10.4 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. 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.

10.4.1 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 M0fV and Mf0V . The transition strengths SV 1V 20f are obtained as a symmetrized combinations of both [130]:

       1 {              (           )*}
S0Vf1V2 = -- M 0V←1f M Vf2←0 +  M V02←f M Vf1←0
       2
(10.21)

Note, that only the transition strengths SV 1V 20f are a well-defined observables but not the transition moments M0fV and Mf0V . For a review of the theory see refs.  [128,130]. 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.  [13,126]. 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 Mμ 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 Nμ 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 20.2.17) 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 O(N6), 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.

10.4.2 Transition moments between excited states

For the calculation of transition moments between excited states a set of Lagrangian multipliers Nμ has to be determined instead of the Mμ 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

           {                       }
M V   = ∑    Dξ ( ŻN fi) + DA (ŻEf ,Ei ) Vˆ .
  f←i         pq         pq           pq
        pq
(10.22)

where Vˆ 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

         {              (           )*}
Sif  =  1- M V1 M V2  +  M V2 M  V1
 V1V2   2    i←f   f←i      i←f  f←i
(10.23)

As for the ground state transitions, only the transition strengths SV 1V 2if are a well-defined observables but not the transition moments MifV and MfiV .

The single-substitution parts of the transition Lagrangian multipliers Nμ 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