The matrix eigenvalue equation

In the context of the HF-LCAO model, we seek a solution of the matrix eigenvalue equation... 

This is known as the Jaffe series [117]. The important recurrence relations (6.468) can be represented by the matrix eigenvalue equation [118]... 

There have been two proposed approaches to the problem of finding optimal step lengths when the straightforward solution of the matrix eigenvalue equation is inadequate. The first is a line search approach , which requires the evaluation of the exact energy at one or more points along the line... 

Subtracting (15) and (16) gives the desired result (10). Substitution of the operator expansion (2) for yields (6) for the excitation energy case. The derivation of (7) for the IP-EA variant of the theory follows in a similar fashion. It is clear that the a s in (11) are fixed by the choice of operator basis because for a given operator basis, the matrix eigenvalue equations (6) and (7) yield unique results. 

We noted earlier that the matrix eigenvalue equation which determines the solutions of the Hartree-Fock equation is equivalent to the condition... 

The one-electron matrix Schrodinger equation itself is a quite practical approach and so we outline its derivation. We need two sets of one-electron matrix elements. The first set consists of those of the Hamiltonian Hjk = f droverlap matrix S Sjk = / dr( (r) fc(r). Often one assumes, not quite correctly, that the overlap matrix is diagonal and so is the identity matrix. The unknown coefficients in the expression for the wave function V i(j ) are the solution of the matrix eigenvalue equation... 

We will illustrate the variational technique by rederiving the matrix eigenvalue equation of the linear variational problem given in Subsection 1.3.2. Given a linear variational trial wave function. 

The PPP CO method > is derived by taking into account explicitly, as in the case of molecules, only the n electrons of the system. Putting S(A ) = 1, equation (3.1) reduces to the matrix eigenvalue equation... 

We now collect some important results relating to the solution of the matrix eigenvalue equation (2.3.4). First we note that the solutions have orthogonality properties analogous to those of the solutions of the operator equation (2.3.1). Thus two sets of coefficients, and corresponding to eigenvalues k and l, satisfy... 

The corresponding generalization of the matrix-eigenvalue equations is straightforward. If the A set contains degenerate electronic functions 0a> admission of nuclear spin yields a manifold of pe x p v4-set products a a > and this will be the dimension of the new. 4.4-block in (11.4.4). The discussion then follows similar lines, except that the matrix elements in the various blocks are labelled by double indices, a— aa. 

From (K18) it follows that the eigenvalues of Goi are 1. These properties, as we shall show below, allow the reduction of the matrix eigenvalue Equation K12 to onedimensional form. 

The problem of separating the matrix eigenvalue Equation K12 with the condition (K17) and (K18) to two one-dimensional equations is quite straightforward and may be carried out with the aid of the projection operators P+ and P defined as... 

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