6.8 Coupled-Cluster Methods 6.8.3 Optimized Orbital Coupled Cluster Doubles (OD)6.8.5 Resolution of the Identity with CC (RI-CC)

6.8.4 Quadratic Coupled Cluster Doubles (QCCD)

The non variational determination of the energy in the CCSD, QCISD, and OD methods discussed in the above subsections is not normally a practical problem. However, there are some cases where these methods perform poorly. One such example are potential curves for homolytic bond dissociation, using closed shell orbitals, where the calculated energies near dissociation go significantly below the true energies, giving potential curves with unphysical barriers to formation of the molecule from the separated fragments.⁹⁴³ The Quadratic Coupled Cluster Doubles (QCCD) method⁹⁴⁴ recently proposed by Troy Van Voorhis at Berkeley uses a different energy functional to yield improved behavior in problem cases of this type. Specifically, the QCCD energy functional is defined as

E_{\mathrm{QCCD}}=\left\langle{\Phi_{0}\left({1+\hat{\Lambda}_{2}+\frac{1}{2}% \hat{\Lambda}_{2}^{2}}\right)\left|{\hat{H}}\right|\exp\left({\hat{T}_{2}}% \right)\Phi_{0}}\right\rangle_{C}

(6.37)

where the amplitudes of both the $\hat{T}_{2}$ and $\hat{\Lambda}_{2}$ operators are determined by minimizing the QCCD energy functional. Additionally, the optimal orbitals are determined by minimizing the QCCD energy functional with respect to orbital rotations mixing occupied and virtual orbitals.

To see why the QCCD energy should be an improvement on the OD energy, we first write the latter in a different way than before. Namely, we can write a CCD energy functional which when minimized with respect to the $\hat{T}_{2}$ and $\hat{\Lambda}_{2}$ operators, gives back the same CCD equations defined earlier. This energy functional is

E_{\mathrm{CCD}}=\left\langle{\Phi_{0}\left({1+\hat{\Lambda}_{2}}\right)\left|% {\hat{H}}\right|\exp\left({\hat{T}_{2}}\right)\Phi_{0}}\right\rangle_{C}

(6.38)

Minimization with respect to the $\hat{\Lambda}_{2}$ operator gives the equations for the $\hat{T}_{2}$ operator presented previously, and, if those equations are satisfied then it is clear that we do not require knowledge of the $\hat{\Lambda}_{2}$ operator itself to evaluate the energy.

Comparing the two energy functionals, Eqs. (6.37) and (6.38), we see that the QCCD functional includes up through quadratic terms of the Maclaurin expansion of $\exp(\hat{\Lambda}_{2})$ while the conventional CCD functional includes only linear terms. Thus the bra wave function and the ket wave function in the energy expression are treated more equivalently in QCCD than in CCD. This makes QCCD closer to a true variational treatment⁹⁴³ where the bra and ket wave functions are treated precisely equivalently, but without the exponential cost of the variational method.

In practice QCCD is a dramatic improvement relative to any of the conventional pair correlation methods for processes involving more than two active electrons (i.e., the breaking of at least a double bond, or, two spatially close single bonds). For example calculations, we refer to the original paper,⁹⁴⁴ and the follow-up paper describing the full implementation.¹³¹ We note that these improvements carry a computational price. While QCCD scales formally with the 6th power of molecule size like CCSD, QCISD, and OD, the coefficient is substantially larger. For this reason, QCCD calculations are by default performed as OD calculations until they are partly converged. Q-Chem also contains some configuration interaction models (CISD and CISDT). The CI methods are inferior to CC due to size-consistency issues, however, these models may be useful for benchmarking and development purposes.

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6.8.4 Quadratic Coupled Cluster Doubles (QCCD)