Derivative Fock operator matrices

All that is required is to directly differentiate Eqn. (68). The simplifications that make this practical require considering the forms of the operators in closer detail than done in the preceding. The derivative of the Fock operator matrix can be broken into derivatives of constituent matrices ... 

There is another widely used method of obtaining the Fock operator, namely to obtain its matrix elements F lv as the derivative of the energy functional with respect to the density. In our case that yields... 

Any Fock operator can be represented as a sum of the symmetric one and of a perturbation which includes both the dependence of the matrix elements on nuclear shifts from the equilibrium positions and the transition to a less symmetric environment due to the substitution. To pursue this, we first introduce some notations. Let hi be the supervector of the first derivatives of the matrix of the Fock operator with respect to nuclear shifts Sq counted from a symmetrical equilibrium configuration. By a supervector, we understand here a vector whose components numbered by the nuclear Cartesian shifts are themselves 10 x 10 matrices of the first derivatives of the Fock operator, with respect to the latter. Then the scalar product of the vector of all nuclear shifts 6q j and of the supervector hi yields a 10 x 10 matrix of the corrections to the Fockian linear in the nuclear shifts ... 

Here we introduce the notation ( . ..) for the scalar product of vectors whose components are numbered by the Cartesian shifts of the nuclei). Next, let h" be the supermatrix of the second derivatives of the matrix of the Fock operator with respect to the same shifts. As previously, we refer here to the supermatrix indexed by the pairs of nuclear shifts in order to stress that the elements of this matrix are themselves the 10 x 10 matrices of the corresponding second derivatives of the Fock operator with respect to the shifts. The contribution of the second order in the nuclear shifts can be given the form of the (super)matrix average over the vector of the nuclear shifts ... 

This is another type of solvated Fock operator in the combination method of RISM and ab initio MO theory. It should be noted that the first order derivatives of radial distribution functions with respect to the effective charges are required to construct the Fock matrix. 

Using Eqs. (4.61) and (4.63), matrix U is calculated to give the response properties in terms of the uniform electric field dipole moments, polarizabilities, hyperpolarizabilities, and so forth. Equation (4.61) is called the coupled perturbed Kohn-Sham equation. Other response properties are calculated by solving Eq. (4.61) after setting the first derivative of the Fock operator, F, in terms of each perturbation. Note, however, that this method has problems in actual calculations similarly to the time-dependent response Kohn-Sham method. For example, using most functionals, this method tends to overestimate the electric field response properties of long-chain polyenes. 

The matrix elements of B that involve the pseudospinor y are zero because it is still orthogonal to the core, but the matrix elements of the Fock operator are not necessarily zero. The second derivative terms that are zeroth order in S are... 

Besides the evaluation of the Fock operator, density matrix, and integral derivatives, the analytical evaluation of energy second derivatives is achieved once the density matrix first derivatives with respect to geometric perturbations are known. The evaluation of the latter terms is the bottleneck of computational procedures because of its cost in terms of CPU time and disk storage. In the common practice, such a quantity is obtained by resorting to the first-order coupled perturbed HF (or KS) technique [4], which conceptually starts from the HF (or KS) equations, expands all the matrices in terms of the perturbation, and, by collecting all the terms at the same order, yields sets of equations which are usually solved iteratively. 

It appears, therefore, that it is possible to obtain accurate expectation values of the spin-orbit operators for diatomic molecules. Matcha et a/.112-115 have provided general expressions for the integrals involved and from their work Hall, Walker, and Richards116 derived the diagonal one-centre matrix elements of the spin-other-orbit operator for linear molecules. Provided good Hartree-Fock wavefunctions are available, these should be sufficient for most calculations involving diatomic molecules. 

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. 

It would be good now to get rid of the non-diagonal Lagrange multipliers in order to obtain a beautiful one-electron equation analogous to the Fock equation. To this end, we need the operator in the curly brackets in Eq. (11.33) to be invariant with respect to an arbitrary unitary transformation of the spinorbitals. The sum of the Coulomb operators (Ucoui) is invariant, as has been demonstrated on p. 406. As to the unknown functional derivative SE c/Sp (i.e., potential Uxc), its invariance follows from the fact that it is a functional of p [and p of Eq. (11.6) is invariant]. Finally, after applying such a unitary transformation that diagonalizes the matrix of Sij,v/e obtain the Kohn-Sham equation (su = Si) ... 

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. 

The expression for the matrix elements F y can be derived in the second quantized formalism in an elegant manner. This derivation relies on the physical picture behind the Hartree-Fock approximation. As known, in this model the electrons interact only in an averaged manner, so correlational effects are excluded. To derive such an averaged operator, we start again with the usual Hamiltonian ... 

Atomic basis functions in quantum chemistry transform like covariant tensors. Matrices of molecular integrals are therefore fully covariant tensors e.g., the matrix elements of the Fock matrix are F v = (Xn F Xv)- In contrast, the density matrix is a fully contravariant tensor, P = (x IpIx )- This representation is called the covariant integral representation. The derivation of working equations in AO-based quantum chemistry can therefore be divided into two steps (1) formulation of the basic equations in natural tensor representation, and (2) conversion to covariant integral representation by applying the metric tensors. The first step yields equations that are similar to the underlying operator or orthonormal-basis equations and are therefore simple to derive. The second step automatically yields tensorially correct equations for nonorthogonal basis functions, whose derivation may become unwieldy without tensor notation because of the frequent occurrence of the overlap matrix and its inverse. 

---