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. Transition moments for excitations from the ground to an excited state are also available for ADC(2), but use an additional approximation (see below). Note, that for transition moments (as for 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) 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_{0←f}^{V } and M_{f←0}^{V }. The transition strengths S_{V 1V 2}^{0f} are obtained
as a symmetrized combinations of both [141]:

| (10.21) |

Note, that only the transition strengths S_{V 1V 2}^{0f} are a well-defined observables but not the
transition moments M_{0←f}^{V } and M_{f←0}^{V }. For a review of the theory see refs. [139,141]. The
transition strengths calculated by coupled-cluster response theory according to Eq. (10.21) have
the same symmetry with respect to an interchange of the operators V _{1} and V _{2} 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. [11,137]. 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 _{μ} 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 _{μ} 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 20.2.18) 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

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 (^{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.

For the calculation of transition moments between excited states a set of Lagrangian multipliers
_{μ} has to be determined instead of the _{μ} 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

| (10.22) |

where 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

| (10.23) |

As for the ground state transitions, only the transition strengths S_{V 1V 2}^{if} are a well-defined
observables but not the transition moments M_{i←f}^{V } and M_{f←i}^{V }.

The single-substitution parts of the transition Lagrangian multipliers _{μ} 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=all fstates=(a1 1) operators=diplen

cc2

$excitations

irrep=a1 nexc=2

irrep=a2 nexc=2

tmexc istates=all fstates=(a1 1) operators=diplen