The eigenvalue equation

We use Greek capital letters almost exclusively to indicate many-electron 

Before considering the matrix equations in detail, we recall certain 

The matrix representing such an operator is thus Hermitian in the matrix sense (2.2.5), H = H. A useful consequence is that the adjoint of a non-Hermitian operator such as A + iB (A and B Hermitian) is 

The most important property of the Hamiltonian operator, for our purposes, is that its eigenfunctions Wi,. .., may be assumed to form a complete orthogonal set (Kato, 1951) if they also belong to class 1 we may normalize in the usual way and write 

We now return to the matrix equation (2.3.3), which in practice is applied in the finite form that arises when only, say, n members of the complete set are taken into account. What is the significance of the solutions of the resultant truncated equation Written out in full, the matrix equation then becomes 

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The exchange matrix, K, is just the rate, k, times the unit matrix. In block fonn, the full matrix for two sites is given in the eigenvalue equation, (B2.4.38). 

With either of these diabatic reference Hamiltonians, the LHSFs satisfy the eigenvalue equation... 

Note that even after requiring normalization, there is still an indeterminaney in the sign of v(3). The eigenvalue equation, as we reeall, only speeifies a direetion in spaee. The sense or sign is not determined. We ean ehoose either sign we prefer. 

VIII. Hermitian Matrices and The Turnover Rule The eigenvalue equation ... 

Consider in more detail the calculation of the determinant in (3.70) for the potential (3.66). The eigenvalue equation... 

That relation follows from the fact that the operator X2 + Y2 has positive eigenvalues. To see this, write the eigenvalue equations for X, Y, and X2 + Y2 ... 

It is clear from the preceding section that, until inequality (2.30) holds, the moments satisfy Eq. (2.24). Using this fact in Eq. (2.36) and the eigenvalue equation... 

The orthogonality theorem can also be extended to cover a somewhat more general form of the eigenvalue equation. For the sake of convenience, we present in detail the case of a single variable, although the treatment can be generalized to any number of variables. Suppose that instead of the eigenvalue equation (3.5), we have for a hermitian operator 4 of one variable... 

For a continuous spectrum of eigenkets with non-degenerate eigenvalues, it is more convenient to write the eigenvalue equation (3.45) in the form... 

In terms of a complete set of solutions (Xj, Vj) of the eigenvalue equation (28), the general solution to the linearized equation of motion (27) can be written... 

The eigenvalue equation of the representation of the effective Hamiltonian operators (28) in the base of the number occupation operator of the slow mode is characterized by the equation... 

The eigenvalue equations of the two diagonal blocks of the effective Hamiltonian matrix is characterized by the equations... 

Fermi coupling operator l is given by (90). Owing to the eigenvalues equation... 

Note that the stability of the spectra with respect to k m and um must be carefully checked. Within this enlarged base, the eigenvalues equation of the total Hamiltonian (104) may be written... 

The important states for atomic systems are those of definite energy E that satisfy the eigenvalue equation (11), which implies that the time-dependence of k for such states is given by... 

In order to solve the electronic structure problem for a single geometry, the energy should be minimized with respect to the coefficients (see Eq. (5)) subject to the orthogonality constraints. This leads to the eigenvalue equation ... 

Although in principle one could choose a set of arbitrary values for the solvent coordinates sm, solve the eigenvalue equation (2.23), and compute the free energy (2.12), in practice a preliminary aquaintance with the equilibrium solvation picture for the target reaction system serves as a computationally convenient doorway for the calculations in the nonequilibrium solvation regime. We show this below in the section dedicated to an illustration of the method for a two state case reported in BH-II. 

The eigenvalue equation corresponding to the Hamiltonian of Eq. [57] can be solved self-consistently by an iterative procedure for each orientation of the spin magnetization (identified as the z direction). The self-consistent density matrix is then employed to calculate the local spin and orbital magnetic moments. For instance, the local orbital moments at different atoms i are determined from... 

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