Off-diagonal terms of the Hamiltonian

Nonadiabatic Feshbach calculations. Using the reaction-path Hamiltonian and invoking an adiabatic separation of the reaction coordinate from all other coordinates, resonance energies and adiabatic partial widths are obtained by neglecting all off-diagonal terms of the Hamiltonian. The most important... 

This formulation includes the lowest-order off-diagonal term of the Hamiltonian matrix, which leads to the production of singular points. However, these singular... 

The. operator X is defined by setting the off-diagonal term of the Hamiltonian to zero, resulting in the operator equation... 

In Hiickel theory, atomic basis functions are assumed to be orthonormal. The interaction between neighbouring functions is included in the off-diagonal term of the Hamiltonian. In the HL-CI framework, these two approximations are done among the local stmctures. 

The next set of approximations concerns the propagator and the evaluation of the matrix elements that are required for Eq. (2.11). At each time step and for each nuclear basis function, numerical integration of Eq. (2.11) requires diagonal and off-diagonal terms of the Hamiltonian matrix elements ((x[ //// x[,) and respectively). The need to evaluate these integrals... 

First, as the molecule on which the chromophore sits rotates, this projection will change. Second, the magnitude of the transition dipole may depend on bath coordinates, which in analogy with gas-phase spectroscopy is called a non-Condon effect For water, as we will see, this latter dependence is very important [13, 14]. In principle there are off-diagonal terms in the Hamiltonian in this truncated two-state Hilbert space, which depend on the bath coordinates and which lead to vibrational energy relaxation [4]. In practice it is usually too difficult to treat both the spectral diffusion and vibrational relaxation problems at the same time, and so one usually adds the effects of this relaxation phenomenologically, and the lifetime 7j can either be calculated separately or determined from experiment. Within this approach the line shape can be written as [92 94]... 

The implementation of the algorithm outlined above is somewhat delicate due to our use of the polar representation of the complex Hermite polynomials that project onto the final states (3 or / . When the complex polynomial is zero, the phase is ill-defined. This is reflected in the expression of the force in (44) by the apparent singularity in the off-diagonal terms. The existence of a divergence in the force, however, depends on the behavior of the gradients of the off-diagonal terms of the electronic Hamiltonian. As they usually are, or go to, zero very rapidly in regions of zero population of the final state, it is... 

Mclver has further analyzed the two-orbital case , which consists of three states in the full expansion. Because of the fact that the off-diagonal element of the Hamiltonian matrix in the two-term pair expansion is equal to an... 

The presence of the collective parameters Ti2 and ft 2 introduces off-diagonal terms in the Hamiltonian II and in the dissipative part of the master equation. This suggests that in the presence of the interaction between the atoms the bare atomic states are no longer the eigenstates of the two-atom system. We can diagonalize the Hamiltonian (32) with respect to the dipole-dipole interaction and find collective states of the two-atom system. 

The beauty of CVPT is that it provides a simple and rapid method whereby one can transform the Hamiltonian from an initial representation to a final one, where the molecular properties of interest are more readily calculated in the final representation than they are in the initial one. In traditional perturbative approaches, one represents the Hamiltonian in some well-known zero-order representation, this yielding a Hamiltonian matrix. One then uses the perturbative analysis to generate a matrix T, such that when one carries out the similarity transformation (cf. the top panel of Fig. 2) all of the contributions to the off-diagonal elements of the Hamiltonian matrix that were of order X before the transformation are of order X2 after the transformation. Although one can then proceed to higher order, here we consider a single transformation in order to obtain results of order X2. The contribution of Van Vleck was to develop his perturbative expressions in terms of S where T = exp(/XS) rather than work directly with the matrix T. 

Compared with the X2C approach, the BSS approach requires a few more matrix multiplications. The off-diagonal terms of the X2C Hamiltonian matrix H are diagonal matrices, whereas those in the BSS Hamiltonian matrix Hq are not. Furthermore, the BSS decoupling transformation matrices are composed of more terms than the X2C expression in Eq. (14.39). 

Configuration state functions (spin-adapted Slater determinants) constructed from excitation-adapted molecular orbitals (EAMOs) possess minimal off-diagonal elements of the Hamiltonian matrix. These orbitals, which result from separate unitary transformations among the occupied and virtual MOs. offer the most concise description of electronic excited states in terms of electron jumps . For example, at the CIS/6-31 +G level of theory, a symmetric combination of Just two singly excited configurations built from EAMOs [0.7049(7 — 9) -I-0.7049(6 -> 8)] suffices to adequately describe the first triplet excited state of N2, whereas several configurations involving MOs [0.5975 (7 9) -I- 0.5975(6 8) -t- 0.3646(7 16) -l-0.3646(6 15) -I- 0.0858(7 23) -I- 0.0858(6 22)J are... 

Equation (1-6) does not show when the off diagonal terms on the right hand side become important. To judge the importance of the nonadiabatic effects it is most convenient to use the perturbation theory37- 40. The Hamiltonian tK can be represented by the sum... 

The H , = Hoo contribution to the effective Hamiltonian Hef / contains only-scaled isotropic chemical-shift terms. The first-order correction to the effective Hamiltonian requires the evaluation of commutators between DD elements, CSA elements and cross-terms DDx CSA. We should remind ourselves that the basic justification for using the van Vleck transformation is that the off-diagonal elements of the interactions are small with respect to the differences between the diagonal elements (see Eqs. 48a and 48b). When that is the case... 

The effective hamiltonian representation of equations [8-10] is readily extensible to systems with more than one TM. The Hdd< matrix for such cases contains terms that account for d-electron delocalization (via the second term in Equation 9) from one metal to another - indirectly via the bridging ligands or directly via the corresponding off-diagonal terms of Hdd [33]. This interaction produces magnetic-exchange coupling, which is reviewed elsewhere [34, 35],... 

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