Brian A. Harkins
05/06/99
Density Functional Theory Calculations of the Structures, Binding
Energies, and Infrared Spectra of Methanol Clusters
Research performed in this paper dealt with the structures, binding energies, and
infrared spectra of methanol clusters. The methanol clusters ranged form 2- 5 monomers.
There were a variety of conformers studied. Trends among families of conformers and
groups of different conformers were discussed in the literature along with comparisons to
experimental data. Calculations were all performed on the Gaussian 92 and Gaussian 94
suite of programs. B3LYP was the method used and the basis set consisted mainly of the
6-31+G(d) set with references to the 6-311++G(3df,2p). My purpose is to explain the
calculations performed in this paper by reproducing their results for the monomer and
dimer.
Perhaps the first and most important aspect of this study concerns the selection of
which Ab Initio Theory to use and along with that what basis set. To understand the
decision making that goes on when choosing these items we must first gain an
understanding of what Ab Initio Theory is and how does this correspond to the data and
calculations obtained. Ab Initio molecular orbital theory is connected with predicting the
properties of atomic and molecular systems. It is based upon the fundamental laws of
Quantum mechanics and uses a variety of mathematical transformations and
approximations.1
In quantum mechanics the Ψ is a description of the amplitude of the matter wave
and is called the wave function of the particle.2 The particle in chemistry is sometimes an
electron. The Shrodinger equation uses this wavefunction by applying the Hamiltonean
operator, H, to find energy, E, and other properties of the particle.
HΨ=EΨ
H=
Equation #1
{(-h2/8π2m)▽2 + V}
In the Hamiltonean operator, the m stands for the mass of the particle (electron),
the del,▽, is the concentration gradient, the, h, is Planck’s constant, and, V, is the
potential field in which the particle is moving.3 The wavefunction is said to be an
eigenfunction of the Hamiltonean operator and the energy is an eigenvalue. For each
wavefunction there corresponds a particular energy.
When we are talking about molecules we are no longer speaking of a single
electron but rather an entire molecule. Shrodinger’s Equation can still be used but certain
Where the wavefunction in the example above stood for an electron, in Molecular
Orbital Theory the wavefunction is described as a collection of molecular orbitals, Ф.
Ψ= Ф1(r1), Ф2(r2),… Фn(rn)
Equation #2
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05/06/99
The (rn) designation stands for the position vectors of the electrons. The position
vector for the nuclei of the molecules would also have to be included if not for the
assumption that we can make because of the Born-Oppenheimer Approximation.
According to this approximation the velocities of the electrons move at such a greater
speed than the nuclei that the of the nuclei appear to be fixed. For this reason we are
allowed to use only the positions vectors of the electrons.
Two electrons that switch positions in a system should result in a sign change.
This is not the case in the Equation #2. To compensate for this discrepancy spin states
are applied to the molecular orbitals. There are two spin states for an electron, spin up,
α(i), and spin down, β(i), where ,i, stand for the which electron. A pair of spin orbitals
is prepared by applying both spin states to a molecular orbital, Ф1(r1)α(1) and Ф1(r1)β(1).
A matrix with n columns and n/2 rows, where n is the number of electrons, is created.
The determinant mixes all the possible orbitals of all of the electrons in the molecular
system to form the wavefunction.4
Next we say that the molecular orbital is defined as:
ФI = ∑CuiXu
Equation #3
where the coefficients Cui are the molecular orbital expansion coefficients and the Xu are
the basis functions.5 The u stands for which basis function the coefficient belongs to and
the i stands for which molecular orbital it belongs to. This is what brings us to the Ab
Initio theory part of out discussion.
Different theories or methods take this information and process it to solve the
wavefunction. One method squares the different coefficients and then takes their sum.
The sum of the squares is known as the density matrix. This density matrix is a very
important part of the energy that is obtained from the operator. The values of the
coefficients are not known going in but Gaussian allows for a guess of the values. Now
the program tries to minimize the energy of the system that is obtained from the
Hamiltonean while at the same time minimizing the molecular coefficients. When this
point is reached the system is optimized and properties such as frequencies, binding
energies, bond enthalpies, and structures can be obtained. This method of Ab Initio
theory is called the Hartree- Fock Theory (HF).
