The virtual EPR laboratory: a user guide to ab initio modelling

The virtual EPR laboratory: a user guide to ab initio modelling
Antonino Polimeno and Vincenzo Barone
The virtual EPR laboratory: a user guide to ab initio modelling
Antonino Polimeno and Vincenzo Barone
1. INTRODUCTION ..............................................................................................................................4
1.1 Modelling cw-EPR spectra ....................................................................................................4
1.2 Fitting and predicting ............................................................................................................6
1.3 Chapter overview ...................................................................................................................8
2. MODELLING TOOLS .......................................................................................................................9
2.1 Theory ....................................................................................................................................9
2.2 Implementation ....................................................................................................................14
3. TUTORIAL & CASE-STUDIES ........................................................................................................17
3.1 Tutorial: tempone in aqueous solition .................................................................................19
3.2 Case study 1: p-(Methylthio)phenyl Nitronylnitroxide in toluene .......................................35
3.3 Case study 2: Fmoc-(Aib-Aib-TOAC)2-Aib-OMe in acetonitrile.........................................37
3.4 Case study 3: tempo-palmitate in 5CB ................................................................................41
4. CONCLUSIONS .............................................................................................................................43
4.1 Perspectives .........................................................................................................................43
4.2 Summary ..............................................................................................................................44
5. ACKNOWLEDGMENTS ..................................................................................................................44
6. REFERENCES ...............................................................................................................................45
The virtual EPR laboratory: a user guide to ab initio modelling
Antonino Polimeno
Dipartimento di Scienze Chimiche, Università degli Studi di Padova
Via Marzolo 1, I-35131 Padova, Italy
Vincenzo Barone
Dipartimento di Chimica and INSTM, Università di Napoli “Federico II”
Complesso Universitario di Monte Sant’Angelo, Via Cintia, I-80126 Napoli, Italy
1. Introduction
1.1 Modelling cw-EPR spectra
In the 1981 science fiction movie “Outland”, Space Marshal O’Neil (Sean Connery) investigates the
mysterious deaths of a number of mine workers on one of Jupiter’s moons. Pretty soon he discovers
that the mine boss has been giving his workers an amphetamine-like, work-enhancing drug that
keeps them productive for months - until they finally snap, go berserk and die. One of the topic
moments of the movie, at least for a computational chemist, is when Marshal O’Neil asks Dr.
Lazarus, the station resident physician, to identify the mysterious drug. She activates a wonderful
panoramic screen, starts punching buttons on a complex console - no doubt attached to a gigantic
computer - and from a tiny dry sample she extracts the killer molecule, visualizes it on the screen,
and calculates all its properties: reactivity, spectroscopic fingerprints, toxicology and so on. All of
this in less than thirty seconds.1
Now, this is Hollywood computational chemistry of the eighties. But, could it become real in a
foreseeable future? Or perhaps, with a number of limitations and approximations that are inherent to
real science, could it be already currently available? Perhaps so. In particular, we are nowadays able
to state that, if Marshal O’Neil’s molecule is a free radical in solution, a full prediction of its
continuous wave Electron Paramagnetic Resonance (cw-EPR) spectrum is available to us. Granted,
we shall probably need a little more than thirty seconds. But, if the molecule is made of, say, less
than 50 atoms, and basic information on the solvent nature are available, there is a fair chance that
Dr. Lazarus’ trick is doable.
This is not surprising. The link between theoretical predictive methodologies and EPR spectroscopy
dates back to several decades, and it is due to a happy coincidence between experimental needs and
available interpretative tools. On one side, the intrinsic resolution of the EPR spectra, together with
the unique role played by paramagnetic probes in providing information about their environment,
make in principle EPR one of the most powerful methods of investigation on the electron
distribution in molecules, and on the properties of their environments. On the other side, EPR
spectroscopy is intrinsically amenable to an advanced theoretical interpretation: the tools needed are
based on quantum chemistry, as far as the parameters of the spin Hamiltonian are concerned, and
on statistical thermodynamics, for the spectral lineshapes.
Nowadays, the introduction of the Density Functional Theory (DFT) has proved to be a turning
point for the calculations of the spin Hamiltonian parameters.2 Reliable methods for the evaluation
of hyperfine tensors are available for several cases and, particularly for radicals in solution, the
agreement between experimental and calculated parameters of the spin Hamiltonian by DFT is
outstanding.1-4
Moreover, because of its favorable time scale, EPR experiments can be highly sensitive to the
details of the rotational and internal dynamics. In the so-called slow motional regime the spectral
line shapes take on a complex form which is found to be sensitive to the microscopic details of the
motional process. This is to be contrasted with the fast motional regime, where simple Lorentzian
line shapes are observed, and only estimates of molecular parameters (e.g. diffusion tensor values)
are obtained independently from the microscopic details of the molecular dynamics.
1.2 Fitting and predicting
The interpretation of slow motional spectra requires an analysis based upon sophisticated theory,
and it is usually carried on via explicit modelization of the paramagnetic probe dynamics, as
predicted for various Markovian models of reorientation. In order to extract useful dynamic
information from EPR experiments, a slow motional theory based on the Stochastic Liouville
Equation (SLE) has been developed, which shows that the more dramatic lineshape changes are
particularly sensitive to microscopic details of the dynamics.
The relationship between EPR spectroscopic measurements and molecular properties can be
gathered only indirectly, that is, structural and dynamic molecular characteristics can only be
inferred by the systematic application of modelling and numerical simulations to interpret
experimental observables. A straightforward way to achieve this goal is the employment of
spectroscopic evidence as the 'target' of a fitting procedure of molecular, mesoscopic and
macroscopic parameters entering the model.
This strategy, based on the idea of a general fitting approach, can be very helpful in providing
detailed characterization of structural parameters (e.g. intramolecular distances) and dissipative
parameters (e.g. diffusion tensors). An intrinsic limitation of this approach is the difficulty of
avoiding uncertainties due to multiple minima in the fitting procedure, and the difficulty, in many
cases, to reconciliate best-fitted parameters with more general approaches or known physical trends
(e.g. temperature dependence).
A more refined methodology is based on an integrated computational strategy (ICS), i.e. the
combination of i) quantum mechanical (QM) calculations of structural parameters and magnetic
tensors possibly including average interactions with the environment (by discrete-continuum
solvent models)1 and short-time dynamical effects; ii) direct feeding of calculated molecular
1
The most promising general approach to the problem of environmental (e.g. solvent) effects can be based, in our
opinion, on a system-bath decomposition. The system includes the part of the solute where the essential of the process
to be investigated is localized together with, possibly, the few solvent molecules strongly (and specifically) interacting
with it. This part is treated at the electronic level of resolution, and is immersed in a polarizable continuum, mimicking
the macroscopic properties of the solvent. The solution process can then be dissected into the creation of a cavity in the
parameters into dynamic models based on molecular dynamics, coarse grain dynamics, and, above
all, stochastic modelling or a combination of the three. Fine-tuning of a limited set of molecular or
mesoscopic parameters via limited fitting can still be employed. In particular, electron spin
resonance measurements are highly informative and they are nowadays becoming particularly
amenable to the integrated strategy, thanks to increasing experimental technological progress,
advancement in computational methods, and refinement of available dynamics models. Nitroxidederived paramagnetic probes allow in principle to detect several information contents at once:
secondary structure information, inter-residual distances, if more than one spin probe is present,
large amplitude protein motions from the overall EPR spectrum shape.5-7
An ab initio interpretation of EPR spectroscopy needs to take into account different aspects
regarding the structural, dynamical and magnetic properties of the molecular system under
investigation, and it requires, as input parameters, the known basic molecular information and
solvent macroscopic parameters. The application of the stochastic Liouville equation formalism
integrates the structural and dynamic ingredients to give directly the spectrum with minimal
additional fitting procedures.8-12
solute (spending energy Ecav), and the successive switching on of dispersion-repulsion (with energy Edis-rep) and
electrostatic (with energy Eel) interactions with surrounding solvent molecules.
