Equation coupled Hartree-Fock

The first two kinds of terms are called derivative integrals, they are the derivatives of integrals that are well known in molecular structure theory, and they are easy to evaluate. Terms of the third kind pose a problem, and we have to solve a set of equations called the coupled Hartree-Fock equations in order to find them. The coupled Hartree-Fock method is far from new one of the earliest papers is that of Gerratt and Mills. 

The coupled Hartree-Fock equations are then solved (Figure 17.5). 

The Hartree-Fock equations have to be solved by the coupled Hartree-Fock method. The following article affords a typical example. 

The simplest polarization propagator corresponds to choosing an HF reference and including only the h2 operator, known as the Random Phase Approximation (RPA). For the static case oj = 0) the resulting equations are identical to those obtained from a Time-Dependent Hartree-Fock (TDHF) analysis or Coupled Hartree-Fock approach, discussed in Section 10.5. 

The Hartree-Fock approximation leads to a set of coupled differential equations (the Hartree-Fock equations), each involving the coordinates of a single electron. While they may be solved numerically, it is advantageous to introduce an additional approximation in order to transform the Hartree-Fock equations into a set of algebraic equations. 

More recently, Caves and Karplus71 have used diagrammatic techniques to investigate Hartree-Fock perturbation theory. They developed a double perturbation expansion in the perturbing field and the difference between the true electron repulsion potential and the Hartree-Fock potential, V. This is compared with a solution of the coupled Hartree-Fock equations. In their interesting analysis they show that the CPHF equations include all terms first order in V and some types of terms up to infinite order. They propose an alternative iteration procedure which sums an additional set of diagrams and thus should give results more accurate than the CPHF scheme. Calculations on Ha and Be confirmed these conclusions. 

The forms of the orbitals ut are determined by demanding that the energy obtained in equation (13) be a minimum with respect to arbitrary variations in the orbitals. This procedure81 leads to a set of n coupled equations, the Hartree-Fock equations, of the form... 

Alternatively, analytical methods can be used such as coupled Hartree-Fock (CHF), where the perturbed HF equations are solved directly. Using standard perturbation theory one can also develop a sum-over-states (SOS) formalism and write... 

By minimizing the energy of d>, in Eq. (3.12), we obtain a set of coupled integro-differential equations, the Hartree-Fock equations, which may be expressed in the following form for closed-shell systems (for open-shell cases see Szabo and Ostlund, 1989) ... 

P is obtained by solving the coupled Hartree-Fock equation ... 

We saw in Section III that the polarization propagator is the linear response function. The linear response of a system to an external time-independent perturbation can also be obtained from the coupled Hartree-Fock (CHF) approximation provided the unperturbed state is the Hartree-Fock state of the system. Thus, RPA and CHF are the same approximation for time-independent perturbing fields, that is for properties such as spin-spin coupling constants and static polarizabilities. That we indeed obtain exactly the same set of equations in the two methods is demonstrated by Jorgensen and Simons (1981, Chapter 5.B). Frequency-dependent response properties in the... 

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. 

The matrix equation for the p-h part of p has the familiar form (l-A)X = B, which occurs in coupled Hartree-Fock theory [34]. 

This leads to the response theory [38,50,51,64,65] or coupled DFT (CDFT) which is the direct analog of the coupled Hartree-Fock (CHF) approach [3,57]. The equations thus obtained are coupled, since the perturbed KS molecular orbitals (MO) are coupled with each other by self-consistently as in the FPT approach. In contrast with FPT, the CDFT equations (18) remain real also in the case of a purely imaginary perturbation because of the lack of dependency on A. The disadvantage is the need to evaluate the linear response of the KS effective potential v cl] analytically. 

In order to construct 4 and one needs the, which are obtained from a coupled Hartree-Fock type equation. 

GIAO FPT for each AO new integrals of higher angular momentum, coupled Hartree-Fock equations, complex MOs... 

The first-order perturbation correction to the molecular orbitals satisfies the coupled Hartree-Fock equations written in the form... 

The first-order correction satisfies the (gauge) modified coupled Hartree-Fock equations... 

The above modified coupled Hartree-Fock equations for Vf 1> should be solved through an iterative procedure. Once they are determined, the second-order energy appears as a sum of four contributions... 

In the system of equation (73) one should observe that because of the occurrence of the unknown functions Acpfm in the integrals (last two terms of the l.h.s. of (73) the coupled Hartree-Fock (HF) equations (as the simple HF ones) are non-linear and, therefore, have to be solved in an iterative way. 

For the numerical implementation of these formulae one can either use a perturbation expansion for the functions coupled Hartree-Fock (CHF) equations for each order.124 This procedure seems to be more advantageous for numerical calculations, but the resulting expressions are quite complicated. For this reason we do not reproduce them here, but refer to the original paper.124 It seems to be an acceptable compromise to use the second order CHF equations for the dynamic as and (Js, but use simple perturbation theory for the dynamic ys using as unperturbed wave functions the results of the solutions of the first and second order CHF equations. One should point out, however, that in our calculations124 the second numerical derivatives... 

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