Young D.C. - Computational chemistry (2001)(en)

232 28 BASIS SET CUSTOMIZATION

create a new basis set, but it is extremely di½cult to create one better than the existing sets. Creating a basis set requires ®rst obtaining an initial set of functions and then optimizing both coe½cients and exponents in order to get the best variational energy. An initial set of functions can be created by using a known approximate mathematical relationship between basis functions or ®tting to high-accuracy results, such as numerical basis set calculations. Even once a low variational energy has been obtained, that does not mean it is a good basis set. A good description of the core electrons will yield a low energy. It is also necessary to give a good description of the valence region, where bonding occurs. In order to describe the valence region well, the basis must have both basis functions in the correct region of space and enough ¯exibility (uncontracted functions) to describe the shift in electron density as bonds are being formed. Then the issues of polarization and di¨use functions must be addressed.

28.3COMBINING EXISTING BASIS SETS

It is possible to mix basis sets by choosing the functions for di¨erent elements from di¨erent existing basis sets. This can be necessary if the desired basis set does not have functions for a particular element. It can result in very good results as long as basis sets being combined are of comparable accuracy. Mixing very large and very small basis sets can result in inaccurate calculations due to uneven spanning of the function space, called an unbalanced basis.

How does one tell if two basis sets have the same accuracy? If the two were de®ned for the same atom, then they would be expected to have a similar number of Gaussian primitives, similar number of valence contractions, and similar total energy's for the atom-only calculation. If they are not de®ned for the same elements, then a similar number of valence contractions is a crude indicator. In this case, the number of primitives should follow typical trends. For example, examining a basis set that has been de®ned for a large number of elements will show general trends, such as having more primitives for the heavier elements and more primitives for the lower-angular-momentum orbitals. These patterns have been de®ned based on studies in which the variational energy lowering for each additional primitive was examined. A well-balanced basis set is one in which the energy contribution due to adding the ®nal primitive to each orbital is approximately equivalent.

What if an unbalanced basis is used? This is tantamount to asking what would happen if the calculation contains two nearby atoms: one of which is described by a large basis and the other by a small basis. In this case, the energy of the atom with a small basis can be variationally lowered if the basis functions of the other atom are weighted in order to describe the electron density around the ®rst atom. This leads to an extreme case of basis set superposition error. The energy of the atom with the smaller basis can be lowered even more if it moves closer to the atom with the large basis. Thus, this leads to errors in both energy and geometry. An unbalanced basis can give results of a poorer accuracy than a small but balanced basis.

FIGURE 28.1 Uncontracted basis input.

28.4CUSTOMIZING A BASIS SET

The most di½cult part of creating a basis set from scratch is optimizing the exponents. In customizing a basis, one must start with a set of exponents at the very least. The steps involved then consist of creating contractions, adding additional functions to the valence contractions, adding polarization functions, and adding di¨use functions. It is possible to add outer primitives (low exponents) that are nearly optimal. It is more di½cult to add near-optimal inner functions (large exponents). Thus, the description of the core region provided by the starting exponents must be acceptable.

Determining contraction coe½cients is something that occasionally must be done. Early calculations did not use contracted basis functions and so older basis sets do not include contraction coe½cients. The user must determine the contractions in order to make optimal use of these basis sets with the current generation of software, which is designed to be used with contracted basis sets. If no contraction coe½cients are available, the ®rst step is to run a calculation on the atom with the basis set completely uncontracted. For example, Figure 28.1 shows a 6s2p nitrogen atom basis created by Duijneveldt and formatted for input in the Gaussian program.

Figure 28.2 shows the molecular orbitals from the uncontracted atom calculation. A reasonable set of contractions will be obtained by using the largestweight primitives from each of these orbitals. There is not a speci®c value that is considered a large weight. However, some guidance can be obtained by examining established basis sets. In this example, the basis will be partitioned into two s and one p contracted functions. Our choice of coe½cients are those printed in bold in Figure 28.2. The px and pz orbitals show some mixing to give an arbitrary orientation in the xz plane, which does not change the energy of a single atom. The py orbitals were used since they mix very little with the x and

236 28 BASIS SET CUSTOMIZATION

8.3954000000E-01 4.7481000000E-01

2.5953000000E-01 6.4243000000E-01

4.1543000000E-01 7.7381000000E-01

Note that the answers have been rounded to three signi®cant digits. Since the even-tempered formula is only an approximation, this does not introduce any signi®cant additional error.

