Electron density:

Electron density:

2

1 1 2 1 2

( ) r N ... | ( , ,..., x x x

) | ds dx ... dx

ρ ^ = ∫ ∫ Ψ ^{    }

Probability of finding one electron of arbitrary spin within a volume element dr₁ (other electrons may be anywhere).

i i i

x   ≡ r s

Properties of electron density (non-negative):

1.

2.

3.

1 1

( )r dr N

ρ =

∫

(r ) 0

ρ  → ∞ = ρ( )r ∝exp^−2 2 | |I r ^

(r R_A)...cusp ρ

∇  → 

(r R_A) max ρ  →  =

lim

2 ( ) 0

r

Z r

r ρ

→

 ∂ +  =

 ∂ 

 

=> exchange-correlation functionals should respect

these conditions

Pair density:

2

2

( , x x

) N N ( 1) ... | ( , x x

,..., x

) | dx

... dx

ρ ^{ } = − ∫ ∫ Ψ ^{    }

Antisymmetric wavefunction requirement => reduced density matrix γ

' ' * ' '

2

( , x x x x

; ,

) N N ( 1) ... ( , x x x

,

..., x

) ( , x x x

,

..., x

) dx

... dx

γ ^{   } = − ∫ ∫ Ψ ^{   } Ψ ^{     }

Variables in Ψ

which are not included in integration are primed.

γ

changes sign when x

and x

(or x

’ and x

’) are interchanged

Diagonal elements of reduced density matrix => pair density (two-electron density matrix)

'

1 1

'

2 2

x x x x

=

=

 

 

2

( , x x

)

( , x x

) ρ   = − ρ  

Probability of finding two electrons with the same spin at the same point is 0 !!!

Probability of finding a pair of two electrons with particular spins within a volume elements dr₁ and dr₂ (remaining N-2 electrons may be anywhere).

Non-negative quantity

Normalized to N(N-1), contains all information about electron correlation.

Symmetric

' ' ' '

2

( , x x x x

; ,

)

( , ; , x x x x

) γ     = − γ    

(

) ( ) ( ) ^x

^x

N 1 x N

,

x   = − ρ  ρ  ρ

Non-correlated motion:

=> “Fermi correlation”, “Exchange correlation” - described already at the HF level Two electrons with the same spin cannot be at the same point in space.

This “correlation” does not depend on the electron charge, purely exchange effect.

Fermi correlation has nothing common with (“Coulomb”) correlation defined for post HF methods!

{ }

[

]

2 1 HF

2

( x , x )

2 det

( r )

( s )

( r )

( s )

ρ    

2 2 2 2 2 2 2 2

2 1 2 1 1 2 2 1 1 2 2 1 2 2 1 2 1 1 2

1 1 2 1 1 2 2 2 1 1 2 1 1 2 2 2

( , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

HF x x r r s s r r s s

r r r r s s s s

ρ φ φ σ σ φ φ σ σ

φ φ φ φ σ σ σ σ

= + −

−

     

   

σ

= σ

( x , x

) 0

HF

ρ

  =

Correlated electron motion

σ

≠σ

(

) ( ) ( )

,

HF ^{1 2}

 r ,  r = ρ  r ρ  r ρ

(

)

,

HF^σ1=σ2

 r ,  r ρ

Completely uncorrelated motion

1 2

,

2^{HFσ σ}

( ,

) ( ) ( )

ρ

ρ

ρ

1 2

1 2

,

2^{HFσ σ}

( ,

)

0 ρ

Small detour: HF pair density:

Correlation factor - defines the difference between uncorrelated and correlated densities:

[ ]

2

( ,

) ( ) (

) 1

( ;

) ρ

ρ

ρ

- completely uncorrelated case =>

=> wrong normalization of ρ₂ (N²)! (due to self-interaction) Introducing Conditional probability - probability of finding electron at position 2 when there is just one electron at position 1.

Integrates to (N-1).

