Hartree-Fock equation matrix, derivation

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. 

It is important that the orbitals r ) that satisfy (5.17) are orthogonal for different eigenvalues We can use them to construct an orthonormal set with which to express the many-electron problem. We have assumed orthogonality in deriving (5.17). We now show that this is consistent by considering (5.17) as an equation for the matrix elements formed for the bra orbital ( from a Schrodinger equation for rj). The set of such equations for N different rj) is called the Hartree—Fock equations. 

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. 

For the moment we simply note that the Hartree-Fock equation ( 3.24) could, more elegantly, have been derived from the vanishing of the first-order variation (SL) in L induced by arbitrary variations in the matrix C. 

Once again, the derivative of the density matrix P can be obtained as solution of first-order coupled-perturbed Hartree-Fock equation with derivar tive Fock matrix given by eq. (1.70), exactly as for the nuclear shielding. 

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... 

The derivatives of the MO coefficients or the density matrix can be obtained by solving the the corresponding derivatives of the Hartree-Fock equations, i.e. the coupled perturbed Hartree-Fock (CPHF) equations. These can be solved either in the MO basis or in the AO basis. After some manipulation, the CPHF equations in the MO basis can be reduced to ... 

Hartree-Fock wave function, and, in fact, the most general derivation of the Hartree-Fock equations is possible through the Brillouin theorem which can be proved directly from the variation principle (Mayer 1971,1973,1974). We shall not prove here the complete equivalence of the Hartree-Fock equations and Eq. (11.1), it will be shown only that the Brillouin theorem is fulfilled for the Hartree-Fock wave function. The proof will make use of second quantization which helps us to evaluate the matrix element easily. To this goal, Eq. (11.1) should be rewritten in the second quantized notation. The ground state is simply represented the Fermi vacuum ... 

With only a few electrons outside a closed shell, the above equations may be solved iteratively in essentially the same manner as the Hartree-Fock equations. Usually an algebraic approximation is used in which every is expressed as a linear combination of basis functions as in Section 6.2 and this leads to matrix equations that are more suitable for computational purposes. The actual techniques of optimization are similar to those used in multiconfiguration versions of MO SCF theory (Chapter 8) the more rapidly convergent procedures require also second derivatives of the energy expression, but these may be obtained in the same way as the first derivatives. 

DERIVATION OF HARTREE-FOCK THEORY Equation (A.46) may be cast as a matrix equation... 

The first derivative of the density matrix with respect to the magnetic induction (dPfiv/dBi) is obtained by solving the coupled-perturbed Hartree-Fock (or Kohn-Sham) equations to which the first derivative of the effective Fock (or Kohn-Sham) operator with respect to the magnetic induction contributes. Due to the use of GIAOs, specific corrections arising from the effective operator Hcnv describing the environment effects will appear. We refer to Ref. [28] for the PCM model and to Ref. [29] for the DPM within either a HF or DFT description of the solute molecule. 

Here, W is an energy-weighted density matrix element as in the Hartree-Fock gradient formula (23). Equation 5 contains two "difficult" terms (the last two), the derivative of density matrix elements and the fitting coefficients e. It turns out that these terms can be eliminated by using the relationship... 

This initial guess may then be inserted on the right-hand sides of the equations and subsequently used to obtain new amplitudes. The process is continued until self-consistency is reached. For the special case in which canonical Hartree-Fock molecular orbitals are used, the Fock matrix is diagonal and the T2 amplitude approximation above is exactly the same as the first-order perturbed wave-function parameters derived from Moller-Plesset theory (cf. Eq. [212]). In that case, the Df and arrays contain the usual molecular orbital energies, and the initial guess for the T1 amplitudes vanishes. 

The unrestricted Hartree-Fock (UHF) case is completely analogous to the closed-shell one. New terms do appear in the open-shell SCF and the few-configuration case. Nevertheless, the preferred technique is quite similar to the closed-shell case. In particular, the two-particle density matrix can be constructed from compact matrices, and the solution of the derivative Cl equations is very simple, due to the small dimension. 

The effective Hartree-Fock matrix equation for a many-shell system has been derived in Chapter 14 and used in several applications open shells and some MCSCF models. So far, it has been seen simply as the formally correct equation to generate SCF orbitals for these many-sheU structures without any interpretation. In particular, the fact that the effective Hartree-Fock matrix (the McWeenyan ) contains many arbitrary parameters has not been addressed, nor has the practical problem of the actual grounds for the choice of values for these parameters been systematised. In looking at this problem we must bear two points in mind ... 

On the contrary, the density derivatives are necessary to get higher order free energy derivatives in particular the second free energy derivatives require the first derivative of the density matrix P , which can be obtained by solving an appropriate coupled perturbed Hartree-Fock (CPHF) or Kohn-Sham (CPKS) equations. 

In this section, then, we first introduce a set of unrestricted spin orbitab to derive the spatial eigenvalue equations of unrestricted Hartree-Fock theory. We then introduce a basis set and generate the unrestricted Pople-Nesbet matrix equations, which are analogous to the restricted Roothaan equations. We then perform some sample calculations to illustrate solutions to the unrestricted equations. Finally, we discuss the dissociation problem and its unrestricted solution. 

Hence, the relativistic analog of the spin-restriction in nonrelativistic closed-shell Hartree-Fock theory is Kramers-restricted Dirac-Hartree-Fock theory. We should emphasize that our derivation of the Roothaan equation above is the pedestrian way chosen in order to produce this matrix-SCF equation step by step. The most sophisticated formulations are the Kramers-restricted quaternion Dirac-Hartree-Fock implementations [286,318,319]. A basis of Kramers pairs, i.e., one adapted to time-reversal s)mimetry, transforms into another basis under quatemionic unitary transformation [589]. This can be exploited not only for the optimization of Dirac-Hartree-Fock spinors, but also for MCSCF spinors. In a Kramers one-electron basis, an operator O invariant under time reversal possesses a specific block structure. 

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