mirror of https://github.com/QMCPACK/qmcpack.git
166 lines
6.3 KiB
TeX
166 lines
6.3 KiB
TeX
\documentclass[letterpaper]{article}
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\special{papersize=8.5in,11in}
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\setlength{\pdfpageheight}{\paperheight}
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\setlength{\pdfpagewidth}{\paperwidth}
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\usepackage{amsmath}
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\title{Notes on Orbital Optimization}
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\author{Ken Esler}
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\begin{document}
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\maketitle
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\newcommand{\vr}{\mathbf{r}}
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\newcommand{\Aopt}{\tilde{A}}
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\section{Trial wave functions in logarithmic form}
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Let us consider a trial wave function, $\psi$, which is a product of
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components,
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\begin{equation}
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\psi = \psi_1 \psi_2 \psi_3 \dots
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\end{equation}
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To avoid the complicated application of the product rule, we work
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rather with the quantity, $\ln(\psi)$.
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\begin{equation}
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\ln(\psi) = \ln(\psi_1) + \ln(\psi_2) + \ln(\psi_3) \dots
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\end{equation}
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\subsection{Optimizing trial wave functions}
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Each of the components of $\psi$ can be parametrized to allow
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optimization of the trial function. Efficient optimization algorithms
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can be employed if we can compute the derivatives of two quantities
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with respect to these parameters. For simplicity, we label one
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parameter as $\alpha$. We then need to compute
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\begin{eqnarray}
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\partial_\alpha \ln(\psi) & = & \partial_\alpha \ln (\psi_1)
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+ \partial_\alpha \ln (\psi_2) + \dots,\ \text{and} \\
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\partial_\alpha \left[ \frac{\nabla^2\psi}{\psi}\right] & = &
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\partial_\alpha \left\{\nabla^2 \ln(\psi) + \left[\nabla
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\ln(\psi)\right]^2 \right\} \\
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& = & \partial_\alpha \nabla^2 \ln(\psi) + 2 \left[\nabla \ln(\psi)
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\right]\cdot \left[ \partial_\alpha \nabla \ln(\psi)\right].
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\end{eqnarray}
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Let us consider the second term in terms of its components.
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\begin{eqnarray}
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\partial_\alpha \left[ \frac{\nabla^2\psi}{\psi}\right] & = &
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\partial_\alpha \nabla^2\ln(\psi_1) + \partial_\alpha
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\nabla^2\ln(\psi_2)+ \dots \\
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& & + 2\left[\nabla \ln(\psi)\right]\cdot
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\left[\partial_\alpha \nabla \ln(\psi_1)\right] +
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2\left[\nabla \ln(\psi)\right]\cdot
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\left[\partial_\alpha \nabla \ln(\psi_2)\right] + \cdots \nonumber
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\end{eqnarray}
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Thus, the contribution of $\psi1$ to the total is given by
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\begin{equation}
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\partial_\alpha \left[ \frac{\nabla^2\psi}{\psi}\right]_{\psi_1} =
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\partial_\alpha \nabla^2\ln(\psi_1) + 2\left[\nabla \ln(\psi)\right]\cdot
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\left[\partial_\alpha \nabla \ln(\psi_1)\right].
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\end{equation}
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\section{Statement of problem}
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We would like to optimize the occupied single-particle orbitals,
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$\left\{\phi_i(\vr)\right\}$ in a determinant in the basis of the
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single-particle excited states, $\left\{\varphi_j(\vr)\right\}$.
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That is, we write the optimized orbitals,
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$\left\{\tilde{\phi}_i(\vr)\right\}$, as
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\begin{equation}
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\tilde{\phi}_i(\vr) = \phi_i(\vr) + \sum_j c_{ij} \varphi_j(\vr).
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\end{equation}
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In this set of notes, we derive the required algebra for computing the
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derivatives of the determinants w.r.t. $c_{ij}$. First, let
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\begin{equation}
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A_{mn} \equiv \phi_m(\vr_n),\qquad \Aopt_{mn} \equiv \tilde{\phi}_m(\vr_n)
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\end{equation}
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If we let $\vr_n \rightarrow \vr'_n$, we can compute the ratio of the
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new determinant to the old as
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\begin{equation}
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\frac{\det(A')}{\det(A)} = \sum_m \left(A^{-1}\right)_{nm} \phi_m(\vr_n').
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\end{equation}
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Similarly,
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\begin{equation}
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\frac{\nabla_n\det(A)}{\det(A)} = \sum_m \left(A^{-1}\right)_{nm} \nabla\phi_m(\vr_n).
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\end{equation}
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Analogously,
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\begin{equation}
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\frac{d}{d c_{ij}} \log\left[\det(\Aopt)\right] = \sum_n
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\left(A^{-1}\right)_{ni} \varphi_j(\vr_n).
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\end{equation}
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We need to also compute the derivative of the local kinetic energy
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with respect to $c_{ij}$. These terms are more complicated.
