Equation Roothaan, modified

Roothaan equations have been modified in a previous work with the aim of avoiding BSSE at the Hartree-Fock level of theory. The resulting scheme, called SCF-MI (Self Consistent Field for Molecular Interactions), underlines its special usefulness for the computation of intermolecular interactions. 

The simplicity of the standard SCF procedure has been preserved. The closed shell Roothaan equations and the Guest and Saunders open shell equations have been modified at the cost of a negligible complication with respect to the usual algorithm. 

The equations require to be modified for open-shell systems, in which some orbitals are doubly occupied and some singly (this is called spin-restricted Hartree-Fock theory). A further extension to the theory involves electrons of a and /3 spin being assigned to different molecular orbitals, type equations are described as unrestricted Hartree-Fock [31]. 

This restriction is not demanded. It is a simple way to satisfy the Pauli exclusion principle, but it is not the only means for doing so. In an unrestricted wavefunction, the spin-up electron and its spin-down partner do not have the same spatial description. The Hartree-Fock-Roothaan procedure is slightly modified to handle this case by creating a set of equations for the a electrons and another set for the p electrons, and then an algorithm similar to that described above is implemented. 

An alternative approach has been proposed by Philipp and Friesner [33] and Murphy et al. [34]. It differs from the previous one by the introduction of modified Roothaan equations to compute the electronic density and energy of the QM part, which avoids the orthogonalization process, and by the treatment of the interaction of... 

The LCAO coefficients are determined by solving the modified Roothaan equations [3]... 

All exact-decoupling approaches can be related to the modified Dirac equation and we closely follow here the work presented in Refs. [16,647]. Two-component electrons-only Hamiltonians can be obtained from block-diagonalizing the four-component (one-electron) modified Dirac equation in matrix representation. As we have discussed in chapters 8 and 10 for four-component Dirac-Hartree-Fock-Roothaan calculations, basis functions for the small component must fulfill certain constraints as otherwise variational instability and a wrong nonrelativistic limit [547] would result. The correct nonrelativistic limit will be obtained if the kinetic-balance condition,

[Pg.533]

Thus, the evaluation of the X-operator in matrix form requires the diago-nalization of the modified Dirac-Roothaan equation, but its diagonalization posed the pitfalls that had led to the derivation of exact-decoupling methods in order to avoid pathologies originating from the negative-energy states in the first place. However, if this step can be accomplished with an approximate potential V —> V, which does not include the full electron-electron interaction, then it can be a very efficient procedure. 

Abstract Some previous results of the present author are combined in order to develop a Hermitian version of the Chemical Hamiltonian Approach. In this framework the second quantized Bom-Oppenheimer Hamiltonian is decomposed into one- and two-center components, if some finite basis corrections are omitted. (No changes are introduced into the one- and two-center integrals, while projective expansions are used for the three- and four-center ones, which become exact only in the limit of complete basis sets.) The total molecular energy calculated with this Hamiltonian can then presented as a sum of the intraatomic and diatomic energy terms which were introduced in our previous chemical energy component analysis scheme. The corresponding modified Hartree-Fock-Roothaan equations are also derived they do not contain any three- and four-center integrals, while the non-empirical character of the theory is conserved. This scheme may be useful also as a layer in approaches like ONIOM. 

Fock modified Hartree s SCF method to include antisymmetrization. Roothaan further modified the Hartree-Fock method by representing the orbitals by linear combinations of basis functions similar to Eq. (16.3-34) instead of by tables of numerical values. In Roothaan s method the integrodifferential equations are replaced by simultaneous algebraic equations for the expansion coefficients. There are many integrals in these equations, but the integrands contain only the basis functions, so the integrals can be calculated numerically. The calculations are evaluated numerically. This work is very tedious and it is not practical to do it without a computer. 

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