Calculation of Excitation Energies

= = | (9.8) |

Since the CC2 Jacobian is a non-symmetric matrix, left and right eigenvectors are different and the right (left) eigenvectors

E^{j} = E^{j}_{ν1} + E^{j}_{ν2} = δ_{ij} . |
(9.9) |

To obtain excitation energies only the right or the left eigenvalue problem needs to be solved, but for the calculation of transition strengths and first-order properties both, left and right, eigenvectors are needed (see below). A second complication that arises from the non-symmetric eigenvalue problem is that in the case of close degeneracies within the same irreducible representation (symmetry) it can happen that instead of two close lying real roots a degenerate complex conjugated pair of excitation energies and eigenvectors is obtained. CC2 (and also other standard coupled-cluster response methods) are thus not suited for the description of conical intersections etc. For the general theory behind coupled cluster response calculations see e.g. ref. [104,105] or other reviews.

The `ricc2` program exploits that the doubles/doubles block of the
CC2 Jacobian is diagonal and the (linear) eigenvalue problem in the
singles and doubles space can be reformulated as a (non-linear)
eigenvalue problem in single-substitution space only:

(*t*, *ω*) = (*t*) - (*t*)( - *ω*)(*t*)

(*t*^{CC2}, *ω*^{CC2})*E*_{ν1} = *ω*^{CC2}*E*_{ν1}

This allows to avoid the storage of the double-substitution part of the
eigen- or excitation vectors
The solution of the CC2 eigenvalue problem can be started from the solutions
of the CCS eigenvalue problem (see below) or the trial vectors or solutions
of a previous CC2 excitation energy calculation.
The operation count per transformed trial vector
for one iteration for the CC2 eigenvalue problem is
about 1.3-1.7 times the operation count for one iteration for the
cluster equations in the ground-state calculation--depending on the
number of vectors transformed simultaneously.
The disk space requirements are about *O*(*V* + *N*)*N*_{x} double precision words
per vector in addition to the disk space required for the ground state
calculation.

CCS excitation energies are obtained by the same approach, but here double-substitutions are excluded from the expansion of the excitation or eigenvectors and the ground-state amplitudes are zero. Therefore the CCS Jacobian,

= = 〈μ_{1}|[H, τ_{ν1}]| HF〉 , |
(9.10) |

is a symmetric matrix and left and right eigenvectors are identical and form an orthonormal basis. The configuration interaction singles (CIS) excitation energies are identical to the CCS excitation energies. The operation count for a RI-CIS calculation is (

The second-order perturbative correction CIS(D) to the CIS excitation energies is calculated from the expression

ω^{CIS(D)} = ω^{CIS} + ω^{(D)} = (t^{MP1}, ω^{CIS}) |
(9.11) |

(Note that