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The HF wavefunction defines a single configuration of the N electrons. By removing electrons from the occupied MOs and placing them into the virtual (unoccupied) MOs, we can create new configurations, new N-electron functions. These new configurations can be indexed by how many electrons are relocated. Configurations produced by moving one electron from an occupied orbital to a virtual orbital are singly excited relative to the HF configuration and are called singles; those where two electrons are moved are called doubles, and so on. A simple designation for these excited configurations is to list the occupied MO(s) where the electrons are removed as a subscript and the virtual orbitals where the electrons are placed as the superscript. Thus, the generic designation of a singles configuration is c_i^a or c_S, a doubles configuration is ψ_ij^ab or ψ_D, and so on. These configurations are composed of spin-adapted Slater determinants, each of which is constructed from the arrangements of the electrons in the various, appropriate molecular orbitals. Basically, there are two approaches: either to follow configuration interaction type methods (CI, MC SCF, CC, etc.), or to go in the direction of explicitly correlated functions. The first means a barrier of more and more numerous excited configurations to be taken into account, the second, very tedious and time-consuming integrals. In both cases, we know the Hamiltonian and fight for a satisfactory wave function (often using the variational principle). It turns out that there is also a third direction (presented in this article) that does not regard configurations (except a single special one) and does not have the bottle neck of difficult integrals. Instead, we have the kind of wavefunction in the form of a single Slater determinant, but we have a serious problem in defining the proper Hamiltonian. The ultimate goal of the DFT method is the calculation of the total energy of the system and the ground-state electron density distribution without using the wave function of the system. The DFT calculations (despite taking electronic correlation into account) are not expensive, their cost is comparable with that of the Hartree–Fock method. Therefore, the same computer power allows us to explore much larger molecules than with other post-Hartree–Fock (correlation) methods.

Density functional theory methods are ultimately derived from the Thomas-Fermi-Dirac model of the 1920s and Slater’s work in quantum chemistry in the 1950s. This approach is based on modeling electron correlation by general functionals of the electron density. This method is a semiempirical method that parametizes the equations to reproduce key experimental data. Most modern Density functional theories are based on the Hohengerg-Kohn theorem of the 1960s. A functional is a function of a function. This assumes that the ground state of a many electron atom or molecule can be exactly expressed as a functional of the electron density. Unfortunately, this theorem does not provide the exact form of the functional. In this method the electronic energy is partitioned into several terms:

E = E^T + E^V + E ^J + E ^XC (3)

In this equation, E^T is the kinetic energy of the electrons, E^V is the sum of the potential energy from the nuclear-electron attraction and the nuclear-nuclear repulsion and E^J is the electron-electron repulsion term. E^XCis the exchange-correlation term that contains the remaining part of the electron-electron interactions. These electron-electron interactions include the exchange energy, E^X, from the antisymmetry of the wave function and the dynamic correlation, E^C, in the motions of the individual electrons. All terms in Equation 3 except the nuclear-nuclear term are functions of the electron density (ρ).

The electron density is obtained from the coordinates of the electrons. The kinetic energy term can be expressed as a functional of the electron density but the general expression is complicated and not completely known. An expression for the kinetic energy of electrons in boxlike potentials has the following form:

T = 3/10 (3π²)^2/3 ∫ρ^5/3dτ (4)

This expression and improvements of this equation that take into account the gradient of ρ are rarely used for the electronic structure and the more traditional expression for the kinetic energy that involves wave functions is normally used.

T = -1/2 ∑∫ψi∇²ψi (5)

Both potential energy terms are written as functionals of the electron density and have the following form.

Vnucl = −ΣZαρ(1)/ r1αdτ1 (6)

Vrep = 1/2∫∫ρ(1) ρ(2)/ r12 d τ1d τ2. (7)

The exchange-correlation term is normally written as a sum of exchange functional and correlation functional:

E^XC(ρ ) = E^X (ρ ) + E^C (ρ ) (8)

Both components of the exchange-correlation term can be written as local functionals that depend on the electron density or gradient-corrected functionals that depend on the electron density and the gradient of electron density. The local density approximation (LDA) is used to indicate any density functional where the E^XC term at some position r can be calculated exclusively from the value of ρ at that position.