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We first need to compute
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\begin{equation}
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\frac{d}{d_{c_{ij}}} \left(\Aopt^{-1}\right)_{nm} =
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\end{equation}
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\begin{equation}
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\left[A + e_k\delta^T\right]^{-1} = A^{-1} - \frac{A^{-1}e_k\delta^T
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A^{-1}}{1 + \delta^T A^{-1}e_k},
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\end{equation}
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\begin{equation}
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\left[A + e_k\delta^T\right]^{-1}_{nm} = \left[A^{-1}\right]_{nm} - \frac{\left[A^{-1}e_k\right]_n\left[\delta^T
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A^{-1}\right]_m}{1 + \delta^T A^{-1}e_k},
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\end{equation}
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\begin{equation}
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\left[A + e_k\delta^T\right]^{-1}_{nm} = \left[A^{-1}\right]_{nm} - \frac{\left[A^{-1}\right]_{nk}\left[\delta^T
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A^{-1}\right]_m}{1 + \delta^T A^{-1}e_k},
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\end{equation}
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\begin{eqnarray}
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\frac{d \left[\Aopt^{-1}\right]_{nm}}{d_{c_{ij}}} & = &
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-\left[\Aopt^{-1}\right]_{ni}\left[\varphi_j(\vr_k)\Aopt^{-1}\right]_m \\
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& = & -\left[\Aopt^{-1}\right]_{ni} \sum_k \left(\Aopt^{-1}\right)_{km} \varphi_j(\vr_k) \\
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& = & -\left[\Aopt^{-1}\right]_{ni} \frac{d}{d c_{mj}} \log\left[\det(\Aopt)\right]
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%-\left[\varphi_j(\vr_m)\Aopt^{-1}\right]_i
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\end{eqnarray}
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Define
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\begin{equation}
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\gamma_{mj} \equiv \frac{d}{d c_{mj}} \log\left[\det(\Aopt)\right].
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\end{equation}
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Now, we can compute
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\begin{eqnarray}
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\frac{d}{dc_{ij}} \left[\frac{\nabla^2_n
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\det(\Aopt)}{\det(\Aopt)}\right] & = & \sum_m \left\{\rule{0cm}{0.6cm}
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-\left[\Aopt^{-1}\right]_{ni}\left[\varphi_j(\vr_i)\Aopt^{-1}\right]_m
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\nabla^2 \phi_m(\vr_n) +
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\left(\Aopt^{-1}\right)_{nm} \nabla^2 \varphi_j(\vr_n)\delta_{i,m} \right\} \nonumber \\
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& = & \sum_m \left\{\rule{0cm}{0.6cm}
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-\left[\Aopt^{-1}\right]_{ni} \gamma_{mj}
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\nabla^2 \phi_m(\vr_n) +
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\left(\Aopt^{-1}\right)_{nm} \nabla^2 \varphi_j(\vr_n) \delta_{i,m}\right\} \\
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& = & \sum_m \left\{\rule{0cm}{0.6cm}
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-\left[\Aopt^{-1}\right]_{ni} \gamma_{mj}
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\nabla^2 \phi_m(\vr_n) \right\}+
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\left(\Aopt^{-1}\right)_{ni} \nabla^2 \varphi_j(\vr_n)
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\end{eqnarray}
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Define
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\begin{eqnarray}
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L_{nm} & \equiv & \nabla^2\phi_m(\vr_n) \\
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\mathcal{L}_{nj} & \equiv & \nabla^2\varphi_j(\vr_n)
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\end{eqnarray}
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\begin{equation}
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\frac{d}{dc_{ij}} \left[\frac{\nabla^2_n \det(\Aopt)}{\det(\Aopt)}\right] =
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\sum_m \left\{\rule{0cm}{0.6cm}
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-\left[\Aopt^{-1}\right]_{ni} L_{nm} \gamma_{mj} \right\} + \left[\Aopt^{-1}\right]_{ni} \mathcal{L}_{nj}
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\end{equation}
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\newpage
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When we change $c_{ij}$, a column of $\Aopt$ changes. Using the Sherman-Morrison formula, we can compute the change in $\Aopt^{-1}$.
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\begin{equation}
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\left[\Aopt^{-1} + \Delta c_{ij} \varphi_j e_i^T\right]^{-1}_{nm} =
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\Aopt^{-1} - \frac{\Delta c_{ij} \left[\sum_k \left(\Aopt^{-1}\right)_{nk} \varphi_j(\vr_k)\right]\Aopt^{-1}_{im}}{1 + \lambda}.
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\end{equation}
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We expand around $\Delta c_{ij}=0$. Taking the derivative,
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\begin{equation}
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\frac{d}{dc_{ij}} \left[\Aopt^{-1}\right]_{nm} = \underbrace{\left[-\sum_k \Aopt^{-1}_{nk} \varphi_j(\vr_k)\right]}_{\equiv \gamma_{nj}}\Aopt^{-1}_{im}.
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\end{equation}
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Now, let us consider how the gradient with respect particle $m$ changes with
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$c_{ij}$. Two terms contribute. First, a term contributes from the
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ratio of
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\end{document}
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