Transition Moments

Transition moments are presently only implemented for excitations out of the ground state and only for the coupled cluster models CCS and CC2. Note, that for transition moments (as excited-state first-order properties) CCS is not equivalent to CCS and CIS transition moments are not implemented in the RICC2 program.

In response theory, transition strengths (and moments) are identified from the first residues of the response functions. Due to the non-variational structure of the coupled cluster models different expressions are obtained for ``left'' and ``right'' transitions moments $M^V_{0\gets f}$ mathend000# and $M^V_{f\gets 0}$ mathend000# and the transition strengths $S^{0f}_{V_1 V_2}$ mathend000# are obtained as a symmetrized combinations of both:

$$S^{0f}_{V_1 V_2} = \frac{1}{2} \left\{ M^{V_1}_{0 \gets f} M^{V_2}_{f \gets 0} + \Big(M^{V_2}_{0 \gets f} M^{V_1}_{f \gets 0} \Big)^\ast \right\}$$ (9.23)

Note, that only the transition strengths $S^{0f}_{V_1 V_2}$ mathend000# are a well-defined observables but not the transition moments $M^V_{0\gets f}$ mathend000# and $M^V_{f\gets 0}$ mathend000#. For a review of the theory see refs. [88,91]. The transition strengths calculated by coupled-cluster response theory according to Eq. (9.23) have the same symmetry with respect to interchange of the operators $V_1$ mathend000# and $V_2$ mathend000# and with respect to complex conjugation as the exact transition moments. In difference to SCF (RPA), (TD)DFT, or FCI, transition strengths calculated by the coupled-cluster response models CCS, CC2, etc. do not become gauge-independent in the limit of a complete basis set, i.e., for example the dipole oscillator strength calculated in the length, velocity or acceleration gauge remain different until also the full coupled-cluster (equivalent to the full CI) limit is reached.

For a description of the implementation in the RICC2 program see refs. [84,13]. The calculation of transition moments for excitations out of the ground state resembles the calculation of first-order properties for excited states: In addition to the left and right eigenvectors, a set of transition Lagrangian multipliers $\bar{M}_\mu$ mathend000# has to be determined and some transition density matrices have to be constructed. Disk space, core memory and CPU time requirements are thus also similar.

The single-substitution parts of the transition Lagrangian multipliers $\bar{N}_\mu$ mathend000# are saved in files named CCME0-s--m-xxx.

To obtain the transition strengths for excitations out of the ground state the keyword spectrum must be added with appropriate options (see Section 14.2.13) to the data group $excitations; else the input is same as for the calculation of excitation energies and first-order properties:

$ricc2
  cc2
$excitations
  irrep=a1 nexc=2
  spectrum states=all operators=diplen,qudlen