝑑𝜌
𝑑𝜌
𝑑𝜌
+𝒋
+𝒌
(24)
𝑑𝑥
𝑑𝑦
𝑑𝑧
Um ponto crítico (PC) na densidade eletrônica é um ponto no espaço em que cada derivada do ∇ρ é zero
(∇ρ = 0). Para se distinguir um ponto crítico de mínimo local, máximo local ou um ponto crítico de sela, são
consideradas as derivadas segundas da densidade eletrônica. Existem nove derivadas segundas de ρ(r) que
podem ser dispostas em uma matriz Hessiana, mostrada na equação 25 (Bader, 1990):
𝜕2𝜌
𝜕𝑥 2
𝜕2𝜌
∇∇𝜌 =
𝜕𝑦 𝜕𝑥
𝜕2𝜌
(𝜕𝑧 𝜕𝑥
𝜕2𝜌
𝜕𝑥 𝜕𝑦
𝜕2𝜌
𝜕𝑦 2
𝜕2𝜌
𝜕𝑧 𝜕𝑦
𝜕2𝜌
𝜕𝑥 𝜕𝑧
𝜕2𝜌
(25)
𝜕𝑦 𝜕𝑧
𝜕2𝜌
𝜕𝑧 2 )
A matriz Hessiana pode ser diagonalizada através da rotação do sistema de coordenadas r (x, y, z)  r
(x’, y’, z’). Os novos eixos coordenados são chamados de eixos principais da curvatura porque a magnitude
das três derivadas da segunda de ρ, calculadas com respeito a esses eixos, são maximizados. A rotação do
sistema de coordenadas é acompanhada de uma transformação unitária, r’ = rU, onde U é uma matriz
unitária construída à partir de uma conjunto de três equações de autovalores (𝛁𝛁𝝆)𝒖𝑖 = 𝜆𝑖 𝒖𝑖 (com 𝑖 = 1, 2,
3) em que 𝒖𝑖 é o inésimo autovetor em U. A equação matricial U-1(𝛁𝛁𝝆)U = Λ transforma a Hessiana na
sua forma diagonal, que pode ser escrita como:
𝜕2𝜌
𝜕𝑥′2
𝛬=
0
0
0
𝜕2𝜌
𝜕𝑦′2
( 0
0
𝜕2𝜌
𝜕𝑧′2 )
0
𝜆1
= (0
0
0
𝜆2
0
0
0 ) (26)
𝜆3
em que λ1, λ2 e λ3 são as curvaturas da densidade em relação aos três eixos principais x’, y’ e z’.
A soma dos autovalores da matriz Hessiana é conhecida como o Laplaciano da densidade eletrônica
2
(∇ ρ), que é dado pela equação 27:
∇2 𝜌 =
𝜕2 𝜌
𝜕2 𝜌
𝜕2 𝜌
𝜕𝑥
𝜕𝑦
𝜕𝑧2
+
2
+
2
= 𝜆 1 + 𝜆2 + 𝜆3
(27)
32
Um ponto crítico pode ser classificado de acordo com o seu ranking, denotado por ω, que é o número
de autovalores não-zero da ρ(r) de um ponto crítico e pela assinatura, denotada por σ, que é a soma
algébrica dos sinais dos autovalores. Assim, um ponto crítico é caracterizado pelo conjunto de valores (ω,σ),
como pode ser exemplificado na Tabela 1. (Firme, 2007) Existem quatro tipo de pontos críticos estáveis,
possuindo três autovalores não-zero. O primeiro deles é o ponto crítico (3,-3), denominado de atrator nuclear
(Nuclear Attractor – NA), que possui todos os autovalores negativos, sendo a sua densidade eletrônica ρ(r)
um máximo local; o segundo ponto crítico é o (3,-1), denominado de ponto crítico de ligação (Bond Critical
Point - BCP), que possui dois autovalores negativos (λ1 e λ2) e um positivo (λ3) com ρ(r) sendo um máximo
no plano definido pelos autovetores correspondentes e um mínimo ao longo dos três eixos que é
perpendicular a esse plano; o terceiro ponto crítico (3,+1) é o ponto crítico do anel (Ring Critical Point RCP) que possui dois autovalores positivos (λ2 e λ3) e um negativo (λ1) com ρ(r) sendo um mínimo e por
último o ponto crítico de gaiola (Cage Critical Point – CCP) (3,+3) que possui todos os autovalores
positivos, sendo a ρ(r) um mínimo local. (Matta et al., 2007) Todos os pontos críticos estão representados na
Figura 2.
Tabela 1. Acrônimos, sinais dos autovalores e denominações dos pontos críticos.
Nome
Atrator Nuclear
Acrônimo
NA
λ1
-
λ2
-
λ3
-
(ω,σ)
(3,-3)
Ponto Crítico da Ligação
BCP
-
-
+
(3,-1)
Ponto Crítico do Anel
RCP
-
+
+
(3,+1)
Ponto Crítico da Gaiola
CCP
+
+
+
(3,+3)
(3,-1)
(3,+1)
(3,+3)
(3,-3)
Figura 2. Gráfico molecular do Ferroceno. Fonte: (Firme et al., 2010).
33
Um ponto crítico (3,-3) age como um atrator do campo vetorial do ∇ρ, ou seja, existe uma vizinhança
aberta do atrator que é invariante ao fluxo de ∇ρ tal que qualquer caminho do gradiente originado nessa
vizinhança aberta termina no atrator. Existem também caminhos de gradiente que terminam ou se originam
nos pontos críticos (3,-1). Os caminhos de gradiente que se originam nos pontos críticos (3,-1) definem os
caminhos de ligação (Figura 2). Os dois caminhos de gradiente definem uma linha através da densidade
eletrônica ligando os núcleos vizinhos ao longo do qual ρ(r) é máximo em relação a qualquer linha vizinha.
(Bader et al., 1979). Essa linha é encontrada entre cada par de núcleos cujas bacias atômicas compartilham
uma superfície interatômica comum e é chamada de interação atômica. A existência do ponto crítico (3,-1) e
a sua linha associada de interação atômica indicam que a densidade eletrônica é acumulada entre os núcleos
que estão ligados. Esse acúmulo de carga é a condição necessária quando as forças de Feyanman que atuam
nos núcleos e elétrons estão em equilíbrio.(Bader e Fang, 2005) Sendo assim, a presença da linha de
interação atômica satisfaz as condições necessárias para que os átomos estejam ligados. Essa linha de
interação é denominada de caminho de ligação e o ponto crítico (3,-1) é chamado de ponto crítico de
ligação. Mais a frente voltaremos a falar do caminho de ligação.
A densidade eletrônica máxima nas posições dos núcleos resulta em uma topologia originada através
das trajetórias da densidade eletrônica são obtidas à partir do vetor gradiente ∇ρ. Essa topologia é
particionada do espaço molecular em regiões mononucleares, Ω, denominadas de bacias atômicas. (Matta et
al., 2007) A superfície de ligação entre duas bacias atômicas é denominada de superfície de fluxo zero
(Figura 3), onde o produto escalar do vetor ∇ρ e o vetor normal [n(r)] se anulam (Equação 28).
Superfície de fluxo zero
Figura 3. Mapa de contorno da distribuição de carga do cloreto de sódio e o gradiente do campo vetorial ∇ρ.
Fonte: (Bader, 1990).
∇𝜌. 𝑛(𝑟) = 0 (28)
34
Existe um ponto na superfície de fluxo zero onde os vetores gradiente se anulam (∇ρ = 0), pois esses
vetores apontam para os núcleos atômicos e, consequentemente, ocorre formação do ponto crítico de ligação
(3,-1).
A presença de uma superfície interatômica de fluxo zero entre dois átomos ligados é sempre
acompanhada por uma linha de densidade local máxima, denominada de caminho de ligação. O caminho de
ligação é um indicador universal de todos os tipos de ligação química: interações fracas, fortes, camada
fechada e aberta.(Bader, 1998)
O conjunto de caminhos de ligação que conectam os núcleos dos átomos ligados em uma geometria
de equilíbrio, assim como os seus pontos críticos de ligação, é denominado de gráfico molecular. (Wiberg et
al., 1987) O gráfico molecular é o resultado direto das propriedades topológicas principais da distribuição de
cargas de um sistema em que o máximo local, os pontos críticos (3,-3), ocorrem nas posições dos núcleos e
os pontos críticos (3,-1) que conectam certos pares de núcleos em uma molécula (Figura 2). (Bader, 1990)
Por outro lado, existe outro tipo de gráfico, que é considerado um “espelho” do gráfico molecular. Esse tipo
de gráfico é definido por um conjunto de linhas com densidade máxima de energia potencial negativa. Em
outras palavras, existe uma única linha com densidade máxima de energia potencial negativa conectando os
mesmos núcleos de um caminho de ligação. (Keith et al., 1996) Essa linha de “máxima estabilidade” no
espaço real é denominada de “caminho virial”. O conjunto de caminhos viriais associados aos pontos
críticos constituem o gráfico virial.
2.3. FUNÇÃO DE LOCALIZAÇÃO ELETRÔNICA (Electron Localization Function - ELF)
A função de localização eletrônica (Electron Localization Function – ELF) descrita primeiramente por
Becke e Edgecombe (Becke e Edgecombe, 1990) como uma função local que demonstra o quanto o
princípio da exclusão de Pauli é eficiente em um dado ponto do espaço molecular. Ou seja, o ELF fornece
uma descrição quantitativa do princípio da exclusão de Pauli. O ELF é definido como:
𝜂 (𝑟 ) =
1
1 + (𝐷⁄𝐷ℎ )2
(29),
em que,
𝑁
1
1 |∇𝜌|2
𝐷 = ∑|∇ѱ𝑖 |2 −
2
8 𝜌
𝑖=1
e
(30)
35
𝑁
3
𝐷ℎ =
(3𝜋 2 )2/3 𝜌5/3 (31), 𝜌 = ∑|ѱ𝑖 |2
10
(32),
𝑖=1
onde as somas são todos os (spin) orbitais ѱ i(r) monocupados.(Savin, 2005) O valor do ELF é confinado ao
intervalo [1,0], onde a tendência para 1 significa que os spins paralelos são amplamente improváveis, e onde
há, portanto, uma alta probabilidade de spins opostos. A tendência para zero indica uma alta probabilidade
de regiões com os mesmos pares de spin. A função de localização eletrônica fornece um rigoroso suporte
para a análise da função de onda e da estrutura eletrônica em moléculas e cristais.(Noury et al., 1999) A
topologia da ELF tem sido amplamente empregada no estudo da ligação química.(Noury et al., 1998;
Chevreau et al., 2001; Chesnut, 2002; Chevreau, 2004; Rosdahl et al., 2005) O particionamento topológico
do campo gradiente da ELF (Häussermann et al., 1994; Silvi e Savin, 1994) fornece bacias de atratores
correspondentes às ligação químicas e aos pares de elétrons isolados. Em uma molécula podemos encontrar
dois tipos de bacias. A primeira como sendo as bacias de caroço que estão em torno dos núcleos com
número atômico Z > 2 denominado de C(A), onde A é o símbolo atômico do elemento. O outro tipo de bacia
é denominado de bacia de valência. As bacias de valência são caracterizadas pelo número de camadas de
valência que a bacia possui, em outras palavras, pelo número de bacias de caroço que os elementos
compartilham nas suas fronteiras. Esse número é chamado de ordem sináptica. Assim, existem bacias
monosinápticas, disinápticas, trisinápticas e assim por diante, especificadas como V(A), V(A, X), V(A, X,
Y) respectivamente. Bacias monosinápticas correspondem aos pares de elétrons isolados do modelo de
Lewis. Bacias polisinápticas correspondem aos pares de elétrons compartilhados do modelo de Lewis.
A função local ELF, η(r), fornece a informação química e ∇η(r) fornece dois tipos de pontos no
espaço real: pontos críticos onde (r)=0 e para os demais pontos, (r)≠0. Os pontos críticos são
caracterizados pelos seus índices que são os números de autovalores positivos da matriz Hessiana de η(r). A
topologia da ELF já foi aplicada no estudo da ligação de hidrogênio.(Fuster e Silvi, 2000; Alikhani e Silvi,
2003; Alikhani et al., 2005; Soler et al., 2005) No complexo FH•••N2 o valor da ELF entre as bacias V(F,H)
e V(N) é 0,047. Enquanto para o complexo FH•••NH3 o valor de (r) é 0.226. Esses valores são atribuídos
às ligações fracas e fortes respectivamente. Nesse trabalho, a ELF foi utilizada como um método
complementar para avaliar a ligação hidrogênio-hidrogênio entre os complexos de alcanos estudados.
36
2.4. MÉTODO HARTREE-FOCK (HF) E TEORIA DA LIGAÇÃO DE VALÊNCIA
GENERALIZADA (Generalized Valence Bond – GVB)
A descrição da estrutura eletrônica de átomos e moléculas em termos de orbitais eletrônicos fornece os
fundamentos básicos para teorias mais complexas.(Petersson, 1974) A energia obtida à partir de cálculos
baseados em orbitais possuem geralmente um erro de cerca de 0,015 a.u. (10 kcal mol-1) para cada elétron.
Embora esses erros compreendam uma pequena fração em relação à energia total, eles são bem
significativos para os padrões químicos. Faz-se, portanto, necessário uma descrição mais adequada da
estrutura eletrônica dos orbitais para se obter dados mais confiáveis.
Uma aproximação comum das funções de onda das moléculas é o método Hartree-Fock,(Hartree, 1947)
em que a função de onda é dada como um produto antissimetrizado (determinante de Slater) das funções
espaciais e spin. O produto antissimétrico assegura que o princípio da exclusão de Pauli esteja satisfeito.
Assim, para a molécula de H2, a função de onda Hartree-Fock é dada por:
Â[𝜙(1)𝛼(1)𝜙(2)𝛽(2)] = 𝜙(1)𝜙(2)[𝛼(1)𝛽(2) − 𝛽 (1)𝛼(2)] (33)
onde Â é o antissimetrizador (operador determinante), 𝜙 é o orbital espacial duplamente ocupado otimizado,
e α e β são as funções spin. Para o H2, o orbital otimizado 𝜙 é gerado(Goddard e Ladner, 1971) como
mostrado na Figura 4A e pode ser expendido como:
𝜙 = 𝜒𝑙 + 𝜒𝑟 (34)
onde 𝜒𝑙 é a função localizada à esquerda do próton e 𝜒𝑟 é a função localizada à direta deste. A expansão da
parte espacial da função de onda total é dada por:
𝜙(1)𝜙(2) = (𝜒𝑟 𝜒𝑟 + 𝜒𝑙 𝜒𝑙 ) + (𝜒𝑟 𝜒𝑙 + 𝜒𝑙 𝜒𝑟 )
(35)
a equação (35) se aplica a todas as distâncias internucleares R, enquanto que para grandes distâncias, a
função de onda toma a forma:
(𝜒𝑟 𝜒𝑙 + 𝜒𝑙 𝜒𝑟 )
(36)
o que significa que existe probabilidade zero de ambos os elétrons estarem simultaneamente próximos no
mesmo núcleo. A função de onda Hartree-Fock se comporta incorretamente para altos valores de R, como
mostrado na Figura 5, que compara o método Hartree-Fock com a energia exata.
37
A
B
Figura 4. Orbital Hartree-Fock da molécula H2 para (A) R = 0,741 Å e (B) R = 6,35 Å. Fonte: (Goddard e
Ladner, 1971).
Assim, o método Hartree-Fock fornece uma descrição pobre com relação à quebra e a formação de
uma ligação que é intolerável em estudos de reações químicas, pois esse método força ambos os elétrons a
ocuparem um único orbital, enquanto, para grandes separações, teremos dois orbitais monocupados. Uma
alternativa à esse problema, é o uso da função de onda da ligação de valência (VB) que já foi mostrada na
equação (36), o que leva a uma função de onda apropriada para R = ∞. Entretanto, essa função de onde
funciona incorretamente para pequenos valores de R (Figura 5).
Figura 5. Energias da molécula H2 obtidas à partir dos métodos Hartree-Fock, MO, VB e GVB, comparadas
com a energia exata não-relativística. Fonte: [(Goddard et al., 1973)].
Assim, nenhum dos métodos (Hartree-Fock e VB), leva a uma descrição satisfatória para todos os
valores de R.(Goddard et al., 1973) Com a finalidade de contornar esses problemas, é usado o formalismo da
função de onda VB, de modo que a mesma se torna:
𝛹 𝐺𝑉𝐵 = (𝜙𝑙 𝜙𝑟 + 𝜙𝑟 𝜙𝑙 ) (37)
38
essa função de onda é apropriada para altos valores de R, quando resolvida para orbitais utilizando o campo
auto-consistente à cada R tal como no método Hartree-Fock. Isso combina as qualidades tanto do método
Hartree-Fock quanto do método VB e leva a uma função de onda que se comporta apropriadamente quando
os átomos estão separados assim como para pequenos valores de R. Essa abordagem é chamada de ligação
de valência generalizada (generalized valence bond – GVB)(Goddard, 1967; Ladner e Goddard, 1969;
Goddard e Ladner, 1971; Hunt et al., 1972) e difere do método Hartree-Fock pois agora existem dois
orbitais, um para cada elétron, ao invés de um par de elétrons por orbital.
2.5. CÁLCULOS DE QM/MM
Métodos híbridos combinam duas técnicas computacionais em um cálculo e permitem o estudo de
grandes sistemas químicos com uma precisão maior que cálculos de mecânica molecular. (Hall et al., 2008;
Senn e Thiel, 2009; Wang et al., 2012) O método QM/MM é uma técnica hibrida que combina um método
quanto-mecânico (QM) com um método de mecânica molecular (MM).(Warshel e Levitt, 1976; Singh e
Kollman, 1986; Field et al., 1990; Maseras e Morokuma, 1995) O ONIOM(Dapprich et al., 1999a) (N-layer
Integrated molecular Orbital molecular Mechanics) é um dos métodos de QM/MM mais empregados e
permite a combinação de diferentes métodos quanto-mecânicos assim como métodos clássicos de mecânica
molecular. No geral, a região do sistema onde o processo químico (interação química ou reação química)
ocorre é tratado com um método de maior acurácia (mecânica quântica ou semi-empírico), e essa região é
denominada de camada alta (high layer). O restante do sistema é tratado com MM (camada baixa – low
layer) (Figura 6). Muitos estudos com o método ONIOM são focados em sistemas biológicos, assim como
em proteínas(Lin e O’malley, 2008; Alzate-Morales et al., 2009; Kitisripanya et al., 2011; Ugarte, 2014)
onde o sítio ativo é tratado na camada alta, usando frequentemente a teoria funcional da densidade (Density
Functional Theory – DFT) e o restante da proteína é tratado como camada baixa, usando frequentemente o
campo de força AMBER.
Figura 6. Representação do método ONIOM de um sistema proteico dividido em duas camadas. Na
camada baixa (low layer), os átomos são representados por estruturas do tipo arame (wireframe) e na
camada alta (high layer) os átomos são representados por estruturas do tipo bola e bastão (ball-and-stick).
39
O ONIOM obtém a energia do sistema simulado através da combinação das energias computadas pelos
diferentes métodos teóricos,(Dapprich et al., 1999b) em que a camada baixa engloba todo o sistema
molecular incluindo a parte pertencente a camada alta. A equação abaixo apresenta, de modo simplificado,
as considerações realizadas em um sistema de dupla camada calculado com o método ONIOM (Equação
38):
𝐸 𝑂𝑁𝐼𝑂𝑀 = 𝐸 𝑙𝑜𝑤 (𝑅) − 𝐸ℎ𝑖𝑔ℎ (𝑆𝑀 ) − 𝐸 𝑙𝑜𝑤 (𝑆𝑀) (38)
Onde a energia final do sistema, EONIOM, corresponde a energia calculada da região com o método de
maior precisão, Ehigh(SM), e da energia dessa mesma região calculada com um método de menor precisão,
Elow(SM), subtraídas da energia do restante do sistema calculado com o método de menor precisão, Elow(R).
O termo “SM” refere-se a expressão small, ou seja, do sistema pequeno tratado com o método QM. O termo
“R” vem da palavra real e refere-se a todo o sistema.
2.6. SIMULAÇÃO MOLECULAR
A simulação por dinâmica molecular (DM) é uma das técnicas mais versáteis no estudo de
macromoléculas biológicas.(Delano, 2002; Alonso et al., 2006) A DM pode ser caracterizada como um
método para gerar as trajetórias de um sistema de N partículas por integração numérica direta das equações
de movimento Newtonianas(Burkert e Allinger, 1982; Höltje e Folkers, 1996), ou seja, os átomos são
tratados como uma coleção de partículas unidas por forças harmônicas ou elásticas. O conjunto completo
dos potenciais de interação entre as partículas é referido como “campo de força”. (Brooks et al., 1990; Van
Gunsteren e Berendsen, 1990) O campo de força empírico é conhecido como uma função energia potencial e
permite que a energia potencial total do sistema, V(r), seja calculada partir da estrutura tridimensional do
sistema. V(r) é descrito como a soma de vários termos de energia, incluindo os termos para os átomos
ligados (comprimentos e ângulos de ligação, ângulos diedros) e não-ligados (interações de van der Waals e
de Coulomb) e assume a forma geral:
𝑉(𝑟) = 𝑉𝑙𝑖𝑔𝑎𝑑𝑜 + 𝑉 𝑛ã𝑜−𝑙𝑖𝑔𝑎𝑑𝑜 (39)
𝑉𝑙𝑖𝑔𝑎𝑑𝑜 =
∑ 𝐾𝑟 (𝑟 − 𝑟0 )2 + ∑ 𝐾𝜃 (𝜃 − 𝜃0 )2 +
𝑙𝑖𝑔𝑎çõ𝑒𝑠
â𝑛𝑔𝑢𝑙𝑜𝑠
𝑑𝑖𝑒𝑑𝑟𝑜
12
𝑉
𝑛ã𝑜−𝑙𝑖𝑔𝑎𝑑𝑜
𝜎𝑖𝑗
= ∑ 4𝜀𝑖𝑗 [( )
𝑟𝑖𝑗
𝑖,𝑗
1
∑ 𝑉𝑛 [1 − (−1)𝑛 cos(𝑛𝜑 + 𝛾𝑛) (40)
2
6
𝜎𝑖𝑗
𝑞𝑖 𝑞𝑗
−( ) ]+
𝑟𝑖𝑗
4𝜋𝜀0 𝑟𝑖𝑗
(41)
40
Vários campos de força são comumente usados em simulações de dinâmica molecular de sistemas
biológicos, incluindo o AMBER(Wang et al., 2004), CHARMM(Brooks et al., 1983) e GROMOS.(Christen
et al., 2005) Estes diferem principalmente no modo de parametrização, mas geralmente fornecem resultados
semelhantes.
