7. DFT Calculations

8/9/2019 7. DFT Calculations

1/24

DENSITY FUNCTIONAL THEORY AND

TOTAL ENERGY CALCULATIONS

Introductory remarks

The precursor of any atomistic modeling is knowledge of the dependence of the

potential energy, E p , of the system studied on the positions, ri , of the particles that

form the system. Depending on the approach used when evaluating E p , this energy

may also be an explicit function of the total volume of the system. Hence, in general,

E p =E p(r1,r2 ,.....,rN ,V). As explained in the Chapter dealing with the general

aspects of computer modeling, due to the translational invariance of the energy, E p

may depend only on the relative positions of the particles forming the system, i. e. onthe vectors ri !rj= rij .

A fundamental description of the potential energy of a system of atoms must be based

on understanding the electronic structure of the system since bonding between atoms

is always mediated by the electrons. In general, E p is composed of the energy

associated with the electrons that provide the bonding and the interaction energy of

the nuclei. The sum of these two contributions is often called the total energy of the

system. In this chapter we discuss the so called ab-initio calculations of the total

energy. Most generally, such calculations are based on the fundamental quantum

mechanics (Schrdinger equation) when determining the energy associated withelectrons. Atomic numbers of the constituent atoms, and possibly some very basicstructural information, are the only empirical input. Such calculations are routinely

performed in the framework of the density functional theory in which the very

complex many-body problem of interacting electrons is replaced by an equivalent but

much simpler problem of a single electron interacting with an effective potential1.

The density functional theory approach is the most fundamental treatment presently

available in atomistic studies. It is applicable to all types of bonding. Any other,

more approximate schemes, are always applicable only to certain types of bonding,

such as metallic, covalent, ionic.

Born-Oppenheimer approximation

Since electrons are much lighter than nuclei their readjustment relative to the nuclei is

always much faster than the movement of the nuclei. Hence, the instantaneous

1The Nobel Prize for Chemistry in 1998 was awarded for the development of this idea to W. Kohnand J. Pople.


2/24


3/24

3

In a system of atoms Vext(r) is the potential describing the interaction between

electrons and nuclei. Its most general form is

Vext (r)=!Zje

2

4"#0 r ! rjj$ . (DFT4)

Zj is the number of unit charges the magnitude of which is equal to the electronic

charge, e, associated with the nucleus j, rj is the position of this nucleus and ! 0 the

dielectric constant of the vacuum3. The summation extends over all the atoms.

However, a variety of approximations, such as the muffin-tin potential, various

pseudopotentials etc., have been used for the potential Vext. The total energy of thesystem of fixed atoms, identified with its potential energy, E p , defined by equation

(G1), is then

Ep =E n(r)!" #

$+12

ZiZ

je2

4%&0 ri' rji, ji(j) (DFT5)

where the second term on the right side of this equation is the interaction energy

between nuclei (ions) i and j. n(r) is determined by the functional minimization of

(DFT2). Note that n(r) depends on the positions of atoms since Vext(r)depends on

these positions.

While E ext in (DF2) is readily determined as a functional of n(r) according to

(DFT3), it is a major problem to ascertain the electron-electron interaction and the

kinetic energy of the electrons as functionals of n r) . This problem is solved

differently in different schemes. In general, the idea is how best to express E eeandE

kinas functionals of n(r) without solving the full many-body problem. If this can be

done, the problem of finding the ground state energy and electron density is reduced

to a problem of functional minimization of E n(r)[ ] with respect to n r) .

Electron-electron interaction

The classical interaction energy between charges of the density n(r) is the Coulomb

type interaction

EH =1

2

n(r)n( !r )

r - !r"" drd !r (DFT6)

3In the following we use atomic units with e2

4!"0

=1,

! = m = 1 ; the lengths are then in atomic

units (5.29x10-11m) and energy in Hartrees = 2Ry (4.36x10-18J).


4/24

4

This is called the Hartree energy. If E eewere identified with EH then the electron-electron interaction energy is overestimated for three reasons.

(i) Owing to the Pauli exclusion principle the electrons are kept out of each other's

way. This leads to the lowering of the electron-electron interaction energy by

the so-called exchange energy.

(ii) The mutual electrostatic repulsion of electrons also keeps the electrons apart.

This repulsion lowers the electron-electron interaction energy even further.The term describing this lowering is called correlation energy.

(iii)

Even interaction of an electron with itself is included in EH while this should

be excluded.

