Resolvent operator approach to many-body perturbation theory. II. Open shells

In this paper, we develop a time‐dependent approach to many‐body perturbation theory for open shells based on the resolvent of the Schrödinger equation. We introduce, analogous to the closed‐shell case, quantities $S_{\mathrm{ij}}{\mathrm{(t)\; =\; i<\phi }}_i{\mathrm{\Vert e}}^{{\mathrm{-i(H-E}}_c\mathrm{)t}}{\mathrm{\hspace{0.17em}\Vert \psi }}_j{\mathrm{>/<\psi }}_j^0{\mathrm{\Vert \psi }}_j\mathrm{>,}$ where ${\psi }_j^0$ and ${\psi }_j$ are the $j\mathrm{th}$ unperturbed and exact functions, respectively. The ${\psi }_j^0’\mathrm{s}$ can be expressed in terms of ’’model space’’ functions ${\phi }_i{\mathrm{=\; \Omega }}_i^{+}\mathrm{\Vert \phi >,}$ where the ${\Omega }_i^{+}$ are appropriate creation/annihilation operator products acting on a conveniently chosen closed‐shell vacuum φ. These ${\phi }_i\mathrm{’s}$ are not necessarily degenerate with respect to the unperturbed Hamiltonian $H_0.$ $E_c$ is the exact (correlated) energy of the vacuum φ. The Fourier transforms $S_{\mathrm{ij}}\mathrm{(\omega )}$ of $S_{\mathrm{ij}}\mathrm{(t)}$ have the form $S_{\mathrm{ij}}{\mathrm{\hspace{0.17em}(\omega )=<\phi }}_i{\mathrm{\Vert (\omega \; +E}}_c{\mathrm{-H)}}^{\text{−1}}{\mathrm{\Vert \psi }}_j{\mathrm{>/<\psi }}_j^0{\mathrm{\Vert \psi }}_j>$ and thus have poles at energy differences ${\mathrm{(E}}_j{\mathrm{-E}}_c\mathrm{),}$ i.e., relative to the exact vacuum energy. Using the time‐dependent perturbation expansion of $\mathrm{S(t),}$ we obtain a Dyson‐like equation ${\mathrm{N̄}}^{\text{−1}}{\mathrm{(\omega )=N̄}}^{\text{0–1}}\mathrm{(\omega )+\Sigma ,}$ where N̄ is defined as ${\mathrm{N̄}}_{\mathrm{ij}}{\mathrm{\hspace{0.17em}(\omega )\; =\; <\phi }}_i{\mathrm{\Vert (\omega +E}}_c{\mathrm{-H)}}^{\text{−1}}{\mathrm{\Vert \phi }}_j>$ and ${\mathrm{N̄}}^0$ is the corresponding unperturbed component. Knowledge of the combining coefficients $C_{\mathrm{ji}}$ in ${\psi }_j^0{\mathrm{=\Sigma }}_J{C}_{\mathrm{ji}}{\phi }_i$ is thus not required for finding the poles. We arrive at the Dyson‐like equation by first eliminating closed diagrams and then regrouping the remaining terms in the perturbation series for S into ’’top’’ and ’’bottom’’ parts. Regrouping appropriate to the Brillouin–Wigner (BW) case together with an associated time‐integration procedure yields ${\Sigma }^{\mathrm{BW}}$ which consists of disconnected and ω‐dependent diagrams. This is shown to yield the open‐shell BW series in the Bloch–Horowitz form. An alternative regrouping procedure and use of the ’’folding technique’’ of Johnson and Baranger leads to a ${\Sigma }^{\mathrm{RS}}$ which is ω‐independent, Hermitian, contains connected diagrams only, and is, thus, size‐consistent.