Constructing "Full-Frequency" Spectra via Moment Constraints for Coupled Cluster Green's Functions

Abstract

We propose an approach to build "full-frequency" quasiparticle spectra from conservation of a set of static expectation values. These expectation values define the moments of the spectral distribution, resulting in an efficient and systematically improvable expansion. By computing these initial moment constraints at the coupled-cluster level, we demonstrate convergence in both correlated state-specific and full spectral quantities, while requiring a fraction of the effort of traditional Green's function approaches. Tested across the GW100 benchmark set for charged excitation spectra, we can converge frontier excitations to within the inherent accuracy of the CCSD approximation, while obtaining a simultaneous representation of the entire excitation spectrum at all energy scales.

Figure 1

Comparison of the convergence in the ionization potential for a CO molecule (bond length of 1.1314 ) in a cc-pVDZ basis between the Hermitian and non-Hermitian block Lanczos recursion, showing the correct convergence of the non-Hermitian form to the exact IP-EOM-CCSD ionization potential, and incorrect convergence or numerical errors for Hermitized versions. Note that the HOMO energy at the level of the Hartree–Fock starting point is 19.12 eV, showing that even at the lowest levels of moment conservation, significant correlation-driven changes to the IP are found.

Figure 2

Comparison of theoretical photoelectron spectra computed using the recursive GF moment conservation (presented in section 3 in black), and a more conventional GF-CCSD approach using the GCROT algorithm (in red under-laying all spectra), for water with a bond length of (a) 1.1 Å and (b) 1.8 Å in a cc-pVDZ basis. The notation GF(n) indicates the number of iterations of the recursion algorithm performed, conserving all GF moments up to order 0 ≤ m ≤ 2n + 1. GF(0′) corresponds to a modified GF(0) approximation where the moments are exactly computed via the reduced density matrices (section 4.3). Also shown is the spectrum at the mean-field (Hartree–Fock) level. The labels also give the number of matrix-vector products per orbitals required to produce the spectrum, where the GCROT result depends on the number of frequency points N_ω, which was selected to be 512 in this example with a broadening parameter η of 1.0 eV (the other GF results are artificially broadened with the same broadening). The Wasserstein metric W₁ between each of the spectra and the GCROT spectrum is shown, indicating the their fit to the true GF-CCSD spectrum. The value of the gap for each method is also included, with the chemical potential at the zero frequency.

Figure 3

Convergence of CCSD core-level ionization energies from K- (1s), L- (2s, 2p) and M- (3s, 3p) shells of a zinc atom with increasing moment constraints, in a cc-pwCVTZ basis with X2C Hamiltonian. The transparency in the shading of points is proportional to the weight of the resulting ionization potential on the atomic orbital of the same character as the desired excitation. This shows that different moment constraints can sometimes result in a splitting of the state across a range of energies. The ADC(2) and ADC(3) results are taken from ref (77) and experimental results from ref (76). We note that experimental values for the 2p and 3p orbitals are obtained by averaging ionization energies for states with the total angular momentum quantum number J = 1/2 and J = 3/2.

Figure 4

Convergence of the moment-conservation approach to approximate the IP/EA-EOM-CCSD excitations for (a) ionization potential and (b) electron affinity over the GW100 benchmark test set with a def2-TZVPP basis set. The white circle defines the mean absolute error at each order, with the standard deviations in these quantities also given below the plot (in units of eV). Excitations with a weight of <0.1 in the physical space were rejected in order to compare only quasiparticle-like excitations. The EOM-CCSD reference values were calculated using the PySCF package.^,