Density Functional Theory Modulo: Funzionale DensitàChimica Computazionale A.A. 2009-2010 Docente: Maurizio Casarin.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Density Functional Theory

Modulo: Funzionale Densità Chimica ComputazionaleA.A. 2009-2010

Docente: Maurizio Casarin



Chimica Computazionale

Academic Year 2009 - 2010 maurizio casarin

Second Semester

DFT allows to get information about the energy, the structure and the molecular properties of molecules at lower costs that traditions approaches based on the wavefunction use.

Density functional theory (DFT) has revolutionized the quantun chemistry development of the last 20 years





Second Semester

DFT Publications

QuickTime™ and a decompressor






Second Semester

DFT Publications









Second Semester

Quantum Methods

Wavefunctions Electron Density

Hartree-Fock DFT

TD-DFTMP2-CI

The HF equations have to be solved iteratively because VHF depends upon solutions (the orbitals). In practice, one adopts the LCAO scheme, where the orbitals are expressed in terms of N basis functions, thus obtaining matricial equations depending upon N4 bielectron integrals.





Second Semester

Correlation energyExchange correlation: Electrons with

the same spin (ms) do not move independently as a consequence of the Pauli exclusion principle. = 0 if two electrons with the same spin occupy the same point in space, independently of their charge. HF theory treats exactly the exchange correlation generating a non local exchange correlation potential.

Coulomb correlation: Electrons cannot move independently as a consequence of their Coulomb repulsion even though they are characterized by different spin (ms). HF theory completely neglects the Coulomb correlation thus generating, in principle, significant mistakes. Post HF treatments are often necessary.





Second Semester

x

y

z

drr1 d

rr2

rr1

rr2

Prr1,

rr2( )d

rr1d

rr2

x1,x2( ) = χ 1 x1( )χ 2 x2( )

χ1 x1( ) = ψ 1

rr1( )α ω1( )

χ 2 x2( ) = ψ 2

rr2( )β ω2( )





Second Semester

x1,x2( ) = χ 1 x1( )χ 2 x2( ) =

1

2!

χ 1 x1( ) χ 1 x2( )

χ 2 x1( ) χ 2 x2( )

2

dx1dx2 =1

2

ψ 1

rr1( )α ω1( )ψ 2

rr2( )β ω2( ) −

ψ 1

rr2( )α ω2( )ψ 2

rr1( )β ω1( )

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

2

dx1dx2





Second Semester

x

y

z

drr1 d

rr2

rr1

rr2

Prr1,

rr2( )d

rr1d

rr2

If the electrons have not the same spin





Second Semester

Prr1,

rr2( )d

rr1d

rr2 = dω1dω2 ∫

2=

12 ψ 1

rr1( )

2ψ 2

rr2( )

2+ ψ 1

rr2( )

2ψ 2

rr1( )

2⎡⎣

⎤⎦drr1d

rr2

if

Prr1,

rr2( ) =ψ 1

rr1( )

2ψ 1

rr2( )

2

ψ 1 =ψ 2

Prr1,

rr1( ) =ψ 1

rr1( )

2ψ 1

rr1( )

2≠0





Second Semester

x

y

z

drr1 d

rr2

rr1

rr2

Prr1,

rr2( )d

rr1d

rr2

If the electrons have the same spin





Second Semester

Prr1,

rr2( ) = 1

2

ψ 1rr1( )

2ψ 2

rr2( )

2+ ψ 1

rr2( )

2ψ 2

rr2( )

2−

ψ 1* r

r1( )ψ 2rr1( )ψ 2

* rr2( )ψ 1

rr2( ) +

ψ 1rr1( )ψ 2

* rr1( )ψ 2

rr2( )ψ 1

* rr2( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

χ1 x1( ) = ψ 1

rr1( )β ω1( )

χ 2 x2( ) = ψ 2

rr2( )β ω2( )

Prr1,

rr1( ) =0





Second Semester

Information provided by is redundant


are needed to see this picture.benzene

Number of terms in the determinantal form : N! = 1.4 1051

Number of Cartesian dimensions: 3N = 126

is a very complex object including more information than we need!

