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# Relativistic corrections

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

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

## Using the Douglas–Kroll–Hess or eXact-2-Component Hamiltonians

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):

Optimum parametrization (OPT, default)`DKHP=1`

:Exponential parametrization (EXP)`DKHP=2`

:Square-root parametrization (SQR)`DKHP=3`

:McWeeny parametrization (MCW)`DKHP=4`

:Cayley parametrization (CAY)`DKHP=5`

:

**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).

## Example for computing relativistic corrections

- 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