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 true 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 few years earlier to describe the positron.

Diagrammatic approaches are all based on the Gell-Mann and Low theorem, which 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 solution of the Kohn-Sham Hamiltonian $\lbrace \epsilon _{n{\bf {k}}},\phi _{n{\bf {k}}}\rbrace$.

Diagrammatic perturbation 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 time $t$ with a finite life time. 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 $\psi ({\rm {r}},t)^{\dagger }$ and annihilation operators $\psi ({\rm {r}},t)$ describing the creation and annihilation of an virtual electron at the space-time coordinates ${\rm {r}},t$, 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; no additional objects are required. This includes the interacting Hamiltonian

$H=\underbrace {\int {\rm {d}}{\bf {r}}{\rm {d}}t\psi ^{\dagger }({\rm {r}},t)\left[-{\frac {\Delta }{2}}+V_{ext}\right]\psi ({\rm {r}},t)} _{H_{0}}+\underbrace {\int {\rm {d}}{\bf {r}}{\rm {d}}t\int {\rm {d}}{\bf {r'}}{\rm {d}}t'\psi ^{\dagger }({\rm {r}},t)\psi ^{\dagger }({\rm {r'}},t'){\frac {1}{|{\rm {r}}-{\rm {r'}}|}}\psi ({\rm {r}},t)\psi ({\rm {r'}},t')} _{V}$

To account for the Pauli-principle canonical anti-commuation relations at are introduced

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

and can be seen as the quantum mechanical analogue to the Poisson bracket of classical field theory.

The main quantity in this formalism, is the time-evolution operator

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

that uses the time-ordering operator $T$.

in the limits $t\to \infty$ and $t_{0}\to -\infty$. Here

One of the main quantities in perturbation theory is the time-evolution operator

The RPA becomes exact at high density and zero-temperature.

They showed These diagrams are a pictorial representation of terms in perturbation theory and are often used to describe a specific approximation for the groundstate.

, where only one specific class of diagrams is accounted for, namely bubble (or ring) diagrams. The RPA has a

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