Although the even tempered function scheme is fairly reasonable far from the nucleus, each function added is slightly further from the energy-optimized value. Generally, two or three additional functions at the most will be added to a basis set. Beyond this point, it is most e½cient to switch to a di¨erent, larger basis.

A di¨erent scheme must be used for determining polarization functions and very di¨use functions (Rydberg functions). It is reasonable to use functions from another basis set for the same element. Another option is to use functions that will depict the electron density distribution at the desired distance from the nucleus as described above.

Having polarization functions of higher angular momentum than the highest occupied orbitals is usually the most polarization that will bene®t HF or DFT results. Higher-angular-momentum functions are important for very-high- accuracy con®guration interaction and coupled-cluster calculations. As a general rule of thumb, uncontracting a valence primitive generally lowers the variational energy by about as much as adding a set of polarization functions.

A new basis for the element can now be created by combining these techniques. The basis in Figure 28.4 was created from the contracted set illustrated in Figure 28.3. Additional even-tempered exponents have been added to both the s and p functions. A polarization function of d symmetry was obtained from the 6ÿ31G(d) basis set. In a realistic scenario, a certain amount of trial-and- error work, based on obtaining low variational energies and stronger chemical bonds, would be involved in this process. This nitrogen example is somewhat arti®cial because there are many high-quality basis functions available for nitrogen that would be preferable to customizing a basis set.

28.5 BASIS SET SUPERPOSITION ERROR 237

The ®nal step is to check the performance of the basis set. This can be done by ®rst doing a single-atom calculation to check the energy and virial theorem value. The UHF calculation for this basis gave a virial theorem check of ÿ1.9802, which is in reasonable agreement with the correct value of ÿ2. The UHF atom energy is ÿ54.10814 Hartrees for this example. This is really not a very good total energy for nitrogen due to the fact that the example started with a fairly small basis set. The 6ÿ31G(d) basis gives a total energy of ÿ54.38544 Hartrees for nitrogen. The basis in this example should probably not be extended any more than has been done here, since it would lead to having a disproportionately well-described valence region and poorly described core.

The ®nal test of the basis quality, particularly in the valence region, is the result of molecular calculations. This basis gave an N2 bond length of 1.1409 AÊ at the HF level of theory and 1.1870 at the CCSD level of theory, in only moderate agreement with the experimental value of 1.0975 AÊ . The larger 6ÿ31G(d) basis set gives a bond length of 1.0783 AÊ at the HF level of theory. The experimental bond energy for N2 is 225.9 kcal/mol. The HF calculation with this example basis yields 89.9 kcal/mol, compared to the HF 6ÿ31G(d) bond energy of 108.6 kcal/mol. At the CCSD level of theory, the sample basis gives a bond energy of 170.3 kcal/mol.

28.5BASIS SET SUPERPOSITION ERROR

Basis set superposition error (BSSE) is an energy lowering of a complex of two molecules with respect to the sum of the individual molecule energies. This results in obtaining van der Waals and hydrogen bond energies that are too large because the basis functions on one molecule act to describe the electron density of the other molecule. In the limit of an exact basis set, there would be no superposition error. The error is also small for minimal basis sets, which do not have functions di¨use enough to describe an adjacent atom. The largest errors occurred when using moderate-size basis sets.

The procedure for correcting for BSSE is called a counterpoise correction. In this procedure, the complex of molecules is ®rst computed. The individual molecule calculations are then performed using all the basis functions from the complex. For this purpose, many ab initio software programs contain a mechanism for de®ning basis functions that are centered at a location which is not on one of the nuclei. The interaction energy is expressed as the energy for the complex minus the individual molecule energies computed in this way. In equation form, this is given as

where the subscripts denote the basis functions being used and the letters in parentheses denote the molecules included in each calculation.

Counterpoise correction should, in theory, be unnecessary for large basis

238 28 BASIS SET CUSTOMIZATION

sets. However, practical applications have shown that it yields a signi®cant improvement in results even for very large basis sets. The use of a counterpoise correction is recommended for the accurate computation of molecular interaction energies by ab initio methods.