The sum of the corrections which need to be added to (DFT6) is called the exchangeand correlation energy, Exc , and the electron-electron interaction energy is, therefore,

E ee = E H + E xc (DFT7)

Some approaches also include so-called self-interaction correction mentioned in (iii).

Kinetic energy

The kinetic energy functional cannot be, in general, determined 'analytically' unlike

other parts of the energy. Approximations can be made, such as the Thomas-Fermi

model (see Appendix). However, within the density functional theory Ekin

n(r)!" #$ is

ascertained using the following argument.

First, the kinetic energy, Ekin

n(r)!

"

#

$, of the system of interacting electrons is

considered to be the same functional of the electron density as in the case of non-

interacting electrons of the same density, Ekin

on(r)!" #$ . The difference

Ekin

n(r)!" #$% Ekino

n(r)!" #$ is usually small and it is assumed that this difference can be

included into the exchange and correlation energy, E xc n(r)[ ] . The energy of the

system of interacting electrons, written as the density functional, is then

E n(r)!" #$ =Ekino n(r)!" #$ + n(r)% Vext (r)dr

+12

n(r)n(r')

r& r'%% drdr' +E

xc n(r)!" #$

(DFT8.1)

We now consider a fictitious system of non-interacting electrons of the same

density n(r) as the system studied, moving in an effective external potentialVeff(r) .

At this stage neither Veff(r) nor n r) are known and the goal is first to determine

Veff(r) . This step means that the electron-electron interaction becomes a part of the


5/24

5

effective potential and no direct interaction between electrons needs to be considered.

When this is done, the next step is to solve the simpler problem of non-interacting

electrons that move in the field defined by Veff(r) .

For the fictitious system of non-interacting electrons the total energy, written as the

density functional, is

Eo

n(r)!" #$ =Ekino

n(r)!" #$ + n(r)% Veff(r)dr (DFT8.2)

The requirement that the ground states of both functionals, E n(r)[ ] (equation

(DFT8.1)) and Eon(r)[ ] (equation (DFT8.2)), have the same charge density leads to

the equation determining Veff by applying the variational principle to both of them.

The variation of E n(r)[ ] (DFT8.1) gives

!E n(r)"# $% =!E kino

n(r)"# $% + !n(r)& Vext(r)dr+!n(r)n(r')

r' r'&& drdr' + !Exc n(r)"# $% (DFT9.1)

where the last term is the functional variation of Exc (n(r)). Similarly, the variation

of Eon(r)[ ] (DFT8.2) yields

!Eo n(r)"# $% =!Ekino

n(r)"# $% + !n(r)& Veff(r)dr (DFT9.2)

The variational problems !E n(r)[ ] = 0 and !Eon(r)[ ] = 0 will be identical and yield

thus the same solution, n(r) , if

!n(r)" Veff(r)dr= !n(r)" V

ext(r)dr+!n(r)n( #r)

r$ r'"" drd #r + !Exc n(r)%& '( (DFT10)

Since equation (DFT9.2) corresponds to the system of non-interacting electrons we

can write the charge density as

n(r)= !!(r)

!

" !! (r) (DFT11.1)

where !!(r) are one-electron wave functions and summation extends over all the

occupied states. If the system studied is not spin polarized, i. e. pairs of electrons

with opposite spins are in the same state,

n(r) = 2 !!(r)

!=1

M

" !! (r) (DFT11.2)

where the summation extends over M lowest energy states; the factor of 2 arisesbecause each state is now occupied by two electrons. In the following we shall

always assume that the system studied is not spin polarized but spin polarization,

leading to magnetic effects, may be taken into account.


6/24

6

The variational condition (DFT9.2) expressed via Euler-Lagrange equation, leads tothe Schrdinger-like equation called Kohn-Sham equation

!1

2"

2#

!(r)+ V

eff(r)#

!(r) =$

!#

!(r) (DFT12)

for the fictitious system of non-interacting electrons moving in an effective potential,

Veff . Its solutions are the one-electron wave functions !! (

r

) and associatedeigenenergies !

!. However, unlike in the Schrdinger equation the potential Veff is

not given but it depends on the wave functions !!(r) and thus (DFT12) needs to be

solved self-consistently, as described below. The energy of this fictitious system of

non-interacting electrons is

Ekin

on(r)!" #$ + n(r)% Veff(r)dr =2 &!

