# 6.10.4 Optimized Orbital Coupled Cluster Doubles (OD)

(June 30, 2021)

It is possible to greatly simplify the CCSD equations by omitting the single substitutions (i.e., setting the $T_{1}$ operator to zero). If the same single determinant reference is used (specifically the Hartree-Fock determinant), then this defines the coupled-cluster doubles (CCD) method, by the following equations:

 $\displaystyle E_{\mathrm{CCD}}$ $\displaystyle=$ $\displaystyle\left\langle\Phi_{0}\left|\hat{H}\right|\left(1+\hat{T}_{2}\right% )\Phi_{0}\right\rangle_{C}$ (6.45) $\displaystyle 0$ $\displaystyle=$ $\displaystyle\left\langle\Phi_{ij}^{ab}\left|\hat{H}\right|\left(1+\hat{T}_{2}% +\frac{1}{2}\hat{T}_{2}^{2}\right)\Phi_{0}\right\rangle_{C}$ (6.46)

The CCD method cannot itself usually be recommended because while pair correlations are all correctly included, the neglect of single substitutions causes calculated energies and properties to be significantly less reliable than for CCSD. Single substitutions play a role very similar to orbital optimization, in that they effectively alter the reference determinant to be more appropriate for the description of electron correlation (the Hartree-Fock determinant is optimized in the absence of electron correlation).

This suggests an alternative to CCSD and QCISD that has some additional advantages. This is the optimized orbital CCD method (OO-CCD), which we normally refer to as simply optimized doubles (OD). The OD method is defined by the CCD equations above, plus the additional set of conditions that the cluster energy is minimized with respect to orbital variations. This may be mathematically expressed by

 $\frac{\partial E_{\mathrm{CCD}}}{\partial\theta_{i}^{a}}=0$ (6.47)

where the rotation angle $\theta_{i}^{a}$ mixes the $i$th occupied orbital with the $a$th virtual (empty) orbital. Thus the orbitals that define the single determinant reference are optimized to minimize the coupled-cluster energy, and are variationally best for this purpose. The resulting orbitals are approximate Brueckner orbitals.

The OD method has the advantage of formal simplicity (orbital variations and single substitutions are essentially redundant variables). In cases where Hartree-Fock theory performs poorly (for example artificial symmetry breaking, or non-convergence), it is also practically advantageous to use the OD method, where the HF orbitals are not required, rather than CCSD or QCISD. Q-Chem supports both energies and analytical gradients using the OD method. The computational cost for the OD energy is more than twice that of the CCSD or QCISD method, but the total cost of energy plus gradient is roughly similar, although OD remains more expensive. An additional advantage of the OD method is that it can be performed in an active space, as discussed later, in Section 6.12.