Adiabatic connection fluctuation dissipation theorem

^[10][11]

The adiabatic connection (AC)^[12] is a perturbation theory along the electron–electron interaction ${\textstyle {\hat {V}}_{\text{ee}}=\sum _{i<j}{\frac {1}{|r_{i}-r_{j}|}}}$ with the coupling strength ${\textstyle 0\leq \alpha \leq 1}$ from the Kohn–Sham (KS) system of non-interacting electrons ${\textstyle \alpha =0}$ towards the real system of interacting electrons ${\textstyle \alpha =1}$ and given by the following perturbative Schrödinger equation

${\displaystyle [\overbrace {{\hat {T}}+{\hat {v}}(\alpha )+\alpha {\hat {V}}_{\text{ee}}} ^{{\hat {H}}(\alpha )}]\Psi (\alpha )=E(\alpha )\Psi (\alpha )}$

${\textstyle {\hat {H}}(\alpha )}$ is the coupling-constant dependent many-body Hamiltonian. ${\textstyle {\hat {T}}=\sum _{i}-{\frac {1}{2}}\Delta _{i}}$ is the many-body kinetic energy operator with the Laplacian ${\textstyle \Delta =\nabla ^{2}}$, where the indices ${\textstyle i,j}$ correspond to the respective electron coordinates, ${\textstyle {\hat {v}}(\alpha )=\sum _{i}v_{i}(\alpha )}$ is the local coupling-strength-dependent potential. Note there that ${\textstyle {\hat {v}}(\alpha =0)={\hat {v}}_{S}}$ is the Kohn–Sham (KS) potential, ${\textstyle {\hat {v}}(\alpha =1)={\hat {v}}_{\text{ext}}}$ the external potential, i.e. electron-nuclei interaction, ${\textstyle \Psi (\alpha =0)=\Phi _{S}}$ the Kohn–Sham (KS) Slater determinant, ${\textstyle \Psi (\alpha =1)=\Psi _{0}}$ the real electronic ground state wave function, ${\textstyle E(\alpha =0)=E_{S}}$ is the energy of the KS system, ${\textstyle E(\alpha =1)=E_{0}}$ is the real electronic ground state energy. Thus accordingly for ${\textstyle \alpha =0}$ the many-body Kohn–Sham (KS) equation is obtained

${\displaystyle [{\hat {T}}+{\hat {v}}_{S}]\Phi _{S}=E_{S}\Phi _{S}}$

while for ${\textstyle \alpha =1}$ the electronic Schrödinger equation is obtained within the Born–Oppenheimer approximation

${\displaystyle [{\hat {T}}+{\hat {v}}_{\text{ext}}+{\hat {V}}_{\text{ee}}]\Psi _{0}=E_{0}\Psi _{0}}$

The coupling-constant-dependent correlation energy is given as difference of the energy of the interacting system minus that of the artificial KS system in bra–ket notation

${\displaystyle E_{c}(\alpha )=\langle \Psi (\alpha )|{\hat {H}}(\alpha )|\Psi (\alpha )\rangle -\langle \Phi _{S}|{\hat {H}}(\alpha )|\Phi _{S}\rangle }$

which can be simplified further with the fact, that the density along the adiabatic-connection stays fixed, and the locality of the potential ${\textstyle \langle \Psi _{0}|v(\alpha )|\Phi _{0}\rangle =\int \rho (r)v(\alpha ,r)dr=\langle \Phi _{S}|v(\alpha )|\Phi _{S}\rangle }$ (This also accounts for the derivative ${\textstyle {\frac {dv(\alpha )}{d\alpha }}}$ which hence cancel out) and apply the Hellmann–Feynman theorem^[13][14] with differentiating the Hamiltonian ${\textstyle {\frac {d{\hat {H}}(\alpha )}{d\alpha }}={\frac {d}{d\alpha }}({\hat {T}}+{\hat {v}}(\alpha )+\alpha {\hat {V}}_{\text{ee}})={\frac {d{\hat {v}}(\alpha )}{d\alpha }}+{\hat {V}}_{\text{ee}}}$

${\displaystyle {\frac {dE_{c}(\alpha )}{d\alpha }}={\bigg \langle }\Psi (\alpha ){\bigg |}{\frac {d{\hat {H}}(\alpha )}{d\alpha }}{\bigg |}\Psi (\alpha ){\bigg \rangle }-{\bigg \langle }\Phi _{S}{\bigg |}{\frac {d{\hat {H}}(\alpha )}{d\alpha }}{\bigg |}\Phi _{S}{\bigg \rangle }}$

${\displaystyle {\frac {dE_{c}(\alpha )}{d\alpha }}=\langle \Psi (\alpha )|{\hat {V}}_{\text{ee}}|\Psi (\alpha )\rangle -\langle \Phi _{S}|{\hat {V}}_{\text{ee}}|\Phi _{S}\rangle }$

Lastly the fundamental theorem of calculus to obtain the correlation energy back is used, which completes the adiabatic-connection (AC) theorem

${\displaystyle E_{c}^{\text{AC}}[\rho ]=\int _{0}^{1}d\alpha {\frac {dE_{c}(\alpha )}{d\alpha }}=E_{c}(\alpha =1)-\underbrace {E_{c}(\alpha =0)} _{0}=\int _{0}^{1}d\alpha \langle \Psi (\alpha )|{\hat {V}}_{\text{ee}}|\Psi (\alpha )\rangle -\langle \Phi _{S}|{\hat {V}}_{\text{ee}}|\Phi _{S}\rangle }$