!=1

M

' (DFT13)

The factor 2 again results from the spin degeneracy. Equation (DFT13) determines

the kinetic energy that can be inserted into equation (DFT8.1) for the energy of thesystem of interacting electrons E n(r)[ ] . However, at this stage Veff is not yet known

explicitly and is only given implicitly by equation (DFT10). The difficulty is that the

exchange-correlation energy functional E xc (n(r)) is non-local and cannot be, in

general, written in the form !n(r)F(r)dr , where F(r) is a function, which would

allow us to calculate Veff explicitly from equation (DFT10). For this reason

additional approximations, such as the local density approximation (LDA), need to be

introduced.

Local density approximation (LDA)

In the framework of LDA we assume that the exchange-correlation energy functionalis local and can be written as

Exc[n(r)]= n(r)! "xc(n(r))dr (DFT14)

where !xc (n(r)) is the exchange and correlation energy per electron determined at a

point r by the electron density n(r) at this point and not non-locally by its

environment. In this case

!E xc(n(r)) = xc(n(r))" !n(r)dr (DFT15.1)

where

xc (n(r)) =d n(r)! xc (n (r)){ }

dn(r) (DFT15.2)


7/24

7

and it follows from equation (DFT10) that the effective potential entering the Kohn-Sham equation (DFT12) is

Veff(r)=Vext(r)+n( !r )r" !r

d !r# + xc(n(r)) (DFT16)

Inserting equations (DFT16), (DFT14) and (DFT13) into equation (DFT8.1), the

energy of the interacting system of electrons is

E n(r)!" #$ =2 %!!=1

M

& '1

2

n(r)n(r')

r' r'(( drdr'

+ n(r)( (%xc (n(r)' xc(n(r))dr (DFT17)

The sum of the second and third term in this equation is commonly called the double

counting correction; it subtracts the Hartree energy and exchange-correlation energy

that are implicitly counted twice in the sum of the one-electron energies.

The exchange-correlation energy functional !xc (n(r)) is not generally known for

spatially varying charge density. However, it can be evaluated highly accurately for a

jellium of uniform electron densityand for this case it is possible to obtain !xc (n(r))

as a function of an arbitrary uniformelectron density. A common approximation is to

employ the exchange-correlation energy determined for the uniform electron density

in the case of non-uniform density.

Extensive calculations for solids, atoms and molecules employing the LDA

demonstrated its success. The reason is that the most important effect of the

exchange and correlation is the formation of an 'exchange-correlation hole' near eachelectron that contains -1 electron (positive charge). This hole is not, in general,

spherically symmetric and its shape and size adjust according to the environment. In

the uniform electron gas this hole is spherical and this shape is, therefore, assumed in

the LDA. Hence, the shape of the hole is an approximation but it still includes -1

electron.

Evaluation of the effective potential Veff(r)

The effective potential Veff(r) (equation (DFT16)), which is needed when solving theKohn-Sham (Schrdinger-like) equation (DFT12), is itself a function of the charge

density (DFT11) determined by one-electron wave functions !!(r) that are solutions

of equation (DFT12). Thus the two equations, the Kohn-Sham equation (DFT12) and

equation (DFT16) for Veff(r) need to be solved self-consistently. This is what is

done in the so-called ab-initio calculations and the following flow chart summarizes

these procedures.


8/24

8

Is the new n(r)the same as the input n(r)?

Construct new electron density n(r)= 2 !!(r)

!=1

M

" !! (r)

Solve ! 12"2#!(r)+Veff(r)#!(r) =$!#!(r)

Construct Veff(r)=Vext (r)+VH +xc(n(r))

Choose a starting electron density n(r)

Construct the Hartree potential VH (r)= n( !r )

r" !rd !r#

This may be done by solving the Poissons equation

!2VH ="4#n(r)

YES Calculation finished

NO


9/24

9

Forces on atoms: The Hellmann-Feynman theorem

The force acting on an atom i is Fi =!grad riE p and thus to calculate the force we

need to evaluate the change of the total energy when an atom i is displaced by !ri .

The atomic coordinates are present explicitly in Vext(r) and in the term describing the

nuclei-nuclei interaction, which we call in the followingE

nn (see equations (DFT2, 4and 5)). The differentiation of E

nn is trivial but differentiation of E ext , which

contains Vext(r) (equation (DFT2)), is not straightforward. For this purpose let us

consider a change of the potential Vext(r) such that

Vext(r)! Vext(r) + "Vext(r)

that induces a change in the charge density, !n(r) . In the framework of the density

functional theory the change in the total potential energy is

!Ep n(r)"#

$%=!E

kin n(r)"#

$%+!E

ee n(r)"#

$%+!E

ext n(r)"#

$%+!E

nn(DFT18.1)

where

!E ext n[ ] = !n(r)" Vext (r)dr + n(r)" !Vext (r)dr (DFT18.2)The variational principle of the density functional theory says that for the groundstate the energy is at a minimum with respect to the variations in the charge (electron)

density and therefore

!Ekin

nGS(r)"# $% +!Eee nGS (r)"# $% + !nGS (r)& Vext (r)dr =0 (DFT19)

where nGS (r) is the ground state charge density. Combining equations (DFT19) and