The so called polarizable continuum model (PCM) offers a unified and well sound framework for the evaluation of all
these contributions both for isotropic and anisotropic solutions. In PCM the solute molecule (possibly supplemented by
some strongly bound solvent molecules, to include short-range effects like, e.g., hydrogen bonds) is embedded in a
cavity formed by the envelope of spheres centered on the solute atoms. The cavity surface is finely subdivided in small
tiles (tesserae), and the solvent reaction field determining the electrostatic contribution is described in terms of apparent
point charges appearing in tesserae and self-consistently adjusted with the solute electron density. The solvation charges
(q) depend, in turn, on the electrostatic potential (V) on tesserae through a geometrical matrix Q related to the position
and size of the surface tesserae, so that the free energy in solution G can be written: G
1
= E[ ρ ] + VNN + V †QV
2
where E[ρ] is the free-solute energy, but with the electron density polarized by the solvent, and VNN is the repulsion
between solute nuclei.
The core of the model is then the definition of the Q matrix, which in the most recent implementations of PCM depends
only on the electrostatic potentials, takes into the proper account the part of the solute electron density outside the
molecular cavity, and allows the treatment of conventional, isotropic solutions, ionic strengths, and anisotropic media
like liquid crystals. Furthermore, analytical first and second derivatives w.r.t. geometrical, electric, and magnetic
parameters have been coded, thus giving access to proper evaluation of structural, thermodynamic, kinetic, and
spectroscopic solvent shifts.
1.3 Chapter overview
Our main objectives in this Chapter are 1) to apply integrated theoretical tools to the modelling of
cw-EPR, 2) to shed light on methodological aspects13,14 and 3) to underline the applicability and
user-friendliness of ICS if a careful implementation is made available, in the form of purposely
tailored software. We shall concentrate on cw-EPR of mono and bi organic radicals in solution for
several reasons, by chiefly for the need of re-address a relatively well studied field on EPR
spectroscopy with modern theoretical and computational tools; and for the availability of numerous
and novel high-quality experimental data, to compare with sophisticate a predictive strategy.
Naturally, most of the methodologies presented here can be extended to other electron paramagnetic
resonance
techniques (e.g. ENDOR, FT-EPR) and to other classes of systems (e.g.
metalloproteins). Some considerations on these topics are presented in the conclusive Section of this
Chapter.
Therefore, our plan-of-work is the following. In Section 2, a summary of the theoretical techniques
is given for the ICS interpretation of cw-EPR spectra of radicals in solutions. Formal derivations
will be kept at a bare minimum. Section 3 is devoted to the actual application of the methodology to
test cases, which are used as introductive tutorials to a general computational software tool
implementing the overall theoretical procedure. Conclusions are presented in Section 4. We shall
adopt, throughout the whole Chapter, a two-level teaching strategy: i.e. ideally the reader will be
able to follow the main presentation and discussion without being distracted by too many formal
and details, but in-sets (in the form of footnotes) devoted to advanced methodological aspects will
be available, should the need arise in the reader to clarify some more technical points. After all,
Space Marshall O’Neill, and most experimental researchers working in the EPR spectroscopy field,
wish to have quick answers and a general understanding of the way the answers are obtained. Only
rarely they need to catch up with messy computational intricacies.
2. Modelling tools
2.1 Theory
We present here some qualitative considerations on the foundation of an ab-initio integrated
computational strategy (ICS) to the interpretation of cw-EPR spectra of free radicals.
The
calculation of EPR observables can be in principle based on the complete solution of Schrödinger
equation for the system made of paramagnetic probe + explicit solvent molecules. The system can
be described by a ‘complete’ Hamiltonian2 which contains i) electronic coordinates of the
paramagnetic probe ii) nuclear coordinates and iii) all degrees of freedom of all solvent molecules.
The basic object of study, to which any spectroscopic observable can be linked, is given by the
density matrix ρ̂ , which in turn is obtained from the Liouville equation.
Solving for ρ̂ in time - for instance via an ab-initio molecular dynamics scheme - allows in
principle the direct evaluation of any molecular property. However, significant approximations are
possible, which are basically rooted in time-scale separation arguments. The nuclear coordinates
can be separated into fast probe vibrational coordinates and slow probe coordinates, i.e.
intermolecular rotation degrees of freedom and, if required, intramolecular ‘soft’ torsional degrees
of freedom, Q , relaxing at least in a picoseconds time scale. Then the probe Hamiltonian is
averaged on i) femtoseconds and sub-picoseconds dynamics, pertaining to probe electronic
coordinates and ii) picoseconds dynamics, pertaining to probe internal vibrational degrees of
freedom. The averaging on the electron coordinates is the usual implicit procedure for obtaining a
spin Hamiltonian from the complete Hamiltonian of the radical. In the frame of Born-Oppenheimer
approximation, the averaging on the picosecond dynamics of nuclear coordinates allows to
Hˆ ({ri } , {R k } , {qα } ) = Hˆ probe ({ri } , {R k } ) + Hˆ probe-solvent ({ri } , {R k } , {qα } ) + Hˆ solvent ({qα } ) where probe
and solvent terms are separated. Hamiltonian Hˆ ({r } , {R } , {q } ) contains i) electronic coordinates {r } of the
2
i
k
α
i
paramagnetic probe (where index i runs on all probe electrons), ii) nuclear coordinates {R k } (where index k runs on
all ro-vibrational nuclear coordinates) and iii) coordinates {qα } , in which we include all degrees of freedom of all
solvent molecules, each labelled by index
α.
introduce in the calculation of magnetic parameters the effect of the vibrational motions, that can be
very relevant in some cases. The dependence upon solvent or bath coordinates can be treated at a
classical mechanical level, either by solving explicitly the Newtonian dynamics of the explicit set
or by adopting standard statistical thermodynamics argument3. This is formally equivalent to
averaging the density matrix with respect to solvent variables.
In this way an effective probe Hamiltonian is obtained characterized by magnetic tensors. By
taking into account only the electron Zeeman and the hyperfine interactions, for a probe with one
unpaired electron and N nuclei we can define an averaged magnetic Hamiltonian:
β
Hˆ ( Q ) = e B 0 ⋅ g ( Q ) ⋅ Sˆ + γ e ∑ Iˆ n ⋅ A n ( Q ) ⋅ Sˆ
=
n
3
(1)
The computation of reliable magnetic properties in solution calls for the consideration of true dynamic effects
connected to the proper sampling of the solvent configurational space. Here we discuss briefly short-time effects
leading essentially to averaged values.
As an illustration let us consider a prototypical nitroxide spin probe molecule, di-tert-butyl nitroxide (dtbn), in aqueous
solution: in order to overcome the limitation of currently available empirical force field parameterizations, we
performed first-principle molecular dynamics simulations of the dtbn aqueous solution and, for comparison, in the gasphase [N. Rega, G. Brancato, V. Barone, Chem. Phys. Lett. 2006, 422, 367; M. Pavone, P. Cimino, F. De Angelis, V.
Barone, J. Am. Chem. Soc. 2006, 128, 4338].
The results can be summarized in three main points: the effect of the solvent on the internal dynamics of the solute, the
very flexible structure of the dtbn-water hydrogen bonding network and the rationalization of the solvent effects on the
magnetic parameters. Magnetic parameters are quite sensitive to the configuration of the nitroxide backbone, and in the
particular case of dtbn, the out-of-plane motion of the nitroxide moiety is strongly affected by the solvent medium.
While the average structure in the gas phase is pyramidal, the behaviour of dtbn in solution presents a maximum
probability of finding a planar configuration: this does not mean that the dtbn minimum in solution is planar, but that
there is a significant flattening of the potential energy governing the out-of-plane motion and that the solute undergoes
repeatedly an inter-conversion among pyramidal positions.
