Spinor Optimization

An efficient approach to improve on the Hartree-Fock Slater determinant is to employ Moller-Plesset perturbation theory, which works satisfactorily well for all molecules in which the Dirac-Hartree-Fock model provides a good approximation (i.e., in typical closed-shell single-determinantal cases). The four-component Moller-Plesset perturbation theory has been implemented by various groups [519,584,595]. A major bottleneck for these calculations is the fact that the molecular spinor optimization in the SCF procedure is carried out in the atomic-orbital basis set, while the perturbation expressions are given in terms of molecular spinors. Hence, all two-electron integrals required for the second-order Moller-Plesset energy expression must be calculated from the integrals over atomic-orbital basis functions like... 

Variational one-center restoration. In the variational technique of one-center restoration (VOCR) [79, 80], the proper behavior of the four-component molecular spinors in the core regions of heavy atoms can be restored as an expansion in spherical harmonics inside the sphere with a restoration radius, Rvoa, that should not be smaller than the matching radius, Rc, used at the RECP generation. The outer parts of spinors are treated as frozen after the RECP calculation of a considered molecule. This method enables one to combine the advantages of two well-developed approaches, molecular RECP calculation in a gaussian basis set and atomic-type one-center calculation in numerical basis functions, in the most optimal way. This technique is considered theoretically in [80] and some results concerning the efficiency of the one-center reexpansion of orbitals on another atom can be found in [75]. 

The large component accounts for most of the electron density of a spinor, and as such will carry the largest weight in basis set optimizations. It also has the larger amplitude, and as such must weigh heavily in any fitting scheme. This is only natural, and for most purposes, including standard chemical applications, creates no problems. However, there are some properties that depend heavily on the quality of the small component description. One of these would be the interaction of a possible electric dipole moment, dg of the electron with an applied external field, S. This interaction is described by the operator [20]... 

Orbital (spinor) energies from AREP-HF (REP-KRHF) calculations at the optimized geometries, and ionization potentials from photoelectron spectroscopy for methyl halides and carbon tetrahalides. Units are in eV, and the degeneracy number is set in parenthesis. 

Fig. 10.7. Illustration of the model space in the multireference Cl method used mainly in the situation when no single Slater determinant dominates the Cl expansion. In the figure the orbital levels of the system are presented. Part of them are occupied in all Slater determinants considered ( frozen spinor-bitals ). Above them is a region of closely spaced orbital levels called active space. In the optimal case, an energy gap occurs between the latter and unoccupied levels lying higher. The model space is spanned by all or some of the Slater determinants obtained by various occupancies of the active space levels.

Large scale optimized-level (OL) calculations corresponding to 39 nonrelativistic configurations (Cai et al. 1992). 38 replacements into spinors of up to symmetry from 4f were considered, leading to 354, 858, 1386, 1579, 1708, 1535 and 1344 relativistic configurations for J = 0 to J=6. 

For basis-set expansion techniques it turned out to be decisive to fulfill the kinetic-balance condition for the basis functions (see again chapter 10 for details and references), whereas fully numerical four-component calculations had already been carried out around 1970 without encountering variational collapse. In numerical approaches it is possible to search for optimized spinors in the vicinity of the nonrelativistic solution with a given number of nodes and associated orbital energy as we shall see in chapter 9. 

The derivation has been general so far, i.e., we have derived the most general form of the self-consistent field equations that one may use to optimize a set of one-electron spinors for truncated Cl expansions in the MCSCF approach. 

The solution of the SCF equations involves a number of technical tricks decisive for actual calculations. In the end, they represent an optimization problem that can be tackled in many different ways. An interesting aspect to mention, though, is that usually only the first derivative (the linear variation) is considered. The set of spinors obtained produces a stationary energy but it is not guaranteed that this is a true minimum rather than a saddle point. Nevertheless, this possibility is usually not tested unless a special type of MCSCF calculations that we introduce in section 10.6 is carried out. 

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