Generalized Matrix Eigenvalue Equation

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

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). 

The Kohn-Sham equation (10) then becomes a generalized matrix eigenvalue equation for C, the A basis x N/2] matrix of linear coefficients,... 

Show how the general matrix-eigenvalue equation (2.3.4) may be reduced... 

The derived EOM equations (6) [(7)] are linear matrix eigenvalue equations for the exact excitation energies (ionization potentials and electron affinities). The eigenvectors give an satisfying conditions (12) and (13) (with o 0 =0 for IPs and EAs). The most general Oj is expressed in terms of the basis operators X >[Pg.12]

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

A transformation can be found that simultaneously diagonalizes several matrices if they commute with each other. This will not necessarily be a unitary transformation. The problem of simultaneous diagonalization sometimes arises just as a need to transform several matrices to diagonal form. It also arises in a special matrix eigenvalue equation that is really a generalization of Equation A.28. 

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. 

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. 

Equation (40) is a generalized eigenvalue equation. The eigenvalue j is interpreted as the excitation energy from the ground state to the Jth excited state. The vectors Xj and Yj are the first-order correction to the density matrix at an excitation and describe the transition density between the ground state and the excited state J. 

The concept of the eigenvalue equation associated with a matrix A is perhaps the most profound extension of matrix algebra beyond the algebra of scalars. The eigenvalue equation for A can generally be written as... 

The eigenvalue equation (S9.1-15) therefore presents an intuitive geometrical picture of how a matrix A operates on a general vector u by differentially stretching its components in different eigen-directions. 

The Jacobi method is generally slower than these other methods unless the matrix is nearly diagonal. In SCF calculations one is faced with the non-orthogonal eigenvalue equation... 

Let s recall how to find eigenvalues and eigenvectors. (If your memory needs more refreshing, see any text on linear algebra.) In general, the eigenvalues of a matrix A are given by the characteristic equation det(A - Af) = 0, where 1 is the identity matrix. For a 2 x 2 matrix... 

Since the Fock matrix is dependent on the orbital coefficients, the Roothaan equations have to be repeatedly solved in an iterative process, the self-consistent field (SCF) procedure. One important step in the SCF procedure is the conversion of the general eigenvalue equation (7) into an ordinary one by an orthogonalization transformation... 

The overlap matrix S is also taken into account in the EHT version of the general eigenvalue equation (7) which can be solved by applying the mentioned orthogonal-ization transformation and matrix diagonalization techniques. Due to the independence of Feht from the orbital coefficients, no SCF procedure has to be performed. This is similar to HMO. 

In a general case of equation (5.62), the necessary condition on the Hopf bifurcation is that the stability matrix has, for the critical value of the control parameter c = c0, one pair of purely imaginary eigenvalues, whereas the remaining eigenvalues must have a non-zero real part. 

One seeks a solution to the linear variational problem in Eq. (3.34) in the sense that for all i the Rayleigh quotient < , >/< > is stationary with respect to variation of the coefficients d. The solution is that the matrix of coefficients d has to satisfy the following generalized eigenvalue equation ... 

Here Fdimer and Sdimer have the same definitions as before, the unitary matrix U is formed from the solutions Uj of the generalized eigenvalue equation... 

Thus, another important property of L is that it diagonalizes the force constant matrix. Also, since L is an orthogonal matrix, Eq. (2.44) can be written in the form of a general eigenvalue equation, that is. 

The matrix representation of a Hermitian operator 0 in an arbitrary basis i> is generally not diagonal. However, its matrix representation in the basis formed by its eigenvectors is diagonal. To show this we multiply the eigenvalue equation (1.72) by [Pg.16]

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. 

With the aid of these equations we can condense the matrix equations (1.122) (by performing matrix multiplications and adding corresponding block matrices) into the generalized eigenvalue equation... 

We want to solve the generalized eigenvalue equation fc = Set. We already know that ( )i, is the lowest eigenfunction with eigenvalue We can shift the energy scale by subtracting from the eigenvalue matrix, i = — e I, and we must therefore... 

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