![Page 1: Hartree-Fock Theory Variational Principlekouichi.hagino/lectures/notes3.pdfMean-Field Approximation 2 body problem r =r 1-2 R=(m 1 r 1 +m 2 r 2)/(m 1 +m 2) 4 body problem 3 body problem](https://reader035.vdocument.in/reader035/viewer/2022062403/60ec50c5725b3d5d8910fa6e/html5/thumbnails/1.jpg)
Hartree-Fock Theory
Variational Principle (Rayleigh-Ritz method)
(note)
(note)
Schrodinger equation:
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Example: find an approximate solution for AHV
Trial wave function:
(note)
bopt
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Mean-Field Approximation
2 body problem
r =r1-r2 R=(m1r1+m2r2)/(m1+m2)
4 body problem
3 body problem Mean Field Approach
treat the interaction with other particles
on average
independent particle motion in an
effective one-body potential
N body problem: necessitates
an approximate method
Variational principle
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Hartree-Fock Method
independent particle motion
in a potential well
Slater determinant: antisymmetrization due to the Pauli principle
(note)
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many-body Hamiltonian:
Variation with respect to
Hartree-Fock equation:
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Density matrix:
(note)
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Remarks
1. Single-particle Hamiltonian:
Direct (Hartree) term
Exchange (Fock) term
[non-local pot.]
2. Iteration
VHF: depends on non-linear problem
Iteration:
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3. Total energy (here, we use the Hatree approximation for simplicity,
but the same argument holds also for the HF approximation.)
Hartree equation:
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Bare nucleon-nucleon interaction
Existence of short range
repulsive core
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Bare nucleon-nucleon interaction
Phase shift for p-p scattering
(V.G.J. Stoks et al., PRC48(’93)792)
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Phase shift:
Radial wave function
Asymptotic form:
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Phase shift: +ve -ve
at high energies
Existence of short range
repulsive core
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Bruckner’s G-matrix Nucleon-nucleon interaction in medium
Nucleon-nucleon interaction with a hard core
HF method: does not work
Matrix elements: diverge
…..but the HF picture seems to work in nuclear systems
Solution: a nucleon-nucleon interaction in medium (effective
interaction) rather than a bare interaction
Bruckner’s G-matrix
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(note) Lippmann-Schwinger equation
or
define (T-matrix)
For a two-particle system in the momentum representation:
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For a two-particle system in the momentum representation:
Analogous equation in nuclear medium
(Bethe-Goldstone equation)
in the operator form: (G-matrix)
Use G instead of v in HF calculations
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Hard core
Even if v tends to infinity, G may stay finite.
Independent particle motion
Healing distance
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Phenomenological effective interactions
ab initio
but, cumbersome to compute (especially for finite nuclei)
qualitatively good, but quantitatively not successful
G-matrix
HF calculations with a phenomenological effective interaction
Philosophy: take the functional form of G, but determine the
parameters phenomenologically
Skyrme interaction (non-rel., zero range)
Gogny interaction (non-rel., finite range)
Relativistic mean-field model (relativistic, “meson exchanges”)
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Skyrme interaction
(note) finite range effect momentum dependence
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Skyrme interactions: 10 adjustable parameters
A fitting strategy:
B.E. and rrms: 16O, 40Ca, 48Ca, 56Ni, 90Zr, 208Pb,…..
Infinite nuclear matter: E/A, ρeq ,….
Parameter sets:
SIII, SkM*, SGII, SLy4,…….
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Iteration
VHF: depends on non-linear problem
Iteration:
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Skyrme-Hartree-Fock calculations for 40Ca
optimize the density by taking into account the
nucleon-nucleon interaction
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Skyrme-Hartree-Fock calculations for 40Ca
optimize the density by taking into account the
nucleon-nucleon interaction
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optimize the density by taking into account the
nucleon-nucleon interaction
Skyrme-Hartree-Fock calculations for 40Ca
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optimized density (and shape) can be determined
automatically
Skyrme-Hartree-Fock calculations for 40Ca
optimize the density by taking into account the
nucleon-nucleon interaction
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Z. Patyk et al.,
PRC59(’99)704
Examples of HF calculations
for masses and radii
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Density Functional Theory
With Skyrme interaction:
Energy functional in terms of local densities
Close analog to the Density Functional Theory (DFT)
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Density Functional Theory Ref. W. Kohn, Nobel Lecture
(RMP 71(’99) 1253) i) Hohenberg-Kohn Theorem
H=H0+Vext
(unique)
Density: the basic variable
ii) Hohenberg-Kohn variational principle
The existence of a functioal , which gives the exact g.s.
energy for a given g.s. density
(note)
a part of the correlation effect is included in the Skyrme
functional through the value of the parameters
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Proof of the Hohenberg-Kohn theorem
Assume that there exist two external potentials, V1 and V2, which give
the same g.s. density (with different g.s. wave functions, and )
(note)
?
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iii) Kohn-Sham Equation
Set
Kohn-Sham equation
(note)
KS: extension of HF