N = 42e-





Second Semester

The use of electron density allows to limit the redundant information

The electron density is a function of three coordinates no matter of the electron number.





Second Semester

• 1920s: Introduction of the Thomas-Fermi model.• 1964: Hohenberg-Kohn paper proving existence of

exact DF.• 1965: Kohn-Sham scheme introduced.• 1970s and early 80s: LDA. DFT becomes useful.• 1985: Incorporation of DFT into molecular dynamics

(Car-Parrinello) (Now one of PRL’s top 10 cited papers).

• 1988: Becke and LYP functionals. DFT useful for some chemistry.

• 1998: Nobel prize awarded to Walter Kohn in chemistry for development of DFT.

Timetable





Second Semester

Quotation: “If I have seen further [than certain other men] it is by standing upon the shoulders of giants.”*

Isaac Newton (1642–1727), British physicist, mathematician. Letter to Robert Hooke, February 5, 1675.

*With reference to his dependency on Galileo’s and Kepler’s work in physics and astronomy.





Second Semester

(a) Thomas, L. H. Proc. Cambridge Philos. Soc. 1927, 23, 542; (b) Fermi, E. Z. Phys. 1928, 48, 73; (c) Dirac, P. A. M. Cambridge Philos. Soc. 1930, 26, 376; (d) Wigner, E. P. Phys. Rev. 1934, 46, 1002.



Thomas, L. H.1903-1992



Fermi, E.1901-1954



Dirac, P.M.A.1902-1984



Wigner, E.1902-1995





Hartree, D.R.1897-1958



Fock, D.R.1898-1974

(a) Hartree, D. R. Proc. Cambridge Phil. Soc. 1928, 24, 89;

(b) ibidem 1928, 24, 111; (c) ibidem 1928, 24, 426; (d) Fock, V. Z. Physic 1930, 61, 126; (e) Slater, J. C. Phys. Rev. 1930, 35, 210.



Slater J. C. 1900 -1976





Second Semester

Definitions

Function: a prescription which maps one or more numbers to another number:

y = f x( ) =x2





Second Semester

DefinitionsOperator: a prescription which maps a

function onto another function:

F =∂2

∂x2

Ff x( ) =∂2 f x( )∂x2





Second Semester

DefinitionsFunctional: A functional takes a function

as input and gives a number as output. An example is:

F[ f ] = f (x)dx–∞

∞

F f x( )⎡⎣ ⎤⎦=y

Here f(x) is a function and y is a number. An example is the functional to integrate x from - to .



Vito Volterra1860 - 1940





Second Semester

Francesco ed Edoardo Ruffini e Fabio Luzzatto (giuristi);Giorgio Levi Della Vida (orientalista);Gaetano De Sanctis (storico dell'antichità);Ernesto Buonaiuti (teologo);Vito Volterra (matematico);Bartolo Nigrisoli (chirurgo);Marco Carrara (antropologo);Lionello Venturi (storico dell'arte);Giorgio Errera (chimico);Piero Martinetti (studioso di filosofia).

In base a un regio decreto emanato il 28 agosto 1931 i docenti delle università italiane avrebbero dovuto giurare di essere fedeli non solo allo statuto albertino e alla monarchia, ma anche al regime fascista.





Second Semester

Nel 1938, con la promulgazione delle Leggi razziali, perdettero il posto i professori considerati di origine ebraica in base alla normativa razziale







Second Semester









Second Semester

ab-initio methods can be interpreted as a functional of the wavefunction, with the functional form completely known!

E [ ] =*∫ x1,L ,xN( )H x1,L ,xN( )dx1L dxN

*∫ x1,L ,xN( ) x1,L ,xN( )dx1L dxN





Second Semester

Electrons are uniformly distributed over the phase space in cells of

2πh( )3

Each cell may contain up to two electrons with opposite spins

Thomas–Fermi-Dirac Model

Electrons experience a potential field generated by the nuclear charge and by the electron distribution itself.





