Eigenvalues orthonormality

Here (1/2%) exp(ikx) is the normalized eigenfunction ofF =-ihd/dx corresponding to momentum eigenvalue hk. These momentum eigenfunctions are orthonormal ... 

For the kind of potentials that arise in atomic and molecular structure, the Hamiltonian H is a Hermitian operator that is bounded from below (i.e., it has a lowest eigenvalue). Because it is Hermitian, it possesses a complete set of orthonormal eigenfunctions ( /j Any function spin variables on which H operates and obeys the same boundary conditions that the ( /j obey can be expanded in this complete set... 

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 ... 

For the hermitian matrix in review exereise 3b show that the pair of degenerate eigenvalues ean be made to have orthonormal eigenfunetions. 

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 ... 

The best wave function of the approximate form (Eq. 11.38) may then be determined by the variational principle (Eq. II.7), either by varying the quantity p as an entity, subject to the auxiliary conditions (Eq. 11.42), or by varying the basic set fv ip2,. . ., ipN subject to the orthonormality requirement. In both ways we are lead to Hartree-Fock functions pk satisfying the eigenvalue problem... 

If the hamiltonian is truly stationary, then the wx are the space-parts of the state function but if H is a function of t, the wx are not strictly state functions at all. Still, Eq. (7-65) defines a complete orthonormal set, each wx being time-dependent, and the quasi-eigenvalues Et will also be functions of t. It is clear on physical grounds, however, that to, will be an approximation to the true states if H varies sufficiently slowly. Hence the name, adiabatic representation. 

Although the eigenvalues 0 and 1 are universal, there are many possible eigenvectors r that depend on the kind of states p is to designate. To each one of an orthonormal set of functions , for instance,... 

This contribution considers systems which can be described with just the Hamiltonian, and do not need a dissipative term so that TZd = 0- This would be the case for an isolated system, or in phenomena where the dissipation effects can be represented by an additional operator to form a new effective non-Hermitian Hamiltonian. These will be called here Hamiltonian systems. For isolated systems with a Hermitian Hamiltonian, the normalization is constant over time and the density operator may be constructed in a simpler way. In effect, the initial operator may be expanded in its orthonormal eigenstates (density amplitudes) and eigenvalues Wn (positive populations), where n labels the states, in the form... 

Along these lines, we can immediately write down the eigenvalues and the orthonormal eigenfunctions... 

Let N be the dimension of a finite-dimensional space H, Xf. be eigenvalues and be orthonormal eigenfunctions of the problem (see Chapter 1, Section 1 and Chapter 2, Section 1)... 

Equation (31.3) defines the eigenvalue decomposition (EVD), also referred to as spectral decomposition, of a square symmetric matrix. The orthonormal matrices U and V are the same as those defined above with SVD, apart from the algebraic sign of the columns. As pointed out already in Section 17.6.1, the diagonal matrix can be derived from A simply by squaring the elements on the main diagonal of A. 

The same applies to the other eigenvectors U2 and Vj, etc., with additional constraints of orthonormality of u, U2, etc. and of Vj, Vj, etc. By analogy with eq. (31.5b) it follows that the r eigenvalues in A must satisfy the system of linear homogeneous equations ... 

If the eigenvalues of N are represented by the parameter X and the corresponding orthonormal eigenfunctions by (pxi( ) or, using Dirac notation, by Xi), then we have... 

In this case, the operator FT(1, 2,. .., A) is obviously symmetric with respect to particle interchanges. For the N particles to be identical, the operators H i) must all have the same form, the same set of orthonormal eigenfunctions and the same set of eigenvalues where... 

The quantity k > is the unperturbed Hamiltonian operator whose orthonormal eigenfunctions and eigenvalues are known exactly, so that... 

We assume in this section and in Section 10.2 that equation (10.6) has been solved and that the eigenfunctions Q) and eigenvalues k(Q) are known for any arbitrary set of values for the parameters Q. Further, we assume that the eigenfunctions form a complete orthonormal set, so that... 

The electronic Hamiltonian and the corresponding eigenfunctions and eigenvalues are independent of the orientation of the nuclear body-fixed frame with respect to the space-fixed one, and hence depend only on qx. The index i in Eq. (9) can span both discrete and continuous values. The v /f, ad(r q J form a complete orthonormal basis set and satisfy the orthonormality relations... 

The eigenvalue-eigenvector decomposition of a Hermitian matrix with the complete orthonormal set of eigenvectors Vi and eigenvalues A, is written as... 

As indicated in Table 4.2, the eigenvalues of the Hessian matrix of fix) indicate the shape of a function. For a positive-definite symmetric matrix, the eigenvectors (refer to Appendix A) form an orthonormal set. For example, in two dimensions, if the eigenvectors are Vj and v2, v[v2 =0 (the eigenvectors are perpendicular to each other). The eigenvectors also correspond to the directions of the principal axes of the contours of fix). 

It is well established that the eigenvalues of an Hermitian matrix are all real, and their corresponding eigenvectors can be made orthonormal. A special case arises when the elements of the Hermitian matrix A are real, which can be achieved by using real basis functions. Under such circumstances, the Hermitian matrix is reduced to a real-symmetric matrix ... 

Due to the hermitian character of the dynamical matrix, the eigenvalues are real and the eigenvector satisfies the orthonormality and closure conditions. The coupling coefficients are given by... 

However, although the locally scaled transformed wavefunctions preserve the orthonormality condition, they fail to comply with Hamiltonian orthogonality. Of course, one can recombine the transformed wavefunctions so as to satisfy the latter requirement, by solving once more the eigenvalue problem... 

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