RI-MP2-F12 Calculations

To obtain the F12 correction to the MP2 energy, the data group
`$rir12` must be added to the `control` file. A typical run
will include the input:

$ricc2 mp2 energy only $rir12

The MP2-F12 ground-state energy is

(9.24) |

where is the conventional MP2 energy and the correction from explicitly-correlated theory. The second term contains contributions from explicitly-correlated geminal basis functions of the form

(9.25) |

where is a two-electron determinant of occupied (semi-)canonical Hartree-Fock spin orbitals, is a correlation factor, which can be either linear (in this case, the approach is denoted MP2-R12 instead of MP2-F12) or a function of , and defines the doubles excitation space covered by the geminals (it also ensures strong orthogonality to the occupied orbitals). Usually is chosen to be , where is the projection operator onto the space spanned by the occupied spin orbitals and is the projector onto the virtual spin orbitals.

The F12 correction is obtained by minimizing the functional

(9.26) |

with respect to the amplitudes collected in the vector . The vectors and the matrices are defined as

(9.27) | ||

(9.28) |

in the spin-orbital formalism ( denote spin orbitals and is a two-electron determinant). is the Fock operator for electron and is a (semi-)canonical Hartree-Fock orbital energy.

A MP2-F12 calculation is defined through a number of choices concerning the
nature of the geminals ( and
), the geminal excitation space
(ijkl or ijij) and approximations in computing the matrix (GBC, EBC,
).
These choices correspond to keywords in the `$rir12` data group, explained below.

To run a MP2-F12 calculation, one has to select the auxiliary basis sets
`cbas`

, `cabs`

and optionally `jkbas`

.
The `ricc2` program uses the robust fitting techniques of Ref. [97]
for the F12 integrals and the `cbas`

basis is used for both the F12
and the usual MP2 Coulomb integrals. For the density fitting of the Coulomb and
exchange matrices of the Fock matrix, the `jkbas`

will be used
instead of the `cbas`

basis if it is included in the control file (this is
recommended and is achieved using the `rijk`

menu in `define`).
For the RI approximation of the 3- and 4-electron integrals as sums of products
of 2-electron integrals, intrinsic to the F12 method, the complementary auxiliary
basis (CABS) approach is used [98]. If `define` is
used to set up the `cabs`

basis, the library
`cabasen`

is searched. This library contains the
optimised `cabs`

basis sets [99] for the
cc-pVXZ-F12 basis sets of Peterson *et al.* [100].
For other basis sets, the auxilliary basis in the library `cabasen`

is identical with the auxilliary basis in the library `cbas`

.

The `$rir12` data group may be set by choosing the `mp2-f12`

option in the `ricc2`

menu when running `define`.
This command activates the `mp2-f12`

menu, where the default
options may be changed if desired:

INPUT MENU FOR MP2-F12 CALCULATIONS ansatz : CHOOSE ANSATZ 2 [1,2*,2] r12model : CHOOSE MODEL B [A,A',B] comaprox : COMMUTATOR APPROXIMATION T+V [F+K,T+V] cabs : CABS ORTHOGONALIZATION svd 1.0d-08 [cho,svd] examp : CHOOSE FORMULATION fixed [inv,fixed,noinv] local : CHOOSE LOCALIZATION METHOD off [off,boys,pipek] corrfac : CHOOSE CORRELATION FACTOR LCG [R12,LCG] cabsingles: CABS SINGLE EXCITATIONS on [on,off] * / end : write $rir12 to file and leave the menu & : go back - leaving $rir12 unchanged...

`ansatz`- corresponds to the choice of
. Almost
all modern MP2-F12 calculations use ansatz
`2`

(default), which gives much improved energies over ansatz`1`

(see Ref. [101] for details). The principal additional cost of using ansatz`2`

over ansatz`1`

is concerned with the coupling between the F12 and conventional amplitudes. This is avoided by choosing`2*`

, which corresponds to neglecting EBC (Extended Brillouin Condition) terms in the Fock matrix elements. `r12model`- is the method of computing the matrices
(see Ref. [101] for details). The cost and
accuracy increases in the progression
`A`

,`A'`

,`B`

. It is recommended to use`B`

(default). The energies computed using`A`

are then also printed out in the output. `comaprox`- is the method for approximately computing the integrals
for the operator
, where the matrix representations of
`F+K`

or`T+V`

are used.`T+V`

(the core Hamiltonian) is recommended and is the default. `cabs`- refers to the method of orthogonalising the orbitals in
the complementary auxiliary basis. Single-value decomposition (
`svd`

