9.5 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 \[\begin{align} E_{\hbox{\tiny MP2-F12}} & = E_{\rm MP2} + E_{\rm F12} , \end{align}\](9.5) where $E_{\rm MP2}$ is the conventional MP2 energy and $E_{\rm F12}$ the correction from explicitly-correlated theory. The second term contains contributions from explicitly-correlated geminal basis functions of the form \[\begin{align} \hat Q_{12} f_{12} \vert ij \rangle, \end{align}\](9.6) where $\vert ij \rangle$ is a two-electron determinant of occupied (semi-)canonical Hartree–Fock spin orbitals, $f_{12}$ is a correlation factor, which can be either linear $r_{12}$ (in this case, the approach is denoted MP2-R12 instead of MP2-F12) or a function of $r_{12}$, and $\hat Q_{12}$ defines the doubles excitation space covered by the geminals (it also ensures strong orthogonality to the occupied orbitals). Usually $\hat Q_{12}$ is chosen to be $\hat Q_{12} = (1-\hat O_1)(1-\hat O_2) - \hat V_1 \hat V_2$, where $\hat O_\mu = \sum_{k} \vert\varphi_k(\mu)\rangle\langle\varphi_k(\mu)\vert$ is the projection operator onto the space spanned by the occupied spin orbitals $\varphi_k$ and $\hat V_\mu = \sum_{a} \vert\varphi_a(\mu)\rangle\langle\varphi_a(\mu)\vert$ is the projector onto the virtual spin orbitals.

The F12 correction is obtained by minimizing the functional \[\begin{align} F_{\rm F12} & = \sum_{i<j} \left\{ {\mathbf c}_{ij}^T {\mathbf B}_{ij} {\mathbf c}_{ij} + 2{\mathbf c}_{ij}^T {\mathbf v}_{ij}\right\} \end{align}\](9.7) with respect to the amplitudes collected in the vector ${\mathbf c}_{ij}$. The vectors ${\mathbf v}_{ij}$ and the matrices ${\mathbf B}_{ij}$ are defined as \[\begin{align} {\mathbf v}_{ij}(kl) & = \langle kl\vert f_{12}\hat Q_{12} r_{12}^{-1}\vert ij\rangle , \\ {\mathbf B}_{ij}(kl,mn) & = \langle kl\vert f_{12}\hat Q_{12} (\hat f_1 + \hat f_2 - \varepsilon_i - \varepsilon_j ) \hat Q_{12} f_{12}\vert mn\rangle , \end{align}\](9.8–9.9) in the spin-orbital formalism ($m,n$ denote spin orbitals and $\vert mn\rangle$ is a two-electron determinant). $\hat f_\mu$ is the Fock operator for electron $\mu$ and $\varepsilon_k$ is a (semi-)canonical Hartree–Fock orbital energy.

The F12 implementation is compatible with ECP, DKH and X2C scalar relativistic treatments. The additional terms that arise in the Fock matrix are included in the evaluation of the F12 contributions.[246, 247]

A MP2-F12 calculation is defined through a number of choices concerning the nature of the geminals ($f_{12}$ and $\hat Q_{12}$), the geminal excitation space (ijkl or ijij) and approximations in computing the $B$ matrix (GBC, EBC, $[\hat T,f_{12}]$). 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. [248] 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 [249]. If define is used to set up the cabs basis, the library cabasen is searched. This library contains the optimised cabs basis sets [250] for the cc-pVXZ-F12 basis sets of Peterson et al. [251]. 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 f12 option in the cc menu when running define. This command activates the f12 menu, where the default options may be changed if desired:

 Orbital basis       : cc-pVTZ-F12
 Cardinal number     : T
 Recommended exponent:   1.0000
 Actual      exponent:   1.0000

