15.1. The Linear Response Module

This module describes the implementation of Linear Response (LR) theory for modeling electronic excitation properties based on a pair Coupled Cluster Doubles (pCCD) reference and its extension to singles (pCCDS = pCCD+S).

Before proceeding, please ensure you’re familiar with the basics of post-Hartree-Fock models in PyBEST (General Remarks concerning Post-Hartree-Fock Calculations).

If you use this module, please cite [ahmadkhani2024].

15.2. Supported Features

The current version of PyBEST supports two different LR models that allow you to compute singlet excitation energies and transition dipole moments using property operators. These include

  • LR-pCCD: Excitations built upon the pair Coupled Cluster Doubles reference.

  • LR-pCCDS: Includes additional single excitations on top of pCCD (formally abbreviated as LR-pCCD+S).

Both methods solve a response eigenvalue problem using the Jacobian of the similarity-transformed Hamiltonian.

15.3. Theoretical Framework

The LR approach solves a generalized eigenvalue problem of the form

\[\mathbf{J} \mathbf{X}_k = \omega_k \mathbf{S} \mathbf{X}_k\]

where:

  • \(\mathbf{J}\) is the Jacobian matrix (the response of the similarity-transformed Hamiltonian)

  • \(\mathbf{S}\) is the overlap metric between excitation operators (usually identity)

  • \(\omega_k\) are the excitation energies

  • \(\mathbf{X}_k\) are the right (CI-like) eigenvectors

Transition properties between states are computed as

\[\langle 0|\hat{A}|k\rangle \langle k|\hat{B}|0\rangle = \lim_{\omega\rightarrow\omega_k} (\omega-\omega_k) \langle\langle \hat{A};\hat{B}\rangle\rangle = \langle k | \hat{B} | 0 \rangle \langle \Lambda |[\hat{A}, \hat{\tau}_k]|\mathrm{pCCD}\rangle\]
\[-\sum_{\nu } \left\{ \sum_{\mu} (-\hat{J} + \omega \mathbf{I})_{\mu\nu}^{-1} \langle \bar{\nu} | \hat{A} | \mathrm{pCCD} \rangle \langle \Lambda | [[\hat{H}_0, \hat{\tau}_\nu], \hat{\tau}_k] | \mathrm{pCCD} \rangle \right\} \langle k | \hat{B} | 0 \rangle\]

where:

  • \(\hat{A}\) and \(\hat{B}\) are property operators (e.g., dipole operator or momentum integrals)

  • \(\hat{J}\) is the Jacobian matrix

  • \(\hat{\tau}_k\) is the excitation operator for state \(k\)

  • \(\nu\) and \(\mu\) denote the index over excitation operators

  • \(\langle \Lambda |\) is the left eigenvector of the response problem

  • \(\hat{H}_0\) is the unperturbed molecular Hamiltonian

15.4. Diagonalization Method

All LR methods currently use Davidson diagonalization to solve the eigenvalue problem. This iterative method efficiently computes a few of the lowest excited states.

15.5. Practical Usage

To perform an LR-pCCD or LR-pCCD+S calculation, you need:

  • A converged pCCD ground-state wavefunction

  • Solution of the Jacobian eigenproblem

  • A property operator (e.g., electric dipole moment) for computing properties

15.6. Computing Transition Properties

To compute transition properties (e.g., transition dipole moments) using the LR-pCCD or LR-pCCD+S methods, you can use the property interface described in Linear Response Transition Dipole Moment and Related Properties.