The local exchange functional that was developed to reproduce the exchange energy of a uniform free electron gas is of the following form:

E ^X_LDA = −(9α/8)(3/ π) ^1/3∫ ρ^4/3 d³r (9)

where α is an empirical constant for the type of system being described and has a value of 2/3 for a uniform free electron gas. This functional by itself is has problems when describing molecular systems. In 1988, Becke formulated the following gradient corrected exchange functional that improves the LDA functional:

E^X_Becke88= E^X_LDA −γ∫ρ^4/3 x²/(1 - 6γ sinh^-1 x) d³ r (10)

where x = ρ^4/3│∇ρ│ and γ is a parameter that is chosen to fit the known exchange energies of an inert gas atom.

The electron density may also be expressed in terms of an effective radius where one electron is contained within a sphere defined by a radius where it would have the same density throughout as its center.

r_s(r) =(3/4πρ(r))^1/3 (11)

The spin of the electrons is dealt with by using individual functionals for the α and β spins. The spin densities at any position are expressed in terms of the normalized spin density (ζ).

ζ(r) = (ρ^α(r) - ρ^β(r))/ ρ(r) (12)

ζ is zero everywhere for an unpolarized system (closed-shell system) and has a value between zero and one for a polarized system (open-shell). The spin density ρ(r) is equal to zero for a for a closed-shell system and is one for an open-shell system. The α spin density is one-half the product of the total spin density ρ and (ζ+1) and the β spin density is the difference between the total spin density and the α spin density. The LDA method can be extended to the spin-polarized regime by using:

ε_r[ρ(r),ζ] = ε_x^o[ρ(r)]+{ ε_x¹[ρ(r)] - ε_x^o[ρ(r)]}[((1 + ζ )^4/3 + (1- ζ)^4/3 -2)/(2(2^1/2 -1)))] (13)

where the superscript-zero exchange energy density is from Equation 9 with the appropriate value of α and the superscript 1 represents the analogous expression for a uniform free electron gas. This is known as the local spin density approximation (LSDA). The relevant theory for the correlation functional is from Vosko, Wilk, and Nusair (VWN). They designed a local functional that is dependent on rs.

ε_cⁱ(r_s)=A/2{ln(r_s/(r_s+b√r_s+c))+2b/√(4c-b²)tan^-1(√(4c-b²)/(2√r_s+b))-bx_o/(x_o²+bx_o+c)[ln((√r_s-x_o)²/(r_s+b√r_s+c))+2(b+2x_o)/√(4c-b²)tan^-1(√(4c-b²)/(2√r_s+b))} (14)

where there are different sets of empirical constants A, x_o, b, and c for i=1 and i=0.

Another type of approximation gives rise to a hybrid method called B-LYP/HF procedure. The mixing of DFT and HF methods developed this method. According to this procedure, the total energy at HF is first determined;

E_HF = E_T+ E_v+ E_j + E_k (15)

where E_j is the kinetic energy, E_v is the potential energy and E_j and E_k are Coulomb and exchange energy parts. The exchange energy E_k is replaced by an exchange-correlation functional from Becke and Lee-Yang-Parr approximations using the electron density from HF. For example, the B3LYP exchange-correlation functional is a hybrid functional and is written as follows:

E_B3LYP^XC = (1− a) E_LSDA^X + aE_HF^X + bΔE_B88^X + (1− c) E_LSDA^C + cE_LYP^C (16)

where a, b, and c were optimized to 0.20, 0.72, and 0.81 respectively.

This hybrid method has been widely tested and found to give good results comparable to those obtained from the MP2 method. However, the time taken for computations using this hybrid method is found to be very less compared to MP2 methods.

In conclusion, the correlation functional, E^C(ρ), has been formulated by Vosko, Wilk and Nusair. This work was based on quantum Monte-Carlo simulations of the uniform electron gas performed by Ceperley and Alder for a range of electron densities. The functional is designed to ensure that E^C(ρ), as defined previously, reproduces the quantum Monte-Carlo results; it is known as the VWN correlation functional. This formulation of the exchange and correlation functionals, which is called the Local Density Approximation, when applied to atoms and molecules via the Kohn-Sham equations it has been found that this approach is not particularly useful for quantum chemical applications, having accuracy, which is comparable with that of Hartree-Fock SCF theory. While the exchange and correlation functionals described above (such as B3LYP) allow DFT to give good descriptions of molecular energies, geometries and related properties, their forms, in particular the presence of fractional powers of the density, mean that the integrals involved cannot be calculated analytically. This necessitates the use of numerical quadrature with a three dimensional grid of points spanning the space of the molecule. Full details of the implementation of such schemes can be found in Reference 13. It is important to note that when such numerical procedures are employed for quantum chemical calculations of energies and their gradients the grids used must be sufficiently fine grained to guarantee adequate precision in the quantities of interest.