Coulomb potential, electronic kinetic energy

The standard quantum chemical model for the molecular hamiltonian Hm contains, besides purely electronic terms, the Coulomb repulsion among the nuclei Vnn and the kinetic energy operator K]. The electronic terms are the electron kinetic energy operator Ke and the electron-electron Coulomb repulsion interaction Vee and interactions of electrons with the nuclei, these latter acting as sources of external (to the electrons) potential designated as Ve]q. The electronic hamiltonian He includes and is defined as... 

Here f is the electron kinetic energy operator, VTe and Vpe are the potential operators for the interactions (coulombic or effective) between the electron and, respectively, the target and the projectile centers. 

This is the single-electron operator including the electron kinetic energy and the potential energy for attraction to the nuclei (for convenience, the single electron is indexed as electron one). The two-electron operators in eq. (2.4) are defined as the Coulomb, J... 

The configuration coordinates of electrons (p) and nuclei (R) in the new frame are related to the laboratory one by rk = u + pk, Qk. = u + Rk, symbolically written as r =u+p, Q=u+R, and T=(p,R). Ke represents the electrons kinetic energy operators Vee (p), VeN(p, R) and Vnn(R) are the standard Coulomb interaction potentials they are invariant to origin translation. The vector u is just a vector in real space R3. Kn is the kinetic energy operator of the nuclei, and in this work the electronic Hamiltonian He(r Z) includes all Coulomb interactions. This Hamiltonian would represent a general electronic system submitted to arbitrary sources of external Coulomb potential. 

Holas and March [11], This physical description further distinguishes between the Pauli and Coulomb correlation components and the correlation-kinetic-energy component of the potential, and thereby provides insights into its structure and of its components. In addition the interpretation helps distinguish between the Pauli, Coulomb and correlation-kinetic-energy contributions to the electron-interaction energy functional. A consistent physical description for both the energy functional and its functional derivative is thereby achieved. 

The basics of DFT are embodied in Eq. 14.54. The total energy is partitioned into several terms. Each term is itself a functional of the electron density. is the electron kinetic energy term (the Bom-Oppenheimer approximation is in place, so nuclear kinetic energy is neglected). The E potential energy term includes both nuclear-electron attraction and nuclear-nuclear repulsion. The term is sometimes called the Coulomb self-interaction term, and it evaluates electron-electron repulsions. It has the form of Coulomb s law. The sum of the first three terms (E + -I- ) corresponds to the classical energy of the charge distribution. 

Just as for an atom, the hamiltonian H for a diatomic or polyatomic molecule is the sum of the kinetic energy T, or its quantum mechanical equivalent, and the potential energy V, as in Equation (1.20). In a molecule the kinetic energy T consists of contributions and from the motions of the electrons and nuclei, respectively. The potential energy comprises two terms, and F , due to coulombic repulsions between the electrons and between the nuclei, respectively, and a third term Fg , due to attractive forces between the electrons and nuclei, giving... 

This expression is exact within our original approximation, where we have neglected relativistic effects of the electrons and the zero-point motions of the nuclei. The physical interpretation is simple the first term represents the repulsive Coulomb potential between the nuclei, the second the kinetic energy of the electronic cloud, the third the attractive Coulomb potential between the electrons and the nuclei, and the last term the repulsive Coulomb potential between the electrons. 

To understand how Kohn and Sham tackled this problem, we go back to the discussion of the Hartree-Fock scheme in Chapter 1. There, our wave function was a single Slater determinant SD constructed from N spin orbitals. While the Slater determinant enters the HF method as the approximation to the true N-electron wave function, we showed in Section 1.3 that 4>sd can also be looked upon as the exact wave function of a fictitious system of N non-interacting electrons (that is electrons which behave as uncharged fermions and therefore do not interact with each other via Coulomb repulsion), moving in the elfective potential VHF. For this type of wave function the kinetic energy can be exactly expressed as... 

The first two tenns in Eq. (2) represent the kinetic energy of the nuclei and the electrons, respectively. The remaining three terms specify the potential energy as a function of the interaction between the particles. Equation (3) expresses the potential function for the interaction of each pair of nuclei. In general, this sum is composed of terms that are given by Coulomb s law for the repulsion between particles of like charge. Similarly, Eq. (4) corresponds to the electron-electron repulsion. Finally, Eq. (5) is the potential function for the attraction between a given electron (<) and a nucleus (j). 

As indicated, we shall denote electrons and nuclei with Roman (/) and Greek (a) indices, respectively. In terms of kinetic-energy operators for electrons ( e) and nuclei (7k) and the Coulombic potential-energy interactions of electron-electron (Dee), nuclear-nuclear (UNN), and nuclear-electron (VW) type, we can write the supermolecule Hamiltonian as... 

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