Electron-nuclear potential

The terms on the right-hand side of eq. (11.41) denote the kinetic energy, the electron-nuclear potential energy, the Coulomb (J) and exchange (K) terms respectively. Together J and K describe an effective electron-electron interaction. The prime on the summation in the expression for K exchange term indicates summing only over pairs of electrons of the same spin. The Hartree-Fock equations (11.40) are solved iteratively since the Fock operator / itself depends on the orbitals iff,. 

Ia) is included in the electronic Hamiltonian since, as we shall see, its most important effects arise from interactions involving electronic motions. The interactions which arise from electron spin, 30(5 ), will be derived later from relativistic quantum mechanics for the moment electron spin is introduced in a purely phenomenological manner. The electron-electron and electron-nuclear potential energies are included in equation (2.36) and the purely nuclear electrostatic repulsion is in equation (2.37). The double prime superscripts have been dropped for the sake of simplicity. We remind ourselves that // in equation (2.37) is the reduced nuclear mass, M M2/(M + M2). 

It is of considerable importance to note that the density-potential relationship (3) of the TF theory follows from a variational principle for the total energy. To see this, we note first that the classical electrostatic potential energy U consists of the sum of two terms in an atomic ion, the electron-nuclear potential energy Ken and the electron-electron potential energy Kee. We can write... 

Figure 5 Quantity 2Vee+ Ven with Vtn the electron-nuclear potential energy against total kinetic energy T for light molecules. Energies are in Hartree units...

As in the case of atoms, one can determine the individual components of the energy from this expression. Thus, using Feynman s theorem one can obtain the electron-nuclear potential energy as... 

The one-electron part of the Hamiltonian has two terms the kinetic energy and the electron-nuclear potential energy. We can treat these two terms for a single electron... 

Figure A10.8 The contributions to the three terms for the electron nuclear potential energy difference between H2 in the first excited state, with the electron in 2a ) and H integrated over planes perpendicular to the molecular axis, (a) The interaction of the density described by each s,) basis function with the corresponding nuclear centre (thin line) is compared with the same interaction for the isolated atoms scaled by a factor of 0.5 (dashed line) the difference gives a negative contribution to the bond formation energy (thick line), (b) The interaction of the density described wholly by the Si> basis function with the potential due to the H2 nucleus the corresponding integral for IS2) with H, is shown as a dotted line, (c) The integral value versus z from the interaction of the overlap density with H the interaction with Hi is shown as a dotted line.

Figure A10.9 The various terms in the electron-nuclear potential energy plotted as a function of Internuclear separation for (a) H2 in the ground state, with theelectron in 1

The potential energy of the nuclear motion is defined through the quadratic part of internuclear potential plus some additive term representing the selfconsistent influence of electron-nuclear potential... 

The only term in the Hamiltonian that refers to the nuclear positions, and hence to orientation in space, is the electron-nuclear potential-energy function (atomic units)... 

This simplification occurs since, in this case, the potential-energy operator scales in a simple fashion characteristic of the Coulomb potential when the coordinates of all particles (electrons and nuclei) are uniformly scaled. It is the failure of the electron-nuclear potential-energy operator (4.2.62) to scale in a simple fashion in response to the electron coordinate scaling that introduces the scaling-force term in the molecular electronic virial theorem. 

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