Second Semester



W

m

Let us consider a free electron gas





Second Semester

Let us consider a free electron gas

N A

V=6.023 ×10 23 atoms

mol ×8.92 g

cm3

63.5 gmol

=8.47 ×10 22 electronscm3

P =82.06 cm3atm

molK( )293K

7.11 cm3

mol( )=3381atm

7.11 cm3

mol( ) =6.023 ×10 23 atoms

mol

8.47 ×10 22 electronscm3

N A

V=6.023 ×10 23 atoms

mol ×Zρ

at.weight





Second Semester

V

N=4π3

rs3

rs =

3V4πN

⎛⎝⎜

⎞⎠⎟

13

rs = 1.72 Å (Li), 2.08 Å (Na), 1.12 Å (Fe)

Na+ radius = 0.95 Å

rs

0.95⎛⎝⎜

⎞⎠⎟3

>10

m =

mvm2

2=

3

2kBT

vm =107 cm/ s





Second Semester



m





Second Semester

Main Assumption: independent electron approximation

V(r) = constant

Hψ = −

h2

2m∇2 +V

rr( )

⎡

⎣⎢

⎤

⎦⎥ψ =Eψ

−

h2

2m∇2ψ = Eψ

ψ =Aeirk⋅

rr

h2

2mk2ψ =Eψ

E =

h2

2mk2

Let see what happens when QM is applied





Second Semester

ψ =Aeirk⋅

rr

pψ ≡−ih

∂∂r

ψ =−ih∂∂r

Aeirk⋅

rr( ) =h

rkψ

De Broglie relation states that

p =hλ

k =

rk =

2πλ





Second Semester

ψ 0( ) =ψ L( ) = 0

ψ n =

2

L⎛⎝⎜

⎞⎠⎟

12

sinnπ x

L⎛⎝⎜

⎞⎠⎟

En =

h2

2mπL

⎛⎝⎜

⎞⎠⎟2

n2



Two kinds of boundary conditions





Second Semester

ψ x( ) = ψ x + L( )

ψ x( ) = Aeikx x ψ x + L( ) = Aeikx xeikx L eikx L =1

kx L =2πm

ψ m x( ) = L− 12 ei 2π mx

L

Em =h2

2mk2 =

h2

2m2πm

L⎛⎝⎜

⎞⎠⎟2

m=0,±1,±2,K





Second Semester

Moving to three dimensions

ψ m x, y, z( ) = L− 32 e

i 2 πL( ) mx x+myy+mzz( )

Em =

h2

2mk2 =

h2

2m2πL

⎛⎝⎜

⎞⎠⎟2

mx2 +my

2 +mz2( )

mx ,my ,mz =0,±1,±2,K

rk =

2πL

rm





Second Semester

ψ m x, y, z( ) = L− 32 e

i 2πL( ) mxx+myy+mzz( )

Em =

h2

2m2πL

⎛⎝⎜

⎞⎠⎟2

mx2 +my

2 +mz2( )

rk =

2πL

rm

ψ k = V − 12 ei

rk⋅

rr

Em =

h2

2mkx2 + ky

2 + kz2( )





Second Semester

Degeneracies of free electron levels

Typical possibilities Orbital degeneracy

Total degeneray mx my mz m2

0 0 0 0 1 2 ±1 0 0 1 6 12 ±1 ±1 0 2 12 24 ±1 ±1 ±1 3 8 16 ±2 0 0 4 6 12

For large m values the degeneracies go up as m2( )12





Second Semester

mF =rmmax

Energy, temperature and velocity of electrons with mF

N =2

4π3

⎛⎝⎜

⎞⎠⎟mF

3 =24π3

⎛⎝⎜

⎞⎠⎟

V2π( )3

kF3

EF =

h2

2mkF2 =

h2

2m2πL

⎛⎝⎜

⎞⎠⎟2

mF2 =

h2

2m3π 2N

V⎛

⎝⎜⎞

⎠⎟

23

rk =

2πL

rm

pψ ≡−ih

∂∂r

ψ =−ih∂∂r

Aeirk⋅

rr( ) =h

rkψ





Second Semester

If we assume that the number of electrons per unit volume is ρ0, then the Fermi momentum pF of a uniform free electron gas is:

