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# Relativistic corrections

There are three ways in Molpro to take into account scalar relativistic effects:

1. Use the Douglas-Kroll-Hess or eXact-2-Component (X2C) relativistic one-electron integrals.
2. Compute a perturbational correction using the Cowan-Griffin operator (see section One-electron operators and expectation values (GEXPEC)).
3. Use relativistic effective core potentials (see section effective core potentials).

For all-electron calculations, the prefered way is to use either the Douglas–Kroll–Hess (DKH) or eXact-2-Component (X2C) Hamiltonians, the former of which is available up to (in principle) arbitrary order in Molpro. DKH is activated by setting any of

SET,DKROLL=1
SET,DKHO=$n$, ($n=2,\dots,99$),
SET,DKHP=$m$, ($m=1,\dots,5$)

or for X2C by setting

SET,DKHO=101
somewhere in the input before the first energy calculation.

Alternatively, these values can be given as options on the INT command:

INT,[DKROLL=1],DKHO=$n$,DKHP=$m$.

or

INT,DKHO=101

The DKH option DKROLL is available for compatibility with earlier versions of Molpro. If only DKROLL=1 is given, the default for DKHO is 2. Setting DKROLL=0 disables DKH and X2C, independently of the setting of DKHO. DKH is also disabled by setting DKHO=0, unless DKROLL=1 is set. In order to avoid confusion, it is recommended only to use DKHO and never set DKROLL.

The value of DKHP specifies the parametrization for the DKH treatment (it has no effect for X2C):

• DKHP=1: Optimum parametrization (OPT, default)
• DKHP=2: Exponential parametrization (EXP)
• DKHP=3: Square-root parametrization (SQR)
• DKHP=4: McWeeny parametrization (MCW)
• DKHP=5: Cayley parametrization (CAY)

Example:

 SET,DKHO=8 ! DKH order = 8 SET,DKHP=2 ! choose exponential parametrization for unitary transformations (recommended)

Up to fourth order (DKHO=4) the DKH Hamiltonian is independent of the chosen parametrization. Higher-order DKH Hamiltonians depend slightly on the chosen paramterization of the unitary transformations applied in order to decouple the Dirac Hamiltonian.

For details on the infinite-order DKH Hamiltonians see
M. Reiher, A. Wolf, JCP 121, 2037–2047 (2004),
M. Reiher, A. Wolf, JCP 121, 10945–10956 (2004).

For details on the different parametrizations of the unitary transformations see
A. Wolf, M. Reiher, B. A. Hess, JCP 117, 9215–9226 (2002).

The current implementation is the polynomial-cost algorithm by Peng and Hirao: D. Peng, K. Hirao, JCP 130, 044102 (2009).
A detailed comparison of the capabilities of this implementation as well as the current implementation of the X2C approach is provided in:
D. Peng, M. Reiher, TCA 131, 1081 (2012).

examples/ar2_rel.inp
***,ar2
geometry={ar1;ar2,ar1,r}   !geometry definition
r=2.5 ang                  !bond distance
{hf;                       !non-relativisitic scf calculation
expec,rel,darwin,massv}    !compute relativistic correction using Cowan-Griffin operator
e_nrel=energy              !save non-relativistic energy in variable enrel
show,massv,darwin,erel     !show individual contribution and their sum

dkroll=1                   !use douglas-kroll one-electron integrals
hf;                        !relativistic scf calculation
e_dk=energy                !save relativistic scf energy in variable e_dk.
show,massv,darwin,erel     !show mass-velocity and darwin contributions and their sum
show,e_dk-e_nrel           !show relativistic correction using Douglas-Kroll