(sec:caspt2.detailed)=
# Complete Active Space Peturbation Theory : CASPT2 and CASPT2-K

The fully internally contracted CASPT2 (FIC-CASPT2) approach is
available with real, imaginary and IPEA
shifts.{cite}`caspt2,caspt2Shift,caspt2IShift`. The ORCA implementation
employs a reformulation of the CASPT2, that completely avoids the fourth
order reduced density matrix, that would appear in the canonical
implementation.{cite}`kollmar_d4efficient_2021` Some concepts are shared by a
recent development reported by Sokolov and
coworkers.{cite}`sokolov_mradc_2020` The modification allows calculations
with large active spaces without approximating the results e.g. with the
cumulant expansion.

It should be noted that the IPEA shift in OpenMOLCAS slightly deviates
from ORCA.{cite}`openmolcas2019`. Here, the IPEA shift, $\lambda$, is added
to the matrix elements of the internally contracted CSFs
$\Phi^{pr}_{qs}=E^p_qE^r_s|\Psi^0>$ with the generalized Fock operator

$$<\Phi^{p'r'}_{q's'}|\hat{F}|\Phi^{pr}_{qs}>+=<\Phi^{p'r'}_{q's'}|\Phi^{pr}_{qs}> \cdot \frac{\lambda}{2}\cdot (4+\gamma^p_p-\gamma^q_q+\gamma^r_r-\gamma^s_s),$$

where $\gamma^p_q=<\Psi^0|E^p_q|\Psi^0>$ is the expectation value of the
spin-traced excitation operator.{cite}`caspt2ipea2` The labels p,q,r,s refer
to general molecular orbitals (inactive, active and virtual).
Irrespective of the ORCA implementation, the validity of the IPEA shift
in general remains questionable and is thus by default
disabled.{cite}`zobel_ipea_2016` ORCA features an alternative formulation,
named **CASPT2-K**, that revises the zeroth order Hamiltonian
itself.{cite}`caspt2k` Here, two additional Fock matrices are introduced for
excitation classes that add or remove electrons from the active space.
The new Fock matrices are derived from the generalized Koopmans'
matrices corresponding to electron ionization and attachment processes.
The resulting method is less prone to intruder states and the same time
more accurate compared to the canonical CASPT2 approach. For more a
detailed discussion, we refer to the paper by Kollmar et al.{cite}`caspt2k`

The CASPT2 and CASPT2-K approaches are called in complete analogy to the
FIC-NEVPT2 approach. Note that the methodology can be combined with the
RI approximation. A detailed example with comments on the output is
given in Section
{ref}`sec:energygradients.caspt2.typical`. Below is concise list
with the accessible options.

```orca
%casscf
  ...

  MULT   1,3 # multiplicity block
  NRoots 2,2 # number of roots for the MULT blocks

  TrafoStep RI      # optional for RI approximation for CASSCF and CASPT2
  PTMethod  FIC_CASPT2  # canonical CASPT2 approach
            FIC_CASPT2K # CASPT2-K with revised H0

  # Detailed settings (this is optional)
  PTSettings
    CASPT2_ishift 0.0    # imaginary level-shift
    CASPT2_rshift 0.0    # real      level-shift
    CASPT2_IPEAshift 0.0 # IPEA shift.
    MaxIter       20     # Maximum for the CASPT2 iterations
    TSmallDenom   1e-2   # printing thresh for small denominators

    # general settings
    NThresh 1e-6  # FIC-CASPT2 cut off for linear dependencies
    D4Tpre  1e-10 # truncation of the 4-pdm
    D3Tpre  1e-14 # trunaction of the 3-pdm
    EWIN  -3,1000 # Energy window for the frozencore setting fc_ewin

    # Option to skip the PT2 correction for a selected multiplicity blocks and roots 
    # (same input structure as weights in %casscf)
    selectedRoots[0]=0,1 # skip the first roots of MULT=1
    selectedRoots[1]=0,0 # skip MULT=3 roots

    #CASPT2-K specific options
    TReg 1e-2 # default for the Tikhonov reguralization
  end

end
```

CASPT2 can also be set using the simple keywords on top of any valid
CASSCF input.

```orca
!CASPT2       # FIC-CASPT2
!CASPT2K      # FIC-CASPT2-K
!RI-CASPT2    # FIC-CASPT2 with RI approximation
!RI-CASPT2-K  # FIC-CASPT2-K with RI approximation
%casscf ...
```