ρ0 =

N

V=

pF3

3π 2h3

EF =

pF2

2m=

h2

2m3π 2N

V⎛

⎝⎜⎞

⎠⎟

23

pF =h

3π 2NV

⎛

⎝⎜⎞

⎠⎟

13





Second Semester

Thomas and Fermi applied such a relation to an inhomogeneous situation as that of atoms, molecules and solids. If the inhomogeneous electron density is denoted by , when the equation defining ρ0 is applied locally at , it yields

ρrr( )

rr

ρ r

r( ) =pF

3 rr( )

3π 2h3





Second Semester

Let us define the density of states g(), i.e. the number of states between and + d

2

4π3

⎛⎝⎜

⎞⎠⎟

m3 = 24π

3⎛⎝⎜

⎞⎠⎟

V

2π( )3 k3 =

V

3π 2

⎛⎝⎜

⎞⎠⎟

2mε

h2

⎛⎝⎜

⎞⎠⎟

32

g ( )d

0

∫ =24π3

⎛⎝⎜

⎞⎠⎟m3 =2

4π3

⎛⎝⎜

⎞⎠⎟

V2π( )3

k3 =V3π 2

⎛⎝⎜

⎞⎠⎟

2mh2

⎛⎝⎜

⎞⎠⎟

32

The number of states with energy up to is

EF =

h2

2mkF2





Second Semester

g ( ) =

32

V3π 2

⎛⎝⎜

⎞⎠⎟

2mh2

⎛⎝⎜

⎞⎠⎟

32

12 = 2

Vπ 2

⎛⎝⎜

⎞⎠⎟

mh2

⎛⎝⎜

⎞⎠⎟

32

12 =C

12

g ( )d

0

∫ =24π3

⎛⎝⎜

⎞⎠⎟m3 =2

4π3

⎛⎝⎜

⎞⎠⎟

V2π( )3

k3 =V3π 2

⎛⎝⎜

⎞⎠⎟

2mh2

⎛⎝⎜

⎞⎠⎟

32





Second Semester

T0 = g ( )d

0

F

∫ =C 32d

0

F

∫ =25

CF

52

F =

pF2

2m=

h2

2m

3π 2 N

V

⎛

⎝⎜⎞

⎠⎟

23

F( )

32 =

h2

2m

⎛

⎝⎜⎞

⎠⎟

32 3π 2 N

V

⎛

⎝⎜⎞

⎠⎟

T0 =25

CF

52 =

25

2Vπ 2

⎛⎝⎜

⎞⎠⎟

mh2

⎛⎝⎜

⎞⎠⎟

32

C1 24 4 34 4

h2

2m⎛

⎝⎜⎞

⎠⎟

32 3π 2N

V⎛

⎝⎜⎞

⎠⎟F =

35

NF

T0 = total kinetic energy

pF =h

3π 2NV

⎛

⎝⎜⎞

⎠⎟

13





Second Semester

T ρ r

r( )⎡⎣ ⎤⎦=3h2

10m38π

⎛⎝⎜

⎞⎠⎟

23

Vρ rr( )

53

T0 ρ0[ ] =25

2Vπ 2

⎛⎝⎜

⎞⎠⎟

mh2

⎛⎝⎜

⎞⎠⎟

32

C1 24 4 34 4

h2

2m3π 2( )

23 ρ0

23

⎛

⎝⎜⎞

⎠⎟

F

E F555555555

52

T0 ρ0[ ] =

3h2

10m38π

⎛⎝⎜

⎞⎠⎟

23

Vρ0

53

T0 =

35

NF T0 =

25

CF

52





Second Semester

One can next write the classical energy equation for the fastest electrons as

μ =

pF2 r

r( )