(DFT18) gives

!E p nGS[ ]= nGS(r)" !Vext (r)dr +!E nn (DFT20)

The first term of the right hand side of this equation is the change in the electrostatic

energy due to the change in the external potential calculated for the fixed ground

state charge densityof the electrons,nGS (r) . Hence, the force acting on the atom i is

Fi =!grad

ri

Ep =! n

GS(r)" gradr

i

(Vext(r))dr! grad

ri

(Enn) (DFT21)

The important result is that the force acting on an atom i does not depend on thechange in the electron density, !n(r) . But this result is only true when n(r) = n

GS(r )

is the correct ground state density of electrons that corresponds to the exact

minimization of the density functional E p n[ ] . If the solution is not exact then

!Ekin

n(r)"# $% +!Eee n(r)"# $% + !n(r)& Vext (r)dr'0 and this term contributes to theforces. These forces are so called Poules forces.


10/24

0

Practical total energy calculations based on the LDA approximation

General formulation

In all methods of solving the Kohn-Sham equation, and subsequent calculation of the

total potential energy of a system of atoms, the one-electron wave functions are

always expanded into some chosen set of basis functions ! J (r) :

!!(r) = C

J

!"

J(r)

J=1

K

# (DFT22)

where CJ

!are expansion coefficients and K is the number of basis functions. At this

point the index J merely numbers the functions of the basis and is not related to atom

positions. When this expansion is inserted into the Kohn-Sham equation (DFT12) we

can transform the problem into the usual matrix formulation

CJ

!H

LJ ! "

!SLJ

( )J=1

K

# = 0 (DFT23)

where

HLJ = !L "1

2#

2+Veff(r)! J (DFT24)

is the matrix element of the Hamiltonian and

SLJ = !L!J (DFT25)

is the overlap matrix3.

The eigenvalue problem (DFT23), which determines K eigenvalues of the

Hamiltonian, !! , and the corresponding coefficients CJ!

(and thus the wave function

!!(r) ), needs to be solved self-consistently since Veff (equation (DFT16)) depends

on the electron density

n(r) =2 !!(r)

!

" !! (r) =2 CL!

CJ

!#

L(r)#

J(r)

J ,L=1

M

"!

" (DFT26)

where ! numbers different solutions of the Kohn-Sham equation and the summation

over ! extends over the occupied states. However, this form of the solution can be

sought for a cluster or a molecule but not for a bulk material containing ! 1023

atoms.

3The meaning of the expressions !1L !

2and !

1 !

2, where !

1and !

2are wave functions

and L an operator, is the usual one when using bra and ket notation

!1L!

2 = !

1(r)L!

2(r)dr

All space

" and !1 !2 = !1(r )!2(r)drAll space

" .


11/24

1

In the case of the bulk, periodic boundary conditions have to be introduced asdescribed below.

Infinite periodic systems - Bloch's theorem

When dealing with a solid or liquid composed of a large number of atoms we employ

periodic boundary conditions and investigate thus a periodic system with a repeat cellcontaining a finite number of non-equivalent atoms.

The repeat cell is either the unit cell of the structure studied (e. g. the repeat cell of

an ideal lattice) or a supercell constructed in accordance with the periodic boundary

conditions applied in the atomistic study. In the latter case the size of the repeat cell

is somewhat arbitrary and often dictated by the computational possibilities and

capabilities.

In periodic structures with translation vectorsTp , i. e. when Veff(r + Tp ) = Veff(r) , the

Bloch's theorem states that for any wave function !(r)

!(r + Tp) = e

ik"Tp!(r) (DFT27)

where k is an arbitrary wave vector. This vector can be limited to the first

Brillouin zone of the periodic structure considered. An example of the first

Brillouin zone for the case of the body-centered-cubic structure in the real space is

shown below in Fig. 1.

Fig. 1. Brillouin zone for the body-centered-cubic structure.


