Differentiation, derivative Hartree-Fock

A. Analytical Differentiation and Derivative Hartree-Fock Theory... 

We shall initially consider a closed-shell system with N electroris in N/2 orbitals. The derivation of the Hartree-Fock equations for such a system was first proposed by Roothaan [Roothaan 1951] and (independently) by Hall [Hall 1951]. The resulting equations are known as the Roothaan equations or the Roothaan-Hall equations. Unlike the integro-differential form of the Hartree-Fock equations. Equation (2.124), Roothaan and Hall recast the equations in matrix form, which can be solved using standard techniques and can be applied to systems of any geometry. We shall identify the major steps in the Roothaan approach. 

These are the glorious Hartree-Fock equations derived in general in the spin orbital basis. But wait - there s a problem. These are coupled integro-differential equations, and while they are not strictly unsolvable, they re a pain. It would be nice to at least uncouple them, so let s do that. 

The KS equations are obtained by differentiating the energy with respect to the KS molecular orbitals, analogously to the derivation of the Hartree-Fock equations, where differentiation is with respect to wavefunction molecular orbitals (Section 5.2.3.4). We use the fact that the electron density distribution of the reference system, which is by decree exactly the same as that of the ground state of our real system (see the definition at the beginning of the discussion of the Kohn-Sham energy), is given by (reference [9])... 

This method is valid only in the static field limit (zero frequency), which is a weakness. However, recent advances of a derived procedure (Coupled Perturbed Hartree-Fock) permit the frequency dependence of hyperpolarizabilities to be computed. The FF method mainly uses MNDO (modified neglect of diatomic differential overlap) semi-empirical algorithm and the associated parametrizations of AM-1 and PM-3, which are readily available in the popular MOPAC software package. ... 

In principle, the differentiation is either done numerically in the so-called finite-field methods, or in an analytical scheme, or a combination of both. Numerical finite-field calculations are limited to derivatives with respect to static fields. Since SFG is an optical process that involves dynamic oscillating fields, it becomes necessary to use an analytical approach, such as the time-dependent Hartree Fock (TDHF) method. 

Under the conditions of our derivation, i.e. S-state atoms A and B with vanishing differential overlap, we can show that the localized orbitals Ip) and Icr) are identical (apart from mixing possibly degenerate orbitals) to the orbitals obtained by solving the monomer Hartree-Fock equations (in the dimer basis)... 

The Hartree-Fock equation of the previous chapter is a non-separable partial differential equation in three dimensions. In this chapter we derive an equation which is satisfied by an approximation to the differential Hartree-Fock equation. This matrix equation provides the techniques and concepts for the vast majority of quantum chemistry. 

Essentially the same result (with S = 1 and real coefficients) was previously obtained in Subsection 1.3.2. The functional variation technique thus leads to the same result as is obtained by differentiating with respect to the coef-.ficients. Functional variation is a more general technique, however, and we now proceed to derive the Hartree-Fock equations using it. 

To solve the unrestricted Hartree-Fock equations (3.312) and (3.313), wc need to introduce a basis set and convert these integro differential equations to matrix equations,just as we did when deriving Roothaan s equations. We thus introduce our set of basis functions = 1, 2,..., X and... 

We now turn to the calculation of f and H in quantum chemistry. A straightforward approach is to simply numerically differentiate the energy. Alternatively, it is possible to obtain these derivatives by analytic derivative methods. Let us illustrate some of the essential ideas in the framework of Hartree-Fock calculations. 

Since we have two different sets of molecular spinor coefficients connected to large- and small-component basis functions, the differentiation of the Dirac-Hartree-Fock energy is more tedious than the differentiation of the nonrelativistic Hartree-Fock energy. For this reason, we proceed with taking the derivative of the Hartree-Fock energy for the sake of simplicity in order to highlight the principles. The Hartree-Fock energy can be written as... 

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