2m+ V

rr( )

μ =

h2

2m

⎛

⎝⎜⎞

⎠⎟3π 2

( )23 ρ

rr( ){ }

23 + V

rr( )

ρ r

r( ) =pF

3 rr( )

3π 2h3

The basic equation of the TF theory. It is a classical expression, and consequently it can be applied only in those cases for which μ – V > 0

h 3π 2( )13 ρ r

r( ){ }13 = pF

rr( )





Second Semester

TTF ρ rr( )⎡⎣ ⎤⎦=CTF ρ

53∫

rr( )d

rr

ETF ρ r

r( )⎡⎣ ⎤⎦=CTF ρ53∫

rr( )d

rr −Z

ρ rr( )rr∫ d

rr +

12

ρ rr1( )ρ

rr2( )

rr1—

rr2

∫∫ drr1d

rr2

T ρ r

r( )⎡⎣ ⎤⎦=3h2

10m38π

⎛⎝⎜

⎞⎠⎟

23

l 3ρ rr( )

53





Second Semester

A results of the free electron gas theory is that the mean kinetic energy per particle is 3/5 of the Fermi energy. The total kinetic energy T0 of a free electron gas constituted by N particles is then:

T0 = 3

5pF2

2m⎛

⎝⎜⎞

⎠⎟N

and hence, t0 (the kinetic energy per unit volume) is

t0 =

T0

V=35

pf2

2m

⎛

⎝⎜⎞

⎠⎟ρ0 =Ckρ0

5/3 , Ck = 3h2

10m 3π 2( )2/3





Second Semester

t =Ck ρ rr( ){ }

5/3

E =Ck ρ r

r( ){ }5/3

drr∫ + ρ r

r( )∫ VMrr( )d

rr +

e2

2ρ r

r( )ρ rr '( )

rr −

rr '∫∫ d

rrd

rr '

The physical meaning of the last equation is that the electronic properties of a system are determined as functionals of the electronic density by applying, locally, relations appropriate to a homogeneous free electron gas. This approximation, known as local density approximation (LDA), is probably one of the most important concept of the modern DFT!





Second Semester

δ ETF ρ[ ] − μTF ρ

rr( )∫ d

rr − N( ){ } = 0

μTF =

δ ETF ρ[ ]δρ

rr( )

= 53 CFρ

23

rr( ) − Φ

rr( )

Φ

rr( ) =

Z

r−

ρrr2( )

rr −

rr2

∫ drr2





Second Semester







Second Semester

v

XHF ≈vXα ρ r

r( )⎡⎣ ⎤⎦;rr( ) =−3

2α3π ρ

rr( )⎡⎣ ⎤⎦

13





Second Semester

A non local operator is characterized by the general equation

rr ' A = d

rr∫ A

rr ',vr( )

vr( ) = ' rr '( )

Arr ',

rr( ) =

rr ' A

rr

rr ' A

rr =A

rr( )δ r

r ' −rr( )





Second Semester

T = qi ψ i

i=1

N

∑ −12 ∇

2 ψ i

The Kohn-Sham method

ρ r

r( ) = qi ψ i

rr , s( )

2

s∑

i=1

N

∑

Ts = ψ i

i=1

N

∑ −12 ∇

2 ψ i

ρ r

r( ) = ψ i

rr , s( )

s∑

i=1

N

∑2





Second Semester

In ogni punto si associa alla densita alla densita ρ(r) l’energia XC che avrebbe un gas elettronico uniforme con la stessa densità. Ciò è ripetuto per ogni punto e i valori usati nelle formule







Second Semester



Theodor von Kàrmàn

George de Hevesy

Michael Polanyi

Leo Szilard

Eugene Wigner

John von Neumann

Edward Teller

The seven majar kings of science





Second Semester



QuickTime™ and a decompressorare needed to see this picture.

