MOLPRO

31 EXPLICITLY CORRELATED METHODS 230
31.1 Reference functions
The MP2-F12, CCSD-F12, and UCCSD-F12 methods must use conventional (non-density fitted)
spin-restricted Hartree-Fock reference functions (HF or RHF). DF-HF cannot be used for
these methods. This restriction is necessary to ensure that the Fock matrix is diagonal and
consistent with the integrals used in these methods. For DF-MP2-F12, DF-LMP2-F12, and
DF-RMP2-F12 either HF or DF-HF reference functions can be used.
Currently, only finite dipole fields can be applied, other perturbations are not yet supported.
ECPs can be used, but this is still experimental and not extensively benchmarked. The Douglas-
Kroll-Hess Hamiltonian cannot be used in combination with F12 methods.
31.2 Wave function Ansätze
The so called ”ansatz” determines the definition of the explicitly correlated wave function. This
is to be distinguished from the various approximations that can be used to approximate the
Hamiltonian matrix elements. Generally, we use ansatz 3 (cf. I), for which the projector has the
form
ˆQ 12 = (1 − ô 1 )(1 − ô 2 )(1 − ˆv 1 ˆv 2 ),
where ô i is a one-electron projector for electron i onto the occupied space, and ˆv i projects onto
the virtual orbital space. In the case that domain approximations are used in local explicitly
correlated wave functions, the operators ˆv are replaced by operators dˆ
i j that project just onto the
domain for the orbital pair i j.
In MOLPRO the following wave function ansätze can be used:
31.2.1 The general ansatz
The conventional external pair functions are augmented by terms of the form
|u F12
i jp 〉 = ∑ ∑T i jp
kl
ˆQ 12 ˆF 12 |kl〉
p =±1 kl
This ansatz is orbital invariant (i.e., the same results are obtained with canonical or localized
orbitals), but it often suffers from geminal basis set superposition errors. Furthermore, singularities
may occur in the zeroth-order Hamiltonian, in particular for larger systems. Therefore, this
ansatz is normally not recommended.
31.2.2 The diagonal ansatz (D)
The sum over kl in equation (63) is restricted to i j. This ansatz is not orbital invariant and
size consistent only when localized orbitals are used. However, geminal basis set superposition
errors are absent and therefore the results are often more accurate than with the general ansatz.