Generalized matrix eigenvalue problem

Given approximations to P and P, at each step of the iteration, we can form F and F, solve the two generalized matrix eigenvalue problems... 

This equation may be written in the well-known compact form as a generalized matrix eigenvalue problem. 

The approaches (i) and (//) are straightforward in the sense that once the basis functions are chosen or optimized, the matrix elements of the Hamiltonian are computed and the generalized matrix eigenvalue problem is solved. 

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

Here, L is a lower triangular matrix (not to be confused with L, the Cholesky factor of the matrix of nonlinear parameters A ), and D is a diagonal matrix. The scheme of the solution of the generalized symmetric eigenvalue problem above has proven to be very efficient and accurate in numerous calculations. But the main advantage of this scheme is revealed when one has to routinely solve the secular equation with only one row and one column of matrices H and S changed. In this case, the update of factorization (117) requires only oc arithmetic operations while, in general, the solution from scratch needs oc operations. 

This is not so for the matrix eigenvalue problem the eigenvalues (and eigenvectors) of real matrices n generally can only be found in the complex plane C (and in C"). The... 

In order to find extrema of E( ui ), subject to the normalization condition, standard moves known as the Lagrange multipliers method are applied, which readily lead us to the well-known form of the generalized matrix eigenvalue/eigenvector problem ... 

For future reference we now reformulate the above theory in a way which might appear unfamiliar at first glance but on closer inspection will turn out to be a generalization of the procedure we have used many times to find the lowest eigenvalue of a matrix. We are now interested in finding the sum of the N lowest eigenvalues. The matrix eigenvalue problem in Eq. (5.89) is equivalent to four equations, two of which are... 

In this section, we discuss briefly the generalized Floqnet formnlation of TDDFT [28,60-64]. It can be applied to the nonperturbative stndy of mnltiphoton processes of many-electron atoms and molecules in intense periodic or qnasi-periodic (multicolor) time-dependent fields, allowing the transformation of time-dependent Kohn-Sham equations to an equivalent time-independent generalized Floquet matrix eigenvalue problems. 

One can easily see from equations (7), (8), (10), and (12), that we have a non-linear problem when we want to solve the generalized matrix eigenvalue equation (7) (in the same way as in HF calculations of atoms and molecules). 

If the functions Oj are orthonormal, then the overlap matrix S reduces to the unit matrix and the above generalized eigenvalue problem reduces to the more familiar form ... 

In general, a matrix equation in the form A x = 0 will have solutions other than x = 0 only if dot A = 0. In the case of vibrations, there will be non-trivial solutions only if det(7 — w M) = 0. This is an example of an eigenvalue problem. 

In deriving this we have used the properties of the integrals Hij = //, and a similar result for Stj. Equation (1.14) is discussed in all elementary textbooks wherein it is shown that a Cy 0 solution exists only if the W has a specific set of values. It is sometimes called the generalized eigenvalue problem to distinguish from the case when S is the identity matrix. We wish to pursue further information about the fVs here. 

The expansion column matrix A from Eq. (26) is the eigenvector of U(s), and the elements Snm of the overlap matrix S are given in Eq. (19). The obtained expression (26) is not an ordinary but a generalized eigenvalue problem involving the overlap matrix S due to the mentioned lack of orthogonality... 

For the numerical study of the whole spectrum (for g R fixed), [79] uses a spectral tau-Chebychev discretization in y and the Arnold method (see [88]) to solve the generalized eigenvalue problem (see [89]). This numerical method is based on the orthonormalization of the Krylov space of the iterates of the inverse of the matrix A B. This method has been used more recently in [90]. It has been proven efficient in the stiff problems arising in the study of spectral stability of viscoelastic fluids. 

---