P. Hohenberg W. Kohn





Second Semester

E

HF= HF H HF = Hi

i=1

N

∑ + 12 J ij −K ij( )

i, j=1

N

∑

H

i= ψ i

* x( ) − 12∇2 + v x( )⎡⎣ ⎤⎦ψ i

x( )dx∫

J

ij= ψ i x1( )ψ i

* x1( )

1

r12

ψj* x

2( )ψ jx

2( )dx1dx

2∫∫

K

ij= ψ i

* x1( )ψ j

x1( )

1

r12

ψix

2( )ψ j* x

2( )∫∫ dx1dx

2





Second Semester

Minimization and orthonomalization conditions

Fψ i x( ) = ε

ijj=1

N

∑ ψjx( )

F =−12 ∇

2 +v + g





Second Semester

• 1920s: Introduction of the Thomas-Fermi model.• 1964: Hohenberg-Kohn paper proving existence of exact DF.• 1965: Kohn-Sham scheme introduced.• 1970s and early 80s: LDA. DFT becomes useful.• 1985: Incorporation of DFT into molecular dynamics (Car-Parrinello)

(Now one of PRL’s top 10 cited papers).• 1988: Becke and LYP functionals. DFT useful for some chemistry.• 1998: Nobel prize awarded to Walter Kohn in chemistry for

development of DFT.

Background





Second Semester







Second Semester

The first HK theorem legitimates ρ as basic variable.

The external potential is determined, within a trivial additive constant, by the electron density.

Ev ρ[ ] =T ρ[ ] +Vem ρ[ ] +Vee ρ[ ] = ρ∫rr( )v

rr( )d

rr + FHK ρ[ ]

FHK ρ[ ] =T ρ[ ] +Vee ρ[ ]

Vee ρ[ ] =J ρ[ ] +non classical terms





Second Semester

The second HK theorem supplies the variational principle for the ground state energy.

%ρ r

r( )∫ drr = N E0 ≤Ev %ρ[ ]

δ Ev ρ[ ] − μ ρ

rr( )∫ d

rr − N⎡

⎣⎤⎦{ } = 0

The ground state energy and density correspond to the minimum of some functional E subject to the constraint that the density contains the correct number of electrons. The Lagrange multiplier of this constraint is the electronic chemical potential μ.





Second Semester

In matematica e in fisica teorica, la derivata funzionale è una generalizzazione della derivata direzionale. La differenza è che la seconda differenzia nella direzione di un vettore, mentre la prima differenzia nella direzione di una funzione. Entrambe possono essere viste come estensioni dell'usuale derivata.

F ρ[ ] = frr,ρ,∇ρ,∇2ρ,K( )∫ d3r

δF ρ[ ]δρ

=δ f

δρ−∇ ⋅

δ f

δ ∇ρ( )+∇2 δ f

δ ∇2ρ( )−K





Second Semester

J ρ[ ] =

12

ρ rr( )ρ r

r '( )rr −

rr '

d3r '⎛

⎝⎜⎞

⎠⎟∫ d3r

δ J ρ[ ]δρ

=δ j

δρ=

ρrr '( )

rr −

rr '∫ d3r '

j =

12

ρ rr( )ρ r

r '( )rr −

rr '∫ d3r '

δ 2J ρ[ ]δρ 2 =

δ

δρ

ρrr '( )

rr −

rr '∫ d3r ' =

δ

δρ

ρrr '( )

rr −

rr '

=1

rr −

rr '





Second Semester

μ =

δEv ρ[ ]δρ

rr( )

= vrr( ) +

δ FHK ρ[ ]δρ

rr( )

Ev ρ[ ] =T ρ[ ] +Vem ρ[ ] +Vee ρ[ ] = ρ∫rr( )v

rr( )d

rr + FHK ρ[ ]





Second Semester

Despite the importance of the HK theorems, it is noteworthy that the result they give is somehow incomplete. Actually, the first HK theorem refers only to the ground state energy and ground state density. Furthermore, as far as the second HK theorem is concerned, it is simply an existence theorem and no information about how to get the ground state energy functional is provided. Nevertheless, the existence of an exact theory justifies the research of new funtionals that, though approximate version of the correct one, are more and more accurate.





