Adiabatic connection fluctuation dissipation theorem

In density functional theory (DFT) the adiabatic-connection fluctuation-dissipation theorem (ACFD) is an exact formula for the Kohn–Sham correlation energy. A connection between noninteracting electrons and interacting electrons (the adiabatic connection (AC)) is combined with the random density fluctuations of molecular or solid systems (fluctuation-dissipation (FD)). It is used as a tool in theoretical chemistry and quantum chemistry to approximate the electronic energy.

The theorem states

${\displaystyle E_{c}^{\text{ACFD}}[\rho ]={\frac {-1}{2\pi }}\int _{0}^{1}d\alpha \iint drdr'f_{H}(r,r')\int _{0}^{\infty }d\omega [\chi _{\alpha }(r,r',\omega )-\chi _{0}(r,r',\omega )]}$

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where ${\textstyle f_{H}(r,r')={\frac {1}{|r-r'|}}}$ is the Hartree kernel, ${\textstyle \chi _{\alpha }(r,r',\omega )}$ the interacting dynamic response function, ${\textstyle \chi _{0}(r,r',\omega )}$ the dynamic Kohn–Sham (KS) response function from time-dependent density functional theory (TDDFT).