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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 ?

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treatment of correlation effects in electron momentum density natural orbital functional theory

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

B. Barbiellini

Northeastern University

overview
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
Density Matrix and Natural Orbitals

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

occupation numbers
Occupation numbers

The occupation numbers are the eigenvalues and

they satisfy the relations:

In the IPA the orbitals are either occupied or empty.

Virtual orbitals appear because of correlation.

electron momentum density emd
Electron Momentum Density (EMD)

The EMD is given by the simple formula:

The occupation number can be obtained by measuring

the EMD.

h2 molecule hartree fock
H2 molecule: Hartree-Fock

Molecular Orbitals

h2 molecule dissociation
H2 molecule: dissociation

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

h2 molecule agp
H2 molecule: AGP

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
AGP Many Body wave-function

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
AGP Natural Orbital Functional

Following Goscinski we obtain an N-representable natural orbital functional:

(Cooper pair)

The amplitudes h change sign at the HF Fermi level.

agp total energy
AGP Total Energy

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

noft various schemes
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
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
Cr: Projected EMD

Nakao et al, KKR calculation

cr folded emd
Cr Folded EMD

PM phase

AF phase

Smearing of FS

Sharp FS

conclusion
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.