Time-Dependent Relativistic Two-Component Equation-of-Motion Coupled-Cluster for Open-Shell Systems: TD-EA/IP-EOMCC

We present a combined imaginary-time/real-time time-dependent (TD) approach for evaluating linear absorption spectra of open-shell systems at the electron attachment (EA) and ionization potential (IP) equation-of-motion coupled-cluster (EOMCC) levels of theory and within the exact two-component relativistic framework. The absorption lineshape is given by the Fourier transform of the electric dipole autocorrelation function, which is obtained from a real-time simulation. Approximations of the lowest-energy EA- and IP-EOMCC eigenstates, which are required as initial states for the real-time simulation, are generated by propagating a Koopman EA/IP state in imaginary time. TD-EA/IP-EOMCC linear absorption spectra of open-shell atomic systems (Na, K, Rb, F, Cl, and Br) closely reproduce those obtained from standard TD-EA/IP procedures carried out in the frequency domain. We find that the existence of low-lying states with non-negligible overlap with the Koopman determinant impacts the length of the imaginary-time propagation required to obtain an initial state that produces correct absolute energies and peak height intensities in spectra extracted from the subsequent real-time TD-EA/IP-EOMCC calculations.

💡 Research Summary

This paper introduces a combined imaginary‑time/real‑time time‑dependent (TD) methodology for computing linear absorption spectra of open‑shell systems at the electron‑attachment (EA) and ionization‑potential (IP) equation‑of‑motion coupled‑cluster (EOM‑CC) levels, within the exact two‑component (X2C) relativistic framework. The central idea is to obtain the absorption lineshape by Fourier transforming the electric‑dipole autocorrelation function, which is generated from a real‑time propagation of suitably prepared initial states.

The authors first review the standard EA‑EOMCCSD and IP‑EOMCCSD formalisms. Starting from a closed‑shell reference, the coupled‑cluster singles‑and‑doubles (CCSD) cluster operator (\hat T) is determined, and non‑particle‑conserving excitation operators (\hat R^{+1}) (EA) and (\hat R^{-1}) (IP) are applied to generate (N + 1)‑ and (N − 1)‑electron states, respectively. Because the similarity‑transformed Hamiltonian (\bar H = e^{-\hat T}\hat H e^{\hat T}) is non‑Hermitian, left‑hand de‑excitation operators (\hat L^{\pm1}) are introduced, leading to a bi‑orthonormal set of right and left eigenvectors.

To move to the time domain, the dipole operator (\mu_\alpha) is similarity‑transformed ((\bar\mu_\alpha)) and used to construct moment vectors (|M_{\alpha}^{(N\pm1)}\rangle = \bar\mu_\alpha \hat R^{\pm1}0 |\Phi\rangle) and (\langle\tilde M{\alpha}^{(N\pm1)}| = \langle\Phi| \hat L^{\pm1}0 \bar\mu\alpha). These moments are essentially the same algebraic objects that appear in the standard EA/IP‑EOMCCSD sigma‑build, so existing EA/IP‑EOMCCSD code can be reused with minimal changes. Real‑time propagation is performed with the propagator (\hat U(t)=e^{i\bar H_N t}), where (\bar H_N = \bar H - E^{(N\pm1)}_0) is the Hamiltonian shifted by the (N ± 1)‑electron ground‑state energy. The dipole autocorrelation function (\langle\tilde M(t)|M(0)\rangle) is evaluated, damped with a Lorentzian factor (\exp(-\gamma|t|)), and Fourier‑transformed to yield the frequency‑dependent oscillator strength (f(\omega)).

A novel aspect of the work is the generation of the required lowest‑energy EA/IP eigenstates via imaginary‑time propagation of a single Koopmans determinant (the highest occupied orbital for IP or the lowest virtual orbital for EA). By evolving (|\Phi^{(N\pm1)}\rangle) under (\exp(-\bar H \tau)) (with (\tau = -i t)), the wavefunction converges to the ground‑state of the (N ± 1)‑electron manifold without solving the left‑right eigenvalue problems explicitly. The propagation is stopped when the energy change falls below a preset threshold, and bi‑orthonormality is enforced at each step.

Computational details: all calculations were carried out with an in‑house Python implementation that leverages the p†q automatic code generator. One‑electron X2C (1eX2C) integrals were obtained from Chronus Quantum, while non‑relativistic two‑electron integrals came from PSI4. The X2C‑SVPall‑2c basis set was employed. EA‑EOMCCSD calculations were performed on the cationic references Na⁺, K⁺, and Rb⁺, while IP‑EOMCCSD calculations used the anionic references F⁻, Cl⁻, and Br⁻. Core electrons were frozen (1s for K⁺/Cl⁻, ten core orbitals for Rb⁺/Br⁻) and, for Rb⁺, the 40 highest virtual orbitals were also frozen to reduce cost. Imaginary‑time steps of Δτ = 0.01 a.u. up to τ = 10 a.u. were used to generate the initial states; real‑time propagation employed a standard Runge‑Kutta integrator.

Results show that the TD‑EA/IP‑EOMCCSD absorption spectra of the six atoms reproduce the corresponding frequency‑domain EA/IP‑EOMCCSD spectra with high fidelity. Peak positions, spin‑orbit split components, and relative intensities match closely, confirming that the X2C relativistic treatment captures the essential spin‑orbit coupling. A key observation is that when low‑lying excited states have non‑negligible overlap with the initial Koopmans determinant, longer imaginary‑time propagation is required to obtain an initial state that yields correct absolute energies and peak heights. Conversely, when the overlap is small, a short imaginary‑time evolution suffices.

The paper’s contributions are threefold: (1) it extends non‑particle‑conserving EA/IP‑EOMCC to a real‑time TD framework, enabling spin‑adapted open‑shell spectroscopy; (2) it introduces an efficient imaginary‑time scheme for generating the required EA/IP ground‑state wavefunctions, bypassing the need to solve left‑right eigenvalue problems; (3) it demonstrates that the combined approach works robustly within an exact two‑component relativistic Hamiltonian, making it suitable for heavy‑element spectroscopy where spin‑orbit effects are pronounced. The methodology opens the door to time‑dependent studies of radicals, ions, and other open‑shell species, including non‑linear spectroscopies and pump‑probe dynamics, with a favorable balance of accuracy and computational cost.