Hamiltonian equations diagonalization techniques

Because

double excitation in the open-shell space, and because we left excitations within this space out of the excitation operators, the second part of the normalization term is zero, and the energy is given by the left side of the equation. This technique can be used for open-shell Kramers pairs belonging to complex or real irreps, but not to quaternion irreps. In the last case, there are four determinants that are composed of the open-shell spinors, and even though they occur in pairs related by time-reversal symmetry, the Hamiltonian operator connects all four. In the case of complex irreps, the absolute value of the off-diagonal matrix element must be taken, because it will in general be complex. 

The natural solution of the operator Eq. 4.38 would be in the momentum space due to the presence of momentum p which replaces the standard configuration space formulation by a Eourier transform. The success of the Douglas-Kroll-Hess and related approximations is mostly due to excellent idea of Bernd Hess [53,54] to replace the explicit Eourier transformation by some basis set (discrete momentum representation) where momentum p is diagonal. This is a crucial step since the unitary transformation U of the Dirac Hamiltonian can easily be accomplished within every quantum chemical basis set program, where the matrix representation of the nonrelativistic kinetic energy T = j2m is already available. Consequently, all DKH operator equations could be converted into their matrix formulation and they can be solved by standard algebraic techniques [13]. 

In this it resembles the techniques used in the large Cl calculations to determine an eigenvalue without having to store the entire Hamiltonian matrix. This approach is rapidly convergent, at least in SCF calculations where the matrix A is diagonally dominant. For a molecule with N nuclei it is necessary to solve 3N CHF equations for the perturbations due to the nuclear displacements (ignoring symmetry for the moment). All of these equations can be solved at the same time. ° ... 

The CEO computation of electronic structure starts with molecular geometry, optimized using standard quantum chemical methods, or obtained from experimental X-ray diffraction or NMR data. For excited-state calculations, we usually use the INDO/S semiempirical Hamiltonian model (Section IIA) generated by the ZINDO code " however, other model Hamiltonians may be employed as well. The next step is to calculate the Hartree— Fock (HE) ground state density matrix. This density matrix and the Hamiltonian are the Input Into the CEO calculation. Solving the TDHE equation of motion (Appendix A) Involves the diagonalization of the Liouville operator (Section IIB) which is efficiently performed using Kiylov-space techniques e.g., IDSMA (Appendix C), Lanczos (Appendix D), or... 

The essential features of variationally-based approaches of MS-MA type are that the matrix elements contain the total Hamiltonian H instead of V (i.e. there is no separation into -h V) that the free-molecule product functions are replaced by antisymmetrized products [44b] that the expansions are truncated, and that the coefficients are obtained by treating the expansion as a linear variation function [31]. When the corresponding matrix equations are solved by partitioning or perturbation techniques [44b] the resultant first-order interaction resembles (3), while the second-order contribution resembles (12) except that the product functions are antisymmetrized and the summation is discrete and finite. The formal reduction of El and E2 for exact free-molecule functions is dealt with elsewhere [16, 18, 19], in the context of a second-order EN expansion. Except in cases of heavy overlap, the antisymmetrizer in the off-diagonal elements may safely be omitted, as E2 is already of second-order. 

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