Hamilton matrix diagonalization

The Hamiltonian operator for the electric quadrupole interaction, 7/q, given in (4.29), coimects the spin of the nucleus with quantum number I with the EFG. In the simplest case, when the EFG is axial (y = Vyy, i.e. rf = 0), the Schrddinger equation can be solved on the basis of the spin functions I,mi), with magnetic quantum numbers m/ = 7, 7—1,. .., —7. The Hamilton matrix is diagonal, because... 

As has been shown in the foregoing subsection, the stationary states T,(r) and Pn(r) are solutions of the full Hamiltonian H with given energies , and . This means their Hamilton matrix is diagonal (the subscript I is used now to indicate all such stationary states characterized by Roman numbers I, II, III,...) ... 

In general, (1.9) must be solved numerically by quantum chemical or so-called ab initio methods (Lowe 1978 Szabo and Ostlund 1982 Daudel et al. 1983 Dykstra 1988 Hirst 1990 ch.2). The pointwise solution of (1.9) for a set of nuclear geometries and the fitting of all points to an analytical representation yields the PES which is the input to the subsequent dynamics calculations. In principle, one expands Ee (q Q) in a suitable set of electronic basis functions and diagonalizes the corresponding Hamilton matrix, i.e., the representation of Hei within the chosen basis of electronic wavefunctions. Since the number of electrons is usually large, even for simple molecules like H2O and C1NO, the solution of... 

From Eq. (5.22) it follows that commutes with/2 and/z and consequently is diagonal in the / and M quantum numbers. Hence the eigenvalues are independent of M and can be found for individual /-values, either by diagonalizing the Hamilton matrix, formed in an appropriate basis, or by numerical integration. The former method has been applied below. 

In defining the harmonic oscillator basis by Eqs. (5.23)—(5.25) the problem of scaling was postponed. Equation (5.27) shows that the three yet undefined parameters, D, E° and a, are interrelated so that they are all determined when anyone has been given a value. A reasonable estimate is most easily obtained for the spacing E°, which should be close to the mean spacing of the levels considered, in order to minimize the dimensions of the Hamilton matrix. Thus, in the present example it was found that with E° = 80 cm-1 the basis could safely be truncated at n = 39 corresponding to matrix dimensions 20 x 20 for the diagonal blocks of H. ... 

However, the theory may be subject to a more serious examination if we calculate the /-dependence of the energy levels directly from the Hamilton matrix as outlined in Sect. 5.6.2, i.e. we may diagonalize the individual/,/-blocks oiH and subsequently treat the /-type resonances by diagonalizing smaller matrices diagonal in v. 

For short Cl expansions one could simply construct and diagonalize the Hamiltonian matrices For longer Cl expansions a direct Cl procedure must be employed. Here the desired eigenvectors are obtained iteratively without explicitly calculating and storing the Hamilton matrix. The basic operation in a direct Cl iteration is the evaluation of the residual vectors ... 

The time-dependent Hamiltonian is separated into a time-independent part (diagonal elements Ha of the Hamilton matrix)... 

In the case of the sodium trimer the Hamilton matrix of the corresponding Schrodinger equation takes a 3 x 3 form. The diagonal elements Ha,... 

It is worth mentioning that the assignment of the many states of benzene above was made possible by using the. symmetry facilities of the VB code, which enable us to diagonalize the Hamilton matrix separately for each irreducible... 

The last equation shows that the unknown coefficients are those solutions which diagonalize the Hamilton energy matrix Hik. Hence, they can be found by a diagonalization procedure where for non-trivial solutions the condition... 

The energies of the unperturbed (sub)states have been defined above. Ha is the configuration interaction Hamiltonian, which describes essentially the electron-electron interaction [128]. The off-diagonal terms are taken as the perturbation. The aim of this discussion is, at least for this simple model, to present the structure of the corresponding perturbational formulas. For example, it will be shown that different energy denominators occur, which are connected to the different states involved. In this model, we neglect any diagonal contributions to the model Hamilton operator and treat the matrix elements as real for simplicity. 

In thermodynamic equilibrium the olf-diagonal density-matrix elements are zero. The diagonal elements are determined by the Boltzmann distribution in the eigenbasis of the Hamilton operator Hz,... 

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