The Hartree- Fock theory is a very poor model. The reason is the way it uses the
density matrix. In Hartree- Fock it takes into account the interaction of an electron and all
its neighboring electrons. This is an acceptable approximation but is not a very accurate
picture of what the surface is actually like. Other models have been developed that take
the Hartree- Fock theory one step further and actually use functions of the electron
density. It looks at the electron correlation of the molecule. In this process a grid is set up
around the molecule and the program traces all around the molecule observing the
electron density at each particular point. This method of calculations is called Density
Functional Theory (DFT) and is a much more accurate method than HF. The BeckeLYP
model is an example of a DFT method.. The DFT is so effective that it actually
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Brian A. Harkins
05/06/99
overshoots the experimental value. For this reasons methods have been created to
compensate for this dilemma.
Since the development of DFT a number of hybrid models have been developed.
Examples are the B1LYP and the B3LYP. The B3LYP is the model used in the research
paper. It incorporates both HF and the BLYP model. The hybrid can be thought of as a
linear combination of both the HF and BLYP including a cross product of the two:
HYBRID = C1(HF) + C2(BLYP) + C3(HF)(BLYP)
A percentage, C, is assigned to each model to determine the weight from each that
is used. The percentages came about based on data that was obtained using the
combination of these two methods and than varying the percentages until they got good
agreement with the experimental data. The number in the models, 1 and 3, correspond to
how many of the percentages are fixed. In the B1LYP only the HF is fixed. In the
B3LYP all three are fixed. These models are quite accurate but there are other models
just as accurate or more so.
Moller-Plesset perturbation theory is a model that is actually applied to the HF
calculation and acquires great accuracy. Examples are the MP2 and MP4 models. This
theory takes electrons that are in the ground state and puts them into higher excited
orbitals. The number designates the number of excited states to use in the calculation.
This theory is a one time correction to the HF energy and are generally less-time
consuming than the approaches that iteratively solve for the actual weights of the higher
configurations in the total wave function.6 This being true it is still rather time
consuming compared to the other two models mentioned above. Graph #1 7 actually
shows the comparison of time it takes for each model to perform a calculation of the
molecule C5H12.
Graph #1
Relative Times for the Calculations of C5H12
70
60
50
40
Relative Times (factors of
6.4 sec.)
HF
B3LYP
MP2
30
20
10
MP2
B3LYP
0
3-21G
Theories
HF
6-31G(d)
6-31+G(d)
Basis Sets
6-311++G(2d,p)
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05/06/99
The values on the left-hand side are relative time factors. The job actually took 6.4
seconds and these are all factors of that time. Methods are listed on the right hand side.
Notice that the times increase greatly going from the HF method to the MP2 method.
Time has to be taken into account when deciding on what model to use for an
experiment. Only cases where extreme accuracy is needed would a time consuming
method be used. The labels along the bottom are the basis sets and they are described in
the following section.
The basis sets are chosen along with the method to solve the wavefunction. They
can be thought of as the molecular orbitals (s,p,d,…) that were discussed in Equation #3.
Larger basis sets mean longer calculations. Basis sets such as the ones listed in
Graph #1 can be broken down to determine the number of basis functions involved in
the calculation. Take for example the basis sets that were discussed in the paper: 631+G(d) and 6-311++G(3df,2p). The table below breaks down each of these basis sets
and explains what each number, letter, and symbol refers to in the set. Total Basis
Functions can be found in the bottom row.
