Hamilton matrix

Hamilton operator or Hamilton matrix (general, electronic, nuclear)... 

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... 

To get more insight into the various effects seen in table 2, the different influences of the excitations on Ahave to be studied. Going from a S-CI to a SD-CI treatment, the double excitations can influence Ais0 in two ways [20]. A direct effect arises from the coefficients of the double excitations themselves, which are not contained in the S-CI wavefunction. A second influence of the double excitations on Aiao is of a more indirect nature. Due to interactions within the SD-CI Hamilton matrix between configurations already included in the S-CI and the double excitations, the coefficients of the RHF determinant and of the singly excited determinants obtained by a SD-CI treatment differ from those obtained by the S-CI treatment. From these differences in... 

S-WF denotes the wavefunction including the RHF determinant and all single excitations, SD-WF is used if all double excitations are also included, etc. Accordingly, the expression S-CI describes the Cl treatment in which the RHF and the single excitations are included in the Hamilton matrix, etc. For further explanation, see text. 

Hamilton operator or Hamilton matrix (general, electronic, nuclear) Matrix element of a Hamilton operator between Slater determinants Exchange type matrix elements in semi-empirical theory x,y = s,p,d) Summation indices for occupied MOs ... 

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. 

The direct Cl method was proposed in 1972 by Roos. The idea of this method is to avoid the explicit construction and storage of the large Hamilton matrix. Instead, the eigenvectors are found iteratively. The basic operation in each iteration is to form the vector g = H c directly from the molecular integrals and the trial vector c. The optimum algorithm to form this product... 

This implies that for the electronic ground state E is the lowest eigenvalue of the Hamilton matrix... 

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 ... 

As examples for the simplifications in the Hamilton matrix elements we consider the contributions of the doubly external integrals to the operators pjp.uq (197)) Depending on the number of excitations from the closed shells one can distinguish six distinct cases. Using the relations 7f 0> = 7f 0> = 0 and the anticommutation rules in Eqs (6)-(8) one obtains the following results (the all-internal integral contributions are omitted). 

HR AND HI NOW CONTAIN THE TRANSFORMED HAMILTON MATRIX APPROPRIATE FOR A STANDARD EIGENVALUE PROBLEM ... 

For the three-center terms of the valence-valence Hamilton matrix we obtain... 

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