12/24

2

Let us consider that the basis functions are centered, though not necessarily localized,

at atomic positions. Owing to the periodicity, every position vector can be written as

rj =rj0+ Tp, where rj

0is the position vector of the corresponding atom in the repeat

cell and Tp is a translation vector of the periodic structure. The basis function

centered at atom J can now be written as ! J (r) = !j,"(r # rj0#Tp ) , where j labels

atoms within the repeat cell, !labels other characteristics of these functions, such astheir symmetries (e. g. s, p, d, f functions) and Tp are various translation vectors.

The expansion of the one electron wave function !!(r) into the basis ! J (r) (DFT22)

is then

!!(r) = C

j,"

!(r

j

0+ T

p)#

j,"(r $ r

j

0$ T

p)

j,"

%Tp

% .

When replacing r! r + T"p, where T

!p is another translation vector of the periodic

structure,

!!(r +T "p) = Cj,#

!(r

j

0+ T

p)$

j,# (r % rj0% T

p + T "p)

j,#

&Tp

&

but since !Tp= T

p!T

"pis also a translation vector of the structure, we can sum over

!Tp

rather than Tp

and, therefore,

!!(r +T "p) = Cj,#

!(r

j

0+

"Tp +T

"p)$j,# (r % rj0% "T "p)

j,#

&"Tp

& .

Following the Bloch's theorem

!!(r +T "p) =e

ik#T "p !!(r)=e

ik#T "p Cj,$

!

(rj

0+ T

p)%

j,$ (r& rj0&T

p)

j,$

'Tp

'

Comparison of these two equations for !!(r +T

"p) leads to

Cj,!

!(r

j

0+ T

p + T

"p) =e

ik#T"p C

j,!

!(r

j

0+ T

p)

This equation is satisfied if

Cj,!

!

(rj

0+ T

p) =c

j,!

!

(k)exp[ik"(rj

0+ T

p)]

where cj,!

!

(k) depends only on the positions of atoms within the repeat cell, numbered

by j. However, it also depends on the vector kand, therefore, there is not just one set

of coefficients Cj,!

!but different sets of coefficients are obtained for different values

of the vector k.


13/24

3

The Bloch's theorem, which is a consequence of the translation symmetry, reduces

the dependence of the expansion coefficients on atom positions to the positions

within the repeat cell but introduces a new quantum number k. Thus, there is a whole

space of one-electron functions corresponding to different k-vectors, which will be

labeled in the following !!

k

.

The expansion of such one electron function is then

!!

k

(r)= cj,"

!(k)exp[ik# (r

j

0+ T

p)]$

j,"(r % r

j

0% T

p)

j,"

&Tp

& (DFT28.1)

We can introduce functions

!j,"

k

(r)= exp[ik# (rj

0+ T

p)]$

j,"(r % rj0% T

p)

Tp

& (DFT28.2)

that are called Bloch's functions and we can write

!!k

(r)= cj,"!

(k)#j,"k

(r)j,"$

Inserting this expansion into the Kohn-Sham equation (DFT12) yields

cj,!

!(k)H"

j,!

k

(r)j,!

# =$!(k) cj,!!

(k)"j,!

k

(r)j,!

# (DFT29)

where the Hamiltonian

H = !1

2"

2+ V

eff(r) . (DFT30)

The usual transformation leads to the eigenvalue problem determining !! (k) and

cj,!

!

(k)

cj,!

!(k)H

i,j

",!

j,!

# (k) =$! (k) cj,! (k)Si, j",!

(k)j,!

# (DFT31)

where the Hamiltonian matrix elements are

Hi,j

!,"(k) = #

i,!

k

(r) H#j,"

k

(r) (DFT32)

and the overlap integrals4

Si,j!,"

(k) = #i,!k

(r) #j,"k

(r) (DFT33)

4More detailed expression for the Hamiltonian matrix and for the overlap integrals are

Hi, j

! ,"(k) = exp[ik# (r

j

0

$ ri

0

+ Tp$ !T

p)]

Tp

, !Tp

% & i ,!(r$ ri0

$ !Tp ) H &

j,"(r$ r

j

0

$ Tp)

Si, j

!,"(k) = exp[ik# (r

j

0

$ ri

0

+ Tp$ !T

p)]

Tp

, !Tp

% &i,!

(r$ ri

0

$ !Tp)&

j,"(r$ r

j

0

$Tp)

If the orbitals are orthonormal then Si , j

! ,"() = #

ij#

!".


14/24

4

If there are Na atoms in the repeat cell and n! basis functions corresponding to

different parameters !, then (DFT31) is the system of Na ! n" homogeneous

equations and the index ! denotes the corresponding solutions for a given vector k.

