Transition Moments

Transition moments are presently implemented for excitations out of the ground state and for excitations between excited states for the coupled cluster models CCS and CC2. Note, that for transition moments (as excited-state first-order properties) CCS is not equivalent to CIS 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←f and M^V_f←0 and the transition strengths S^0f_V1V2 are obtained as a symmetrized combinations of both:

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

Note, that only the transition strengths S^0f_V1V2 are a well-defined observables but not the transition moments M^V_0←f and M^V_f←0. For a review of the theory see refs. [105,108]. 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 V₁ and V₂ 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. [103,13]. 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 $\bar{{M}}_{\mu}^{}$ 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 $\bar{{N}}_{\mu}^{}$ 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 15.2.13) 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

For the calculation of transition moments between excited states a set of Lagrangian multipliers $\bar{{N}}_{\mu}^{}$ has to be determined instead of the $\bar{{M}}_{\mu}^{}$ for the ground state transition moments. From these Lagrangian multipliers and the left and right eigenvectors one obtaines the ``right'' transition moment between two excited states i and f as

$$\sum_{{pq}}^{} \left\{\vphantom{D^{\xi}_{pq}(\bar{N}^{fi})+D^{A}_{pq}(\bar{E}^f,E^{i})}\right. \bar{{N}}^{{fi}}_{} \bar{{E}}^{f}_{} \left.\vphantom{D^{\xi}_{pq}(\bar{N}^{fi})+D^{A}_{pq}(\bar{E}^f,E^{i})}\right\} \hat{{V}}_{{pq}}^{}$$ (9.24)

where $\hat{{V}}$ are the matrix elements of the perturbing operator. A similar expression is obtained for the ``left'' transition moments. The ``left'' and ``right'' transition moments are then combined to yield the transition strength

$${\frac{{1}}{{2}}} \left\{\vphantom{ M^{V_1}_{i \gets f} M^{V_2}_{f \gets i} + \Big(M^{V_2}_{i \gets f} M^{V_1}_{f \gets i} \Big)^\ast }\right. \Big( \Big)^{\ast}_{} \left.\vphantom{ M^{V_1}_{i \gets f} M^{V_2}_{f \gets i} + \Big(M^{V_2}_{i \gets f} M^{V_1}_{f \gets i} \Big)^\ast }\right\}$$ (9.25)

As for the ground state transitions, only the transition strengths S^if_V1V2 are a well-defined observables but not the transition moments M^V_i←f and M^V_f←i.

The single-substitution parts of the transition Lagrangian multipliers $\bar{{N}}_{\mu}^{}$ are saved in files named CCNE0-s--m-xxx.

To obtain the transition strengths for excitations between excited states the keyword tmexc must be added to the data group $excitations. Additionally, the initial and final states must be given in the same line; else the input is same as for the calculation of excitation energies and first-order properties:

$ricc2
  cc2
$excitations
  irrep=a1 nexc=2
  irrep=a2 nexc=2
  tmexc istates=(a1 1) fstates=all operators=diplen