Neon Dimer Results
Examine a simple, weakly interacting, 20 electron, water dimer analog system (i.e. the Neon dimer) for preliminary selection of Potential Energy Surface (PES) tools and methods screening.
As we discussed previously, we are now using the recently updated Windows NT version of GAMESS, called PC-GAMESS () by Alex Granovsky and co-workers in Moscow. Some of the work presented here was done with Gaussian 94W ().
Results were calculated using standard methods (i.e. Hartree-Fock (HF) and HF with correlation corrections from Moller-Plesset perturbation theory (MP4)). We also used a number of Density Functional methods which explicitly include correlation effects at the SCF level of calculation.
Various basis sets were used , i.e. 6-311+G(3df) - a triple zeta basis with multiple polarization functions and extra diffuse functions and Dunning's correlation consistent basis sets (triple zeta through quintuple zeta, augmented by associated diffuse functions - see  for references). A summary of types of functions within these basis sets is as follows:
Note: BF = basis functions and for the MP4 calculations we used the entire space of occupied and virtual orbitals for the calculation. No orbitals were omitted.
The PESs are calculated from the Binding Energy at many points in space. Binding Energy (BE) is defined as:
- Energy(atom1 + basis set of atom2 (located at atom2's position))
- Energy(atom2 + basis set of atom1 (located at atom1's position))
Our definition of Binding Energy takes into account basis set superposition errors (BSSE). These errors arise from the fact that there are more basis functions used to represent the combined system than there are to represent the individual components.
For example, in the Neon dimer using a 6-311+G(3df) basis, we have 39 basis functions representing the electron density of each atom. For the dimer we have 2*39 or 78 basis functions. The basis functions on atom2 will provide an additional polarization opportunity for the electrons on atom 1 that is not present in the plain atomic calculation. Thus, the energy of the atom in the atomic calculation is artificially kept too high relative to the dimer causing the binding energy of the dimer to be too large.
Huber et al  have published a thorough analysis of the Neon dimer PES. We'll use that as a metric in our own analysis and compare our results to his to determine the quality and accuracy of our computations.
Note: for the purposes of this work, a chemically useful PES is one who's accuracy is within 10% to 20% of an RT thermal energy unit. R is the Gas Constant and T is the temperature at which the fluid of choice is in its liquid state. For Neon, T=26 degrees K so that 10% to 20% of RT = 8 to 16 micro au.
1) Basis Set Superposition Errors
To ensure our Binding Energy results are correct, we first had to address the issue of BSSE. In the figure below we show an example of the effect of BSSE:
As we can see, the BSSE effect can be quite large and, in some cases, yields unreasonable results, e.g. the prediction of negative HF Binding Energies. It's also interesting to note that the size of the effect increases as the two atoms come closer. This is precisely what one would expect given the description of BSSE given above.
The size of the effect is dependent on the size of the basis set. Indeed, one measure of the completeness of the basis set could be the size of the counterpoise effect. The size of the effect also varies by level of calculation. The following chart shows this diminution of counterpoise corrections as the basis set increases in quality:
We see that the as the basis set is improved from 3-11+G(3df) to AUG-cc-pVQZ the counterpoise correction for the HF calculations is improved by a factor of 14 and for the MP4SDTQ method, by a factor of 6. Note, however, that the magnitude of the correction is still significant and cannot be ignored.
From this we see that to achieve accuracy that is required to yield chemically meaningful results, BSSE must be taken into account.
2) Choosing the proper method
A summary of our Binding Energy results, for various methods, are charted below:
From these we derive the following conclusions:
- As expected, the HF results are inadequate.
- The best method was the MP4SDTQ.
- The addition of single and triple excitations (the S and T in SDTQ) produces significant improvement in the Binding Energy results.
We observed a 10% - 20% increase in attraction in the Binding Energy across the entire range of Ne-Ne separation arising from the inclusion of Single and Triple excitations in the correlated methods (compare the results of MP4DQ vs MP4SDTQ). Although this behavior has been observed previously for covalent bonds, it was unexpected in this case (especially for weakly interacting close shell systems that do not covalently bond).
