Hartree-Fock limit, properties

The theoretical results provided by the large basis sets II-V are much smaller than those from previous references [15-18] the present findings confirm that the second-hyperpolarizability is largely affected by the basis set characteristics. It is very difficult to assess the accuracy of a given CHF calculation of 2(ap iS, and it may well happen that smaller basis sets provide theoretical values of apparently better quality. Whereas the diagonal eomponents of the eleetrie dipole polarizability are quadratic properties for which the Hartree-Fock limit can be estimated with relative accuracy a posteriori, e.g., via extended calculations [38], it does not seem possible to establish a variational principle for, and/or upper and lower bounds to, either and atris-... 

For a the agreement between these CPHF/TDHF results performed with near-Hartree-Fock limit basis sets and our corresponding results (3rd columns of Tables 1-6) obtained with our largest basis set is rather good. It proves the adequacy of adding just one set of diffuse functions to reach the basis set saturation for these properties. 

Almost all contemporary ab initio molecular electronic structure calculations employ basis sets of Gaussian-type functions in a pragmatic approach in which no error bounds are determined but the accuracy of a calculation is assessed by comparison with quantities derived from experiment[l] [2]. In this quasi-empirical[3] approach each basis set is calibrated [4] for the treatment of a particular range of atoms, for a particular range of properties, and for a particular range of methods. Molecular basis sets are almost invariably constructed from atomic basis sets. In 1960, Nesbet[5] pointed out that molecular basis sets containing only basis sets necessary to reach to atomic Hartree-Fock limit, the isotropic basis set, cannot possibly account for polarization in molecular interactions. Two approaches to the problem of constructing molecular basis sets can be identified ... 

If W were known exactly, the value of a first-order property calculated from equation (12) would be exact. In practice, only an approximation to W is known, and it is important to know how the expectation value differs from the exact value. Since errors in calculated dipole moments due to the breakdown of the Bom-Oppenheimer approximation are likely to be small8 (typically 0.002 a.u.), and for most molecules relativistic effects can be ignored,6 there are two separate remaining problems in practice. The first concerns the likely accuracy when the wavefunction is at the Hartree-Fock limit, the second the effect of using a truncated basis set to obtain a wavefunction away from the Hartree-Fock limit. 

For any variational wavefunction which is not near the Hartree-Fock limit the Brillouin theorem is irrelevant, and even for those of Hartree-Fock accuracy low-lying important excited states may invalidate the conclusions drawn from it. The statement that values of one-electron properties are expected to be good because of the Brillouin theorem should therefore be regarded with caution. 

Of paramount importance in this latter category is the Hartree-Fock approximation. The so-called Hartree-Fock limit represents a well-defined plateau, in terms of its methematical and physical properties, in the hierarchy of approximate solutions to Schrodinger s electronic equation. In addition, the Hartree-Fock solution serves as the starting point for many of the presently employed methods whose ultimate goal is to achieve solutions to equation (5) of chemical accuracy. A discussion of the Hartree-Fock method and its associated concept of a self-consistent field thus provides a natural starting point for the discussion of the calculation of potential surfaces. 

When comparing STO and CGTF basis sets of a particular size, it should be taken into account that with STO basis sets the results are only due to exponents of STO s, whereas with the CGTF basis sets also the effects of the number of primitives and their contractions are involved. Furthermore it should be kept in mind that the total energy is a rather insensitive test of the quality of wave functions, so that also some other molecular properties should be considered. Finally, a rigorous comparison should also include calculations going beyond the Hartree-Fock limit. 

In cases where one is restricted to smaller basis sets by computational limitations, a double-C set which has been augmented by a single (well chosen) set of polarization functions offers an attractive alternative. Many of the above properties of H-bonded systems can be computed to excellent accuracy. It was shown above that a basis set containing only 50 functions was able to approach within 0.1 kcal/mol of the Hartree-Fock limit for the interaction energy in the water dimer. Moreover, a basis set of this type is sufficiently flexible to permit meaningful inclusion of electron correlation effects. Again, application of counterpoise correction is advised whenever possible. 

D. Feller, C. M. Boyle, and E. R. Davidson,/. Chem. Phys., 86, 3424 (1987). One-Electron Properties of Several Small Molecules Using Near-Hartree-Fock Limit Basis Sets. 

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