2 1 2

2 1

1

( , ) ( ; )

( )

x

ρ

Ω   =

ρ



Pair density for completely uncorrelated motion:

(

) ( ) ( ) ^x

^x

N 1 x N

,

x   = − ρ  ρ  ρ

Formulation of pair density in terms of electron density and (whatever) is the correlation:

1 2

( ; ) 0 f x x  =

2 1 2

( ; )

1

Ω = −

∫

Exchange-correlation hole:

The difference between conditional probability Ω and uncorrelated (unconditional) probability of finding electron at x₂.

h_xc accounts for: exchange and coulomb correlation and self-interaction

2 1 2

2 1

2 1 2

( , )

( ) ( )

( ) ( ; )

x x x

x

x f x x

ρ ρ

ρ ρ

= −

=

  



  

Correlation - typically leads to depletion of electron density

1 2 2

( ; ) 1

h

x x dx = −

∫ ^{  }

1 2 1 2 2

( ; ) ( ; ) ( )

ρ

Schrödinger equation in terms of spin-independent pair density (two-electron part):

2 1 2 1 12

2 1 2 N

i N

i j ij

ee

d r d r ds ds

r

) x , x ( 2

1 r

E 1    

∑∑ ^{Ψ =} ∫∫ ^ρ

Ψ

=

>

Probability of finding a pair of electrons at x₁, x₂ Ingegration over 8 variables only !

) x

; x ( h ) r ( ) r ( ) r ( ) x , x

( _{1 2 1 2 1 XC 1 2}

2













 =ρ ρ +ρ

ρ

2 1 2 1 12

2 1 XC 1 2

1 12

2 1

ee

d r d r ds ds

r

) x

; x ( h ) r ( 2

r 1 d r r d

) r ( ) r ( 2

E 1

∫∫

∫∫

=

J[ ρ] Exchange, correlation, SIC

1 1 2

1 2

1 2 1 2

12 12

( ) ( ; ) ( ) ( )

1 1

2 2

XC ee

r h r r

r r

E dr dr dr dr

r r

ρ ρ ρ

=

∫∫

∫∫

Classical J[ρ] QM contribution (correlation) + self-interaction FORMALLY - exchange-correlation hole can be split into the Fermi hole and Coulomb hole

1 2 1, 2

1 2 1 2 1 2

( ; ) ( ; ) ( ; )

XC X C

h r r   = h

r r   + h

r r  

Fermi hole - dominates coulomb hole - contains self-interaction - integrates to -1

- equals to minus density of electrons (same spin) at the position of this electron (at the same point) - negative everywhere

- depends also on the density at r₂ - no spherical symmetry

) r ( )

r

; r r

(

h



→ 



= − ρ 

(

)

X 2 2

1

X

( r ; r ) ( r ) f r ; r h   = ρ   

Coulomb hole - integrates to 0

- negative at the position of reference electron - mirrors the cusp condition

0 r d ) x

; x (

h

∫

2

1 2 1

1 1

( ; ) ( )

2 2

hX^α r r  = −

ρ

Example – H

molecule:

Exchange hole – only SIC

⇒It is just half of the density!

⇒It does not depend on the electron position (r₁)

⇒h_X leads to depletion of electron density

⇒HF method considers only h_X and it results in too diffuse one-el.

Functions => underestimating V_en, low T_e, and also J_ee is underestim.

Coulomb hole

⇒Changes with the position of reference electron

⇒For r_HH→∞ h_C removes halves the electron from one atom and puts it to the other one

Baerends & Gritschenko

J. Phys. Chem. A 101 (1997) 5383

1 1 2

1 2

1 2 1 2

12 12

( ) ( ; ) ( ) ( )

1 1

2 2

XC ee

r h r r

r r

E dr dr dr dr

r r

ρ ρ ρ

=

∫∫

∫∫

Hamiltonian only contains parts depending on (i) one electron or (ii) on two electrons

=> Schrödinger equation can be rewritten in terms of one- and two-particle density matrices

Knowledge of

1 1 2

( ) ( ; )

XC

r h r r ρ ^

 

“Easy” solution of Schrödinger equation in terms of spin-densities (8 variables) Is it possible?

Does the density contains all the information?

Answer: Hohenberg-Kohn Theorem (1964)

First attempts:

Thomas-Fermi Model (1927)

Properties of electron density (non-negative):

1.

