Chemistry 460 Spring 2015 Dr. Jean M. Standard April 20, 2015

Chemistry 460
Spring 2015
Dr. Jean M. Standard
April 20, 2015
QUANTUM CHEMISTRY PROJECT 4: DENSITY FUNCTIONAL METHODS
OUTLINE
This project involves additional applications of quantum mechanical methods. In particular, you will employ
density functional theory (DFT) methods to first learn about the DFT methodology, then apply it to the study of
hydrogen bonding.
You will be able to access Gaussian and Avogadro in the computer lab located in Julian Hall Room 216. Please
note that the computers in the JH 216 are usually available at any time during the day; however, the room is
occasionally reserved for classes as noted on the schedule posted on the door. You may use any of the Macintosh
computers in JH 216 to access the Avogadro/Gaussian programs (instructions are provided later in this handout). In
fact, you can use one Macintosh for one part of the project, and another Macintosh for another part of the project
since none of the data is stored on the Macs. All the calculations actually will be carried out on Linux workstations
that you will connect to from the Macintoshes through the internet.
DUE DATE
The project is due on WEDNESDAY, APRIL 29, 2015.
GRADING
Project 4 is worth 50 points and consists of two parts. Part A is worth 25 points and Part B is worth 25 points.
2
OVERVIEW: DENSITY FUNCTIONAL THEORY METHODS
Hohenberg-Kohn Theorems
Two scientists, Walter Kohn and Pierre Hohenberg, developed a pair of theorems in 1964 related to the electron
density. Unlike the quantum mechanical wavefunction, Ψ( x1 , y1,z1 , x 2 , y 2 ,z 2 , … x N , y N ,z N ) , which is not
physically observable and depends upon the coordinates of all N electrons of a system, the electron density ρ( x, y,z )
is physically observable and depends upon only three spatial coordinates.
€
Technically, the electron density can be obtained from the electronic wavefunction using the relation
€
ρ ( x, y,z ) =
∫ dr2∫ d r3…∫ drN
Ψ * r 1 ,r 2 , … r N Ψ r 1,r 2 , … r N .
(
) (
)
(1)
Notice that the integration is over all the electron coordinates except one set. The electron density ρ( x, y,z )
therefore is€
the probability density for finding an electron at the point ( x, y, z ) . The choice to integrate over all the
electron coordinates except those of electron 1 is arbitrary. Since the electrons are indistinguishable, the expression
above represents the probability density for finding any one electron at the point ( x, y, z ) .€
€
The first Hohenberg-Kohn theorem states that the energy eigenvalue (i.e., the electronic energy of the system) is a
functional of the electron density,
€
E = F [ ρ( x, y,z )] .
(2)
Thus, if we have the electron density and we know the specific functional that relates the density to the energy, we
can determine exactly the energy eigenvalue.
Note that this theorem technically applies to electronic ground states
€
only, so it does not treat electronically excited states. This theorem tells us that a functional exists, but it does not
tell us how to find it.
The second Hohenberg-Kohn theorem provides a Variation Principle for density functional theory. Since we do not
know the exact electron density, we can place bounds on any approximate or trial electron density, ρ t ( x, y, z ) ,
E [ ρ t ( x, y,z ) ] ≥ E [ ρ exact ( x, y,z ) ] .
(3)
€
The bounding property only works for the exact functional, though. Since we do not know it, we have to come up
with an empirical functional and
€ an approximate electron density.
A variety of empirical functionals are employed in density functional theory (DFT) calculations today. Some
common functionals are, for example, B3LYP, BLYP, BP86, CAM-B3LYP, PBE, M062X, and so on.
Kohn-Sham Equation and Kohn-Sham Orbitals
In order to obtain an electron density from DFT, some assumptions are made that are analogous to those made in
solving the Schrödinger equation. For the Schrödinger equation, we employed the LCAO-MO approximation to
obtain a trial wavefunction. In DFT, we employ a similar approximation for the trial electron density,
ρ t ( x, y, z ) =
∑
2
φ iKS ( x, y, z ) .
(4)
i
KS
Here, φ i ( x, y,z ) are Kohn-Sham orbitals. These are similar to the molecular orbitals that appear in our approach
€ approximately using the Hartree-Fock method.
for solving the Schrödinger equation
€
3
Each Kohn-Sham orbital holds two electrons and can be represented using a linear combination of atomic orbital
basis functions,
K
φ iKS ( x, y, z ) =
∑ c µi f µ ( x, y, z) .