Hence, there are Na ! n" eigenvalues !! (k) and corresponding sets of coefficients

cj,!

!

(k) and therefore there are Na ! n" one-electron wave functions

!k

!(r)= c

j,"

!(k)#

j,"

k

(r)j,"

$ (DFT34)

for each value of the wave vector k.

The eigenvalue problem must be solved for every vector kwithin the Brillouin zone5.

In practice we construct in the first Brillouin zone a mesh composed of Nk k-points,

km

, and the solution is obtained for these k-vectors. The index ! then numbers

bands and !!(k) is the k dependence of the energy of the ! th band. Plot of

!!(k) vs kdisplays the band structure for the studied case. However, such plot would

be surface in the four-dimensional space since kis a vector in the three-dimensional

reciprocal space. Hence, cross-sections of this plot along certain paths in the

Brillouin zone are always presented. An example is shown in Fig. 2.

Fig. 2. Calculated energy vs kdependence (band structure) for the bcc iron (with allspins up) for paths "H, HN, N"

and "P shown in Fig. 1. EFis the Fermi energy and it

is set as zero level of energy.

5When solving the eigenvalue problem for different kvalues we use the symmetry of the Brillouin

zone to reduce the amount of calculations. The simplest symmetry corresponds to the time reversal

and says that !!("k) = !

!(k) .


15/24

5

Occupied states, charge density, Fermi energy, density of states

The occupied states, characterized by the quantum numbers kand ! , are those states the

energy of which is below the Fermi energy, EF , which is the highest energy

corresponding to an occupied state. The charge density is determined by the wave

functions !k

!(r) of the occupied states, i. e.

n(r)= 2 !k

!

(r)!,k

occupied

" !k!

(r) (DFT35)

or, when expressed via the Bloch basis functions !j,"

k

(r) and corresponding expansion

coefficients cj,!

!

(k) ,

n(r) =2 cj,!

!

(k)ci,"

!

(k)#j,!

k

(r)i,"

$ # i,"k

(r)j,!

$!,k

occupied

$ (DFT36)

However, the Fermi energy,EF , is not known a priori and needs to be determined. This

can be done most efficiently using the concept of the density of states.

The density of states, !( ) , is defined as the number of electronic states for which

!!(k) = E where k-vectors are continuously filling the first Brillouin zone. Hence,

mathematically it is defined as

!(E)=2

VBZ

"[E# $!(k)]dk

Over BZ

%!

& (DFT37)

where VBZ is the volume of the Brillouin zone, #the Dirac #-function and the factor 2again arises due to the spin degeneracy; the integration extends over all the k-vectors

in the first Brillouin zone.

If Neis the total number of electrons in the repeat cell then the Fermi energy is defined

by the relation

Ne = !(E)dE

"#

EF

$ (DFT38)

Equation (DFT38) must then be solved for EF .

Example: Non-interacting, i. e. free electrons and the repeat cell in the form of a

parallelepiped.

In this case !(k) depends only on the magnitude of k and it is equal to the kinetic

energy of the electrons:

!(k) =k2

2


16/24

6

The Brillouin zone is in this case also a parallelepiped and its volume is

VBZ

= (2! )3V , where V is the volume of the repeat cell. Following equation (DFT37)

!(E)=V

4" 3 # E $

k2

2

%

&'(

)*dk

Over BZ

+

and by introducing spherical coordinates whendk = k

2sin

!dkd

!d" we obtain

!(E)=V

4" 3 sin#d# d$ % E &

k2

2

'

()*

+,k

2dk

0

kmax

-0

2"

-0

"

- =V

" 2 % E&

k2

2

'

()*

+,k

2dk

0

kmax

-

where kmax

is the largest magnitude of the vector k in the first Brillouin zone.

Substitutingk

2

2= x so that dk=

dx

2x

!(E)=2V

" 2 #E $x( ) x

dx

2x=

0

xmax

% 2V

" 2 #E $x( ) xdx

0

xmax

% and by definition of the Dirac delta function6

!(E) =V

"2

2E( )1/ 2

Since !(E) = 0 for E < 0, it follows from equation (DFT38) that

N e =2 2V

3!2

E F3/2

and, therefore,

EF =1

23!

2"e( )

2/3

where !eis the density of electrons in the unit cell equal to Ne / V.