The vibrational averaging effects of this large amplitude internal motion have been taken into account by computing the
EPR parameters along the trajectories. The hydrogen bonding network embedding the nitroxide moiety in aqueous
solution presents a very interesting result: the dynamics of the system points out the presence of a variable number of
hydrogen bonds, from zero to two, with an highest probability of only one genuine H-bond. Such feature of dtbn-water
interaction is actually system-dependent, the high flexibility of the NO moiety and the steric repulsion of the tert-butyl
groups decreases the energetically accessible space around the nitroxide oxygen. As a matter of fact, simulations carried
out in the same conditions and level of theory, for a more rigid five-ring nitroxide (proxyl), in aqueous solution
provided a different picture with an average of two nitroxide water H-bonds. In this case the substituents embedding the
NO moiety are constrained in a configuration where methyl groups are never close to the nitroxide oxygen, and also the
backbone of the nitroxide presents an average value of the CNC angle which is lower than in the case of the dtbn, thus
evidencing a better exposition of the NO moiety to the solvent molecules in the case of the proxyl radical. Nevertheless,
the behavior of the closed ring nitroxide in water could not be generalized to all the protic solvents: a similar simulation
of the proxyl molecule in methanol solutions presents, in average, only one genuine solute-solvent H-bond, possibly
because the more crammed H-bonded methanol molecule prevents an easy access to the NO moiety for other solvent
molecules. Once again the reliable description of solvent dynamics plays a crucial role for an accurate prediction of
spectroscopic data. Eventually, the discrete-continuum approach allowed the decoupling of the different contributions
and also the quantification of their effect on each of the molecular parameters: the hydrogen bonding interaction and the
dielectric contribution of the solution bulk, taken independently, have a roughly comparable effect, the dielectric
contribution decreasing when going from dtbn to dtbn-water adducts.
The modified time evolution equation for ρˆ ( Q,t ) can efficiently been interpreted within the
framework of explicit stochastic modelling according to the so-called Stochastic Liouville Equation
(SLE) formalism, defined by the direct inclusion of motional dynamics in the form of stochastic
(Fokker-Planck / diffusive) operators in the Liouvillean governing the time evolution of the system
∂
ρˆ ( Q, t ) = −i ⎡⎣ Hˆ ( Q ) , ρˆ ( , t ) ⎤⎦ − Γˆ ρˆ ( Q, t ) = −Lˆ ( Q ) ρˆ ( Q, t )
∂t
(2)
where g , A n are the averaged magnetic tensors ,while Γ̂ is the stochastic (Fokker-Planck or
Smoluchowski) operator modelling the dependence of the reduced density matrix on relaxing
processes described by stochastic coordinates Q .
This is a general scheme, which can allow for additional considerations and further approximations.
First, the average with respect to picoseconds dynamic processes is carried on, in practice, together
with the average with respect to solvent coordinates to allow the QM evaluation of magnetic tensors
corrected for solvent effects. Secondly, time-separation techniques can also be applied to treat
approximately relatively faster relaxing coordinates included in the relevant set Q , like restricted
(local) torsional motions. Thirdly complex solvent environments like e.g. highly viscous fluids, can
be described by an augmented set of stochastic coordinates, to be included in Q , which describes
slow relaxing local solvent structures. In the case of a rigid paramagnetic probe freely rotating, the
set of stochastic relevant coordinates is usually restricted to the set of orientational coordinates
Q ≡ Ω ; these are described in terms of a simple formulation for a diffusive rotator, characterized by
a diffusion tensor D. The diffusion tensor is determined by the shape of the molecule, deriving from
the minimum energy conformations obtained from the QM calculations. This choice is formalized
by adopting the following simple form for Γ̂ 4
4
For instance, in the case of a rotating probe with one conformational degree of freedom, the internal dynamics can be
described by an extended stochastic model which includes explicitly the torsional angle α; torsional potential and
diffusion properties for the internal rotation are obtained straightforwardly from QM and hydrodynamic estimates,
ˆ = Jˆ ( Ω ) ⋅ D ⋅ Jˆ ( Ω ) + D
respectively, while the modified stochastic operator is Γ
∂
∂ −1
Peq (α )
Peq (α ) in its
∂α
∂α
Γ̂ = Jˆ ( Ω ) ⋅ D ⋅ Jˆ ( Ω )
(3)
where Jˆ ( Ω ) is the angular momentum operator for body rotation.15.
Once the effective Liouvillean is defined, the direct calculation of the cw-EPR signal is possible by
evaluating the spectral density from the expression
I (ω − ω0 ) =
1
π
Re v | [i (ω − ω0 ) + iLˆ ]−1 | vPeq
(4)
where the Liouvillean L̂ acts on a starting vector which is defined as proportional to the x
component of the electron spin operator Sˆx .5 Basic parameters for the direct evaluation of Eq. (4)
are therefore the following: principal values and orientation of hyperfine tensors A n ; principal
values and orientation of Zeeman tensor g; finally the knowledge of the rotational diffusion tensor
D is required.
We can now summarize the ICS as follows. Modeling based on the SLE approach requires the
characterization of magnetic parameters (e.g. hyperfine for
14
N nuclei and Zeeman tensors).
Integration among 1) evaluation of magnetic tensor parameters via QM calculation, with corrections
based on averaging of fast motions, 2) explicit modelling of slow motional processes via stochastic
treatment and 3) evaluation of EPR spectra via SLE is the basic strategy behind a sound ab-initio
approach to interpretation of EPR data. Notice that shape dependent dissipative parameters (e.g.
rotational diffusion tensor) included in stochastic models can be obtained via a simple but effective
hydrodynamic model, directly based on the molecular geometry. The overall strategy is sketch in
Figure 1.
simplest form neglecting coupling in the diffusion tensor and assuming a constant diffusion coefficient
conformational dynamics.
5
1/ 2
For instance, if only one nucleus (e.g. nitrogen) is coupled to the electron one has vPeq
where
I = 1 ; Peq is the Boltzmann distribution in Ω -space, ω
ω0 = g 0 β e B0 / = = γ e B0 , where g 0 = Tr(g) / 3 .
D for
= [ I ]−1/ 2 Sˆx ⊗ 1I Peq1/ 2 ,
is the sweep frequency and
DFT optimization &
characterization / tensors g, A
(1)
Other
interaction
tensors (e.g.
spin-spin
dipolar
interaction)
CW-ESR spectrum in solution
via SLE equation
Diffusion
properties via
HD model /
tensor D
(4)
(2)
(3)
Comparison between calculated
and experimental cw-EPR
spectra
Figure 1. Chart of the integrated computational approach to simulation of the cw-EPR spectra in solution. Steps (2) and (3) are based on the optimized
geometry and electronic structure obtained in step (1).
2.2 Implementation
Without delving too deeply, at least at first level, into the actual implementation of this scheme,
which requires a number of detailed steps and should be discussed both from the theoretical
(formalism, approximations) and computational point of view (numerical implementation, code
structure), let us make the following consideration: steps (1), (2), (3) and (4) of the ICS scheme can
be considered, in a utilitarian philosophical mind-frame, as ‘black boxes’ in which suitable input
should be inserted and from which suitable output should be obtained. Which kind of black boxes
do we have around to be employed? How reliable are they? How can they be customized and
adapted to case of experimental interest? We shall address partially in the final part of this Section,
and moreover in the next one, dedicated to tutorials and test-cases, some of these questions.
Basically, steps (1), (2) are nowadays answered (partially or totally depending upon the specific
cases) by some up-to-date quantum chemistry programs. The optimized structure of the free radical
is in fact obtained by DFT calculations in a solvated environment. Hyperfine and g tensors can been
computed directly. Notice that while dipolar hyperfine terms and g tensors are negligibly affected
by local vibrational averaging effects, this is not the case for the isotropic hyperfine term especially
concerning those large amplitude vibrations which modify hybridization at the radical center.
Naturally, despite ongoing progress, the quantitative agreement between computed and
experimental values is not always sufficient for a fully satisfactory interpretation of the spectrum,
especially concerning isotropic hyperfine splittings (1/3 of the trace of the corresponding hyperfine
tensors). A minimal adjustment of this term from the computed value in the simulation of the EPR
spectrum can therefore be allowed. All these magnetic terms are local in nature, so that they are
scarcely dependent on conformational modifications. Other magnetic terms, relevant for bi-radicals,
(J and spin-spin dipolar interaction) have a long-range character and provide a signature of different
molecular structures. Although the computation of J is, in principle, quite straightforward by e.g.
the so called broken symmetry approach,16 currently available density functionals are not always
sufficiently reliable for the distances characterizing the systems under investigation (>5-6 Å).17
While work is in progress in our laboratory to this end, it is often still preferable to use an
experimental estimate of J. The situation is different for the spin-spin dipolar term, which is the
most critical long-range contribution. Usually, this tensor is calculated by assuming that the two
electrons are localized and placed at the centre of the N – O bond. In this view, the two electrons
are considered just as two point magnetic dipoles. Complete quantum mechanical computations
starting from the computed spin density are available.
