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(June 30, 2021)

Brueckner orbitals (BOs) are highly desirable when one is unsure whether
artificial symmetry breaking occurs at the Hartree-Fock (HF) level. It is
*artificial* because this symmetry breaking merely reflects the lack of
dynamic correlation at the HF level, not the lack of strong correlation. On
the other hand, it is possible for a single-reference approach to attempt to
describe strongly correlated systems by *essential* symmetry breaking.
Therefore, it is often crucial to distinguish these two by obtaining orbitals
in the presence of electron correlation.
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632
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Phys. Chem. Chem. Phys.

(2019),
21,
pp. 47638.
Link

Since orbital-optimized coupled-cluster doubles (OOCCD) can be computationally
demanding ($\mathcal{O}({o}^{2}{v}^{4})$), Rohini Lochan working with Martin
Head-Gordon proposed to obtain orbitals by optimizing the MP2 correlation
energy. To this end, BOs are introduced into SOSMP2 and MOSMP2 methods to
resolve the problems of symmetry breaking and spin contamination that are often
associated with Hartree-Fock orbitals. The molecular orbitals are optimized
with the mean-field energy plus a correlation energy taken as the opposite-spin
component of the second-order many-body correlation energy, scaled by an
empirically chosen parameter. This “optimized second-order opposite-spin”
(O2) method
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690
}
J. Chem. Phys.

(2007),
126,
pp. 164101.
Link
requires fourth-order computation on each
orbital iteration. O2 is shown to yield predictions of structure and
frequencies for closed-shell molecules that are very similar to scaled MP2
methods. However, it yields substantial improvements for open-shell molecules,
where problems with spin contamination and symmetry breaking are shown to be
greatly reduced.

$molecule 1 2 F H 1 1.001 $end $rem EXCHANGE HF BASIS sto-3g UNRESTRICTED TRUE JOBTYPE FORCE Options are SP/FORCE/OPT DO_O2 1 O2 with O(N^4) SOS-MP2 algorithm SOS_FACTOR 1000000 Opposite Spin scaling factor = 1.0 SCF_ALGORITHM DIIS_GDM SCF_GUESS GWH AUX_BASIS rimp2-vdz SCF_CONVERGENCE 8 THRESH 14 SYMMETRY FALSE PURECART 1111 $end

$molecule 1 2 F H 1 1.001 $end $rem UNRESTRICTED TRUE JOBTYPE FORCE Options are SP/FORCE/OPT EXCHANGE HF DO_O2 2 O2 with O(N^4) MOS-MP2 algorithm OMEGA 600 Omega = 600/1000 = 0.6 a.u. SCF_ALGORITHM DIIS_GDM SCF_GUESS GWH BASIS sto-3g AUX_BASIS rimp2-vdz SCF_CONVERGENCE 8 THRESH 14 SYMMETRY FALSE PURECART 1111 $end

Although O2 (or OOMP2) was successful in numerous applications, there are two
limitations of this model. First of all, the energy optimization often runs
into a numerical instability caused by the singularity of the MP2 energy due to
a small energy denominator. Secondly, the disappearance of Coulson-Fischer
point hinders the use of essential symmetry breaking. This led David
Stück and Martin Head-Gordon to regularized OOMP2 where they employed a
linear level shift parameter, $\delta $, to stabilize small energy
denominators.
^{
1057
}
J. Chem. Phys.

(2013),
139,
pp. 244109.
Link
The thermochemistry performance of
$\delta $-OOMP2 was found to be disappointing when one wishes to keep $\delta $
large enough to recover the Coulson-Fischer point.
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923
}
Mol. Phys.

(2017),
115,
pp. 21029.
Link

Joonho Lee working with Martin Head-Gordon developed a new regularized OOMP2
suite of methods that utilizes an energy-dependent regularizer
($\kappa $-regularizer) unlike the $\delta $-regularizer.
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631
}
J. Chem. Theory Comput.

