Excited State Properties, Gradients and Geometries

Also for excited states presently unrelaxed and relaxed first-order properties are available in the RICC2 program. These are implemented for CCS and CC2. Note, that in the unrelaxed case CIS and CCS are not equivalent for excited-states first-order properties and no first-order properties are implemented for CIS in the RICC2 program.

The unrelaxed first-order properties are calculated from the variational excited states Lagrangian [89], which for the calculation of unrelaxed properties is decomposed into a ground state contribution and Lagrange functional for the excitation energy which leads to expressions for difference densities (or changes of the density matrix upon excitations):

$$L^{\rm ur,ex}(\bar{N},\bar{E},E,\bar{t},t,\beta) = L^{\rm ur,gs}(\bar{t},t,\beta) + L^{\rm ur,diff}(\bar{N},\bar{E},E,\bar{t},t,\beta)$$ (9.18)

$$L^{\rm ur,diff}(\bar{N},\bar{E},E,\bar{t},t,\beta) = \sum_{\mu\nu} \bar{E}_{\mu} {\bf A}_{\mu\nu}(t,\beta) E_{\nu}$$ (9.19)

$$+ \sum_{\mu_2} \bar{N}_{\mu_2} \langle\mu_2\vert\hat{H} +[F_0+\beta\hat{V},T_2]\vert\mathrm{HF}\rangle ~~,$$

$$\langle V \rangle^{\rm ur,ex} = {\cal R} \left(\frac{\partial L^{ur,ex}(\bar{E},E,\bar{t},t,\beta) }{\partial \beta} \right)_0$$ (9.20)

$$= \sum_{pq} D^{\rm ur,ex}_{pq} V_{pq} = \sum_{pq} \Big(D^{\rm ur}_{pq} + \Delta D^{\rm ur,ex}_{pq}\Big)V_{pq} ~~,$$ (9.21)

with $H = H_0 + \beta V$ mathend000# and $\Re$ mathend000# indicating that only the real part is taken, $D^{\rm ur}$ mathend000# is the unrelaxed ground-state density and $\Delta D^{\rm ur,ex}_{pq}$ mathend000# the difference density matrix. The unrelaxed excited-state properties obtained thereby are equivalent to those identified from the second residues of the quadratic response function and are related in the same way to the total energy of the excited states as the unrelaxed ground-state properties to the energy of the ground state. For a detailed description of the theory see refs. [89,88]; the algorithms for the RI-CC2 implementation are described in refs. [84,12]. ref. [84] also contains a discussion of the basis set effects and the errors introduced by the RI approximation.

In the present implementation, the ground-state and the difference density matrices are evaluated separately. The calculation of excited-state first-order properties thus requires also the calculation of the ground-state density matrix. In addition, the left ( $\bar{E}_\mu$ mathend000#) and right ($E_\mu$ mathend000#) eigenvectors and the Lagrangian multipliers $\bar{N}_\mu$ mathend000# need to be determined for each excited state. The disk space and CPU requirements for solving the equations for $\bar{E}_\mu$ mathend000# and $\bar{N}_\mu$ mathend000# are about the same as those for the calculation of the excitation energies. For the construction of the density matrices in addition some files with ${\cal O}(n_{root} N^2)$ mathend000# size are written, where $n_{root}$ mathend000# is the number of excited states.

The single-substitution parts of the excited-states Lagrangian multipliers $\bar{N}_\mu$ mathend000# are saved in files named CCNE0-s--m-xxx.

For the calculation of first-order properties for excited states, the keyword exprop must be added with appropriate options to the data group $excitations; else the input is same as for the calculation of excitation energies:

$ricc2
  cc2
$response
  fop unrelaxed_only operators=diplen,qudlen
$excitations
  irrep=a1 nexc=2
  exprop states=all operators=diplen,qudlen

Because for calculation of excited-states first-order properties also the (unrelaxed) ground-state density is evaluated, it is recommended to specify also ground-state first-order properties in the input, since they are obtained without extra costs.

To obtain orbital-relaxed first-order properties or analytic derivatives (gradients) the Lagrange functional for the excited state in Eq. (9.18) is--analogously to the treatment of ground states--augmented by the equations for the SCF orbitals and external perturbations are (formally) included in the SCF step, i.e. also in the Fock operator. Since relaxed densities or often computed in connection with geometry optimizations for individual states (rather than simultaneously for many states) a Lagrangian for the total energy of the excited state is used. This has the advantage the only one equation for Lagrangian multipliers for cluster amplitudes ($\bar{N}$ mathend000#) needs to be evaluated instead of two (one for the ground state and one for the energy difference):

$$L^{\rm rel,ex}(\bar{E},E,t) = \langle\mathrm{HF}\vert H\vert\mathrm{CC}\rangle + \sum_{\mu\nu} \bar{E}_{\mu} {\bf A}_{\mu\nu}(t) E_{\nu}$$ (9.22)

$$+ \sum_{\mu_2} \bar{N}_{\mu_2} \langle\mu_2\vert\hat{H} +[F,T_2]\vert\mathrm{HF}\rangle + \sum_{\mu_0} \bar{\kappa}_{\mu_0} F_{\mu_0} ~~.$$

Again the construction of gradients requires the same variational densities as needed for relaxed one-electron properties and the solution of the same equations. The construction of the gradient contributions from one- and two-electron densities and derivative integrals takes approximately the same time as for ground states (approx. 3-4 SCF iterations) and only minor extra disk space. The implementation of the excited state gradients for the RI-CC2 approach is described in detail in Ref. [90]. There also some information about the performance of CC2 for structures and vibrational frequencies of excited states can found.

The following is an example for the CC2 single point calculation for an an excited state gradient (not that in the present implementation it is not possible to compute gradients for several excited states at the same time):

$ricc2
  cc2
$excitations
  irrep=a1 nexc=2
  exprop states=all operators=diplen,qudlen
  xgrad states=(a1 2)

A different input is again required for geometry optimizations: in this case the model and excited state for which the geometry should be optimized have to be specified in the data group $ricc2 with the keyword geoopt:

$ricc2
  geoopt model=cc2 state=(a1 2)
$excitations
  irrep=a1 nexc=2
  exprop states=all operators=diplen,qudlen