Table #1
Basis
Set
ORBITAL
Basis
Set
ORBITAL
6
1s
6
1s
3
2s, 2p
3
2s, 2p
1
3s, 3p
1
3s, 3p
+
4s, 4p
1
4s, 4p
d
3d
+
5s, 5p
(non-hydrogen)
+
2s (hydrogen)
3df
3d, 4d, 5d, 4f
(non-hydrogen)
2p
3s, 3p, 4s, 4p
(hydrogen)
Total
Basis
Functions
19
Total
Basis
Functions
54
The ,+, in the basis set is like the normal s and p orbitals except it allows the
orbitals to occupy more space. This is called a diffuse function and is good for systems
with lone pairs as there are in methanol. The 6-31+G(d) has a total basis function value of
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05/06/99
19. This means that there would be 19 basis functions used for each atom, not to mention
the 19 coefficients that go along with those functions. That would mean a wavefunction
consisting of 1,805 (19X19X5) molecular orbtials for methanol. The size and complexity
of these calculations would be far to great to solve without the advent of modern
computers with speeds and memory size capable of performing such calculations. For as
P.A.M. Dirac realized in 1929:
The underlying physical laws necessary for the mathematical theory of a large part of
physics and the whole of chemistry are thus completely known, and the difficulty is only
that the exact application of these laws leads to equations much too complicated to be
soluble.8
Referring back to Graph #1 it is clear to see that the basis set is a major
contributor to the time it takes to perform a calculation of a molecule. Notice also that
the larger basis set, 6-311++G(3df,2p), that the researchers compared their findings to is
not even listed on the graph. It would be an even larger set and take an even greater
amount of time. It is for these reasons that the researchers might have chosen to use the
B3LYP model along with the 6-31+G(d) basis set. It should be noted that these might
have not been the original model and basis sets planned for use in the experiment, but
rather the ones that were in most agreement with experimental findings and the larger
basis set, 6-311++G(3df,2p).
Understanding the background involved in Ab Initio Molecular Theory brings us
to the next step. That step is the explanation of the work done in the paper and how they
came about their findings. I have taken two of the simplest molecules used in the paper
to explain how Gaussian is used and how the data is collected in this paper. The two
molecules are the methanol monomer and the methanol dimer cluster. I used the Gaussian
94 program with the B3LYP model and 6-31+G(d) basis set, as done in the paper.
The first calculation that I reproduced too much success was the binding energy
of the dimer. The binding energy that they are referring to in the paper involves the
hydrogen bonds between the hydrogen of the -OH group on one methanol with the
oxygen of an –OH group on an adjacent methanol.
Figure #1
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Brian A. Harkins
05/06/99
The dotted lines in Figure #1 shows the hydrogen bond. An important part of
solving for these energies is feeding the coordinates of each atom into the Gaussian
program. There are a number of ways to do this step. I have chosen to build the
molecule in a program called Hyperchem. The steps are as follows:
I.
Build the molecule by double clicking on the uppermost left button that looks
like a target. This brings up a menu with all the elements of the periodic table that
are used to build the molecule. Begin by picking the element and then clicking on
the screen. For my methanol monomer I chose the hydrogen atom and then
clicked on the screen in four different places, one for each hydrogen. I then chose
the carbon and clicked once on the screen and did the same for oxygen. By
clicking on one atom on the screen and then dragging it to another you can form
the bonds. Figure #2 is a snapshot of my monomer:
Figure #2
II.
Select the second button that resembles a bull’s eye and then by clicking on two
or three atoms at a time, bond lengths or bond angles can be highlighted for the
molecule. After the atoms have been highlighted, go to the menu bar at the top of
the screen and click on Select. Scroll down to Set Bond Length or Set Bond
Angle and input the measurements. The OH bond length and C-O-H bond angle
were 0.969 A and 109.2 respectively. These were the only two values I was given
from the literature. These values are all I need because Gaussian will optimize
the rest of the coordinates. Figure #3 shows the bond angle input:
Figure #3
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Brian A. Harkins
05/06/99
III.
After all the bond lengths and bond angles are set, it is time to save the molecule.
In order for Gaussian to be able to read the file, the image must be saved as a
*.pdb file. Now start up Gaussian and under the Utilities selection choose
NewZMat. This will convert the *.pdb file into a *.gjf file that the Gaussian
program can read. Once this is done go under File and click Open. Choose the
gjf file and open it. Now all the coordinates are loaded. In the route section
choose the appropriate model and basis set. In the experiment they used B3LYP
and 6-31+G(d). The route section must also contain the operations you want it to
perform. The researchers did calculations for energies, frequencies, and normal
modes. For that reason I chose the command that would have Gaussian run a
frequency and optimization job. Figure #4 is the input commands:
Figure #4
The Guess=Read command in the route section is there because I ran the monomer first with
the 6-31G(d) basis set and then used those optimized conditions to run the 6-31+G(d) basis
set. Without the diffuse function my binding energy deviated by 81% from the
experimental value. With the diffuse function it deviated only by 14%. This demonstrates
the importance of diffuse functions in molecules with lone pairs.