From this we conclude that the PESs must be computed using the MP4SDTQ method.
3) Selection of a Basis Set
Calculating the Binding Energy for a variety of basis sets, we obtained the results shown:
We have deliberately included a large number of diffuse functions in the basis sets used in our study. The following chart shows why this is important:
The cc-pVQZ basis set is the AUG-cc-pVQZ basis set with the diffuse functions removed. From these results we see the key role played by the diffuse functions in the determination of the PES. The primary effects of the diffuse functions are to place the potential energy well at shorter Ne-Ne distances and increase its depth. Both effects are due to the diffuse functions allowing a proper polarization of the electron density to proper distances around the atomic nuclei. This behavior is consistent with previously observed effects for systems of negative ions and lone pair electrons. Lack of diffuse functions would lead to a PES which is far too repulsive and whose minimum is not properly located.
Given our definition of chemically useful PES accuracy, we conclude that basis set of at least AUG-cc-pVQZ or better is required to achieve an accurate PES.
4) Examination of Density Functional Methods
It is well documented that Density Functional Methods can provide quite good representations of some properties for chemical systems (including correlation effects) at a lower cost than the HF plus MP methods. We wanted to see if that was the case for weakly interacting systems of the type in which we have an interest.
We performed a series of calculations using a variety of Density Functional methods available in G98W. We calculated the Neon Dimer PES to see if any produced results comparable to the standard methods. Our results for these calculations are displayed below:
None of the functionals yield results that are correct. Only one (MPW1PW91) is even physically reasonable. It is clear the Density Functional methods are not appropriate for determining these type of PES.
From these results we see that some necessary conditions to achieve an accurate representation of a PES are:
- Use the MP4SDTQ method. All of S,D,T and Q are required to get accurate results.
- Use a substantial basis set with extensive polarization and diffuse functions, i.e. at least AUG-cc-pVQZ or better.
- Counterpoise corrections are essential.
- Density functional methods do not work well for these types of systems.
Sounds kind of tautological since everyone knows the bigger the basis set and the higher the level of theory the better the answer gets. However, when we started this it wasn't clear that we would have to go to such extremes to get chemically meaningful answers, that counterpoise was so important, and that Density Functional methods would work so poorly for these systems.
My sincere thanks to Professor Doctor Hanspeter Huber for sending me copies of his papers and providing me with important insights into the methods necessary to develop quality Potential Energy Surfaces.
 Gaussian 98, Revision A.6,
M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria,
M. A. Robb, J. R. Cheeseman, V. G. Zakrzewski, J. A. Montgomery, Jr.,
R. E. Stratmann, J. C. Burant, S. Dapprich, J. M. Millam,
A. D. Daniels, K. N. Kudin, M. C. Strain, O. Farkas, J. Tomasi,
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S. Clifford, J. Ochterski, G. A. Petersson, P. Y. Ayala, Q. Cui,
K. Morokuma, D. K. Malick, A. D. Rabuck, K. Raghavachari,
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M. W. Wong, J. L. Andres, C. Gonzalez, M. Head-Gordon,
E. S. Replogle, and J. A. Pople,
Gaussian, Inc., Pittsburgh PA, 1998.
 GAMESS VERSION = 6 MAY 1998
FROM IOWA STATE UNIVERSITY
M.W.SCHMIDT, K.K.BALDRIDGE, J.A.BOATZ, S.T.ELBERT,
M.S.GORDON, J.H.JENSEN, S.KOSEKI, N.MATSUNAGA,
K.A.NGUYEN, S.J.SU, T.L.WINDUS,
TOGETHER WITH M.DUPUIS, J.A.MONTGOMERY
J.COMPUT.CHEM. 14, 1347-1363(1993)
********Intel x86 (WIN32, OS/2, DOS32) VERSION********
PC GAMESS version 5.1, build number 1519
Compiled on Tuesday, 06-10-1998, 16:26:20
Partial Copyright (c) 1994, 1998 by
Alex. A. Granovsky, Moscow State University
Intel specific optimization, bug fixes,
and multiple enhancements.
 Eggenberger R., Gerber S., Huber H., and Welker M., 1994, Molecular Physics, 82, 689.