Variational Interaction Terms

The general equations for the variational parameters ${q}$ in a parameterized density matrix, from Eq. 3.19, are

$$\frac{1}{2}\frac{\partial H}{\partial \vec{q}}\; + \;\, ... ...\textstyle \leftrightarrow}}{{\mathcal{N}}}\: \dot{\vec{q}}= 0$$ (230)

where

$$H \equiv \int \rho {\mathcal{H}}\rho\;{\bf d}{\bf R}= \int \rho {\mathcal{H}}\rho_{I}{\bf d}{\bf R}$$ (231)

and the norm matrix

$${\mathcal{N}}_{ji} \equiv \int p_j \: p_i \: \rho^2 \, {\bf ... ...rightarrow q} \frac{\partial^2 N}{ \partial q_j \partial q'_i}$$ (232)

with

$$N \equiv \int \rho({\bf R},\vec{q}\,;\beta) \; \rho({\bf R},\vec{q}\:'\,;\beta)\;{\bf d}{\bf R}\;\;.$$ (233)

The subscript $I$ in Eq. A.2 indicates that only one $\rho$ needs to be antisymmetric and the identity permutation can be used in the other. This appendix contains the detailed formulae for these equations for a parameterized Gaussian density matrix applied to a Coulomb system.

Repeating Eq. 3.56, the parameterized variational density matrix is an anti-symmetrized product of one-particle density matrices,

$$\rho({\bf R},{\bf R'},\beta) = \sum_{\cal{P}}\epsilon_{\cal{P}} \prod_{k} \rho_1({\bf r}_k,{\bf r}'_{{\cal{P}}_k},\beta)$$ (234)

$$= \sum_{\cal{P}}\epsilon_{\cal{P}} e^D \prod_{k} \:(\pi w_{{\cal{P}... ...{ -\frac{1}{w_{{\cal{P}}_k}} ({\bf r}_k-{\bf m}_{{\cal{P}}_k})^2 \right\}\quad,$$ (235)

where the amplitude $D$ and the widths $w_{k}$ and means ${\bf m}_{k}$ are the variational parameters. We also dropped $1/N!$ prefactors which are the same for the norm matrix and thus cancel out. The permutation sum is over all permutations of identical particles (e.g. same spin electrons) and $\epsilon_{\cal{P}}=\pm 1$ is the permutation signature. The initial conditions are $w_k=0$, ${\bf m}_k = {\bf r}_k'$, and $D=0$.

For this ansatz the generator of the norm matrix,

$$N= \sum_{\cal{P}}\epsilon_{\cal{P}}\prod_{k} [\pi (w_{k}+w_{{\cal... ..._{k}-{\bf m}_{{\cal P}_k} ')^{2}/(w_{k}+w_{{\cal P}_k}')\right\} \exp( D+D')\;.$$

For a periodic system the above equation also is summed over all periodic simulation cell vectors, ${\bf L}$, with ${\bf m}_{k}-{\bf m}_{{\cal P}_k}'\rightarrow {\bf m}_{k}-{\bf m}_{{\cal P}_k}'+{\bf L}$. Using this the components of the norm matrix are then:

$${\mathcal{N}}_{DD} = \sum_{\cal{P}}\epsilon_{\cal{P}} N_{\cal P}$$ (236)

$${\mathcal{N}}_{{\bf m}_{i}D} = \sum_{\cal P}\epsilon_{\cal P} \left[ {-2({\bf m}_{i}-{\bf m}_{{\cal P}_i})\over w_{i}+w_{{\cal P}_i}} \right] N_{{\cal P}}$$ (237)

$${\mathcal{N}}_{w_{i}D} = \sum_{\cal P}\epsilon_{\cal P} \left({-1\over w_{i}+w_{{\cal P}_i... ... m}_{i}-{\bf m}_{{\cal P}_i})^{2} \over w_{i}+w_{{\cal P}_i}} \right]N_{\cal P}$$ (238)

$${\mathcal{N}}_{{\bf m}_{i}{\bf m}_{j}} = \sum_{\cal P}\epsilon_{\cal P} \left[ {2\delta_{j,{\cal P}_i}\sta... ...m}_{{\cal P}_{j}^{-1}}) \over (w_{j} +w_{{\cal P}_j ^{-1}})} \right] N_{\cal P}$$ (239)

$${\mathcal{N}}_{{\bf m}_{i}w_{j}} = \sum_{\cal{P}}\epsilon_{\cal{P}} \left[ {\delta_{j,{\cal P}_i}\ov... ... m}_{i}-{\bf m}_{{\cal P}_i})\over w_{i}+w_{{\cal P}_i}} \right] \!\!N_{\cal P}$$

$${\mathcal{N}}_{w_{i}w_{j}} = \sum_{\cal{P}}\epsilon_{\cal{P}} \left\{ {\delta_{j,{\cal P}_i}\o... ...}} \right]\right. +{1\over (w_{i}+w_{{\cal P}_i})(w_{j}+w_{{\cal P}_{j}^{-1}})}$$

