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Green functions for open shell systems

30 pages
Green functions for open-shell systems Christian Brouder Institut de Mineralogie et de Physique des Milieux Condenses (Paris)

  • green functions

  • physique des milieux condenses

  • low state

  • degenerate ground

  • low theorem

  • ground state

  • interacting system

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Green functions for open-shell systems
Christian Brouder
Institut de Mineralogie´ et de Physique des Milieux Condenses´

Layout of the talk
• The quantum Graal
• The Gell-Mann and Low theorem
• The Green functions for open shells
• The generating function
• The Hopf algebra of derivations
• Non-perturbative equations
• The hierarchy of Green functions
• The hierarchy of connected Green functions
• Open questions
• Conclusions

The quantum Graal
• H|Ψ = E|Ψ
• H is the nonrelativistic Hamiltonian
N N2 ∆i
H = − + V (r )+ V (|r −r |)n i c i j
i=1 i=1 i=j

• Density n(r)= Ψ|δ(r−r )|Ψ.ii
• Atoms or small molecules: direct diagonalization
• Solids: Density functional theory
• Solids: Green functions with non-perturbative
approximations (GW approximation, Bethe-Salpeter

The Gell-Mann and Low theorem
• Perturbation theory H = H + H0 1
N N ∆i
H =− + V (r ),H = V (|r −r |)0 n i 1 c i j
i=1 i=1 i=j
• Adiabatic switching H(t )=H + f(t )H0 1
int iH t −iH t0 0• Interaction picture H (t)=f(t )e H e1
• Evolution operator
∞ n (−i) int intU(t, t)= dt ...dt T H (t )...H (t )1 n 1 n
• If|Φ is the non-degenerate ground state of H :0 0
U(0,−∞)|Φ 0
|Ψ = lim is an eigenstate of HGL
→0Φ |U(0,−∞)|Φ 0 0
• Proof by Nenciu and Rasche (Helv. Phys. Acta 62 (1989)
1p.372-88) if f, f and f are in L and if H and H are0 1
self-adjoint, H is bounded from below and H is bounded0 1
with respect to H : for|ψ in the domain of H ,0 0
||H |ψ|≤ a||H |ψ| + b|||ψ| witha<1.1 0

The Gell-Mann and Low theorem: discussion
•| Ψ is an eigenstate of H but not necessarily the groundGL
state of H.
• Rule of thumb: The Gell-Mann and Low state is the
ground state of the interacting system if the energy
difference between the ground state and the first excitated
state of the non-interacting system is large compared with
the interaction energy.
• This condition rarely satisfied in practice
• The Gell-Mann and Low theorem is not valid for
degenerate or quasi-degenerate non-interacting systems
• The standard Green function (i.e. many-body) theory does
not work for open shell systems.

Solution: Green functions for open shells
• Start from a set of low-energy states|i of the
non-interacting system
• Transform them into states of the interacting system by
|Ψ = U(0,−∞)|ii
• Calculate the energy matrix of the interacting system

intH =i|U(+∞, 0) H + H (0) U(0,−∞)|jij 0
• Diagonalize it
• The matrix to diagonalize is very small (it contains only
the lowest energy states of the non-interacting system)
• Powerful non-perturbative methods of the Green function
theory can be used to calculate H .ij
• Describes the degeneracy splitting due to the interaction

Green functions for open shells: density matrix

• Start from a density matrix ρˆ = ρ |i j| of theijij
non-interacting system
• Calculate the energy E(ρ) of the interacting system
• Minimize E(ρ) with respect to ρ
• Preserves the symmetry of the system
• Green functions
– Self-consistent determination of the orbitals
– Resummation of infinite families of terms of the
perturbative expansion
– Detailed description of the electron-hole interactions
through the Bethe-Salpeter equation
– Beyond the coupled-cluster method
• The many-body theory is recovered for closed shells
• The crystal field equations are recovered as the first term
of the perturbative expansion of the equations for the
Green functions for open shells

The Green functions
• One-body operators

f = ρ Ψ | f(r )|Ψ = tr ρfˆ (r ) .nm m i n i
mn i i
• Examples

– The electron densityn(r) = tr ρδˆ (r−r ) .ii

– The velocityv = tr ρˆ∇ .ii
• The one-body Green function G(x, y) with x=(r,t) is
such that, for any one-body operator f

f = − G(x, x) f(r)dr
• Two-body operators (Coulomb energy, dielectric function)

g = tr ρgˆ (r ,r ) .i j
• The expectation value of two-body operators can be
calculated from the two-body Green function
G (x ,x ,y ,y )2 1 2 1 2

The generating function for the Green functions
• There is a function Z(j), with j =(j ,j ), that generates+ −
all the Green functions. For example
Z(j)2G(x, y)=(−i) | .j =j =0+ −δj(x)δj(y)
• A(ϕ) is the interacting Hamiltonian
2 e ϕ¯(x)¯ϕ(x )δ(t− t )ϕ(x )ϕ(x) A(ϕ)= dxdx
8π |r−r|0

δ δ−iD• Z(j)=e Z (j) with D = A − A0 iδj iδj+ −
W (j)0and Z (j)=e with0

1 0W (j)=− j(x)G (x, y)j(y)+K (j + j )0 ρ + −
i dxϕ(x)k(x)K (k)=logtr ρˆ:e :ρ
• Cumulants of ρˆ (initial correlations)

(1)K (k)= dx dy K (x ,y )k(x )k(y )+ρ 1 1 1 1 1 1ρ

(2)dxdyK (x ,x ,y ,y )k(x )k(x )k(y )k(y )+...1 2 1 2 1 2 1 2ρ

Correlation as propagators
• One-body correlation function

(1) †K (x, y)= ρ j|:ψ(x)ψ (y):|i =ijρ xy
(1)0 0• One-body propagator: G = G + K .ρρ
• Two-body correlation function

† † x y x y x y1 1 1 1 1 1ρ j|:ψ (x )ψ (x )ψ(y )ψ(y ):|i = + +ij 1 2 2 1
x y x y x yij 2 2 2 2 2 2
• Two-body propagator: K (x ,x ,y ,y )ρ 1 2 1 2
– it is zero for a single Slater determinant|i
– it is the source of the multiplets
• Three body propagator
x y1 1(3)K (x ,x ,x ,y ,y ,y)=1 2 3 1 2 3ρ x y2 2
x y3 3
– beyond the multiplets
• For a M-fold degenerate system, up to M-body