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Complex-Energy Second-Order Approximate Coupled-Cluster Methods for Electronic Resonances

Cansu Utku, Garrette Pauley Paran, Thomas-C. Jagau

TL;DR

The paper develops two independent complex-energy RI-CC2 implementations (RI-EA-CC2 and RI-EE-CC2) incorporating CAP and CBF approaches to characterize electronic resonances, including EA and IP variants within EOM formalisms. It demonstrates that RI-CC2 offers substantial computational savings ($O(N^5)$ scaling and $O(N^3)$ memory) while producing resonance energies and widths that are broadly in good agreement with EOM-EA-CCSD, with systematic underestimation of widths and slightly less negative affinities that spin-scaling can ameliorate for larger systems. Benchmarking on a set of temporary anions (N$_2^-$, C$_2$H$_4^-$, CH$_2$O$^-$, HCOOH$^-$, uracil, naphthalene, cyan-naphthalenes, and pyrene) shows that EA-CC2 often aligns better with experimental trends than EOM-EA-CCSD, particularly when spin scaling is applied, and that CAP and CBF approaches generally agree within about $0.1$ eV for energies and widths. The work enables efficient exploration of resonances in large molecules (e.g., pyrene) and sets the stage for extending complex-energy CC2 to properties and gradients, broadening its applicability in spectroscopy and radiation-chemistry contexts.

Abstract

Electronic resonances are metastable states with finite lifetimes, encountered in processes such as photodetachment, electron transmission, and Auger decay. Resonances appear in Hermitian quantum mechanics as increased density of states in the continuum rather than as discrete energy levels. To describe resonances accurately, including their coupling to the continuum, methods based on non-Hermitian quantum mechanics can be used, which yield complex energies. In this work, we combine the complex absorbing potential (CAP) and complex basis functions (CBF) techniques with the RI-CC2 method. The second-order coupled cluster method (CC2) offers a good balance between accuracy and computational cost by approximating equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) theory, making it suitable for studying of electronic resonances in larger molecules. The resolution-of-the-identity (RI) approximation further reduces computational demands without significant loss in accuracy. We investigate the numerical performance of the new complex-energy RI-CC2 methods focusing on temporary anions. Negative electron affinities and decay widths can be computed using the electron-attachment (EA) variant of RI-CC2. For N2, C2H4, CH2O, and HCOOH, EA-CC2 yields affinities about 0.1-0.2 eV smaller than EOM-EA-CCSD, while deviations reach 0.5 eV for larger anions such as uracil, naphthalene, cyanonaphthalene, and pyrene. As a result of these trends, EA-CC2 is in better agreement with experiment for the negative electron affinities than EOM-EA-CCSD for all studied anions. The corresponding resonance widths from EA-CC2 calculations are about 0.05-0.25 eV smaller compared to EOM-EA-CCSD. Semi-empirical spin-scaling increases electron affinities by 0.3-0.5 eV and broadens resonance widths, improving the agreement with EOM-EA-CCSD but worsening the agreement with experiment.

Complex-Energy Second-Order Approximate Coupled-Cluster Methods for Electronic Resonances

TL;DR

The paper develops two independent complex-energy RI-CC2 implementations (RI-EA-CC2 and RI-EE-CC2) incorporating CAP and CBF approaches to characterize electronic resonances, including EA and IP variants within EOM formalisms. It demonstrates that RI-CC2 offers substantial computational savings ( scaling and memory) while producing resonance energies and widths that are broadly in good agreement with EOM-EA-CCSD, with systematic underestimation of widths and slightly less negative affinities that spin-scaling can ameliorate for larger systems. Benchmarking on a set of temporary anions (N, CH, CHO, HCOOH, uracil, naphthalene, cyan-naphthalenes, and pyrene) shows that EA-CC2 often aligns better with experimental trends than EOM-EA-CCSD, particularly when spin scaling is applied, and that CAP and CBF approaches generally agree within about eV for energies and widths. The work enables efficient exploration of resonances in large molecules (e.g., pyrene) and sets the stage for extending complex-energy CC2 to properties and gradients, broadening its applicability in spectroscopy and radiation-chemistry contexts.

Abstract

Electronic resonances are metastable states with finite lifetimes, encountered in processes such as photodetachment, electron transmission, and Auger decay. Resonances appear in Hermitian quantum mechanics as increased density of states in the continuum rather than as discrete energy levels. To describe resonances accurately, including their coupling to the continuum, methods based on non-Hermitian quantum mechanics can be used, which yield complex energies. In this work, we combine the complex absorbing potential (CAP) and complex basis functions (CBF) techniques with the RI-CC2 method. The second-order coupled cluster method (CC2) offers a good balance between accuracy and computational cost by approximating equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) theory, making it suitable for studying of electronic resonances in larger molecules. The resolution-of-the-identity (RI) approximation further reduces computational demands without significant loss in accuracy. We investigate the numerical performance of the new complex-energy RI-CC2 methods focusing on temporary anions. Negative electron affinities and decay widths can be computed using the electron-attachment (EA) variant of RI-CC2. For N2, C2H4, CH2O, and HCOOH, EA-CC2 yields affinities about 0.1-0.2 eV smaller than EOM-EA-CCSD, while deviations reach 0.5 eV for larger anions such as uracil, naphthalene, cyanonaphthalene, and pyrene. As a result of these trends, EA-CC2 is in better agreement with experiment for the negative electron affinities than EOM-EA-CCSD for all studied anions. The corresponding resonance widths from EA-CC2 calculations are about 0.05-0.25 eV smaller compared to EOM-EA-CCSD. Semi-empirical spin-scaling increases electron affinities by 0.3-0.5 eV and broadens resonance widths, improving the agreement with EOM-EA-CCSD but worsening the agreement with experiment.

Paper Structure

This paper contains 20 sections, 29 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Theory
  3. Complex-energy methods
  4. Complex absorbing potentials
  5. Complex basis functions
  6. The RI-CC2 reference state
  7. Computation of excitation energies with EOM-EE-CC2 theory
  8. Computation of electron-attachment and ionization energies with the EOM-EA-CC2 and EOM-IP-CC2 methods
  9. Spin scaling
  10. Modifications for complex-energy CC2 methods
  11. Implementation
  12. Code structure
  13. Solution of the CC2 pseudoeigenvalue equation for excitation, attachment, or ionization energies
  14. Computational Details
  15. Results and Discussion
  16. ...and 5 more sections

Figures (3)

  • Figure 1: Algorithm for solving the CC2 pseudoeigenvalue equation (Eq. \ref{['eq:Aeff1']}). Macro iterations follow the dashed lines and micro iterations follow the dotted lines. Different from the standard Davidson algorithm are indicated in red.
  • Figure 2: Orientation of the molecules in our calculations. From top left to bottom right: formaldehyde, ethylene, naphthalene, pyrene, 1-cyanonaphthalene, and 2-cyanonaphthalene.
  • Figure 3: Electronic configurations of the low-lying anion states of pyrene and corresponding Hartree-Fock orbitals computed in the cc-pVTZ basis without f-shells on carbon atoms and d-shells on hydrogen atoms. Electron affinities were extracted from 2D photoelectron spectroscopy. Figure adapted from Ref. lietard2021temporary with permission.