Inverse Hartree-Fock theory

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 $H$ by a one-body Hartree-Fock Hamiltonian $H^{HF}$ = $\sum_{kl} h_{kl} a^\dagger_ka_l$ with matrix elements $h$ defined, for example in coordinate representation, as

$$h_{xx^\prime} = T_{xx^\prime} + \sum_k^N <\!x \varphi_k\,\vert\,V\,\vert\,x^\prime \varphi_k\!> ,$$ (70)

the $\varphi_k$ being the $N$-lowest (orthonormalized) eigenstates of $h$. The corresponding eigenvalue equation

$$h\varphi_k = \epsilon_k \varphi_k ,$$ (71)

is nonlinear, due to the $\varphi_k$-dependent definition (70) of $h$, and has to be solved by iteration. The Hartree-Fock ground state is given by the Slater determinant $\mbox{$\vert\,\Phi_0\!>$}$ = $\det\{\mbox{$\vert\,\varphi_k\!>$}\}$ made from the $N$-lowest orbitals, and has energy $E_0^{HF}$ = $\sum_k^N t_{kk} +\frac{1}{2}\sum_{kl}^N v_{klkl}$ = $\sum_k^N \epsilon_{k} -\frac{1}{2}\sum_{kl}^N v_{klkl}$. Considering now the case of zero temperature, the many-body likelihood for the true ground state $\psi_0$,

$$p(x_i\vert\hat x,\rho(v)) = \mbox{$<\!\psi_0\,\vert\,x_i\!>$}\mbox{$<\!x_i\,\vert\,\psi_0\!>$} ,$$ (72)

becomes in Hartree-Fock approximation

$$p(x_i\vert\hat x,\rho_{HF}(v))= \mbox{$<\!\Phi_0\,\vert\,x_i\!>$}\mbox{$<\!x_i\,\vert\,\Phi_0\!>$} .$$ (73)

The scalar product of the Hartree-Fock ground state $\Phi_0$ and the many-body position eigenfunction $\mbox{$\vert\,x_i\!>$}$ corresponding to the measured vector $x_i$ is a determinant and can be expanded in its cofactors $M_{kl;i}$

$$\mbox{$<\!x_i\,\vert\,\Phi_0\!>$} = \det\{\mbox{$<\!x_{i,l}\... ...t\,\varphi_k\!>$}\} = \det B_i = \sum_l^N M_{kl;i} B_{kl;i} ,$$ (74)

$B_i$ being the matrix of overlaps with elements $B_{kl;i} = \mbox{$<\!x_{i,l}\,\vert\,\varphi_k\!>$}$ = $\varphi_k(x_{i,l})$. (For the generalization to non-hermitian $h$ 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]

$$\delta_{v(x)} p(x_i\vert\hat x,\rho_{\rm HF}(v)) = \mbox{$<... ...ert\,x_i\!>$}\mbox{$<\!x_i\,\vert\,\delta_{v(x)} \Phi_0\!>$} .$$ (75)

Here the factors

$$\mbox{$<\!x_i\,\vert\,\delta_{v(x)} \Phi_0\!>$} = \sum_{kl}^... ... {\varphi_k}\!>$} = \sum_{kl}^N M_{kl;i} \, \Delta_{kl;i}(x) ,$$ (76)

can be expressed by single particle derivatives $\Delta_{kl;i}(x)$ = $\mbox{$<\!x_{i,l}\,\vert\,\delta_{v(x)} {\varphi_k}\!>$}$ = $\delta_{v(x)} {\varphi_k}(x_{i,l})$. Analogously to Sect. 3 the functional derivatives $\delta_{v(x)} {\varphi_k}$ can be obtained from the functional derivative of Eq. (71)

$$(\delta_{v(x)} h) \,\varphi_k + h\, \delta_{v(x)} \varphi_k ... ...psilon_k) \,\varphi_k + \epsilon_k \,\delta_{v(x)} \varphi_k .$$ (77)

Projecting onto $\mbox{$<\!\varphi_k\,\vert$}$ and using the hermitian conjugate of Eq. (71) we find the Hartree-Fock version of Eqs. (68) and (69)

$$\delta_{v(x)} \epsilon_k = \frac{<\!\varphi_k\,\vert\,\delta_{v(x)} h\,\vert\,\varphi_k\!>} {\mbox{$<\!\varphi_k\,\vert\,\varphi_k\!>$}} ,$$ (78)

$$\mbox{$\vert\,\delta_{v(x)} \varphi_k\!>$} = \sum_{l\atop \epsilon_l\ne \epsilon_k} \frac{1}{\epsilon_k-\epsil... ...rphi_l\!><\! \varphi_l \, \vert$} \delta_{v(x)} h \mbox{$\vert\,\varphi_k\!>$},$$ (79)

where we, as done before, have fixed orthonormalization and phases by choosing $\mbox{$<\!\delta_{v(x)}\varphi_k\,\vert\,\varphi_l\!>$}$ = 0 for orbitals with equal energy. In contrast to Sect. 3, however, $h$, and thus $\delta_{v(x)} h$, now obey a nonlinear equation. Indeed, from Eq. (70) it follows

$$\delta_{v(x)} h_{x^\prime x^{\prime\prime}} = \sum_k^N \Big( <\!x^{\prime} \,\varphi_k\,\vert\,\delta_{v(x)} V\,\vert\,x^{\prime\prime}\,\varphi_k\!>$$ (80)

$$+ <\!x^{\prime} \,\delta_{v(x)} \varphi_k\,\vert\,V\,\vert\,x^{\p... ...\varphi_k\,\vert\,V\,\vert\,x^{\prime\prime}\,\delta_{v(x)} \varphi_k\!> \Big).$$