(2018),
14,
pp. 5203.
Link
The $\kappa $-regularizer modifies the MP2 correlation energy as follows:

$${E}_{\kappa -\text{MP2}}=-\frac{1}{4}\sum _{ijab}\frac{{|\u27e8ij||ab\u27e9|}^{2}}{{\mathrm{\Delta}}_{ij}^{ab}}{\left(1-\mathrm{exp}(-\kappa {\mathrm{\Delta}}_{ij}^{ab})\right)}^{2}$$ | (6.23) |

where the energy denominator ${\mathrm{\Delta}}_{ij}^{ab}={\u03f5}_{a}{\u03f5}_{b}-{\u03f5}_{i}-{\u03f5}_{j}$ and $\kappa $ controls the regularization strength.
Evidently, $\kappa =0$ gives zero correlation energy (*i.e.*, HF) and $\kappa \to \mathrm{\infty}$ recovers the unregularized MP2 energy expression. In
$\kappa $-OOMP2, orbitals are then determined as a minimizer for ${E}_{\text{HF}}{E}_{\kappa -\text{MP2}}$. The $\kappa $ value of 1.45 ${E}_{h}^{-1}$ is recommended
due to its good balance between the Coulson-Fischer point recovery and
thermochemistry performance. It should be noted that $\kappa $-OOMP2 runs
through Q-Chem’s new SCF library, libgscf, and new MP2 library,
libgmbpt. The older OOMP2 code (written by Rohini Lochan and David
Stück) is no longer supported and should be used with a greater caution.
Furthermore, the new OOMP2 code can handle restricted (R), complex, restricted
(cR), unrestricted (U), generalized (G), and complex, generalized (cG) orbital
types. The complex, unrestricted (cU) orbital type is not yet supported due to
its limited applicability.

Summary of rem variables relevant to run $\kappa $-OOMP2:

CORRELATION | None (default) |
---|---|

JOBTYPE | sp (default) single point energy evaluation; force (force supported); opt (geometry optimization supported) |

BASIS | user’s choice (standard or user-defined: GENERAL or MIXED) |

GEN_SCFMAN_FINAL | TRUE (default if $\kappa $-OOMP2 is requested) |

FALSE (default for other SCF jobs) | |

SCF_ALGORITHM | GDM (default) |

DIIS | |

GDM-LS | |

AUX_BASIS | corresponding auxiliary basis (standard or user-defined: |

AUX_GENERAL or AUX_MIXED) | |

REGULARIZED_O2 | 0 (no regularizer; default) |

1 ($\delta $-regularizer) | |

2 ($\kappa $-regularizer; recommended) | |

3 ($\sigma $-regularizer) | |

REG_PARAMETER | regularization parameter multiplied by 1e${}^{3}$; no default |

1450 (Recommended value for $\kappa $-OOMP2) | |

N_FROZEN_CORE | 0 (Code supports this functionality but it is not |

recommended due to some convergence issues) | |

N_FROZEN_VIRTUAL | 0 (Code supports this functionality but it is not |

recommended due to some convergence issues) | |

SCS | 0 (default) |

1 Turns on spin-component scaling with SCS-OOMP2, | |

2 SOS-OOMP2, | |

3 arbitrary SCS-OOMP2 | |

SSS_FACTOR | 1000000 (default) Specify same-spin-component scaling factor (multiplied by 1e${}^{6}$) |

SOS_FACTOR | 1000000 (default) Specify opposite-spin-component scaling factor (multiplied by 1e${}^{6}$) |

DO_S2 | 0 (default) |

1 (Compute $\u27e8{S}^{2}\u27e9$ at the MP2 level) |

$molecule 0 2 O -2.766559046 0.187082886 0.566917837 H -3.696304300 1.179189102 -0.642506882 $end $rem BASIS cc-pvdz AUX_BASIS rimp2-cc-pvdz EXCHANGE hf THRESH 14 INPUT_BOHR true SCF_CONVERGENCE 8 SCF_ALGORITHM gdm MAXSCF 1000 SYMMETRY false SCF_GUESS sad GEN_SCFMAN true GEN_SCFMAN_FINAL true N_FROZEN_CORE 0 no frozen core N_FROZEN_VIRTUAL 0 no frozen virtual DO_O2 3 run OOMP2 REGULARIZED_O2 2 use kappa-regularizer REG_VARIABLE 1450 set kappa = 1.45 SCS 3 use arbitrary SCS SOS_FACTOR 883532 use cos = 0.883532 SSS_FACTOR 883532 use css = 0.883532 DO_S2 1 compute s^2 at the MP2 level UNRESTRICTED true use unrestricted GHF true use generalized COMPLEX true use complex $end

$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 BASIS cc-pvdz AUX_BASIS_CORR rimp2-cc-pvdz EXCHANGE hf THRESH 14 INPUT_BOHR true SCF_CONVERGENCE 8 SCF_ALGORITHM gdm MAXSCF 1000 SCF_GUESS sad SYMMETRY false GEN_SCFMAN true UNRESTRICTED false use restricted DO_O2 3 run OOMP2 REGULARIZED_O2 2 use kappa-regularizer REG_VARIABLE 1450 set kappa = 1.45 $end