(5)
µ =1
In this equation, c µi correspond to numerical coefficients and the functions f µ ( x, y,z ) are atomic orbital basis
functions. The coefficients c µi are
€ obtained by solving a set of equations called the Kohn-Sham equations. These
are analogous to the Hartree-Fock equations employed when obtaining a solution of the Schrödinger equation.
€
€
Advantages and €
Disadvantages of DFT
One advantage of DFT is that unlike the Hartree-Fock method, DFT includes some electron correlation. The
accuracy of DFT methods tends to be between that of Hartree-Fock and MP2.
Another advantage is that DFT methods are cost efficient. The CPU time for DFT calculations tends to be similar to
that of Hartree-Fock calculations. Thus, some electron correlation can be included using DFT for much less cost
than it takes to carry out an MP2 calculation.
One disadvantage of DFT is that there are many functionals available. It is not always clear what the best choice of
functional is for the system of interest. Many DFT methods perform poorly for hydrogen-bonded systems and other
weakly-bound intermolecular complexes. In addition, DFT sometimes has difficulties representing transition states
in reaction paths. Therefore, care must be taken in selecting the DFT functional.
Another disadvantage of DFT is that it cannot be systematically improved. With the Hartree-Fock method, we know
that we can systematically improve the results by carrying out an MP2 or MP4 calculation, for example. With DFT,
there is only one level of theory. The only improvements that can be made are in the basis set.
4
USING THE COMPUTERS AND THE AVOGADRO SOFTWARE PACKAGE
This section contains instructions for logging in to the Macintosh computers in JH 216 and accessing the Linux
computers on which the calculations will be performed. In addition, it provides instructions on how to start the
Avogadro software package. The use of the Gaussian software package will be described later.
Logging In
To log in to one of the Macintosh computers in JH 216, enter your ISU ULID and password.
Starting X11
In order to access the Linux workstations to run the Avogadro/Gaussian software packages, you must run a software
package on the Mac called X11. To start the X11 application, go to the Applications folder on the Mac hard drive
and click on the "X" icon. A single white window should appear on the screen; this is called an "xterm" window, or
simply a terminal window.
Logging on to the Linux Computers
You may use any of the six available Linux computers for the project. To log on to a Linux computer, type the
following command in the xterm window:
ssh
–Y
che460@hostname.che.ilstu.edu
Here, "ssh" stands for secure shell; this application provides a secure connection to the remote Linux workstation.
Also, hostname is the name of a Linux computer [select one of: frodo, samwise, gandalf, aragorn, gimli, or legolas].
The password is "erwin". Enter the password as prompted. Once you have done this, you are connected to the
Linux computer.
Starting the Avogadro Program
To start the Avogadro program, type "avogadro" in the xterm window and hit the return key. A window for the
Avogadro software package should appear on your screen. At this point, you are ready to begin the project.
5
PART A (25 points) – DFT METHODS AND BENCHMARKING FUNCTIONALS
Introduction
In this part of the project, you will carry out DFT calculations on the water molecule and water dimer to benchmark
a variety of functionals for the use in systems involving hydrogen bonds. The basis set to be employed in all the
calculations is a triple-zeta quality basis set with added polarization and diffuse functions on all the atoms,
6-311++G(2df,2pd). This basis set was selected because it has been shown to perform well not only for oxygencontaining species like water and water dimer, but also for sulfur-containing compounds like those to be studied in
Part B [see, for example, J. M. Standard, I. S. Buckner, and D. H. Pulsifer, Theochem 673, 1 (2004)].
In order to benchmark the different functionals, you will compare a variety of properties to the known experimental
values for water and water dimer. These properties include bond lengths, bond angles, dipole moments, binding
energies, and vibrationally-corrected binding energies. From the relative errors in the properties computed with the
different functionals, you will determine which functional performs the best overall. This functional will then be
employed for the rest of the project.