2.

3.

1 1

( )r dr N

ρ =

∫

(r ) 0

ρ  → ∞ = ρ( )r ∝exp^−2 2 | |I r ^

(r R_A)...cusp ρ

∇  → 

(r R_A) max ρ  →  =

lim 2 ( ) 0

r

Z r

r ρ

→∞

 ∂ +  =

 ∂ 

 

Small detour: Uniform Electron Gas

Hypothetical system, “Homogeneous Electron Gas”

Electrons move on the positive background charge Overall system charge is 0

Volume

Number of electrons

Electron density (constant) _/

el

V N

N V ρ

→ ∞

→ ∞

=

“So – so” model for simple metals; constant density is far from reality for molecules !

Only system for which we know exchange-correlation functional exactly.

Thomas-Fermi Model

Partially classical

Neglects exchange and correlation contribution

Crude approximation for kinetic energy (far from real molecules)

⇒Poor performance!

NEVERTHELESS – Energy is given as a functional of electron density !

∫

π

=

ρ (3 ) (r)dr 10

)] 3 r ( [

T_TF  ^{2 2/3 5/3}  

2 1 12

2 3 1

/ 5 3 / 2 2

TF drdr

r ) r ( ) r ( 2 r 1 r d

) r Z ( r d ) r ( )

3 10( )] 3 r ( [

E         

∫∫

∫

∫

π

= ρ

Solution – variational principle under the constraint of number of electrons.

Numerous extentions and improvments:

• “chemical” accuracy never reached (by a distance!)

• Even when V_ee description improved problems stay – due to kinetic energy description.

• It was rigorously proofed that withing T-F model all molecules will dissociate into their fragments!

Slater’s approximation for electron exchange

1951 – to find an approximate way to calculated exchange in HF

2 1 12

2 1 X 1

X drdr

r

) r

; r ( h ) r ( 2

E 1     

∫∫

=

One needs a good approximation to h

1. Assuming spherically symmetric hole centered around the reference electron.

2. Assuming that density is constant within the hole and that it integrates to -1.

⇒ Sphere radius (Wigner-Seitz radius)

⇒ Simple interpretation – average distance between electrons

⇒ Approximate solution:

⇒ Density functional for exchange energy!

3 / 1 1 3 / 1

S (r )

4

r 3  ρ ⁻



 





= π 

1 3 / 4 1 X

X

[ ] C ( r ) d r

E

∫

≅ ρ

Original work – Hartree-Fock-Slater (HFS) method known also as X

method:

Exchange integrals replaces by

( α is a parameter between 2/3 and 1)

1 3 / 4 1 3

/ 1

X

3 ( r ) d r

8 ] 9

[

E

^{ρ = − } _ ^ _π ^ _ ^{ α} ∫ ^ρ  

Used before by Dirac:

Thomas-Fermi-Dirac model

Density functional theory

Traditional ab initio: finding the N-electron wavefuntion Ψ(1,2,…,N) depending on 4N coord.

DFT: finding the total electron spin-densities depending on 8 coordinates

Hohenberg & Kohn:

Theorem I: Energy of the system is unique functional of electron density

{ } ^ˆ

( ) r N

, R Z

,

V

H E ,

ρ  => => ⇒ ⇒ Ψ

Theorem II: Variational principle

E

[ ] ρ ≤ E [ ] ρ 

0

[

]

( )

[

]

[

]

universally valid system dependent

E ρ = ∫ _ ρ _ r V dr ^ _ ^ + T _{} ρ + E ρ

0

[

]

( )

[

]

E ρ = ∫ ρ r V dr ^{ } + F ρ

Hohenberg-Kohn functional:

Kinetic energy of electron Coulomb repulsion

Non-classical interaction

(self-interaction, exchange, and correlation)

All properties (defined by V_ext) are determined by the ground state density H&K only proofed that F_HK exist, however, we do not know it

H&K do not give a direction how can we find density

H&K theorems allow us to construct the rigorous many-body theory using density as a fundamental properties

[ ( )] [ ( )] [ ( )] _ncl[ ( )]

F ρ r =T ρ r + J ρ r + E ρ r

The Second Hohenberg-Kohn Theorem:

“Density functional F

[ ρ] will give the lowest energy of the system only if the ρ is a true ground state density.”