INPUT MENU FOR MP2-F12 CALCULATIONS

 ansatz    : CHOOSE ANSATZ               2      [1,2*,2]
 r12model  : CHOOSE MODEL                B      [A,B]
 comaprox  : COMMUTATOR APPROXIMATION    F+K    [F+K,T+V]
 cabs      : CABS ORTHOGONALIZATION      svd  1.0D-08   [cho,svd]
 examp     : CHOOSE FORMULATION          fixed noflip [inv,fixed,noinv, flip,noflip]
 r12orb    : CHOOSE GEMINAL ORBITALS     hf     [hf,rohf,boys,pipek]
 corrfac   : CHOOSE CORRELATION FACTOR   LCG    [R12,LCG]
 cabsingles: CABS SINGLE EXCITATIONS     on     [on,off]
 pairenergy: PRINT OUT PAIRENERGIES      off    [on,off]
 slater    : SLATER EXPONENT             1.0000

 * / end    : write $rir12 to file and leave the menu
 &          : go back - leaving $rir12 unchanged...

ansatz: corresponds to the choice of $\hat Q_{12}$. Almost all modern MP2-F12 calculations use ansatz 2 (default), which gives much improved energies over ansatz 1 (see Ref. [252] 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 ${\mathbf B}_{ij}$ (see Ref. [252] for details). The cost and accuracy increases from A to 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 $[\hat T,f_{12}]$, where the matrix representations of F+K or T+V are used. F+K (the core Hamiltonian plus Coulomb term) is recommended and is the default.
cabs: refers to the method of orthogonalising the orbitals in the complementary auxiliary basis. Singular-value decomposition (svd) or Cholesky 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 cc menu.
examp: refers to the choice of excitation space. inv is the orbital-invariant merhod of Ref. [253], with amplitudes $c_{ij}(kl)$. noinv is the original orbital-dependent diagonal "ijij" method of Ref. [253], with amplitudes $c_{ij}(ij)$ (not recommended, unless in combination with localised orbitals). fixed is the (diagonal and orbital-invariant) rational generator approach of Ref. [254], where the F12 amplitudes are not optimised but predetermined using the coalescence conditions (default). An additional keyword noflip suppresses the use of spin-flipped geminals in open-shell calculations; by default spin-flipped geminals are used as described in Ref. [23].
r12orb: controls which orbitals are used in the F12 energy contribution. hf means that (semi-)canonical Hartree–Fock orbitals are used (default). rohf means that ROHF orbitals are used (any frozen orbitals will then also implicitly be ROHF). For calculations on closed-shell systems, localised orbitals may be used. Both the Boys [255] and Pipek–Mezey [256] methods are available for localisation of the orbitals.
corrfac: corresponds to the choice of correlation factor $f_{12}$ in the geminal basis functions. R12 results in a calculation using linear-$r_{12}$ and LCG results in a calculation using the Slater-type correlation factor with exponent 1.4 $a_0^{-1}$, represented as a linear combination of six Gaussians (see Ref. [257]). Note that the exponents 0.9, 1.0 and 1.1 $a_0^{-1}$ are recommended for use with the cc-pVXZ-F12 basis sets [251].
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.
pairenergy: controls whether or not the F12 contribution to the MP2 pair energies appear in the output (default off).

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  1.4000

The 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 $a_0^{-1}$, represented as a linear combination of three Gaussians, use

$lcg
  nlcg    3
  slater  1.0000

Alternatively, the exponents and coefficients of the fit may be given explicitly:

$lcg
  nlcg    3
  expo1  coef1
  expo2  coef2
  expo3  coef3

MP2-F12 calculations may be combined with Grimme’s SCS approach (S. Grimme, J. Chem. Phys. 118 (2003) 9095) by inserting scs in $ricc2,

$ricc2
  mp2 energy only
  scs

In 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 10.8.

For open-shell calculations, two choices of the examp fixed noflip 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 $\alpha\beta$ 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 [251] 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 [250] 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 [258, 246, 259].

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@kit.edue).