The functionals to be investigated include B3LYP, which is the most common functional used today (the name for
B3LYP is based upon the names of the developers, Becke, Lee, Yang, and Parr). This functional has been shown to
perform very well in the description of common organic molecules, for example. Two other functionals to be
investigated are BLYP and BP86, which are similar to B3LYP but less commonly used. Finally, two more recent
functionals also will be explored. The first is CAM-B3LYP, which is a variant (a correction, perhaps) of B3LYP in
order to better describe weakly-bound systems such as hydrogen bonds. The second, M062X is a functional
designed to provide improved descriptions of reaction paths, including transition states, for a variety of reactions.
Procedure: Water Molecule Calculations
1.) First, begin by building a water molecule using the Avogadro software package. You should be able to easily
create a water molecule by starting Avogadro and selecting the Drawing Tool,
. Set the Element to
Oxygen, Bond Order to Single, and make sure that "Adjust Hydrogens" is turned ON. Click with your mouse in
the drawing window to place a single oxygen atom; the hydrogens should be added automatically to create a
water molecule.
Next, carry out a geometry optimization and vibrational frequency calculation of the water molecule at the
B3LYP/6-311++G(2df,2pd) level using the Gaussian software package. From the Avogadro menus at the top
of the window, select "Extensions → Gaussian". Use the following settings:
Calculation: Frequencies
Theory: B3LYP
Charge: 0
Processors: 4
Basis: 6-31G(d,p)
Multiplicity: 1
In addition, you must change one line in the setup window as shown in Figure 1.
6
Figure 1. Setup window for a Gaussian calculation of the water molecule.
You should see on the second line in the white input box the following text:
#n B3LYP/6-31G(d,p) Opt Freq
Edit the line so that it has the following form (you are replacing the 6-31G(d,p) basis set with the
6-311++G(2df,2pd) basis set):
#n B3LYP/6-311++G(2df,2pd) Opt Freq
In addition, you should add a line at the beginning of the white input box that has the form:
%mem=1GB
Once the settings are correct, click "Compute". Enter a filename and click "Save". The calculation will begin
and you should see a popup window appear that says "Running Gaussian calculation…". When the popup
window disappears, your calculation of the water molecule should be complete. The optimized structure should
appear in the Avogadro window. Use the Measure Tool,
bond angle and record them.
, to measure the equilibrium bond distances and
To obtain the electronic and zero-point vibrational energies of the water molecule for use in later analysis, you
will have to search the Gaussian results file directly. To do this, first close Avogadro by selecting "File →
Quit". You may be asked to save changes, but you can safely discard any changes because the file you need is
already saved.
7
Next, in the terminal window, open the H2O results file with the 'gedit 'command (the file extension should be
'.log'). Search in the file for the string 'Sum of electronic'. You should see a block of text similar to the
following:
Zero-point correction=
Thermal correction to Energy=
Thermal correction to Enthalpy=
Thermal correction to Gibbs Free Energy=
Sum of electronic and zero-point Energies=
Sum of electronic and thermal Energies=
Sum of electronic and thermal Enthalpies=
Sum of electronic and thermal Free Energies=
0.021348 (Hartree/Particle)
0.024183
0.025127
0.003709
-76.442194
-76.439358
-76.438414
-76.459832
The electronic + vibrational zero-point energy (sometimes referred to as the total energy, Etot , such that
Etot = Eel + ZPE ) is listed in bold above (and is given in atomic units. Record this value. In addition, the
first entry in the block above gives the vibrational zero-point energy itself (listed as the "Zero-point
correction"). Record this value as well; the difference between the two values is the electronic energy, Eel .
You also should obtain the dipole moment from the Gaussian results file and record it for use in later analysis.
To do this, search in the results file for the string 'Dipole moment'. There are multiple occurrences of this
string; make sure you get the one that appears last in the file. You should see some lines of text similar to the
following:
Dipole moment (field-independent basis, Debye):
X=
0.0000
Y=
0.0000
Z=
1.9480
Tot=
1.9480
The last value shown is the total dipole moment in units of Debye. Record this value.
2.) Next, repeat the calculations you performed in step 1 for the water molecule using four additional density
functionals. Build a new copy of the water molecule for each calculation, and set up the Gaussian calculation
for each molecule using the same settings as in step 1, including the change of basis set to 6-311++G(2df,2pd)
and addition of the line %mem=1GB. This time however, you will change the functional from B3LYP to one
of four other functionals: BP86, BLYP, CAM-B3LYP, and M062X.
