Transcript Pharmacogenomics I - Northeastern University

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
 (r, r)  N   (r,  ) (r,  )d
*

ˆ  2 ni  i  i
i 1
The NO’s diagonalize the 1st order density matrix.
Occupation numbers
The occupation numbers are the eigenvalues and
they satisfy the relations:
0  ni  1

Tr ( ˆ )  2 ni  N
i 1
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:
 (p)  2 ni p  i
2
The occupation number can be obtained by measuring
the EMD.
The (, e) experiment
(Compton scattering)
d / d2 d e d   const   (p)
(Kaplan, Barbiellini & Bansil)
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
    g1  1s 1s  g2  
*
1s
*
1s
d 
g1   g2  1/ 2
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
N /2
ˆ (  (r , r ))
 (r1 ,..., rN )  A
 2i 1 2i
i 1

 (r1 , r2 )   gi i* (r1 ) i (r2 )
i 1
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[hi , i ]  EHF [ ˆ ]  EBCS [hi , i ]  O(1/ N )
EBCS  C | V12 | C
C (r1 , r2 )   hi (r1 ) i (r 2 )
*
i
(Cooper pair)
i
hi   ni (1  ni )
The amplitudes h change sign at the HF Fermi level.
AGP Total Energy


E  2 n h   (aij J ij  bij K ij )
i 1
0
i ii
ij
aij  2ni n j
bij  ni n j  hi h 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
Sharp FS
AF phase
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.