Brueckner orbitals (BOs) are shown to largely ameliorate the artificial
symmetry breaking that occurs at the Hartree-Fock (HF) level.
Phys. Chem. Chem. Phys.
(2019), 21, pp. 47638. 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 690 J. Chem. Phys.
(2007), 126, pp. 164101. due to the similarity of orbital optimized coupled cluster doubles (OOCCD) and BD. Another possible source of approximate BD orbitals is Bruckner CC2 (BCC2). 31 Chem. Phys. Lett.
(2011), 506, pp. 276.
Rather than optimizing the energy as in OOMP2, BCC2 optimizes the orbitals in order to reduce the CC2 t amplitudes to 0. In the absence of t amplitudes, the CC2 doubles amplitudes are exactly the same as MP2, and the singles amplitude equations are as follows:
BCC2 has a computational cost scaling as - 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.
$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
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 regularizer
originally proposed by Joonho Lee and Martin Head-Gordon
J. Chem. Theory Comput.
(2018), 14, pp. 5203. 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:
As in -OOMP2, the parameter sets the regularization strength, with yielding HF, and yielding unregularized BCC2. A value of is suggested for most applications. -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 -BCC2:
|JOBTYPE||sp (default) single point energy evaluation|
|BASIS||user’s choice (standard or user-defined: GENERAL or MIXED)|
|SCF_ALGORITHM||DIIS (gradient based algorithms currently unsupported)|
|AUX_BASIS||corresponding auxiliary basis (standard or user-defined:|
|AUX_GENERAL or AUX_MIXED)|
|DO_BCC2||3 (run BCC2)|
|REGULARIZED_BCC2||0 (no regularizer; default)|
|2 (-regularizer; recommended)|
|REG_PARAMETER||regularization parameter multiplied by 1e; no default|
|1200 (Recommended value for -BCC2)|
|N_FROZEN_CORE||0 (frozen core currently unsupported)|
|N_FROZEN_VIRTUAL||0 (frozen core currently unsupported)|
|1 (Compute at the MP2 level)|
$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
$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