$$\hspace*{.5in} \left. \left[{3 \over 2}- {({\bf m}_{i}- {\bf m}_{... ...P}_{j}^{-1}})^{2}\over w_{j}+w_{{\cal P}_{j}^{-1}}} \right] \right\} N_{\cal P}$$ (240)

$$\mbox{where}$$

$$N_{\cal P} = e^{2D}\prod_{j} \frac{ \exp \left\{ -\frac{({\bf m}_{j}-{\bf m}_{... ...al P}_j})} \right\}}{(\pi (w_{j}+w_{{\cal P}_j}))^{3/2}} = N_{{\cal P}^{-1}}\;.$$ (241)

The Hamiltonian for a periodic system of electrons and ions is given by,

$${\mathcal{H}}=-\frac{1}{2}\sum_{i=1}^{N_{e}} {\bf\nabla}^{2... ...i}\sum_{I} Z_{I}\psi({\bf r}_{iI}) +\sum_{i}U_{Mad} +U_{ions}$$ (242)

where the purely ionic terms are,

$$U_{ions} = \sum\sum_{I<I'} Z_{I}Z_{I'}\psi({\bf r}_{II'}) +\sum_{I} Z_{I}^{2}U_{Mad} \;.$$ (243)

The Ewald potential, $\psi({\bf r})$, which includes interactions with periodic images and incorporates charge neutrality reads,

$$\psi({\bf r})=\sum_{{\bf L}}\begin{array}{c}\underline{\mbox{erfc... ...neq 0}{4\pi\over{V^{\!\!\!\!\!\!\:^\diamond}}k^{2}} \exp(i{\bf k}\cdot {\bf r})$$

where ${V^{\!\!\!\!\!\!\:^\diamond}}$ is the periodic cell volume and $G$ an arbitrary constant. The Madelung term in ${\mathcal{H}}$ is the interaction energy of an electron with it's periodic images and neutralizing background (e.g. $U_{Mad}=-1.41865/L$ for a simple cubic simulation cell, the usual case). To do the integrals, we represent the Gaussians by their Fourier series

$$\left({2\over \pi w}\right)^{3/2}\sum_{{\bf L}} e^{-{\frac{2... ...\:^\diamond}}}e^{-k^{2}w/8}e^{i{\bf k}\cdot ({\bf r}-{\bf m})}$$ (244)

and in the interaction terms use the Fourier representation for $\psi({\bf r})$. This finally gives

$$H=\sum_{\cal P} \epsilon_{\cal P} \left\{K_{\cal P}+U_{\cal P}\right\} N_{\cal P}$$ (245)

with

$$K_{\cal P}\!\! = \sum_{i}\left[ \frac{3}{w_{i}+w_{{\cal P}i}} - 2\frac{({\bf m}_{i}-{\bf m}_{{\cal P}i})^{2}}{(w_{i}+w_{{\cal P}i})^{2}}\right ]$$ (246)

$$U_{\cal P}\!\! = \sum_i\sum_{j>i} W(\tilde{\bf m}_{i}-\tilde{\bf m}_{j}, \tilde{w}... ...lde{\bf m}_{i}-{\bf R}_{I}, \tilde{w}_{i}) +\!\sum_{i} U_{\rm Mad}+U_{\rm ions}$$

where $\tilde{w}_{i}\equiv w_{i}w_{{\cal P}i}/(w_{i}+w_{{\cal P}i})$ and $\tilde{{\bf m}}_{i}\equiv ({\bf m}_{i}w_{{\cal P}i}+{\bf m}_{{\cal P}i} w_{i})/ (w_{i}+w_{{\cal P}i})\;$. The interaction integral

$$W({\bf r},w)\equiv \sum_{k\neq 0}\frac{4\pi}{{V^{\!\!\!\!\!\!\:^\diamond}}k^{2}} e^{-k^2 w/4} e^{i{\bf k}\cdot {\bf r}}$$ (247)

is symmetric in ${\bf r}$ when the periodic cell has inversion symmetry. Continuing, the left hand side of Eq. A.1 is

$$H_{D} \equiv \frac{1}{2}\frac{\partial H}{\partial D}=H$$ (248)

$$H_{w_{i}} \equiv \frac{1}{2}\frac{\partial H}{\partial w_{i}}= \frac{1}{2}\sum_{\c... ... P}+ (K_{\cal P}+U_{\cal P}) \frac{\partial N_{\cal P}}{\partial w_{i}}\right\}$$ (249)

$$H_{{\bf m}_{i}} \equiv \frac{1}{2}\frac{\partial H}{\partial {\bf m}_{i}}= \frac{1}{2}\s... ...K_{\cal P}+U_{\cal P}) \frac{\partial N_{\cal P}}{\partial {\bf m}_{i}}\right\}$$ (250)

