Attraction, electron-nuclear

Note the stnicPiral similarity between equation (A1.6.72) and equation (Al.6.41). witii and E being replaced by and the BO Hamiltonians governing the quanPim mechanical evolution in electronic states a and b, respectively. These Hamiltonians consist of a nuclear kinetic energy part and a potential energy part which derives from nuclear-electron attraction and nuclear-nuclear repulsion, which differs in the two electronic states. 

Consider now the Hamilton operator. The nuclear-nuclear repulsion does not depend on electron coordinates and is a constant for a given nuclear geometry. The nuclear-electron attraction is a sum of terms, each depending only on one electron coordinate. The same holds for the electron kinetic energy. The electron-electron repulsion, however, depends on two electron coordinates. 

As mentioned in the start of Chapter 4, the correlation between electrons of parallel spin is different from that between electrons of opposite spin. The exchange energy is by definition given as a sum of contributions from the a and /3 spin densities, as exchange energy only involves electrons of the same spin. The kinetic energy, the nuclear-electron attraction and Coulomb terms are trivially separable. 

Relativistic corrections to the nuclear-electron attraction (V e) are of order 1 /c (owing to the much smaller velocity of the nuclei) and are normally neglected. 

If this is combined with the classical expression for the nuclear-electron attractive potential and the electron-electron repulsive potential we have the famous Thomas-Fermi expression for the energy of an atom,... 

Here v(x) denotes the external potential of the molecule for an isolated molecule in the absence of external electric fields, this is simply the potential due to nuclear-electron attraction. 

The sharing of electrons between two atoms is called a covalent bond. Such bonds owe their stability to the interaction of the shared electrons with both positive nuclei. The nuclei will be separated by a certain distance — termed the bond distance -that maximizes the nuclear-electron attractions balanced against the nuclear-nuclear repulsion. A molecule is a neutral species of two or more atoms held together by covalent bonds. 

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... 

Vc, an exchange-correlation term, Exc(p), and an external potential, [V , which arises primarily from nuclear-electron attraction but could include extramolecular perturbations, such as electric and magnetic fields. If the electronic wave function were expressed as a determinantal wave function, as in HF theory, then a set of equations functionally equivalent to the HF equations (A.40) emerges [324]. Thus... 

Different pieces in Eq. (12) can be chosen to be the above perturbation potential, resulting in different LR functions,293"301 which are closely related to the second functional derivatives of corresponding EDF s. For example, the (static) external LR function of only the nuclear-electron attraction potential is given by... 

On the right in Fig. 32 is an electron-domain representation of Lin-nett s model of an Octet-Rule satisfying atom in field-free space. For domains of (i) fixed size and distribution of charge, (ii) fixed distances from the nucleus, and (iii) fixed tetrahedral disposition with respect to other domains of the same spin-set, three of the four contributions to the total energies of the two structures in Fig. 32 are identical, namely the energies arising from (i) electronic motion, (ii) nuclear-electron attractions, and (iii) electron-electron repulsions between electrons of the same... 

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... 

It is also necessary to consider the electrostatic nuclear-electron attraction or nuclear-nuclear repulsion effects brought about by the various substituents and which may differ greatly from one substituent to another. 

Vne (nuclear-electron attraction potential) - -Vnn (nuclear-nuclear repulsions) +... 

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 ... 

---