IV.
When Gaussian is finished running it creates an output file that contains all the
information that is needed to calculate the properties discussed in the paper.
Only the frequencies and the normal modes of vibration can be viewed at this
time. The binding energy cannot be calculated until the Gaussian runs the dimer.
In order to see the frequencies and the normal modes the output file is edited so
that the program Gaussview can read it. Once the information is loaded the
vibrations and the modes can be viewed, Figure #5. I went through each of the
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Brian A. Harkins
05/06/99
modes of vibrations until I found the one that corresponded to the OH stretch
discussed in the literature.
Figure #5
The OH stretch frequency is shown in the lower right hand corner. The frequency
is 3753.5 cm –1 compared to the literature value that they found to be 3762.9 cm –1. The
shift of the dimer frequency from the monomer frequency is what they compared to the
experimental findings and the larger basis set. In order to do this I had to build and run
the dimer as I had done with the monomer. Again I started in Hyperchem and built the
dimer cluster. The procedure was as follows:
i. The molecules for the methanol dimer was built in the same way as shown in
Figure #2, except that once I had built the monomer I made a copy of it. Going
under Select in the top menu bar and then clicking on the Molecule option did this.
I then clicked on the second button on the left hand side that resembles a bull’s eye.
Clicking on the methanol monomer highlighted the entire molecule. I then went
under Edit and clicked on Copy this produced another monomer exactly as the first.
Figure #6 shows the window that contains the two monomers:
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Brian A. Harkins
05/06/99
Figure #6
ii. Now that I had the two monomers it was time to set up the cluster to agree with
the geometry that was given in the literature. The first part was to arrange them
as the molecules appear in Figure #1. Clicking on one of the molecules
highlights it. Then by clicking on the selection on the left hand side that
resembles a bent paper clip you can rotate it. To move it left , right , up or
down, choose the selection that looks like four arrows pointing outward. When
the two monomers are in the correct arrangement, the hydrogen bond must be
formed. Go under Build and click on Allow Ions. Then go back under Select
and choose the Atom setting. Now simply click on the hydrogen of one
monomer and the oxygen of the other and connect the two. The bond will be
highlighted. Figure #7 shows an example:
Figure #7
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05/06/99
iii. The bond angle and lengths are entered the same as they were in Figure #3.
The bond angle and lengths are given in the Table #2.
Table #2
Methanol Dimer
MeOH
Unit
0-H
Bond
Length
C-O-H
Bond
Angle
O-O
separation
O-H-O
Bond
Angle
1
0.969 A
109.4
deg
##
##
2
0.977 A
109.3
deg
2.862
174.2
The Values for the table come from Zwier*, Timothy S., et al, Journal of Physical Chemistry,
102, 82-84.
The O-O separation is the distance from one oxygen on a methanol to the
oxygen on the adjacent methanol. To do this the hydrogen bond
must be broke and in its place a oxygen- oxygen formed. Figure #8
shows the bond angle of the O-H-O bonds.
Figure #8
iv. Now that all the coordinates have been set, it is time to feed it into Gaussian as
we did in step III for the monomer. Figure #9 shows the Gaussian input
command. Notice that beside the FOPT FREQ command we also show
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Brian A. Harkins
05/06/99
Figure #9
{CalcFC} and Guess=Read. When I originally tried to run the dimer using just the
FOPT FREQ command the linked died. The reason is that when Gaussian is trying to
minimize the geometric coordinates and the energy of the system, it is unable to locate
the minimum because the interactions of the hydrogen bonds are so much softer than
those of actual bonds. The result is a parabola that is extremely wide and Gaussian cannot
predict the minimum, Graph #2. For normal bonds the parabola is narrow and Gaussian
simple walks down the graph till the minimum is found, Graph #3.