Second Semester

• 1920s: Introduction of the Thomas-Fermi model.• 1964: Hohenberg-Kohn paper proving existence of exact DF.• 1965: Kohn-Sham scheme introduced.• 1970s and early 80s: LDA. DFT becomes useful.• 1985: Incorporation of DFT into molecular dynamics (Car-Parrinello)

(Now one of PRL’s top 10 cited papers).• 1988: Becke and LYP functionals. DFT useful for some chemistry.• 1998: Nobel prize awarded to Walter Kohn in chemistry for

development of DFT.

Background





Second Semester

The Kohn-Sham method.

The massive usage of DFT is tightly bound to its use

in orbitalic theories. This is not very surprising

because of the role played by these theories, in

particular the HF one, in quantum chemistry. Thus,

the major DFT developments have implied either the

improvement of existing orbitalic theories, for

instance the X method [Slater, 1951a-b], or the

proposal of new approaches [Kohn & Sham, 1965].





Second Semester







Second Semester

Interacting electrons + real potential

Non-interacting fictitious particles + effective potential





Second Semester

H s = −1

2 ∇i2( ) +

i

N

∑ vsrri( )

i

N

∑

s = 1N !

det ψ 1ψ 2 L ψ N[ ]

hsψ i = −12 ∇

2 +vsrr( )⎡⎣ ⎤⎦ψ i = iψ i

Ts ρ[ ] = s −1

2 ∇i2( )

i

N

∑ s = ψ i −12 ∇

2 ψ ii

N

∑





Second Semester

F ρ[ ] =Ts ρ[ ] + J ρ[ ] + Exc ρ[ ] J ρ[ ] = 1

21r12∫∫ ρ r

r1( )ρrr2( )d

rr1d

rr2

Exc ρ[ ] ≡T ρ[ ] −Ts ρ[ ] +Vee ρ[ ] −J ρ[ ]

μ =veff

rr( ) +

δTs ρ[ ]δρ

rr( )

veff

rr( ) =v

rr( ) +

δ J ρ[ ]δρ r

r( )+δExc ρ[ ]δρ r

r( )=v

rr( ) +

ρ rr '( )

rr −

rr '

drr '∫ +vxc

rr( )

μ =

δEv ρ[ ]δρ

rr( )

= vrr( ) +

δ FHK ρ[ ]δρ

rr( )

vxc

rr( ) =

δExc ρ[ ]δρ r

r( )





Second Semester

With reference to the single Euler - Lagrange equation, the introduction of N orbitals allows us to treat exactly Ts, the

dominant part of the true kinetic energy T[ρ]. The cost we have to pay is the needed of N equations rather than one expressed in terms of the total electron density. The KS equations have the same form of the Hartree equations unless the presence of a more general local potential, . The computational effort for their solution is comparable to that required for the Hartree equations and definitely smaller than that pertinent to the HF ones. HF equations are characterized by a one-electron Hamiltonian including a non local potential and for this reason they cannot be considered a special case of the KS equations.

veff

rr( )





Second Semester

Relative magnitudes of contributions to total valence energy (in eV) of the Mn atom





Second Semester

The exchange-correlation potential

While DFT in principle gives a good description of ground state properties, practical applications of DFT are based on approximations for the so-called exchange-correlation potential. The exchange-correlation potential describes the effects of the Pauli principle and the Coulomb potential beyond a pure electrostatic interaction of the electrons. Possessing the exact exchange-correlation potential means that we solved the many-body problem exactly.A common approximation is the so-called local density approximation (LDA) which locally substitutes the exchange-correlation energy density of an inhomogeneous system by that of an electron gas evaluated at the local density.