When each calculation completes, measure and record the equilibrium bond lengths and angle, record the total
and zero-point energies, and record the dipole moment from the Gaussian results file.
8
Procedure: Water Dimer Calculations
3.) To create the water dimer, start with the optimized structure of the water molecule by opening the log file from
your first calculation in Avogadro. Position the water molecule on the screen in roughly the position shown in
Figure 2. To do this, make sure the Navigation Tool,
and the right mouse button to translate the molecule.
, is selected and use the left mouse button to rotate
Figure 2. Water molecule positioned on screen for initial step in building the water dimer.
To add a second water molecule to form the dimer, select "Build → Insert → Fragment". In the fragment
listing, scroll down to the bottom, select 'water.cml', and click "Insert Fragment" to insert a second water
molecule. The second water molecule will appear to be highlighted in blue and will probably be on top of the
first one. Use the mouse buttons to rotate (RIGHT mouse button) and translate (LEFT mouse button) the
second water molecule into a position similar to that shown in Figure 3.
Figure 3. Initial structure of water dimer built using Avogadro.
9
The oxygen atom of the second water molecule should be placed approximately collinear with one O-H bond of
the first water molecule. The H----O distance separating the two molecules should be roughly twice that of an
OH bond in the water molecule, and the hydrogens of the second water molecule should be tilted up. There will
be no hydrogen bond shown between the two water molecules even though the two molecules are interacting;
Avogadro is set to automatically display only covalent bonds.
Once the creation of the water dimer structure is complete, carry out a geometry optimization and vibrational
frequency calculation at the B3LYP/6-311++G(2df,2pd) level using the Gaussian software package. From the
Avogadro menus at the top of the window, select "Extensions → Gaussian". You may get a message asking if
you would like to update the preview text; if you do, click "YES". Make sure the following settings are used:
Calculation: Frequencies
Theory: B3LYP
Charge: 0
Processors: 4
Basis: 6-31G(d,p)
Multiplicity: 1
In addition, you may need to edit one line in the setup window as before. Make sure that the second line in the
white input box includes the following text:
#n B3LYP/6-311++G(2df,2pd) Opt Freq
Edit the line in the setup window so that it is given as shown above. In addition, you should add a line at the
beginning of the white input box that has the form:
%mem=1GB
Once the settings are correct, click "Compute". When the calculation is complete (it may take a few minutes),
the optimized structure of the water dimer should appear in the Avogadro window. Using the Measure tool,
measure and record the OH bond distances and HOH angle of each water molecule. Also measure the O-H---O
angle and the O---O separation, referred to as Re(O-O) or RO-O, as shown in Figure 4.
Figure 4. Definitions of geometrical parameters for the water dimer [from S. S. Xantheas and T. H.
Dunning, J. Chem. Phys. 99, 8774 (1993)].
Next, close Avogadro and open the water dimer results file using the 'gedit' command as you did in step 1. Find
and record the sum of electronic and vibrational energies and the zero-point correction. Note that it is not
necessary to record the dipole moment in this case.
10
4.) Repeat the calculations you performed in step 3 for the water dimer using four additional density functionals:
BP86, BLYP, CAM-B3LYP, and M062X. All the other settings should be as in step 3. When each
calculation completes, measure and record the equilibrium geometrical parameters. Also, record the total and
zero-point energies from the Gaussian results file.
Results, Analysis, and Discussion
When you are finished with the procedure for Part A, you should have five different calculations of the water
molecule and water dimer, all with the same basis set, summarized in the table below.
Run
DFT
Basis Set
A
B3LYP
6-311++G(2df,2pd)
B
BP86
6-311++G(2df,2pd)
C
BLYP
6-311++G(2df,2pd)
D
CAM-B3LYP
6-311++G(2df,2pd)
E
M062X
6-311++G(2df,2pd)
1.) For the water molecule, obtain experimental values of the bond lengths, bond angle, and gas phase dipole
moment. Remember to cite your source(s). Tabulate these along with the DFT results for the same quantities.
Compute % errors relative to the experimental values.
Discuss the performance of the different functionals for describing the geometry of the water molecule. Which
performs best for the geometry? To decide, you might want to calculate the sum of the absolute values of the %
errors for the bond length and bond angle.
