# 6.6.6 Brueckner CC2

(May 16, 2021)

Brueckner orbitals (BOs) are shown to largely ameliorate the artificial symmetry breaking that occurs at the Hartree-Fock (HF) level. This can lead to improved results even in higher order wavefunction theories that typically use HF orbitals as a starting point. Unfortunately, the cheapest traditional Brueckner theory, Brueckner doubles (BD), is still quite computationally demanding. Orbital optimized MP2 (OOMP2) was proposed as a low scaling approximation to BD due to the similarity of orbital optimized coupled cluster doubles (OOCCD) and BD. Another possible source of approximate BD orbitals is Bruckner CC2 (BCC2).

Rather than optimizing the energy as in OOMP2, BCC2 optimizes the orbitals in order to reduce the CC2 t${}_{1}$ amplitudes to 0. In the absence of t${}_{1}$ amplitudes, the CC2 doubles amplitudes are exactly the same as MP2, and the singles amplitude equations are as follows:

 $\Omega_{i}^{a}=f_{ai}+\sum_{jb}f_{jb}t_{ij}^{ab}+\frac{1}{2}\sum_{jbc}\langle ja% ||cb\rangle t_{ij}^{bc}+\frac{1}{2}\sum_{jkb}\langle jk||bi\rangle t_{jk}^{ab}$ (6.24)

BCC2 has a computational cost scaling as $\mathcal{O}(o^{2}v^{3})$ - a significant improvement over OOCCD/BD, and can greatly improve the quality of orbitals over HF in open-shell molecules due to the reduction of artificial spin contamination.

Example 6.15  Example of BCC2 methodology

$molecule 1 2 F H 1 1.001$end

$rem UNRESTRICTED TRUE JOBTYPE SP EXCHANGE HF GEN_SCFMAN_FINAL TRUE DO_BCC2 3 run BCC2 SCF_ALGORITHM DIIS SCF_GUESS GWH BASIS sto-3g AUX_BASIS rimp2-vdz SCF_CONVERGENCE 8 THRESH 14 SYMMETRY FALSE PURECART 1111$end


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As CC2 is a perturbative approach similar to MP2, breakdowns occur in the presence of orbital degeneracy. While BCC2 does not explicitly optimize the energy and thereby drive orbitals toward degeneracy, results are still severely hindered when orbital energy differences become small. The $\kappa$ regularizer originally proposed by Joonho Lee and Martin Head-Gordon can be used in to further improve the results of BCC2 in this case. The regularizer is applied by modifying the t amplitudes as follows:

 $t_{ij}^{ab}=\frac{-\langle ab||ij\rangle}{\Delta_{ij}^{ab}}\left(1-\exp(-% \kappa\Delta_{ij}^{ab})\right)^{2}$ (6.25)

As in $\kappa$-OOMP2, the parameter $\kappa$ sets the regularization strength, with $\kappa=0$ yielding HF, and $\kappa=\infty$ yielding unregularized BCC2. A value of $\kappa=1.2$ $E_{h}^{-1}$ is suggested for most applications. $\kappa$-BCC2 runs through libgscf and libgmbpt. Currently, DIIS should be used to converge BCC2 orbitals, as the singles residual is not an orbital gradient and cannot be used with gradient-based algorithms. The BCC2 code can currently handle restricted (R) and unrestricted (U) orbital types.

Summary of rem variables relevant to run $\kappa$-BCC2:

CORRELATION JOBTYPE None (default) sp (default) single point energy evaluation user’s choice (standard or user-defined: GENERAL or MIXED) TRUE DIIS (gradient based algorithms currently unsupported) corresponding auxiliary basis (standard or user-defined: AUX_GENERAL or AUX_MIXED) 3 (run BCC2) 0 (no regularizer; default) 1 ($\delta$-regularizer) 2 ($\kappa$-regularizer; recommended) 3 ($\sigma$-regularizer) regularization parameter multiplied by 1e${}^{3}$; no default 1200 (Recommended value for $\kappa$-BCC2) 0 (frozen core currently unsupported) 0 (frozen core currently unsupported) 0 (default) 1 (Compute $\langle S^{2}\rangle$ at the MP2 level)

Example 6.16  Example of $\kappa$-BCC2 with the R orbital type applied to a water dimer

$molecule 0 1 O -2.766559046 0.187082886 0.566917837 H -3.696304300 1.179189102 -0.642506882 H -3.395837846 -1.509891173 0.389283582 O 2.587035064 0.275900014 -0.746441819 H 3.579141280 0.918406897 0.633058252 H 0.852266482 0.311804811 -0.156847268$end

$rem EXCHANGE hf BASIS cc-pvdz AUX_BASIS_CORR rimp2-cc-pvdz THRESH 14 INPUT_BOHR true SCF_CONVERGENCE 8 MAXSCF 1000 SCF_GUESS sad SYMMETRY false GEN_SCFMAN_FINAL true UNRESTRICTED false use restricted DO_BCC2 3 run BCC2 REGULARIZED_O2 2 use kappa-regularizer REG_VARIABLE 1450 set kappa = 1.45$end


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Example 6.17  Example of $\kappa$-BCC2 with the U orbital type applied to an OH radical

$molecule 0 2 O -2.766559046 0.187082886 0.566917837 H -3.6963043 1.179189102 -0.642506882$end

$rem EXCHANGE hf BASIS cc-pvdz AUX_BASIS_CORR rimp2-cc-pvdz THRESH 14 INPUT_BOHR true SCF_CONVERGENCE 8 MAXSCF 1000 SCF_GUESS sad SYMMETRY false GEN_SCFMAN_FINAL true UNRESTRICTED true use unrestricted DO_BCC2 3 run BCC2 REGULARIZED_O2 2 use kappa-regularizer REG_VARIABLE 1450 set kappa = 1.45$end


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