2.7. DOCAGEM MOLECULAR
É amplamente aceito que a atividade farmacológica é obtida através da interação de uma molécula
(ligante) em um sítio (pocket) de outra molécula (receptor), comumente uma proteína.(Vieth et al., 1998) O
processor computacional capaz de predizer a orientação geométrica de um determinado ligante candidato à
fármaco dentro do sítio ativo de uma enzima é denominado de docking molecular.(Cavasotto e Orry, 2007)
A ideia geral das técnicas de docking é de gerar um leque de conformações do ligante e ordená-las através
de um score com base em suas estabilidades com o receptor.(Alonso et al., 2006)
41
3. CAPÍTULO 1 – Artigo publicado no periódico World Journal of Chemical Education.
Teaching thermodynamic, geometric and electronic aspects of Diels-Alder
Norberto K. V. Monteiro* and Caio L. Firme
Institute of Chemistry, Federal University of Rio Grande do Norte, Natal-RN, Brazil, CEP 59078-970.
*Corresponding author. Tel: +55-84-3215-3828; e-mail: norbertokv@gmail.com
ABSTRACT
We have applied an undergraduate minicourse of Diels-Alder reaction in Federal University of Rio
Grande do Norte. By using computational chemistry tools (Gaussian, Gaussview and Chemcraft) students
could build the knowledge by themselves and they could associate important aspects of physical-chemistry
with Organic Chemistry. They have performed a very precise G4 method for the quantum calculations in
this 15-hour minicourse. Students were taught the basics of Diels-Alder reaction and the influence of
frontier orbital theory on kinetics. They learned how to use the quantum chemistry tools and after all
calculations they organized and analyzed the results. Afterwards, by means of the PhD student and teacher
guidance, students discussed the results and eventually answered the objective and subjective questionnaires
to evaluate the minicourse and their learning. We realized that students could understand the influence of
substituent effect on the electronic (and kinetic indirectly), geometric and thermodynamic aspects of DielsAlder reactions. Their results indicated that Diels-Alder reactions are exergonic and exothermic and that
although there is an important entropic contribution, there is a linear relation between Gibbs free energy and
enthalpy. They learned that electron withdrawing groups in dienophile and electron donating groups in diene
favor these reactions kinetically, and on the other hand, the opposite disfavor these reactions. Considering
thermodynamically controlled Diels-Alder reactions, the EDG decrease the equilibrium constant, in relation
to reference reaction, unlike the EWG. We believe that this minicourse gave an important contribution for
Chemistry education at undergraduate level.
42
GRAPHIC FOR MANUSCRIPT
HOMOdiene
≡
LUMOdienophile
KEYWORDS: Diels-Alder reaction; Diene; Dienophile; Computational Chemistry; G4; Frontier Orbital
Theory
INTRODUCTION
Pericyclic reactions[1,2] are one of three major types of reactions, together with polar and radical
reactions. Pericyclic reactions can be divided into four classes, cycloadditions, electrocyclic reactions,
sigma-tropic rearrangements and the less common group transfer reactions. All these categories share a
common feature that is a concerted mechanism involving a cyclic transition state with a concerted
movement of electrons. Concerted periclyclic cycloaddition involves the reaction of two unsaturated
molecules with resulting in the formation of a cyclic product with two new σ-bonds.[3] Examples of
cycloaddition reactions include the 1,3-dipolar cycloaddition (also referred as Huisgen cycloaddition),[4]
nitrone-olefin cycloaddition[5] and cycloadditions like the Diels-Alder reaction which is the most frequently
encountered and most studied of all pericyclic reactions (Scheme 1). The number of π-electrons involved in
the components characterizes the cycloaddition reactions, and for the three abovementioned cases this would
be [2+3], [3+2] and [4+2] respectively.
43
In 1950 the Nobel Prize of chemistry was granted to Otto Diels and Kurt Alder by describing an
important reaction that involves the formation of C-C bond in a six membered ring with high stereochemical
control, named Diels-Alder reaction.[6] Since 1928, the Diels-Alder reaction has been used in the synthesis
of new organic compounds, providing a reasonable way for forming a 6-membered systems.[7–13]
The simplest Diels-Alder reaction involves the reagents butadiene and ethane (dienophile), but it needed
200°C under high pressure although it has low yield[14] (about 18%) (Scheme 1A). Butadiene reacts more
easily with acrolein[15], (Scheme 1B). A methyl substituent on the diene, on C1 or C2, the reaction rate
increases, e.g. trans-piperylene reacting with acrolein[16,17] at 130°C (Scheme 1C). These last two
reactions achieved a yield Close to 80%.
Scheme 1
The standard state thermodynamic functions for Diels-Alder reactions are represented by enthalpy
change, Gibbs free energy change and entropy. The exothermic nature of these reactions are the result of
converting two weaker π-bonds into two stronger σ-bonds.[18] Because it is an addition reaction it has
negative entropy change but Diels-Alder reactions are exergonic due to the high negative entalphy change.
44
The reactivity of Diels-Alder reactions can be reasoned by molecular orbital (MO) theory[19] in
which the HOMO of diene donates electrons to the LUMO of the dienophile by overlapping these orbitals.
The addition of an EDG to the diene results in a better electron donor (and a better nucleophile) by
increasing its HOMO energy. Likewise, a EWG-substituted dienophile generates a better electron acceptor
(and a better electrophile) by decreasing its LUMO energy. [20] As a consequence, the frontier molecular
orbital theory accounts for the dependence of the Diels-Alder reaction rate on the substituent effect. [21] In
the transition state of the Diels-Alder reactions, the HOMO of diene interacts with LUMO of dienophile.
The Scheme 2A shows the energy diagram of butadiene and ethene. The Scheme 2B shows the energy
diagram of butadiene and a substituted dienophile with an EWG whose energy gap is lower than that in
Scheme 2A. Even lower energy gap is reached when an EDG is attached to the C1 or C2 of diene (Scheme
2C). It is well-known that the lower the energy gap between HOMO and LUMO from diene and dienophile,
respectively, the faster the corresponding reaction rate.
Scheme 2
LUMO
LUMO
LUMO
HOMO
HOMO
LUMO
LUMO
LUMO
HOMO
HOMO
HOMO
HOMO
(A)
(B)
(C)
One great challenge for chemistry education research is to improve students' understanding of
physical chemical properties of chemical reactions. One very important tool for chemistry education practice
is computational chemistry which improves the understanding of microscopic properties and/or provides
macroscopic properties more easily. Several successful chemistry education papers focused on using
computational chemistry as the main tool for the study of organic reactions. Hessley[22] used the molecular
modeling to study reaction mechanisms for the formation of alkyl halides and alcohols in order to prevent
the students from memorizing these reactions. Other works used computational chemistry for obtaining the
Hammett plots;[23] for studying the controversial Wittig reaction mechanism;[24] for analyzing the origin of
solvent effects on the rates of organic reactions;[25] and for investigating the kinetic and thermodynamic
aspects of endo vs exo selectivity in the Diels-Alder reactions.[26] However, except for ref. 26 none of these
studies have associated the aspects of physical-chemistry of organic reactions of these reactions.
45
In this work, we used the computational chemistry tool in an undergraduate minicourse in order to
provide chemistry undergraduate students from Federal University of Rio Grande do Norte better
understanding of the influence of substituent groups on the kinetic and thermodynamic properties of DielsAlder reactions. All results from this minicourse were obtained by the students. Due to relatively short time
of the minicourse, each student or pair of students calculated just one reaction out of seven selected reactions
and afterwards all results were collected and analyzed as it is presented in this work.
COMPUTATIONAL DETAILS
Seven Diels-Alder reactions were previously selected and the geometries of all studied species were
optimized by using standard techniques.[27] Frequency calculations were performed to analyze vibrational
modes of the optimized geometry in order to determine whether the resulting geometries are true minima or
transition states. The geometry optimization, frequency calculations and the generation of the HOMO and
LUMO molecular orbitals of all studied species were calculated by G4 method[28] from Gaussian 09
package.[29] Every molecule´s initial geometry was done with GaussView 5[30] and final geometry was
taken from Chemcraft.[31] The G4-based computational method has been successfully used in the study of
pericyclic reactions before.[32]
DESCRIPTION OF THE COMPUTATIONAL CHEMISTRY MINICOURSE
We offered a 15-hour minicourse (divided into 3 hours per day) in July, 2015. Ten students participated
in the minicourse and according to pre-minicourse questionnaire (see Supporting Information) all
participants were undergraduate chemistry students in different grades from Institute of Chemistry of
Federal University of Rio Grande do Norte. All students have completed Organic Chemistry 1 which is the
main prerequisite for this minicourse. About 50% of students did not have any experience with
computational chemistry. This minicourse was part of a teaching project in the Institute of Chemistry in
which the first week at the beginning of the every semester is devoted to a series of minicourses where all
chemistry students must attend and can choose the minicourse(s) that is (are) most convenient. This
minicourse is taught once a year at the beginning of the first or second semester. This project can be part of a
regular Organic Chemistry discipline which includes Diels-Alder reaction or it can be similarly given as
minicourse. All data shown in this paper were provided by the students from calculations performed in the
minicourse. Some time-consuming calculations have lasted nearly one day.
In the first class, the students received a pre-minicourse questionnaire with short-answer questions
(Supporting information) in order to obtain student's academic background prior to the minicourse; After
46
pre-minicourse questionnaire, the students were taught the theoretical content about the Diels-Alder
reactions; the thermodynamic and kinetic properties for analyzing the Diels-Alder reactions; and they were
presented Gaussian 09, GaussView 5 and Chemcraft software. In the second class, the students were
grouped (four pairs of students) and each group or individual student performed the optimization and
frequency calculations of the reagents and products of one selected Diels-Alder reaction shown in Scheme 2
(The G4 calculations occurred during the second class and in the interval between the second and third
classes). In the third class the students analyzed all thermodynamic data in order to obtain student-generated
data in Tables 1-2 and the plots in Figures 1 and 3. In the first half of fourth class the students generated the
HOMO and LUMO molecular orbitals in order to obtain similar MO representations as shown in Figure 2
and to get their energy values and corresponding energy gaps as it is shown in Table 3. In the second half of
fourth class the students analyzed the bond lengths of all reagents and products of the Diels-Alder reactions
in order to obtain data in the Table 4 and Figure 4. In the fifth class, all the results were discussed as
presented in sections 5 to 8 and the students answered the post-minicourse questionnaire (objective and
subjective questions) and the evaluation which are shown in Supporting Information. Eventually, a statistics
(in Supporting Information) were generated with the rates for answers given by the students for objective
questions and student’s evaluation. Three filled post-minicourse subjective questionnaire forms are shown in
Supporting Information.
RESULTS
The Scheme 3 shows all studied Diels-Alder reactions where R1 and R2 represent the substituents of the
diene and dienophile respectively and the products cyclohexene derivatives formed which resulted in seven
Diels-Alder reactions. We used two substituents: one electron donating group (EDG) (-NH2) and one
electron withdrawing group (EWG) (-NO2).
47
Scheme 3
1.
R1 = R2 = H;
2.
R1 = NH2 and R2 = H
3.
R1 = H and R2 = NO2
4.
R1 = NH2 and R2 = NO2
5.
R1 = NO2 and R2 = H
6.
R1 = H and R2 = NH2
7.
R1 = NO2 and R2 = NH2
In Table 1 it is depicted the absolute values of enthalpy (H), Gibbs free energy (G) and entropy (S) of all
reagents (dienes and dienophiles) and products (cyclohexene and cyclohexene derivatives) and the
corresponding values of enthalpy change (∆H), Gibbs free energy change (∆G), temperature and entropy
change product (T∆S) and logarithm of equilibrium constant (ln K) for Diels-Alder reactions (1)-(7) are
shown in Table 2. In Figure 1A it is shown the linear relation involving ∆H and ∆G for Diels-Alder
reactions (1)-(7) whereas in Figure 1B there is the linear relation of ln K and ∆H for Diels-Alder reactions
(1)–(7).
Figure 2 depicts the HOMO of buta-1,3-diene, buta-1,3-dien-2-amine and 2-nitrobuta-1,3-diene and
LUMO of ethene, nitroethene and etheneamine (Figure 2A, 2B and 2C respectively). The energy values of
HOMO and LUMO of the reagents of the Diels-Alder reactions (1)–(7), along with the corresponding
energy gap values, are shown in Table 3.
ΔH (kcal mol-1)
-50
-47.5
-45
-42.5
-40
ln K
-37.5
-35
R² = 0.9839
A
ΔG (kcal mol-1)
-20
-22
-24
-26
-28
-30
-32
-34
-36
-38
-40
35.0
-30
40.0
45.0
50.0
55.0
60.0
65.0
70.0
-35
ΔH (kcal mol-1)
-52.5
R² = 0.9839
-40
-45
-50
-55
B
48
Figure 1. (A) Plot of enthalpy change, ∆H, versus Gibbs free energy change, ∆G, of Diels-Alder reactions
(1)-(7). (B) Plot of ln K versus ∆H of Diels-Alder reactions (1)-(7).
Table 1. Values of enthalpy, H, Gibbs free energy, G, in kcal mol-1, and entropy, S, in cal mol-1 K-1 of all
studied molecules.
Molecule
G/kcal mol-1
H/kcal mol-1
S/cal mol-1 K-1
Ethene
Ethenamine
nitroethene
Buta-1,3-diene
Buta-1,3-dien-1-amine
1-nitrobuta-1,3-diene
Cyclohexene
Cyclohex-2-en-1-amine
Cyclohex-3-en-1-amine
4-nitrocyclohexene
3-nitrocyclohexene
6-nitrocyclohex-2-en-1-amine
2-nitrocyclohex-3-en-1-amine
-49286.36
-84010.42
-177588.33
-97825.36
-132549.78
-226121.56
-147141.70
-181859.99
-181861.41
-275447.53
-275446.12
-310164.56
-310164.17
-49270.75
-83991.90
-177567.34
-97806.25
-132527.46
-226098.50
-147119.59
-181835.93
-181837.44
-275420.66
-275419.62
-310136.10
-310135.86
52.31
62.02
70.25
63.98
74.63
77.14
73.92
80.38
80.09
89.78
88.53
95.04
94.52
49
Table 2. Gibbs free energy change, ∆G, Enthalpy change, ∆H, entropy change and temperature product, in kcal mol -1 K-1, T∆S, entropy change, in kcal mol-1, ∆S,
and equilibrium constant logarithm, ln K, of Diels-Alder reactions (1)–(7).
Reaction
(1) Buta-1,3-diene + Ethene  Cyclohexene
(2) Buta-1,3-dien-1-amine + Ethene  Cyclohex-2en-1-amine
(3) Buta-1,3-diene + Nitroethene  4nitrocyclohexene
(4) Buta-1,3-dien-1-amine + Nitroethene  6nitrocyclohex-2-en-1-amine
(5) 1-nitrobuta-1,3-diene + Ethene  3nitrocyclohexene
(6) Buta-1,3-diene + Ethenamine  Cyclohex-3-en1-amine
(7) 1-nitrobuta-1,3-diene + Ethenamine  2nitrocyclohex-3-en-1-amine
∆G/kcal mol-1 ∆H/kcal mol-1
T∆S/kcal mol-1
∆S/kcal mol-1 K-1
ln K
-29.98
-42.59
-12.63
-0.0424
50.61
-23.85
-37.72
-13.88
-0.0466
40.26
-33.84
-47.07
-13.25
-0.0445
57.12
-26.45
-41.29
-14.86
-0.0498
44.65
-38.21
-50.37
-12.20
-0.0409
64.49
-25.63
-39.29
-13.69
-0.0459
43.26
-32.19
-45.46
-13.31
-0.0446
54.34
50
A
B
C
Figure 2. HOMO and LUMO of buta-1,3-diene and ethene (A); buta-1,3-dien-2-amine and nitroethene (B);
and 2-nitrobuta-1,3-diene and ethenamine (C).
Table 3. Energy values of HOMO (EHOMO) and LUMO (ELUMO) and energy gap between HOMO and
LUMO orbitals, |EHOMO-ELUMO|, in a.u., of the reagents of the Diels-Alder reactions (1) – (7).
Reaction
EHOMO/a.u.
ELUMO/a.u.
|EHOMO-ELUMO|/a.u.
(1)
-0.2293
0.0173
0.2466
(2)
-0.1867
0.0173
0.2040
(3)
-0.2293
-0.0923
0.1370
(4)
-0.1867
-0.0923
0.0944
(5)
-0.2522
0.0173
0.2695
(6)
-0.2293
0.0509
0.2801
(7)
-0.2522
0.0509
0.3031
Figure 3 shows the C-C bond length of the diene and C-C bond length of the dienophile of reactions
(4) (Figure 3A) and (7) (Figure 3B) in which these bond lengths are represented in the Scheme 4 (r1, r2, r3,
r4, r’1, r’2, r’3 and r’4). The values of C-C bond length for Diels-Alder reactions (1)–(7) are listed in Table 4.
51
Scheme 4
A
+
B
+
Figure 3. Optimized geometries of corresponding reactants and products from Diels-Alder reactions 4 (A)
and 7 (B). Selected bond lengths, in Angstroms, are depicted.
52
Table 4. Bond lengths of reagents (r1, r2, r3 and r4) and products (r’1, r’2, r’3 and r’4), in Angstroms, of the
studied Diels-Alder reactions.
Reaction
r1/Å
r2/Å
r3/Å
r4/Å
r’1/Å
r’2/Å
r’3/Å
r’4/Å
(1)
1.336
1.468
1.336
1.327
1.507
1.333
1.507
1.534
(2)
1.345
1.458
1.339
1.327
1.510
1.332
1.506
1.532
(3)
1.336
1.468
1.336
1.323
1.508
1.332
1.506
1.532
(4)
1.345
1.458
1.339
1.323
1.509
1.334
1.504
1.524
(5)
1.330
1.465
1.335
1.327
1.506
1.333
1.504
1.533
(6)
1.336
1.468
1.336
1.335
1.507
1.332
1.507
1.532
(7)
1.330
1.465
1.335
1.335
1.506
1.333
1.504
1.540
THERMODYNAMIC DATA AND EQUILIBRIUM CONSTANT
Enthalpy change values from Table 2 indicate that all reactions are exothermic because two weaker 
bonds in reactants are converted to two stronger  bonds in the product. Cycloaddition reactions have
negative entropy change owing to the smaller translational and rotational contributions of entropy in
cyclohexene adducts compared to those values from reactants (diene and dienophile). The negative entropy
does not contribute to the spontaneity of Diels-Alder reacdrptions. However, according to Gibbs free energy
change values in Table 2 these reactions are exergonic due to low entropy change values in modulus (see
Table 2) and high enthalpy change values in modulus and one can see that the modulus of TS is nearly
three times smaller than the modulus of H, in average. Experimental data show that Diels-Alder reactions
are exothermic and exergonic, e.g., the reaction of buta-1,3-diene and ethene to form cyclohexene[33] which
has an H and G of -40.0 and -27.0 kcal mol-1 respectively; the reaction of cyclopentadiene with itself to
form dicyclopentadiene (H and G of -18.4 and -8.2 kcal mol-1 respectively);[34] and the reaction of
cyclopentadiene and maleic anhydride (H = -24.8 kcal mol-1).[35] Although the reactions are not the same
as those students have calculated, their thermodynamic data is within the same range.
The tendency for a reaction to reach equilibrium at the standard conditions is driven by the standard
Gibbs free energy, ΔG°, which is related to the equilibrium constant [36] according to Eq. 1.
∆𝐺° = −𝑅𝑇𝑙𝑛𝐾 (1)
There is no relationship between the kinetics and thermodynamics for most of chemical reactions,
except for some electron transfer and some radical reactions. [37–39] As previously described, an EDG
attached to the diene and a dienophile substituted with a EWG result in an increase in the rate of a Diels-
53
Alder reaction. However, these effects are not observed for the thermodynamic properties as shown in Table
2. The EDG (amino group) thermodynamically disfavors the Diels-Alder reactions (2), (4) and (6) resulting
in less exothermic and exergonic reactions and hence lower ln K values compared to the reaction (1). On the
other hand, the reactions with EWG (nitro group) substituent [reactions (3), (5) and (7)] have more negative
values of ∆G and ∆H and higher values of ln K which thermodynamically favor these reactions.
The plot from Figure 1A has a very good correlation coefficient indicating that the trend in enthalpy
change for formation of cyclohexene and derivatives is nearly the same as that from corresponding Gibbs
free energy change, which means that entropy change for these cycloaddition reactions does not alter
significantly both trends, though TS values are not negligible (see Table 2).
Since G for all reactions is negative, there is a direct relation between modulus of G and
equilibrium constant, K, according to expressions in Eq. 2.
∆𝐺
(−∆𝐺)
𝑅𝑇
𝐾 = 𝑒 −𝑅𝑇 ∴ ∆𝐺 < 0 ∴ 𝐾 = 𝑒 −
∆𝐺
∴ 𝐾 = 𝑒 𝑅𝑇 (2)
Then, for exergonic Diels-Alder reactions, there is a direct relation between K and |ΔG°|, i.e., the
smaller ΔG° (in modulus) the smaller K. Moreover, there is direct relation between K and reaction yield.
The very good correlation coefficient of the Figure 1B indicates that ΔH can be associated with the K
as well. Thus, we can associate K with stability factors instead of exergonicity. Then, there is a direct
relation between modulus of ΔH (in modulus) and K.
The data in Table 2 indicate that reaction (5) has the highest K value and hence the highest negative
values of ΔG and ΔH. On the other hand, the reactions (2) and (6) have the lowest K values and the lowest
negative values of ΔG and ΔH. The direct proportionality between K and |ΔG| and between K and |ΔH| can
be observed in the decreasing order of K, |ΔG| and |ΔH|: (5) > (3) > (7) > (1) > (4) > (6) > (2).
YIELD AND KINETIC PROPERTIES
The efficiency of a chemical reaction is usually related to reaction yield. In a thermodynamically
controlled reaction, the reaction yield depends on the equilibrium constant and not on the reaction rate. In a
kinetically controlled reaction, yield depends on the reaction rate and the reaction time. In Diels-Alder
reactions, according to frontier molecular orbital theory, the electron-rich diene acts as a nucleophile. The
most important orbital in the diene is the Highest Occupied Molecular Orbital (HOMO). For the formation
of chemical bond, this orbital interacts with the Lowest Unoccupied Molecular Orbital (LUMO) of the
dienophile, which acts as an electrophile. As most Diels-Alder reactions are kinetically controlled, the
54
smaller the HOMO-LUMO energy difference, the greater the charge transfer, the faster the reaction and the
higher its yield.2 The energy diagram (Figure 2A) shows the interaction between the HOMO of the diene
and the LUMO of the dienophile. The energies of HOMO and LUMO depend on the nature of the
substituents in the diene and in the dienophile, in which an EWG decreases the LUMO energy of the
dienophile (Figure 2B) bringing it closer to the HOMO energy of the diene. An EDG-substituted diene
(Figure 2C) further decreases the energy gap between HOMO and LUMO. Thus, EWG-substituted
dienophiles and EDG-substituted dienes result in faster reactions and higher yields in comparison with
unsubstituted dienophiles and dienes.
