Exchange-Correlation Energy

The reason the Hartree-Fock theory gives worse answers than the Hartree theory is simply that there is another piece of physics, which we are still ignoring. To some extent it cancels out with the exchange effect and so when we use the Hartree approach (i.e. we ignore both effects) we obtain reasonable results. On the other hand the Hartree-Fock approach includes the exchange effect but ignores the other effect,

which balances it somewhat, completely. This new effect is the electrostatic correlation of electrons.

Ignoring the Pauli exclusion principle generated exchange hole for the moment, we can also visualize a second type of hole in the electron distribution caused by simple electrostatic processes. If we consider the region immediately surrounding any electron (spin is now immaterial) then we should expect to see fewer electrons than the average, simply because of their electrostatic repulsion. Consequently each electron is surrounded by an electron-depleted region that known either the Coulomb hole (because of its origin in the electrostatic interaction) or the correlation hole (because of it origin in the correlated motion of the electrons). Just as in the case of the exchange hole the electron-depleted region is slightly positively charged. The effect of the correlation hole is twofold. Thus any other interaction effects, such as exchange, will tend to be reduced by the correlation hole [Stephen, 1997].

Clearly we can now see why the Hartree-Fock approach fails for solids: firstly the exchange interaction should be screened by the correlation hole rather than acting in full, and secondly the binding between the correlation hole and electron has been ignored. At this point, the Hartree-Fock approach gives quite creditable results for small molecules. This is because there are far fewer electrons involved than in a solid, and so correlation effects are minimal compared to exchange effects.

2.1.5 Hohenberg & Kohn Theorem

The discussion above has established that direct solution of the Schrödinger equation is not currently feasible for systems of interest in condensed matter science – this is a major motivation for the development and use of density functional theory.

DFT is based upon a fundamental observation that the total energy of an assembly of atoms is a function of the total electron charge density [Hohenberg and Kohn, 1964], which is a function of space and time. The electron density is used in DFT as the fundamental property unlike Hartree-Fock theory, which deals directly with the many-body wave function. Using the electron density significantly speeds up the calculation. Whereas the many-body electronic wave function is a function of 3N variables (the coordinates of all N atoms in the system) the electron density is only a function of x, y, z only three variables.

In 1964 Hohenberg and Kohn proved the two theorems:

1. For a non-degenerate ground state Ψ of the system the external potential V^ext(r) is determined, within a trivial additive constant, as a functional of the electronic density n(r).

2. Given an external potential Vext(r), the correct ground-state density n(r) minimizes the ground-state energy E0, which is a functional uniquely determined by n(r). It holds,

[ ]

E

E _v ~

0 ≤ (2.9)

where n~(r) is any trial density fulfilling n~(r)≥0 and

∫

They considered the ground state of the system to be defined by that electron density distribution which minimizes the total energy. Furthermore, they showed that all other ground state properties of the system (e.g. lattice constant, cohesive energy, etc) are functional of the ground state electron density. That is, that once the ground state electron density is known all other ground state properties follow (in principle, at least).

The theorem – which has a remarkable short proof (Fig. 2.1) – guarantees the existence of an energy functional E[n] that reaches its minimum for the correct density n(r) yet gives no explicit prescription for its construction [Stephen, 1997]. In order to determine E[n] it is useful to separate the various known contributions to the total energy, like Ts[n], the kinetic energy of a non-interacting electron gas, Eext[n], the classical Coulomb energy of the electrons moving in the external potential Vext(r), and

ECoul[n], the classical energy due to the mutual Coulomb interaction of the electrons:

[ ]

[ ]

[ ]

[ ]

[ ]

E = _s + _ext + _Coul + _xc (2.10)

The last term Exc[n] contains the quantum-mechanical exchange and correlation energy and – in principle – the difference between the true kinetic energy, T[n], and

Ts[n], the kinetic energy of the gas of non-interacting Kohn-Sham electrons. But since this difference is very small it is typically neglected.

2.1.6 Kohn-Sham Equations

Due to the second part of the H&K theorem, namely that the total energy is minimized by the true ground-state density, the variational principle can now be utilized. With the standard functional derivatives and the additional definition of the so-called exchange-correlation potential,

[ ]

the following set of equations can be derived

)

where the effective potential – as a functional of the electronic density – is given by

[ ]

∫

[ ]

and the electronic density as

∑

The set of equations (2.12) to (2.14) are the famous Kohn-Sham (KS) equations. They have to be solved self-consistently, i.e., starting from some initial density a potential veff [n(r)] is obtained for which the Eq. (2.12) are solved and a new electronic density Eq. (2.14) is determined. From the new density an updated effective potential can be

calculated and this process is repeated until self-consistency is reached, i.e., until the new electronic density equals the previous one [Schöne].

In fact, the K-S equations give an exact description of the many-electron system since up to this point no approximations have been made. Nevertheless, the method has reasonable precision from the past experience. First-principle DFT methods can currently predict binding energies to within a tenth of an electron volt and bond lengths to within 0.02 Å. It is becoming relatively straightforward to use this method to analyze the kinetics of relevant surface processes, including adsorption, chemical reaction and diffusion. However, there exists an exchange-correlation functional Exc

due to electrons in general DFT, which often requires some ad hoc assumptions (e.g., local density approximation (LDA) assuming uniform electron gas, [Seminario and Politzer, 1995]) to close the problem.

The success of the density functional theory (DFT) in reproducing measurable physical quantities of many-electron systems is rather remarkable, and may be attributed to the very fact that in the configuration space where exists a density distribution that is unique, and corresponding to the lowest energy state, no matter how complex and populous the system might be. There are, however, physical properties that will require the phase space information, at the minimum the pair correlation in order to quantify the electron-electron interaction energy.