where ρ, n and ψ are the density, number and wavefunction of electrons respectively. To realize the theory, we firstly consider a box of stationary and uniform electrons. States must be occupied in turn from the ground state to the highest occupied state correspond to the wavenumber kF
which is related to the density,
3π2ρ = k3F, (2.7)
and the kinetic energy of the electrons is
T = Ω π2
¯h2
10mk2F, (2.8)
where Ω is the volume of the box. Now we go back to the real system. A non-uniform system is seen locally uniform. The density and wavenumber are functions of the position. Thus the kinetic energy of electrons within the Thomas-Fermi theory is
T[ρ] =
Here we must notice that this kinetic term is associated with the non-interacting electrons. The Coulomb potential energy of the nucleus-electron and the electron-electron interactions are
Unu−e[ρ] = A many-body Schrodinger equation can successfully be simplifed into a one-body Schrodinger equation. It is a large step of achievement that the equation is simple enough to be solved. How-ever, the accuracy of the method above is limited. Because the kinetic energy term is oversim-plified that it consider the electrons as the non-interacting Fermi gas, the exchange-correlation of electrons are completely neglected. There are interactions between electrons respect to spin called the exchange interaction. It lowers the total energy further, since it makes electrons more apart. The physist Dirac tried to add an exchange term into the equation, but still failled because of that the kinetic term which see electrons as a uniform gas is conceptually a big error, and the correlation of electrons is completely neglected.
2.3 Density Functional Theory
Although the concept of the functional of the electronic density can be traced back to the Thomas-Fermi theory in 1927 [4], the confirmed theoretical foundation, the Hohenberg-Kohn Theorem, is not proposed till 1964 by Hohenberg and Kohn [1]. However, it does not provide a way of finding the electronic density in the ground state. By employing the Kohn-Sham equation
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proposed in 1965 [2], the ground state density is able to be found and the density functional theory gains the incomparable achievement in physics and chemistry from this moment on. The complete discussion will be in the following subsections.
2.3.1 Hohenberg-Kohn Theorem
There are two theorems proposed and proved by Hohenberg and Kohn as follows.
The First Hohenberg-Kohn theorem
The first theorem demonstrates that the Hamiltonian and all the properties of the system are uniquely determined by the electron density. The proof is derived in the following. Assume that there are two different external potentials generated by a set of nuclei, V1(r) and V2(r), we would have two different Hamiltonians, H1 and H2, each leads to the same ρ(r) for its ground state, but the two corresponding wavefunctions, ψ1(r) and ψ2(r), are different. Now we take ψ2
as a trial wave function for the problem of the Hamiltonian H1, E1< hψ2|H1| ψ2i = hψ2|H2| ψ2i + hψ2|H1− H2| ψ2i = E2+ Z
ρ (r)[V1(r) −V2(r)]dr, (2.12) where E1 and E2 are eigenenergies of H1 and H2 respectively. Similarly, we take ψ1as a trial wave function for the problem of the Hamiltonian H2,
E2< hψ1|H2| ψ1i = hψ1|H1| ψ1i + hψ1|H2− H1| ψ1i = E1+ Z
ρ (r)[V2(r) −V1(r)]dr. (2.13) Adding Eq. (2.12) and (2.13), we would obtain a contradictory relation,
E1+ E2< E2+ E1. (2.14)
The first Hohenberg-Kohn theorem is proved that no two different external potentials can give the same electron density for the ground state. The total energy is a unique functional of the electron density which can be written as
E[ρ] = T [ρ] +Un−e[ρ] +Ue−e[ρ], (2.15) where T and Un−e are kinetic energy and nucleus-electron Coulomb potential energy. And the Ue−ehere that differs from the term of Thomas-Fermi theory is the energy related to the electron-electron interaction including the electron-electron-electron-electron Coulomb repulsion, exchange interaction and Coulomb correlation.
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As follows , the second Hohenberg-Kohn theorem, the variation principle, enable us to make sure that the ground state density is what we are looking for. Recall that any trial electron density ρ1defines its own Hamiltonian H1 and hence its own wavefunction ψ1. The wavefunction can now be taken as the trial wavefunction for the Hamiltonian H0generated by the external potential Vext. Thus,
hψ1|H0| ψ1i = T [ρ1] +Unu−e[ρ1] +Ue−e[ρ1] = E[ρ1] ≥ E[ρ0] = hψ0|H0| ψ0i = E0, (2.16) the functional E[ρ] has its minimum relative to variations δ ρ of the density at the equilibrium density ρ0,
δ E[ρ ]
δ ρ |ρ =ρ0= 0. (2.17)
2.3.2 Kohn-Sham Equation
The kinetic energy has a large contribution to the total energy. The failure of the Thomas-Fermi theory is due to that it makes the wrong approximation of the term of kinetic energy.