~ VARIATIONAL PRINCIPLE

~ ] [ E

~ ] [ E

~ ] [ T

~ ] [ E

E

≤ ρ = ρ +

ρ +

ρ

Proof – literally trivial

1. Trial density defines its own Hamiltonian, thus, wave function:

2. Applying variational theorem for this trial wava function: ρ( )r → Hˆ → Ψ

0 0

0 0 ext

ee

[ ~ ] ~ ( r ) V d r E [ ~ ] E [ ] H ˆ

V

~ ] [

~ T H ˆ

~

Ψ

∫

NOTE:

Strictly VP holds only for “exact” functional.

Approximate functionals can easily give energies below a true minimum (different from HF).

Mathematical vs. Physical meaning of VP.

Hartree-Fock method from a different point of view:

• Slater determinant – approximation to the true N-electron wave function

• It can be viewed as the exact wave function of fictitious system of N non-interacting electrons moving in the effective potential V_HF (“electrons” viewed as uncharged fermions not explicitly interacting via Coulomb repulsion)

• Kinetic energy of such system is then exactly

• One electron functions, spin-orbitals, obtained from variational principle Kohn-Sham Approach – A Basic Idea

Kohn-Sham:

• Most problems of Thomas-Fermi type approaches come from kinetic energy

• Kohn-Sham – establishing a similar strategy as used in Hartree-Fock method HK theorems ^{E = min}_ρ→

(

_∫

)

0 N





)]

r ( [ E )]

r ( [ J )]

r ( [ T )]

r ( [

F ρ  = ρ  + ρ  + _ncl ρ 

SD ee Ne

N SD

HF min Tˆ Vˆ Vˆ

E

SD

Φ +

+ Φ

=Φ →

∑

−

= ^N

i

i 2 i

HF 2

T 1

• In analogy with above “non-interacting” electrons K&S introduced a non-interacting reference system for particles interacting via effective local potential V_S, that in some way includes desired interactions between particles.

⇒Hamiltonian with effective local potential V_S:

⇒ Slater determinant is then exact wave function:

⇒ One electron functions obtained (in analogy with Fock equations) by solving Kohn-Sham equations, using a one-electron Kohn-Sham operator f^KS:

⇒ Resulting orbitals ~ Kohn-Sham (KS) orbitals

⇒ Effective potential V_S is such that the density constituted from KS orbitals exactly equals the ground state density of “real” system with interacting electrons

∑

∑

−

= ^N

i

N

i

i S 2

i

S V (r)

2

Hˆ 1 

( ) ( ) ( )

( ) ( ) ( )

( )

( )

( )

1

2 N 2

2 2

1

1 N 1

2 1

1

S

x x

x

x x

x

x x

x

! N 1

 

















 





ϕ ϕ

ϕ

ϕ ϕ

ϕ

ϕ ϕ

ϕ

= Θ

i i i

fˆKS ϕ =ε ϕ

) r ( 2 V

fˆ^KS =−1∇² + _S 

) r ( )

s , r ( )

r

( ₀

N

i s

2 i

S





 = ϕ =ρ

ρ

∑∑

Kohn-Sham Approach

Adopting a better expression for kinetic energy:

Using exact kinetic energy of the non-interacting reference system that has the same density as a real one.

Such kinetic energy cannot be the same as a true one; it is expected to be close.

Residual part of kinetic energy (T_C) is shifted to the functional.

Kohn-Sham functional is then:

∑

−

= ^N

i

i 2 i

S 2

T 1

T_S ≠ T

(

) (

)

] [

E_XC ρ ≡ ρ − _S ρ + _ee ρ − ρ = _C ρ + _ncl ρ )]

r ( [ E )]

r ( [ J )]

r ( [ T )]

r ( [

F ρ  = _S ρ  + ρ  + _XC ρ 

Kinetic energy of non-interacting reference system

Coulomb repulsion of uncorrelated densities

Exchange-correlation functional Includes:

Electron exchange Electron correlation

Residual part of kinetic en.