ΩMF→GF
GF
B0
ΩLF→MF
LF
MF
ΩMF→AnF
AnF
Figure 2 Laboratory (inertial) frame LF and magnetic field B0 in red; molecule-fixed diffusion MF and magnetic
frames GF and AnF in blue
The evaluation of the diffusion properties, i.e. step (3) can be based on a hydrodynamic approach.
We may start from a simplified view of the molecule under investigation as an ensemble of N
fragments, each formed by spheres representing atoms or groups of atoms, immersed in a
homogeneous isotropic fluid of known viscosity. By assuming a form for the friction tensor of nonconstrained atoms, ξ , one can calculate the friction for the constrained atoms, Ξ . We may assume
for simplicity the basic model for non-interacting or weakly interacting spheres in a fluid, namely
that matrix ξ has only diagonal blocks of the form ξ (T ) 13 where ξ (T ) is the translational friction
of a sphere of radius R0 at temperature T and given by the Stokes law ξ (T ) = CRη (T ) π , where
η (T ) is the solvent viscosity at the given temperature T and C depends on hydrodynamic boundary
conditions. The system friction is then given as Ξ = ξ (T ) B tr B , where B is a rectangular matrix
depending on atomic coordinates only. The diffusion tensor (which can be conveniently partitioned
in translation, rotational, internal and mixed blocks) can now be obtained as the inverse of the
friction tensor
⎛ DTT
⎜ tr
D = ⎜ DTR
⎜ Dtr
⎝ TI
DTR
D RR
DtrRI
DTI ⎞
⎟
D RI ⎟ = k BT Ξ −1
D II ⎟⎠
(5)
and neglecting off-diagonal couplings, an estimate of the rotational diffusion tensor is given by
D RR ≡ D , which depends directly from the atomic coordinates, temperature, and the solvent
viscosity.
Finally the numerical implementation of the SLE is usually based on a symmetrized equivalent of
Eq. (4), in the form I (ω − ω0 ) =
1
−1
Re vPeq1/ 2 ⎣⎡i (ω − ω0 ) + L ⎦⎤ vPeq1/ 2 , where L = Peq−1/ 2Lˆ Peq1/ 2 is the
π
symmetrized stochastic operator. The cw-EPR spectrum is obtained by numerically evaluating the
spectral density defined adopting iterative algorithms, like Lanczos or conjugate gradients8. In
particular, Lanczos algorithm is a recursive procedure to generate orthonormal functions which
allow a tridiagonal matrix representation of the system Liouvillean. Software implementing
different flavours of numerical solutions to the SLE for cw-EPR are available. Here we remind first
off all the open-source ACERT software18 by Freed and coworkers for simulation and analysis of
EPR spectra, which includes basic simulation program EPRLL and non-linear least-square fitting
code NLSL.6.Another example of implementation of (partial) solutions of the SLE is EasySpin, a
free simulation toolbox for the Matlab package.19.
Other codes are based on a simplified description in terms of Lorentzian or Lorentzian-Gaussian
function. Among others we can quote commercial codes Xsophe20, which is intended for
simulating cw-EPR spectra in a user friendly way thanks to an intuitive graphical user interface;
Molecular Sophe20 for the simulation other spectra than cw like FT-EPR, 2 and 3 pulse
ESEEM, HYSCORE and pulsed ENDOR spectra; SimFonia20 which includes several kinds of
magnetic interactions, i.e. Electronic Zeeman, nuclear quadrupole, nuclear hyperfine, nuclear
Zeeman and zero field splitting. Finally eprfit21 is available at the Institute fűr Chemie und
Biochemie of the University of Berlin, while EWSim22 is intended for organic anion radicals in
solution and in the absence of anisotropy effects.
3. Tutorial & case-studies
This Section is essentially devoted
to the description of some paradigmatic cases chosen to
illustrate the potentialities as well as some current limitations of an integrated computational
strategy, whose fundamental building blocks are: i) an accurate determination of structural and
electronic properties of different electronic states via DFT and TD-DFT methods - including
evaluation of solvent effects via PCM approaches and inclusion of short-time dynamic; ii)
evaluation of the diffusion tensor via application of hydrodynamic approach; iii) numerical solution
of SLE. As tutorial, we consider a basic examples: tempone in aqueous solution. Several casestudies are discussed afterwards.
The software employed throughout the Section is a novel outcome of the ICS approach, since it is
based on the idea of full integration between quantum mechanical tools (for structural and magnetic
6
Other codes available from the same source include NL2DC, NL2DR, to simulate 2D-FT EPR spectra with different
methods for the diagonalization of the matrix (conjugate gradients and Rutishauer methods respectively). Both
programs implement a basis pruning algorithm and include a least squares fitting procedures; and NLSL_SRLS, a code
to simulate and fit cw spectra within the so-called Slowly Relaxing Local Structure model.
properties evaluation, in the presence of solvent effects and fast motion averaging) and stochastic
approach to the lineshape evaluation (including a full evaluation of diffusion tensor properties via
the hydrodynamic approximation). Although still in beta version, the new software protocol
(Electron SPIn Resonance Simulation, E-SPIRES) has some attractive features, and can be
thought as a reasonable realization of a user-friendly, wide-purpose virtual cw-EPR spectrometer, at
least for mono and biradicals in solution, with the inclusion of rotational and internal
(conformational) dynamics:
1. the code is highly modular, built (in the C language) according to an object-oriented structure;
changes, additions, inclusions of new features are easy and transparent
2. a fully integrated graphical interface (written in Java) allows the user to set up the numerical
experiment in a natural way, setting up a project, and adding up information as needed (probe
molecular structure, solvent physico-chemical properties, etc.)
3. the code calls independent software for quantum chemistry calculation (currently Gaussian),
creating input files customized automatically and reading output files without any manual
intervention from the user
4. the code is parallelized and can be used on multiprocessor systems (cluster) under OS Linux
As a general rule, the user can proceed by leaving most technical parameters (e.g. matrix
dimensions, Lanczos step for generating cw-EPR spectrum) at their default value, chosen by the
program; but access to refined choices is always available.
The reader can peruse through the tutorial without bothering to consider the footnotes which will be
used liberally throughout the whole Section: here we shall confine theoretical and computational
detailed information.
3.1 Tutorial: tempone in aqueous solition
In the first tutorial we calculate the cw-EPR spectrum of Tempone in
water at 298.15 K. The numerical experiment has been performed on
a single node of a quadriprocessor Linux cluster.
First of all, as in all computational chemistry programs, the user has
the need to define the molecular structure of the probe. This is
currently done by defining the so-called Z-matrix, which define the
topology and initial structure of the paramagnetic molecule of Figure 3. Initial tempone molecular
structure
interest. This is usually generated via existing standard software tools, like Molden, GaussView
etc., or manually. We shall assume, in the case under investigation, that a suitable Z-matrix files has
been already generated when starting to use E-SPIRES. Most chemists are nowadays familiar with
standard molecular drawing tools, which can generate easily a Z-matrix file.
We shall now proceed step-by-step in uploading the data, generating the necessary structural and
magnetic information and finally calculating the spectrum.
Step 1
Step 2
Step 3
Step 4
Figure 4. Steps 1 through 4: loading up the paramagnetic probe Z-matrix
Steps 1 through 4: the user calls the program (1). Technically, the E-SPIRES graphical
java graphical interface is activated; a main control panel and a 3D space is opened where
molecules are drawn (2): by clicking the “SetProject” button a tagged window called “Parameter
Selector” appears, where all the physical properties of the system under study can be set . First, the
user clicks on ''Load Z-Matrix” button to load the molecule (3). In the tutorials/tempone directory
the user selects the ''tempone.zmt” file which contains the Z-matrix, generated in this case by the
Molden package (4).
Figure 5. Step 5. Tempone initial structure is loaded
Step 5: the molecule is loaded. A 3D representation appears in the 3D Space; a reference frame is
drawn. This is the inertial laboratory frame (LF). In the “Parameter Selector” window the Z-matrix
is written in the white area. This area is reactive to mouse clicking, i.e. when a row is clicked, the
corresponding atom is highlighted in green.