The Table 3 shows the energy values of HOMO (E HOMO) and LUMO (ELUMO) of the reagents of the
Diels-Alder reactions (1)-(7), and the energy gap between HOMO and LUMO (|EHOMO-ELUMO|). The
reference reaction (1) has 0.2466 a.u. energy gap. The reactions with EDG in diene [reaction (2)] or EWG in
dienophile [reaction (3)] have lower energy gaps as expected and additive effect of both EDG in diene and
EWG in dienophile [reaction (4)] imparts further decrease in energy gap. From the analysis of energy gaps
in Table 3 and assuming that frontier molecular orbital theory can indicate relatively faster or slower
reactions, we could predict lower energy barriers for reactions (2), (3) and (4) which would provide higher
yields for these reactions in comparison with reaction (1). On the other hand, reactions (5), (6) and (7) have
the highest |HOMO-LUMO| values (0.2695, 0.2801 and 0.3031 a.u. respectively) due to inverse electron
demand, i.e. EWG-substituted dienes and/or EDG-substituted dienophiles. Since these reactions have the
highest energy gap, they are the slowest reactions. Therefore, the decreasing order of |HOMO-LUMO| is: (7)
> (6) > (5) > (1) > (2) > (3) > (4).
Figure 2 shows the HOMO of buta-1,3-diene, buta-1,3-dien-2-amine and 2-nitrobuta-1,3-diene and
LUMO of ethene, nitroethene and etheneamine (Figures 2A, 2B and 2C respectively). According to Figures
2B (HOMO of buta-1,3-dien-2-amine and LUMO of nitroethene) and 2C (HOMO of 2-nitrobuta-1,3-diene
and LUMO of etheneamine) some changes occur in the size and shape of their corresponding HOMO and
LUMO in comparison with those orbitals of their parent molecules depicted in Figure 2A (HOMO of buta1,3-diene and LUMO of ethene). When an EDG (-NH3) is attached to the diene, its HOMO includes the
nitrogen lone pair participation resulting in a better nucleophile while LUMO in nitroethene includes oxygen
lone pair participation and –NO2 vicinal carbon orbital distortion towards nitrogen atom (Figure 2B). On the
other hand, HOMO of 2-nitrobuta-1,3-diene (Figure 2C) has no nitrogen lone pair participation and 2nitrobuta-1,3-diene is probably a weaker nucleophile than buta-1,3-dien-2-amine.
GEOMETRIC CHANGES
Scheme 4 represents a general Diels-Alder reaction in which selected bond lengths of reactants and
product are assigned (r1, r2, r3 for dienes, r4 for dienophiles, and r’1, r’2, r’3, r’4 for cyclohexenes). Figure 3
depicts the optimized geometries of reactants and product of reactions (4) (Figure 3A) and (7) (Figure 3B).
55
Surprisingly, data from Table 4 indicate that amino group increases the double bond length of reactants
more than nitro group. Moreover, data from Table 4 also confirm that double bonds r1, r3 and r4 in reactants
become single bonds in products while the single bond r 2 becomes double bond in products.
Except for r’4 in reactions (4) and (7), no other selected bond length of all cyclohexene derivatives is
significantly affected by substituent group. Then, the variations of enthalpy change in the reactions (1) to (7)
are probably accounted for by the reactants than the product. Figure 3 also shows that cyclohexane
derivatives lost the carbon chain planarity from their precursors due to the lost of  bonds in the product.
CONCLUSIONS
The computational chemistry allowed to teach the thermodynamic, electronic (indirectly kinetic) and
geometric aspects of Diels-Alder reactions. All of these reactions were exergonic and exothermic. Entropy
change has an important contribution on the Gibbs free energy change, however it did not affect the linear
relation between the latter and enthalpy change. Then, the thermodynamics of these reactions can be
reasoned by the influence of electronic effects on the stability of reactants and product. However, from the
analysis of geometrical parameters, stability is more affected by reactants than by products because the bond
lengths of reactants vary more significantly in comparison with reference compound due to the substituent
effect than those in the corresponding product.
Students could also understand that substituent effects on thermodynamics and kinetics (indirectly
from frontier orbital theory) are distinguished. The EDG in diene makes Diels-Alder reaction less favorable
thermodynamically, whereas EWG substituent in diene results in more favorable reaction in
thermodynamics. On the other hand, EDG in diene and EWG in dienophile favor the Diels-Alder reaction
kinetically.
From the statistics of post-minicourse questionnaire and students evaluation (Supporting information)
of these minicourse we realized that students could better understand thermodynamic and kinetic aspects of
a reaction from the use of computational chemistry. This minicourse also contributed for the Organic
Chemistry knowledge by giving students practical and simple tools to develop the knowledge by
themselves, in which these tools can also be employed in the teaching of Diels-Alder reactions in the
classroom. This minicourse also contributed for the association between physical-chemistry knowledge
(quantum chemistry, thermodynamics and kinetics, indirectly) with organic chemistry by means of DielsAlder reaction.
ASSOCIATED CONTENT
Supporting Information
56
Pre-minicourse, Post-minicourse questionnaires (objective and subjective questions) and students
evaluation. This material is available via the Internet at http://www.sciepub.com/
ACKNOWLEDGMENTS
Authors thank Fundação de Apoio à Pesquisa do Estado do Rio Grande do Norte (FAPERN),
Coordenação de Ensino de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Conselho Nacional
de Desenvolvimento Científico e Tecnológico (CNPq) for financial support.
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Carey FA, Sundberg RJ. Advanced Organic Chemistry Part A: Structure and Mechanisms. 2007.
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Nicolaou KC, Snyder S a, Montagnon T, Vassilikogiannakis G. The Diels-Alder reaction in total
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Juhl M, Tanner D. Recent applications of intramolecular Diels-Alder reactions to natural product
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laboratory experiment, J Chem Educ, 1968, 45, 55.
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[12] Parsons BA, Dragojlovic V. Demonstration of a Runaway Exothermic Reaction: Diels–Alder
Reaction of (2 E ,4 E )-2,4-Hexadien-1-ol and Maleic Anhydride, J Chem Educ, 2011, 88, 1553–7.
[13] Celius TC. Fast Hetero-Diels-Alder Reactions Using 4-Phenyl- 1,2,4-triazoline-3,5-dione (PTAD) as
the Dienophile, J Chem Educ, 2010, 87, 10–1.
[14] Joshel LM, Butz LW. The Synthesis of Condensed Ring Compounds. VII. The Successful Use of
Ethylene in the Diels—Alder Reaction1, J Am Chem Soc, 1941, 63, 3350–1.
[15] Olah GA, Molnár Á. Hydrocarbon Chemistry. Wiley, 2003.
[16] Anh NT. Frontier Orbitals: A Practical Manual. 2007.
[17] Fleming I. Molecular Orbitals and Organic Chemical Reactions. Wiley, 2010.
[18] Williamson K, Masters K. Macroscale and microscale organic experiments. Cengage Learning, 2011.
[19] Palmer DRJ. Integration of Computational and Preparative Techniques To Demonstrate Physical
Organic Concepts in Synthetic Organic Chemistry: An Example Using Diels-Alder Reactions, J
Chem Educ, 2004, 81, 1633.
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[21] Houk KN. Frontier molecular orbital theory of cycloaddition reactions, Acc Chem Res, 1975, 8, 361–
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Wittig Reaction, J Chem Educ, 2014, 91, 2182–5.
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Organic Reactions in Solution. Transit. State Model. Catal., vol. 721, American Chemical Society,
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a Diels–Alder Reaction, J Chem Educ, 2009, 86, 199.
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Revision A.02, Gaussian Inc Wallingford CT, 2009, 34, Wallingford CT.
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59
for
Teaching aspects of organic chemistry reaction from Diels-Alder cycloadditions by
using computational chemistry – an undergraduation experiment.
Norberto K. V. Monteiro* and Caio L. Firme
*Corresponding author. Tel: +55-84-3215-3828; e-mail: norbertokv@ufrnet.br
PRE-MINICOURSE QUESTIONNAIRE
FEDERAL UNIVERSITY OF RIO GRANDE DO NORTE
CHEMISTRY INSTITUTE
STUDENT´S NAME:_________________________________________________ DATE:____________
1)
What course are you majoring in?
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
2) What is your grade?
______________________________________________________________________________________
3) Which organic chemistry disciplines have you done before?
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
60
4) Do you have any previous experience in computational chemistry?
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
POST-MINICOURSE QUESTIONNAIRE – OBJECTIVE QUESTIONS.
1. Did this minicourse help you to understand Diels-Alder reaction? (a) Very much; (b) Considerably;
(c)Moderately; (d) Hardly ever
2. Did this minicourse help you to understand the frontier orbital theory? (a) Very much; (b)
Considerably; (c)Moderately; (d) Hardly ever
3. Did this minicourse help you to understand the relation between frontier orbital theory, kinetic
approach and yield? (a) Very much; (b) Considerably; (c)Moderately; (d) Hardly ever
4. Did this minicourse help you to understand the relation between thermodynamic approach and yield?
(a) Very much; (b) Considerably; (c)Moderately; (d) Hardly ever
5. How do you evaluate the importance of computational chemistry for understanding chemical
reactions? (a) Very important; (b) Important; (c) Moderately important; (d) Not important
STUDENTS EVALUATION
Grades (1-minimum to 5-maximum) that correspond to students opinion about features of the
Diels-Alder minicourse.
Features
Grade (% of students)
1
I had the prerequisite knowledge to complete
this minicourse?
Depth of content
Backgrounds need
2
3
4
5
61
Visually attractive
Multidisciplinary
Easy to be performed
The objectives of this minicourse were clear?
The computational tool really helped in
understanding?
POST-MINICOURSE QUESTIONNAIRE – SUBJECTIVE QUESTIONS.
FEDERAL UNIVERSITY OF RIO GRANDE DO NORTE
CHEMISTRY INSTITUTE
STUDENT´S NAME:_________________________________________________ DATE:___________
1) List the strong points of the minicourse. Give reason(s) in few words.
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
2) Do you agree that the computational chemistry contribute for better understanding thermodynamics
and kinetics (by means of Frontier Orbital Theory) of organic chemistry reactions? Why?
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
3) What is the relation between HOMO-LUMO energy difference and kinetics in DA reactions? What
is the relation between Gibbs free energy difference and equilibrium constant?
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
62
______________________________________________________________________________________
4) How do the EWG and EDG affect thermodynamic and kinetics of the DA reactions?
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
5) What are the main changes in bond lengths and planarity of reactants and products in DA reactions?
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
STATISTICAL
RESULTS
OF
POST-MINICOURSE
QUESTIONNAIRE
(OBJECTIVE
QUESTIONS)
The figures below show de plots of students percentage choice versus multiple choice questions
1. Did this minicourse help you to understand Diels-Alder reaction?
100
Students (%)
80
60
40
20
0
Very much
Considerably
Moderately
Hardly ever
2. Did this minicourse help you to understand the frontier orbital theory?
63
60
Students (%)
50
40
30
20
10
0
Very much
Considerably
Moderately
Hardly ever
3. Did this minicourse help you to understand the relation between frontier orbital theory, kinetic
approach and yield?
60
Students (%)
50
40
30
20
10
0
Very much
Considerably
Moderately
Hardly ever
4. Did this minicourse help you to understand the relation between thermodynamic approach and yield?
60
Students (%)
50
40
30
20
10
0
Very much
Considerably
Moderately
Hardly ever
5. How do you evaluate the importance of computational chemistry for understanding chemical
reactions?
64
120
100
Students (%)
80
60
40
20
0
Very important
Important
Moderately
important
Not important
STATISTICAL RESULTS OF STUDENTS EVALUATION
The figures below show de plots of students percentage versus Grade (1 to 5)
Students (%)
I had the prerequisite knowledge to complete this minicourse?
45
40
35
30
25
20
15
10
5
0
1
2
3
4
5
Grade
Depth of content
70
Students (%)
60
50
40
30
20
10
0
1
2
3
Grade
Backgrounds need
4
5
65
Students (%)
45
40
35
30
25
20
15
10
5
0
1
2
3
4
5
Grade
Visually attractive
90
80
Students (%)
70
60
50
40
30
20
10
0
1
2
3
4
5
Grade
Multidisciplinary
70
Students (%)
60
50
40
30
20
10
0
1
2
3
Grade
4
5
66
Easy to be performed
70
Students (%)
60
50
40
30
20
10
0
1
2
3
4
5
4
5
Grade
The objectives of this minicourse were clear?
90
80
Students (%)
70
60
50
40
30
20
10
0
1
2
3
Grade
The computational tool really helped in understanding?
120
Students (%)
100
80
60
40
20
0
1
2
3
Grade
4
5
67
POST-MINICOURSE QUESTIONNAIRE ANSWERS – SUBJECTIVE QUESTIONS.
68
69
70
4. CAPÍTULO 2 – Artigo publicado no periódico The Journal of Physical Chemistry A.
Monteiro, N. K. V.; Firme, C. L. The Journal of Physical Chemistry A 2014, 118, 1730.
Hydrogen-hydrogen bonds in highly branched alkanes and in alkane complexes: a DFT, ab initio,
QTAIM and ELF study
Norberto K. V. Monteiro and Caio L. Firme*
*E-mail: caiofirme@quimica.ufrn.br
ABSTRACT
The hydrogen-hydrogen (H-H) bond or hydrogen-hydrogen bonding is formed by the interaction between a
pair of identical or similar hydrogen atoms that are close to electrical neutrality and it yields a stabilizing
contribution to the overall molecular energy. This work provides new, important information regarding
hydrogen-hydrogen bonds. We report that stability of alkane complexes and boiling point of alkanes are
directly related to H-H bond, which means that intermolecular interactions between alkane chains are
directional H-H bond, not nondirectional induced dipole-induced dipole. Moreover, we show the existence
of intramolecular H-H bonds in highly branched alkanes playing a secondary role on their increased
stabilities in comparison with linear or less branched isomers. These results were accomplished by different
approaches: density functional theory (DFT), ab initio, quantum theory of atoms in molecules (QTAIM) and
electron localization function (ELF).
Keywords: hydrogen-hydrogen bond; H-H bond, hydrogen-hydrogen bonding; alkane; QTAIM; ELF.
INTRODUCTION
Bader defined bond path as an atomic interaction line (AIL) which is mirrored by its corresponding
virial path, where the potential energy density is maximally negative 1,2. Bader3,4 has stated that bond paths
71
are indicative of bonded interactions, which encompass all kinds of chemical interactions. Bond paths or
virial paths can be used to characterize four types of bonded interactions involving hydrogen atom:
dihydrogen bond, hydrogen bond, hydride bond and hydrogen-hydrogen bond.
The dihydrogen bond is a bonded interaction consisting of electrostatic interaction between two
hydrogen atoms of opposite charges and its formation is restricted to systems that possess a hydridic
hydrogen wherein transition metals and boron atom are typical elements that can interact with this hydridic
hydrogen5. On the other hand, the hydrogen bond involves only one hydrogen atom and it can be
represented by X-H•••Y interaction, where X and Y are electronegative atoms; X-H is named as a protondonating bond, and Y is an acceptor center.6
An unconventional type of hydrogen bond is named hydride bond, also known as “inverse hydrogen
bond”. In the hydride bond, a negatively charged hydrogen atom will be electron donor for non-hydrogen
acceptors. The hydride bond consists of a hydride placed between two electrophilic centers. 7,8
Recently,9 a new kind of interaction involving hydrogen atoms has been discovered by means of the
quantum theory of atoms in molecule (QTAIM): the hydrogen-hydrogen (H-H) bond or hydrogen-hydrogen
bonding, wherein a bond path links a pair of identical or similar hydrogen atoms that are close to electrical
neutrality, opposing the dihydrogen bonding with positively and negatively charged hydrogens. 5,9 It is
demonstrated that these interactions are found to link hydrogen atoms bonded to both unsaturated and
saturated carbon atoms in hydrocarbon molecules and contribute to molecular stabilization, where each H-H
bond makes a stabilizing contribution of up to 10 kcal/mol to the energy of the molecule.9 These interactions
are found between the ortho-hydrogen atoms in planar biphenyl, between the hydrogen atoms bonded to the
C1-C4 carbon atoms in phenanthrene and other angular polybenzenoids, and between the methyl hydrogen
atoms in the cyclobutadiene, tetrahedrane and indacene molecules corseted with tertiary-tetra-butyl groups.9
Some experimental10–14 and computational15–18 studies exemplify interactions between two
hydrogen atoms, although not differentiating H-H bond from dihydrogen bond. Nonetheless, the interactions
cited in these works occur between similar or identical hydrogen atoms close to electrical neutrality. Some
studies used the term “non-bonding repulsive interaction” to describe the H-H bond.
The H-H bond in planar biphenyl was questioned. 19 However. no other studied molecular system
with hydrogen-hydrogen bond was questioned to the date. Bader 20 argued that there is no repulsion in
rotamers such as ethane or biphenyl, only decrease of nuclear attraction. In reply 21 to Bader´s statements, it
was used an arbitrary set of MO´s (which is obviously not univocal22). Nonetheless, QTAIM is dependent
upon the electron density which, in turn, arises from a variational calculation and it represents “the
distribution that minimizes the energy for any geometry and thus the presence of bond paths represent
spatial accumulation of density that lower the energy of the system”, as stated by Bader. 20 Then, bond path
72
is an universal indicator of bonded interactions. 3 Accordingly, QTAIM has become a method that has
already been successfully applied to understand conventional hydrogen bonds, 23,24 C-H•••O bonds25 and
dihydrogen bonds.26,27
In this work, we show that there are very good linear relations between H-H bond in alkane
complexes and their stabilities and between H-H bond in alkane complexes and their boiling points. We also
demonstrate that there are H-H bonds in highly branched alkanes and we show that the H-H bonds play a
secondary role in the stability of highly branched alkanes. All results in this work were obtained by means of
different approaches: the topology of the virial field function based on QTAIM; the critical points of
electron localization function (ELF); ab initio and density functional theory.
COMPUTATIONAL DETAILS AND THEORETICAL METHODS
The geometries of the studied species were optimized by using standard techniques.28 Vibrational
modes of the optimized geometry were calculated in order to determine whether the resulting geometries are
true minima or transition states. Calculations were performed at ωB97XD29, G430, G4/CCSD(T),31 M062X32
and B3LYP33,34 methods with the 6-311++G(d,p) basis set by using Gaussian 09 package, 35 including the
electron density which was further used for QTAIM calculations. The 6-311++G(d,p) basis set was applied
because the inclusion of the diffuse components in the basis set is desired to describe properly the hydrogenhydrogen bond36. Among the density functionals accounting for dispersion, ωB97XD37,38 contains empirical
dispersion terms and also long-range corrections, exhibiting a van der Waals correction. All topological
information were calculated by means of AIM2000 software39. The algorithm of AIM2000 for searching
critical points is based on Newton-Raphson method which relies heavily on the chosen starting point. 40
Every available alternative for increasing calculation accuracy and choosing different starting points was
done. Integrations of the atomic basins were calculated in natural coordinates in default options of
integration. All integrations yielded 10 -3 to 10-4 order of magnitude for the Laplacian of the calculated
atomic basin. Atomic energies were calculated from the atomic virial approach. All calculated bond paths
are mirrored by their corresponding virial paths (where maximum negative potential energy exists between
two atoms). According to atomic virial approach, the bond paths mirrored by virial paths are indicative of
bonded interaction. The evaluation of the electron localization function (ELF) values, (r), and topological
information of bond critical point (BCP) have been carried out with the TopMoD program. 41,42 For each
alkane complex, the QTAIM coordinates of bond critical point of a selected H-H bond were used to locate
the corresponding ELF BCP. All ELF BCP ´s were confirmed as an index 1 critical point since their
corresponding [r(QTAIM BCP coordinates)] is 0.0000. Fuster and Grabowski24 obtained a 0.9935 coefficient of
determination involving ELF H--- BCP distance and QTAIM H--- BCP distance, which reinforces our
approach for finding ELF bond critical point of H-H bond from QTAIM coordinates.
73
QTAIM AND VAN DER WAALS INTERACTIONS
The QTAIM is an extension of quantum mechanics to an open system which is made up of atomic
basins in a molecule and it can be developed by deriving Heisenberg´s equations of motion from
Schrödinger´s equation wherein any atomic property in a molecule can be calculated. 43
The quantum theory of atoms in molecules was developed by Bader to study the electronic
structure and especially the chemical bond by trespassing the limits of orbital theories when approaching to
the physics underlying atoms in a molecule1,2. This quantum model is considered innovative in the study of
chemical bonding and characterization of intra- or intermolecular interactions. 44,45 According to the QTAIM
quantum-mechanic concepts,46,47 the calculation of gradient of charge density, ∇ρ, of the studied molecular
system is performed through the electron density's Hessian matrix in order to find critical points of the
charge density.48 The critical points of the charge density are obtained where ∇ρ = 0. The bond critical point
is a saddle point on an electron density curvature being a minimum in the direction of the atomic interaction
line and a maximum in the two directions perpendicular to it. 49–51 The BCP’s give information about the
character of different chemical bonds or chemical interactions, for example, metal-ligand interactions, 49,52,53
covalent bonds1,2,54, different kinds of hydrogen bonds, 24,55,56 including dihydrogen bonds26,57,58 and H-H
bond.36,59 The Laplacian of the charge density (∇2ρ) is obtained from the sum of the three eigenvalues of the
Hessian matrix (λ1, λ2 and λ3) of the charge density1,2. The negative (∇2ρ < 0) or positive (∇2ρ > 0) sign of
the Laplacian represents concentration and depletion of the charge density, respectively60.
The H-H bonds exhibit characteristics of closed-shell interaction since they have: low value of
charge density (ρb) (< 6 x 10-2 au); the ratio │λ1│ / λ3 < 1; ∇2ρ>0; and the positive value for the total energy
density Hb that is close to zero. The H-H bond presents BCP topological values similar to those from weak
hydrogen bonds61–63.
