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To tackle the inverse many-body problem
we will treat it in Hartree-Fock approximation
[74-77]. Thus, we replace the full many-body Hamiltonian
by a one-body Hartree-Fock Hamiltonian
=
with matrix elements defined,
for example in coordinate representation, as
|
(70) |
the being the -lowest
(orthonormalized) eigenstates of .
The corresponding
eigenvalue equation
|
(71) |
is nonlinear,
due to the -dependent definition (70) of ,
and has to be solved by iteration.
The Hartree-Fock ground state
is given by the Slater determinant
=
made from the -lowest orbitals,
and has energy
=
=
.
Considering now the case of zero temperature,
the many-body likelihood
for the true ground state ,
|
(72) |
becomes in Hartree-Fock approximation
|
(73) |
The scalar product
of the Hartree-Fock ground state
and the many-body position eigenfunction
corresponding
to the measured vector
is a determinant
and can be expanded in its cofactors
|
(74) |
being the matrix of overlaps
with elements
=
.
(For the generalization to non-hermitian see for example
[76,78].)
To maximize the posterior,
we have to calculate
the functional derivative of the Hartree-Fock likelihood
with respect to the potential
[79]
|
(75) |
Here the factors
|
(76) |
can be expressed by single particle derivatives
=
=
.
Analogously to Sect. 3
the functional derivatives
can be obtained from the functional derivative of
Eq. (71)
|
(77) |
Projecting onto
and using the hermitian conjugate of Eq. (71)
we find the Hartree-Fock version of
Eqs. (68) and (69)
where we, as done before, have fixed orthonormalization and phases by
choosing
= 0
for orbitals with equal energy.
In contrast to Sect. 3, however,
, and thus
, now obey a nonlinear equation.
Indeed, from Eq. (70) it follows
Inserting
Eq. (78)
and Eq. (80)
into Eq. (79),
we obtain the inverse Hartree-Fock equation
for
Recalling the definition of the antisymmetric matrix elements of
we finally arrive at
This linear equation can be solved directly
(where for Hamiltonian with real matrix elements in coordinate space
the orbitals, and thus their functional derivatives,
can be chosen real)
or, quite effectively, by iteration, starting for example with initial guess
= 0.
As the
,
which are only required for the lowest orbitals,
depend on two position variables , ,
Eq. (82) has essentially the dimension of a two-body equation.
Having calculated
=
(for , , , )
from Eq. (82)
the likelihood terms in the stationarity equation
(44)
follow as
recalling that =
and defining
analogously to
the matrix
with elements
.
The freedom to linearly
rearrange orbitals within
the Slater determinants
=
(for each data point , analogously for
),
makes it possible to diagonalize
the matrix of overlaps
in new orbitals
,
which are then linear combinations of the
[80,78].
Next: Numerical example of an
Up: Inverse many-body theory
Previous: Systems of Fermions
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Joerg_Lemm
2000-06-06