Density Functional Theory

DFT was formulated in the 1960s based on the Hohenberg-Kohn

theorem

27,28

_{, which states that the energy of a system is a functional of the}

electron density, and therefore the electron density is the only requirement

to calculate the energy of the system. This reduces the dimensionality of

the problem (the dimension of the wavefunction is 3N while the density is

three-dimensional), but the exact functional is not known, for which

reason approximations are derived by assuming that the electronic part of

the system can be seen locally as a uniform electron gas, from which all

properties can be derived.

This initial approach, called Local Density Approximation

29

(LDA) did not

consider different α and β spin densities (open shell systems), and was

later replaced by the Local Spin Density Approximation

30

(LSDA). In the

Generalised Gradient Approximation (GGA) methods

31

the exchange and

correlation energy depends on the electron density and its first derivative.

Second order derivatives of the electron density (represented by the

Laplacian, ∇

2

_{ρ) have also been implemented in the so-called “meta-GGA}

methods”, which also include the orbital kinetic energy density

32

. Hybrid

functionals are built to improve the exchange term. They contain a mix of

GGA and LSDA exchange and correlation terms plus exact HF exchange.

One example of this is the very popular B3LYP functional

33,34

_which

combines the Becke88 exchange functional and the Lee-Yang-Parr

correlation functional with 20% of exact HF exchange. Although DFT was

initially found to perform well for a variety of circumstances

35

, failure has

since become evident when calculating energies (see for example

Izgorodina

et al.

36

and references therein) although they remain good

methods for geometries. The main reason why DFT is inaccurate for

energy calculations is the exchange and correlation (XC) functional, for

which no exact form is known and therefore needs to be approximated.

There are two main problems of the approximations. The first of these is

the self-interaction (SI) error

37

, which is a spurious interaction of each

electron with itself that causes large errors in the computation of energies.

The second problem is the improper description of the tail of the electronic

wavefunction

38,39

. The first problem arises from the construction of the

Coulombic energy functional and results in unreliable energies

32

_{, the}

second affects also geometries, and will be discussed next.

Dispersion forces, also known as London forces, are attractive forces that

originate when a fluctuation in the electronic density in one molecule

creates an instantaneous dipole, which in turn induces a dipole moment in

another molecule. Although weak, these interactions play an important

role in the structure of bigger molecules and complexes, such as proteins

and nucleic acids. They decay as R

-6

_{and are therefore long-range forces}

(Grimme et al.

40

reviews dispersion-corrected DFT-D methods and the

importance of dispersion in biomolecules). Recent work has pointed out

the importance of dispersion energy when analyzing protein-substrate

binding

41

and protein folding

42

. There is evidence

43

that the recently

developed DFT-D methods provide satisfactory results for the interaction

energies of biologically relevant groups.

Pauli’s exclusion principle states that two fermions cannot occupy the

same space at a time, which implies that the two-particle wavefunction

must be antisymmetric to exchange of the particles. This is the origin of

the exchange energy. This is part of the HF method but it needs to be

added in the case of DFT. HF, however, includes no electron correlation –

which DFT does by construction. Because of the lack of dynamic effects,

the electron repulsion is underestimated by HF. These effects give a

reduced probability of finding a second electron close to where a first

electron is, and a higher chance for this event far from the first electron in

HF methods. This is known as the exchange-correlation hole, which is

non-local – that is to say, it is not limited to one atom but rather

delocalized over the entire system. The exchange and correlation (XC)

functional in DFT is local, and so classical DFT methods fail to describe

long-range interactions properly. This inability of DFT methods to

reproduce dispersion effects was noted at the beginning of this century

39,44

_.

Since then, new functionals with this added capability have been created

45- 48

_{which show improved performance}

49

_{. In these methods, the total energy}

E

DFT-D

is the DFT (Kohn-Sham, KS) energy plus a correction for the

dispersion energy E

disp47

.

E

DFT-D

= E

DFT(KS)

+ E

disp

(2.3)

where E

disp

is calculated as

Edisp = -s6

!!!" !_!"! ! !!!!! !!! !!!

fdamp(Rij)

(2.4)

The dispersion correction is scaled by the scaling factor s

6

, which is

empirically determined for each density functional. N is the number of

atoms, !

_!!"

is the dispersion coefficient for a given atom pair (ij) and R

ij

is

the distance between atoms i and j. The analytical form of the dispersion

energy was shown by Alonso et al

50

_{and to have asymptotic behaviour,}

which will produce unphysical effects at short distances. This was

addressed long time ago by Ahlrichs

et al.

51

_{who added dispersion}

corrections to a semi-empirical method, and who proposed the use of a

damping function to eliminate the spurious effects of adding dispersion

energies at short distances

51

_{. The damping function f}

_damp

_{is as follows:}

!

_!"#$

(!)

=

! !!!!!(

!

!_!!!)

(2.5)

An analytical expression for the dispersion forces has been derived for use

with DFT functionals

50

although the empirical correction just discussed is

the one of choice for dispersion-corrected DFs so far – likely due to its

lower cost as indicated in Chai et al.

46

.

In this work, the following functionals are used. ωB97XD

46

is a hybrid

functional based on the ωB97X long-range corrected functional

52

(that was

developed from B97

53

with long-range corrections) with an unscaled

dispersion correction, which is equivalent to setting the factor s

6

to 1. This

functional is claimed to be free of long-range self-interaction, although

there is some degree of self-interaction at short range. It is important to

note that the long-range correlation is only based on empirical dispersion

corrections and therefore the dispersion effects do not enter the KS

orbitals. The M06 suite of methods

54

are hybrid meta-GGAs with exchange

and correlation functionals based on the M05 suite

55

_{. All these include a}

percentage of HF exchange, which is around 25-27% for M05 and M06 and

double that for the -2X version of the functional. DFT-D is still a work in

progress, with more accurate and widely applicable functionals expected

in the future.

In document Computational studies of the E3 carboxylesterase from Lucilia cuprina (Page 43-47)