Graph #2
Energy
Hydrogen Bond
Hydrogen Bond
Geometric Coordinates
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Brian A. Harkins
05/06/99
Graph #3
Energy
Normal Bond
Normal Bond
Geometric Coordinates
The values in the charts are arbitrary. The graph is just to show what Gaussian is
trying to work with and how in Graph #2 it can’t locate the minimum and ends up
bouncing back and forth on the peak until the link dies. The {CalcFc} command
compensates by telling it to calculate the Force Constants instead of guessing at them.
This would normally take a long amount of time but since I had already optimized my
geometry it took only a few hours.
v. Gaussian again creates an output file for the dimer and it can be converted to a gjf
file as was done in IV. Now it is ready to be ran in Gaussview. Figure #10 shows
the frequency spectrum and structure for the dimer. The frequencies that I
calculated were in good agreement with those from the literature and with the
experimental. Going through each normal mode of vibration and finding the one
that corresponded to the OH stretch identified the frequency. The vibrations were
numbers 29 and 30 on Figure #10.
Figure #10
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05/06/99
The frequencies that correspond to the vibrational modes are 3616.7 cm-1
and 3765.5 cm-1, respectively. The literature does not compare the frequencies
directly with those of the experimental and larger basis set but rather the shifts of
the dimer frequencies from the monomer. In the case of my calculations, the OH
stretch of my monomer was 3753.5 cm-1. This means that the shifts from the
monomer's frequencies are –136.8 cm-1 and 12 cm-1, respectively. These are
in good agreement with the experimental. In fact it is almost more accurate
than the findings in the literature. I have created a table at the end of the
discussion on my findings, Table #3. The table lists my results, the researcher’s
results, as well as the experimental data.
Before continuing onto the binding energies, I should mention that the normal
modes of vibration in the above section are the same as those that would be obtained by
doing a complete symmetry analysis of the molecule(s). Gaussian performs these
operations and the vibrations that I view are the different stretches and bending of the
molecule(s). The table that is given in the book assigns a value to the size of the stretch
of the one OH bond on a methanol unit in comparison to the OH bond on the adjacent
methanol. The stretch value of one OH bond is seven times larger than that of the
adjacent methanol in both vibrational modes.9 Consequently, the same held true in my
findings.
When Gaussian performed the calculations for the monomer and dimer it created
an output file. In the previous section the file was edited and fed into Gaussview to get
the frequencies. Binding energies can be found directly from the output file. The paper
makes several comparisons of different energies and their significance against larger
basis sets. As an example I chose only to use the “Sum of electronic and
zero-point Energies” from the output file and compare it to the ZPE corrected
energies in the paper. In order to do this the energy from monomer must be subtracted
from the dimer. Since the dimer is made up of two methanol units, the monomer must be
multiplied by a factor of two. The energy from the monomer was –115.673892 hartrees
and the energy from the dimer was –231.355378 hartrees. The difference must then be
multiplied by a factor of 627.5095.10 This is how many kcal/mol are in one hartree. The
ZPE corrected energy that I found was 4.76 kcal/mol. Applying the scaling factor11 of
0.7863071 to the binding energy gave me 3.74 kcal/mol.
Table #3 Comparison of Frequencies, Frequency Shifts, and Binding Energies
My Calculations
B3LYP
B3LYP
Experimental
12
13
B3LYP 6-31+G(d)
6-31+G(d)
6-311++G(3df,2p)
Freq. Freq.
Bind Freq. Freq. Bind Freq. Freq. Bind Freq Freq14
Shift
Ener
Shift Ener
Shift Ener
Shift
Monomer 3753 0.0
#
3762 0.0
#
#
0.0
#
#
0.0
Unit 1
Dimer
Unit 1
3616
-137
#
Dimer
Unit 2
3765
12
-4.76 3758 -4.3
-3.74
3612 -150
#
#
-4.82 #
-3.79
-148
#
#
-3
-3.55 #
-3.08
Bind15
Ener
#
-170
#
+3
3.2
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Brian A. Harkins
05/06/99
Table #3 shows the data I obtained from my calculations, the literature’s data, those of
the larger basis set, and the experimental. All frequencies are in reciprocating
centimeters and all binding energies are in kcal/mol. All my calculations were very close
to those obtained in the literature. In fact my value for the binding energy was one
percent closer to the experimental than the literatures. The importance of this comparison
is that although the larger basis set produces values closer to the experimental, the
smaller basis set is close enough that it is an acceptable model as long as great accuracy
is not required.