with

$$\frac{\partial N_{\cal P}}{\partial w_{i}} = \left[-\frac{3}{w_{i}+w_{{\cal P}i}} + 2\frac{({\bf m}_{i}-{\bf m}_{{\cal P}i})^{2}}{(w_{i}+w_{{\cal P}i})^{2}}\right]N_{\cal P}$$ (251)

$$\frac{\partial N_{{\cal P}}}{\partial {\bf m}_{i}} = \left[ -4\frac{({\bf m}_{i}-{\bf m}_{{\cal P}i})}{w_{i}+w_{{\cal P}i}}\right] N_{\cal P}$$ (252)

$$\frac{\partial K_{\cal P}}{\partial w_{i}} = \left[-\frac{6}{(w_{i}+w_{{\cal P}i})^{2}}+ 8\frac{({\bf m}_{i}-{\bf m}_{{\cal P}i})^{2}}{(w_{i}+w_{{\cal P}i})^{3}}\right]$$ (253)

$$\frac{\partial K_{\cal P}}{\partial {\bf m}_{i}} = \left[ -8\frac{{\bf m}_{i}-{\bf m}_{{\cal P}i}}{(w_{i}+w_{{\cal P}i})^{2}}\right]\;.$$ (254)

where we have used the fact that terms in ${\cal P}i$ and ${\cal P}^{-1}i$ give the same contribution under the permutation sum and so combined them. The derivatives of the interaction integral are,

$$\frac{\partial U_{\cal P}}{\partial {\bf m}_{i}}\!\!\! = \frac{2 w_{{\cal P}i}}{w_{i}+w_{{\cal P}i}} \left[ \sum_{j\neq i}... ...})-\sum_{I}Z_{I} {\bf W}^{[1]}(\tilde{m}_{i}-{\bf R}_{I},\tilde{w}_{i}) \right]$$

$$\frac{\partial U_{\cal P}}{\partial w_{i}}\!\!\! = \frac{2 w_{{\cal P}i}}{(w_{i}+w_{{\cal P}i})^{2}} \left[ w_{{\cal... ...um_{I} Z_{I} W^{[2]}(\tilde{m}_{i}- {\bf R}_{I}, \tilde{w}_{i}) \right) \right.$$

$$\hspace*{-0.05in} + \left. ({\bf m}_{{\cal P}i}-{\bf m}_{i})\cdot... ..._{I}Z_{I} {\bf W}^{[1]}(\tilde{m}_{i}-{\bf R}_{I},\tilde{w}_{i})\right) \right]$$

where ${\bf W}^{[1]}$ and $W^{[2]}$ denote the derivatives of $W$ with the first and second argument. Comparing equation A.18 and Eq. A.15 the interaction integral may be written as

$$W({\bf r},w)= \psi({\bf r})-\sum_{\bf L} \begin{array}{c}\un... ...\vert \end{array}+\frac{\pi w }{{V^{\!\!\!\!\!\!\:^\diamond}}}$$ (255)

and its derivatives as:

$${\bf W}^{[1]}({\bf r},w) = {\bf\nabla}\psi({\bf r})+ \sum_{\bf L} \frac{{\bf r}+{\bf L}}{\ve... ...\bf r}+{\bf L}\vert}{\sqrt{\pi w}}\exp(-\vert{\bf r}+{\bf L}\vert^{2}/w)\right)$$

$$W^{[2]}({\bf r},w) = -\sum_{\bf L} \frac{\exp(-\vert{\bf r}+{\bf L}\vert^{2}/w)}{w^{3/2}\sqrt{\pi}} +\frac{\pi}{{V^{\!\!\!\!\!\!\:^\diamond}}}$$ (256)

For an isolated system ( ${\bf L}\rightarrow\infty$) and these would simplify to,

$$W({\bf r},w) = \frac{\mbox{erf}\;[r/\sqrt{w}\;]}{ r}$$ (257)

$${\bf W}^{[1]}({\bf r},w) = -\frac{{\bf r}}{r^{3}}\left( \mbox{erf}\;[r/\sqrt{w}\;]-\frac {2 r}{\sqrt{\pi w}}e^{-r^{2}/w}\right)$$ (258)

$$W^{[2]}({\bf r},w) = -\frac{1}{w \sqrt{\pi w}}e^{-r^{2}/w}$$ (259)

At $\beta=0$, the initial derivatives for the variational parameters reduce to

$$\dot{w}_{i} = 2$$ (260)

$$\dot{\bf m}_{i} = 0$$ (261)

$$\dot{D} = -U_{I}$$ (262)

For large numbers of electrons, the computational requirements to treat all exchange terms increase drastically. Here the approximation discussed in section 3.7 is used where the kinetic pair exchange corrections given there are added to the identity permutation term derived here.