Diagonal local approximation Hamiltonian

This approach was employed by Peralta et al. [649,661] and explored in detail by Thar and Kirchner [668] for the low-order DKH method. We refer to it as the diagonal local approximation to the Hamiltonian (DLH). It is clear that the DLH approximation can be applied to all relativistic exact-decoupling approaches. Obviously, the DLH approximation will work best at large interatomic distances. For example, it is a good approximation for heavy-atom molecules in solution for which it was conceived in Ref. [668]. Then, A represents the group of atoms that form one of the solute and/or solvent molecules. 

The harmonic approximation consists of expanding the potential up to second order in the atomic or molecular displacements around some local minimum and then diagonalizing the quadratic Hamiltonian. In the case of molecular crystals the rotational part of the kinetic energy, expressed in Euler angles, must be approximated, too. The angular momentum operators that occur in Eq. (26) are given by... 

A straightforward idea for a local approximation to the heavy-heavy blocks is to transform only the heavy-atom block-diagonal blocks of the relativistic Hamiltonian matrix H,... 

The obstacle to simultaneous quantum chemistry and quantum nuclear dynamics is apparent in Eqs. (2.16a)-(2.16c). At each time step, the propagation of the complex coefficients, Eq. (2.11), requires the calculation of diagonal and off-diagonal matrix elements of the Hamiltonian. These matrix elements are to be calculated for each pair of nuclear basis functions. In the case of ab initio quantum dynamics, the potential energy surfaces are known only locally, and therefore the calculation of these matrix elements (even for a single pair of basis functions) poses a numerical difficulty, and severe approximations have to be made. These approximations are discussed in detail in Section II.D. In the case of analytic PESs it is sometimes possible to evaluate these multidimensional integrals analytically. In either case (analytic or ab initio) the matrix elements of the nuclear kinetic energy... 

In this contribution, within the asymptotic approach, we have elaborated the basis for the calculation either of adiabatic channel potentials (diagonalization of the full Hamiltonian in a body-fixed frame at given interfragment distances) or of axially-nonadiabatic channel potentials (diagonalization of the full Hamiltonian in a space-fixed frame at given interfragment distances). As a by-product, we have compared our asymptotic PES on different levels of approximations with available local ab initio data. In subsequent work, we envisage the calculation of low temperature rate constants for complex-formation of the title reactions. 

As mentioned earlier, it is possible to calculate the interference signal directly from (3.5) without resorting to adiabatic or locally quadratic approximations, by diagonalizing the Longuet-Higgins model Hamiltonian with the standard procedure [10]. Figure 1 shows the exact interfer-... 

In FDE-ET we seek a method capable of computing the Hamiltonian matrix in the basis of charge-localized states generated with FDE. First, we have to define the needed matrix elements. As diabatic states are not the eigenfunctions of the molecular Hamiltonian, the off-diagonal elements of such Hamiltonian are not zero and can be approximated by the following formula [114, 115] if and y/ are slater determinants representing the donor and acceptor diabats ... 

In order to derive consistent expressions for relativistically transformed off-diagonal Hamiltonian blocks, we need to consider atomic approximations to the ingredients of the transformation matrices. As a first option, a local construction scheme for the matrix representation of the X-operator can be considered. 

The one-electron part of the Hamiltonian can be approximated by keeping only diagonal elements h. This corresponds to the neglect of charge transfer terms of type a an (m n) which is consistent to the localization of electrons at the sites. Then the one-electron part reduces to ... 

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. 

Cortona developed a method to calculate the electronic structure of solids by calculating individually the electron density of atoms in a unit cell with a spherically averaged Hamiltonian as the local Hamiltonian. The tests of the method have been successful for alkali halides where the density around each nucleus can be well approximated by a spherical description. Goedecker proposed a scheme closely related to the divide-and-conquer approach. The local Hamiltonian is also constructed by truncation in the atomic orbital space. Instead of the matrix diagonalization for the local Hamiltonian described in equation (34) in the divide-and-conquer approach, Goedecker used an iterative diagonalization based on the Chebyshev polynomial approximation for the density matrix. Voter, Kress, and Silver s method is related to that of Goedecker with the use of a kernel polynomial method. 

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