RPA/ACFDT: Correlation energy in the Random Phase Approximation

Diagrammatic approach to the correlation energy

The correlation energy $E_{c}$ is defined as the missing piece of the Hartree-Fock energy $E_{x}$ to the total energy, that is $E_{tot}=E_{x}+E_{c}$. The exact form of $E_{c}$ is unknown and can be calculated only approximately for a realistic system. The Random Phase Approximation (RPA) is such an approximation that provides access to $E_{c}$. The RPA was first studied by Bohm and Pines for the homogeneous electron gas and was later recognized by Gell-Mann and Brueckner as an approximation of $E_{c}$ that can be expressed in the same language as Feynman used a few years earlier to describe the positron.

Diagrammatic approaches are formulated with quantum field theory, which in turn is based on the Gell-Mann and Low theorem. This theorem states that the eigenstate of an interacting Hamiltonian can be expressed in terms of the eigenstates of the non-interacting one. For this reason, each diagrammatic calculation, like the RPA or GW, requires the solution of the non-interacting Hamiltonian $H_{0}$ of the system, like for instance the Hartree-Fock energies and orbitals or the solutions of the Kohn-Sham Hamiltonian $\epsilon _{n{\bf {k}}},\phi _{n{\bf {k}}}$.

Quantum field theory is commonly formulated in the Dirac (also known as interaction) picture, where the dynamics described by the interaction part $V$ of the fully interacting Hamiltonian $H=H_{0}+V$ are singled out via time-dependent operators like $V(t)=e^{iH_{0}t}Ve^{-iH_{0}t}$. These operators act on states like the non-interacting groundstate of the system $|\Psi (t)\rangle$ causing fluctuations at a specific point in time $t$. The main idea of quantum field theory is to understand observations, which can be measured by an observer, as a collective phenomena of all possible fluctuations.

For a system of electrons and holes, fluctuations are subdivided into creation $\psi ^{\dagger }({\rm {r}},t)$ and annihilation operators $\psi ({\rm {r}},t)$ describing the creation and annihilation of an virtual electron (negatively charged fluctuation) at $({\rm {r}},t)$, respectively. In fact, all operators that describe a measurable quantity of a system of interacting electrons can be represented in terms of $\psi ^{\dagger }({\rm {r}},t)$ and $\psi ({\rm {r}},t)$ alone; additional objects are not necessary. One, soley imposes (equal-time) fermionic anti-commutator relations to describe the effect of exchanging two operators

$\lbrace \psi ({\rm {r}},t),\psi ^{\dagger }({\rm {r}},t)\rbrace =\psi ^{\dagger }({\rm {r}},t)\psi ^{\dagger }({\rm {r}},t)+\psi ^{\dagger }({\rm {r}},t),\psi ({\rm {r}},t)=i\delta ({\rm {r}}-{\rm {r}}').$

$\lbrace \psi ^{\dagger }({\rm {r}},t),\psi ^{\dagger }({\rm {r}},t)\rbrace =0=\lbrace \psi ({\rm {r}},t),\psi ({\rm {r}},t)\rbrace$

The anti-commutator $\lbrace A,B\rbrace =AB+BA$ accounts for the Pauli-principle and excludes contributions, where two virtual fermions are created (or annihilated) at the same space-time point.

The anti-commuator relations reappear implicitly in the definition of the time-ordering operator

$TA(t)B(t')=\Theta (t-t')A(t)B(t')-\Theta (t'-t)B(t')A(t)$

and the time-evolution operator

$S(t,t_{0})=Te^{-i\int _{t_{0}}^{t}V(t'){\rm {d}}t'},$

Then the Gell-Mann and Low theorem states that the vectors ${\frac {|\Omega _{\nu }\rangle }{\langle \Omega _{\nu }|\Psi _{\nu }\rangle }}=\lim _{\eta \to 0}{\frac {S_{\eta }(0,-\infty )|\Psi _{\nu }\rangle }{\langle \Omega _{\nu }|\Psi _{\nu }\rangle }}$ are the eigenstates of the interacting Hamiltonian. Consequently the interacting groundstate energy reads $E_{tot}=E_{0}=\langle \Omega _{0}|H|\Omega _{0}\rangle ={\frac {\langle \Psi _{0}|S(\infty ,-\infty )H|\Psi _{0}\rangle }{\langle \Psi _{0}|S(\infty ,-\infty )|\Psi _{0}\rangle }}$

This compact expression is exact and can be used to derive all possible approximations by expanding the time-evolution operator $S$ into a series. The resulting perturbation series is evaluated term by term using Wick's theorem, where each term is expressed by only two quantities, the non-interacting Feynman propagator

$G_{0}({\rm {r,r'}},t,t')=-i\langle \Psi _{0}|T\lbrace \psi ^{\dagger }({\rm {r}},t),\psi {({\rm {r'}}}t')\rbrace |\Psi _{0}\rangle$

and the Coulomb interaction

$V({\rm {r,r'}},t,t')={\frac {\delta (t-t')}{|{\rm {r}}-{\rm {r'}}|}}.$

Then, each term in the series corresponds to an integral over $({\rm {r}},t)$ that can be visualized with specific rules for the mapping of $G_{0}$ and $V$ and results in a specific diagram.

The Random Phase Approximation

The RPA is obtained from neglecting all terms in the perturbation series of the groundstate energy, except of those which can be expressed in terms of the independent particle polarizability

$\chi _{0}({\rm {r,r'}},t,t')=-iG_{0}({\rm {r,r'}},t,t')G_{0}({\rm {r',r}},t',t)$

From a diagrammatic point of view, this corresponds to ring or bubble diagrams for the correlation part.