2.3 Quantum Mechanics
2.3.3 Density Functional Theory
2.3.3.2 The Exchange-Correlation Functional
The performance of DFT is reliant on the exchange-correlation functional, VXC, as
all other terms in the total energy expression are well defined. The choice of ap- proximation used for this functional is dependent on the material in question and the properties of interest. Below we outline two different approximations; the local density approximation (LDA) and the generalised gradient approximation (GGA). Methods have also been developed to treat the exchange-correlation functional us- ing the local parts of DFT (e.g. a LDA or GGA functional) combined with non-local quantum mechanical techniques (e.g. Hartree-Fock); this technique, known as Hy- brid DFT, will also be outlined briefly in this section.
Local Density Approximation
The local density approximation (LDA) is the simplest approach to approximating the exchange-correlation functional, proposed by Kohn and Sham[230] when out- lining their approximation for E[ρ(r)]. The approximation is based on the uniform electron gas model in which the electron density is constant throughout all space, which allows the exchange-correlation energy to be evaluated by integrating over all space.
EXCLDA[ρ(r)] =
Z
ρ (r)εXC(ρ(r))dr (2.44)
where εXCis the exchange correlation energy per electron as a function of the elec-
tron density of the uniform electron gas, for which there are very accurate formulas determined by Monte Carlo simulations[231] and conveniently parametrised.[232] It is assumed that the exchange correlation energy of the uniform electron gas, in which the electron density does not depend on position, can be applied to a non- uniform system where the electron density does depend on position.
The exchange correlation functional is obtained by differentiating equation 2.44 with respect to the electron density:
VXCLDA[r] = ρ(r)δ εXC(ρ(r))
δ ρ (r) + εXC(ρ(r)) (2.45) At each point r, in a system with electron density ρ(r), the values of εXC(ρ(r)) and
VXC[ρ(r)] have the same value as if point r were in a uniform electron gas. As the
value of εXCis known accurately for all densities of interest, it can be expressed in
an analytical form and solved computationally. It is usual practice to split εXC(ρ(r))
into its exchange contribution and correlation contribution and consider these sep- arately. There are a number of different ways to calculate these contributions, and the method implemented depends on the modelling package and the basis set. For solid state systems, plane waves are often used, as opposed to localised atomic basis sets which are common in molecular systems. Plane waves will be covered in more detail later in Section 2.3.5.
It is possible to generalise equation 2.44 for spin-polarised systems, leading to a local spin density approximation (LSDA):[233, 234]
EXCLSDA[ρ(r)] =
Z
ρ (r)εXC(ρ↑(r), ρ↓(r))dr (2.46)
where εXC(ρ↑(r), ρ↓(r)) is the exchange correlation energy per particle of a homo-
geneous, spin-polarised electron gas, where ρ↑ and ρ↓ refer to spin up and spin
LDA has been found to perform well for a wide range of materials and problems. In the context of solid state modelling, it is able to reproduce lattice constants rea- sonably well, however it fails when studying energetic properties - due to electron densities generally being strongly inhomogeneous in real materials. Therefore, al- ternative methods for calculating the exchange correlation functional have been de- veloped, including the generalised gradient approximation.
Generalised Gradient Approximation
Due to the inhomogeneity of electron density in a real system, it was proposed that the exchange correlation energy should depend on both electron density, ρ, and the gradient of the density, 5ρ, at each point in space.
EXCGGA[ρ(r)] =
Z
f(ρ, 5ρ) (2.47)
where f is a parameterised analytical function. The exchange-correlation functional is split and the exchange functional and correlation functional treated separately, as different variations with density are observed for each functional,
EXC= EX[ρ, 5ρ] + EC[ρ, 5ρ] (2.48)
The form each of these functionals will take depends on the type of GGA being used, which will differ by the approximations and assumptions used by the authors when developing it. One of the first GGAs was proposed by Langreth and Mehl [235, 236] which took the form:
EXC= (ELDARPA)XC+ a Z [5ρ(r)]2 [ρ(r)]4/3(2e −F−7 9)d 3r (2.49) where F= b| 5 ρ(r)| [ρ(r)]7/6 (2.50)
and a and b are constants. As equation 2.49 shows, the gradient approximation is treated as a correction to the exchange-correlation functional of LDA. Langreth and Mehl used the exchange-correlation functional from the random phase approxima-
tion (RPA) of LDA, noting that it was not appropriate to use any other LDA EXCas the generalised gradient approximation correction was derived based on this form of LDA and therefore a large amount cancellation between the two terms would be lost if an alternative form was used.[236]
There were a number of issues with the GGA proposed by Langreth and Mehl, the most problematic being the behaviour of the exchange potential which fails to re- produce the asymptotic behaviour of the exchange energy density with 1/r, making the approximation difficult to implement. Therefore, a number of adaptations were developed, including those by Becke [237] and Perdew and Wang[238–242]. The GGAs considered in the following work are PBE[243] and PBEsol.[244]
PBE and PBEsol The functional of Perdew-Burke-Ernzerhof (PBE)[243] was de- veloped as a simple GGA with all parameters in the GGA correction (i.e. all param- eters other than the LDA-based exchange-correlation functional) being fundamen- tal constants. It is based on the Perdew-Wang 1991 GGA (PW91)[238, 245] which was developed with the aim of keeping all the best features of the LDA exchange- correlation functional while incorporating some inhomogeneity effects.