BIBLIOGRAPHY

Text books containing detailed basis set discussions are

J.B. Foresman, A. Frisch, Exploring Chemistry with Electronic Structure Methods Second Edition Gaussian, Pittsburgh (1996).

A. R. Leach Molecular Modelling Principles and Applications Longman, Essex (1996). I. N. Levine, Quantum Chemistry Fourth Edition Prentice Hall, Englewood Cli¨s (1991). W. J. Hehre, L. Radom, P. v. R. Schleyer, J. A. Pople, Ab Initio Molecular Orbital

Theory John Wiley & Sons, New York (1986).

Review articles and more detailed sources are

T. H. Dunning, K. A. Peterson, D. E. Woon, Encycl. Comput. Chem. 1, 88 (1998).

F.B. van Duijneveldt, J. G. C. M. van Duijneveldt-van de Rijdt, J. H. van Lenthe, Chem. Rev. 94, 1873 (1994).

A. D. Buckingham, P. W. Fowler, J. M. Hutson, Chem. Rev. 88, 963 (1988).

S.Wilson, Ab Initio Methods in Quantum Chemistry-I 439 K. P. Lawley, Ed., John Wiley & Sons, New York (1987).

E. R. Davidson, D. Feller, Chem. Rev. 86, 681 (1986).

J.Andzelm, M. Kobukowski, E. Radzio-Andzelm, Y. Sakai, H. Tatewaki, Gaussian Basis Sets for Molecular Calculations S. Huzinaga, Ed., Elsevier, Amsterdam (1984).

Basis set superposition error is reviewed in

N. R. Kestner, Rev. Comput. Chem. 13, 99 (1999).

F.B. van Duijneveldt, Molecular Interactions S. Scheiner, Ed., 81, John Wiley & Sons, New York (1997).

240 29 FORCE FIELD CUSTOMIZATION

29.2ORIGINAL PARAMETERIZATION

Understanding how the force ®eld was originally parameterized will aid in knowing how to create new parameters consistent with that force ®eld. The original parameterization of a force ®eld is, in essence, a massive curve ®t of many parameters from di¨erent compounds in order to obtain the lowest standard deviation between computed and experimental results for the entire set of molecules. In some simple cases, this is done by using the average of the values from the experimental results. More often, this is a very complex iterative process.

The ®rst step in creating a force ®eld is to decide which energy terms will be used. This determines, to some extent, the ability of the force ®eld to predict various types of chemistry. This also determines how di½cult the parameterization will be. For example, more information is needed to parameterize anharmonic bond-stretching terms than to parameterize harmonic terms.

The parameters in the original parameterization are adjusted in order to reproduce the correct results. These results are generally molecular geometries and energy di¨erences. They may be obtained from various types of experimental results or ab initio calculations. The sources of these ``correct'' results can also be a source of error. Ab initio results are only correct to some degree of accuracy. Likewise, crystal structures are in¯uenced by crystal-packing forces.

Many parameterizations are merely a massive ®tting procedure to determine which parameters will best reproduce these results. This procedure may be done with automated software or through the work and understanding of the designers. Often, a combination of both gives the best results. In recent years, global search techniques, such as genetic algorithms, have been used. This is usually an iterative procedure as parameters are adjusted and results computed for the test set of molecules.

A second procedure is called a rule-based parameterization. This is a way of using some simple relationship to predict a large number of parameters. For example, bond lengths might be determined as the geometric mean of covalent bond radii multiplied by a correction factor. In this case, determining one correction factor is tantamount to determining all needed bond lengths. This procedure has the advantage of being able to create a force ®eld describing a large variety of compounds. The disadvantage is that the accuracy of results for a speci®c compound is not usually as good as that obtained with a force ®eld parameterized speci®cally for that class of compounds.

29.3ADDING NEW PARAMETERS

A measure of sophistication is necessary in order to obtain a reasonable set of parameters. The following steps are recommended in order to address the concerns above. They are ranked approximately best to worst, but it is advisable to use all techniques for the sake of doublechecking your work. Step 9 should