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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 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 The NO’s diagonalize the 1st order density matrix.
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) The EMD is given by the simple formula: The occupation number can be obtained by measuring the EMD.
The (, e) experiment (Compton scattering) (Kaplan, Barbiellini & Bansil)
H2 molecule: Hartree-Fock Molecular Orbitals
H2 molecule: dissociation One must occupy the anti-bonding orbital to get correct dissociation.
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 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: (Cooper pair) The amplitudes h change sign at the HF Fermi level.
AGP Total Energy 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 Smearing of FS Sharp 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.