Matrices eigenvalue equation

The value of detennines how much computer time and memory is needed to solve the -dimensional Sj HjjCj= E Cj secular problem in the Cl and MCSCF metiiods. Solution of tliese matrix eigenvalue equations requires computer time that scales as (if few eigenvalues are computed) to A, (if most eigenvalues are... 

Roothaan actually solved the problem by allowing the LCAO coefficients to vary, subject to the LCAO orbitals remaining orthonormal. He showed that the LCAO coefficients are given from the following matrix eigenvalue equation ... 

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

Knowing that H and S are hermetian, this matrix eigenvalue equation can be rewritten as... 

To advance further, the resonance property of the exact density response at the true excitation energies (0k [9, 20, 32] is used, namely, that a finite external perturbation 8vext(ri,OJk) leads to an infinite change in the density. Then Eq. 29 can be satisfied only if the matrix in the square brackets possesses a zero eigenvalue at cok. This leads, after a unitary transformation, to the master matrix eigenvalue equations for the excitation energies cok [9]... 

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

The matrix Bis numerically much less important than A and accordingly y << x. This suggests a simplification of the foregoing equations by setting B = 0. This Tamm-Dancoff approximation [11] leads to a symmetric matrix eigenvalue equation... 

Since there are electrons, there are at most terms in the expansion (4-4). The L group electrons now only enter (3-7 a) through the matrix elements of the density operator derived from their ground state wave function Vlo, pKo - i> 2)— 2) for short. The stationary value condition applied to Eq. (3-6 a) can be transformed in the usual way into a matrix eigenvalue equation for the eigenvalues Ewm and the expansion coefficients C in Eq. (4-4)... 

The discussion of the various aspects of the MCSCF method requires the discussion of some background material. In this section, some of the elementary concepts of linear algebra are introduced. These concepts, which include the bracketing theorem for matrix eigenvalue equations, are used to define the MCSCF method and to discuss the MCSCF model for ground states and excited states. The details of V-electron expansion space represent-... 

This is called the first-order variational space which is spanned by the reference state, the orthogonal complement states and the single excitation states (10>, ln>, T 10> or equivalently by the set of expansion CSFs and the single excitation states n>,r 0>. The parameters K and p may be determined by minimizing the expectation value of the Hamiltonian operator within this non-orthonormal basis. This results in the non-orthogonal matrix eigenvalue equation... 

This equation is neither a pure matrix eigenvalue equation nor a pure linear equation. However, its solution may be found using iterative methods that are no more complicated than the iterative solution of the pure matrix eigenvalue equation of Eq. (193) or the pure linear equation of Eq. (186). The iterative solution of these equations may use the effective operator methods discussed previously. 

The relation between the wavefunction corrections determined by the WSCI and PSCI methods and the corresponding Newton-Raphson methods may be determined using the matrix partitioning approach. For example, the solution to the PCI matrix eigenvalue equation is equivalent to the solution of the linear equation... 

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... 

As the dimension of the blocks of the Hessian matrix increases, it becomes more efficient to solve for the wavefunction corrections using iterative methods instead of direct methods. The most useful of these methods require a series of matrix-vector products. Since a square matrix-vector product may be computed in 2N arithmetic operations (where N is the matrix dimension), an iterative solution that requires only a few of these products is more efficient than a direct solution (which requires approximately floating-point operations). The most stable of these methods expand the solution vector in a subspace of trial vectors. During each iteration of this procedure, the dimension of this subspace is increased until some measure of the error indicates that sufficient accuracy has been achieved. Such iterative methods for both linear equations and matrix eigenvalue equations have been discussed in the literature . 

The dimensions of these subspace representations are small enough so that Eqs (274) and (275) may be solved using direct linear equation an matrix eigenvalue equation methods. The computation of the residual vector uses the same matrix-vector products as is required for the construction of the subspace representations of the Hessian blocks. If either of the components of the residual vector are too large, a new expansion vector corresponding to the large residual component may be computed from one of the equations... 

Substitutions of Eqs. (78-85) in Eq. (73) lead to three-term recurrence relations of the expansion coefficients contained in Eqs. (32-39) in Ref. [6] and not reproduced here for the sake of space. The three-term recurrence relations can be cast onto the form of matrix eigenvalue equations with eigenvalues li and the eigenvectors, just like in the case of Section 2.3. 

Matrix-eigenvalue equation. 103-105 Molecular diagrams. 119-121 Molecular orbitals as a linear combination of atomic orbitals, 8-9... 

The use of atomic symmetry block-diagonalizes the matrix representation of the DHFB Hamiltonian of (95) giving the generalized matrix eigenvalue equations... 

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