Ground state energy, definition

Fig. 1. The ground state energies of a Z = 90 hydrogen-like atom obtainedfrom the Dirac (D) andfrom the Levy-Leblond (L) equations as functions of the nonlinear parameters. In the upper-rowfgures (Dl and LI) the a (abscissa) and (3 (ordinate) dependence ofE is displayed when L and S are set equal to the values corresponding to the exact solutions. In the lower-row figures (D2 and L2 ) the L (abscissa) and S (ordinate) dependence of E is displayed when a and 5 are set equal to the exact value. The arrows show directions of the gradient their length is proportional to the value of the gradient The solid line crossing the saddle corresponds to the functions andLl) or S = Sj (L) D2 and L2. For the definitions of thesefunctions see text.

To gain an understanding of this mechanism, consider the Hamiltonian operator (H — Egl) with only two-body interactions, where Eg is the lowest energy for an A -particle system with Hamiltonian H and the identity operator I. Because Eg is the lowest (or ground-state) energy, the Hamiltonian operator is positive semi-definite on the A -electron space that is, the expectation values of H with respect to all A -particle functions are nonnegative. Assume that the Hamiltonian may be expanded as a sum of operators G,G,... 

Enlarging the domain of definition of Eqn (1) to all positive n, one could assume the minimum of the density functional E lp], i.e. the ground-state energy Eo n) for a given external potential u(r), to be a continuous and even a differentiable function of the number of electrons n. From the Lagrange multiplier theory, it would further follow that... 

The energy of the three separated particles on the right side of Eq (8.1) is, by definition, zero. Therefore the ground-state energy of the helium atom is given by () = -(l + h) = -79.02 cV = -2.90372 hartrees. We will attempt to reproduce this value, as closely as po.ssiblc, by theoretical analysis. 

It is not yet possible to draw definite conclusions concerning this kind of approach to the theory of metals many more numerical computations will be necessary and eventually contributions of order A should also be retained in the ground-state energy. 

The motivation for the definitions (15.115), (15.116), and (15.117) is to express [p] in terms of three quantities, the first three terms on the right side of (15.118), that are easy to evaluate from p and that include the main contributions to the ground-state energy, plus a fourth quantity E, which, although not easy to evaluate accurately, will... 

Eq. (51), the i-th optimal, ground-state wavefunction q G True, generally speaking, the latter is the approximate ground-state wavefunction that yields an upper bound to the exact ground-state energy 0 which is attained, by definition, only at the Hohenberg-Kohn orbit that is, = Eq. 

Many years after, a new emerging form of quantum mechanics, the DFT, appears as the modem quantum frame in which a chemical system (an atom, an ion, a radical, a molecule or several molecules) can be treated in a state of interaction (Parr Yang, 1989). In this modem context, the cornerstone EN definition of Parr as the minus of the chemical potential ( ) of a system in a grand canonical ensemble at zero temperature (7) was formulated (in atomic units), see (Parretal., 1978),j = -/r asinEq. (3.1), when the ground state energy E is assumed to be a smooth function of the total number of electrons N. 

The differential definition more appropriate to atomic system is on the basis that for a system of N electrons with ground state energy E [N,v],... 

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