10.6 Ground State third-order Properties with CC2

Third-order properties or quadratic response functions for one-electron perturbations are only implemented for CC2 using the orbital-unrelaxed response formalism. The implementation is currently limited to a closed-shell Hartree-Fock reference and the sequential and OMP-parallel versions of the program. Note that second-order properties are also not yet available for the spin-component scaled variants of CC2. Furthermore, non-Abelian points groups and point groups with complex irreducible representations are not implemented for third-order properties.

The input for third-order properties is similar as for second-order properties: In addition to the standard input for the CC2 ground-state calculation, they require that the data group for the numerical Laplace transformation $laplace is set. Third-order ground-state properties are requested with the top option in the data group $response:

$ricc2
  cc2
$laplace
  conv=4
$response
  top operators=(diplen,diplen,diplen) freq1=0.077d0 freq2=-0.077d0

The above input will request the full dipole-dipole-dipole tensor for the frequency-dependent hyperpolarizability $\beta_{ijk}(0.077,-0.077,0.0)$. The frequencies specified with freq1 are associated to the first and those specified with freq2 with the second operator in the operator triple specified with operators. The frequencies have to be given in atomic units. If not given, a value of 0.0 is used. The frequencies for the third operator are set to the negative of the sum of the frequencies for the first two operators. In general, the operators option can be followed by a comma-separated list of operator triples and freq1 and freq2 by comma-separated lists of frequencies, e.g.

  top operators=(xdiplen,xdiplen,xdiplen),(zdiplen,zdiplen,zdiplen) 
                     freq1=0.077d0,0.088d0 freq2=-0.077d0,0.088d0

(without linebreak!) will request the four hyperpolarizability vaues $\beta_{xxx}(0.077,-0.077,0.0)$, $\beta_{zzz}(0.077,-0.077,0.0)$, $\beta_{xxx}(0.088,0.088,0.176)$, and $\beta_{zzz}(0.088,0.088,0.176)$. Note that the $i$-th frequency in the list freq1 is (only) combined with the $i$-th frequency in the list freq2, while each operator triple is combined with each frequency combination.

10.6.1 Damped Third-order Properties with CC2

Damped quadratic response is available at the CC2 level. In addition to the third-order property input, the flag cpp=on must be set along with a damping factor gamma, in atomic units. The following input will compute the damped hyperpolarizability with $\gamma=4.5563\cdot10^{-3}$ a.u.:

$response
  cpp=on gamma=4.5563d-3
  top operators=(diplen,diplen,diplen) freq1=0.077d0 freq2=-0.077d0

As for damped second-order properties, it is possible to set a frequency interval in the input

$response
  cpp=on gamma=4.5563d-3
  top operators=(diplen,diplen,diplen) frqscan=0.3d0, 0.4d0, 0.005d0

To build an MCD spectrum using damped response over a grid of frequencies, use the mcdcpp option along with frqscan

$response
  cpp=on gamma=4.5563d-3
  top mcdcpp frqscan=0.3d0, 0.4d0, 0.005d0

The operator combination for the MCD property is (diplen,diplen,angmom).

It is also possible to compute nuclear spin-induced CD (NSCD) from damped quadratic response. For this, the operator combination is (diplen,diplen,psoXXXX) where XXXX is the nucleus’ number as it appears in the coord file, i.e., the PSO operator of the first atom is pso0001. Its damped NSCD spectrum is build from the input

$response
  cpp=on gamma=4.5563d-3
  top operators=(diplen,diplen,pso0001) frqscan=0.3d0, 0.4d0, 0.005d0