Integrals nuclear-electron attraction

The Fock integrals first encountered in equation (A.45) are constructed from kinetic energy integrals, nuclear-electron attraction integrals, and two-electron repulsion integrals, as follows, continuing from equation (A.49) ... 

The nuclear-electron attraction integrals are collected as the matrix V" whose elements are defined by... 

The above equation implicitly defines our use of pA. Conversely, vAB is an integral over the nuclear-electron attraction operator where the electrons belong to fragment B and the nuclei to fragment A, and it is therefore given by... 

Cusachs reported (4) that the repulsive terms in the W-H model which assumes that electron repulsion and nuclear repulsion cancel nuclear-electron attraction, consist of one-electron antibonding terms only. Cusachs noted Ruedenberg s observation that the two-center kinetic energy integral is proportional to the square of the overlap integral rather than the first power. Cusachs used this to develop the approximation ... 

Because variational Coulomb fitting with charge neutrality involves a combination of electron-electron repulsion and nuclear-electron attraction Coulomb integrals, we consider the following prototypical combination of those integrals... 

The first sum is taken over all atomic nuclei in the molecule and ft v) denotes the nuclear-electron attraction integrals on the given atomic basis. The nuclear-electron attraction energy between given two atomic species (atoms A and B) in the molecule is expressed as follows ... 

The term "coulomb integral for a is unfortunate since the same name is used for repulsion integrals of the form f xi(1)X2(2)(1/ 12)xi(1)X2(2)< u. The quantity a also contains kinetic energy and nuclear-electronic attraction energy. 

Check first the interaction operator in scheme (i). It is very simple to evaluate the Hamiltonian for the dissociated system from Eq. (15.9). Since the orbitals do not move with R, only the nuclear-electron attraction integrals change ... 

In this sense, electronic spin-spin contact integrals can be computed as simple bilinear functions of overlap integrals, and electron repulsion integrals are expressible as bilinear fimctions of some sort of two electron nuclear attraction integrals. 

It remains to specify the elements of the one-electron core Hamiltonian, Hj y, containing the kinetic energy and nuclear attraction integrals. 

The P matrix involves the HF-LCAO coefficients and the hi matrix has elements that consist of the one-electron integrals (kinetic energy and nuclear attraction) over the basis functions Xi - Xn - " h matrix contains two-electron integrals and elements of the P matrix. If we differentiate with respect to parameter a which could be a nuclear coordinate or a component of an applied electric field, then we have to evaluate terms such as... 

The two potential energy integrals, electron repulsion and nuclear attraction, are derived quite similarly. Both involve the Gaussian transformation... 

---