The PBE exchange correlation functional takes the form:
EXCGGA[ρ(r)] =
Z
ρ (r)εXuni f(ρ(r))FXC(rs, ζ , s)d3r (2.51)
where εXuni f is the exchange energy per particle of the uniform electron gas and FXC
is an enhancement factor of the LDA uniform electron gas exchange, dependent on; rs, the Seitz radius, ζ , the relative spin polarisation and s, a density gradient. PBEsol[244] was developed by Perdew et al. in order to improve the equilibrium properties of closely packed solids, compared to those produced using PBE. In order to outline the differences between the two functionals it is necessary to present the exchange and correlation functionals of PBE separately.
Firstly, the exchange energy of PBE is presented as:
EXGGA[ρ(r)] =
Z
where euni fX (ρ(r)) is the exchange energy density of a uniform electron gas, and FX(s(r)) is the enhancement factor.[243]
For a GGA that recovers the uniform gas limit, such as PBE;
Fx(s) = 1 + µs2+ ... (s → 0) (2.53) For PBE, µ = 0.2195 which is roughly equal to 2µGE, where µGE is the gradient
expansion value, a parameter used when defining GGA-based functionals. PBEsol, however, uses µ = µGE; this value is accurate for slowly varying densities - such
as densities in solids - whereas the value used in PBE is used to ensure accurate exchange energies for neutral atoms - which violates expansion for slowly varying densities.
Moving on to the correlation functional, within PBE it is expressed as:
ECGGA[ρ(r)] =
Z
ρ (r){εCuni f(ρ(r)) + β t2(r) + ....} d3r (2.54) where β is a coefficient and t is the reduced density gradient for correlation. The link between the exchange and correlation functions is that, in order to retain the LDA’s response of a uniform gas to a weak potential - which is very accurate - then,
µ =π
2β
3 (2.55)
since the value of µ has been altered in PBEsol, so must the value of β . In PBEsol, β = 0.046 is used, which doesn’t completely satisfy equation 2.55, which would require β = 0.0375, but is closer to that value than the value used in PBE; β = 0.0667.
These revisions to PBE make PBEsol closer to the LDA exchange-correlation func- tional compared to PBE. The slowly varying uniform electron gas, on which LDA is based, is a better approximation to the electronic properties of a solid than that of a free atom or molecule, leading to the improved performance of this functional for solid state systems.
version of the exchange-correlation functional, which, when used, is referred to as Hybrid DFT. This approximation aims to combine the exact treatment of exchange that the Hartree-Fock theory provides (referred to as ”exact-exchange” in the con- text of Kohn-Sham DFT[229, 230]), with the LDA treatment of correlation, which can be evaluated easily and quickly, compared to post-HF methods which scale poorly with system size.[246]
The adiabatic connection formula[247–250] of DFT is used to combine the two approaches. The adiabatic connection formula connects the potential energy of the non-interacting reference system, UXC0 , to the fully interacting system, UXC1 ,
EXC=
Z 1
0
Uλ
XCdλ (2.56)
through increasing λ , the interelectronic coupling-strength parameter.
A first approximation to this, is a simple linear interpolation between the reference system and the fully interacting one:
EXC= 1 2U 0 XC+ 1 2U 1 XC (2.57)
The first term of this expression, UXC0 , is the Kohn-Sham exchange energy, EX, and
can be evaluated exactly. The second term, UXC1 , can be approximated as the LDA exchange correlation energy, UXCLDA. Therefore, the exchange correlation functional with hybrid DFT becomes:
EXC' 1 2EX+ 1 2U LDA XC (2.58)
The appropriateness of this exchange-correlation functional approximation for the system in question, LaFeO3, along with the other functionals mentioned in this
section, will be discussed in Chapter 3.