Step 6: clicking on the “Set Dynamics” button a new window appears. Here the user chooses the
form of the diffusive operator Γ̂ . Tempone is a small and rigid molecule, so the “One Rigid Body”
model is chosen.
Figure 6. Step 6. Tempone dynamic model is chosen
Step 7: mouse action on the “Spin Probes” button opens a new window to graphically set the spin
Hamiltonian of the molecule. The window has 3 tags, the first to decide how many spin probes are
present and in which position of the molecule they are located via the “Choose Atom” button.
Choose the O-N bond to set the probe of tempone. The O atom becomes green and a frame appears
for the g tensor.
Step 8 : the second and third tags allows to add spin active nuclei to the probe(s). In the “Spin
Probe 1” tag click on “Choose Atom” and select the N atom in the 3D space. The atom becomes
green and a reference frame appears for the hyperfine A tensor; set to 1 the spin number of the
nucleus.
Tutorial 1. Step 7
Tutorial 1. Step 8
Figure 7. Steps 7 and 8. Magnetic tensors g and A are localized
Steps 9 through 12: in the “Physical Data” tag of the “Parameter Selector” a number of
relevant parameters are set. The “B0” button sets the magnetic field to 3197.3 Gauss (9); the “B
sweep” button sets the field sweep to 75.7 Gauss (10); the “Viscosity” button sets the viscosity of
the solvent (water in this tutorial) to 0.89 cP (11); the “Temperature” button sets the temperature to
298.15 K (12).
Step 13: the user selects the “Additional Data” tag to set the intrinsic linewidth to 2.4 Gauss, to
take into account the unresolved super-hyperfine coupling of the electron with the twelve
surrounding hydrogen atoms.
Step 14: the user clicks on the “Diffusion” button in the “Main Control Panel” to enter in the
“diffusion environment”. The diffusion tensor of the molecule is automatically calculated and a new
frame appears in the 3D space. The molecule changes its colour: atoms assume different colours if
they belong to different fragments. In this case, there is only one fragment and so all the atoms look
the same.7
7
Before proceeding further, we address here the general problem of determining diffusion tensor values. In most cases,
description of molecular probes as macroscopic objects immersed in a fluids, i.e. a purely hydrodynamic view, is
sufficient to allow the determination of friction and/or diffusion tensors, by means of a relatively simple description
linking the overall molecular shape directly to roto-translational (and internal) friction properties. The resulting
friction/diffusion tensors are surprisingly – given the limits of a macroscopic description of a molecular system – close
to available experimental values and they are at least a very good starting point for refining fitting procedures, which
can be employed to tune the simulated spectrum to the measured one.
Let us briefly summarize the overall procedure, in its simplest implementation, to estimate diffusion properties of
molecular systems, with internal degrees of freedom, based on a hydrodynamic approach. We may start from a
simplified view of the molecule under investigation as an ensemble of N fragments, each formed by spheres
representing atoms or groups of atoms, immersed in a homogeneous isotropic fluid of known viscosity. Let us assume
that the i-th fragment is composed by Ni spheres (extended atoms) and that the torsional angle θ i defines the relative
orientation of fragments i and i+1. We denote by ui the unitary vector for the corresponding bond. A total of N-1
torsional angles/bonds are present - only non-cyclic and non-ramified topologies are considered here - and each
fragments has ni atoms. For convenience, each ui points from an atom in fragment i to an atom in fragment i+1 for
i ≥ ν and p points from an atom in fragment i+1 to an atom in fragment i for i < ν .
Notice that the definition of the MF in a flexible system is somewhat arbitrary, and can be essentially left to
convenience arguments. For sake of simplicity we may assume that MF is fixed on generic fragment ν. In Figure 11 we
show a scheme of the system, with an assumed MF fixed on the second fragment (ν=2).
By definition, in the MF, atoms of fragment ν have only translational and rotational motions while atoms of all other
fragments have additional internal rotational motions. Let us no associate the set of coordinates
( r, Ω, θ ) , which
describe the translational, rotational and internal torsional motions respectively, with the velocities
( V, ω, θ )
representing the molecule translational velocity, angular velocity around an inertial frame, and associated torsional
momenta. In the presence of constraints in and among fragments, the generalized force, made of force F, torque N and
the internal torques Nint, is related to the generalized velocity by the relation
F = −ΞV
⎛ F ⎞
⎛V⎞
⎜
⎟
⎜ ⎟
F = ⎜ N ⎟ ,V = ⎜ ω ⎟
⎜ N int ⎟
⎜ θ ⎟
⎝
⎠
⎝ ⎠
while in the absence of constraints a similar relation hold for each single extended atom between its velocity and the
force acting on it, which in compact matrix form can be written
f = −ξ v
⎛ f11 ⎞
⎛ v11 ⎞
⎜ ⎟
⎜
⎟
f = ⎜ ... ⎟ , v = ⎜ ... ⎟
⎜ f nN ⎟
⎜ v nN ⎟
⎝ N⎠
⎝ N⎠
i
where f j is the force acting on j-th atom of i-th fragment ( 1 ≤ j ≤ ni ) etc. Constrained and unconstrained forces and
velocities can be related via geometric considerations
F = Af
v = BV
and one can show easily by inspection that A = B . It follows that Ξ = B ξB . By assuming g a form for the friction
tr
tr
tensor of non-constrained atoms, ξ , one can calculate the friction for the constrained atoms, Ξ . We may assume for
simplicity the simplest model for non-interacting sphere in a fluid, namely that matrix ξ has only diagonal blocks of the
ξ0 13 where where ξ 0 is the translational friction of a sphere of radius R0 given by the Stokes law: ξ0 = CR0ηπ
where η is the solvent viscosity and C depends on hydrodynamic boundary conditions. The system friction is then
tr
given as Ξ = ξ 0 B B . The diffusion tensor (which can be conveniently partitioned in translation, rotational, internal
form
and mixed blocks) can now be obtained as the inverse of the friction tensor
⎛ DTT
⎜ tr
D = ⎜ DTR
⎜ Dtr
⎝ TI
DTR
D RR
DtrRI
DTI ⎞
⎟
D RI ⎟ = k BTΞ−1
D II ⎟⎠
i
Finally, let us show how to evaluate matrix B for the system of linearly connected fragments. Let r j be the vector of
the j-th atom of the i-th fragment in the MF. Thus for atoms belonging to the fragment ν, velocities are
vνj = V + ω × rνj
while
for
all
other
atoms
velocities
are
v ij = V + ω × r ji + ∑ θk u k × r ij ,k v ij = T Bij v + R Bij ω + ∑ I Bij ,kθk where r ij ,k is the difference between the
k
k
vector of the j-th atom and the atom at the origin of the unit vector
reference fragment ν to the fragment i;
matrix
T
B = 13 , B = − r
i
j
R
i
j
u k and the sum is taken over fragments that link the
i× I
j ,
Bij = −r ji ,k u k or 0; and finally for vector a, 3 × 3
a× is defined such that, for a generic vector b, the relation a × b = a×b holds.
Step 9
Step 10
Step 11
Step 12
Figure 8. Steps 9 to 12. Definition of physico-chemical parameters
Step 13
Step 14
Figure 9. Steps 13 and 14 of tutorial 1. Intrinsic linewidth and diffusion tensor evaluation
Steps 15 to 17 : to evaluate the magnetic tensors via quantum mechanical calculations the
user enters in the “Gaussian Environment”. Here the user has the possibility to edit the Gaussian
input file generated by E-SPIRES, launch Gaussian or load a pre-calculated Gaussian output file
(15); clicking “Edit input” a simple editor is loaded, using which one can personalize the Gaussian
input file. In this tutorial no changes are introduced (16); clicking the “Launch” button, the input
file is submitted to Gaussian (17). The user can choose in the OPTIONS to run Gaussian
interactively (on a local computer) or to append the job to PBS (in a cluster).8
8 Let us first consider electron-field interactions. It is convenient, as far as the g tensor is concerned, to refer absolute
values to shifts with respect to the free-electron value (ge=2.002319). Namely we consider ∆g = g − g e 13 where 13 is
the 3x3 unit matrix. Let us dissect ∆g into three main contributions ∆g = ∆g
+ ∆g + ∆g
where
the
first two terms are first order contributions, which take into account relativistic mass (RMC) and gauge (GC)
RMC
corrections, respectively. The first term can be expressed as: ∆g
α −β
fine structure constants, S the total spin of the ground state, Pµν
the
kinetic
by: ∆g
1
=
Pµνα − β ϕ µ
∑
2S µν
GC
energy
operator.
