Fock operator matrix representation

Since the Hartree-Fock wavefunction 0 belongs to the totally symmetric representation of the symmetry group of the molecule, it is readily seen that the density matrix of Eq. (10) is invariant under all symmetry operations of that group, and the same holds, therefore, for the Hartree-Fock operator 7. 

On the basis of AOs the Fock operator eq. (1.147) acquires a matrix representation of the form ... 

The problem is now solved again by an iterative process, which starts with a choice of the x set and the expansion (6.58). The Hartree-Fock operator F and the matrix representation Fx are calculated, (6.64) is solved for the orbital energies, and these are used to compute a new set of coefficients in (6.63). If these are different from the starting set, the cycle is repeated until the self-consistent-field limit is reached. The total electronic energy is obtained by adding the SCF energy to the core energy for the lowest occupied n/2 levels ... 

Electronic energy Ea = 2Tr(RF) - Tr(RG) R—first-order density matrix F—matrix representation of the Hartree-Fock operator G—matrix representation of the electron repulsion energy 57... 

The DIRAC package [55,56], devised by Saue and collaborators, rather than exploiting the group theoretical properties of Dirac spherical 4-spinors as in BERTHA, treats each component in terms of a conventional quantum chemical basis of real-valued Cartesian functions. The approach used in DIRAC, building on earlier work by Rosch [80] for semi-empirical models, uses a quaternion matrix representation of one electron operators in a basis of Kramers pairs. The transformation properties of these matrices, analysed in [55], are used to build point group transformation properties into the Fock matrix. 

The matrix representation of the Fock operator for closed-shell systems may be separated into one-body and two body parts... 

The main computational attraction of the SCF method is that it reduces to a one-electron matrix problem the manipulations are those of matrices of the size of the number of basis functions. This means, for example, that any orbital transformations which may have to be done involve transformations of matrix representations of one-electron operators no time-consuming transformations of the four-index electron-repulsion integrals have to be done explicitly. The electron-repulsion terms are all contained in the Hartree-Fock matrix via G or,... 

All of the above hinges on the assumption that /i is a representation of an operator h which is invariant with respect to the operations of the molecular point group. Obviously the one-electron Hamiltonian and the unit operator satisfy this condition and so the matrices h and S of the LCAO method can be symmetry blocked in this way. We have seen that, in general, the matrix representation of the Hartree-Fock operator will not satisfy this condition, so that it is a constraint on the LCAO method to make the assumption that the matrix can be treated in this way. As we have seen, this constraint consists of generating self-consistent symmetries which in certain critical cases may prevent us from obtaining the lowest-energy determinant. 

The computational procedure involves obtaining the matrix representation of the symmetry operators of the (2n + 2) site chain in the direct product basis. The matrix representation of both J and P for the new sites in the Fock space is known from their definitions. Similarly, the matrix representations of the operators J and P for the left (right) part of the system at the first iteration are also known in the basis of the corresponding Fock space states. These are then transformed to the density matrix eigenvectors bcisis. The matrix representation of the symmetry operators of the full system in the direct product space are obtained as the direct product of the corresponding matrices ... 

However, the solution given by Eq. (4.277) is based on the form of effective independent-electron Hamiltonians that can be quite empirically constructed - as in Extended Huckel Theory (Hoffmann, 1963) such arbitrariness can be nevertheless avoided by the so-called self-consistentfield (SCF) in which the one-electron effective Hamiltonian is considered such that to depend by the solution of Eq. (4.266) itself, i.e., by the matrix of coefficients (C) this way we identify the resulted Hamiltonian as the Fock operator, while the associated eigen-problem rewrites the Hartree-Fock equation (4.267) under the mono-electronic wave-function representation ... 

Multiplying (3.70) by matrix representation of the Fock operator in the basis of spin orbital eigenfunctions is diagonal with diagonal elements equal to the orbital energies. 

The unitary transformation required for the block diagonalization of the relativistic Fock operator can be obtained in one step if a matrix representation of the Fock operator is available this is achieved by the so-called eXact-2-Component (X2C) approach [725-728,731-734]. An important characteristic of the X2C approach is its noniterative construction of the key operator X of Eq. (11.2). In this noniterative construction scheme, the matrix operator X is obtained from the electronic eigenvectors of the relativistic (modified) Roothaan Eq. (14.13),... 

Large-scale SCF treatments have been made possible with the direct SCF method developed by Almlof and co-workers. This procedure is based on the evaluation of two-election integrals on die fly whenever they are needed in the construction of the Fock operator (in the matrix representation required in SCF treatments). The efficiency of direct SCF method.s results from the fact that many integrals give negligible contributions and can be safely neglected. This requires reliable estimates for the integral values, TURBOMOLE employs the mathematically best separable bounds for this purpose. ... 

The matrix representation of this Fock operator can be written... 

Since the Wannier functions are not eigenfunctions of the Fock-operator, F, switch to a Bloch representation to calculate the matrix elements of and Hj. This gives... 

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