All properties of matter are somewhat related to its corresponding density of electronic charge ρ(r),
where one can associate to its topology delineating atoms and the bonding between them. 9 In molecules, the
nuclei of bonded atoms are linked by a line along which the electron density is a maximum with respect to
any neighboring gradient path and it is termed bond path.9 In truth, according to Bader, a bond path has to
be mirrored by its corresponding virial path, otherwise it cannot be regarded as a bond path. In a virial path
the potential energy density is maximally stabilizing. 64 When the molecular system is also in a stable
electrostatic equilibrium, it ensures both a necessary and a sufficient condition for bonding. 65 Bader also
defined bond path as a measured consequence of the action of the density operator along the trajectories that
originate in bond critical point and end in both interacting nuclei. 66 The bond path is based on theorems of
quantum mechanics that govern the interactions between atoms. Only two forces operate in a molecular
system: Feynman and Ehrenfest, which act on the nuclei and electrons, respectively. They both are united in
the virial theorem,67 which relates these forces to the energy of the molecule. 68 This is the essential physics
74
of all bonded interactions that when combined with the properties of the bond critical point, provides all the
features of a chemical bond.
Many studies involving critical points and bond paths in van der Waals and weak interactions were
performed.69–71 The van der Waals interactions were identified as directional bonded interactions because of
the existence of bond paths in solid chloride and N4S4 which are important to understand the structure of
these crystals. Hydrogen-hydrogen bond is also a directional interaction, different from induced dipoleinduced dipole interaction which is not directional. Wolstenholme and Cameron72 showed that crystal
structures of tetraphenylphosphonium squarate, bianthrone, and bis (benzophenone) azine contains a variety
of C-Hδ+•••+δH-C interactions, as well as a variety of C-H•••O and C-H•••Cπ interactions. This study
established similarities and differences between these weak interactions, leading to a better understanding of
H-H bonds. Slater has shown in his paper73 that there are no fundamental differences between van der Waals
and covalent bonding, because they have the same quantum mechanism. 74 According to Feynman,75 the
origin of chemical bond is derived from electron density between nuclei, at the same time the requirement
for occurrence of bond path.
ELF
The electron localization function introduced by Becke and Edgecombe 76 is measured from the
Fermi hole curvature in terms of the local excess kinetic energy density due to Pauli repulsion. It has been
shown that ELF is confined to the [1,0] interval, where the tendency to 1 means that parallel spins are highly
improbable, otherwise opposite spin pairs are highly probable; while the tendency to zero indicates a high
probability of same spin pairs. This interpretation gives ELF function a deep physical meaning. The electron
localization function provides a rigorous basis for the analysis of the wave function and electronic structure
in molecules and crystals.77 The ELF topology has been widely applied for study of chemical bonding.78–82
The topological partitioning of ELF gradient field 83,84 provides two types of basins of attractors which can
be thought as corresponding to atomic cores, bonds and lone pairs: core basins surrounding nuclei with
atomic number Z > 2 and labeled by C(A) where A is the atomic symbol of the element, and valence basins
which are characterized by its synaptic order 85 which is the number of the atomic valence shells in which
they participate. Therefore, there are monosynaptyc, disynaptyc, trisynaptyc basins etc, labeled V(A), V(A,
X), V(A, X, Y), etc, respectively. Monosynaptyc basins correspond to the lone pairs of the Lewis model.
Disynaptyc and trisynaptyc basins correspond to two-center and three center bonds respectively.
The ELF local function, (r), provides the chemical information and (r) provides two types of
points in real space: wandering points where (r)≠0 and critical points where (r)=0. The critical points
are characterized by their index which is the number of positive eigenvalues from hessian matrix of (r).
The topology of the ELF has been applied in the study of hydrogen bond. 86–89 In FH•••N2 complex, the value
75
of ELF at the index 1 critical point between the V(F,H) and V(N) basins is 0.047, while for FH•••NH 3 the
corresponding (r) is 0.226. These values are attributed to weak and strong hydrogen bonds, respectively.
In this work, the ELF was used as a complementary method for evaluating hydrogen-hydrogen bond for
alkane complexes from methane-methane until hexane-hexane.
RESULTS AND DISCUSSION
We analyzed the H-H bonds as intramolecular interactions in some C8H18 isomers and their
contribution to molecular stabilization. Afterwards, we analyzed the H-H bonds as intermolecular
interactions in the case alkane-alkane complexes and its relation with complex stability and boiling point.
Octane isomers: We investigated the possible relation between the amount of H-H bonds and
branched alkane stabilization based on heat of combustion (ΔHcomb) of the corresponding alkanes. The
isomer with the lowest ΔHcomb is the most stable. It is well known that branched alkanes are more stable than
their linear isomers.90 Among some C8H18 isomers (Table 1), the highly branched isomer, 2,2,3,3tetramethylbutane, is the most stable and the linear isomer octane is the least stable. The heats of combustion
were calculated at ωB97xd/6-311++G(d,p) level, since this functional has signiﬁcantly lower absolute errors
when considering the noncovalent attractions and empirical dispersion terms. 91
Table 1. Amount of branching, amount of H-H bonds and heats of combustion (ΔHcomb) in branched
alkanes.
ΔHcomb(kcal/mol) ΔΔHcomb(kcal/mol)
Octane isomers*
Branching
H-H bonds
Octane
0
0
-1647.6
2.6
3-methyl-heptane
1
0
-1647.3
2.3
3,3-dimethyl-hexane
2
0
-1646.9
1.9
2,2,3-trimethyl-pentane
3
2
-1645.8
0.8
2,2,3,3-tetramethyl-butane
4
3
-1645.0
0
*Optimized at ωB97XD/6-311++(d,p) level of theory.
Figure 1 shows the virial graphs of some C8H18 isomers (octane, 3-methylheptane, 3-3dimethylhexane, 2-2-3-trimethylpentane and 2-2-3-3-tetramethylbutane). The averaged value of charge
density of the bond critical point, ρb, from the H-H bonds, is 0.0098 au. which is typical for weak hydrogen
bonds and higher than that for van der Waals interactions.61
76
(A)
(B)
(C)
(D)
1
2
(E)
Figure 1. Virial graphs of C8H18 isomers. (A) Octane, (B) 3-methylheptane, (C) 3,3 dimethylhexane, (D)
2,2,3-trimethylpentane and (E) 2,2,3,3-tetramethylbutane. The geometry optimization of the molecules were
performed at ωB97xd/6-311++G(d,p) level of theory.
On going from octane, 3-methyl-heptane to 3,3-dimethyl-hexane, there is a unitary increase of
methyl branching accompanying an increasing branching stabilization on C 8H18 system (Table 1). However,
no H-H bond was found in this set of alkanes. Then, other factors contribute for 3-methyl-heptane and 3,3dimethyl-hexane stabilizations relative to octane, which is not the focus of this work. In literature, one may
find different explanations for the alkane branching stabilization. Gronert’s work 92 suggests that germinal
steric interactions in linear alkane are relieved on branching. Opposing Gronert´s hypothesis, Wodrich and
Schleyer93 attributed branching stabilization to 1,3-alkyl-alkyl stabilizing interaction (so called
77
protobranching). In a more recent work, Kennitz and collaborators94 attribute alkane branching stabilization
to a greater electron delocalization in branched alkanes relative to linear alkanes.
Nonetheless, when comparing 3,3-dimethyl-hexane, 2,2,3-trimethyl-pentane and 2,2,3,3-tetramethylbutane, there is an increasing trend of intramolecular H-H bond and branching stabilization (Table 1). It was
already been shown that these interactions contribute to energy stabilization in hydrocarbons. 9,36 The
2,2,3,3-tetramethylbutane is the most stable branched C8H18 isomer and it has three H-H bonds, while the
second most stable C8H18 isomer, 2,2,3-trimethylpentane, has two intermolecular H-H bonds. Moreover, in
the series 3-methyl-heptane to 3,3-dimethyl-hexane, the average stabilization, ΔΔHcomb, is 0.35 kcal/mol,
while for 3,3-dimethyl-hexane to 2,2,3,3-etramethyl-butane series, the average stabilization is 0.95 kcal/mol,
which implies the influence of an additional stabilization factor when leaving the first series and going to the
latter.
Additional evidence on stabilization due to H-H bond of highly branched C8H18 isomer was given
by comparing hydrogen atomic energy belonging or not to H-H bond. The atomic energy of hydrogen atoms
belonging to H-H bond (H1 in Figure 1E) are in average 0.80 kcal/mol lower when compared to the
hydrogen atoms not belonging to H-H bond (H2 in Figure 1E). Then, for highly branched alkanes, H-H bond
can be regarded as secondary stabilization factor.
One possible reason for the non-existence of H-H bond in octane, 3-methylheptane and 3,3dimethylhexane can be ascribed to the lack of necessary alignment involving C-H---H-C bonds of vicinal or
germinal methyl groups. Further details on this geometrical requirement will be in the following subsection.
Alkane complexes: In this section we evaluated the H-H bonds in linear bimolecular alkane
complexes: methane-methane, ethane-ethane, propane-propane, butane-butane, pentane-pentane and hexanehexane. These alkane complexes were obtained from two different optimization steps: firstly, the
optimization of each single alkane molecule (the geometry of each single molecule is in Supporting
Information Figure S1); secondly, the optimization of the whole bimolecular complex built from an initial
geometry where each single optimized molecule was arbitrarily positioned. The optimizations were done in
different levels of theory: ωB97XD/6-311++G(d,p), G4/CCSD(T)/6-311++G(d,p), B3LYP/6-311++G(d,p)
(Figure S2, Supporting Information), and M062X/6-311++G(d,p) (Figure S3, Supporting Information).
Only optimized structures of alkane complexes obtained from G4/CCSD(T)/6-311++G(d,p) and ωB97XD/6311++G(d,p) are depicted in Figures 2 and 3, respectively. We emphasize that the single molecules in
Figures 2 and 3 comes from optimized geometries, but both Figures contain randomized non-optimized
initial geometry (to the left) and optimized geometry (to the right) for the complex where individual
molecules are optimized as above-mentioned. Both (columns) are shown for comparison reasons which
depict differences of some geometrical parameters after optimization of the whole complex affording better
understanding of H-H bond.
78
In Figure 2, it is depicted the CH---HC dihedral angle for methane-methane, ethane-ethane,
propane-propane, butane-butane from G4/CCSD(T)/6-311++G(d,p) level of theory. In this case, geometry
optimization proceeded through G4 method. The CCSD(T)/6-311++G(d,p) was used for self-consistent field
and density matrix calculations from the G4-optimized geometry (Figure S4, Supporting Information). Due
to infrastructure limitations, it was not possible to calculate pentane-pentane and hexane-hexane complexes
from G4/CCSD(T)/6-311++G(d,p) level of theory. From Figure 2, one can see that all CH---HC dihedral
angle decreases from non-optimized to optimized structures. Except for ethane-ethane complex, this
decrease is reasonably considerable: in methane-methane (from 17.53 to 0.34o), in propane-propane (from
30.51 to 8.02o), and in butane-butane (from 65.18 to 18.56o), which tends to point both hydrogen atoms in
nearly the same direction. This is a clear indication that hydrogen atoms tend to form an intramolecular
interaction.
RANDOMIZED, NON-OPTIMIZED GEOMETRY
Figure 2.
OPTIMIZED GEOMETRY
Randomized, non-optimized and optimized geometries of methane-methane, ethane-ethane,
propone-propane and butane-butane complexes depicting their corresponding selected CH--HC dihedral
angles, in degrees. The geometry optimization of the molecules were performed
311++G(d,p) level of theory. The dashed lines represent the selected dihedral angles.
at G4/CCSD(T)/6-
79
In Figure 3, it is depicted selected H-H interatomic distances for methane-methane, ethane-ethane,
propane-propane, butane-butane, pentane-pentane and hexane-hexane from ωB97XD/6-311++G(d,p) level
of theory where all molecular pairs are closer after optimization. Except from methane-methane complex,
the optimized structures of the alkane complexes shown in Figure 3 are notoriously different from those in
Figure 2. This is a consequence in the usage of different methods for optimization calculation. The
optimization with ωB97XD functional tends to impose a conformation which increases the number of
intramolecular H-H bonds in the alkane complex. This is confirmed when comparing the virial graphs of G4
and ωB97XD optimized structures in Figures 4 and 5, respectively.
RANDOMIZED, NON-OPTIMIZED GEOMETRY
OPTIMIZED GEOMETRY
80
Figure 3. Randomized non-optimized initial geometry and optimized geometry of methane-methane,
ethane-ethane, propone-propane, butane-butane, pentane-pentane and hexane-hexane complexes depicting
their corresponding selected H-H interatomic distances, in angstroms. The geometry optimization of the
molecules were performed at ωB97XD/6-311++G(d,p) level of theory. The dashed lines represent the
selected interatomic distances.
Figure 4 shows the virial graphs of the optimized alkane complexes using G4/CCSD(T)/6311++G(d,p) level of theory. There are one H-H bond in methane-methane complex and two H-H bonds in
other alkane complexes. The values of charge density of BCP (ρb) and the Laplacian of ρb (∇2 𝜌) indicate
that all H-H bond paths from alkane complexes exhibit the characteristics of closed-shell interactions.
81
Figure 4. Virial graphs of Alkane complexes optimized at G4/CCSD(T)/6-311++G(d,p) level of theory.
Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro) and butane-butane (ButBut) along with the charge density, ρ(r), and Laplacian of the electron density, ∇2 ρ(r) at H-H bond critical,
in au. The others complexes were not optimized due to lack of computational resources.
Figure 5 shows the virial graphs of the optimized alkane complexes using ωB97XD/6-311++G(d,p)
level of theory. There is an increasing number of H-H bond as the number of carbon atoms increase in the
alkane complex. The number of H-H bond increase by a unit from methane-methane to butane-butane. From
butane-butane to hexane-hexane, the increase is two units of H-H bond. In ethane-ethane complex there are
also two C-H intramolecular bonds whose virial paths are curved, meaning less attractive and stable
interactions, with a low value of ρb (about 8.5 × 10-6 at 6.7 × 10-3 ua range values). Accordingly, Table 2
shows the number of H-H bonds in the alkane complexes calculated at ωB97XD/6-311++G(d,p) level of
theory, along with the experimental values of the boiling points of the corresponding alkanes and the socalled complex stability, G(complex-2*isolated), derived from Gibbs energy of complex and two-fold Gibbs
energy of the corresponding isolated alkane, whose equation is: G= Gcomplex - 2Gisolated.
82
Table 2. Number of H-H bonds, boiling point, in °C, Gibbs energy of alkane complexes (G complex), two-fold
Gibbs energy of the corresponding isolated alkane (2*G isolated), and complex stability (G(complex-2*isolated)) of
the alkane complexes.
Complexes*
Number of
H-H bonds
Boiling Point
(°C)
Gcomplex
(kcal/mol)
2*Gisolated
(kcal/mol)
G(complex2*isolated)
(kcal/mol)
Met-Met
1
-162
-50848.0
-50847.8
-0.2
Eth-Eth
2
-89
-100184.2
-100182.2
-2.0
Pro-Pro
3
-42
-149522.3
-149520.2
-2.1
But-But
4
0
-198860.6
-198858.2
-2.3
Pen-Pen
6
36
-248201.1
-248196.5
-4.6
Hex-Hex
8
69
-297540.0
-297534.5
-5.4
*Optimized at ωB97XD/6-311++(d,p) level of theory.
Met-Met
Met-Met
ρ = 0.0000085
0.0032
2ρ = −0.0023
0.0000060
But-But
But-But
Eth-Eth
Eth-Eth
0.0050
0.0043
Pen-Pen
Pen-Pen
Pro-Pro
Pro-Pro
0.0051
0.0034
Hex-Hex
Hex-Hex
0.0057
0.0045
0.0059
0.0045
0.0067
0.0049
Figure 5. Virial graphs of Alkane complexes optimized at ωB97XD/6-311++G(d,p) level of theory.
Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro), butane-butane (But-
83
But), pentane-pentane (Pen-Pen) and hexane-hexane (Hex-Hex) complexes along with the charge density,
ρ(r), and Laplacian of the electron density, ∇2 ρ(r) at H-H bond critical, in au.
We can see that as the number of H-H bonds increase from methane-methane to hexane-hexane, the
G(complex-2*isolated) increases as well. This linear relation is plotted in Figure 6, with a reasonably good
coefficient of determination. In another words, the hydrogen-hydrogen bonds influence on the stability of
alkane complexes.
G(complex-2*isolated) (kcal/mol)
6.40
R² = 0.9428
5.60
4.80
4.00
3.20
2.40
1.60
0.80
0.00
0
2
4
6
8
10
H-H bond paths
Figure 6. Plot of number of H-H bond paths versus ΔG (complex-2*isolated), in kcal/mol, of selected
alkanes at ωB97XD/6-311++G(d,p) level of theory.
From Table 2 there is another linear relation: between alkanes’ boiling point and number of H-H
bonds, which is plotted in Figure 7, whose coefficient of determination is 0.9052. The boiling point is
directly dependent upon the magnitude of the intermolecular interactions which, in turn, is directly related to
the complex stability. Each H-H bond in these complexes contribute to decrease on complex electronic
energy (Ecomplex) compared to the sum of electronic energy of the two corresponding isolated alkanes
(Eisolated), leading to an increased complex stability and boiling point. As a consequence, as the number of H-
84
H bonds increase, the complex stability increases, which leads to an expected increase in the alkane boiling
point. These trends are confirmed with the reasonably good coefficient of determination in Figure 7.
150
150
R² = 0.9052
point (( C)
Boiling
C)
point
Boiling
100
100
R² = 0.9052
Met-met
50
50
0
Eth-Eth
00
-50
2
0
1
4
2
3
6
4
8
5
6
-50
10
7
8
Pro-Pro
• But-But
-100
Pen-Pen
-150
Hex-Hex
-100
-150
-200
-200
H-H bond paths
Number of H-H bond paths
Figure 7. Plot of number of H-H bond paths, from ωB97XD/6-311++G(d,p) wave function, versus boiling
point, in °C, of selected alkanes.
It is well known the correlation between boiling/melting point in linear alkanes with increasing
number of carbon atoms.90,95 The intermolecular attractive forces acting in neutral species are known as van
der Waals forces based on non-directional induced dipole-induced dipole interactions. However, the results
from this work strongly indicate that the intramolecular forces in alkane complexes are consequence of
directional attractive H-H bonds.
In order to demonstrate the stabilizing effect of H-H bond, the atomic energies of a hydrogen atoms
belonging and not to H-H bond were obtained from atomic virial theorem of QTAIM. Figures 8 and 9 show
the energy of a hydrogen atom belonging (E H(H-H)) and not belonging (EH) to H-H bond, respectively, versus
energy of the ethane-ethane complex in some selected optimization steps, which includes the first and the
last steps of optimization plus five other points (steps 1, 22, 27, 33, 43, 49 and 52). Figure S5, in
Supporting Information, shows the plot of energy, E, in hartrees, versus optimization steps of ethane-ethane
85
complex. The decrease of EH(H-H) of hydrogen atom belonging to H-H bond versus E (Figure 8) reveals the
stabilizing effect of this type of interaction in the ethane-ethane complex. On the other hand, there is no
decrease in overall EH of a hydrogen atom not belonging to H-H bond in ethane-ethane complex (Figure 9).
Then, the decreasing trend in alkane complex energy is accompanied with the decreasing trend of hydrogen
atoms from H-H bonds, while no overall change in hydrogen energy not belonging to H-H bond occurs
when alkane complex reaches the minimum in the potential energy surface.
E (hartrees)
-159.654
-159.653
-159.652
-159.651
-159.650
-159.649
-0.6200
-0.6210
-0.6215
EH(H-H) (hartrees)
-0.6205
-0.6220
-0.6225
Figure 8. Plot of energy, in hartrees, of a hydrogen atom belonging to H-H bond (EH(H-H)) in ethane-ethane
complex versus the energy (E) in ethane-ethane complex of selected steps in the optimization steps.
86
E (hartrees)
-159.654
-159.653
-159.652
-159.651
-159.650
-159.649
-0.6204
-0.6206
-0.6208
-0.6212
-0.6214
-0.6216
EH (hartrees)
-0.6210
-0.6218
-0.6220
-0.6222
-0.6224
Figure 9. Plot of energy, in hartrees of a hydrogen atom not belonging to H-H bond (EH) in ethane-ethane
complex versus the energy (E) in ethane-ethane complex of selected steps in the optimization steps.
A complementary ELF analysis of H-H bond in alkane complexes (from methane-methane until
hexane-hexane) was carried out. The coordinates of QTAIM bond critical point from a selected H-H bond in
each alkane complex was used for locating the corresponding ELF BCP. The calculation of gradient of ELF
function at QTAIM BCP coordinates confirmed the ELF BCP for all alkane complexes. These BCP’s are
index 1 critical points, (r), whose values are 0.010, 0.015, 0.021, 0.017, 0.026 and 0.025 for the methanemethane, ethane-ethane, propane-propane, butane-butane, pentane-pentane and hexane-hexane complexes,
respectively (Table 3). These (r) values are nearly half the (r) values for weak hydrogen bond.89
Furthermore, the electron density and Laplacian of electron density values at these points have topological
features of closed-shell interactions. Then, ELF is capable to identify H-H bond as well as QTAIM,
indicating that each H-H bond in alkane complexes is weaker than a weak hydrogen bond.
87
Table 3. ELF value, η(r), electron density, ρ(r), and Laplacian of the electron density, ∇2 ρ(r) at bond
critical point of a selected H-H bond in alkane complexes.
Complex
η(r)
ρ(r)
𝛁2 ρ(r)
Methane-methane
0.010
0.0032
0.0093
Ethane-ethane
0.015
0.0053
0.0180
Propane-propane
0.021
0.0051
0.0137
Butane-butane
0.017
0.0050
0.0158
Pentane-pentane
0.026
0.0067
0.0193
Hexane-hexane
0.025
0.0067
0.0197
CONCLUSIONS
This work gives new insights and understanding about intramolecular and intermolecular
interactions in alkanes.
There are H-H bonds in highly branched C8H18 isomers, but not in linear or less branched alkanes.
Then, the hydrogen-hydrogen bond plays a secondary role on the stability of branched alkanes which partly
accounts for the increased stability of highly branched alkanes when compared with linear or less branched
isomers.
We found very good linear relations between H-H bond in complex alkanes and their stabilities and
between H-H bond in alkanes and their boiling points. For all studied molecular systems, the hydrogenhydrogen bonds perform a stabilizing interaction. The hydrogen-hydrogen bond plays an important role on
the boiling point of linear alkanes. Intermolecular interactions in alkane complexes are related to H-H bond,
not to non-directional induced dipole-induced dipole.
The electron localization function can also be used for identification of hydrogen-hydrogen bond.
The ELF topological approach enables a clear identification of hydrogen-hydrogen bonds in alkane
complexes whose (r) are smaller than that from a weak hydrogen bond.
ACKNOWLEDGMENTS
Authors thank, Fundação de Apoio à Pesquisa do Estado do Rio Grande do Norte (FAPERN),
88
ASSOCIATED CONTENT
Optimized geometries of alkane molecules; Virial graphs of Alkane complexes optimized at
B3LYP/6-311++G(d,p), M062X/6-311++G(d,p) levels and G4 method; plot of energy optimization of
ethane-ethane complex. This material is available free of charge via the Internet at http://pubs.acs.org.