∑ ξ (r ) (r r
n
n
n 0
RMC
=−
α2
S
GC
Pµνα β
∑
µν
−
OZ / SOC
ϕ µ Tˆ ϕ µ
where
α
is the
is the spin density matrix, {ϕ} the basis set and Tˆ is
The
− rn ,r r0,s ) Tˆ ϕν
second
term
is
given
where rn is the position vector of the electron
relative to the nucleus n, r0 the position vector relative to the gauge origin and ξ(rn), depending on the effective charge
of the nuclei, will be defined below. These two terms are usually small and have opposite signs so that their
contributions tend to cancel out.
The last term is a second-order contribution arising from the coupling of the Orbital Zeeman (OZ) and the Spin-Orbit
Coupling (SOC) operators. The OZ contribution in the system Hamiltonian is: Hˆ OZ =
β ∑ B ⋅ ˆl ( i ) . It shows a gauge
i
origin dependence, arising from the angular momentum of the ith electron, ˆl ( i ) . In our calculations a Gauge Including
Atomic Orbital (GIAO) approach is used to solve this dependence.
Finally the SOC term is a true two-electron operator, but here it will be approximated by a one-electron operator
involving adjusted effective nuclear charges. This approximation has been proven to work fairly well in the case of
light atoms, providing results close to those obtained using more refined expressions for the SOC operator16. The oneelectron approximate SOC operator reads: Hˆ SOC =
∑ ξ ( r ) ˆl ( i ) ⋅ sˆ ( i )
i,n
n
where ˆl n ( i ) is the angular momentum
n ,i
operator of the ith electron relative to the nucleus n and sˆ ( i ) its spin-operator. The function
( )
as: ξ ri , n =
α2
Z effn
2 ri − R n
ξ ( ri ,n )
is defined
n
3
where Z eff is the effective nuclear charge of atom n at position Rn.
In our general procedure, spin-unrestricted calculations provide the zero-order Kohn-Sham (KS) orbitals and the
magnetic field dependence is taken into account using the coupled-perturbed KS formalism. Solution of the coupled
perturbed KS equation (CP-KS) leads to the determination of the OZ/SOC contribution.
The second term is the hyperfine interaction contribution, which, in turn, contains the so-called Fermi-contact
interaction (an isotropic term), which is related to the spin density at the corresponding nucleus n
by An ,0 =
8π g e
g n β n ∑Pµα,ν− β ϕ µ δ (rkn ) ϕν
3 g0
µ ,ν
and an anisotropic contribution, which can be derived from the
Step 15
Step 16
Tutorial 1. Step17
Figure 10 Steps 15, 16 and 17: evaluating magnetic tensors.
classical expression of interacting dipoles An ,ij =
ge
g n β n ∑Pµα,ν− β ϕ µ rkn−5 ( rkn2 δ i , j − 3rkn ,i rkn , j ) ϕν . A tensor
g0
µ ,ν
components are usually given in Gauss (1 G = 0.1 mT); to convert data to MHz one has to multiply by 2.8025.
From a computational point of view, evaluation of the A tensor (a first-order property) should be simpler than that of
the g tensor. This is true for the anisotropic term, but evaluation of the Fermi-contact contribution involves a number of
difficulties, related to the local quality of basis functions at the nuclei. In the following examples we will use the
purposely tailored NO7D basis sets together with B3LYP or PBE0 hybrid functionals, which have been proven to be
very effective, especially for non hydrogen atoms.
Step 18: the user needs to inform the program that magnetic tensors and structural information is
superseded by output from Gaussian by checking the “Use output” checkbox. When the Gaussian
output is loaded, all the tensors are updated.
Step 19: tensors can be modified manually, in the “Diffusion” environment. To modify a tensor,
just click on it and change the desired quantity. In this tutorial, the Gaussian output (based on
slightly inefficient basis set) gives a value of the trace of the hyperfine constant about 2 Gauss less
than the experimental one. Thus one needs to edit the hyperfine tensor by setting the isotropic value
(“Iso” slide bar in the setting window) to 16.14 Gauss. WARNING: changes are effective only by
pressing the “Apply” button.
Step 20: further adjustments can be obtained by fitting (although, as a general rule, only small
corrections should be necessary), in the “Refine” environment. In this case, after entering the
Refine environment, the user adjusts the traces of g, A and the intrinsic linewidth, checking the
proper boxes.
Step 21: next the user loads a reference experimental spectrum, by clicking the “Load Spectrum”
button and choosea the file tutorials/tempone/experimental_spectra/exp.dat.
Step 18
Step 20
Figure 11. Steps 18 to 21 of tutorial 1. Refining data.
Step 19
Step 21
Steps 22 through 25 : now the user is ready to enter the “ESR environment”, where
spectra can be calculated with or without fitting and then plotted; to refine three parameters, the
user checks the “Fit Mode” checkbox (22); by clicking on the “Calculate” button the spectrum is
obtained by solving the stochastic Liouville equation. (23). Notice that it is possible to run the
calculation interactively (choosing the number of dedicated processors) or via PBS. In this case the
calculation was performed parallelizing the job on one 4 CPU-nodes (four processors); after 15
seconds the calculation ends (24). In the present case, very small corrections (less than 0.1 %) to the
refined parameters are obtained. The theoretical and experimental spectra can be visualized by
clicking the “Plot” button (25).9
9 Naturally, the computational task of solving the SLE is usually carried on in finite arithmetic, by projecting the
+ iHˆ × = iL and the starting vector vP1/ 2
symmetrized time evolution operator Γ
eq
case can be initially defined as
on a suitable basis set that in our
Σ = p S p I ⊗ l = σ , l . The basis set is given by the direct product of spin
S
S
I
I
operators of the nitroxide , defined by electron and nuclear spin quantum numbers p , q , p , q , and of a complete
(usually orthonormal for sake of simplicity) basis set in the functional space in the generic set of stochastic coordinates
Q, which is indicated here generically by l . One needs to define the matrix operator and starting vector elements
( L )Σ ,Σ ' =
Σ iL Σ ' ,
( v )Σ =
Σ |1
and the matrix-vector counterpart of the tri-diagonal coefficient are
β n +1 v n +1 = (L − α n 1) v n − β n v n −1 , α n = v n ⋅ v n , β n = v n ⋅ v n −1 .
Symmetry arguments can be employed to
significantly reduce the number of basis function sets required to achieve convergence, together with numerical
selection of a reduced basis set of functions based on ‘pruning’ of basis element with negligible contribution to the
spectrum. Further details are given below, where two practical examples are presented.
Definition of the stochastic coordinates Q and time evolution operator depends, naturally, from the system specifics. As
a first example, let us consider the description of a rigid molecule in solution. No conformational degrees are included
and only rotational motion is taken into account. In this simple case, the dynamics is characterized by the set of degrees
of freedom identified by the Euler angles specifying the orientation of the molecules with respect to the laboratory
frame LF. By adopting the minimal view of purely diffusive behavior - i.e. neglecting inertial effects due to fast
relaxation of conjugate momenta - a convenient definition of the stochastic coordinates is Q = Ω LF → MF , where
Ω LF → MF is the set of Euler angles defining the instantaneous orientation of frame molecular frame MF, which is the
principal frame of reference for the rotational diffusion tensor D . In isotropic solvents we may write
Γˆ = Jˆ ( Ω LF → MF ) ⋅ D ⋅ Jˆ ( Ω LF → MF ) where Jˆ ( Ω LF → MF ) is the angular momentum operator for body rotation. The
2
Boltzmann distribution (equilibrium solution) is simply P = 1/ 8π . By defining Jˆ ( Ω
) and D in the MF, a
eq
LF → MF
convenient form of Eq. (35) is obtained which is directly written in terms of the diffusion tensor principal values
Γˆ = D1 Jˆ12 ( Ω LF → MF ) + D2 Jˆ22 ( Ω LF → MF ) + D3 Jˆ32 ( Ω LF → MF )
The molecule-fixed properties are evidenced by frames MF, GF and AnF; the magnetic tensors orientation is given now
with respect to the molecular frame MF, which is defined with respect to the laboratory frame. Parameters of the SLE
equation for the case of a rigid paramagnetic probe dissolved in an isotropic medium are then defined as the principal
values of the diffusion tensor D, the principal values of the g and An tensors, and Euler angles Ω MF →GF , Ω MF → An F
which give the relative orientation of the magnetic tensors with respect to the diffusion tensor.
