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Quantum Chemistry Methods for Molecular Disordered Materials
Published in Alexander Bagaturyants, Vener Mikhail, Multiscale Modeling in Nanophotonics, 2017
Alexander Bagaturyants, Vener Mikhail
In the problems that require a more accurate consideration, the nondiagonal (cross) terms in Eq. 3.58, which are neglected in the adiabatic and BO approximations, are commonly treated as nonadiabatic corrections perturbatively or in a variational calculation. The coefficients in these cross terms are matrix elements between the electronic wave functions of a given electronic term l and all other electronic terms k. A matrix element of this type can be estimated from perturbation theoryg as ⟨Ψl∂∂QΨk⟩=⟨Ψl|∂VeN∂Q|Ψk⟩El(Q)-Ek(Q). $$ \langle {\mathbf{\Psi }}_{l} \frac{\partial }{{\partial Q}}{\mathbf{\Psi }}_{k} \rangle = \frac{{\langle {\mathbf{\Psi }}_{l} |\frac{{\partial V_{{eN}} }}{{\partial Q}}|{\mathbf{\Psi }}_{k} \rangle }}{{E_{l} (Q) - E_{k} (Q)}}. $$
Laser Excitation
Published in F.J. Duarte, Quantum Optics for Engineers, 2017
where Hmn is the matrix element of the Hamiltonian. For a simple diatomic molecule, the dependence of this matrix element on the Franck–Condon factor (qv″,v″) and the square of the transition moment (|Re|2) are described by Chutjian and James (1969).
Introduction to Lasers
Published in F.J. Duarte, Tunable Laser Optics, 2017
where: Hmnis the matrix element of the Hamiltonian
Analytic gradients for compressed multistate pair-density functional theory
Published in Molecular Physics, 2022
Jie J. Bao, Matthew R. Hermes, Thais R. Scott, Andrew M. Sand, Roland Lindh, Laura Gagliardi, Donald G. Truhlar
CMS-PDFT energies and wave functions are the solutions to an eigenequation for an effective Hamiltonian defined in a model space of multideterminantal wave functions: where where is a matrix element of the molecular electronic Hamiltonian and is the MC-PDFT energy expression: for the internuclear Coulomb repulsion; h and g respectively for the one- and two-electron Hamiltonian interaction elements; , , and respectively for the one-body reduced density matrix (1-RDM), density, and on-top pair density for state P; and for the on-top energy functional that provides the energetic contributions due to electron correlation and exchange.
High-order contact transformations of molecular Hamiltonians: general approach, fast computational algorithm and convergence of ro-vibrational polyad models
Published in Molecular Physics, 2022
Vladimir Tyuterev, Sergey Tashkun, Michael Rey, Andrei Nikitin
In the standard Rayleigh–Schrödinger perturbation theory, one often applies the Wigner theorem: if a wavefunction is known up to n-th order approximation , then the diagonal matrix element of the Hamiltonian over gives the energy valid to (2n + 1)-order approximation: . Using remarkable mathematical properties of the operations < … >, [6], the relations (9) and (12) and the formal solutions (18), one can extend this result to effective Hamiltonian transformations. For any decomposition (13a) satisfying the conditions of Section 2.2, the diagonal parts for many recurrent commutator contributions in (12) exactly vanish leading to: This means that n-times transformed Hamiltonian is directly expressed via results of [[n/2]] transformations only. This result helps economising a lot of computations, and the solutions (18) of CT equations take the form [6]
An alternative formulation of vibrational heat-bath configuration interaction
Published in Molecular Physics, 2021
Abuzar U. Bhatty, Kurt R. Brorsen
In this study, we construct a dictionary for each set of excitations involving the same number of modes, but possibly a different number of quanta excitations, which we then use to select additional Hartree products to include in the variational space. The keys to each dictionary are unique integer indexes that depend on the identity of the modes, the number of quanta in these modes, and the maximum mode excitation level. Each dictionary value is a list with list elements consisting of a tuple with two elements. The first element of the tuple contains another list of length equal to the number of modes in the excitation with each list element equal to the change in the number of quanta in the modes. This change can be either positive or negative. The second element of the tuple is the Hamiltonian matrix element for the excitation. The list for each dictionary value is sorted by the absolute magnitude of the Hamiltonian matrix element to allow the screening procedure to terminate whenever the selection criterion is not satisfied such that most possible excitations are never checked when enlarging the variational space.