Finite analytic basis set

In 1951, Hall [6] and, independently, Roothaan [7] put the Hartree-Fock equations - the ubiquitous independent particle model - in their matrix form. The Hartree-Fock equations describe the motion of each electron in the mean field of all the electrons in the system. Hall and Roothaan invoked the algebraic approximation in which, by expanding molecular orbitals in a finite analytic basis set, the integro-differential Hartree-Fock equations become a set of algebraic equations for the expansion coefficients which are well-suited to computer implementation. 

LG. Kaplan, Symmetry of Many-Electron Systems, translated by J. Gerratt, Academic Press, New York London 1975 (p. 269, the term algebraic approximation is employed to describe the use of finite analytic basis sets in molecular electronic structure calculations)... 

Preliminary results from the analytic basis set method indicate it to be a factor of 60 faster than the finite element method on the LiFH problem. 

The most time consuming part of the calculation, the determination of the surface functions, has been improved by almost two orders of magnitude. The DVR and analytic basis set methods are preferred over finite elements methods. We have designed an efficient numerical method for studying reactive scattering that can produce exact and approximate results for systems and energies which have heretofore been impossible to study. 

Again, nonequilibrium calculations were performed using a DFT/B3LYP model, and employing for all systems (see below) experimental gas phase geometries and an aug-cc-pVDZ basis set. The linear response functions afso(0), aajg(—w to) and aa 8(—to to) were obtained analytically, whereas the higher order polarizabilities, Papy(—to to, 0) and 0), were determined via a finite electric field technique. The results published in ref. [30] are shown in Table 2.11. 

Figure 4. Static hypermagnetizability anisotropy. Atj(O), computed with the d-aug-cc-pV5Z basis set (Neon) and d-aug-cc-pVQZ basis set (Argon). Orbital-relaxed results obtained with a finite field approach from analytically evaluated magnetizabilities are compared to those obtained from orbital-unrelaxed quadratic and cubic response functions...

The non-relativistic PolMe (9) and quasirelativistic NpPolMe (10) basis sets were used in calculations reported in this paper. The size of the [uncontractd/contracted] sets for B, Cu, Ag, and Au is [10.6.4./5.3.2], [16.12.6.4/9.7.3.2], [19.15.9.4/11.9.5.2], and [21.17.11.9/13.11.7.4], respectively. The PolMe basis sets were systematically generated for use in non-relativistic SCF and correlated calculations of electric properties (10, 21). They also proved to be successful in calculations of IP s and EA s (8, 22). Nonrelativistic PolMe basis sets can be used in quasirelativistic calculations in which the Mass-Velocity and Darwin (MVD) terms are considered (23). This follows from the fact that in the MVD approximation one uses the approximate relativistic hamiltonian as an external perturbation with the nonrelativistic wave function as a reference. At the SCF and CASSCF levels one can obtain the MVD quasi-relativistic correction as an expectation value of the MVD operator. In perturbative CASPT2 and CC methods one needs to use the MVD operator as an external perturbation either within the finite field approach or by the analytical derivative schems. The first approach leads to certain numerical accuracy problems. 

The self-consistent field procedure in Kohn-Sham DFT is very similar to that of the conventional Hartree-Fock method [269]. The main difference is that the functional Exc[p] and matrix elements of Vxc(r) have to be evaluated in Kohn-Sham DFT numerically, whereas the Hartree-Fock method is entirely analytic. Efficient formulas for computing matrix elements of Vxc(r) in finite basis sets have been developed [270, 271], along with accurate numerical integration grids [272-277] and techniques for real-space grid integration [278,279]. 

Fullerenes. - Jonsson et al.234 have carried out analytical Hartree-Fock calculations, expected to be near the basis set limit, of a, and the magnetiz-ability for the C70 and C84 fullerenes. The results are compared with earlier calculations on ) and the electronic structures of the molecules discussed. Moore et al.235 have made semi-empirical AMI finite field calculations of the static y-hyperpolarizability of Qo, C70, five isomers of C78 and two isomers of C84. The results are interpreted in terms of bonding and structural features. 

We applied the finite-temperature FCI and MPO to the FH, N2, and F2 molecules with the STO-3G basis set using the determinant-based implementations described above. Additionally, the MPO calculations were performed using the semi-analytical expressions based on the Fermi-Dirac statistics. In the N2 and F2 calculations, two U core orbitals were excluded from FCI or MPO. The bond lengths were 0.9168 A (FH), 1.0977 A (N2), and 1.41193 A (F2). The reference wave functions were obtained by the HF calculations at 0 K in all cases. 

---