Primitive basis function matrix elements

Next, we shall consider four kinds of integrals. The first is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at that nucleus. The second is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at a different point (usually another nucleus). Then, we will consider the matrix element of a Coulomb term between two primitive basis functions at different centers. The third case is when one basis function is centered at the nucleus considered. The fourth case is when both basis functions are not centered at that nucleus. By that we mean, for two Gaussian basis functions defined in Eqs. (73) and (74), we are calculating... 

It is also possible to perform a basis set transformation from primitive basis functions to symmetry combinations of the KS MOs of the atoms or larger fragments that constitute a system. In that case the population matrix elements P v become more meaningful, because they reflect the involvement of the fragment MOs in the orbitals of the total system. A Mulliken population analysis in... 

The magnetic hyperfine interaction terms were given in equation (8.351) and the electric quadrupole interaction in equation (8.352). We extend the basis functions by inclusion of the 7Li nuclear spin I, coupled to J to form F the value of / is 3/2. We deal with each term in turn, first deriving expressions for the matrix elements in the primitive basis set (8.353), and then extending these results to the parity-conserved basis. All matrix elements are diagonal in F, and any elements off-diagonal in S and / can of course be ignored. 

It is now a simple matter to use the above results for the primitive basis functions to generate matrix elements for the parity-conserved basis. For the positive-parity states the hyperfine matrix is as follows. 

E Matrix elements of the truncated harmonic potential F Matrix elements in Gaussian primitive basis functions... 

We turn next to examine the general structure of the matrix elements of the Hamiltonian operator H in terms of the basis functions of the first zone. A discussion of the matrix elements of H in terms of the primitive basis functions is found in Appendix F. 

Matrix elements of the other primitive basis functions are readily obtained by differentiation. It is possible to demonstrate that the form of the matrix elements is very much the same as for the simpler matrix elements that arise with the Morse potential. Thus, one finds for... 

Va . Then, in terms of the ground state primitive basis functions, the single center matrix element is given by... 

The cell that contains the origin of coordinates for the whole system is used as the reference cell. The two-center matrix element that involves only the ground state primitive basis functions < >0 is... 

The two-center matrix elements in the second primitive basis functions are... 

F Matrix elements in Gaussian primitive basis functions... 

An expansion of the Morse potential, for example, in a set of Gaussian functions is given by eq (C4) in Appendix C. Matrix elements of the Morse potential in terms of the Gaussian primitive basis functions are therefore simply three center overlap integrals [49], These matrix elements can be evaluated for each term in the sum and then converted to the final expression in a straightforward manner. 

These equations are solved at each SCF iteration. Though the matrix A involves the electron density associated with a given iteration (via n eut), that contribution can be evaluated swiftly in terms of precomputed matrix elements with respect to the primitive basis functions for the orbitals and the charge fitting along with elements of the density matrix fP with respect to the orbital basis. Eqs. 1 and 5 combined give... 

Approximate linear dependence of AO-based sets is always a numerical problem, especially in 3D extended systems. Slater functions are no exceptions. We studied and recommended the use of mixed Slater/plane-wave (AO-PW) basis sets [15]. It offers a good compromise of local accuracy (nuclear cusps can be correctly described), global flexibility (nodes in /ik) outside primitive unit cell can be correct) and reduced PW expansion lengths. It seems also beneficial for GW calculations that need low-lying excited bands (not available with AO bases), yet limited numbers of PWs. Computationally the AOs and PWs mix perfectly mixed AO-PW matrix elements are even easier to calculate than those involving AO-AO combinations. 

The primitive VB model is defined in terms of overlap and Hamiltonian matrix elements over the basis states of eqn. (2.1.3). For fixed there are 2N possible spin-product functions so that this gives the dimension of the model s space. Indeed (though not originally formulated in this manner) the model may be mathematically represented entirely in spin space, despite the fundamental spin-free nature of the interactions. One may introduce a spin-space overlap operator by integrating out the spin-free coordinates... 

We shall calculate the matrix elements of the effective Hamiltonian within the basis of these six primitive states in due course. These states do not, however, have definite parities. Since parity is conserved (except in the presence of an applied electric field), we construct a basis set of six functions, three of each parity type, so that for a given J level we are left with the diagonalisation of 3 x 3 matrices, rather than 6x6. More importantly, we are aiming to understand the electric dipole radio frequency and microwave spectra, and know that transitions must occur between states of opposite parity. 

The simplest approach to the book-keeping problem of calculating all the nuclear spin-dependent matrix elements is first to evaluate all the terms in the primitive basis set, leaving only J, I and F as variables. We therefore construct the following 6x6 matrix, using the functions listed in (8.353). The rotational levels are widely spaced compared with the hyperfine terms, so that we also confine attention to matrix elements diagonal in J. The required 6x6 matrix is as follows. 

The single-center matrix element for the second primitive (first nominal excited state) basis function < >, is... 

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