Ground to excited state transition moments

In response theory, transition strengths (and moments) for transitions from the ground to excited state are identified from the first residues of the response functions. Due to the non-variational structure of coupled cluster different expressions are obtained for the CCS and CC2 ``left'' and ``right'' transitions moments M^V_0←f and M^V_f←0. The transition strengths S^0f_V1V2 are obtained as a symmetrized combinations of both[109]:

$${\frac{{1}}{{2}}} \left\{\vphantom{ M^{V_1}_{0 \gets f} M^{V_2}_{f \gets 0} + \Big(M^{V_2}_{0 \gets f} M^{V_1}_{f \gets 0} \Big)^\ast }\right. \Big( \Big)^{\ast}_{} \left.\vphantom{ M^{V_1}_{0 \gets f} M^{V_2}_{f \gets 0} + \Big(M^{V_2}_{0 \gets f} M^{V_1}_{f \gets 0} \Big)^\ast }\right\}$$ (9.21)

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. [106,109]. The transition strengths calculated by coupled-cluster response theory according to Eq. (9.21) have the same symmetry with respect to an 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. [104,13]. 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 ADC(2) model, which is derived by a perturbation expansion of the expressions for exact states, the calculation of transition moments for excitations from the ground to an excited state would require the second-order double excitation amplitudes for the ground state wavefunction, which would lead to operation counts scaling as $\cal {O}$($\cal {N}$⁶), if no further approximations are introduced. On the other hand the second-order contributions to the transition moments are usually not expected to be important. Therefore, the implementation in the ricc2 program neglects in the calculation of the ground to excited state transition moments the contributions which are second order in ground state amplitudes (i.e. contain second-order amplitudes or products of first-order amplitudes). With this approximation the ADC(2) transition moments are only correct to first-order, i.e. to the same order to which also the CC2 transition moments are correct, and are typically similar to the CC2 results. The computational costs for the ADC(2) transition moments are (within this approximation) much lower than for CC2 since the left and right eigenvectors are identical and no lagrangian multipliers need to be determined. The extra costs (i.e. CPU and wall time) for the calculations of the transitions moments are similar to the those for two or three iterations of the eigenvalue problem, which reduces the total CPU and wall time for the calculation of a spectrum (i.e. excitation energies and transition moments) by almost a factor of three.