The calculations that I performed above could be applied to any one of the
different methanol clusters in the paper. The only difference would be the amount of
time the program would take to complete the task and the amount of memory needed to
store all the results. What is the importance of being able to do these calculations and
obtain this data? The researchers in this paper use the findings from the different clusters
to show trends and relationships between their sets of data. The data, if correct, should
show agreement between structural data, binding energies, and infrared spectra.
One example is when the researchers took the binding energies that they had
calculated for cyclic methanol clusters and compared them to the structural data. It
appeared that as the ring increased (trimer- pentamer) the binding energies did so as well.
Structural data indicated that steric interference due to the hydrogens caused the O-H-O
bond angles to approach 180 degrees and the oxygen-oxygen distance to decrease greatly
from the trimer to the pentamer.16 The two trends are in excellent agreement. As the
distance of a bond decreases (to an extent) and the rigidity of the molecule increases, the
strength of that bond increases. Take for an example a single bond compared to a double
bond (ethane vs. ethene). The sigma bond is longer than the double bond and is free to
rotate. The double bond is stuck in place due to the pi bonds. The amount of energy
needed to break that double bond is extremely higher than the energy needed to break the
single bond.
Another example from the paper is that, even though the average H bond energy
in the chain conformer is higher than that of the cyclic with the same amount of
methanols, the cyclic geometry is favored. The increased cooperativity of the extra H
bond stabilizes the cyclic form so much that it is more stable than the chain. The result is
a higher total binding energy for the cyclic form than the chain. The total binding
energies from the paper show a higher value for the cyclic conformer than the chain.
Frequency comparisons can be made directly from experimental spectra. The researchers
found that the results collected for the different calculations were in agreement with each
other and with experimental data.17
In this paper I explained Ab Initio Theory, how the calculations could have been
obtained, and briefly how the results are used to validate the whole of the data. My
purpose, along with creating a clearer understanding of the processes involved in this
type of work, was to bestow upon the reader a greater appreciation of the time, patience,
and complexity of this type of work.
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References
1) Foresman, James B.; Frisch A.E.; Exploring Chemistry with Electronic Structure
Methods; Gaussian Inc.: Pittsburgh, 1993, 253.
2) McQuarrie, Donald A.; Quantum Chemistry; University Science Books.: 1983, 79.
3) Foresman and Frisch, 253.
4) Foresman and Frisch, 260.
5) Foresman and Frisch, 261.
6) Foresman, James B.; Ab Initio Techniques in Chemistry: Interpretation and
Visualization; American Chemical Society.: 1997, 249.
7) Foresman and Frisch, 123.
8) Foresman and Frisch, 258
9) Hagemeister, Frederick C.; Gruenloh, Christopher J.; Zwier, Timothy S. J. Phys.
Chem. 1998, 102, 89.
10) Foresman and Frisch.
11) Boys, S. F.; Bernardi, F. Mol. Phys. 1970, 19, 553.
12) Hagemeister, Gruenloh, and Zwier, 90-91.
13) Mo, O.; Yanez, M.; Elguero, J. J. Chem. Phys. 1997, 107, 3592- 3601.
14) Huisken, F.; Kulcke, A.; Laush, C.; Lisy, J. M. J. Chem. Phys. 1991, 95, 3924-29.
15) Bizzari, A.; Stolte, S.; Reuss, J.; Rijdt, J. G. C. M. V. D.-V.D.; Duijneveldt, F.B.V.
Chem. Phys. 1990, 143, 423-35.
16) Hagemeister, Gruenloh, and Zwier, 86.
17) Hagemeister, Gruenloh, and Zwier, 92.
15