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Manu: (hopefully) found proper notations for density matrices and Coulomb-exchange integrals

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Emmanuel Fromager 2020-02-13 20:18:16 +01:00
parent 005326057c
commit 3a480b0adc
1 changed files with 27 additions and 5 deletions
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32
Manuscript/eDFT.tex
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@ -415,7 +415,7 @@ integrals.
\blue{ \blue{
\beq \beq
&&\dfrac{1}{2}\sum_{PQRS}\langle PQ\vert\vert &&\dfrac{1}{2}\sum_{PQRS}\langle PQ\vert\vert
RS\rangle\Gamma_{PR}\Gamma_{QS} RS\rangle\Gamma^{(K)}_{PR}\Gamma^{(L)}_{QS}
\nonumber\\ \nonumber\\
&& &&
=\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}RS} =\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}RS}
@ -423,19 +423,41 @@ RS\rangle\Gamma_{PR}\Gamma_{QS}
&&\Big(\langle p^\sigma\sigma q^\tau\tau\vert RS\rangle -\langle &&\Big(\langle p^\sigma\sigma q^\tau\tau\vert RS\rangle -\langle
p^\sigma\sigma q^\tau\tau p^\sigma\sigma q^\tau\tau
\vert SR\rangle \vert SR\rangle
\Big)\Gamma_{p^\sigma\sigma,R}\Gamma_{q^\tau\tau, S} \Big)\Gamma^{(K)}_{p^\sigma\sigma,R}\Gamma^{(L)}_{q^\tau\tau, S}
\nonumber\\ \nonumber\\
&& &&
=\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}} =\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}}
\nonumber\\ \nonumber\\
&&\Big(\sum_{r^\sigma s^\tau}\langle p^\sigma q^\tau\vert r^\sigma s^\tau\rangle &&\Big(\sum_{r^\sigma s^\tau}\langle p^\sigma q^\tau\vert r^\sigma s^\tau\rangle
\Gamma^\sigma_{p^\sigma r^\sigma}\Gamma^\tau_{q^\tau s^\tau} \Gamma^{(K)\sigma}_{p^\sigma r^\sigma}\Gamma^{(L)\tau}_{q^\tau s^\tau}
\nonumber\\ \nonumber\\
&& -\sum_{s^\sigma r^\tau}\langle && -\sum_{s^\sigma r^\tau}\langle
p^\sigma q^\tau p^\sigma q^\tau
\vert s^\sigma r^\tau\rangle \vert s^\sigma r^\tau\rangle
\delta_{\sigma\tau}\Gamma^\sigma_{p^\sigma \delta_{\sigma\tau}\Gamma^{(K)\sigma}_{p^\sigma
r^\sigma}\Gamma^\sigma_{q^\sigma s^\sigma}\Big) r^\sigma}\Gamma^{(L)\sigma}_{q^\sigma s^\sigma}\Big)
\nonumber\\
&&=\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}}
\nonumber\\
&&\left(\langle p^\sigma q^\tau\vert p^\sigma q^\tau\rangle
n_{p^\sigma}^{(K)\sigma}n_{q^\tau}^{(L)\tau}
-\delta_{\sigma\tau}\langle p^\sigma q^\sigma\vert q^\sigma p^\sigma \rangle
n_{p^\sigma}^{(K)\sigma}n_{q^\sigma}^{(L)\sigma}\right)
\nonumber\\
&&=\dfrac{1}{2}\sum_{\mu\nu\lambda\omega}\sum_{\sigma,\tau}\Big(\langle{\mu}{\lambda}\vert{\nu}{\omega}\rangle
\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\tau}
-\delta_{\sigma\tau}\langle\mu\lambda\vert\omega\nu\rangle\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\sigma}
\Big)
\nonumber\\
&&=\dfrac{1}{2}\sum_{\mu\nu\lambda\omega}\sum_{\sigma,\tau}\Big(\langle{\mu}{\lambda}\vert{\nu}{\omega}\rangle
-\delta_{\sigma\tau}\langle\mu\lambda\vert\omega\nu\rangle
\Big)
\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\tau}
\nonumber\\
&&=\dfrac{1}{2}\sum_{\mu\nu\lambda\omega}\sum_{\sigma,\tau}\Big[({\mu}{\nu}\vert{\lambda}{\omega})
-\delta_{\sigma\tau}(\mu\omega\vert\lambda\nu)
\Big]
\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\tau}
\eeq \eeq
} }
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