Which functional best describes the dipole moment? If there is a difference in which functional best describes
the geometry and dipole moment, determine the overall best functional by computing the sum of absolute
values of errors for the bond length, bond angle, and dipole moment. Discuss your findings.
2.) For the water dimer, the known geometrical parameters from experiment may be found in the paper by
Xantheas and Dunning, among other places [S. S. Xantheas and T. H. Dunning, J. Chem. Phys. 99, 8774
(1993)]. While certain angles have been measured experimentally, they all are reported with large uncertainties.
The best known experimental value for the geometry of the water dimer is RO-O, or Re(O-O). Tabulate the
values you obtained for Re(O-O) from the DFT calculations.
Compute and report % errors relative to the experimental value of Re(O-O) for each DFT method. Discuss your
findings. Which is the best functional for describing the water dimer O-O separation?
3.) Calculate and report the water dimer binding energy in kcal/mol for each DFT method. The binding energy,
referred to in the Xantheas and Dunning article as De , is defined as the electronic energy of the dimer minus
the electronic energies of the separated monomers,
De = Eel ( H 2O − H 2O) − 2 Eel ( H 2O).
(6)
Note that the electronic energy for each species is the difference between the total energy and the zero-point
correction,
Eel = Etot − ZPE .
(7)
11
Compute and report % errors relative to the experimental value of De for each functional. Discuss your
findings. Is the best functional for prediction of De the same as the best for prediction of the O-O separation?
4.) Calculate and report the vibrational zero-point-corrected binding energy of the water dimer in kcal/mol for each
DFT method. The vibrational zero-point corrected binding energy, referred to in the Xantheas and Dunning
article as D0 , is defined as the total energy of the dimer minus the total energies of the separated monomers,
D0 = Etot ( H 2O − H 2O) − 2 Etot ( H 2O)
(8)
Compute and report % errors relative to the experimental value of D0 for each functional. Is the best
functional for prediction of D0 the same as the best for prediction of the uncorrected binding energy, De ?
What is the overall best functional for description of the water dimer structure and energies? Use the sum of the
absolute % errors in Re(O-O) and D0 to make your selection. Compare this to the best functional for the water
monomer. Discuss your findings.
12
PART B (25 points) – QUANTUM CHEMISTRY OF HYDROGEN BONDING
Introduction
Hydrogen bonding is an interaction that is important in a wide variety of chemical and biological systems. For
example, the structure and dynamics of water in both the liquid and solid phases are governed by the hydrogen
bonding between water molecules. It is the strength of the hydrogen bond in water that leads to an unusually high
melting point and boiling point compared with other hydrides in the same series, such as H2S, H2Se, and H2Te,
Figure 5.
Figure 5. The melting points and boiling points of hydrides (from L. Pauling, The Nature of the
Chemical Bond, 3rd ed., Cornell University Press, 1960, p. 455).
The hydrogen bond is typically weaker than an ordinary chemical bond, with a bond strength ranging from a few
kcal/mol up to tens of kcals/mol. Because hydrogen bonding is a weak interaction, computational modeling of
hydrogen bonding is sensitive to the basis set choice and the level of theory employed. Typically, to quantitatively
describe a hydrogen bond, a large basis set and electron correlation corrections must be employed. The literature is
filled with examples of very high-level calculations on one of the simplest hydrogen-bonded systems, the water
dimer. [For a reference that contains a review of a variety of results, see S. S. Xantheas and T. H. Dunning, J.
Chem. Phys. 99, 8774 (1993).]
In this part of the project, you will look at systems that involve hydrogen bonds using DFT methods. The first
system that you will study is the water dimer structure that you previously computed. You will then look at dimers
formed between H2S and H2Se, and compare the binding energies to that of the water dimer. Finally, you will also
study the water trimer and compare the structure and binding energy to that of the water dimer. The DFT functional
to be employed is the one that you determined to be the best functional for the water dimer in Part A4, and the basis
set is 6-311++G(2df.2pd).
13
Procedure: H2S Dimer
1.) Using Avogadro and the procedure that you followed in Part A1 for H2O, construct the H2S molecule. Optimize
the geometry and calculate the vibrational frequencies of H2S using the DFT/6-311++G(2df,2pd) level of
theory; use the functional that was determined to be the best for the water dimer from Part A4.
