Local Hamiltonian matrix

Fleming, P. R., and Hutchinson J. S. (1988), Representation of the Hamiltonian Matrix in Non-Local Coordinates for an Acetylene Bond-Mode Model, Comp. Phys. Comm. 51, 59. 

The total potential energy of the localized, VB-like configuration in solution is thus computed by solving the seeular equation by diagonalizing the Hamiltonian matrix, if, to yield... 

The local ground-state correlation potential is defined in RDFT as the functional derivative of Eq.(7) with respect to p. When infinitesimal variation of occupation numbers is allowed, a more practical definition follows from the fact that the unsymmetrical energy formuala used to construct Eq.(7) is itself a Landau functional of the occupation numbers [19]. Correlation energies of Landau quasiparticles, expressed as diagonal elements of a one-electron Hamiltonian matrix, are defined by differentiating with respect to occupation numbers to give... 

For a quantum center where the perturbing electric field is almost constant, at least neglecting local atomic interactions typically described by short-range potentials such as the Lennard-Jones one, we can write [26] the perturbed Hamiltonian matrix H of the quantum center on the Born-Oppenheimer (BO) surface as... 

Hamiltonian operators. We shall see that with such approximate operators, we shall be able to deduce new and larger groups, which shall permit to introduce additional simplifications into the Hamiltonian matrix solutions. These new groups will be called Local Restricted Non-Rigid Groups [21,22]. The idea of using approximate Hamiltonian operators was already for warded by Bunker [8. ... 

The symmetry eigenvectors of any local NRG may be advantageously used as basis functions in a pre-diagonalization of the complete Hamiltonian matrix. For this purpose, the local NRG has only to be a larger group than the complete NRG. In the present case we have for example ... 

In the present paper, symmetry eigenvectors which factorize the Hamiltonian matrix into boxes are given for the single rotation in phenol (24) for double rotations in benzaldehyde (29), pyrocatechin (34) and acetone (44-46), for double rotation and inversion in non-planar pyrocatechin (40) and pyramidal acetone (49-51). In the same way, symmetry eigenvectors deduced in the local approach are deduced for some of these non-rigid systems (79), (83), and (89). Symmetry eigenvectors for the double internal Czv rotation in molecules with frame of any symmetry are given in reference [36]. 

We may now to point out that the use of Local Groups, more S3munetric than the complete groups, permits a still larger simplification. As an example, the above Hamiltonian matrix for acetone will be factorized in 20 boxes, the largest of which will be of order 60, if the local NRG (95) is used. 

For moderate-sized molecules with tens of vibrational modes, vibrational energy flow is conveniently described in a vibrational quantum number space. A statistical theory for the vibrational Hamiltonian, called Local Random Matrix Theory (LRMT), exploits the local coupling in the state space. LRMT predicts... 

After constructing the Kohn-Sham potential, one must construct the electron density, p(r ), the Hamiltonian matrix, Eq. (86), and the overlap matrix, Eq. (83). Because the basis functions are localized and the Kohn-Sham Hamiltonian is a local operator [cf. Eq. (91)], most of the matrix elements... 

In molecular calculations one-valence electron PPs and CPPs for alkaline atoms have often been tested for homo- and heteronuclear (neutral and mono-positive) dimers [186,203,220,228-230] as well as for (neutral and monopositive) monohydrides [137,186,203]. Since a total of one or two valence electrons is present, exact results within a given one-particle basis set are easily available by means of diagonalising the one-particle Hamiltonian matrix or standard CI(SD) calculations. Table 6 lists results obtained with the fully-relativistic PPs and CPPs discussed above. Again, the steepest cutoff-function augmented by a local potential (CPP III) is seen to yield the best results. In general calculations with large-core PPs tend to yield too strongly bound molecules,... 

The coupling of the local microstates into the proper molecular states is provided by the diagonalisation of the spin Hamiltonian matrix. For the zero-field case the diagonalisation matrix represents an orthogonal transformation and its matrix elements relate to the combination of the Clebsch-Gordan coefficients. 

The calculation of the matrix elements (38) and (39) is for small elementary cells the most time-consuming part of the (R)FPLO approach. For the overlap matrix S, one- and two-center integrals have to be provided while the Hamiltonian matrix requires the calculation of one-, two- and three-center integrals. As both the orbital and potential functions involved are well localized, only a limited number of multi-center integrals have to be calculated. The one- and two-center-integrals are further simplified by the application of angular momentum rules to one- and two-dimensional integrations, respectively. There are however two points which make the calculation of these matrix elements (in principle) much more involved for the relativistic approach. At first, the... 

If we start from the Hamiltonian operator (3.40), in which we formally replace Cuj (2) with M,2, the eigenvalue problem for the first two local modes is expressed in terms of the Hamiltonian matrix,... 

As emphasized, one of the advantages of this model is that it provides explicit wavefunctions which can be used in the computation of expectation values for various operators of interest. Due to limitations of space, we cannot reproduce here the complete set of vibrational wavefunctions obtained in the HCN calculation [76]. However, the typical outcome of the algebraic procedure can be outlined. We obtain a polyad of levels labeled by the numbers Vj and Ig of Eq. (4.56). Each polyad contains a number of local states, such as those listed in Eq. (4.57). The numerical diagonalization of the Hamiltonian matrix is performed separately for each polyad. Thus the eigenvectors derived represent the vibrational wavefunctions in the local basis. A possible outcome of the analysis of the HCN molecule could therefore be given by the following sequence of numbers ... 

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