Ground state energy description

The way in which the spin factor modifies the wave-mechanical description of the hydrogen electron is by the introduction of an extra quantum number, ms = Electron spin is intimately linked to the exclusion principle, which can now be interpreted to require that two electrons on the same atom cannot have identical sets of quantum numbers n, l, mi and rns. This condition allows calculation of the maximum number of electrons on the energy levels defined by the principal quantum number n, as shown in Table 8.2. It is reasonable to expect that the electrons on atoms of high atomic number should have ground-state energies that increase in the same order, with increasing n. Atoms with atomic numbers 2, 10, 28 and 60 are... 

By addition these two inequalities yield E+Ex external potentials cannot generate the same electron density. But E is uniquely determined by the external potential and hence one deduces that the ground-state energy is a unique functional of the electron density p(r). This result, known as the Hohenberg-Kohn theorem, was assumed in all the early work on the density description (see refs. 4 and 16). Of course, the problem of finding the energy functional remains to date it is a matter of judicious approximation for the problem under consideration. 

An alternative model description of dication energy curves has been given by several groups 11 17>. Based directly on the VB method, this model attributes the origin of the barrier in the ground state energy curve of diatomic dications AB2+ to an avoided crossing between a purely repulsive covalent... 

The one-dimensional quadratic potential V = kx2 has been used for the description of covalent binding. The ground-state wave functions for a simple harmonic oscillator, /t and iR, have been used to describe the proton in the left and right wells. The force constant k has been determined from the stretch-mode vibrational transitions for water occurring at 3700 cm-1. The ground-state energy for the proton is 0.368 x 10-19 J. The tunneling barrier is AE = 4 x 10-19 J. 

The conclusion of these comments is that the trial functions and the results reported in [3] do not provide a reliable description of the two-dimensional hydrogen atom ground-state energy for confinements by an angle and by a hyperbola. The solutions of the Schrodinger equation for the hydrogen atom confined by a hyperbola can also be constructed transparently and accurately using standard methods. 

The Density Functional theorem states that the total ground state energy is a unique functional of the electron density, p [40]. This simple but enormously powerful result means that it is possible, in principle, to provide an exact description of all electron correlation effects within a one-electron (i.e. orbital-based) scheme. Khon and Sham (KS) [41] have derived a set of equations which embody this result. They have an identical form to the one-electron Hartree Fock equations. The difference is that the exchange-correlation term, Vxc, is not the same. 

In this first Appendix we consider the quantum field theoretical description of noninteracting spin-1/2 particles. In particular, we sununarize the quantization procedure, emphasizing the close relation between the minimum principle for the ground state energy and the renormalization scheme. At the same time this Appendix provides the background for the field theoretical treatment of the KS system, i.e. Eqs.(50)-(66). 

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