When the H2S calculation completes, measure the equilibrium bond distances and angle, and then from the
Gaussian results file reecord the total energy (sum of electronic and zero-point energies) and zero-point
correction. Use the same procedure as outlined in Part A1.
2.) Construct the H2S dimer. To do this, it is easiest to begin with the optimized water dimer structure, so open the
log file for the water dimer in Avogadro. Now, to change the atoms from O to S, first select the Drawing Tool,
, and make sure that "Adjust Hydrogens" is clicked OFF. Then, select the Element as Sulfur and click on
each of the oxygen atoms of the water dimer to convert them to sulfur atoms. The bond distances are now a
little too small, and the bond angles are a little too large, but they should be close enough to obtain a converged
result.
Perform a geometry optimization and vibrational frequency calculation of the H2S dimer at the DFT/6311++G(2df,2pd) level of theory. Measure the pertinent geometrical parameters; refer to Figure 4. From the
Gaussian results file, ecord the total energy and zero-point correction. Use the same procedure as outlined in
Part A3.
Procedure: H2Se Dimer
3.) Using Avogadro, construct the H2Se molecule. Note that to select the element Se in the builder, select "Other"
and then pick Se from the periodic table window. Using the same procedure as in step B1, carry out a geometry
optimization and vibrational frequency calculation of H2Se at the DFT/6-311++G(2df,2pd) level.
When the H2Se calculation completes, measure the equilibrium bond distances and angle, and then from the
Gaussian results file reecord the total energy (sum of electronic and zero-point energies) and zero-point
correction. Use the same procedure as in step B1.
4.) Follow a similar procedure as that given in step B2 to construct H2Se dimer. This time, though, start with the
optimized H2S dimer structure and change the atoms from S to Se.
Perform a geometry optimization and vibrational frequency calculation of the H2Se dimer at the DFT/6311++G(2df,2pd) level of theory. Measure the pertinent geometrical parameters. From the Gaussian results
file, record the total energy and zero-point correction. Use the same procedures as outlined in step B2.
Procedure: H2O Trimer
5.) Using the structure of the water trimer shown in Figure 5 of the paper by Xantheas and Dunning as a guide,
build the water trimer using Avogadro. It is best to start with the optimized water dimer structure, so first open
the Gaussian results file for the water dimer computed with the best functional. Position the water dimer on
screen to look something like the two water molecules on the left side of the trimer (as shown in Fig. 5 of
Xantheas and Dunning).
To add a third water molecule to form the trimer, select "Build → Insert → Fragment". In the fragment listing,
scroll down to the bottom, select 'water.cml', and click "Insert Fragment" to insert the third water molecule. The
third water molecule will appear to be highlighted in blue. Use the mouse buttons to rotate and translate the
third water molecule into a position similar to that shown for the trimer.
14
Perform a geometry optimization and vibrational frequency calculation of the water trimer at the DFT/6311++G(2df,2pd) level of theory using the best functional as determined in Part A. Measure the pertinent
geometrical parameters, including O-H, O---H, and O-O distances, as well as H-O-H, O-O-O, and O-H---O
angles. From the Gaussian results file, record the total energy and zero-point correction.
Results, Analysis, and Discussion
1.) Tabulate your results for the geometrical parameters of H2S and H2Se. Compare and contrast the calculated
bond lengths and bond angles of the H2O, H2S, and H2Se monomers. Discuss qualitatively the trends observed
on the basis of atomic size and other factors that may be expected to play a role.
2.) It is generally accepted that for a given hydrogen bond type, linear hydrogen bonds–that is, those bonds with the
XH---X (X=O, S, Se) portion of the dimers arranged in a linear fashion–produce the strongest interactions.
Report the calculated XH---X angles for the H2O, H2S, and H2Se dimers at the DFT/6-311++G(2df,2pd) level,
using the best functional. Are the hydrogen bonds formed in your dimers linear? If not, how far are they from
linearity? Discuss any trends you notice.
For the water dimer, also compare with the experimental result: is the hydrogen bond linear? In addition to the
results of water dimer calculations given in the paper by Xantheas and Dunning, a recent study investigated the
water monomer and dimer at even higher levels of theory, up to CCSDTQ with 5Z basis sets [J. R. Lane, J.
