**2.3 Density Functional Theory (DFT)**

**2.3.3 Exchange-Correlation energy E xc [n]**

A commonly used scheme to find approximations for Exc[n] is the adiabatic connec-tion [18]. The relaconnec-tionship between the real system and the KS reference system can be studied by a continuum of fictitious systems. These systems with a relative strength of the electron-electron interactionλWˆee have the same electron density. Forλ= 1, one gets the real system, for λ = 0, the KS reference system. The Hamiltonian of the fictitious systems reads

Hˆ^{λ} = ˆT + ˆVext+λWˆee+ ˆV_{Hxc}^{λ} [n], (2.105)
where ˆV_{Hxc}^{λ} [n] is the Hartree-exchange-correlation potential operator that keeps the density
constant with variations of λ.

The exchange-correlation energy of the real system can be given exactly in terms of an integral over the coupling constant λ [18–20]

Exc[n] = 1 2

Z Z n(r)h_{xc}(r,r^{′})

|r−r^{′}| drdr^{′}, (2.106)
wherehxc(r,r^{′}) is the exchange-correlation hole averaged over all coupling constants, which
takes into account kinetic correlations

h_{xc}(r,r^{′}) = n(r^{′})
Z 1

0

dλ[g^{λ}(r,r^{′})−1], (2.107)
with g^{λ}(r,r^{′}) the pair correlation function of the system with density n(r) and
electron-electron interaction λWˆee. The function g(r,r^{′}) takes into account the fact that the
presence of an electron at r reduces the probability of finding a second electron at a
position r^{′}, in the vicinity of r.

One can define an exchange-correlation energy density per particle, εxc[n] = 1

2

Z h_{xc}(r,r^{′})

|r−r^{′}| dr^{′}, (2.108)
so that the exchange-correlation energy (2.106) can be expressed as

Exc[n] = Z

n(r)εxc[n(r)]dr. (2.109) Finding approximations of εxc[n] that allow for a better description of chemical sys-tems, is one of the most important aspects of research in DFT. Although there exists a large variety of exchange-correlation functionals, we can recast most of the current ap-proximations into 3 different categories: the Local, the Gradient Corrected and the Hybrid ones.

2.3.3.1 Local Density Approximation

The essence of the local density approximation (LDA) [17] is to treat real inhomogeneous electronic systems as locally homogeneous, and then to replace the exchange-correlation hole (2.107) by the one for an homogeneous electron gas (HEG), which is characterized by only one density

h^{LDA}_{xc} (r,r^{′}) = n(r)h

g^{HEG}(|r−r^{′}|, n(r))−1i

, (2.110)

where g^{HEG}(|r−r^{′}|, n(r)) is the pair correlation function of HEG, which depends only
on the distance between r and r^{′}, evaluated at n(r). The exchange-correlation energy
becomes

The exchange energy density ε^{HEG}_{x} can be given by the Slater-Dirac [21, 22] form
ε^{HEG}_{x} n

For spin-polarized systems, the local spin density approximation (LSDA) can be
ob-tained by using the two spin densities, n↑(r) and n↓(r), such that n(r) = n↑(r) +n↓(r),
andζ(r) = (n↑(r)−n↓(r))/n(r) is the spin polarization. The exchange-correlation energy
becomes
where the exchange energy (as the Kohn-Sham kinetic energy T_{s}[n↑, n↓]) obeys the exact
spin scaling relation [23]

Ex[n↑, n↓] = 1

2Ex[2n↑] +1

2Ex[2n↓]. (2.114)

However, the correlation energy has no simple spin scaling relation. The correlation
en-ergy densityε^{HEG}_{c} , as a function of spin-polarization, can be obtained by using expressions
based on either the Random-Phase Approximation [24] or on Quantum-Monte-Carlo
sim-ulations [25] and there are popular parametrizations for both methods [26–28].

The LDA works well for the systems where the electronic density is quite uniform such as bulk metals, however, it is a very crude approximation for the systems where the density varies rapidly in dependence of position r (e.g., atoms and molecules). It gives, in general, reasonable geometries for covalent, ionic, and metallic bonds while it yields too short bond lengths for weakly bound systems. The incorrect asymptotic behavior of the exchange-correlation potential (it decays exponentially, not as -1/r) affects the dissociation limit, ionization energies and metallic surfaces. LDA fails to cancel exactly the electronic self-interaction included in the Hartree term of the energy.

2.3.3.2 Generalized Gradient and Meta-Generalized Gradient Approxima-tions

The exchange-correlation energy in the Generalized Gradient Approximation (GGA), has
the following form:
where the function Fxc is constructed to satisfy a number of known conditions for the
exchange-correlation hole (e.g., sum rules, long-range decay, etc.). There are many
differ-ent prescriptions for choosingF_{xc}, each one leads to distinct GGA. Among the numerous
GGA functionals [29–46] here are some of the commonly used ones. The Becke 88
ex-change functional (B88) [30] which takes the form of an additional correction to the LDA
exchange. It contains one adjustable parameter which was optimized so that the sum of
the LDA and Becke exchange terms reproduces the exchange energy of six noble gas atoms.