Inserting Eq. (78) and Eq. (80) into Eq. (79), we obtain the inverse Hartree-Fock equation for $\delta_{v(x)} \varphi_j$

$$\delta_{v(x)} \varphi_k (x^\prime ) = \sum_{l\atop \epsilon_l\ne \epsilon_k} \frac{1}{\epsilon_k-\epsil... ...ig( <\!\varphi_l\varphi_j\,\vert\,\delta_{v(x)} V\,\vert\,\varphi_k\varphi_j\!>$$ (81)

$$+ <\!\varphi_l\delta_{v(x)}\varphi_j\,\vert\,V\,\vert\,\varphi_k\va... ...!\varphi_l\varphi_j\,\vert\,V\,\vert\,\varphi_k\delta_{v(x)}\varphi_j\!> \Big).$$

Recalling the definition of the antisymmetric matrix elements of $V$ we finally arrive at

$$\delta_{v(x)} \varphi_k (x^\prime ) = \sum_{l\atop \epsilon_l\ne ... ... \; \frac{1}{\epsilon_k-\epsilon_l}\; \varphi_l(x^\prime) \; \sum_j^N \; \times$$ (82)

$$\Bigg(\int\! dz\, \varphi^*_l(z)\varphi^*_j(z-x) \Big( \varphi_k(z) \varphi_j(z-x) -\varphi_k(z-x)\varphi_j(z) \Big)$$

$$+\int\! dz\, \varphi^*_l(z)\varphi^*_j(z+x) \Big( \varphi_k(z) \varphi_j(z+x) -\varphi_k(z+x)\varphi_j(z) \Big)$$

$$+\int\!dz\,dz^\prime\, \varphi^*_l(z) \Big(\delta_{v(x)}\varphi^*... ...t) \Big( \varphi_k(z) \varphi_j(z^\prime) -\varphi_k(z^\prime)\varphi_j(z)\Big)$$

$$+\int\!dz\,dz^\prime\, \Big(\varphi^*_l(z) \varphi^*_j(z^\prime) ... ... v(\vert z-z^\prime\vert) \varphi_k(z) \delta_{v(x)}\varphi_j(z^\prime) \Bigg).$$

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 $\delta_{v(x)}\varphi_j(z^\prime)$ = 0. As the $\delta_{v(x)}\varphi_k(x^\prime)$, which are only required for the $N$ lowest orbitals, depend on two position variables $x$, $x^\prime$, Eq. (82) has essentially the dimension of a two-body equation. Having calculated $\delta_{v(x)} \varphi_k (x_{i,l})$ = $\Delta_{kl;x,i}$ (for $x>0$, $1\le k\le N$, $1\le l\le N$, $1\le i\le n$) from Eq. (82) the likelihood terms in the stationarity equation (44) follow as

$$\delta_{v(x)} \ln p(x_i\vert\hat x,\rho_{\rm HF}(v)) = \frac{\sum_{kl}^N M_{kl;i}\Delta_{kl;i}(x)}{\det B_i} + \frac{\sum_{kl}^N M^\dagger_{kl;i} \Delta^\dagger_{kl;i}(x)}{\det B^\dagger_i}$$

$$= {\rm Tr} (B_i^{-1}\Delta_{i}(x)) +{\rm Tr} ({B_i^\dagger}^{-1}\Delta_{i}^\dagger(x)) ,$$ (83)

recalling that $M_{kl,i}$ = $(B_i)^{-1}_{lk}\det B_i$ and defining analogously to $B_i$ the matrix $\Delta_{i}(x)$ with elements $\Delta_{kl;i}(x)$. The freedom to linearly rearrange orbitals within the Slater determinants $\det \{\mbox{$\vert\,x_{i,l}\!>$}\}$ = $\det \{\mbox{$\vert\,\tilde x_{i,l}\!>$}\}$ (for each data point $i$, analogously for $\det\{\mbox{$\vert\,\varphi_k\!>$}\}$), makes it possible to diagonalize the matrix of overlaps $\mbox{$<\!\varphi_k\,\vert\,\tilde x_{i,l}\!>$}$ in new orbitals $\mbox{$\vert\,\tilde x_{i,l}\!>$}$, which are then linear combinations of the $\mbox{$\vert\,x_{i,l}\!>$}$ [80,78].