L^{ur CC2, ex}(E,, t,, β) |
= | 〈HF| H| CC〉 + (t, β)E_{ν} |
(9.18) |

+ 〈μ_{1}| + [, T_{2}]| HF〉 |
|||

+ 〈μ_{2}| + [F_{0} + β, T_{2}]| HF〉 |

where it is assumed that the left and right eigenvectors are normalized such that 〈

〈V〉^{ur, ex} |
= | = D^{ur, ex}_{pq}V_{pq} , |
(9.19) |

(Again ℜ indicates that the real part is taken.) The unrelaxed excited-state properties obtained thereby 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 and the differences between excited- and ground-state unrelaxed properties are identical to those identified from the second residues of the quadratic response function. For a detailed description of the theory see refs. [114,113]; the algorithms for the RI-CC2 implementation are described in refs. [111,12]. ref. [111] also contains a discussion of the basis set effects and the errors introduced by the RI approximation.

The calculation of excited-state first-order properties thus requires
the calculation of both the right (*E*_{μ}) and left (
)
eigenvectors and of the excited state Lagrangian multipliers
.
The disk space and CPU requirements for solving the equations for
and
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
(*n*_{root}*N*^{2}) size are written,
where *n*_{root} is the number of excited states.

The single-substitution parts of the excited-states Lagrangian
multipliers
are saved in files named
`CCNL0-`

*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