# Navigation Area

# DKH/X2C

# Main content

## DKH - arbitrary-order scalar-relativistic Douglas-Kroll-Hess module

Exact decoupling of positive and negative energy states of the Dirac Hamiltonian is possible in the framework of unitary transformation techniques. The resulting two-component Hamiltonians feature exactly the same spectrum of energy eigenvalues as the original Dirac operator.

The DKH protocol, employing an expansion of the Dirac Hamiltonian in even terms of ascending order in the external potential, yields well-defined and regular expressions for any order; it is the unique analytic unitary transformation scheme for the Dirac Hamiltonian with this salient features.

Its exact infinite-order version DKHoo cannot be realized by a straightforward numerical iterative extension of the original DKH procedure to arbitrary order within one-component electronic structure programs. A more sophisticated ansatz based on analytic evaluation of the DKH operators up to any pre-defined order via a suitable symbolic parser routine has to be employed instead. Its algorithmic principles and its efficient scalar-relativistic implementation are presented.

The necessary order for exact decoupling (i.e., decoupling up to machine precision) can be determined prior to any quantum chemical calculation. Once this maximum order has been determined, the spectrum of the positive-energy part of the decoupled Hamiltonian, e.g., for electronic bound states, cannot be distinguished from the corresponding part of the spectrum of the Dirac operator.

However, for ordinary chemical purposes low-order DKH calculations are sufficient. Already the original second-order DKH scheme invented by Hess provides almost converged results for relative energies and valence-shell-dominated properties.

For practical calculations we recommend fourth-order DKH, DKH4, since up to fourth order the DKH Hamiltonian is independent of the chosen parameterization of unitary transformation. Higher-order DKH Hamiltonians depend slightly on the chosen parameterization 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, J. Chem. Phys. 121 2004, 2037-2047.

M. Reiher, A. Wolf, J. Chem. Phys. 121 2004, 10945-10956.

For details on the different parametrizations of the unitary transformations see

A. Wolf, M. Reiher, B. A. Hess, J. Chem. Phys. 117 2002, 9215-9226.

For a conceptual overview see:

M. Reiher, Theor. Chem. Acc. 116 2006, 241-252.

The Douglas-Kroll-Hess unitary transformation technique can also be applied to the calculation of molecular properties in a scalar-relativistic framework.

However, applying the non-relativistic expression for the property operator in DKH theory leads to an artefact called picture-change error (PCE). Therefore, if a consistent treatment of molecular properties within the DKH framework is sought for, a correction for the PCE has to be taken into account.

A general way how to obtain picture-change corrected molecular properties within (high-order) DKH theory has been established and implemented in our arbitrary-order DKH property module. Our results show that the field independent unitary transformation is perfectly suited to reproduce four-component results.

For details on the infinite-order DKH property transformations see

A. Wolf, M. Reiher, J. Chem. Phys. 124 2006, 064102.

A. Wolf, M. Reiher, J. Chem. Phys. 124 2006, 064103.

### Availability of our DKH code in other programs

- MOLCAS version 7 (arbitrary-order DKH Hamiltonian and property modules)
- MOLPRO version 2006.2 (arbitrary-order DKH module)
- ORCA (old 1st-5th-order DKH Hamiltonian, 2nd-order property subroutine)
- TURBOMOLE (inofficial version available at ETH, Düsseldorf (Marian group) and Karlsruhe (Klopper group))
- CP2k (old 1st-5th-order DKH Hamiltonian revised by J. Thar, Prof. B. Kirchner, Leipzig)