Determination of the density of states and evaluation of EF in numerical

calculations

To evaluate the density of states we integrate equation (DFT37) numerically. For thispurpose we employ the same mesh composed of Nk k-points, km , in the first Brillouin

zone as when solving equations (DFT31). Similarly, we choose a small step in the

energy, !E , such that the energy will be considered the same within the interval

(E ! "E /2 ,E + "E /2 ) . Since the total volume of the Brillouin zone is VBZ

, the

6 ! a" x( ) f(x)dx = f (a)#


17/24

7

volume per one point of the k-mesh isVBZ

Nk

. Equation (DFT37) can then be re-written

as

!(E) =2

Nk

Number of"!(k

m) for which E# $E / 2


18/24

8

energy, EF , is then identified with the value of the highest !! (km ) from this set of

occupied states.

Example of non-interacting, i. e. free electrons. Since !(k) = k2/ 2 the occupied states

are all within the sphere of radius kF

for which EF = k

F

2/ 2 ; k

F is called the Fermi

vector. The number of such states is1

VBZ

4!

3 kF

3

and since VBZ = (2! )3

V ,

1

VBZ

4!

3kF

3=

V

6!2k

F

3. This must be equal to N

e/ 2 and thus k

F = (3!

2"

e)1/3

and

EF = 1 2 (3!

2"e)2/3

Methods of DFT type calculations

The main distinguishing features of various methods of total energy calculations withinthe density functional theory are the choice of the basis functions ! J (r) and potentials,

Vext, in which the electrons move. The 'best choice' depends on the type and size of the

system studied. For example, different choices are the best for simple metals, transition

metals, noble metals, semiconductors and insulators.

Examples of crystal potentials

Full potential: Complete crystal potential with no approximation of its shape (equation

DFT4).

Muffin-tin potential: A flat potential field into which is inserted a lattice of non-

overlapping spheres; inside the spheres the potential has an atomic-like form.

Pseudopotentials: The pseudopotential replaces the potential of an atom such that

outside the atomic core, where the electrons are tightly bound to the nucleus, it

reproduces the same electron density as the true atomic potential. The core electrons

are not treated explicitly and valence electrons move in the field of the pseudopotential.

These pseudopotentials are not weak and cannot be treated using the perturbation

theory.

Commonly used bases

(i) Plane waves and/or plane waves-like functions i.e. de-localized basis functions.

(ii) Atomic-like basis functions which are localized in the vicinity of individual atoms.

(iii) Wave functions developing self-consistently by analyzing scattering of an incident

wave by the crystal potential (KKR method).


19/24

9

Plane wave type basis functions

These are usually used in conjunction withpseudopotentials.

OPW: Orthogonalized plane waves are wave functions constructed from plane

waves such that they are orthogonal to the core functions.

APW: The plane waves in the constant potential region are augmented by atomic-

type functions to match the atomic-like solutions inside the muffin-tin

spheres.

Localized basis functions

LCAO: Linear combination of atomic orbitals employs basis functions identical or

very similar to those obtained for the hydrogen atom.

LMTO: Linearized muffin-tin orbitals are atomic-like orbitals of the muffin-tin

spheres rather than isolated atoms.

Some methods of calculation

FP-LMTO The potential is not approximated and muffin-tin orbitals and Hankel

functions are used as the basis.

FP-LAPW The potential is not approximated and linearized APWs are used as

basis.

LMTO-ASA Linearized muffin-tin orbital method in the atomic sphere

approximation

It is assumed that each atom is at a center of a spherical effective potential well, the

atomic sphere, the volume of which is such that the volume of all the spheres just fills

the space. These spheres are, in general, overlapping so that the geometry is violated.

Relaxation calculations

When evaluating the total energy of a system for fixed positions of the nuclei (or atomiccores) we find the ground state (minimum energy state) for the electrons. However, the

relaxation of the positions of atoms also lowers the energy of the system. This is in fact

the relaxation sought in atomistic studies which is the main theme of this course.

In principle, this can always be done by evaluating the Hellmann-Feynman forces,

which are then used in a simulation technique as forces acting on atoms. However,


20/24

0

direct calculation of these forces is not easy since the electron density must correspondvery accurately to the ground state i. e. Veff(r) has to be very accurately self-consistent.

It is common to carry out minimizations 'by hand' and the following are a few examples:

Calculations of the total energy as a function of atomic volume and/or other appliedstrains. These calculations allow to determine the equilibrium lattice parameter and

elastic moduli. For example, the bulk modulus

B =!od2E p

d!2

where ! is the atomic volume and !othe atomic volume in equilibrium. For instance,

the total energies of a number of transition metals was calculated as a function of the

volume per atom for various structures (Paxton, A. T., Methfessel, M. and Polatoglou,

H. M., Phys. Rev. B 41, 8127, 1990).

Calculations of the total energy as a function of several other parameters such as c/a

ratio in the hexagonal and tetragonal structures, which allow evaluation of equilibriumcharacteristics of these lattices.