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94
TABLE OF CONTENTS
150
150
R² = 0.9052
point (( C)
Boiling
C)
point
Boiling
100
100
R² = 0.9052
Met-met
50
50
0
Eth-Eth
00
-50
2
0
1
4
2
3
6
4
8
5
6
-50
10
7
8
Pro-Pro
• But-But
-100
Pen-Pen
-150
Hex-Hex
-100
-150
-200
-200
H-H bond paths
Number of H-H bond paths
95
Hydrogen-hydrogen bonds in highly branched alkanes and in alkane complexes: a DFT, ab initio,
QTAIM and ELF study
Norberto K. V. Monteiro 1, and Caio L. Firme1*
6
5
1
2
3
4
b)
a)
c)
d)
e)
Figure S1. Optimized geometries of alkane molecules. The alkane molecules were optimized at (a)
B3LYP/6-311++G(d,p), (b) M062X/6-311++G(d,p), (c) G4 method, (d) G4/CCSD/6-311++G(d,p) (T) and
(e) wB97XD/6-311++G(d,p). (1) Methane, (2) ethane, (3) propane, (4) butane, (5) pentane and (6) hexane.
96
Met-Met
Met-Met
Eth-Eth
Eth-Eth
Pro-Pro
Pro-Pro
But-But
But-But
Pen-Pen
Pen-Pen
Hex-Hex
Hex-Hex
Figure S2. Virial graphs of Alkane complexes optimized at B3LYP/6-311G++(d,p) level of theory.
Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro), butane-butane (ButBut), pentane-pentane (Pen-Pen) and hexane-hexane (Hex-Hex) complexes.
97
Pro-Pro
Met-Met
Met-Met
Eth-Eth
Eth-Eth
Pro-Pro
But-But
But-But
Pen-Pen
Pen-Pen
Hex-Hex
Hex-Hex
Figure S3. Virial graphs of Alkane complexes optimized at M062X/6-311G++(d,p) level of theory.
Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro), butane-butane (ButBut), pentane-pentane (Pen-Pen) and hexane-hexane (Hex-Hex) complexes.
98
Met-Met
Met-Met
Eth-Eth
Eth-Eth
But-But
But-But
Pen-Pen
Pen-Pen
Pro-Pro
Pro-Pro
Hex-Hex
Hex-Hex
Figure S4. Virial graphs of Alkane complexes optimized at G4 method. Methane-methane (Met-Met),
ethane-ethane (Eth-Eth), propane-propane (Pro-Pro), butane-butane (But-But), pentane-pentane (Pen-Pen)
and hexane-hexane (Hex-Hex) complexes.
99
Step N
0
10
20
30
40
50
60
-159.6500
-159.6505
E (Hartree)
-159.6510
-159.6515
-159.6520
-159.6525
-159.6530
-159.6535
-159.6540
Figure S5. Plot of energy (E), in Hartree, of ethane-ethane complex versus of optimization steps.
70
100
5. CAPÍTULO 3 – Artigo publicado no periódico New Journal of Chemistry.
Monteiro, N. K. V.; de Oliveira, J. F.; Firme, C. L. New Journal of Chemistry 2014, 38, 5892.
Stability and Electronic Structure of Substituted Tetrahedranes, Silicon and Germanium Parents – a
DFT, ADMP, QTAIM and GVB study
Norberto K. V. Monteiro, José F. de Oliveira and Caio L. Firme*
*Corresponding author. Tel: +55-84-3215-3828; e-mail: caiofirme@quimica.ufrn.br
Graphical Abstract
REF = 0
G= -15.5 kcal.mol-1
ΔGf = 20.5 kcal.mol-1
G = -35.9 kcal.mol-1
Abstract
The electronic structure and the stability of tetrahedrane, substituted tetrahedranes and silicon and
germanium parents have been studied at ωB97XD/6-311++G(d,p) level of theory. The quantum theory of
atoms in molecules (QTAIM) was used to evaluate the substituent effect on the carbon cage in the
tetrahedrane derivatives. The results indicate that electron withdrawing groups (EWG) have two different
behaviors, ie., stronger EWG makes the tetrahedrane cage slightly unstable while slight EWG causes a
greater instability in the tetrahedrane cage. On the other hand, the sigma electron donating groups, σ-EDG,
stabilizes the tetrahedrane cage and π-EDG leads to tetrahedrane disruption. NICS and D3BIA indices were
used to evaluate the sigma aromaticity of the studied molecules, where EWGs and EDGs results in decrease
and increase, respectively, of both aromaticity indices, showing that sigma aromaticity plays an important
role in the stability of tetrahedrane derivatives. Moreover, for tetra-tert-butyltetrahedrane there is another
stability factor: hydrogen-hydrogen bonds which imparts a high stabilization in this cage. Generalized
101
valence bond (GVB) was also used to explain the stability effect of the substituents directly bonded to the
carbon of the tetrahedrane cage. Moreover, the ADMP simulations are in accordance with our
thermodynamic results indicating the unstable and stable cages under dynamic simulation.
Keywords: Tetrahedrane; QTAIM; NICS; D3BIA; GVB; ADMP
1. Introduction
The development of new high energy molecules has aroused great interest of theoreticians and
experimentalists, with applications in the military industry and others high-tech fields.1 Cage strained
organic molecules can be used as potential explosives in response to various external stimuli such heat,
shock or impact.1–4 The highly-strained cage compounds have raised interest due to their high density and
high energy. As a consequence, they can also be used as energetic materials, such as
hexanitrohexaazaisowurtzitane
1,5
and octanitrocubane.6 Similarly, tetrahedrane is also a high-energy
molecule. Despite many attempts to isolate the parent tetrahedron (unsubstituted – C4H4) were not
successful,7 Maier et al8 in 1978 were able to synthesize the first tetrahedrane derivative, the tetra-tertbutyltetrahedrane.
Tetrahedrane is one of the most strained organic molecule, with highly symmetrical structure and
unusual bonding being known as a Platonic solid. It has a strain energy of 586 kJ.mol-1 according to Wiberg
et al
9
which makes this compound thermodynamically unstable. As a consequence, their synthesis and
isolation tends to be extremely difficult. Conversely, tetrahedrane has σ-aromaticity10 because it has a large
negative NICS (Nucleus Independent Chemical Shift ) value, which can somehow contribute for decreasing
the high instability of tetrahedrane cage.
Tetrahedrane and its derivatives can be obtained from the corresponding cyclobutadiene or
derivatives. The anti-aromatic and highly-reactive cyclobutadiene has only been isolated at low temperature
in inert matrices11 and immobilized more recently at room temperature in hemicarcerand.12 On the other
hand, structural parameters of substituted cyclobutadienes have been reported showing pronounced bond
alternations,
13,14
although only slight CC bond length alternations (1.464 and 1.482 Å), under room
temperature, of tetrakis (tert-butyl) cyclobutadiene
15
have been reported. In fact, measurements at -150 °C
showed noticeably larger bond alternation (1.441 and 1.526 Å) for tetrakis (tert-butyl) cyclobutadiene as
well.
Tetra-tert-butyltetrahedrane can be formed from tetra-tert-butylcyclobutadiene by photolysis using noctane as solvent while the reverse reaction takes place thermally with an activation barrier of 26 kcal mol-1
16
. Similarly, an ab initio study of the interconversion of the C4H4 system predicts an activation energy of
about 30 kcal mol-1.17 Maier and co-workers succeeded in synthesizing tetrakis (tert-butyl)tetrahedrane by
102
photochemical isomerization of the corresponding cyclobutadiene.8 The reason for its stability is attributed
to the voluminous tBu-substituents that avoid the tetrahedrane skeleton from ring-opening due to the van der
Waals strain among them, the so-called “corset effect”. However, this effect is lost if one of the tert-butyl
substituents is replaced with a smaller group. Indeed, phenyl- and methyl-substituted tetrahedrane
derivatives were not detected, even under matrix isolation conditions.18
Recently, it was synthesized the
tetrakis (trimethylsilyl) tetrahedrane ((Me3Si)4THD) by
photochemical isomerization of the corresponding cyclobutadiene ((Me3Si)4CBD),19,20 where (Me3Si)4THD
has enhanced thermal stability due the four σ-donating trimethylsilyl groups that electronically stabilize the
highly
strained
tetrahedrane
skeleton.21,22
Indeed,
it
was
also
reported
the
synthesis
of
perfluoroaryltetrahedranes with extended σ-π conjugation between the strained tetrahedrane core and the
aromatic ring.23 In addition, the insertion of a heteroatom (N, O, S, Ge, Si or a halogen atom) into the
tetrahedrane core would be interesting because the tetrahedrane σ framework has the potential to interact
with the nonbonding orbitals on the heteroatom, according to molecular orbital theory. However, such
heteroatom-substituted tetrahedrane derivatives have remained elusive because of the synthetic difficulty in
preparing such molecules.24
In this work tetrahedrane and its derivatives where studied from different tools such as theory of
density functional (DFT), quantum theory of atoms in molecules (QTAIM), atom-centered density matrix
propagation (ADMP) and generalized valence bond (GVB) in order to investigate their electronic structures,
thermochemical and dynamical stabilities that eventually lead to the substituents that enhance the stability
of tetrahedrane cage. The results from QTAIM, GVB and the aromaticity indices provided new and
important information about the main structural and electronic effects associated with the tetrahedrane cage
instability and the reasons why some substituents increase its stability. Moreover, this paper also provides
information about the stability of cages containing germanium and silicon parents of tetrahedrane and
tetrakis(trifluoromethyl) tetrahedrane.
2. Computational details
The geometries of the studied species were optimized by using standard techniques. 25 Frequency
calculations were performed to analyze vibrational modes of the optimized geometry in order to determine
whether the resulting geometries are true minima or transition states. Calculations were performed at
ωB97XD/6-311++G(d,p)26 level of theory by using Gaussian 09 package, 27 including the electronic density
which was further used for QTAIM calculations. ωB97XD functional incorporates an empirical dispersion
term (D) that improves treatment of non-covalent complexes, it uses 100% Hartree Fock (HF) exchange for
long-range interactions and it has an adjustable parameter (X) to include short-range exact exchange.28,29
103
Mohan and co-workers30 have shown that ωB97XD functional exhibited better performance in the
description of bonded and nonbonded interactions in comparison with other DFT methods. All topological
information
31,32
were calculated by means of AIM2000 software. 33 The algorithm of AIM2000 for
searching critical points is based on Newton-Raphson method which relies heavily on the chosen starting
point.34 Integrations of the atomic basins were calculated in natural coordinates with default options of
integration.
The valence bond package VB200035, version 2.5, was used for all generalized valence bond, GVB,
calculations. The GVB singly occupied orbitals were calculated from VB/6-31G level theory and they were
generated by means of Molekel visualization program.36 The package VB2000 generates nonorthogonal
singly occupied orbitals from a general implementation of group function theory (GFT) 37 and modern VB
methods based on high efficiency of the algebrant algorithm.35,38
Nucleus Independent Chemical Shift (NICS)39 and Density, Degeneracy, Delocalization-Based Index
Aromaticity (D3BIA)40 were applied to determine the aromaticity in tetrahedrane and its derivatives. D3BIA
is based on density of (homo)aromatic ring, degeneracy and uniformity of the electron density among the
atoms of the aromatic ring. We will use the same D3BIA formula for homoaromatic species 41 where the
electron density is associated with the charge density of the ring critical point (for 3c-2e bonding systems,
for example) or of the cage critical point (for tridimensional 4c-2e bonding system). The degeneracy (𝛿) is
associated with the atomic energy of the constituent atoms of the aromatic ring of the molecular system.
Another important electronic feature in D3BIA is the uniformity of the electronic density calculated from
the delocalization index of atoms of the aromatic ring. Then, D3BIA for homoaromatic especies is defined
as:
𝐷3𝐵𝐼𝐴 = 𝜌(3, +3) ∙ 𝐷𝐼𝑈 ∙ 𝛿
(1)
Where ρ is the ring density factor of the cage critical point (3,+3), for cage structures, and 𝛿 is the
degree of degeneracy where its maximum value (𝛿=1) is given when the difference of energy between the
atoms of the cage is less than 0.009 au. The formula of 𝛿 is given in Eq. (2).
𝛿=
𝑛
(2)
𝑁
Where n is the number of atoms whose ∆𝐸 (Ω) ≤ 0.009𝑎𝑢. and N is the total number of atoms in the
homoaromatic circuit. The minimum value for n is 1, where there is no atomic pair whose ∆𝐸 (Ω) ≤
0.009𝑎𝑢.
The DIU is the delocalization index uniformity among bridged atoms given by Eq. (3):
𝐷𝐼𝑈 = 100 −
100𝜎
𝐷𝐼
(3)
104
Where σ is mean deviation and 𝐷𝐼 is the average of delocalization index between C-C, Si-Si or GeGe bonds of the cage. The DI gives the amount of electron(s) between any atomic pair. 31
The NICS values were calculated with the B3LYP/6-311++G(d,p) through the gauge-independent
atomic orbital (GIAO) method42 implemented in Gaussian 09. The magnetic shielding tensor was calculated
for ghost atoms located at the geometric center of the cage.
The ab initio molecular dynamics simulations involves quantum chemistry calculations aiming to
obtain the potential energy and nuclear forces.43–46 The simulations performed in this work involved the
atom-centered density matrix propagation (ADMP)47–50 in order to study the stability of tetrahedrane and
selected substituted tetrahedranes. The ADMP calculations were performed at B3LYP/6-31G(d) level of
theory in Gaussian 09. A time step of 0.1 fs was used for the ADMP trajectories. The Nosé-Hoover
thermostat51,52 was employed to maintain a constant temperature at 298.15K. The default fictitious electron
mass was 0.1 amu. A maximum number of 50 steps were used in each trajectory.
3. Results and discussion
Scheme 1 shows the reaction route (diazocompounds 1a-1i  cyclobutadienes 2a-2i 
tetrahedranes 3a-3i) to synthesize tetrahedrane (and derivatives) from its precursors diazocompound and
cyclobutadiene (and derivatives).
Scheme 1
R1=R2=R3=R4 = -H (1a)
= -CH3 (1b)
= -C(H3C) 3 (1c)
= -CF3 (1d)
= -SiF3 (1e)
R1=R2=R3=H and R4 = -NO2 (1f)
= -NH2 (1g)
= -OH (1h)
= -NO2 (1i)
Different
substituted
3
2
1
R1=R2=R3=R4 = -H (2a)
= -CH3 (2b)
= -C(H3C) 3 (2c)
= -CF3 (2d)
= -SiF3 (2e)
R1=R2=R3=H and R4 = -NO2 (2f)
= -NH2 (2g)
= -OH (2h)
= -NO2 (2i)
tetrahedranes
tetramethyltricyclo[1.1.0.02,4]butane,
3b,
were
studied:
R1=R2=R3=R4 = -H (3a)
= -CH3 (3b)
= -C(H3C) 3 (3c)
= -CF3 (3d)
= -SiF3 (3e)
R1=R2=R3=H and R4 = -NO2 (3f)
= -NH2 (3g)
= -OH (3h)
= -NO2 (3i)
tricyclo[1.1.0.02,4]butane,
3a,
1,2,3,4-
1,2,3,4-tetrakis(trifluormethyl)tricyclo[1.1.0.02,4]butane,
3c,
1,2,3,4-tetrakis(trifluorosilyl)tricyclo[1.1.0.02,4]butane, 3d, tetra-tert-butyltricyclo[1.1.0.02,4]butane, 3e, 1-
105
nitro tricyclo[1.1.0.02,4]butane, 3f, 1,2,3,4-tetraamino tricyclo[1.1.0.02,4]butane, 3g, 1,2,3,4-tetrahidroxy
tricyclo[1.1.0.02,4]butane, 3h, and 1,2,3,4-tetranitro tricyclo[1.1.0.02,4]butane, 3i.
In addition, Ge4 and Si4
parents of tetrahedrane, 4a-4b, and tetrakis(trifluoromethyl) tetrahedrane, 5a-5b, were also studied (Scheme
2).
Scheme 2
4
5
R1=R2=R3=R4 = -H (4a)
= -CF3 (4b)
R1=R2=R3=R4 = -H (5a)
= -CF3 (5b)
In scheme 3 shows the structures of tetraamino-substituted tetrahedrane, 3g, tetrahydroxi-substituted
tetrahedrane, 3h, tetranitro-substituted tetrahedrane, 3i, and tetratrifluoromethyl-substituted silicon parent of
tetrahedrane, 4b, that were optimized and yielded the corresponding open structures 3g’, 3h’, 3i’ and 4b’,
which prevented any structural study with these molecules.
Scheme 3
3g
3h
3i
3g’
3h’
3i’
4b
4b’
106
The Scheme 3 indicates that 3g, 3h, 3i and 4b do not have stable tetrahedrane cage. However, no
unique pattern can be established to account for this fact. The –NH2 and –OH groups are electron
withdrawing groups (EWG) by inductive effect and electron donating groups (EDG) by resonance effect. On
the other hand, the –NO2 and –CF3 do not have dual behavior and they are EWGs. Moreover, for
tetrahedrane cage four –CF3 do not disrupt the tetrahedrane cage while four –NO2 groups do so. Conversely,
in Si4 tetrahedrane parent, -CF3 disrupts the corresponding cage.
The equilibrium geometries and selected bond lengths, in Angstroms, of tetrahedrane and
derivatives are depicted in Figure 1. The corresponding virial graphs are shown in Figure 2, along with
several topological data of selected bond critical points. Electronic density, ρ(r), Laplacian of the electron
density, ∇2 ρ(r), local energy density, H(r) and relative kinetic energy density, G(r)/ρ values are shown for
critical points cage-substituent and electronic density, ρ(r), Laplacian of the electron density, ∇2 ρ(r), local
energy density, H(r), relative kinetic energy density, G(r)/ρ, delocalization index, DI, bond order, n, and
ellipticity are shown for critical points of carbon, silicon or germanium atoms in the tetrahedrane (or parent)
cage. From Figure 1, we can see that the alkyl and silyl groups impart the shortening of C-C bonds from the
cage while the –CF3 and –NO2 groups lengthen the corresponding C-C bonds. As a consequence, based only
on geometrical information, we can assume that alkyl and silyl groups behave as EWGs. These EWGs and
EDGs do not disrupt the tetrahedrane cage.
We have analyzed the topology of the electron density of all species using the quantum theory of
atoms in molecules (QTAIM).31,53,54 According to the topological analysis of QTAIM, when ρ of the critical
point is relatively high (×10-1 au.) and ∇2ρ < 0, the chemical interaction is defined as shared shell and it is
applied to covalent bond. However, other parameters are required to describe the nature of the chemical
interaction such as the local energy density, H(r), which is the sum of local kinetic energy density, G(r), and
local potential energy density, V(r), respectively, and the ratio G(r)/ ρ. Cremer and Kraka55 suggested that
H(r) < 0 and G(r)/ρ < 1 are indicative of shared shell (or covalent) interaction. In Figure 2 the values of ρ,
∇2ρ, H(r) and G(r)/ρ of C-C, Si-Si and Ge-Ge bonds of the tetrahedrane cage (or parent) indicate that all CC, Si-Si and Ge-Ge bond paths are shared shell interactions for 3a – 3f, 4a, 5a and 5b. The ellipticity (ε)
values in the tetrahedranes indicate no cylindric symmetry on the single C-C bond, while for Si-Si and GeGe cages the ε is closer to zero indicating symmetry around the distribution of electronic density of these
bonds.
107
5
1
8
4
6
2
3
7
(3a)
A
(3b)
B
(3c)
C
(3d)
D
(4a)
G
(3e)
E
(5a)
H
(3f)
F
(5b)
I
Figure 1: Optimized geometries of the substituted tetrahedranes (A-F), silicon and germanium cages
(G-I) with corresponding C 1 -C4, C1-C5, C1-Si, C1 -N, Si-Si, Si-H, Ge-Ge, Ge-H and Ge-C bond lengths
(in Angstroms); values of selected angles (in degree).
108
ρ = 0.2539
2ρ = 0.1515
H(r) = -0.217
G(r)/ρ = 0.256
ρ = 0.2801
2ρ = 0.2442
H(r) = -0.280
G(r)/ρ = 0.129
ρ = 0.2230
2ρ = -0.015
H(r) = -0.174
G(r)/ρ = 0.847
DI = 1.025
n = 1.02
ε = 0.105
(3a)
A
ρ = 0.2748
2ρ = 0.1835
H(r) = -0.244
G(r)/ρ = 0.222
ρ = 0.2241
2ρ = -0.017
H(r) = -0.175
G(r)/ρ = 0.857
DI = 0.953
n = 0.94
ε = 0.109
(3b)
B
(3d)
D
ρ = 0.2202
2ρ = -0.016
H(r) = -0.169
G(r)/ρ = 0.838
DI = 1.016
n = 1.01
ε = 0.213
(4a)
G
ρ = 0.2316
2ρ = 0.0007
H(r) = -0.190
G(r)/ρ = 0.819
DI = 1.012
n = 1.00
ε = 0.099
ρ = 0.2996
2ρ = 0.1685
H(r) = -0.277
G(r)/ρ = 0.362
(3f)
F
(3e)
E
ρ = 0.1469
2ρ = 0.0683
H(r) = -0.062
G(r)/ρ = 0.595
ρ = 0.0699
2ρ = -0.0034
H(r) = -0.026
G(r)/ρ = 0.420
DI = 0.951
n = 1.25
ε = 0.063
ρ = 0.2191
2ρ = -0.017
48
H(r) = -0.168
G(r)/ρ = 0.842
47 DI = 0.948
n = 0.94
ε = 0.092
(3c)
C
ρ = 0.1567
2ρ = 0.0748
H(r) = -0.115
G(r)/ρ = 0.259
ρ = 0.1379
2ρ = 0.0853
H(r) = -0.110
G(r)/ρ = 0.181
ρ = 0.0783
2ρ = 0.0145
H(r) = -0.036
G(r)/ρ = 0.280
DI = 0.918
n = 1.60
ε = 0.039
ρ = 0.2198
2ρ = -0.017
H(r) = -0.168
G(r)/ρ = 0.845
DI = 0.980
n = 0.97
ε = 0.089
ρ = 0.2531
2ρ = 0.1494
H(r) = -0.213
G(r)/ρ = 0.252
ρ = 0.0695
2ρ = -0.0035
H(r) = -0.026
G(r)/ρ = 0.420
DI = 0.912
n = 1.18
ε = 0.043
ρ = 0.1339
2ρ = 0.0290
H(r) = -0.072
G(r)/ρ = 0.325
(5a)
H
(5b)
I
Figure 2: Virial graphs of substituted tetrahedranes (A-F), silicon and germanium cages (G-I) with the
electronic density, ρ(r), Laplacian of the electron density, ∇2 ρ(r), local energy density, H(r) and relative
kinetic energy density, G(r)/ρ at cage-substituent bond critical, in au. Values of electronic density, ρ(r),
Laplacian of the electron density, ∇2 ρ(r), local energy density, H(r), relative kinetic energy density, G(r)/ρ,
delocalization index, DI, bond order, n, and ellipticity, ε, from carbon, silicon or germanium atoms in the
tetrahedrane (or parent) cage for 3a – 3f, 4a, 5a and 5b.