We may now choose a specific basis set for the specific ensemble of stochastic coordinates, i.e. Ω LF → MF ; usually
normalized Wigner matrix functions are employed l ≡
1
( 8π )
1/ 2
J *
D MK
( Ω LF →MF ) = JMK . Symmetry is also
typically employed by adopting a linearly transformed basis set which accounts for invariance of the Liouvillean for a
rotation around y -molecular axis: Σ
s K = ( −1)
L+ K
, with K ≥ 0 , and j
K
K
= σ , j K LMK = ⎡⎣ 2 (1 + δ K ,0 ) ⎤⎦
−1/ 2
e − iπ ( j
K
−1) / 4
⎡⎣ + + j k s K − ⎤⎦ where
= ±1 for K > 0 , (−1) L for K = 0 ; ket symbols + , − stand for Σ with
positive K and Σ with corresponding opposite K, respectively. Matrix elements of the stochastic Liouvillean in the
symmetrized basis set are real. A symmetric matrix representation of the Liouville operator is given as:
Σ1 Lˆ Σ 2
K
−1/ 2
1
⎡(1 + δ K ,0 )(1 + δ K ',0 ) ⎤
⎦
2⎣
× ⎡δ j K1 , j K2 Re { + Lˆ + + j K2 s K2 + Lˆ - } + δ j K1 ,− j K2 Im { + Lˆ + + j K2 s K2 + Lˆ - }⎤
⎣
⎦
=
To evaluate explicitly symmetrized or unsymmetrized matrix elements, one needs to make explicit the dependence of
ˆ
the superhamiltonian iH
×
from magnetic and orientational parameters. Following the established route we adopt a
spherical irreducible tensorial representation Hˆ =
×
l
∑µ ∑ ∑
l = 0,2 m , m '=− l
where
µ
( l , m ')*
µ , MF
runs over all possible interactions F
l
( l , m ')* ˆ ( l , m )
D mm
' (Ω LF → MF ) Fµ , MF Aµ , LF
ˆ (l , m ) is obtained
is build from elements of g and A in the MF, A
µ , LF
from spin operators. Next the Liouvillean matrix elements are straightforwardly calculated in the unsymmetrized basis
set and the symmetrized matrix is built. The starting vector is also easily calculated, since Σ v
K
∝ δ j K ,1 Σ v .
Explicit matrix element in the unsymmetrized set are obtained following standard arguments reported elsewhere9.
Step 22
Step 24
Figure 12. Steps 22 to 25 of tutorial 1. Evaluating the spectrum.
Step 23
Step 25
3.2 Case study 1: p-(Methylthio)phenyl Nitronylnitroxide in toluene
In the first case study we address the interpretation, via an ab-initio integrated computational
approach, of cw-EPR spectra of p-(methyl-thio) phenyl-nitronyl-nitroxide (MTPNN) dissolved in
toluene for a wide range of temperatures (155-292 K) with minimal resorting to fitting procedures,
proving that the combination of sensitive EPR spectroscopy and sophisticate modelling can be
highly helpful in providing structural and dynamic information on molecular systems. The system
geometry is summarized in Figure 13. A set of Euler angles Ω defines the relative orientation of a
molecular frame (MF), fixed rigidly on the nitroxide ring, with respect to the LF; the local
magnetic frames are in turn defined with respect to MF by proper sets of Euler angles.
In particular, the search of new materials with tailored magnetic properties has intensified in recent
years. In this field the most popular stable radicals are nitronyl nitroxide (NIT) free radicals. They
exhibit a large variety of magnetic behaviour: paramagnetism down to very low temperature,
ferromagnetism, antiferromagnetism.23 Moreover, the nitronyl nitroxides have also been known as
bidentate ligands for various transition and rare-earth metal ions. Ferromagnetic ground states have
been observed also in these complexes.24 For these particular magnetic properties NIT radicals are
particularly appealing as molecular units for composite new materials. In the path towards new
magnetic materials, the characterization of the electronic distributions and magnetic properties of
isolated radicals is of primary interest. Theoretical predictions of the spin distribution on the
radicals by DFT calculations are necessary in order to understand the radical-radical interactions in
bulk and composite materials. On the other hand, the spin density depends strongly on the
interaction with the environment that can be very complex in a composite material. Here, we show
that for a prototypical nitronyl nitroxide like MTPNN25 in a simple environment as a toluene
solution, starting simply from the formula of the radical and the physical parameters of the solvent,
it is possible to calculate EPR spectra showing afterwards an exceptionally good agreement with the
experimental ones, from room temperature to a temperature very near to the glassy transition.
The effective Hamiltonian for the system is given
by Eq. (1): two nuclei are explicitly coupled with
the paramagnetic center. Within E-SPIRES, the
users will modify accordingly steps 7 and 8, to
identify the two nuclei. The spectra are then
calculated
without
further
adjustments
of
temperature-dependent fitted parameters. In Figure
14, we compare the experimental (full line) and
simulated (dashed line) cw-EPR spectra of MTPNN
in toluene in the temperature range 155-292 K.
Since experimental spectra at different temperatures
have
been
measured
at
slightly
varying
Figure 13 Reference frames and geometry of MTPNN
frequencies ν 0 , in Figure 18 spectra are reported
relative to their respective central field B0 , for the reader’s convenience. Notice however that no
adjustment is required in the absolute position of the spectra. In fact the measured value of g 0 at
room temperature (g0 = 2.00681) is matching perfectly the predicted theoretical value, obtained as
1/3 of the trace of the g tensor, g 0calc = 2.00686 .13
Figure 14 Experimental (full line) and simulated (dashed line) cw-EPR spectra of MTPNN in toluene in the
temperature range 155-292 K.
3.3 Case study 2: Fmoc-(Aib-Aib-TOAC)2-Aib-OMe in acetonitrile
Next we consider cw-EPR spectra of the double spin labelled, 3,10-helical, peptide Fmoc-(Aib-AibTOAC)2-Aib-OMe dissolved in acetonitrile. The system is now described by a stochastic Liouville
equation for the two electron spins interacting with each other and with two 14N nuclear spins, in the
presence of diffusive rotational dynamics. Parameterization of diffusion rotational tensor is
provided again by a hydrodynamic model. The system Hamiltonian is defined as
β
Hˆ = e
=
G
∑B
0
i
⋅ g i ⋅ Sˆi + γ e ∑ Iˆi ⋅ A i ⋅Sˆi − 2γ e JSˆ1 ⋅ Sˆ2 + Sˆ1 ⋅ T ⋅ Sˆ2
(6)
i
Where the two radicals are explicitly accounted for by the first term and J and T
10
terms are
included.
