Kohn-Sham Hamiltonian, matrix element

In calculating matrix elements of the Kohn-Sham Hamiltonian of Eq. (2), the greatest problem is posed by the electronic Coulomb repulsion. To render this term tractable, it is convenient to cast the electron density p(r) in a model form, so as to calculate the potential by one-dimensional integrations. This is accomplished by approximating p by a multicenter overlapping multipolar expansion pu [37] ... 

In order to tackle large and complex structures, new methods have recently been developed for solving the eleetronie part of the problem. These are mostly applied to the pseudopotential plane wave method, because of the simplicity of the Hamiltonian matrix elements with plane wave basis functions and the ease with which the Hellmann-Feynman forces can he found. Conventional methods of matrix diagonalization for finding the energy eigenvalues and eigenfunctions of the Kohn-Sham Hamiltonian in (9) can tackle matrices only up to about 1000 x 1000. As a basis set of about 100 plane waves per atom is needed, this restricts the size of problem to... 

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... 

With the basis functions cp. (A + G, E, r), a variational solution is sought to the Kohn-Sham equation, equation (B3.2.4). Since the Hamiltonian matrix elements now depend nonlinearly upon the energy due to the energy-dependent basis functions, the resulting secular equation is solved by finding the roots of the determinant of the E) - E E) matrix. (The problem cannot be treated by the eigenvalue routines of linear algebra.)... 

To solve the Kohn-Sham equations with pseudopotentials, the standard approach is to expand the electron wavefunctions by a plane wave set in reciprocal space lattice vectors. The electron structure is obtained by diagonalization of the Hamiltonian matrix. This basis set has been mostly employed for semiconductor studies because of the relatively smooth pseudopotentials and delocalized electron wavefunctions of these systems. There are several advantages for using plane waves. The Hamiltonian matrix elements are simple to evaluate. Test of convergence in the basis expansion can be done by simply increasing the number of plane waves used. Moreover, the calculation of Hellmann-Feynman forces is the less involved in a plane wave basis. 

In this scenario, diabatic states can be generated with FDE by performing at least two simulations, one featuring a hole/electron on the donor while the acceptor is neutral and one calculation in which the charge hole/electron is on the acceptor. The result is two charge localized states, whose, densities and Kohn-Sham orbitals are used in a later step in order to build the diabatic Hamiltonian and overlap matrices, needed to compute the diabatic coupling matrix element. 

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