The virial graph of tetra-tert-butyltetrahedrane, 3c, in Figure 2C indicates a different reason for its
stability rather than the so-called “corset effect” which is a steric repulsion between bulk substituents
preventing the breaking of the cage. 8 There are 23 intramolecular hydrogen-hydrogen (H-H) bonds between
tert-butyl groups according to the virial graph of 3c where the average charge density of the corresponding
critical point, ρH-H, is 0.0041 au. The augmented cooperative effect of these small interactions contributes for
109
the stabilization of the molecular system. 56 An important study based on QTAIM, ELF and ab initio
calculations demonstrated the stability effect of the H-H bonds in alkane complexes and highly branched
alkanes.
57
As to the derivative 3c, according to the values of the hydrogen atomic energy belonging or not
to H-H bond, the hydrogens atoms belonging to H-H bond (e.g., H48 in Figure 2C) are in average 2.65 kcal
mol-1 lower in energy than those not belonging to H-H bond (e.g., H47 in Figure 1C). This result shows the
stability effect of H-H bond in 3c.
Figure 3 shows the Gibbs free energy difference along the coordination reaction from diazo
compound (as a reference) to cyclobutadiene and to tetrahedrane structures (plus nitrogen), as depicted in
Scheme 1.
ΔG2= -15.5 kcal.mol-1
ΔG1= -44.7 kcal/mol
(A)
(B)
ΔG1= -32.3
kcal.mol-1
ΔGf = -2.9 kcal.mol-1
(C)
(D)
Figure 3: Gibbs free energy difference, in kcal mol-1, from diazocompound (as a reference) to
cyclobutadiene and tetrahedrane structures, along with nitrogen, with respect to the syntheses of 3a (A), 3b
(B), 3f (C) and 3e (D).
Data of G1 in Figure 3 indicate that the diazocompounds 1a, 1b, 1f and 1e are higher in Gibbs free
energy than the corresponding cyclobutadiene (or cyclobutadiene derivative), 2a, 2b, 2f and 2e, respectively.
Their corresponding formation enthalpy (H1) follows the same trend (Table 1). In fact, all cyclobutadiene
derivatives (2a – 2f), plus nitrogen molecule, are more stable than the corresponding diazocompound (Table
1). The same trend is shown for predecessors of silicon tetrahedrane parent 4a, but nothing can be said about
110
the predecessors of germanium tetraedrane parents 5a and 5b because their corresponding diazocompounds
were not formed. Likewise, the tetrahedrane and all tetrahedrane derivatives, 3a – 3f, are lower in energy
(both enthalpy and Gibbs free energy) than the corresponding diazocompound, as indicated by G2 and H2,
except for H2 related to 3f. The silicon tetrahedrane parent 4a exhibits the same behavior.
Table 1 depicts the values of Gibbs free energy difference and enthalpy difference from
diazocompound to cyclobutadiene (and derivatives), G1 and H1, from diazocompound to tetrahedrane
(and derivatives), G2 and H2, and from cyclobutadiene (and derivatives) to tetrahedrane (and derivatives),
Gf and Hf, in kcal mol-1, respectively. Table 1 also shows the delocalization index, DI, bond order, n, and
ellipticity, ε, of carbon-carbon, silicon-silicon or germanium-germanium bond in the tetrahedrane (or parent)
cage, besides the bond path angle, in degree, in the tetrahedrane (or parent) cage for 3a – 3f, 4a, 5a and 5b.
The values of n were obtained from the linear relation between formal bond order (n) and delocalization
index,58 which yields very good correlations for carbon-carbon and germanium-germanium bonds, but
presents moderate correlation for Si-Si bond. The ellipticity, ε, of a bond critical point indicate whether the
corresponding bond has elliptical symmetry (when ε=0, for single or triple bonds) or not.31
Table 1:
Values of Gibbs free energy difference and enthalpy difference for the corresponding
tetrahedranes 3a – 3f, 4a, 5a and 5b from diazocompound to cyclobutadiene (and derivatives), G1 and H1,
from diazocompound to tetrahedrane (and derivatives), G2 and H2, and from cyclobutadiene (and
derivatives) to tetrahedrane (and derivatives), ΔGf and ΔHf, in kcal mol-1, respectively. Values of
delocalization index, DI, bond order, n, and ellipticity, ε, from carbon, silicon or germanium atoms in the
tetrahedrane (or parent) cage, bond path angle, in degree, from bonds in the tetrahedrane (or parent) cage for
3a – 3f, 4a, 5a and 5b.

ΔG1 /
ΔG2 /
ΔH1 /
ΔH2 /
ΔGf /
ΔHf /
kcal mol-1
kcal mol-1
kcal mol-1
kcal mol-1
kcal mol-1
kcal mol-1
3a
-35.92
-15.45
-27.43
-6.24
20.47
21.19
1.025
1.02
0.105
76.9
3b
-44.65
-12.30
-33.77
-2.31
32.35
31.46
0.980
0.97
0.089
79.1
3c
-12.26
-27.72
-2.90
-16.48
-15.46
-13.58
0.948
0.94
0.092
78.8
3d
-37.51
-15.81
-27.04
-5.50
21.70
21.54
0.953
0.94
0.109
78.6
Molecule
DI / e
n
Bond path
angle/o
111
3e
-30.59
-33.48
-20.97
-22.71
-2.89
-1.74
1.016
1.01
0.213
3f
-32.28
-9.72
-20.76
3.45
22.55
24.21
4a
-10.27
-8.92
-4.88
-1.11
1.35
3.77
0.918
1.60
0.039
80.6
5a
-(a)
-(a)
-(a)
-(a)
7.24
10.60
0.951
1.25
0.063
74.0
-(a)
-(a)
-(a)
-(a)
-3.23
11.24
5b
(a) The diazocompound was not obtained.
(b) Values from carbon(from cage)-nitrogen(from nitro group).
(c) Average value
0.912
1.18
0.043
75.3
1.012(b) 1.00(b) 0.099(b)
72.2
80.2(c)
Table 2 shows the topological values (AIM atomic charge, virial atomic energy, atomic volume and
localization index) of the carbon, silicon and germanium atomic basins, , of the tetrahedrane (or parent)
cage for 3a – 3f, 4a, 5a and 5b. These values are obtained from the integration of the corresponding atomic
basin.
Table 2: Values of AIM atomic charge, q(Ω), atomic energy, E(Ω), atomic dipole moment, M(Ω), atomic
volume, V(Ω) and localization index, LI of carbon, silicon and germanium atoms in tetrahedrane, silicon and
germanium derivatives, respectively.
Molecule
q(Ω)
E(Ω)
M(Ω)
V(Ω)
LI
3a
-0.121
-38.0936
0.435
6.09
4.059
3b
-0.109
-38.1522
0.292
6.10
3.968
3c
-0.089
-38.1732
0.300
6.09
3.934
3d
0.037
-38.0509
0.202
6.10
3.844
3e
-0.697
-38.2994
1.48
6.69
4.732
3f
0.235(a)
-37.9671(a)
0.305(a)
5.76(a)
3.691(a)
4a
0.655
-289.3407
0.889
13.30
11.44
5a
0.283
-2074.3391
0.076
31.66
29.74
5b
0.387
-2074.3590
0.097
31.57
29.66
(a) Values of carbon atom attached to the nitro group.
112
Despite the antiaromatic nature of cyclobutadiene or its derivatives and the σ-aromaticity10 of
tetrahedrane cage, in most cases, tetrahedrane and its derivatives are higher in energy (both enthalpy and
Gibbs free energy) than the corresponding cyclobutadiene as it is indicated by Gƒ and Hƒ. The fact that
the σ-aromaticity10 of tetrahedrane (or derivatives) does not impart smaller energy in comparison to the
corresponding cyclobutadiene (which is antiaromatic) can be attributed to the highly strained cage structure
of tetrahedrane and derivatives, except for 3c and 3e which are more stable than 2c and 2e. As for 3c,
QTAIM analysis indicates that its relative stability is ascribed to a large amount of stabilizing hydrogenhydrogen bonds, as aforementioned, while the relative stability of 3e is possibly reasoned by the great
charge density donator effect of the –SiF3 group, according to the more negative values of q(C) and E(C)
plus higher values of V(C) and LI(C) compared to those from 3a. Nonetheless, the relative stability imparted
by hydrogen-hydrogen bond in 3c is nearly 12 kcal mol-1 higher than that bestowed by the donation effect of
–SiF3 group in 3e.
Taking values of any topological property of tetrahedrane 3a as a reference, the DI values of C-C
bonds in the tetrahedrane cage of 3e indicates that trifluorosilyl group partially removes the charge density
of each C-C bond (Table 1 and Figure 2) where ρ = 0.2801 u.a. and 0.2539 u.a. for 3a and 3e respectively.
However, from Table 2, the AIM atomic charge of carbon atom in 3e is 6 times more negatively charged
than that from 3a, which is corroborated by a greater localization index and greater atomic volume of carbon
atom in 3e. In addition, the virial atomic energy of each carbon atom in 3e is 0.2058 au. smaller than that
from 3a. Then, –SiF3 pronouncedly donates charge density to the carbon atoms of the tetrahedrane cage in
3e, although removes partially the charge density in the C-C bonding region. There is also a greater atomic
dipole moment in the carbon atoms of 3e in comparison with those from all other studied tetrahedrane
derivatives or parents, which can be partially related to a larger amount of charge density within carbon
atomic basins in 3e, besides the influence of the electropositive Si(F3) atom bonded to the carbon atom.
Then, –SiF3 in 3e increases the charge density of C atom attached to it but removes the charge density of the
adjacent C-C bond.
Conversely, the methyl and t-butyl groups, in 3b and 3c, respectively, are probably EWGs
according to carbon atomic charge and localization index, where these values are slightly less negative, than
those from 3a. Moreover, methyl and t-butyl groups also partially remove charge density from C-C region
bond of tetrahedrane cage, as indicated by the corresponding charge density of BCP (see Figure 2B and 2C),
the DI, bond order and ellipticity (see Table 1) in comparison with those values from 3a.
The nitro group in 3f has a more pronounced electron withdrawing effect than alkyl groups as can
be easily observed in the positive atomic charge, low LI and higher atomic energy of the carbon atom
attached to the nitro group in 3f with respect to those values for the carbon atoms in 3a (Table 2).
Nonetheless, 3b is more unstable than 3f, but 3f is slightly more unstable than 3a. Then, one nitro group
113
stabilizes the tetrahedrane cage in comparison with four methyl groups, but four hydrogen substituents (in
3a) still generate a more stable structure than one nitro and three hydrogen substituents (in 3f).
Figure 4 shows the doubly occupied GVB oxygen lone pair of nitro group, LP1 and LP2, and singly
occupied C-C sigma orbitals, VB(C-C)’, VB (C-C)1 and VB (C-C)2, of 3f. Table 4 shows the overlap
integrals between the GVB orbitals of Figure 4. From GVB orbitals of 3f (Figure 4), one of the oxygen lone
pair, named LP2, is directed towards the cage. The average overlap integral between LP2 and C-C bonds in
the cage (LP2-CC1) are three-fold greater than the average overlap integral between LP1 (which is not
directed towards the cage) and C-C bonds in the cage (LP1-CC1) as indicated in Table 4. Then, we can infer
that the influence of both oxygen LP2 lone pairs from four nitro groups may cause a great instability and
may lead to the disruption of tetrahedrane cage in 3i yielding 3i’.
VB(C-C)1
VB(C-C)’
LP2
LP1
LP1’
VB(C-C)2
LP2’
(A)
(B)
Figure 4: Doubly occupied generalized valence bond (GVB) orbitals of lone pairs of oxygen of nitro (-NO2)
functional group substituent (LP1, LP1’, LP2 and LP2’) (A and B) and singly occupied GVB orbitals of C-C
sigma bond [VB(C-C)1, VB(C-C)2 and VB(C-C)’] (B).
Table 4: Overlap matrix of selected GVB orbitals of molecule 3f of two oxygen lone pairs (LP1, LP2, LP1’
and LP2’) and of two C-C sigma bonds [VB(C-C)1, VB(C-C)’ and VB(C-C)2].
LP1
LP1’
LP2
LP2’
VB(C-C)1
0.007
0.019
0.054
0.045
VB(C-C)2
0.005
0.005
0.005
0.006
VB(C-C)’
0.008
0.006
0.034
0.009
114
When comparing topological data between 3d and 3f in Table 2, we observe that the
trifluoromethyl group in 3d has a less electron withdrawing effect than the nitro group in 3f, since carbon
atomic charge is less positive in 3d, carbon atomic energy is more stable in 3d, and LI is greater in 3d. In
addition, the –CF3 group also decreases the charge density in the C-C bonding region as noted from its DI
and bond order (Table 1). In terms of molecular energy, the stability of 3d is quite similar to that from 3a
and it is slightly higher than that from 3f.
To sum up the energetic influence of the studied substituents in tetrahedrane cage we can note that
EWGs such as –CF3 do not affect the relative stability of the tetrahedrane cage, while with a stronger EWG
(e.g., –NO2) the tetrahedrane cage becomes slightly more unstable. On the other hand, slight EWG such as –
CH3 imparts a higher instability in tetrahedrane cage. Conversely, a -EDG such as –SiF3 makes the
tetrahedrane cage more stable, while a -EDG (e.g. –NH2 and –OH) disrupts the tetrahedrane cage. These
trends can be observed in the decreasing order of Gƒ (3c > 3e > 3a > 3d > 3f > 3b) and Hƒ (3c > 3e > 3a
> 3d > 3f > 3b), without the silicon and germanium parents.
The integration of the atomic basins gives the value of the bond path angle of the electron density
function. The bond path angle involving the atoms in the cage of all tetrahedrane derivatives and Si/Ge
parents is higher than the corresponding geometric bond angle (Table 1). The increase of the bond angle in
cyclopropane ring (which is part of the tetrahedrane cage) decreases its ring strain (or cage strain) which is
so-called banana bond.
59
However, there is no linear relation between bond path angle and Gƒ for the
studied series.
The tetrahedrane cage with silicon and germanium atoms, 4a and 5a, were also obtained, as well as
trifluoromethyl tetrasubstituted germanium cage, 5b. However, the trifluoromethyl tetrasubstituted silicon
parent disrupted during optimization procedure (Scheme 3). Formation enthalpy, Hƒ, indicates that they
have intermediate stability between the most stable (3c and 3e) and least stable (3a, 3b, 3d and 3f)
substituted tetrahedranes. Nonetheless, this stability order does not match the aromaticity order, according to
NICS and D3BIA, where the set 4a, 5a and 5b are the least aromatic.
̅̅̅ , DIU, δ and ρ(3,+3)) associated with the D3BIA formula
Table 3 shows the topological values (𝐷𝐼
for homoaromatic and cage structures, along with NICS values in the geometrical center of the tetrahedrane
(or parent) cage for 3a, 3b, 3c, 3d, 3e, 3f, 4a, 5a and 5b. The aromaticity of the substituted tetrahedranes,
germanium and silicon parents were investigated with two aromaticity indices: NICS and D3BIA. Hereafter
we mention the increase or decrease of NICS with respect to the aromaticity where aromaticity increases
with higher negative values of NICS and decreases with lower negative values (or higher positive values) of
NICS.
115
Figure 5 shows the linear relations between NICS and D3BIA where Figure 5A lacks only one
molecule, 3f, and Figure 5B has all the studied molecules (3a-3f, 4a, 4b and 5a).
NICS X D3BIA
NICS X D3BIA
20
D3BIA
20
15
15
10
10
R² = 0.9142
R² = 0.7664
5
5
0
0
-60
-40
-20
D3BIA
0
20
40
-60
-40
-20
0
20
40
NICS /ppm
NICS / ppm
(B)
(A)
Figure 5: (A) Plot of NICS versus D3BIA for 3a-3e, 4a, 4b and 5a; (B) Plot of NICS versus D3BIA for 3a3f, 4a, 4b and 5a.
Figure 5A shows a very good linear relation between NICS and D3BIA where 3f was excluded
because NICS in 3f is higher (more negative) than it is supposed to be since 3c is 10.81 kcal mol-1 more
stable than 3f but they have the same NICS value. Probably, this is the influence of lone pairs of oxygen
atoms from nitro group that interact with the tetrahedrane cage of 3f (Figure 4 and Table 4) which enhances
its NICS value.
Table 3: Average DI, ̅̅̅
𝐷𝐼, of the carbon or silicon or germanium atoms in the tetrahedrane (or parent) cage,
the corresponding DIUs, the degree of degeneracy (δ) involving the atoms of the tetrahedrane (or parent)
cage, the charge density of the cage critical points [ρ(3,+3)], D3BIA and NICS values for 3a, 3b, 3c, 3d, 3e,
3f, 4a, 5a and 5b.
Molecule
̅̅̅̅
𝑫𝑰
DIU
δ
ρ(3,+3)/au
D3BIA
NICS / ppm
3a
1.025
99.97
1.0
0.182
18.24
-48.53
3b
0.979
99.89
1.0
0.176
17.61
-45.31
3c
0.954
99.99
1.0
0.177
17.68
-44.95
3d
0.961
99.98
1.0
0.182
18.21
-47.45
3e
1.029
99.97
1.0
0.187
18.67
-54.54
3f
0.997
96.77
0.5
0.181
8.77
-45.30
4a
0.933
99.95
1.0
0.046
4.57
-1.03
5a
0.951
99.99
1.0
0.040
4.00
1.85
5b
0.911
99.93
1.0
0.039
3.88
32.33
116
The alkyl groups (in 3b and 3c) decrease both NICS and D3BIA in comparison with 3a. Likewise,
the trifluoromethyl and nitro groups in 3d and 3f, respectively, also decrease both aromaticity indices with
respect to 3a. On the other hand, the trifluorosilyl group increases both NICS and D3BIA in 3e. Then, we
may assume that EWGs decrease the aromaticity of the tetrahedrane cage while the EDG increases its
aromaticity. The decreasing order of formation enthalpy and NICS for 3a-3f is nearly the same (except for
3c where hydrogen-hydrogen bonds play an important role on its stabilization): 3e > 3a > 3d > 3b  3f (for
NICS) and 3e > 3a > 3d > 3f > 3b (for Hƒ). Then, we may assume that the aromaticity (influenced by
EWGs or EDGs) is an important stability factor in the tetrahedrane cage besides the intramolecular
hydrogen-hydrogen bonds in 3c.
As aforementioned, regarding the set 4a, 5a and 5b, there is no relation between their stability and
aromaticity orders. According to NICS, they are not aromatic molecules, but their stability is higher than
that in 3a, 3b, 3d and 3f. Probably, the stabilities of 4a, 5a and 5b are influenced by their predecessors’
stabilities. Regarding 4a, its H1 is the least negative, which means that its corresponding silicon
cyclobutadiene parent is the least stable of all set of substituted cyclobutadienes. Then, the high instability of
predecessor of 4a plays an important role on the large relative stability of 4a.
Figure 6 shows the singly occupied GVB orbitals of sigma C-C bonds, VB(C-C)’, VB (C-C)1 and
VB (C-C)2, of 3a, 3b, 3f and cubane. Table 5 shows the overlap integral between the singly occupied GVB
orbitals represented in Figure 6. Tetrahedrane and cubane are platonic hydrocarbons where only cubane was
synthetically obtained.60 The overlap between vicinal GVB orbitals in cubane, VB(C-C)1 – VB(C-C)2, is
0.224 – nearly four times smaller than the overlap between GVB orbitals of each C-C bond, VB (C-C)1 –
VB(C-C)’. The overlap between vicinal GVB orbitals in tetrahedrane is higher than that from cubane, 0.279,
where we can infer that vicinal C-C bonds tend to increase the electronic repulsion in the tetrahedrane cage,
which could partially explain the instability of tetrahedrane with respect to cubane. The electron
withdrawing effect of alkyl group can also be noticed by the slightly smaller overlap between vicinal GVB
orbitals in 3b (Table 5). Likewise, the higher electron withdrawing effect of the nitro group can also be
noticed from overlap of GVB orbital because this value in 3f is similar to that from cubane. However, as
aforementioned 3f is more unstable than 3a. Then, the electronic repulsion from vicinal C-C bonds within
the cage is not the main instability factor of tetrahedrane cage which means that the main reason for
tetrahedrane instability should be attributed to its angle strain.
117
VB(C-C)’
VB(C-C)1
VB(C-C)2
(A)
(C)
(B)
(D)
Figure 6: Singly occupied generalized valence bond (GVB) orbitals of C-C sigma bond [VB(C-C)1, VB(CC)2 and VB(C-C)’] of 3a (A), 3b (B), 3f (C) and cubane (D).
Table 5: Overlap matrix of selected GVB orbitals, VB(C-C)’, VB (C-C)1 and VB (C-C)2, of 3a, 3b, 3f and
cubane.
VB(C-C)1
3a
3b
3f
Cubane
VB(C-C)2
0.279
0.272
0.224
0.224
VB(C-C)’
0.828
0.826
0.823
0.810
a
VB orbitals of 3a
VB orbitals of 3b
c
VB orbitals of 3f
d
VB orbitals of cubane
b
The Figures 7, 8 and 9 shows the plots of total energy (in Hartree) versus the time of trajectory of 5
fs using atom-centered density matrix propagation (ADMP) method for tetrahedrane and substituted
tetrahedranes (3a, 3c and 3e) where the starting structure was previously optimized at ωB97XD/6311++G(d,p) level of theory. We used ADMP calculations to test the stability of selected tetrahedrane cages
under dynamic simulation. Some corresponding structures from selected points of each trajectory are also
depicted in Figures 7, 8 and 9. In Figure 7, we can see that no significant changes occur in 3a structure at the
118
beginning of the trajectory (0.2 to 0.6 fs). However, at about 1.3 fs the C2-H7 bond is broken. At 2.8 fs the
C3-H5 bond is broken and C4-H8 bond length increases. Moreover, the C-C bond lengths of three
cyclopropane rings of tetrahedrane cage shorten. At about 4.2 fs, the C4-H8 bond is broken and at 5.0 fs the
C3-H5 bond is restored. Then, during the ADMP trajectory, 3a undergoes structural changes in its structure
but it is not restored to its starting optimized geometry at any point. On the other hand, there is no significant
change in the total energy of 3c during the course of its trajectory (Figure 8). Then, 3c has a thermochemical
and dynamical relative stability mainly due to the large amount of stabilizing H-H bonds. Likewise, 3e
presented a similar behavior (Figure 9), but its relative stability is given by –SiF3 group that acts as σ-EDG,
where in 2.5 fs and 4.4 fs occurs a decreasing in two C-Si bond lengths (indicated by red arrows) according
to the nature of donating electrons of trifluorosilyl group.