The system geometry is summarized in Figure 15
10
The spin-spin dipolar term is the most critical long-range contribution. Usually, this tensor is calculated by assuming
that the two electrons are localized and placed at the centre of the N – O bond. In this view, the two electrons are
considered just as two point magnetic dipoles and the interaction term is given simply by:
⎡
⎛ rx2
µ g β ⎢
3⎜
T= 0
13 − 2 ⎜ ry rx
⎢
r ⎜
4π =r ⎢
⎝ rz rx
⎣
2
e
2
e
3
rx ry
ry2
rz ry
rx rz ⎞ ⎤
⎟⎥
ry rz ⎟ ⎥
rz2 ⎟⎠ ⎥⎦
where r is the distance between the two localized electrons, that is the distance between the centres of the N – O bonds
of the two TOAC nitroxides. Obviously, this is only an approximation because the electrons are not fixed in one point
of space but delocalized in a molecular orbital. A complete quantum mechanical computation starting from the
computed spin density is still lacking for large molecules. Thus we resorted to the following computational strategy
based on the well known localization of nitroxide SOMOs ( π orbitals) on the NO moiety (see Figure 6).30 As a
consequence, the corresponding electron density can be fitted by linear combinations (with equal contributions) of
effective 2 pZ atomic orbitals of nitrogen and oxygen:30,36
*
(
(
)
)
(
(
)
)
N1
O1
r − R N1 − φ210
r − R O1 ⎤
Ψ ' = N ' ⎡φ210
⎣
⎦
O2
''
'' ⎡ N 2
Ψ = N φ210 r − R N2 − φ210 r − R O2 ⎤
⎣
⎦
4 5 −α r
α re Y1,0 (θ , ϕ ) where
3
if the effective nuclear charge; standard Clementi-Raimondi values of Z eff = 3.83
Next we represent the AO’s by Slater type orbitals (STO’s) of the form
α = Z eff / 2
Hartree-1 and Z eff
φ210 ( r ) =
for nitrogen and Z eff = 4.45 for oxygen were used. The molecular geometry allows us to conclude that only the
T ( 2,0) component contributes significantly to the dipolar tensor. As expected, at high distances the point approximation
converges to the exact approach, while increasing differences are found when the distance is less than about 7 Å.
Figure 15 Reference frames and geometry of Fmoc-(Aib-Aib-TOAC)2-Aib-OMe
Again, using E-SPIRES for determining the cw-EPR of this biradical is comparatively simple:
proper definition of paramagnetic centers and coupled nuclei proceeds as in previous examples. We
simulated and compared the spectra of the peptide dissolved in MeCN in the temperature range 270
K to 330 K. Figure 16 shows four theoretical spectra and their relative experimental counterparts.
Figure 16 Experimental (solid lines) and theoretical (dashed lines) cw-EPR spectra of heptapeptide 1 in MeCN at the
temperatures of 330, 310, 290 and 270 K.
We allowed for a limited adjustment δ A of the scalar component, Tr ( A ) / 3 , of the theoretical
hyperfine tensor A. The best agreement is obtained for δ A = 0.3 Gauss, which is well within the
estimated uncertainty of 0.5 Gauss. The overall agreement between the theoretical and experimental
spectra, in the considered range of temperature, is good. Only at the lowest temperature examined
(270 K), the set of parameters employed in the simulations seems to be slightly less effective. It
should be stressed that no internal dynamics model has been employed to describe collective
motions in the heptapeptide, which has been treated again as a simple Brownian rotator, with
diffusive properties predicted only on the basis of fixed molecular shape and solvent viscosity.
Nevertheless, a reasonable prediction of the change in linewidth and change of intensity is observed
in the whole range of temperature considered, thus confirming that the molecular structure is
essentially rigid in solution.14
3.4 Case study 3: tempo-palmitate in 5CB
In the last example we present an example of the ICS applied to the case of nematic liquid
crystalline environments, by performing simulations of the EPR spectra of the prototypical
nitroxide probe 4-(hexadecanoyloxy)-2,2,6,6-tetramethylpiperidine-1-oxy in isotropic and nematic
phases of 5-cyanobiphenyl. The procedure runs as 1) determination of geometric and local magnetic
parameters by quantum-mechanical calculations taking into account solvent and, when needed,
vibrational averaging contributions; 2) numerical solution of a stochastic Liouville equation in the
presence of diffusive rotational dynamics, based on 3) parametrization of diffusion rotational tensor
provided by a hydrodynamic model. Notice that an internal degree of freedom is explicitly taken
into account (meaning that step 6 in E-SPIRES is modified), and that the necessary conformational
potential is evaluated through the QM approach.
Figure 17 Geometry, reference frames and internal degree of freedom for the case of tempo-palmitate in 5CB
We simulate the cw-EPR spectra of the tempo-palmitate in 5-cyanobiphenyl in the range of
temperatures from 316.92 K (isotropic phase) to 299.02 K (nematic phase). In Figure 18 five
simulated spectra are reported, superimposed to experimental spectra taken from the literature. The
results show that again it is possible to apply the ICS even in the quite demanding playground
represented by large nitroxides in nematic phases. In particular, the spectra at different temperatures
and in different phases are reproduced with a very limited number of fitting parameters (ordering
potential and isotropic parts of magnetic tensors), which could be possibly replaced by a priori
computations in the near future. As a matter of fact, the computed value for the isotropic hyperfine
splitting (15.3 G) nicely fits the experimental value in isotropic phase. However, at lower
temperature local effects come into play which cannot be reproduced by the continuum solvent
model employed in our computations.26
Figure 18 Calculated and experimental spectra of tempo-palmitate in 5CB
4. Conclusions
4.1 Perspectives
Perspectives of developments, in our opinion, can be envisaged along three main lines in order of
increasing complexity: 1) setting up an on-line grid-oriented version of existing software: this is
currently being done in our laboratories as part of a general project devoted to setting up userfrindly, grid-based tools for virtual chemistry applications; 2) extension of the ICS to advanced
EPR spectroscopies (ENDOR, ELDOR, DEER, FT-EPR) and extension to paramagnetic metallic
species: these are essentially either a technical redefinition of calculated observables (i.e essentially
upgrade of Eq. 4) or generalization of present code, limited to consider exactly to coupled nuclei, to
exact or approximate multi-nuclei treatments- although relativel trivial, these upgrades could
impose significant additional computational burdens to the whole procedure, requiring for instance
a complete diagonalization of the Liouvillean matrix instead of its reduction to tridiagonal form; 3)
inclusion of mixed dynamic approaches to account for multiscale processes in large biological
molecules.
But even at its present stage of development, the application of the ICS to EPR is to be considered a
success. After all, the relationship between EPR measurements and molecular properties can be
gathered only indirectly, that is, structural and dynamic molecular characteristics can be inferred by
the systematic application of modelling and numerical simulations to interpret experimental
observables. A straightforward way to achieve this goal is by considering the spectrum as the
‘target’ of a fitting procedure of molecular, mesoscopic and macroscopic parameters entering the
model. This strategy, based on the idea of a general fitting approach, can be very helpful in
providing detailed characterization of molecular parameters. But the ICS, i.e. the combination of
quantum mechanical calculations of structural parameters possibly including environmental and fast
vibrational and librational averaging, and direct feeding of calculated molecular parameters into
dynamic models based on dynamic modelling, provides a much more refined methodology.
4.2 Summary
Our main objective in this work has been to discuss the degree of advancement of the ICS to the
interpretation of cw-EPR of organic radicals in solvated environments, via combination of advanced
quantum mechanical approaches and stochastic modelling of relaxation processes. The ICS ab
initio prediction of cw-EPR spectra is able to assess molecular characteristics entirely from
computational models and direct comparison with the experimental data. The sensitivity of the
integrated methodology to the overall molecular geometry is demonstrated, in all the cases
discussed above, by the significant change in the calculated spectrum when significant changes are
arbitrarily introduced in the molecular geometry or in the dynamic description: for instance, in the
case of the heptapetide bi-radicals the ICS is sensitive enough to distinguish between different helix
conformations.14
Some adjustment of computed magnetic tensors is probably unavoidable for a quantitative fitting of
experimental spectra, especially for large systems where only DFT approaches are feasible.
However, the number of free parameters (if any) is limited enough that convergence to the true
minimum can be granted. At the same time the allowed variation of parameters from their QM
value is well within the difference between different structural models. Thus, pending further
developments of DFT models, the ICS is already able to predict cw-EPR spectra of large molecular
systems in solvents starting only from the chemical structure of the solute and some macroscopic
solvent properties. Implementation in a user-friendly package finally could spread the systematic
usage of the ICS in current real-life EPR laboratories, as much as standard QM packages for
structural molecular properties are already diffuse in most modern chemistry research facilities.
5. Acknowledgments
This work was supported by the Ministry for University and Research of Italy (projects FIRB and
PRIN ex-40%) and the National Institute for Materials Science and Technology (project PRISMA
2005). Computational resources have been employed based on the Laboratorio di Chimica
Computazionale (LICC, Padova) and Laboratory of Structure and Dynamics of Molecules (LSDM,
Napoli) computational chemistry facilities, within the CR-INSTM initiative Virtual Italian
Laboratory for Large-scale Applications in a Geographically distributed Environment (VILLAGE).
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