Chem. Theory Comput. 9, 316 (2013)]. Table 4 in that paper gives the best estimate of the optimized structure
of the water dimer from these high-level ab initio studies. At this level, is the water dimer linear? Refer to
Figure 1 and note that that the angle α reported in Table 4 corresponds to the deviation from linearity of the
OH--O angle.
3.) In order to look more deeply at the DFT representation of the water dimer, use the high-level results from Table
4 of the paper by Lane to compare additional structural parameters in addition to those available from
experiment.
Using results from the best functional, tabulate and compare the values of the OH bond lengths from the best
DFT calculation with those of Lane. Note that the OH bond lengths are labeled r1donor, r2donor, and Racceptor in
Figure 4 above; that these labels correspond with R(OH)b, R(OH)f, and R(OH)a in Figure 1 and Table 4 of
Lane's paper.
Also report and compare the HOH angles from your best DFT calcuation with those recommended by Lane.
The HOH angles are labeled ψdonor and ψacceptor in Figure 4 above, which correspond to θ(HOH)d and
θ(HOH)a in Lane's paper.
Finally, report and compare the O-O distance (RO-O) from your best DFT calculation with the value
recommended by Lane. Also discuss how Lane's result compares with the experimental O-O distance. Would
there be any difference in the functional chosen as "best" if Lane's result for RO-O was used instead of the
experimental result?s
4.) How much have the geometries of the water molecules in the water dimer been perturbed by the interactions
between them? Compare quantitatively the O-H bond distances and bond angles of the water molecules in the
dimer to those of the monomer (for example, you might compute and report differences in bond distances and
angles between the complexed and uncomplexed water molecules). Use the results from the best functional.
Which is more perturbed as a result of the hydrogen bond interaction, the water molecule acting as proton donor
or the water molecule acting as proton acceptor?
15
5.) Compare and contrast the structures that you obtained for the H2S and H2Se dimers to the structure that you
obtained for the water dimer. For example, some values for comparison might include X-X distances (X=O, S,
or Se), the X-H bond distance of the proton donor (r1donor), and the linearity of the XH---X angle.
6.) Compare the geometries of the dimers of H2S and H2Se to their respective momomers, as you did in step 4 for
the H2O dimer. Are the perturbations of H2S and H2Se due to their hydrogen-bonded interaction larger or
smaller than those observed for H2O? What does this suggest about the strengths of the interactions? Do the
results follow any expected trend?
7.) Calculate the zero-point corrected binding energy ( D0 ) for the H2S and H2Se dimers using the formula given in
Equation (8). Compare these values to the zero-point corrected binding energy of the water dimer. Discuss
your findings in light of the expected strengths of the hydrogen-bonding interactions that would result due to
differences in electronegativities between the elements.
Do the binding energy results correlate with the experimental melting and boiling points shown in the figure
above? Discuss any trends observed and provide some possible reasons for any correlations (or lack of
correlations) with the melting points and boiling points.
8.) Tabulate your calculated geometrical parameters of the water trimer. Compare with the MP2/aug-cc-pVTZ
results of Xantheas and Dunning (given in Table IV of the paper), as well as to available experimental results.
Discuss your findings. How does the "best" functional perform for this larger system?
Using your results, what can you say about the linearity of the hydrogen bonds in the water trimer? In the
trimer, there are three hydrogen bonds and thus three OH---O angles that should be close to linear for the
strongest OH---O hydrogen bonds. Are the trimer OH---O angles closer to linearity than the OH---O angle for
the dimer? Discuss what this potentially means with respect to the stability of the trimer.
9.) Using the total energies of the water monomer and trimer, compute the zero-point corrected binding energy,
D0 , using the equation
D0 = Etot (( H 2O)3 ) − 3Etot ( H 2O).
(9)
Report this result. To compare on an equal basis to the binding energy of the dimer, compute the binding
energy per water molecule. That is, for the dimer, divide D0 by 2, and for the trimer divide D0 by 3. These
values are binding energies per water molecule. Report and compare the magnitudes of these values for the
water dimer and trimer. Discuss how these values provide information about the strengths of the hydrogen
bonds in the dimer and trimer. Are there correlations with the OH---O angles that you reported in question 7?