General-order degenerate coupled-cluster theory

TL;DR

This work introduces ΔCC, a general-order, degenerate extension of coupled-cluster theory that remains size-extensive and black-box while handling both degenerate and nondegenerate references. By formulating an internal M-by-M reference space with a C-condition, it unifies the treatment of excited, ionized, and electron-attached states under a single framework, recovering full FCI in the limit. ΔCC$(S)$ and ΔCC$(SD)$ provide highly accurate energies that, in many benchmark cases, surpass EOM-CC in accuracy and reliability, especially for multi-electron excitations and challenging degenerate references. The authors develop determinant-based and algebraic implementations and demonstrate that ΔCC offers a viable, complementary alternative to MBGF and traditional EOM-CC methods with promising future extensions to derivatives, higher-order corrections, and periodic or finite-temperature regimes.

Abstract

A size-extensive, converging, black-box coupled-cluster ($Δ$CC) ansatz is introduced that computes the energies and wave functions of stationary states from any degenerate or nondegenerate Slater-determinant references with any numbers of $α$- and $β$-spin electrons, any patterns of orbital occupancy, any spin multiplicities, and any spatial symmetries. For a nondegenerate reference, it is identical to the single-reference, projection coupled-cluster ansatz. For degenerate references, it is a natural coupled-cluster extension of degenerate RS perturbation ($Δ$MP) theory, and is closely related to, but distinct from Li and Paldus's state-universal multireference coupled-cluster (SUMRCC) theory. For ionized and electron-attached references, it can be viewed as a coupled-cluster Green's function, although the present theory is convergent toward the full-configuration-interaction (FCI) limits, while many-body Green's function (MBGF) theory generally is not. Its single-excitation instance is a projection (nonvariational) Hartree-Fock theory for a degenerate or nondegenerate reference as per the Thouless theorem, whose practical utility seems rather limited except for core ionizations, high-spin states, and possibly electron affinities. A determinant-based, general-order algorithm is implemented, generating $Δ$CC energies through connected octuple excitations, which are compared with the results from CI, equation-of-motion coupled-cluster (EOM-CC), and SUMRCC theories up to the FCI limits as well as from $Δ$MP and MBGF theories up to the 19th order. An algebraic, optimal-scaling, order-by-order algorithm is also computer-synthesized at the levels of single excitations only and of single and double excitations. The order of performance is: $Δ$CC $>$ EOM-CC $>$ CI at the same order or $Δ$CC $>$ $Δ$MP $>$ MBGF at the same cost scaling.

Figures (39)

• Figure 1: The diagonal energy, $H_{II}$, of $\Delta$CCSD [Eq. (\ref{['DCCSD3']})]. The dashed line with circled $I$ denotes $(f_I)^{\text{out}}_{\text{in}}$, while the dashed line translates to $(v)^{\text{left out, right out}}_{\text{left in, right in}}$. The solid line with $I$ designates $(t_I)^{\text{out}}_{\text{in}}$ or $(t_I)^{\text{left out, right out}}_{\text{left in, right in}}$.
• Figure 2: The left-hand side, $\langle I^{a}_{i} |\hat{H}_Ie^{\hat{T}_I}|I\rangle$, of the $t$-amplitude equation of $\Delta$CCSD [Eq. (\ref{['DCCSD2']})] in the external single-excitation space ($\epsilon_a -\epsilon_i \neq 0$). The double line with $I$ ( 2a) designates the CC effective Hamiltonian $(\hat{H}_I e^{\hat{T}_I})^\text{out}_\text{in}$, which is unlinked through diagram 2h.
• Figure 3: The left-hand side, $\langle I^{a}_{i} |\hat{H}_Ie^{\hat{T}_I}|I\rangle$, of the $t$-amplitude equation of $\Delta$CCSD [Eq. (\ref{['DCCSD2']})] in the internal single-excitation space, i.e., the off-diagonal energy, $H_{JI}$, with $|J\rangle = |I^a_i\rangle$. The red oblong denotes the fictitious resolvent line that demands $\epsilon_a -\epsilon_i = 0$. The double line with $I$ ( 3a) now designates the connected CC effective Hamiltonian $(\hat{H}_I e^{\hat{T}_I})^\text{out}_\text{in}$. See Fig. 10.2 of Shavitt and Bartlettshavitt for a complete list of diagrams.
• Figure 4: The C-condition, $S_{JI}=0$, of $\Delta$CCSD [Eq. (\ref{['DCCSD5']})], where $|J\rangle = |I^a_i\rangle$ with $\epsilon_a -\epsilon_i = 0$.
• Figure 5: The left-hand side, $\langle I^{ab}_{ij} |\hat{H}_Ie^{\hat{T}_I}|I\rangle$, of the $t$-amplitude equation of $\Delta$CCSD [Eq. (\ref{['DCCSD1']})] in the external double-excitation space ($\epsilon_a +\epsilon_b -\epsilon_i -\epsilon_j \neq 0$). The double line with $I$ ( 5a) designates the CC effective Hamiltonian $(\hat{H}_I e^{\hat{T}_I})^\text{left out, right out}_\text{left in, right in}$, which is unlinked through diagrams 5i and 5j.