Treatment of Correlation Effects in Electron Momentum Density: Natural Orbital Functional Theory

B. Barbiellini Northeastern University

Overview  Beyond the IPA: many-body wave function  .

 Is the concept of one-electron orbital still meaningful ?

 Yes, but one must consider virtual orbital and occupation amplitudes.

 Natural Orbital Functional Theory & Antisymmetrized Geminal Product (AGP) give occupation amplitudes.

 Applications and Conclusions.

Density Matrix and Natural Orbitals  

ˆ

 

2

i

   1

n i N

  *  

i i

 

r

   The NO’s diagonalize the 1 st order density matrix.

Occupation numbers The occupation numbers are the eigenvalues and they satisfy the relations : 0 

n i Tr

(  1 )  2

i

   1

n i



N

In the IPA the orbitals are either occupied or empty.

Virtual orbitals appear because of correlation.

Electron Momentum Density (EMD) The EMD is given by the simple formula:   2 

n i

p



i

2 The occupation number can be obtained by measuring the EMD.

The (  , e  ) experiment (Compton scattering)

d

 /

d

 2

d



e d

(Kaplan, Barbiellini & Bansil)

const

 

H2 molecule:Molecular Orbitals

H2 molecule: Hartree-Fock Molecular Orbitals

H2 molecule: dissociation One must occupy the anti-bonding orbital to get correct dissociation.

H2 molecule: AGP    

g

1   1

s

1

s d

 

g

1

g

2 1/ 2 

g

2 1

s

* 1

s

The amplitudes g change sign at the HF Fermi level.

The correct Heitler London limit is obtained.

A similar limit is obtained with LSDA, which also mixes bonding and anti-bonding orbitals => AF ground state.

AGP Many Body wave-function 

r

1

r

N

)   ( , 1 2 )  (

N i

/ 2   1  (

r

2

i

 1 ,

r

2

i

))

i

   1

g i



i

*

r

1 

i

r

2 The generating geminal g is in a spin singlet and has a diagonal expansion in the natural orbitals.

(Blatt, Cooper, Bessis et al.,Linderberg & Öhrn and other authors).

AGP Natural Orbital Functional Following Goscinski we obtain an N-representable natural orbital functional:

E h i



i E BCS



C

r r

1 2 

E HF

 | 12 

i

 |

h i



C i

* 

E BCS

r

1 

i

r

2

h i



i



O

(Cooper pair)

h i

 

n i

(1 

n i

) The amplitudes h change sign at the HF Fermi level.

AGP Total Energy

E

 2

i

   1

n h i ii

0  

ij

 (

a J ij ij



b K ij ij

)

a ij b ij

 2

n n i j

 

n n i j



h h i j

h(0) one-body, J Coulomb, K exchange

NOFT:various schemes  Goedecker & Umrigar, Csányi and Arias proposed NOFT but non N-representable: over-correlation.

 AGP is a N-representable NOFT. It gives 40 % to 50% of the correlation in some molecules (Bessis et al.).  Estimation of the occupations for EMD studies. Barbiellini & Bansil.

Metallic Cr: modification of the occupation number Cr like H2 has a AF transition. In the PM phase there are also spin correlations that can change the occupations as in the H2 molecule.

Fermi surface

Cr: Projected EMD Nakao et al, KKR calculation

Cr Folded EMD PM phase AF phase Sharp FS Smearing of FS

Conclusion (1) Natural orbital are an important concept for EMD studies.

(2)AGP is a N-representable NOFT that provides occupation numbers as variational parameters.

(4) Formally AGP is very similar to BCS theory.

(3)AGP explain the renormalization of the occupation in term of spin correlations between pairs of electrons.