Approximate Formulations of the Fock Equations

Relativistic quantum chemistry is currently an active area of research (see, for example, the review volume edited by Wilson [102]), although most of the work is beyond the scope of this course. Much of the effort is based on Dirac s relativistic formulation of the Schrodinger equation this results in wave functions that have four components rather than the single component we conventionally think of. As a consequence the mathematical and computational complications are substantial. Nevertheless, it is very useful to have programs for Dirac-Fock (the relativistic analogue of Hartree-Fock) calculations available, as they can provide calibration comparisons for more approximate treatments. We have developed such a program and used it for this purpose [103]. 

The structure of the contribution is as follows. In Section 1.5.2 we discuss the structure of effective nonlinear Hamiltonians for the solute. In Section 1.5.3 we present a two-step formulation of the QM problem, with the corresponding Hartree-Fock (HF) equation. In Section 1.5.4 we introduce the fundamental energetic quantity for the QM solvation models while in Section 1.5.5 extensions beyond the HF approximation are presented and discussed. 

Semiempirical techniques are the next level of approximation for computational simulation of molecules. Compared to molecular mechanics, this approach is slow. The formulations of the self-consistent field equations for the molecular orbitals are not rigorous, particularly the various approaches for neglect of integrals for calculation of the elements of the Fock matrix. The emphasis has been on versatility. For the larger molecular systems involved in solvation, the semiempirical implementation of molecular orbital techniques has been used with great success [56,57]. Recent reviews of the semiempirical methods are given by Stewart [58] and by Rivail [59],... 

The basic concept is that instead of dealing with the many-body Schrodinger equation, Fq. (2.1), which involves the many-body wavefunction P ( r ), one deals with a formulation of the problem that involves the total density of electrons (r). This is a huge simplification, since the many-body wavefunction need never be explicitly specified, as was done in the Hartree and Hartree-Fock approximations. Thus, instead of starting with a drastic approximation for the behavior of the system (which is what the Hartree and Hartree-Fock wavefunctions represent), one can develop the appropriate single-particle equations in an exact manner, and then introduce approximations as needed. 

RPA, and CPHF. Time-dependent Hartree-Fock (TDFIF) is the Flartree-Fock approximation for the time-dependent Schrodinger equation. CPFIF stands for coupled perturbed Flartree-Fock. The random-phase approximation (RPA) is also an equivalent formulation. There have also been time-dependent MCSCF formulations using the time-dependent gauge invariant approach (TDGI) that is equivalent to multiconfiguration RPA. All of the time-dependent methods go to the static calculation results in the v = 0 limit. 

In practical calculations, it is never possible to obtain the exact solution to eqn (1). Instead, approximate methods are used that are based on either the Hartree-Fock approximation or the density-functional formalism in the Kohn-Sham formulation. In the case of the Hartree-Fock approximation one may add correlation elfects, which, however, is beyond the scope of the present discussion (for details, see, e.g., ref. 1). Then, in both cases the problem of calculating the best approximation to the ground-state electronic energy is transformed into that of solving a set of singleparticle equations,... 

Because of its computational simplicity and other obvious qualities the random-phase approximation has been used in many calculations. Reviews of RPA calculations include one on chiroptical properties by Hansen and Bouman (1980), one on the equation-of-motion formulation of RPA (McCurdy et al, 1977) and my own review of the literature through 1977 (Oddershede, 1978, Appendix B). Ab initio molecular RPA calculations in the intervening period are reviewed in Table I. Coupled Hartree-Fock calculations have not been included in the table. Only calculations which require diagonalization of both A -I- B and A — B and thus may give frequency-dependent response properties and excitation spectra are included. In CHF we only need to evaluate either (A -I- B) or (A — B) Mn order to determine the (static) response properties. 

Approximating the Schrodinger Equation. - With the approaches we have discussed so far one attempts to solve the exact Schrodinger equation as accurately as possible. All quantities are formulated in terms of the many-particle wavefunctions and, consequently, these approaches are called wave-function-based methods. Their main disadvantages are that the equations are highly complicated and, therefore, that the solutions become more and more approximate the more complex the system of interest is. Typically, the computational efforts scale as N4 for the Hartree-Fock approximation and up to N1 for the methods that include correlation effects. 

The Algebraic Approximation. - For atoms the use of spherical polar coordinates ( , , ) facilitates the factorization of the Hartree-Fock equations and reduces the problem to one involving a single radial coordinate r and an angular part which can be treated analytically. For diatomic molecules prolate spheroidal coordinates ( , , ) separate the non-relativistic Hartree-Fock equations into a two-dimensional part which can be solved numerical and a -dependent part which can be treated analytically. For arbitrary molecular systems there is no suitable coordinate system in which the problem can be formulated and hence it is usual to resort to the algebraic approxima-... 

We also want to mention that a Dyson-equation approach for propagators like the polarisation and the particle-pcurticle propagator has been formulated and used to derive a self-consistent extension of the RPA, also called cluster-Hartree-Fock approximation, that has been applied in the fields of plasma and nuclear physics [12-14], This formalism, however, has similar problems like Feshbach s theory and does not yield a universal, well-behaved optical potential because the two-particle space has to be restricted in order to make the approach well-defined [14]. 

The usual starting point in the formulation of a QM molecular problem is given by the HF theory applied within the approximation of clamped nuclei. Using an expansion on a discrete basis set and adopting a matrix formulation, the minimization of G (eq.7) can be reduced to the solution of the following Fock equation... 

The ideas and concepts concerning the use of basis sets in relativistic calculations which have been described in the previous subsections allow the Dirac-Fock equations for many-electron systems to be formulated within the algebraic approximation. A discussion of these equations lies outside the scope of the present chapter. 

A simple application of the very general approach used in earlier sections leads to the time-dependent generalization of Hartree-Fock theory. The time-dependent Hartree-Fock (TDHF) equations (Dirac, 1929) were first formulated variationally by Frenkel (1934) they are also widely used in nuclear physics (see e.g. Thouless, 1%1) under the name random-phase approximation (RPA). Since the equations describe response to a perturbation, as in Section 11.9 but now time-dependent, they will... 

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