79 lines
2.9 KiB
TeX
79 lines
2.9 KiB
TeX
%=====================
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\section{
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\label{sec:FI}
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Fundamental integrals}
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%=====================
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Following Persson and Taylor \cite{Persson96}, the $\sexpval{\bo}^{\bm{m}}$ are derived starting from the momentumless integral \eqref{eq:def4} using the following Gaussian integral representation for the Coulomb operator
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\begin{equation}
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C_{12} = \frac{2}{\sqrt{\pi}} \int_0^\infty \exp(-u^2 \ree^2) du.
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\end{equation}
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After a lengthy derivation which is not presented here for the sake of simplicity, one can show that the closed-form expression of the FIs is
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\begin{equation}
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\label{eq:Fund0m}
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\sexpval{\bo}^{m}
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= \frac{2}{\sqrt{\pi}} \sexpval{\bo}_{G} \sqrt{\frac{\delta_0}{\delta_1-\delta_0}} \qty(\frac{\delta_1}{\delta_1-\delta_0} )^{m} F_m \qty[ \frac{ \delta_1 \qty( Y_1-Y_0 )}{\delta_1-\delta_0} ],
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\end{equation}
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where $m$ is an auxiliary index, $F_m(t)$ is the generalised Boys function, and
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\begin{equation}
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\label{eq:Fund0GGGG}
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\sexpval{\bo}_{G}
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= \qty( \prod_{i=1}^4 S_{i} ) \qty( \frac{\pi^4}{\delta_0} )^{3/2} \exp(-Y_0)
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\end{equation}
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is the FI of the ``pure'' GG operator $G_{13}G_{14}G_{23}G_{34}$ from which one can easily get the FI of the 3-chain operator $G_{13}G_{23}$ by setting $\la_{14} = \la_{34} = 0$.
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While the FIs involving a Coulomb operator contain an auxiliary index $m$, the FIs over ``pure'' GG operators (like $G_{13}G_{23}$) do not, thanks to the factorisation properties of GGs \cite{GG16}.
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This is a a major computational saving as the computation of these auxiliary integrals can take a significant fraction of the CPU time, even for two-electron integrals.
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The various quantities required to compute \eqref{eq:Fund0m} are
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\begin{equation}
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\bm{\delta}_u
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= \bm{\zeta} + \bm{\la}_u = \bm{\zeta} + \bG + u^2 \bC,
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\end{equation}
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where
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\begin{subequations}
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\begin{align}
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\bm{\zeta} & =
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\begin{pmatrix}
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\zeta_1 & 0 & 0 & 0 \\
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0 & \zeta_2 & 0 & 0 \\
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0 & 0 & \zeta_3 & 0 \\
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0 & 0 & 0 & \zeta_4 \\
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\end{pmatrix},
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\quad
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\bC =
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\begin{pmatrix}
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1 & -1 & 0 & 0 \\
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-1 & 1 & 0 & 0 \\
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0 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 \\
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\end{pmatrix},
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\\
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\bG & =
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\begin{pmatrix}
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\la_{13}+\la_{14} & 0 & -\la_{13} & -\la_{14} \\
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0 & \la_{23} & -\la_{23} & 0 \\
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-\la_{13} & -\la_{23} & \la_{13}+\la_{23}+\la_{34} & -\la_{34} \\
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-\la_{14} & 0 & -\la_{34} & \la_{14}+\la_{34} \\
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\end{pmatrix},
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\end{align}
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\end{subequations}
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and
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\begin{subequations}
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\begin{align}
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\bm{\Delta}_u & = \bm{\zeta} \cdot \bm{\delta}_u^{-1} \cdot \bm{\zeta},
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&
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\bY^k & =
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\begin{pmatrix}
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0 & \bY_{12}^k & \bY_{13}^k & \bY_{14}^k \\
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0 & 0 & \bY_{23}^k & \bY_{24}^k \\
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0 & 0 & 0 & \bY_{34}^k \\
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0 & 0 & 0 & 0 \\
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\end{pmatrix},
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\\
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\delta_u & = \det(\bm{\delta}_u),
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&
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Y_u & = \Tr( \bm{\Delta}_u \cdot \bY^2).
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\end{align}
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\end{subequations}
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The generalised Boys function $F_m(t)$ in Eq.~\eqref{eq:Fund0m} can be computed efficiently using well-established algorithms \cite{Gill91, Ishida96, Weiss15}.
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