The introduction of Becke’s gradient exchange correction in 1988 was an important step for the acceptance of DFT as a valuable tool for computational chemistry. However, B88 exchange functional is responsible for repulsive potential energy curves for many rare gas diatomic molecules [47]. The Lee-Yang-Parr (LYP) [29, 48] correlation functional which has been derived as a reformulation of the Colle-Salvetti [49] expression for the correla-tion energy of Helium, in term of n and ∇n. This correlation functional is one-electron self-interaction error free, but can not reproduce the correct uniform electron gas like the functionals based on (2.115). A popular recipe is to use B88 exchange complemented with LYP correlation. The Perdew-Burke-Ernzerhof (PBE) exchange-correlation func-tional [31] verifies many of the exact properties for the exchange-correlation hole and does not contain any fitting parameters. It retains the correct features of LDA and combines them with the inhomogeneity aspects that are supposed to be important for describing real systems. GGA sometimes overcorrects LDA for bond lengths, binding energies and also atomic energies. It cannot describe long-range effects. However, hydrogen bonds are usually well accounted for. GGA functionals still do not eliminate the spurious electron self-interaction completely.

The Meta-Generalized Gradient Approximations (meta-GGA) that go beyond GGA,
have been suggested to overcome the shortcomings of LDA and GGA methods. These
functionals employ, in addition to the density and its gradient, either the Laplacian of the
density (∇^{2}n(r)) or the kinetic energy density

τ(r) = 1

Different meta-GGA functionals have been developed [50–63], for example, the Perdew-Kurth-Zupan-Blaha (PKZB) [53] exchange-correlation functional which depends on two fitted parameters, the nonempirical Tao-Perdew-Staroverov-Scuseria (TPSS) [57] exchange-correlation functional. The TPSS exchange functional can be combined with the Krieger-Chen-Iafrate-Savin (KCIS) [64] correlation functional. These meta-GGA correlation func-tionals are one-electron self-interaction error free. The results obtained using meta-GGA are quite improved with respect to PBE, especially when atomisation energy and metal surface energy are considered.

2.3.3.3 Hybrids

Becke [65] employed the adiabatic connection to suggest that the functional should contain
some of the exact HF exchange^{1}

E_{x}^{HF}=−1
Letting Ψ^{λ}[n] be the unique antisymmetric ground state wave function of ˆH^{λ} (2.105), the
exchange-correlation energy can be expressed in terms of the coupling constant
integra-tion [66]
A possible scheme to construct functionals from the adiabatic connection is to approximate
the integrand Uxc,λ by a model function of λ [65, 67–69]. Integrating (2.118) using the
linear model [65], U_{xc,λ}[n] =a[n] +b[n]λ, gives rise to

Exc[n] = a+ 1

2b =Uxc,0 +1

2(Uxc,1 −Uxc,0), (2.119)
where Uxc,0 = E_{x}^{HF} is the exchange energy of the KS determinant. The potential
en-ergy contribution to the exchange-correlation enen-ergy of the fully interacting system was
approximated by Becke as U_{xc,1} ≈U_{xc}^{LDA}, leading to the half-and-half functional

Exc[n] = 1

2(E_{x}^{HF}+E_{x}^{LDA}) + 1

2U_{c}^{LDA}. (2.120)

The last term in (2.120) is often approximated as U_{c}^{LDA} ≈E_{c}^{LDA} in other versions of the
half-and-half functional. Successively, Becke [67] proposed to parametrize the functional
form to chosen sets of experimental data, leading to the family of the three-parameter
hybrid functionals

E_{xc}^{B3PW91}[n] = aE_{x}^{HF}+ (1−a)E_{x}^{LDA}+b∆E_{x}^{B88}+E_{c}^{V W N} +c∆E_{c}^{P W}^{91}. (2.121)
Alternatively, the most widely used B3LYP [70] reads as

E_{xc}^{B3LYP}[n] = aE_{x}^{HF}+ (1−a)E_{x}^{LDA}+b∆E_{x}^{B88}+cE_{c}^{LY P} + (1−c)E_{c}^{V W N}, (2.122)
where the parameters (a= 0.2,b = 0.72,c= 0.81) in (2.121) and (2.122) were determined
by a least square fit to atomization energies, electron and proton affinities, and the
ioniza-tion potentials of the atomic species and molecules in the G2 [71] test set of experimental
data. B3LYP is successful as a result of some cancellation of errors, despite its known
shortcomings [72]. Many single hybrid functionals have been suggested [46, 73–83]. Single
hybrid functionals are known to improve significantly the accuracy of DFT results in
par-ticular for geometries and thermochemistry. They are supposed to compensate partially
the many-electron self-interaction error by using the exact exchange [72]. So for kinetics,
good hybrid functionals tend to have a larger percentage of exact exchange.

1Computed with the Kohn-Sham orbitals and eigenvalues.

Range-separated single-hybrid functionals [84, 85] are characterized by the separation of the electron-electron interaction into two parts which may be treated separately

1 r12

= erf(µr12) r12

+erfc(µr12) r12

, (2.123)

where the parameterµcontrols the separation between the long-range (erf(µr_{12})/r_{12}) and
the short-range (erfc(µr12)/r12) interactions. The long-range corrected functionals [86–95]

in which we mix long-range HF with short-range DFT, are known to improve properties such as charge-transfer and Rydberg excitations in TDDFT. The screened functionals [96–

99] are obtained when we use short-range HF and long-range DFT, and these hybrids are more commonly used in solid-state calculations (band gaps, lattice constants, etc).

Besides the HF exchange, unoccupied orbitals and eigenvalues are also used through the perturbative double-excitation correlation in the double hybrid functionals. These functionals are the subject of two articles [100, 101] presented in this thesis, where the theoretical foundation of types of double hybrid functionals has been provided.

Another class of hybrids, where we combine MCSCF and DFT, was explored through an article [102] exposed in this thesis. This method is called multiconfigurational hybrid functional.