Calculations of energies of alternate crystal structures. For instance the total energies

for Si and Ge were calculated as functions of volume per atom for various structures

(Yin, M. T. and Cohen, M. L., Phys. Rev. B 26, 5668, 1982). This is one of the very

first LDA calculations of structural energy differences.

Calculation of the energy of an interface as a function of the relative displacement of the

adjoining grains which determines the most important relaxation mode.

Problems: The cohesive energies are consistently overestimated in the LDA,

probably due to the error in calculation of the energy of free atoms.

The lattice parameter is consistently underestimated.

The bulk modulus is consistently overestimated.

However these under and overestimates are only up to 15% and usually much smaller.


21/24

1

EXAMPLES OF DFT-LDA STUDIES

Total energy (Ry/atom) for silicon and germanium calculated as a function of volume

(normalized to the experimental volume) for seven different crystal structures: diamond

cubic, hexagonal diamond (wurtzite), $-tin, simple cubic, face-centered-cubic, body-centered-cubic and hexagonal close-packed. The dashed line is the common tangent for

the diamond and $-tin phases. At high pressures Si and Ge attain $-tin phase. (Yin, M.T. and Cohen, M. L., Phys. Rev. B 26, 5668, 1982; Phys. Rev. B 29, 6996, 1984).


22/24

2

The dependence of the binding energy (eV) on volume calculated for 3d transition

metals (Paxton, A. T., Methfessel, M. and Polatoglou, H. M. Phys. Rev. B 41, 8127,

1990) for the following structures: face-centered-cubic, body-centered-cubic, hexagonal

close-packed, simple hexagonal, simple cubic and diamond cubic. The volume % is

normalized to the experimentally observed volume.


23/24

5

APPENDIX

THOMAS-FERMI MODEL

An approximate approach for determination of the kinetic energy as a function (not

functional) of the electron density is the Thomas-Fermi model that effectively usesthe LDA.

The kinetic energy is written as

E k n[ ]= !k n(r)[ ]dr"

where !k n(r)[ ] is the local kinetic energy. In the Thomas-Fermi approximation thelocal kinetic energy is assumed to be equal to that of a homogeneous gas of non-

interacting free electronsof density n.

Free electrons are described by plane waves and in a cubic box with the edge of sizeL these waves are

!k (r) =1

L3/2exp(ik "r)

where L3/2

is the normalization factor. Owing to the quantization, the volume of the

k-space occupied by one state is 2! L( )3

. The energy associated with a given state

of free electrons is just the kinetic energy

E(k) =k 2

2

The electrons occupy all the states up to the Fermi energy, EF , i. e. up to a maximum

value of k , called the Fermi vector, kF . Obviously

EF =kF

2

2

and the total kinetic energy of all the electrons in the box is (using spherical

coordinates)

E k = 2 4!k

2

2k2dk =

4

!5

0

kF

" k F5

The number of electrons inside the Fermi sphere is then

Ne = 24!kF

3

3

L

2!"#$

%&'3


24/24

6

The factor of two is in both cases introduced owing to the spin degeneracy. Since Ne

is the number of electrons in the cube of the volume L3the electron density is in this

case

n =Ne

L3=

k F3

3!2

and, therefore, the kinetic energy density, !k , is related to the electron density asfollows

!k =12"

3

53"

2( )2/3

n5/3

In this model the exchange-correlation is neglected and the total potential energy is

Ep n(r)!" #$ = %k(n(r))dr& + n(r)& Vext (r)dr + 12

n(r)n( 'r)

r( 'r&& drd 'r +E

nn

This potential energy has to be functionally minimized with respect to the charge

density n(r) under the condition that the total number of electrons is conserved

n(r)! dr =N e

This leads to the Thomas-Fermi equation for n(r)

4! 3 3!2( )2 /3

n(r)2 /3

+Vext(r) +

n( "r )

r # "r$ d "r # % =0

where &is the Lagrange multiplier, employed when minimizing E p n(r)[ ] ; it has the

meaning of the Fermi energy.

This equation has a moderate success, in particular for high densities of electrons.

The main drawbacks are the neglect of exchange-correlation and very approximate

estimate of the kinetic energy of electrons.

7. DFT Calculations

Documents

Transcript of 7. DFT Calculations