Hamiltonian matrix operator

The eigenvalues of Eq. (14.24) are identical to the positive energy eigenvalues of Eq. (14.13) if exact decoupling has been achieved. The two-component electrons-only one-electron Hamiltonian matrix operator is a matrix function of only four matrices. 

We now show what happens if we set up tire Hamiltonian matrix using basis functions i ), tiiat are eigenfiinctions of Fand with eigenvalues given by ( equation A1.4.5) and (equation Al.4.6). We denote this particular choice of basis fiinctions as ij/" y. From (equation Al.4.3). (equation A1.4.5) and the fact that F is a Hemiitian operator, we derive... 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

Yarkoni [108] developed a computational method based on a perturbative approach [109,110], He showed that in the near vicinity of a conical intersection, the Hamiltonian operator may be written as the sum a nonperturbed Hamiltonian Hq and a linear perturbative temr. The expansion is made around a nuclear configuration Q, at which an intersection between two electronic wave functions takes place. The task is to find out under what conditions there can be a crossing at a neighboring nuclear configuration Qy. The diagonal Hamiltonian matrix elements at Qy may be written as... 

The evaluation of the action of the Hamiltonian matrix on a vector is the central computational bottleneck. (The action of the absorption matrix, A, is generally a simple diagonal damping operation near the relevant grid edges.) Section IIIA discusses a useful representation for four-atom systems. Section IIIB outlines one aspect of how the action of the kinetic energy operator is evaluated that may prove of general interest and also is of relevance for problems that require parallelization. Section IIIC discusses initial conditions and hnal state analysis and Section HID outlines some relevant equations for the construction of cross sections and rate constants for four-atom problems of the type AB + CD ABC + D. 

Operating on the triplet wave functions as before, we get the Hamiltonian matrix ... 

As in the previous section, by connected we mean all terms that scale linearly with N. Wedge products of cumulant RDMs can scale linearly if and only if they are connected by the indices of a matrix that scales linearly with N transvec-tion). In the previous section we only considered the indices of the one-particle identity matrix in the contraction (or number) operator. In the CSE we have the two-particle reduced Hamiltonian matrix, which is defined in Eqs. (2) and (3). Even though the one-electron part of scales as N, the division by A — 1 in Eq. (3) causes it to scale linearly with N. Hence, from our definition of connected, which only requires the matrix to scale linearly with N, the transvection... 

The Hamiltonian matrix in Equation (15) is obtained from appropriate products of representations of second-quantized operators that act within the left block, right block, or partition orbital. For example, in the case of where... 

The application of G-spinor basis sets can be illustrated most conveniently by constructing the matrix operators needed for DCB calculations. The DCB equations can be derived from a variational principle along familiar nonrelativistic lines [7], [8, Chapter 3]. It has usually been assumed that the absence of a global lower bound to the Dirac spectrum invalidates this procedure it has now been established [16] that the upper spectrum has a lower bound when the trial functions lie in an appropriate domain. This theorem covers the variational derivation of G-spinor matrix DCB equations. Sucher s repeated assertions [17] that the DCB Hamiltonian is fatally diseased and that the operators must be surrounded with energy projection operators can be safely forgotten. 

The different sets of coefficients that cause to satisfy Eq. 1, and the energy that corresponds to each wave function can be obtained by computing the energies of the interactions between each pair of configurations in Eq. 5, due to the Hamiltonian operator, H. If these interaction energies are displayed as a matrix, the coefficients that result in the MC wave function in Eq. 5 satisfying Eq. 1 are those that diagonalize this Hamiltonian matrix. 

The majority of polarizability calculations use the FFT, perhaps primarily because it is easy to incorporate into standard SCF computer programs in the presence of a perturbation A which is a sum of one-electron operators, the Hartree-Fock SCF hamiltonian hF—h+G(R)z becomes h+A + G R), and the SCF equations are solved by any standard technique. Thus, all that is involved is to add an extra array into the Hartree-Fock hamiltonian matrix hF every iteration. The method can be extended to higher polarizabilities, and a review by Pople et al.73b gives a good introduction to the method, including a discussion of the computational errors likely to be involved. 

The primitive VB model is defined in terms of overlap and Hamiltonian matrix elements over the basis states of eqn. (2.1.3). For fixed there are 2N possible spin-product functions so that this gives the dimension of the model s space. Indeed (though not originally formulated in this manner) the model may be mathematically represented entirely in spin space, despite the fundamental spin-free nature of the interactions. One may introduce a spin-space overlap operator by integrating out the spin-free coordinates... 

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