6.18 Simplified Coupled-Cluster Methods Based on a Perfect-Pairing Active Space 6.18.1 Introduction 6.18.3 Coupled Cluster Valence Bond (CCVB)

6.18.2 Perfect Pairing (PP)

(September 1, 2024)

To be more specific, the coupled cluster PP wave function is written as

\left|\Psi\right\rangle=\exp\left({\sum_{i=1}^{n_{\mathrm{active}}}{t_{i}\hat{% a}_{i\ast}^{{\dagger}}\hat{a}_{\bar{i}\ast}^{{\dagger}}\hat{a}_{\bar{i}}\hat{a% }_{i}}}\right)\left|\Phi\right\rangle

(6.64)

where $n_{\mathrm{active}}$ is the number of active electrons, and the $t_{i}$ are the linear number of unknown cluster amplitudes, corresponding to exciting the two electrons in the $i$ th electron pair from their bonding orbital pair to their anti-bonding orbital pair. In addition to $t_{i}$ , the core and the active orbitals are optimized as well to minimize the PP energy. The algorithm used for this is a slight modification of the GDM method, described for SCF calculations in Section 4.5.7. Despite the simplicity of the PP wave function, with only a linear number of correlation amplitudes, it is still a useful theoretical model chemistry for exploring strongly correlated systems. This is because it is exact for a single electron pair in the PP active space, and it is also exact for a collection of non-interacting electron pairs in this active space. Molecules, after all, are in a sense a collection of interacting electron pairs! In practice, PP on molecules recovers between 60% and 80% of the correlation energy in its active space.

If the calculation is perfect pairing (CORRELATION = PP), it is possible to look for unrestricted solutions in addition to restricted ones. Unrestricted orbitals are the default for molecules with odd numbers of electrons, but can also be specified for molecules with even numbers of electrons. This is accomplished by setting GVB_UNRESTRICTED = TRUE. Given a restricted guess, this will, however usually converge to a restricted solution anyway, so additional REM variables should be specified to ensure an initial guess that has broken spin symmetry. This can be accomplished by using an unrestricted SCF solution as the initial guess, using the techniques described in Chapter 4. Alternatively a restricted set of guess orbitals can be explicitly symmetry broken just before the calculation starts by using GVB_GUESS_MIX, which is described below. There is also the implementation of Unrestricted-in-Active Pairs (UAP), ⁷³¹ Lawler K. V., Small D. W., Head-Gordon M.
J. Phys. Chem. A
(2010), 114, pp. 2930. Link which is the default unrestricted implementation for GVB methods. This method simplifies the process of unrestriction by optimizing only one set of ROHF MO coefficients and a single rotation angle for each occupied-virtual pair. These angles are used to construct a series of $2\times 2$ Givens rotation matrices which are applied to the ROHF coefficients to determine the $\alpha$ spin MO coefficients and their transpose is applied to the ROHF coefficients to determine the $\beta$ spin MO coefficients. This algorithm is fast and eliminates many of the pathologies of the unrestricted GVB methods near the dissociation limit. To generate a full potential curve we find it is best to start at the desired UHF dissociation solution as a guess for GVB and follow it inwards to the equilibrium bond distance.

GVB_UNRESTRICTED

GVB_UNRESTRICTED
       Controls restricted versus unrestricted PP jobs. Usually handled automatically.
TYPE:
       LOGICAL
DEFAULT:
       same value as UNRESTRICTED
OPTIONS:
       TRUE/FALSE
RECOMMENDATION:
       Set this variable explicitly only to do a UPP job from an RHF or ROHF initial guess. Leave this variable alone and specify UNRESTRICTED = TRUE to access the new unrestricted-in-active-pairs GVB code which can return an RHF or ROHF solution if used with GVB_DO_ROHF

GVB_DO_ROHF

GVB_DO_ROHF
       Sets the number of Unrestricted-in-Active Pairs to be kept restricted.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       $n$ User-Defined
RECOMMENDATION:
       If $n$ is the same value as GVB_N_PAIRS returns the ROHF solution for GVB, only works with the UNRESTRICTED = TRUE implementation of GVB with GVB_OLD_UPP = 0 (its default value)

GVB_OLD_UPP

GVB_OLD_UPP
       Which unrestricted algorithm to use for GVB.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Use Unrestricted-in-Active Pairs described in Ref. 731 Lawler K. V., Small D. W., Head-Gordon M.
J. Phys. Chem. A
(2010), 114, pp. 2930. Link 1 Use Unrestricted Implementation described in Ref. 103 Beran G. J. O. et al.
J. Phys. Chem. A
(2005), 109, pp. 9183. Link
RECOMMENDATION:
       Only works for Unrestricted PP and no other GVB model.

GVB_GUESS_MIX

GVB_GUESS_MIX
       Similar to SCF_GUESS_MIX, it breaks alpha/beta symmetry for UPP by mixing the alpha HOMO and LUMO orbitals according to the user-defined fraction of LUMO to add the HOMO. 100 corresponds to a 1:1 ratio of HOMO and LUMO in the mixed orbitals.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       $n$ User-defined, $0\leq n\leq 100$
RECOMMENDATION:
       25 often works well to break symmetry without overly impeding convergence.

Whilst all of the description in this section refers to PP solved via projection, it is also possible, as described in Section 6.18.3 below, to solve variationally for the PP energy. This variational PP solution is the reference wave function for the CCVB method. In most cases use of spin-pure CCVB is preferable to attempting to improve restricted PP by permitting the orbitals to spin polarize.

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6.18.2 Perfect Pairing (PP)