] [ E ] [ E ] [ J ] [ T )]

r ( [

E ρ  = _S ρ + ρ + _XC ρ + _Ne ρ

∫

∫∫

+ ρ

= drdr E [ ] V (r)dr r

) r ( ) r ( 2 ] 1 [

T _{1 2 XC Ne}

12 2 1 S







 



∑∫∑

( )

− ρ

+ ^N

i

1 2 1 i M

A 1A A

XC x dx

r )] Z

r ( [

E   

Contains all problematic terms

( ) ( )

∑∑∫∫

∑

−

= ^N

i N

j

2 1 2 2 j 12 2 1 i N

i

i 2

i x dx dx

r x 1 2

1 2

1    

Putting things together:

Applying vatiational principle

i i i M

A 1A A 1

XC 2

12 2 2

r ) Z r ( V r r d

) r ( 2

1 ϕ =ε ϕ









 



 



 ρ + −

+

∇

−

∫

∑

i i i 1 eff

2 V (r)

2

1 ϕ =ε ϕ



 



− ∇ +

= 

Satisfying the conditions stated for non-interacting reference system

∫

∑

=

≡ ^M

A 1A A 1

XC 2

12 2 eff

S r

) Z r ( V r r d

) r ) (

r ( V ) r (

V      Iterative solution.

What is V_XC ? δρ

≡ δ ^XC

XC

V E

Kohn-Sham method – in a nut-shell:

1 2

ˆ ( )

2

N N

S i S i

i i

H = −

∑

∑

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 1 2 1 1

1 2 2 2 2

1 2

1

!

N N S

N N N N

x x x

x x x

N

x x x

ϕ ϕ ϕ

ϕ ϕ ϕ

ϕ ϕ ϕ

Θ =

   

  

  

   

ˆ

i i i

f ϕ ε ϕ =

Kohn-Sham equations

1

ˆ ( )

2

KS

f = − ∇ +V rS 

[ ( )]

[ ( )] [ ( )]

[ ( )]

F ρ r  = T ρ r  + J ρ r  + E ρ r 

( ) ( )

[ ] [ ] [ ] [ ] [ ] [ ] [ ]

XC S ee C ncl

E ρ ≡ T ρ − T ρ + E ρ − J ρ = T ρ + E ρ

Problematic kinetic energy term is divided between non-interacting system and exchange- correlation functional:

T_C=T-T_S ... the residual part of kinetic energy

Non-interacting reference system with effective local potential V_s introduced:

[ ( )] _S[ ] [ ] _XC[ ] _Ne[ ]

E ρ r =T ρ +J ρ +E ρ +E ρ 1 2

1 2 12

( ) ( )

[ ] 1 [ ] ( )

S 2 XC Ne

r r

T dr dr E V r dr

r

ρ ρ

ρ ρ ρ

= +

∫∫

∫

( )

( )

2

1 2 1 2

12

1 1 1

2 2

N N N

i i i j

i i j

x x dx dx

ϕ ϕ ϕ r ϕ

= −

∑

∑∑∫∫

( )

1

[ ( )]

N M

A

XC i

i A A

E r Z x dx

ρ r ϕ

+ ^ −

∑ ∑ ∫

How does it work:

Applying variational principle we solve Kohn-Sham equations in iterative way Everything “unknown” is in E_XC

If we know E_XC we have an EXACT method

E_XC is not known => we have to rely of approximate exchange correlation functionals

Exchange-correlation functionals:

I. local density approximation - E_XC[ρ]

II. Generalized Gradient approx. - E_XC[ρ,∇ρ]

III. Hybrid density functionals - E_XC[ρ,∇ρ] + combines with “exact”=HF exchange

Local Density Approximation: (LDA ~ LSD ~ SVWN) E_XC derived for the model of uniform electron gas

3 3 3 ( )

X 4

ρ r ε ^{= −} π

[ ] ( ) ( ( ))

LDA

XC XC

E

ρ

∫ ρ ε

ρ

XC r X r C r

ε ρ

ε ρ

ε ρ

ε_C (fit of QMC data) VWN