6.8 Size-Consistent Brillouin-Wigner Perturbation Theory 6.8.1 Introduction 6.9 Direct Random Phase Approximation Methods

6.8.2 BW-s2 Job Control

(September 1, 2024)

The BW-s2 approach has been implemented with the resolution-of-the-identity approximation and is invoked with METHOD = RIBWS2. The BW-s2 equation is iterative much like BW2, and keywords that give more precise control over the self-consistent determination of the BW-s2 energy are housed within a separate $bws2 section. Finally, the underlying algorithm that is used to solve the BW-s2 equation can be used to solve for other forms of the correlation energy such as MP2, BW2, size-extensive xBW2 (which is not size-consistent), or second-order Bethe-Goldstone (BGE2)/independent electron pair approximation (IEPA). While these methods are available for academic purposes, it is advised to only use the BW2, xBW2, and BGE2/IEPA codes for comparison as the first two are not size-consistent and the latter is not orbital invariant. For MP2 calculations, the standard RI-MP2 code should be used.

METHOD
       Defines the correlation energy expression to be used.
INPUT SECTION: $bws2
TYPE:
       STRING
DEFAULT:
       BWS2
OPTIONS:
       BWS2, BW2, xBW2, BGE2, IEPA, or MP2
RECOMMENDATION:
       BW-s2 is the recommended method, but if comparisons with non-size-consistent methods like BW2 or xBW2 or non-orbital-invariant methods like BGE2/IEPA are desired, those options are available.

ALPHA
       Controls the extent of BW-s2( $\alpha$ ) regularization.
INPUT SECTION: $bws2
TYPE:
       INTEGER
DEFAULT:
       100
OPTIONS:
       $n$ where $n/100$ is the value of $\alpha$ .
RECOMMENDATION:
       The default is set to the original BW-s2 theory where $\alpha=1$ (corresponding to $n=100$ ) is exact for a two-electron, two-orbital problem. For general applications $\alpha=4$ ( $n=400$ ) is a broadly successful choice. Other values can be set based on the chemical problem at hand.

ENERGY_CONVERGENCE
       Defines the convergence threshold for the correlation energy.
INPUT SECTION: $bws2
TYPE:
       INTEGER
DEFAULT:
       $n$ = SCF_CONVERGENCE
OPTIONS:
       $n$ where the convergence threshold is set to $10^{-n}$
RECOMMENDATION:
       Use the default unless the cost of the calculation becomes prohibitive, then consider reducing to a slightly lower value.

MAX_CYCLES
       Defines maximum number of iterations to be used in determining the correlation energy.
INPUT SECTION: $bws2
TYPE:
       INTEGER
DEFAULT:
       $n$ = SCF_MAX_CYCLES
OPTIONS:
       $n>0$
RECOMMENDATION:
       Use the default unless convergence is particularly slow, then increase the number of iterations accordingly.

Example 6.22 Example of RI-BW-s2 on an acetylene pyridine complex

$molecule
0 1
N   -0.083032490   0.000714589   1.055199987
C   -0.202853764  -1.141725850   0.364933690
H   -0.098485626  -2.055097952   0.937432624
C   -0.446781438  -1.191763671  -1.004512261
H   -0.533649208  -2.145855109  -1.504171553
C   -0.574682087   0.003439534  -1.704309480
H   -0.763683910   0.004480103  -2.768726700
C   -0.453456753   1.197242545  -1.000916473
H   -0.545630798   2.152272643  -1.497795085
C   -0.209311108   1.144507590   0.368367296
H   -0.110167072   2.056697259   0.943573961
C    0.471836022  -0.006058187   5.541718960
H    0.587246066  -0.005483995   6.596732782
C    0.339766259  -0.006607922   4.335471660
H    0.221618143  -0.006345494   3.270966186
$end

$rem
method      ribws2
basis       aug-cc-pvdz
aux_basis   rimp2-aug-cc-pvdz
scf_convergence 8
thresh      14
integral_symmetry   false
$end

! METHOD keyword is not required for BW-s2
! MAX_CYCLES keyword is manually set to the default here
$bws2
alpha       400
max_cycles  50
$end

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6.8.2 BW-s2 Job Control