2

The fluctuation-dissipation theorem, first proven by Herbert Callen and Theodore A. Welton in 1951,^[15] can be reformulated in a modern way for density functional theory to incorporate random fluctuations in the density. The full proof in detail is rather complicated and given in reference.^[2] Some key features will be pointed out here. The response functions are integrated along the frequencies

${\displaystyle \int _{0}^{\infty }d\omega \ \chi (r,r',\omega )=-2\sum _{n\neq 0}\underbrace {\int _{0}^{\infty }{\frac {E_{n}-E_{0}}{(E_{n}-E_{0})^{2}+\omega ^{2}}}d\omega } _{{\bigg [}\arctan {\bigg (}{\frac {\omega }{E_{n}-E_{0}}}{\bigg )}{\bigg ]}_{0}^{\infty }={\frac {\pi }{2}}}\langle \Psi _{0}|{\hat {\rho }}(r)|\Psi _{n}\rangle \langle \Psi _{n}|{\hat {\rho }}(r')|\Psi _{0}\rangle }$

where ${\textstyle {\hat {\rho }}(r)=\sum _{i}\delta (r_{i}-r)}$ is the density operator, a sum of Dirac delta functions, the indices ${\textstyle 0}$ correspond to the ground state, ${\textstyle n}$ to excited states, letting the sum start from ${\textstyle n=0}$, rather than ${\textstyle n\neq 0}$ with the identity operator ${\textstyle \sum _{n=0}|\Psi _{n}\rangle \langle \Psi _{n}|=1}$ and with introducing the 2-electron pair density

${\displaystyle \rho _{2}(r,r')={\frac {1}{2}}\langle \Psi _{0}|{\hat {\rho }}(r){\hat {\rho }}(r')|\Psi _{0}\rangle _{N-2}=\langle \Psi _{0}|{\hat {V}}_{\text{ee}}|\Psi _{0}\rangle }$

after some tedious algebra obtains the fluctuation-dissipation (FD) theorem

${\displaystyle E_{c}^{\text{FD}}[\rho ]=-{\frac {1}{2\pi }}\iint drdr'f_{H}(r,r')\int _{0}^{\infty }d\omega [\chi _{\alpha }(r,r',\omega )-\chi _{0}(r,r',\omega )]=\langle \Psi (\alpha )|{\hat {V}}_{\text{ee}}|\Psi (\alpha )\rangle -\langle \Phi _{S}|{\hat {V}}_{\text{ee}}|\Phi _{S}\rangle }$

3

Combination of the adiabatic-connection (AC) theorem eq. (2) with the fluctuation-dissipation (FD) theorem eq. (3) yields finally the adiabatic-connection fluctuation-dissipation (ACFD) theorem eq. (1).

Only the Kohn–Sham (KS) response function is explicitly known in terms of occupied (denotes as ${\textstyle i}$) and unoccupied (denotes as ${\textstyle a}$) Kohn–Sham (KS) orbitals ${\textstyle \varphi }$ and KS eigenvalues ${\textstyle \varepsilon }$ and is given by

${\displaystyle \chi _{0}(r,r',\omega )=4\sum _{i}\sum _{a}{\frac {\varepsilon _{i}-\varepsilon _{a}}{(\varepsilon _{i}-\varepsilon _{a})^{2}+\omega ^{2}}}\varphi _{i}(r)\varphi _{a}(r)\varphi _{a}(r')\varphi _{i}(r')}$

The interacting response function is calculated from the Petersilka–Gossmann–Gross TDDFT Dyson equation^[16]

${\displaystyle \chi _{\alpha }(r,r',\omega )=\chi _{0}(r,r',\omega )+\iint dr''dr'''\chi _{0}(r,r'',\omega )f_{\text{Hxc}}^{\alpha }(r'',r''',\omega )\chi _{\alpha }(r,r''',\omega )}$

4

while the exchange-correlation (xc) kernel dependens nonlinearly on the coupling strength and the Hartree (H) kernel linearly. Invoking the random phase approximation (RPA)^[17][18][19] i.e. ${\textstyle f_{\text{Hxc}}^{\alpha }(r',r'',\omega )\approx \alpha f_{H}(r,r');~f_{\text{xc}}^{\alpha }(r',r'',\omega )\approx 0}$. That means approximating the Hartree-exchange-correlation (Hxc) kernel with the Hartree kernel or neglecting the exchange-correlation kernel entirely, one obtains the RPA correlation energy while introducing a basis set in matrix notation, if the TDDFT Dyson equation eq. (4) is plugged into the ACFD theorem eq. (1). The coupling constant integration can then be carried out analytically.

${\displaystyle E_{c}^{\text{RPA}}[\{\varphi _{s},\varepsilon _{s}\}]=-{\frac {1}{2\pi }}\int _{0}^{1}d\alpha \int _{0}^{\infty }d\omega \operatorname {Tr} [[[1-\alpha \chi _{0}(\omega )f_{H}]^{-1}-1]\chi _{0}(\omega )f_{H}]=-{\frac {1}{2\pi }}\int _{0}^{\infty }d\omega \operatorname {Tr} [\ln[1-\chi _{0}(\omega )f_{H}]+\chi _{0}(\omega )f_{H}]}$

5

where the trace operator ${\textstyle \operatorname {Tr} \equiv \iint drdr'}$ corresponds to carrying out the spatial integrations, the index ${\textstyle s}$ stands for both occupied and unoccupied KS orbitals. Note here that the RPA correlation energy is a highy KS orbital-dependent functional and is one of the most sophisticated approximations to the correlation energy. It is mostly done in a post-SCF manner. That means the KS orbitals and eigenvalues from a preceding KS calculation such as a generalized gradient approximation like e.g. PBE or hybrid calculation like PBE0 and B3LYP are used.