Figure 7: Plot of total energy, in Hartree, versus the time of trajectory, in femtoseconds, of tetrahedrane (3a)
using ADMP method. Some structures are depicted at the corresponding selected points of the trajectory.
119
Time in Trajectory (femtosec)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
-783.58070
Total Energy (Hartree)
-783.58072
-783.58074
-783.58076
-783.58078
-783.58080
-783.58082
-783.58084
Figure 8: Plot of total energy, in Hartree, versus the time of trajectory, in femtoseconds, of 3c using ADMP
method. The structure at the end of trajectory is also depicted.
Time in Trajectory (femtosec)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
-1366.426550
-1366.426555
Total Energy (Hartree)
-1366.426560
-1366.426565
-1366.426570
-1366.426575
-1366.426580
-1366.426585
-1366.426590
-1366.426595
-1366.426600
-1366.426605
Figure 9: Plot of total energy, in Hartree, versus the time of trajectory, in femtoseconds, of 3e using ADMP
method. A couple of structures are depicted at the selected corresponding points of the trajectory.
120
Conclusions
The influence of substituents on the stability of the tetrahedrane cage reveals that strong electron
withdrawing groups do not affect significantly the relative stability in the tetrahedrane cage, unlike weak
EWG that generate a large instability in tetrahedrane cage. However, substituents that act as sigma electron
donating groups (σ-EDG) stabilize the tetrahedrane cage, although π-EDGs leads to disruption of the
tetrahedrane cage.
According to NICS and D3BIA and their relations with the corresponding formation enthalpy, the
aromaticity is an important factor for the stability of the tetrahedrane cage. The -EDGs increase both
aromaticity and stability of the tetrahedrane cage, whereas the EWGs decrease both aromaticity making and
cage stability, except for 3c in which its stability is due to the intramolecular hydrogen-hydrogen bonds.
Although there is no relation between aromaticity and stability for germanium and silicon tetrahedrane
parents 4a, 5a and 5b their NICS and D3BIA values indicate that these molecules are not aromatic.
According to generalized valence bond method, one of the oxygen lone pair has an electronic
repulsion with C-C bonds in its tetrahedrane cage and probably when four nitro groups are attached to the
tetrahedrane cage the higher electronic repulsion may lead to disruption of corresponding substituted
tetrahedrane. Moreover, the GVB analysis indicate that the instability of the tetrahedrane cage is mainly
imparted by its high strain energy while the C-C electronic repulsion within the cage has a minor influence.
The ADMP analysis indicates that tetrahedrane 3a is dynamically unstable where some C-H bonds
break during the simulation trajectory. Conversely, 3c and 3e remain its structure during the whole
trajectory. Therefore, thermodynamic and dynamic results are in accordance for the relatively stable and
unstable cage structures.
Acknowledgments
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6. CAPÍTULO 4 – Formatação de artigo a ser submetido para o periódico The Journal of Physical
Chemistry B.
Docking, Molecular Dynamics, ONIOM, QTAIM: theoretical investigation of inhibitory effect of
Abiraterone on CYP17.
Norberto K. V. Monteiro, Sérgio R. B. Silva and Caio L. Firme*
*Corresponding author. Tel: +55-84-3215-3828; e-mail: caiofirme@quimica.ufrn.br
ABSTRACT
After skin cancer, prostate cancer (PC) is the most common cancer among men in the western world. Once
metastasized, the prostate cancer treatment becomes palliative. The initial treatment is performed with
androgen deprivation therapy (ADT) leading to a decrease in the level of serum testosterone (chemical
castration). However, resistance to castration-therapy can develop, resulting in men with metastatic
castration-resistant prostate cancer (mCRPC). Recently, a novel hormonal agent, Abiraterone acetate
(abiraterone) has been shown to increase the survival rate by 4.6 months when compared to placebo in men
with mCRPC, by inhibiting the key enzyme for the biosynthesis of androgens [17α-hydroxylase-17,20-lyase
(CYP17)]. In this paper, we present a new computational protocol in order to quantify the affinity of
abiraterone (ABE) compared to pregnenolone (PREG). Molecular dynamics and molecular docking were
used to obtain the CYP17-ABE and CYP17-PREG complexes and to obtain the interaction energies of
ligands ABE and PREG. From the ONIOM (B3LYP:AMBER) method, we find that the interaction
electronic energy of ABE is 21.38 kcal mol-1 lowest than PREG. The quantum theory of atoms in molecules
(QTAIM) method was used to quantify the interactions between ABE (or PREG) and CYP17. The results
indicate that ABE has a higher electronic density in their interactions compared to PREG.
Keywords: Prostate cancer; CYP17; Abiraterone; Pregnenolone; Molecular dynamics; ONIOM; QTAIM.
INTRODUCTION
Prostate cancer (PC) is the most common type of cancer in the US and Europe.1 Its development occurs
mainly by the action of androgens testosterone (T) and dihydrotestosterone (DHT), which are synthesized by
125
the testis and adrenal glands through the androgen receptor (AR). The androgen action consists of the
following steps: (a) the entrance of testosterone in the prostatic cells which is converted to
dihydrotestosterone by 5α-reductase; (b) in presence of DHT, AR is released from heat shock protein 90
(HSP90); (c) the AR further undergoes dimerization, phosphorylation and translocation to the nucleus 2,3
(Figure 1). The translocated receptor binds to the androgen response elements in the DNA, thereby
activating transcription of AR target genes and thus leading to cell proliferation. 4,5 In a reciprocal manner,
after androgen withdrawal, the prostatic cells cease proliferation and undergo apoptosis.6 Then, there is a
dependence of prostatic cells on AR signaling in neoplastic transformation, which is the basis for metastatic
PC therapy.
Figure 1. Inside the cell, testosterone is converted to DHT and binds to AR. AR-DHT complex
undergoes phosphorylation and is translocated to the nucleus in the form of dimer. Inside the nucleus, ARdimer binds to DNA coregulators leading to transcription activation.
According to the above description, there is a biological close relationship between testosterone and
7
PC. However, epidemiologic studies have shown that the association between testosterone and PC is not
clearly understood.7,8 Solania,9 formulated the time-dependency theory which postulates that the endocrine
biology of prostatic tissue is dependent on the exposure time at a given concentration of sex steroid. This
postulate is a consequence of the results obtained by Solania and collaborators10 where serum testosterone
levels could be stable over time in PC patients.
The most common initial treatment of PC is the androgen deprivation therapy (ADT) resulting in
temporary regression of disease in most patients. This therapy is accomplished by the chemical castration
with luteinizing-hormone releasing-hormone (LHRH) analogs, or surgical castration (orchiectomy). 1,11,12
The ADT results in a reduction of about 90% of serum testosterone. 13 However, ADT do not affect the
126
production of adrenal and intra-tumoral androgens, which may be relevant in restoring disease14 resulting in
patients with metastatic castration-resistant prostate cancer (mCRPC). 15 As aforementioned, androgens T
and DHT play an important role in development of PC and then, potent and specific compounds that inhibit
androgen synthesis may be effective for the treatment of PC. 16 A promising target is the 17α-hydroxylase17,20-lyase (CYP17) involved in the biosynthesis of T and DHT through of two key reactions, which act
sequentially. First, by converting pregnenolone (PREG) to 17α-hydroxypregnenolone and progesterone to
17α-hydroxyprogesterone; and second by converting 17α-hydroxypregnenolone to dehydroepiandrosterona
and 17α-hydroxyprogesterone to androstenedione. 17,18 Furthermore, Auchus19 described an alternative
pathway in the biosynthesis of DHT testosterone-independent wherein CYP17 also acts in this pathway,
emphasizing its importance as a therapeutic target, because its inhibition exerts control over androgen
synthesis. Ketoconazole is known as antifungal which inhibits nonspecifically the CYP17 and reduces
androgen levels, however, its use is associated with several side effects including hepatotoxicity,
gastrointestinal toxicity, adrenal insufficiency and their effects on other cytochrome enzymes.14,20,21
Therefore, is very important to search and development of more potent and specific novel CYP17 inhibitors
as well as the identification of these inhibitors.
Many experimental and computational studies were reported on inhibitory molecules against
CYP17.1,20–24 Recently, several promising agents with different mechanisms of action and therapeutic targets
have demonstrated efficacy and four new drugs (cabazitaxel, sipuleucel-T, denosumab, and abiraterone
acetate) were approved by FDA for the treatment of patients with mCRPC.25–27
Abiraterone (Figure 2) (ABE) is a high selective irreversible CYP17 inhibitor (IC50, 2 to 4 nmol/L),
more potent and selective and less toxic than ketoconazole. 12,28,29 Moreover abiraterone reduces serum
testosterone levels below 1.0 ng/dl. 30,31 In phase 3 trials conducted in 1195 patients, abiraterone increased
the survival rate to 15.8 months compared with 11.2 months in the control (placebo). 32
Figure 2. Chemical structure of ABE
The recognition of ligands by proteins is a key process of many biological functions and systems that
involve the enzymatic catalysis, cell signaling and protein conformational switching.33,34 The binding
affinity is a reflection of the molecular recognition. Prediction of the binding affinity are targets of several
127
computational studies.35–38 Molecular dynamics simulations have been used extensively to predict the
affinity of protein-ligand interactions.39–41 In the present work, molecular dynamics was used to predict the
affinity of the ligands ABE and PREG against CYP17 enzyme.
The main focus of our study is to quantify the affinity of ABE and PRE with the CYP17 enzyme and
investigate the specific interactions protein-ligand. We performed the geometric optimization of the CYP17ABE and CYP17-PREG complexes and CYP17 enzyme with ONIOM (B3LYP/6-311G(d,p):AMBER)
method (Figures 10A, 10B and 10C, respectively).
Experimental methods for evaluating the affinity of a specific ligand may require a long working
time and a high monetary cost. The objective of the present work was investigate the inhibitory effect of the
ABE on CYP17. For this study, we compared the stabilities between CYP17-ABE and CYP17-PREG
complexes by using classic molecular dynamics simulations, molecular docking, the ONIOM hybrid method
and, QTAIM method. Our comparative study will provide theoretical information for distinguishing
inhibitors and natural substrates pointing out the direction of predicting new CYP17 inhibitors.
COMPUTATIONAL DETAILS AND THEORETICAL METHODS
PREPARATION OF THE STRUCTURE AND CLASSIC MOLECULAR DYNAMICS SIMULATIONS
The X-ray crystal structure of human CYP17 in complex with abiraterone was obtained from Protein
Data Bank (PDB ID code: 3RUK)42. Protons were added to the protein structure using GROMOS96 43a1
force field43 assuming typical protonation states at pH 7.0. On the other hand, the imidazole side chain of the
histidine residue (His) contains two nitrogen atoms that can be protonated (ND1 and NE2) at pH 7.0, called
HID, HIE and HIP44 (Figure 3). The protonation state of histidine residue close to the heme group (His373),
was systematically varied. The value of the CA-CB-CG-ND1 dihedral angle (χ1) (Figure 3) was also
evaluated because an incorrect protonation of a His residue can result in large fluctuations and rotations of
the His side chain. Different studies about the analysis of the histidine protonation state have been
performed.44–46 In all these studies, the root-mean-square deviation (RMSD) is used to analyze the state of
protonation. To determine the proportion of neutral (HID and HIE) and positively charged (HIP) His on
CYP17 structure, we used the Henderson-Hasselbalch equation.47 Molecular dynamics (MD) simulations
were performed by GROMACS 4.6.5 program48 for each protonation state using GROMOS96 43a1 force
field with SPC water molecules.49,50 CYP17 has 26 aspatic acids, 24 glutamic acids, 34 lysines, 20 arginines
and 1 positively charged histidine (His160), totaling a +5 charge. The histidine close to the heme group
(His373) is hydrogenated in nitrogen NE2 (HIE). Counter ions Na+ and Cl- were added to neutralize charge
128
and maintain the system at a concentration of 150 mM. The system was immersed in a rhombic
dodecahedron box and the minimal distance between the protein and the wall of the cell was 1.2 nm.
Covalent bonds in the protein were constrained using the LINCS algorithm.51 Electrostatic forces were
calculated with the PME52 implementation of the Ewald summation method, and the Lennard–Jones
interactions were calculated within a cut-off radius of 1.2 nm. The system was equilibrated according to
the following protocol: the solvated CYP17-abiraterone complex geometry was minimized twice by the
steepest descent algorithm for 5000 steps with tolerances of 1000 and 500 KJ mol-1 nm-1 followed by two
gradient conjugated algorithm for 50000 steps with tolerances of 250 and 100 KJ mol-1 nm-1. In order to
adapt the water to the complex due the greatest strain dissipated from the system and couple the system to
the heat bath using NVT ensemble at temperature of 310K, several short MD simulations were performed.
Initially, 50, 100, 150 and 200-ps molecular dynamics were performed where the non-hydrogen atoms of the
complex were kept fixed and the water can move freely. After that, 50, 50 and 37.5-ps MD simulations were
performed where the water and abiraterone move freely, following of 25, 50 and 75-ps MD simulations
where the heme group is released. Then, two unrestrained MD simulations of 50 and 75-ps were performed.
In order to control the pressure coupling of system, 200-ps MD simulation were performed at pressure of 1
bar using isotropic pressure bath with Berendsen pressure coupling algorithm. 53 Finally, 50-ns production
MD simulation using NVT ensemble was performed for each protonation state in order to assess the stability
of the system. The RMSD of C-α of His 373 residue and CYP17 structure was measured with respect to the
respective X-ray structure. Snapshots were collected every 10-ns MD simulation and RMSD and dihedral
angle χ1 average values of the His 373 residue of each protonation state were analyzed. All data represent
five snapshots and were expressed as mean ± SD (standard deviation). Differences between groups were
compared by using ANOVA54 and Tukey Test.55 Differences were considered significant when p value was
less than 0.05. Statistical data were analyzed by GraphPad Prism 5.0 software.
(CB)
(CA)
(CG)
A
(ND1)
B
C
Figure 3. The protonation states of histidine at pH 7.0. HID (A) HIE (B) and HIP (C).
129
MOLECULAR DOCKING
The automated docking was conducted in order to find the best conformations and energy ranking of
PREG inside the active site of the human CYP17 (previously equilibrated from MD simulation as
aforementioned). The docking simulation was carried out using AutoDock Vina 1.1.2. 56 The threedimensional structure of PREG was obtained from Protein Data Bank (http://www.rcsb.org). Polar
hydrogens were added and Gasteiger-Marsili partial charges57 were assigned to the PREG and human
CYP17 with the AutoDock Tools (ADT). 58 All torsions to PREG were allowed to rotate during docking.
Grid box with size 38 Å was centered on the iron of the heme group. The AutoDock Vina parameter,
Exhaustiveness, that determined how the program searches exhaustively for the lowest energy conformation
was set to 16 (the default value is 8). After docking, resulting conformations of PREG were ranked based on
the interaction energy. The best conformation of PREG in complex with human CYP17 was chosen with the
lowest docked energy and then was submitted to 50-ns MD simulation.
The ABE was also subjected to the same docking methodology as described above for PREG on
CYP17. We found that the conformation of the docked ABE was very similar to that found in the CYP17ABE complex in the Protein Data Bank (PDB ID code: 3RUK).
TOPOLOGY
The topology contains all the necessary information to describe a molecule in a molecular
simulation. All this information includes atom types and charges (nonbonded parameters) as well as bonds,
angles and dihedrals (bonded parameters). 48 In addition, exclusion and constraints parameters are also
generated by the force field. For this work, the topology was generated for the protein without ligands (ABE
and PREG) through the GROMOS96 43a1 force field. The ligands topology was generated by the
PRODRG59 2.5 automated server.
MOLECULAR DYNAMICS FLEXIBLE FITTING (MDFF) METHOD
In order to analyze conformational changes in the three-dimensional structure of CYP17 complexes,
before and after the molecular dynamics, the molecular dynamics flexible fitting (MDFF) 60 method was
applied. The MDFF method consists of a molecular dynamics that includes information from an
experimental or simulated electron microscopy (EM) map for fitting an initial model of biomolecule in the
target density map. The analysis of MDFF were performed with Visual Molecular Dynamics (VMD) 61
software.
130
THE TEMPERATURE FACTOR
The temperature factor (or B-factor) describes the displacement of the atomic positions in threedimensional protein structures from an average value. 62 The B-factor will reflect the flexibility of atoms in
relation to their average positions. In other words, B-factor is an important indicator of the flexibility of the
protein structure. In this work, we used the B-factor to analyze the regions of the CYP17 complex structures
which have undergone flexibility during the course of molecular dynamics.
CALCULATION OF MOLECULAR MECHANICS INTERACTION ENERGIES
The nonbonded interaction energies involves running two MD simulations: one for the ligand in the
𝑝
𝑤
protein biding site (𝐸𝑏𝑖𝑛𝑑 ) and the other for ligand in water (𝐸𝑏𝑖𝑛𝑑
). The binding energies are composed of
two separate energy terms: The van der Waals (vdW) interaction energy (VLJ) given by the Lennard –Jones
(LJ) potential and the electrostatic interaction energy (VC) given by Coulomb potential.63–66 The computed
interaction energy is the sum of the LJ and Coulomb interaction energies which is given by:
𝑝
𝐸𝑏𝑖𝑛𝑑 = (〈𝑉𝐿𝐽 〉𝑝 + 〈𝑉𝐶 〉𝑝 ) (1)
and
𝑤
𝐸𝑏𝑖𝑛𝑑
= (〈𝑉𝐿𝐽 〉𝑤 + 〈𝑉𝐶 〉𝑤 ) (2)
where < > denotes MD averages of the nonbonded interactions between ligand and its receptor binding site
or just water. 〈𝑉𝐿𝐽 〉𝑝 is the LJ term for ligand-protein interaction; 〈𝑉𝐿𝐽 〉𝑤 is the LJ term for ligand-water
interaction; 〈𝑉𝐶 〉𝑝 is the electrostatic term for ligand-protein interaction; and 〈𝑉𝐶 〉𝑤 is the electrostatic term
for ligand-water interaction. For this study, we analyse the nonbonded interaction energies between CYP17
𝑝
𝑤
and its ligands ABE and PREG. Two simulations are required to obtain the 𝐸𝑏𝑖𝑛𝑑
and 𝐸𝑏𝑖𝑛𝑑 : one with ligand
bound to solvated protein (as described above) and one with the ligand free in solution. The latter was
performed with 1.0-ns using NVT ensemble.
QM/MM CALCULATIONS
131
Hybrid methods combine two computational techniques in one calculation and allow the study of large
chemical systems.67–69 The QM/MM method is a hybrid technique that combine a quantum mechanical
(QM) method with a molecular mechanics (MM) method.70–73 ONIOM74 (N-layer Integrated molecular
Orbital molecular Mechanics) is one of the QM/MM methods most employed and allows the combination of
different quantum mechanical methods as well as classical methods of molecular mechanics. In general, the
region of the system where the chemical process occurs is treated with a more accurate method as well as a
QM theory, and the remainder of the system at low level. Many ONIOM studies are focused on biological
systems, such as proteins,75–78 treating the active site by a high level method, using often density functional
theory (DFT), and the rest of the protein by MM. In a two layer ONIOM calculation, the total energy of the
system is obtained from three independent calculations:
𝐸 𝑂𝑁𝐼𝑂𝑀 = 𝐸 𝑚𝑜𝑑𝑒𝑙,𝑄𝑀 + 𝐸 𝑟𝑒𝑎𝑙,𝑀𝑀 − 𝐸 𝑚𝑜𝑑𝑒𝑙,𝑀𝑀 = 𝐸 𝑚𝑜𝑑𝑒𝑙,ℎ𝑖𝑔ℎ + 𝐸 𝑟𝑒𝑎𝑙,𝑙𝑜𝑤 − 𝐸 𝑚𝑜𝑑𝑒𝑙,𝑙𝑜𝑤 (3)
The real system contains all the atoms (including both QM and MM regions) and is calculated only at
the MM level. The model system is the QM region treated at QM level. To obtain the ONIOM energy, the
model system also needs to be treated at MM and be subtracted from real system MM energy. 79,80 In this
work, the ONIOM method was applied in geometric optimization of the complexes (ABE or PREG in
complex with human CYP17) from 50-ns production MD simulation. The high layer atoms were treated at
the B3LYP/6-311G(d,p) level81–83 and the low layer was treated by means of the AMBER9484 force field
implemented in the Gaussian 09 package.85
The Restrained electrostatic potential (RESP) 86 atomic charges of the abiraterone and pregnenolone
were derived at the HF/6-31G* level by Gaussian03 and AmberTools.87 The heme group, ABE (or
PREG) and Cys442 were included in the high layer region, while the remaining residues were in the low
layer.
The interaction energy of ABE (or PREG) was calculated as below:
𝑐𝑜𝑚𝑝𝑙𝑒𝑥
𝑖𝑛𝑡
𝐸𝐶𝑌𝑃17+𝐴𝐵𝐸 - 𝐸𝐶𝑌𝑃17 - 𝐸𝐴𝐵𝐸 = 𝐸𝐴𝐵𝐸
(4)
𝑖𝑛𝑡
𝐸𝐶𝑌𝑃17+𝑃𝑅𝐸𝐺 - 𝐸𝐶𝑌𝑃17 - 𝐸𝑃𝑅𝐸𝐺 = 𝐸𝑃𝑅𝐸𝐺
(5)
where 𝐸𝐶𝑌𝑃17+𝐴𝐵𝐸 and 𝐸𝐶𝑌𝑃17+𝑃𝑅𝐸𝐺 are the electronic energies of CYP17 in complex with ABE and PREG,
respectively. 𝐸𝐶𝑌𝑃17 is the electronic energy of CYP17; 𝐸𝐴𝐵𝐸 is the electronic energy of ABE; 𝐸𝑃𝑅𝐸𝐺 is the
𝑖𝑛𝑡
𝑖𝑛𝑡
electronic energy of PREG; 𝐸𝐴𝐵𝐸
and 𝐸𝑃𝑅𝐸𝐺
are the interaction electronic energies of ABE and PREG,
respectively. The electronic density of the region enclosed by a cut-off radius of 3Å from ABE or PREG
(binding pockets) which was used for QTAIM calculations, was performed at B3LYP/6-31G(d,p) level by
means of Gaussian09 package, where all topological information were calculated using the AIM2000
132
software.88 The binding pockets in the CYP17 active site were generated through VMD software and are
shown in Figure 4.
