Elucidating Many-Body Effects in Molecular Core Spectra through Real-Time Approaches: Efficient Classical Approximations and a Quantum Perspective

Abstract

Accurately resolving many-body satellite features in molecular core-level spectra requires theoretical approaches that capture electron correlation both efficiently and systematically. The recently developed time-dependent double coupled-cluster (TD-dCC) ansatz achieves this by combining correlation effects from the N- and (N-1)-electron sectors, but its exact formulation remains computationally demanding. Here we introduce a hierarchy of cost-effective approximate TD-dCC ansatzes derived from truncated Baker-Campbell-Hausdorff (BCH) expansions, which preserve a single-similarity-transformation structure while retaining the essential correlation diagrams responsible for satellite formation. We further develop a detailed component analysis that isolates hole-mediated excitation pathways, which are correlated processes arising from the coupling between ground-state and ionized-state amplitudes. We use it to interpret quasiparticle and satellite features across the hierarchy. Applications to the single-impurity Anderson model and molecular systems (H2O and CH4) demonstrate that the approximate TD-dCC methods closely and efficiently reproduce exact many-body spectral features and quasiparticle weights. In parallel, we construct a fault-tolerant quantum signal processing algorithm for the core-hole Green's function, providing a scalable quantum route for simulating correlated core-level dynamics. Together, these developments establish complementary classical and quantum methodologies for quantitative, many-body-accurate core spectroscopy.

Figures (10)

• Figure 1: (a) Systematic hierarchy of approximations to the time-dependent many-body ansätzes employed in the RT-EOM-CC cumulant Green's function approach. (b) Diagrams included in the commutator expansion corresponding to the effective correction $\Delta T^{(nb)}_{\rm eff}$ in the approximate TD-dCC-1($n$b) ansatz. The $s_{i}^{a}(t)$ and $s_{ij}^{ab}(t)$ correspond to the time-dependent single and double cluster amplitudes in the $N{-}1$ sector, while $t_{i}^{a}$ and $t_{ij}^{ab}$ correspond to the $N$-electron cluster amplitudes. The index $c$ denotes the ionized core spin-orbital. This hierarchy illustrates how progressively including commutator terms restores $N$–$(N{-}1)$ correlation pathways important for satellite formation.
• Figure 2: Connected and disconnected (but linked) diagrams contributing to the overlap $\tilde{O}^{\text{HM}}_{(i,j,a,b)}(t)$, where the HM transitions are described through the TD-dCC ansätzes. The core hole is located in spin-orbital $c$. The connected diagram corresponds to two symmetry-equivalent HM transitions due to the antisymmetry of $t_{ic}^{ab}$ with respect to virtual indices $a$ and $b$, whereas the disconnected diagram (linked to the left eigenvector) corresponds to a specific HM transition. These diagrams reveal the algebraic origin of HM coupling channels captured by the TD-dCC framework.
• Figure 3: Spectral function $A(\omega)$ of the four-site SIAM model obtained using various TD-dCC approximation schemes. (a) $U=2$ and (b) $U=3$. The corresponding quasiparticle weight $Z$ is reported for each spectrum. The main quasiparticle peak lies at 0.167 a.u. for $U=2$ and 0.586 a.u. for $U=3$. TD-dCC-based approximations systematically recover satellite positions and spectral weights missed by TD-CC, especially in the strongly correlated $U=3$ regime.
• Figure 4: Most significant HM transition in the four-site SIAM model ($U=2$), shown in terms of its individual spectral function and the corresponding time-dependent Green's function. The TD-dCC hierarchy captures this correlated channel with increasing fidelity as higher-order commutator terms are included.
• Figure 5: Spectral functions of H$_2$O obtained from TD-CC, TD-dCC-1, and the exact approaches. (a) Equilibrium geometry, where the electron in the $2A_1$ orbital is ionized. The highlighted region marks a spurious extra satellite feature in the TD-CC spectrum not present in the exact or TD-dCC results. (b) Stretched geometry with one O--H bond extended to 1.5 Å, where the electron in the $2A'$ orbital is ionized. Only TD-dCC-1 is shown since higher-level TD-dCC results are nearly indistinguishable. Stretching the bond enhances correlation, magnifying differences between TD-CC and TD-dCC.