This tutorial briefly demonstrates the keywords and their usage to invoke the exact two-component (X2C) Hamiltonian and a local X2C variant of it in Dirac. The current available implementation and other features which will be part of the next release Dirac2014 are described in [Knecht2014].

A recent and comprehensive review on the topic of relativistic Hamiltonians for chemistry can be found in [Saue2011] by Trond Saue. The central idea of any decoupling approach is to generate a two-component Hamiltonian that reproduces the positive-energy spectrum of the parent four-component Hamiltonian in the best possible way. This goal may be achieved by a unitary transformation \(\mathbf{U}\) that will formally decouple the large and small components by block-diagonalization:

\[\begin{split}{\hat{h}_{bd}} =
\mathbf{U}^{\dagger}
\left[\begin{array}{cc}
\hat{h}_{LL} & \hat{h}_{LS} \\
\hat{h}_{SL} & \hat{h}_{SS} \end{array}\right]
\mathbf{U} =
\left[\begin{array}{cc}
\hat{h}_{++} & 0 \\
0 & \hat{h}_{--} \end{array}\right],\end{split}\]

Popular choices to obtain a unitary transformation matrix \(\mathbf{U}\) are in particular discussed in Section 3 of [Saue2011] while a detailed abstract of the actual implementation of the X2C approach in Dirac is given in the last part of Section 3.3 in the same review paper. Property operators are transformed automatically on the fly to two-components within the X2C approach in Dirac in order to avoid the so-called picture-change errors. For a detailed discussion on the latter topic with the example of charge densities at a heavy nucleus see for example [Knecht2011] by Knecht and co-workers. Transforming the the electronic Hamiltonian based on the above prescription

\[{\hat{h}^{X2C}} = {\hat{h}_{++}} =
\left[\mathbf{U}^{\dagger}
\hat{h}^{4c}\mathbf{U}\right]_{++}\]

it is mandatory to also transform the parent four-component property-operator \(\Omega^{4c}\) using the same unitary transformation matrix \(\mathbf{U}\)

\[{\Omega^{2c}} =
\left[\mathbf{U}^{\dagger}
\Omega^{4c}\mathbf{U}\right]_{++}\]

since otherwise we would introduce (significant) errors (‘picture-change errors’) by simply using the approximate relation

\[{\Omega^{2c}} \approx \left[\Omega^{4c}\right]_{LL} .\]

To invoke the X2C Hamiltonian \(\hat{h}^{X2C}\) in Dirac which is based on the diagonalization of the 1-electron bare-nucleus Dirac Hamiltonian, use the keyword .X2C under the input deck **HAMILTONIAN:

```
**HAMILTONIAN
.X2C
```

This includes by default atomic-mean-field two-electron spin-same-orbit corrections from the AMFI module.

To add spin-other-orbit corrections (Gaunt term of the Breit interaction) use:

```
**HAMILTONIAN
.X2C
.GAUNT
```

For large systems the diagonalisation of the Dirac Hamiltonian — which is an essential step in the derivation of the
unitary transformation matrix \(\mathbf{U}_{mol}\) — may become a bottleneck for the full molecule (hence to stress
this fact we have introduced the explicit label *mol*).
This bottleneck can be avoided by extracting the coupling at the level of individual atoms as proposed in a working protocol by Peng
and Reiher [Peng2012]. In Dirac2013 we have implemented the *diagonal local approximation to the unitary decoupling
transformation* (DLU scheme) of [Peng2012]. The central idea behind this local scheme is to approximate \(\mathbf{U}_{mol}\) by a superposition
of \(i=1,...,N_A\) atomic (or in principle also fragment) matrices \(\mathbf{U}_{i}\) where a decoupling matrix is calculated for each *unique* atom type.

\[\mathbf{U}_{mol} \approx \sum\limits_{i=1}^{N_{A}} \mathbf{U}_{i} \oplus \mathbf{U}_{i+1} \oplus \mathbf{U}_{i+2} \oplus \ldots,\]

In order to benefit from the above DLU approach in Dirac which at present works *only for C1 symmetry* we need to first carry
out the atomic calculations and save the atomic \(\mathbf{U}_{i}\) matrices. This can be done as follows.
We run an atomic X2C transformation for each *unique* atom in our molecule (here we assume a cc-pVDZ basis):

```
**DIRAC
.TITLE
atomic X2C - U matrix construction
**HAMILTONIAN
.X2C
**WAVE FUNCTION
*SCF
**MOLECULE
*BASIS
.DEFAULT
cc-pVDZ
*SYMMETRY
.NOSYM
**END OF
```

Note the use of .NOSYM to enforce C1 symmetry in the atomic run even for an atom. A possible corresponding xyz input for an atom (here C) would look like:

```
1
carbon atom XYZ file
C 0.0 0.0 0.0
```

We save the file X2CMAT containing the atomic \(\mathbf{U}_{i}\) matrix and rename it according to the atomic number of the respective atom, here X2CMAT.006 for carbon (Z=6). For tungsten (Z=74) the appropriate ending would be X2CMAT.074 and similar for other atoms:

```
$ ./pam --inp=atom.inp --mol=carbon.xyz --get=X2CMAT
$ mv X2CMAT X2CMAT.006
```

After we have created all atomic \(\mathbf{U}_{i}\) matrices for each *unique* atom type in our molecule — let’s
assume to work on W(CH \(_{3}\) ) \(_{6}\) where we would need a transformation matrix for W, C and H — we are ready to setup
the molecular X2C calculation based on the local/atomic approach of the
decoupling unitary \(\mathbf{U}_{i}\) matrices. The relevant part of the input would then read:

```
**DIRAC
.TITLE
local X2C based on atomic U matrices
.WAVE FUNCTION
**HAMILTONIAN
.X2C
*X2C
.fragX2C
**WAVE FUNCTION
.SCF
*SCF
...
...
**MOLECULE
*BASIS
.DEFAULT
cc-pVDZ
*SYMMETRY
.NOSYM
**END OF
```

This input should be used with the following pam line:

`$ ./pam ... --put=X2CMAT.*`

In this DLU approach proposed by Peng and Reiher [Peng2012] the molecular \(\mathbf{U}_{mol}\) matrix will then be approximated as described above. It works with spinfree as well as all spin-dependent (spin-orbit) Hamiltonian schemes and includes a picture-change transformation of all molecular four-component property operators \(\Omega^{4c}_{mol}\).