The density functional theory, therefore, was ignored until Kohn and Sham introduced a method which treated the majority of the kinetic energy exactly. The theory begins by considering the non-interacting reference system that all electrons move in an effective potential with no interaction, in other words, a mean field single particle picture. The ground state is a Slater determinant of orthogonal orbitals ψiwhich satisfies the equations
[− ¯h2
2m∇2+Ve f f(r)]ψi(r) = Ei(r)ψi(r) with
n i=1∑
|ψi(r)|2= ρ(r), (2.18)
where Ve f f is the effective potential including the nucleus-electron, electron-electron Coulomb term and the ill-defined exchange-correlation term, and n is the number of electrons. Due to the Hohenberg-Kohn theorem, the kinetic energy and total energy are given by
T[ρ] =
The functional T [ρ] is just a particular case, the ground state electron density now can equiva-lently be obtained by the solution of th Euler-Lagrange equations,
0 = δ where the Lagrange multiplier µssatisfies the constraint that the density integrates to the correct number of electrons. It is very important to realize that if the exact form of exchange-correlation term in the effective potential can be defined, the Kohn-Sham strategy would lead to the exact
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energy. This will be briefly discussed in the next section.
2.3.3 Exchange-Correlation Energy
Since the work of Kohn and Sham, the remaining question is the available form of the exchange-correlation term of the effective potential,
Ve f f(r) = Vext(r) +
where EXCis the exchange-correlation energy. Some successful methods of exchange-correlation approximation like the local density approximation (LDA) [5] and generalized gradient approx-imation (GGA) [6] are briefly introduced in the following. By the way, the Perdew-Burke-Ernzerhof exchange-correlation functional (PBE) [7] is used in our calculations.
The Local Density Approximation (LDA)
Local density approximation [5], the basis of all approximate exchange-correlation function-als, is the simplest and the most widely used one. The central idea of this approximation is the well-defined limit of exchange-correlation energy of the uniform electron gas. A non-uniform system can be seen as a locally uniform system, and the exchange-correlation energy is
EXCLDA[ρ(r)] = Z
ρ (r)εXC[ρ(r)]dr, (2.22)
where εXC[ρ(r)] is the exchange-correlation energy per electron of an uniform electron gas of density ρ(r). The quantity εXC[ρ(r)] can be further seperated into two parts, exchange and correlation terms,
εXC[ρ(r)] = εX[ρ(r)] + εC[ρ(r)]. (2.23) The exchange part was originally derived by Bloch and Dirac where
εX[ρ(r)] = −3
4(3ρ(r)
π )1/3. (2.24)
However, there has no explicit form for the correlation part, but the numerical quantum Monte-Carlo simulations of the uniform electron gas with good accuracy. It is able to introduce the spin to the LDA (LSDA), and the form becomes
EXCLSDA[ρ↑(r), ρ↓(r)] = Z
ρ (r)εXC[ρ↑(r), ρ↓(r)]dr. (2.25) The accuracy of the L(S)DA is limited. Only materials with slowly varying electron density can be describe well. However, some bulk and surface of solids exist the rapid varying density.
For some materials that have strongly correlated d or f electrons, L(S)DA even predicts them to
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be metallic instead of insulating ground state. It tends to underestimate the ground state energy, band gap and the ionization energies, but overestimates the bonding energy.
The Generalized Gradient Approximation (GGA)
To promote the accuracy, not only ρ(r) but also the gradient of ρ(r) are considered to ac-count the exchange-correlation energy since the fact that the electron density of a real system is non-uniform. The form of exchange-correlation energy in the generalized gradient approxima-tion [6] can be expressed as
EXCGGA[ρ↑(r), ρ↓(r)] = Z
ρ (r)εXC[ρ↑(r), ρ↓(r), ∇ρ↑(r), ∇ρ↓(r)]. (2.26) The GGA has reduced the errors of LDA, but the variation of electron density still must be slow enough.