The electronic energies of CYP17, ABE, PREG, CYP17-ABE, CYP17-PREG complexes, as well as
the electronic interaction energies of ABE and PREG are shown in Table 3. The electronic energy of
complexes were obtained after optimization of its geometries. To obtain the electronic energy of the CYP17,
the ligands (ABE and PREG) were previously removed from complexes and submitted to the single point
calculation at B3LYP/6-311G(d,p) level. Already to obtain ligands electronic energies, these were removed
from their complexes and submitted to the single point calculations with the same level of theory described
earlier.
QUINTET AND TRIPLET STATE OF HEME
The Density Functional Theory (DFT) method was used in this paper because it is already well
employed to porphyrins in general89–93 and B3LYP functional predicts the energy separation of the of Fe IIporphyrin of deoxyheme between quintet and triplet states, with the quintet state 12 kJ mol -1 lowest in
energy94 and this is agreement with experimental data. A study conducted in 1936 by Pauling and Coryell95
measured the magnetic moments of deoxyhemoglobin and oxyhemoglobin. The results showed that the
ferroheme of deoxy state contains four unpaired electrons. In this work, we checked the electronic energy
differences of ferroheme of triplet and quintet states using six functionals (B3LYP, 96,97 BP86,98,99 BVWN,100
M062X,101 ωB97XD102 and PBE1PBE103–105) at 6-311G(d,p) basis set through a simple Fe(NH3)2(NH2)2
complex. We found that the quintet state is about 5.85 kcal mol-1 more stable compared to triplet state using
B3LYP functional (Table S1 and Figure S1, Supporting Information). Therefore, we used the quintet spin
state (S=2) for heme group of CYP17 at the B3LYP/6-311G(d,p) level on the high layer.
QTAIM RATIONALE
The Quantum Theory of Atoms in Molecules (QTAIM) is a quantum model considered efficient in the
study of chemical bond106 and characterization of intra- and intermolecular interactions.107–109
According to the quantum-mechanical concepts of QTAIM, the observable properties of a chemical
system are contained in its electron density, ρ. The trajectories of the electron density are obtained from the
gradient vector of the electron density (∇ρ)110 and the set of these gradient trajectories form the atomic
basins (Ω). The bond critical points [BCP´s, (3,-1)] of the charge density are obtained where ∇ρ = 0111 and
its location is performed through the Laplacian of the charge density (∇2ρ) that is given by the trace of the
Hessian matrix of the electronic density.112 BCPs may give information about the character of different
133
chemical bonds or chemical interaction, for example, metal-ligand interactions,113–115 covalent bonds,112,116
different kinds of H-bonds,117–119 so-called hydrogen-hydrogen bonds108,120–123 as well as other weak
noncovalent interactions.124–126 In addition, several studies show that there is a direct relation between
electronic density of a BCP and the interaction energy between two atomic centers forming the bond, where
there is a linear relation for small ranges of electronic density values. 127–130
A
B
Figure 4. Schematic representations of pockets of the CYP17-ABE (A) and CYP17-PREG (B) complexes.
RESULTS
Molecular dynamics was used in order to achieve the stability of the CYP17-ABE complex and to detect
the protonation state of His 373. The plots of RMSD of the Cα of the CYP17-ABE complex of each
protonation state of His 373 are shown in Figures 5A, 5B and 5C. The RMSD of the Cα atoms of the
134
complex for each protonation state first increased as far as plateau at 5.0 ns, indicating that equilibrium had
been reached (Figures 5A, 5B and 5C) wherein the superposition of C-α of CYP17 complexes onto the Xray coordinates reaching convergence within 50-ns MD simulation. The plateau reached had an average
value of 0.39 nm. There was no difference observed between RMSD´s, since the only difference between
simulations is the protonation state of the His373.
The Figures 6A and 6B shows the plots of RMSD and χ average values, respectively, of His373 for each
protonation state. These average values were obtained every 10-ns snapshots from 50-ns production MD of
CYP17-ABE complex.
In order to obtain the CYP17-PREG interaction, and then the appropriate enzyme-substrate binding
pose, molecular docking using AutoDock Vina was performed. The binding affinity was characterized by
binding energy (ΔG) value (Table 1). The docking poses of PREG within the active site of the CYP17
enzyme are shown in Figures 7A, 7B, 7C and 7D. According to the Autodock Vina binding energies, the
best docking pose (pose 1) has a ΔG value of -12.2 kcal mol-1 (Figure 7A) and then was submitted to 50-ns
MD simulation (Figure 8) where the plateau was reached at 20 ns with superposition of C-α RMSD value
onto the X-ray coordinates of 0.38 nm. The structure obtained after a period of 50 ns simulation was named
0.500
0.400
0.400
0.300
0.300
nm
0.500
0.200
0.100
0.000
0.00
0.200
0.100
10.00
20.00 30.00
Time (ns)
40.00
50.00
0.000
0.00
10.00
20.00 30.00
Time (ns)
(A)
(B)
0.500
0.400
nm
nm
CYP17-PREG complex.
0.300
0.200
0.100
0.000
0.00
10.00
20.00 30.00
Time (ns)
(C)
40.00
50.00
40.00
50.00
135
Figure 5. Plots of RMSD (in nm) versus time of simulation (in ns) for CYP17-ABE complex of each
protonation state of His 373. HISD (A), HISE (B) and HISH (C).
0.16
0.14
χ (degree)
0.12
nm
0.1
0.08
0.06
0.04
0.02
0
HID
HIE
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
HIP
HID
HIE
HIP
(B)
(A)
Figure 6. Plots of RMSD (A) and χ1 (B) of the His373 of each protonation state. The data represent the
average of five 10-ns snapshots from 50-ns production MD of CYP17-ABE complex for each protonation
state expressed as mean standard deviation. It was considered a p value less than 0.05.
Table 1. Binding energies of CYP17 with PREG.
Poses
Binding Energy / kcal mol-1
1
-12.2
2
-10.6
3
-10.2
4
-9.5
136
A
B
C
D
Figure 7. The binding poses of PREG in the active site of the enzyme CYP17.
0.500
nm
0.400
0.300
0.200
0.100
0.000
0.00
10.00
20.00 30.00 40.00
Time (ns)
50.00
Figure 8. Plot of RMSD (in nm) versus time of simulation (in ns) for CYP17-PREG complex
The Figure 9A shows the ABE, heme group and residue CYS442 of the CYP17-ABE. Whereas the
Figure 9B shows the PREG, heme group and residue CYP442 of the CYP17-PREG complex. Both
structures were obtained after 50-ns MD simulations. We have found that after 50-ns MD simulations for
CYP17-ABE and CYP17-PREG complexes, there is no formation of a complex hexacoordinate with
octahedral geometry which is imposed by Fe+2 atomic d-orbitals. Conversely, hexacoordinated states are
137
formed after geometric optimization of complexes by ONIOM calculations (Figures 10A and 10B). This
leads us to infer that classical MD simulations does not adequately describe the d-orbitals geometry. In
addition to the limitations of MD simulations, we can see in Figure 9A and 9B that the porphyrin ring of
heme group is nonplanar where the opposite occurs with the quantum mechanical calculations (Figures 10A
and 10B).
(C17)
(N22)
(Fe)
(SG)
(SG)
A
B
Figure 9. ABE, heme group and CYS442 of CYP17-ABE complex (A) and PREG, heme group and
CYS442 of CYP17-PREG complex (B).
The Table 2 shows the values of Lennard-Jones potential for protein-ligand (CYP17-ABE/PREG)
and ligand-water interactions (〈𝑉𝐿𝐽 〉𝑝 and 〈𝑉𝐿𝐽 〉𝑤 ) and Coulomb potential for protein-ligand and ligand-water
interactions ( 〈𝑉𝐶 〉𝑝 and 〈𝑉𝐶 〉𝑤 ), as well as 〈𝑉𝐿𝐽 〉𝑝 + 〈𝑉𝐶 〉𝑝 and 〈𝑉𝐿𝐽 〉𝑤 + 〈𝑉𝐶 〉𝑤 are referred as the binding
energies between ligand (ABE or PREG) and CYP17 and between ligand and water respectively.
According the Eq. 1 and Eq. 2 and the data of Table 2, the interaction energy for ABE and the
CYP17 binding site and for ABE and water are -53.00 kcal mol-1 and -35.50 kcal mol-1 respectively.
Likewise, the interaction energies for PREG are -68.90 kcal mol-1 and -60.20 kcal mol-1 (Table 2). The
𝑝
𝑤
difference between interaction energies of ligand for enzyme binding site and for water (𝐸𝑏𝑖𝑛𝑑 − 𝐸𝑏𝑖𝑛𝑑
)
indicates its diffusion trend of the water  enzyme binding site. The more negative energy difference value,
the greater the diffusion trend of the water  enzyme binding site.
138
Table 2. Values of Lennard-Jones potential (VLJ) and Coulomb potential (VC), in kcal mol-1, between CYP17
(p)/water (w) and ligands ABE and PREG.
Potentials
Interaction ABE /
kcal mol-1
Interaction PREG /
kcal mol-1
〈𝑉𝐿𝐽 〉𝑝
-52,49
-39,55
〈𝑉𝐶 〉𝑝
-0,51
-29,35
〈𝑉𝐿𝐽 〉𝑝 + 〈𝑉𝐶 〉𝑝
-53.00
-68.9
〈𝑉𝐿𝐽 〉𝑤
-33,53
-23,90
〈𝑉𝐶 〉𝑤
-1,52
-36,30
〈𝑉𝐿𝐽 〉𝑤 + 〈𝑉𝐶 〉𝑤
-35.5
-60.20
The optimized structures of the QM (high layer) and MM (low layer) of the CYP17-ABE and
CYP17-PREG complexes and enzyme CYP17 are shown in Figures 10A, 10B and 10C respectively. The
electronic energy values of CYP17, ABE, PREG, CYP17-ABE and CYP17-PREG complexes are shown in
Table 3. As expected, the interaction energy results of molecular mechanics and QM/MM calculations for
ABE and PREG are not in agreement. This is primarily due to the description of the spin state of the iron
heme from ONIOM calculations, which does not occur in molecular mechanics calculations (more detail see
section discussion).
Table 3. Values of electronic energy of CYP17 enzyme, ABE, PREG, CYP17-ABE and CYP17-PREG
complexes.
a
a
Molecule
Ee / kcal mol-1
ABE
-667313.50
Interaction energy /
kcal mol-1
-61.75a
PREG
-608804.64
-40.37b
CYP17
-2249240.31
CYP17-ABE complex
-2916615.57
CYP17-PREG complex
-2858085.32
Interaction energy derived from Eq. (4)
Interaction energy derived from Eq. (5)
139
(Fe)
(C17)
(SG)
(N22)
(SG)
(SG)
C
B
A
Figure 10. Optimized structures of the QM (highlighted) and MM (gray lines) layers of the CYP17-ABE
(A) and CYP17-PREG (B) complexes and CYP17 enzyme (C) at the ONIOM (B3LYP:AMBER) level.
The virial graphs of binding pockets of the abiraterone and pregnenolone in the CYP17 active site are
shown in Figures 11A and 11B, respectively. Table 4 shows the charge densities (ρ) of the bond critical
points (BCPs) of the bond paths between abiraterone (or pregnenolone) and CYP17 active site.
B
A
Figure 11. Virial graphs of binding pockets of CYP17-ABE (A) and CYP17-PREG (B) complexes.
Table 4. Electronic density (ρ), in au., of the critical points between abiraterone (or pregnenolone) and
CYP17 active site.
Complex
CYP17-ABE
BCP
ρ (au.)
26
0.0003
31
0.0058
34
0.0011
Complex
CYP17-PREG
BCP
ρ (au.)
28
0.0018
31
0.0006
36
0.0012
140
42
0.0027
39
0.0007
47
0.0001
49
0.0017
51
0.0001
51
0.0004
53
0.0021
53
0.0005
60
0.0010
56
0.0008
68
0.0017
58
0.0006
72
0.0026
59
0.0005
75
0.0013
62
0.0002
76
0.0014
65
0.0002
81
82
0.0012
79
0.0073
0.0007
82
0.0026
92
0.0010
86
0.0017
97
0.0033
90
0.0010
110
0.0003
101
0.0005
127
0.0017
107
0.0010
149
0.0009
122
0.0009
150
0.0033
126
0.0012
154
0.0008
127
0.0002
163
0.0012
130
0.0059
168
0.0016
131
0.0023
182
0.0013
142
0.0012
195
0.0003
143
0.0018
196
0.0012
155
0.0021
200
0.0012
155
0.0021
204
0.0015
165
0.0010
206
0.0008
174
0.0004
208
0.0020
185
0.0016
212
0.0003
190
0.0010
219
0.0004
199
0.0014
225
0.0022
202
0.0006
231
0.0018
212
0.0015
233
0.0004
213
0.0004
238
0.0012
217
0.0009
246
0.0012
220
0.0001
258
0.0017
232
0.0010
141
Total
286
0.0022
233
0.0005
300
0.0012
237
0.0010
311
0.0012
253
0.0006
323
0.0015
258
0.0006
458
0.0011
287
0.0007
294
0.0007
300
0.0008
362
0.0005
0.0610
0.0560
DISCUSSION
From Henderson-Hasselbalch equation the proportion found between positively charged and neutral
His was 1:23. Since CYP17 has fifteen His in its structure, only one histidine was protonated on nitrogens
ND1 and NE2, the histidine 160. Molecular dynamics were employed to assess the stability of the CYPABE complex and define the protonation state of His373 situated close to the heme group. According to the
plot shown in Figure 6A, there is no significant difference between RMSD average values of the three
protonation states. The RMSD indicated that there have been no large variations in the positions of C-α of
HID, HIE and HIP protonation states compared to X-ray structure. Likewise, the average values of dihedral
angle showed no significant differences between protonation states (Figure 6B). Therefore, subsequent
theoretical studies involving CYP17 enzyme, its histidine residue close to the heme group (His373), can
assume any protonation state.
There are no significant differences between structures before and after the production MD
simulation of CYP17-ABE and CYP17-PREG complexes as shown in structural fitting (Figures S2A and
S2B, Supporting Information). These data are in accordance with the B-factor62 (that reflects the fluctuation
of an atom about its average position) of complexes (Figures S3A and S3B, Supporting Information).
According to the above results, the differences between the interaction energies of the ABE and PREG
are -17.50 kcal mol-1 and -8.70 kcal mol-1 respectively. Although the PREG has a more negative interaction
energy with the CYP17 binding site (-68.90 kcal mol-1) compared with ABE (-53.00 kcal mol-1), PREG also
has a more negative energy interaction energy with water compared to the ABE (-60.20 kcal mol-1 and -35.5
kcal mol-1 respectively). In the other words, ABE has a higher diffusion trend water  enzyme binding site
compared to PREG.
According to Eqs. (4) and (5), the electronic interaction energies of ABE and PREG have a value of 61.75 kcal mol-1 and -40.37 kcal mol-1 respectively (Table 3). In other words, the interaction electronic
142
energy of ABE is 21.38 kcal mol-1 lowest in comparison to the PREG, indicating that the drug for metastatic
prostate cancer has highest affinity. This more negative value of interaction electronic energy of ABE is
reflected in a smaller distance with respect to iron heme (2.70 Å) compared to PREG (5.80 Å) (Figures 10A
and 10B, respectively).
According to the above description, the interaction energy results between ligands (ABE or PREG)
and CYP17 using molecular dynamics and ONIOM are not in agreement. PREG interaction energy by MD
simulations resulted in a more stable interaction with the CYP17 compared to ABE. In contrast, ONIOM
calculations showed that ABE has a higher affinity for the enzyme then PREG. Metalloproteins like
Cytochrome P450 are hard to be characterized computationally due to the presence of the metal cation
which exerts a strong Coulombic forces acting on structure backbone. 131,132 Metals with unpaired electrons
of d-shells of atomic orbitals have preferred coordination geometries as well as tetrahedral, octahedral, or
square planar. This preference directly influences the position of the amino acids and therefore in the protein
tertiary structure. In addition, the electronic structure of a metal is closely related to its coordination
geometry. Different electronic configurations may be taken for a metal depending on the nature of its ligand.
Therefore, there is a close relationship between electronic structure of a metal and the tertiary structure of
the protein.132 For that reason, a classical description of a metal through molecular dynamics would be a
failure, because the coordination geometry could undergo changes during the simulation. This is viewed in
the distances between ABE or PREG and iron heme after molecular dynamics (Figures 9A and 9B,
respectively) and after ONIOM calculations (Figures 10A and 10B, respectively) of CYP17-ABE and
CYP17-PREG complexes. After ONIOM calculations, the bond length between aromatic nitrogen of ABE
and iron heme (2.70 Å) and the bond length between gamma-sufur (SG) of CYS442 and iron heme (2.40 Å)
were shortened compared to bond lengths after molecular dynamics (4.72 Å and 4.61 Å, respectively). The
same occurred to the bond lengths between C17 of PREG and iron heme and between SG of CYS442 and
iron heme. Furthermore, classical force fields are parameterized for a metal in specific coordination
geometry and ligated to a specific ligand. The spin state of iron porphyrins is determined by the coordination
geometry and the nature of ligands. 133 In heme proteins, iron to be found in different coordination states:
four, five or six-coordinate and exhibits formal charge of +2 or +3, 134 where unligated ferroheme can have
three spin states as a singlet, triplet or quintet (S = 0, 1 and 2, respectively). All these aspects about the metal
cations, particularly iron porphyrins, should be approached in terms of electronic structure through the
quantum mechanics calculations.135 The study of heme systems through the quantum-mechanical
computational methods, as well as Hartree-Fock and density functional theory (DFT) contributed
satisfactorily in the interpretation and understanding of the electronic structure of active site of
hemoproteins.136,137 Thereby, hybrid quantum mechanical and molecular mechanical (QM/MM) methods
can be very effective in the study of hemoproteins, as well ONIOM method. The difference in accuracy
143
between classical and hybrid methods was the main reason for the disagreement between interaction energy
results.
The molecular graphs of binding pockets of the ABE and PREG, in Figures 11A and 11B
respectively, shows all the interactions between the ligand (ABE or PREG) and the CYP17 active site. There
are 43 intramolecular interactions between ABE and active site according to the virial graph of ABE binding
pocket where the charge densities (ρ) values of the corresponding critical points are shown in Table 4. On
the other hand, PREG binding pocket has 46 intramolecular interactions (Figure 11B and Table 4). The sum
of all electronic density of interactions of ABE and PREG binding pockets were 0.0610 au. and 0.0560 au.
respectively. This result shows the overall stabilizing effect of the electronic density of interactions of
pockets on the ABE binding pocket.
CONCLUSIONS
In the present study, the molecular dynamics revealed that the histidine residue situated close to
heme group (His373) in the CYP17 enzyme can display any protonation state since it showed no significant
differences both in RMSD average values of Cα as the variation of dihedral angle (χ). With the protonation
state of His373 defined, we have obtained the two protein-ligand complexes used in this study: CYP17-ABE
and CYP17-PREG complexes.
Although the molecular mechanics interaction energy showed that ABE has a higher diffusion trend
(hydrophobicity) water  CYP17 binding site compared to the PREG, classical descriptions cannot be
employed for enzymes containing heme group due to the limitations observed in Figures 9A and 9B. The
ONIOM hybrid (B3LYP:AMBER) methods showed that ABE has a higher affinity for the CYP17 because
it has a more negative interaction energy value compared to PREG.
To investigate the intramolecular interactions of ligands ABE and PREG, we used the quantum
theory of atoms and molecules (QTAIM). QTAIM analysis indicates that ABE has a higher electronic
density in their interactions with the active site of the CYP17 enzyme compared to PREG.
By using molecular mechanics (Docking and MD simulations) and quantum mechanics calculations
(ONIOM and QTAIM), we have investigated the inhibitory effect of Abiraterone through the interaction
energies and electronic density. This detailed study proves that the rationality evaluation inhibition
performance may grant valuable information about CYP17-ABE recognition providing new insights to
design the new homologous CYP17 inhibitors.
144
ACKNOWLEDGMENTS
ASSOCIATED CONTENT
Table with the electronic energy, bond length Fe-NH2 and bond length Fe-NH3 of triplet and quintet
states at B3LYP, BP86, BVWN, M062X, ωB97XD and PBE1PBE methods with the 6-311G(d,p) basis set
of the Fe(NH3)2(NH2)2. Optimized geometries of Fe(NH3)2(NH2)2 at B3LYP/6-311G(d,p) level in the triplet
and quintet states. Fitting structures of CYP17-ABE and CYP17-PREG complexes. B-factors for the
CYP17-ABE and CYP17-PREG complexes. This material is available free of charge via the Internet at
http://pubs.acs.org.
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156
TABLE OF CONTENTS
157
Supporting Information for “Docking, Molecular Dynamics, ONIOM, QTAIM: theoretical
investigation of inhibitory effect of Abiraterone on CYP17”.
TABLE S1. Electronic energy, in a.u., bond length Fe-NH2 and bond length Fe-NH3, in angstroms, of triplet
and quintet states of the Fe(NH3)2(NH2)2 with different functionals using 6-311G(d,p) basis set.
METHOD
Multiplicity
Electronic Energy
(a.u.)
Triplet
Quintet
Triplet
Quintet
Triplet
Quintet
Triplet
Quintet
Triplet
Quintet
Triplet
Quintet
-1488.82340086
-1488.83272142
-1489.00912021
-1488.99952264
-1492.23892216
-1492.23690248
Error
Error
-1488.74019588
-1488.74695927
-1488.28486454
Error
B3LYP
BP86
BVWN
M062X
ωB97XD
PBE1PBE
B3LYP - Triplet
Bond
length
Fe-NH2
(Å)
1.92212
1.88714
1.87322
1.87293
1.90792
1.89844
1.87885
1.88793
1.90899
-
Bond
length
FeNH3(Å)
2.02763
2.35259
1.98187
2.31036
2.04101
2.41968
2.01482
2.31236
2.00701
-
B3LYP - Quintet
Figure S1. Optimized geometries of Fe(NH3)2(NH2)2 at B3LYP/6-311G(d,p) level in the triplet and quintet
states.
158
A
B
Figure S2. Fitting structures of CYP17-ABE (A) and CYP17-PREG (B) complexes before (yellow) and
after (red) molecular dynamics simulation. The picture was rendered using VMD. 1
A
B
Figure S3. B-factors for the CYP17-ABE (A) and CYP17-PREG (B) complexes. The picture was rendered
using VMD1, “blue” indicates high, “white” indicates intermediate and “red” low values. High temperature
factors imply exibility and low ones rigidity.
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