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Numerical example of an inverse Hartree-Fock calculation

Figure 9: Inverse Hartree-Fock Approximation: The exact two-body likelihood has been calculated for two one-dimensional particles with many-body Hamiltonian $H$, given in Eq. (84), a local one-body potential $V_1(x,x^\prime )$ = $\delta (x-x^\prime ) a (x/10)^2$, $a=10^{-3}$ breaking translational symmetry, mass $m$ = $10^{-3}$, and a given local two-body potential $V_{\rm true}$ of form (65) with $v_{\rm true}(\vert x-x^\prime\vert)$ = $b/(1+e^{-2 \gamma (x-k/2)/k})$, $b$ = $100$, $\gamma $ = 10, $k$ = 21, (thin line on the r.h.s.). As training data 100 pairs $\{(x_{i,1},x_{i,2})\vert 1\le i\le 100\}$ have been sampled according to that exact likelihood. The corresponding exact (thin line) and empirical (bars) likelihoods for inter-particle distances $\vert x_{i,1}-x_{i,2}\vert$ are shown on the l.h.s. of the figure. To reconstruct the potential a Gaussian prior has been used with $\lambda{\bf K}_0$ = $\lambda ({\bf I}-\Delta)/2$, $\lambda $ = $0.5$ $10^{-3}$, and reference potential (dashed on r.h.s) $v_{0}(\vert x-x^\prime \vert)$ = $b/(1+e^{-2 \gamma (x-k/2)/k})$, $b$ = $100$, $\gamma $ = 1, $v_0(0)$ = 0. The related reference likelihood of inter-particle distances is shown on the l.h.s.(dashed). The reconstructed potential $v$ has been obtained by iterating with ${\bf A}$ = ${\bf K}_0$ according to Eq. (50) and solving Eqs. (71) and (82) within each iteration step. The problem has been studied at zero temperature, $v$ fulfilling the boundary conditions $v(0)$ = 0 and $v$ = constant beyond the right boundary. No energy penalty term $E_U$ had to be included. Note, that the number of data is not only small for large inter-particle distances where the potential is large, but also for small distances. This effect is due to antisymmetry which does not allow particles to be at the same place. Thus, the reconstructed potential $v$ is nearly equal to the reference potential $v_0$ for large and for small distances.
\begin{figure}\begin{center}
\psfig{file=r10-4.eps, width= 67mm}$\!\!\!$\epsfig{...
...){\makebox(0,0){inter--particle distance}}
\end{picture}\end{center}\end{figure}

To test the numerical implementation of an inverse Hartree-Fock approach, we study a two-body problem, defined by the Hamiltonian

\begin{displaymath}
H = -\frac{1}{2m}\Delta+V_1+V_{\rm true}
.
\end{displaymath} (84)

Herein we assume the local one-body potential $V_1(x,x^\prime )$ = $\delta (x-x^\prime ) v_1(x)$ to be given and the two-body potential $V$ to be unknown, but local as in Eq. (65). Hence, our aim is to approximate the function $v(\vert x-x^\prime\vert)$, defining the matrix elements of $V$, by using empirical data in combination with appropriate a priori information.

Fig. 9 shows the results of a corresponding inverse Hartree-Fock calculation. (The prior process and parameters are given in the figure caption, computational details will be presented elsewhere). For this two-body problem it is possible to calculate the exact solution and corresponding likelihood numerically. Hence, we was able to sample training data using the exact likelihood. Note that, besides the problem of simulating realistic data, an inverse Hartree-Fock calculation for more than two particles is not much more complex than for two particles. It only requires to add one single particle orbital for every additional particle. Thus, an analogous inverse Hartree-Fock calculation is clearly computationally feasible for many-body systems with three or more particles.

We have already discussed in previous sections that, in regions where the potential is large, the reconstruction of a potential is essentially based on a priori information. Training data are less important in such regions, because finding a particle there is very unlikely. In Fig. 9, for example, a priori information is thus especially important for large distances. A new, similar phenomenon occurs now when dealing with fermions: The antisymmetry, we have to require for fermions, forbids different particles to be at the same location. Hence, antisymmetry reduces the number of training data for small distances, and a priori information becomes especially important. This effect can clearly be seen in the figure, where the reconstructed potential $v$ is influenced by the data mainly for medium distances. For large, but also for small inter-particle distances, the reconstructed potential is quite similar to the reference potential.

Summarizing, we note that for inverse Hartree-Fock problems in addition to the direct Hartree-Fock Eq. (71) a second equation (81) has to be solved determining the change of Hartree-Fock orbitals under a change of the potential. Despite this complication it was possible to solve the inverse Hartree-Fock equations numerically for the example problem considered in this section.


next up previous contents
Next: Conclusions Up: Inverse many-body theory Previous: Inverse Hartree-Fock theory   Contents
Joerg_Lemm 2000-06-06