Second Semester

The Local Density Approximation (LDA)

The LDA approximation assumes that the density is slowly varying and the inhomogeneous density of a solid or molecule can be calculated using the homogeneous electron gas functional.While many ground state properties (lattice constants, bulk moduli, etc.) are well described in the LDA, the dielectric constant is overestimated by 10-40% in LDA compared to experiment. This overestimation stems from the neglect of a polarization-dependent exchange correlation field in LDA compared to DFT. The method can be improved by including the gradient of the density into the functional. The generalized gradient approximation GGA is an example of this type of approach.





Second Semester

The Slater exchange functionalThe predecessor to modern DFT is Slater’s X method. This method was formulated in 1951 as an approximate solution to the Hartree-Fock equations. In this method theHF exchange was approximated by:

The exchange energy EX is a fairly simple function of the electron density .

The adjustable parameter was empirically determined for each atom in the periodic table. Typically is between0.7 and 0.8. For a free electron gas = 2/3.

EXα[ρ] = – 94α

34π

1/3

ρ4/3(r)dr0

∞





Second Semester

The VWN Correlation Functional

In ab initio calculations of the Hartree-Fock type electron correlation is also not included. However, it can be includedby inclusion of configuration interaction (CI). In DFT calculations the correlation functional plays this role. The Vosko-Wilk-Nusair correlation function is often added to the Slater exchange function to make a combination exchange-correlation functional.

Exc = Ex + Ec

The nomenclature here is not standardized and the correlation functionals themselves are very complicated functions. The correlation functionals can be seen on the MOLPRO website http://www.molpro.net/molpro2002.3/doc/manual/node146.html.





Second Semester

Application of the LDAApplication of LDA methods to semi-conductor materials and insulators gives good agreement for the lattice constant and bulk modulus.The lattice constants are typically accurate to within 1-2%up the second row in the periodic table. Since the crystalvolume V is accurately calculated the density is, of course,also obtained.

The bulk modulus is:

Bulk moduli are calculated by systematically varying the lattice parameters and plotting the energy as a function of V.The curvature at the minimum of the E(V) plot is proportionalto the lattice constant.

B = – V ∂P∂V =–V ∂2E

∂V2





Second Semester

Generalized gradient approximation (GGA). Take density gradient into account. Useful for molecules.

Spin density functional theory. Two independent variables:density and magnetization.

Exact exchange density functional theory. Calculate exchange exactly and correlation approximately using DFT.

Generalized density functional theory. Modify K-S energy partitioning to obtain a non-local hamiltonian.

Extensions of the LDA approach

m(r) = – μ0 ρ↑ – ρ↓





Second Semester

The GGA approach takes into account variations in the density by including the gradient of the density in the functional. One commonly used GGA functional is that of Becke.

This functional has only one adjustable parameter, . The value of = 0.0042 was determined based on the best fit to the energies of six noble gas atoms using the sum of the LDA and GGA exchange terms.

The GGA option in DMol3 is that of Perdew and Wang.

Generalized Gradient Approach (GGA)

VxcB = – βρ1/3 x2

1 + 6βxsinh– 1x, x=

∇ρρ4/3





Second Semester

As was discussed above for the Slater exchange functional(no gradient), the VWN correlation functional provides a significant improvement in the calculation of the energies and properties such as bulk modulus, vibrational frequencies etc. In a similar manner the Becke exchange functional (including a gradient correlation) and the Lee-Yang-Parr functional are used together. The Lee-Yang-Parror LYP correlation functional is quite complicated. It can be viewed on the MOLPRO website.

Thus, two of the most commonly used functionals are: S-VWN Slater exchange - VWN correlation (no gradients) B-LYP Becke exchange - LYP correlation (gradients)

Lee-Yang-Parr Correlation Functional

Density Functional Theory Modulo: Funzionale DensitàChimica Computazionale A.A. 2009-2010 Docente: Maurizio Casarin.

Documents

maurizio casarin slide

semester slide

spin slide

semester information

timetable slide

e slide

semester dft publications

semester quotation