Electron nucleus attraction energy

The first term in Eq. [3] is the kinetic energy operator, the second term is the electron-nucleus attraction energy operator and the third term stands for the electron-electron repulsion energy operator. In HF theory Eq. [2] is solved by means of the orbital approximation (1) the many-electron wavefunction P is represented with a determinant of one-electron wavefunctions r, (Slater determinant) satisfying the antisymmetric behavior of the many-electron wavefunc-... 

The first and the second integrals in equation (2.16) are equal to the sum of the kinetic energy and the electron-nucleus attraction energy for electrons 1 and 2 respectively. Each of the two integrals is equal to (Z — IZZ ). The third integral in (2.16) represents the electron-electron repulsion energy and is equal to 1.25 Z R. 

The electron-nucleus attraction energy is calculated as a three-dimensional integral. 

The electron-nucleus attraction energy is related by the virial theorem to the mean value of r (the electron-nucleus separation) for the different kinds of electrons. Fig. 15.6 shows (non-relativistic) Hartree-Fock solutions for the radiail distribution P (r) for plutonium as a typical actinide. The abscissa is chosen as (in atomic units) in order to show more detail at small r and less at large r. The... 

Note that actually consists of four contributions the electron-electron Coulomb energy y e o i, the electron-nucleus attraction energy, denoted V, its dual and the nucleus-nucleus repulsion, y. Summing these four terms yields the full electrostatic interaction between two atoms A and B, or... 

The first two terms are the kinetic energy and the potential energy due to the electron-nucleus attraction. V HF(i) is the Hartree-Fock potential. It is the average repulsive potential experienced by the i th electron due to the remaining N-l electrons. Thus, the complicated two-electron repulsion operator l/r in the Hamiltonian is replaced by the simple one-electron operator VHF(i) where the electron-electron repulsion is taken into account only in an average way. Explicitly, VHF has the following two components ... 

The GEM force field follows exactly the SIBFA energy scheme. However, once computed, the auxiliary coefficients can be directly used to compute integrals. That way, the evaluation of the electrostatic interaction can virtually be exact for an perfect fit of the density as the three terms of the coulomb energy, namely the nucleus-nucleus repulsion, electron-nucleus attraction and electron-electron repulsion, through the use of p [2, 14-16, 58],... 

Even in atoms in molecules which have no permanent dipole, instantaneous dipoles will arise as a result of momentary imbalances in electron distribution. Consider the helium atom, for example. It is extremely improbable that the two electrons in the Is orbital of helium will be diametrically opposite each other at all times. Hence there will be instantaneous dipoles capable of inducing dipoles in adjacent atoms or molecules. AnothCT way of looking at this phenomenon is to consider the electrons in two or more "nonpolar" molecules as synchronizing their movements (at least partially) to minimize electron-electron repulsion and maximize electron-nucleus attraction. Such attractions are extremely short ranged and weak, as are dipole-induced dipole forces. The energy of such interactions may be expressed as... 

In Section 1.4, we discussed the history and foundations of MO theory by comparison with VB theory. One of the important principles mentioned was the orthogonality of molecular wave functions. For a given system, we can write down the Hamiltonian H as the sum of several terms, one for each of the interactions which will determine the energy E of the system the kinetic energies of the electrons, the electron-nucleus attraction, the electron-electron and nucleus-nucleus repulsion, plus sundry terms like spin-orbit coupling and, where appropriate, other perturbations such as an applied external magnetic or electric field. We now seek a set of wave functions P, W2,... which satisfy the Schrodinger equation ... 

C(>I1J does not change because the SCF procedure refines the electron-electron repulsion (till the field each electron feels is consistent with the previous one), but H le in contrast represents only the contribution to the kinetic energy plus electron- nucleus attraction of the electron density associated with each pair of basis functions [Pg.212]

In Equation 6.12, the first term represents the kinetic energy of the electrons, the second term represents the Coulomb electron-nucleus attraction, and the third term represents the electron-electron repulsion. Note further that the present form of Hel does not address relativistic effects (which are neglected throughout this chapter), and that it refers to time-independent states of the molecule (as is also assumed throughout this chapter, and which is indicated implicitly by omitting the time variable t in all expressions and equations). 

Even when confining the variation of the trial wavefunction to the LCAO-MO coefficients c U, the respective approximate solution of the Schrodinger equation is still quite complex and may be computationally very demanding. The major reason is that the third term of the electronic Hamiltonian, Hel (Equation 6.12), the electron-electron repulsion, depends on the coordinates of two electrons at a time, and thus cannot be broken down into a sum of one-electron functions. This contrasts with both the kinetic energy and the electron-nucleus attraction, each of which are functions of the coordinates of single electrons (and thus are written as sums of n one-electron terms). At the same time, orbitals are one-electron functions, and molecular orbitals can be more easily generated as eigenfunctions of an operator that can also be separated into one-electron terms. 

Electrostatic electron-nucleus attractive Coulomb interaction energy ... 

In Eq. (3.2), nie and Wn represent the electron mass and the mass of the nucleus, respectively V is the potential energy corresponding to the electron-nucleus attraction ... 

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