Calculation of Excitation Energies

With the RICC2 program excitation energies can at present be calculated with the RI variants of the methods CCS/CIS, CIS(D), CIS(D$_\infty$ mathend000#), ADC(2) and CC2. The CC2 excitation energies are obtained by standard coupled-cluster linear response theory as eigenvalues of the Jacobian, defined as derivative of the vector function with respect to the cluster amplitudes.

$${\bf A}^{\rm CC2}_{\mu\nu} = \frac{d \Omega_{\mu}}{d t_{\nu}} = \... ...gle \mu_{2} \vert [F,\tau_{\nu_2}] \vert\mathrm{HF} \rangle \end{array} \right)$$ (9.8)

Since the CC2 Jacobian is a non-symmetric matrix, left and right eigenvectors are different and the right (left) eigenvectors $E^i_{\nu}$ mathend000# ( $\bar{E}^i_{\mu}$ mathend000#) are not orthogonal among themselves, but form a biorthonormal basis (if properly normalized):

$$\bar{E}^{i} E^{j} = \bar{E}^i_{\mu_1} E^j_{\nu_1} + \bar{E}^i_{\mu_2} E^j_{\nu_2} = \delta_{ij} ~~.$$ (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. [87,88] 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:

$${\bf A}^{eff}_{\mu_1\nu_1}(t,\omega) = {\bf A}^{\rm CC2}_{\mu_1... ...2}(t) ({\bf A}_{\gamma_2\gamma_2}-\omega) {\bf A}^{\rm CC2}_{\gamma2\nu_1}(t)$$

mathend000#

$${\bf A}^{eff}_{\mu_1\nu_1}(t^{\rm CC2},\omega^{\rm CC2}) E_{\nu_1} = \omega^{\rm CC2} E_{\nu_1}$$

mathend000#

This allows to avoid the storage of the double-substitution part of the eigen- or excitation vectors $E_{\nu_2}$ mathend000#, $\bar{E}_{\nu_2}$ mathend000#. The algorithms are described in refs. [10,11], about the RI error see ref. [84].

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 $1.3--1.7$ mathend000# 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 $O(V+N)N_x$ mathend000# 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,

$${\bf A}^{\rm CCS}_{\mu\nu} = \frac{d \Omega_{\mu}}{d t_{\nu}} = \langle\mu_{1} \vert [H,\tau_{\nu_1}]\vert\mathrm{HF}\rangle ~~,$$ (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 ${\cal O}(ON^2N_x)$ mathend000# per iteration and transformed trial vector.

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

$$\omega^{\rm CIS(D)} = \omega^{\rm CIS}+\omega^{(D)} = {\bf E}^{\rm CIS} {\bf A}^{eff}(t^{\rm MP1},\omega^{CIS}) {\bf E}^{\rm CIS}$$ (9.11)

(Note that $t^{\rm MP1}$ mathend000# are the first-order double-substitution amplitudes from which also the MP2 ground-state energy is calculated; the first-order single-substitution amplitudes vanish for a Hartree-Fock reference due to the Brillouin theorem.) The operation count for a RI-CIS(D) calculation is similar to that of a single iteration for the CC2 eigenvalue problem. Also disk space requirements are similar.