) or Choleski decomposition (`cho`

) are available.`svd`

is recommended and is the default, with a threshold of`1.0d-08`

. The basis set used for CABS is set from the`ricc2`

menu. `examp`- refers to the choice of excitation space.
`inv`

is the orbital-invariant merhod of Ref. [102], with amplitudes .`noinv`

is the original orbital-dependent diagonal "ijij" method of Ref. [102], with amplitudes (not recommended, unless in combination with localised orbitals).`fixed`

is the (diagonal and orbital-invariant) rational generator approach of Ref. [103], where the F12 amplitudes are not optimised, but predetermined using the coalescence conditions (default). `local`- controls which orbitals are used in the calculation.
`off`

means that (semi-)canonical Hartree-Fock orbitals are used (default). For calculations using linear- as correlation factor, and r12model`A`

or`A'`

, localised orbitals may be used. Both the Boys [104] and Pipek-Mezey [105] methods are available for localisation of the orbitals. `corrfac`- corresponds to the choice of correlation factor
in the geminal basis functions.
`R12`

results in a calculation using linear- and`LCG`

results in a calculation using the Slater-type correlation factor with exponent 1.4 , represented as a linear combination of six Gaussians (see Ref. [106]). Note that the exponents 0.9, 1.0 and 1.1 are recommended for use with the cc-pVXZ-F12 basis sets [100]. `cabsingles`- switches on/off the calculation of a second-order
correction to the Hartree-Fock energy by accounting for single excitations
into the complementary auxiliary basis set (CABS).
The single excitations into the CABS basis can be computed without extra costs
if the CABS Fock matrix elements are required anyway for the F12 calculation
(
*i.e.*, for ansatz`2`

, approximation`B`

or comaprox`F+K`

). The computation of CABS singles cannot be switched off if it is free of costs.

Further options:

`corrfac` `LCG`

refers to a further data group for the definition
of the correlation factor. When `define` is used, the default is

$lcg nlcg 6 slater 1.4000The nature of the LCG correlation factor may be changed by editing this data group in the control file. For example, to use a Slater-type correlation factor with exponent 1.0 , represented as a linear combination of three Gaussians, use

$lcg nlcg 3 slater 1.0000Alternatively, the exponents and coefficients of the fit may be given explicitly:

$lcg nlcg 3 expo1 coef1 expo2 coef2 expo3 coef3

`pairenergy` controls whether or not the F12 contribution to
the MP2 pair energies appear in the output (default `off`

),

pairenergy off [on,off]MP2-F12 calculations may be combined with Grimme's SCS approach (S. Grimme,

$ricc2 mp2 energy only scsIn this case, the SCS parameters cos=6/5 and css=1/3 are used. Also individual scaling factors for the same-spin and opposite-spin contributions may be defined, see Section 9.7.

For open-shell calculations, two choices of the `examp` `fixed`

method
are available. These are controled by a keyword in the `$rir12` data group

ump2fixed full [diag,full]These differ in the treatment of the block, where either only the diagonal excitations enter (with amplitude 0.5)

`diag`

, or the
equivalent of the spin-adapted singlet and triplet pair excitations enter
(as far as possible) `full`

. Note that the `diag`

method with UMP2-F12
yields a result different to that of `fixed`

MP2-F12, even for identical
RHF and UHF determinants. However, the `diag`

method is somewhat less
expensive than the `full`

method.
Recommendations for orbital and auxiliary basis sets:

The best orbital basis sets to use for MP2-F12 calculations are probably
the cc-pVXZ-F12 basis sets, specially
optimised for MP2-F12 calculations [100]
for the atoms H, He, B-Ne and Al-Ar.
In conjunction with these cc-pVXZ-F12 basis sets,
we recommend to use the optimised cc-pVXZ-F12 sets of Yousaf and
Peterson [99] as `cabs`

. Furthermore,
`cbas`

and `jkbas`

basis sets can be selected from the
`cbasen`

and `jkbasen`

libraries, respectively, using the
alias cc-pVXZ-F12 (a `jkbas`

is currently not available for
He, Ne and Ar). This alias points to the
corresponding aug-cc-pwCV(X+1)Z `cbas`

and aug-cc-pV(X+1)Z `jkbas`

. These recommendations are on the side
of caution and are likely to be refined as more experience is
gained [88,107,108].

For atoms other than H, He, B-Ne and Al-Ar, optimised F12
basis sets are not yet available. In this case, basis sets must
be selected and/or optimised carefully. It is advised to
contact the Theoretical Chemistry Group in Karlsruhe for support
(e-mail to: `klopper@chem-bio.uni-karlsruhe.de`

).