Local density functional Hamiltonian matrix elements

Several approaches are available in the literature to generate and evaluate Hamiltonian matrix elements with wavefunctions of charge-localized, diabatic states. They differ in the level of theory used in the calculation and in the way localized electronic structures are created [15, 25, 26, 29-31]. When wavefunction-based quantum-chemical methods are employed, the framework of the generalized Mulliken-Hush method (GMH) [29, 32-34], is particularly successful. So far, it has been used in conjunction with accurate electronic structure methods for small and medium sized systems [35-37]. As an alternative to GMH and other derived methods [38, 39], additional methods have been explored for their applicability in larger systems such as constrained density functional method (CDFT) [25, 37, 40, 41], and fragmentation approaches [42-47], which also include the frozen density embedding (FDE) method [48, 49]. 

In the case of delocalized basis states tpa(r), the main matrix elements are those with 0 = 7 and f3 = 6, because the wave functions of two different states with the same spin are orthogonal in real space and their contribution is small. It is also true for the systems with localized wave functions tpa(r), when the overlap between two different states is weak. In these cases it is enough to replace the interacting part by the Anderson-Hubbard Hamiltonian, describing only density-density interaction... 

The RASSI method can be used to compute first and second order transition densities and can thus also be used to set up an Hamiltonian in a basis of RASSCF wave function with separately optimized MOs. Such calculations have, for example, been found to be useful in studies of electron-transfer reactions where solutions in a localized basis are preferred [43], The approach has recently been extended to also include matrix elements of a spin-orbit Hamiltonian. A number of RASSCF wave functions are used as a basis set to construct the spin-orbit Hamiltonian, which is then diagonalized [19, 44],... 

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... 

The use of the Lanczos recursive method to define the local Hamiltonian and to calculate the density matrix is a common feature shared by several groups. Baroni and Giannozzi suggested a method based on a finite-difference representation of the Hamiltonian, and a recursive Green s function approach to calculate the electron density in real space." For the density at each point, the truncated finite length of the recursion defines the local Hamiltonian for that point. Aoki et al. constructed a bond-order potential method which determines each density matrix element with a recursion in the Green s function. ... 

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