# ECP calculation¶

In DIRAC, the effective core potential (ECP) method is implemented and various subsequent correlation methods are available within the two-component or one-component effective Hamiltonian.

From the inclusion (exclusion) of spin-orbit potential parameters in the input file, molecular spinors (orbitals) are obtained and this is the starting point of several ground and excited state calculation with the ECP method.

The first example is the SCF calculation of Hydrogen Iodide.

## SOREP calculation¶

The simplest SCF input file (`HF.inp`) for ECP calculation is

together with the molecular input (`HI_sorep.mol`)

This is the two-component spin-orbit relativistic effective core potential (SOREP) calulation. (For the detail of input, see How to specify ECP parameters in mol files)

In the calculation output, the two-component molecular spinors (see also the `full output`) are listed as

```* Fermion symmetry E1
* Closed shell, f = 1.0000
-0.91849028398625  ( 2)    -0.53547268879208  ( 2)    -0.40006484189588  ( 2)    -0.37307836548798  ( 2)
* Virtual eigenvalues, f = 0.0000
0.03426904787073  ( 2)     0.05875502261229  ( 2)     0.08417741978544  ( 2)     0.08661765422042  ( 2)     0.08729608075284  ( 2)
0.13645335183505  ( 2)     0.20146339739788  ( 2)     0.20209264550882  ( 2)     0.23363818798538  ( 2)     0.23500980856689  ( 2)
0.26681617711053  ( 2)     0.33365537629186  ( 2)     0.34280937791647  ( 2)     0.34322959805083  ( 2)     0.45833702543722  ( 2)
0.48328741534790  ( 2)     0.50649128877099  ( 2)     0.51736684713002  ( 2)     0.61447310303433  ( 2)     0.61538818610103  ( 2)
0.61831033924315  ( 2)     0.61894855200797  ( 2)     0.63389639932323  ( 2)     0.63480881100894  ( 2)     0.63698525034339  ( 2)
0.67886357619745  ( 2)     0.68325244000339  ( 2)     0.75638614817881  ( 2)     0.75745722311601  ( 2)     0.82161646959771  ( 2)
0.92201376113948  ( 2)     0.98455089264787  ( 2)     0.98468688973278  ( 2)     1.01721143371678  ( 2)     1.01778243972275  ( 2)
1.19559448686781  ( 2)     1.32033135104656  ( 2)     1.32175612476556  ( 2)     1.47246800149637  ( 2)     1.47312441361493  ( 2)
1.48573327678144  ( 2)     1.48673373305796  ( 2)     1.52583503659726  ( 2)     1.66340941889701  ( 2)     1.66434543208265  ( 2)
1.68305757276943  ( 2)     1.99128262601425  ( 2)     1.99405394098767  ( 2)     2.00507423191293  ( 2)     2.00643391158823  ( 2)
2.23720017057984  ( 2)     2.96761200756551  ( 2)     3.86994566085372  ( 2)     3.87002312971009  ( 2)     4.03165192324828  ( 2)
4.03197238122657  ( 2)     4.26133522183310  ( 2)     4.26149442194385  ( 2)     4.35237329819399  ( 2)     4.68664685801378  ( 2)
6.91601218905285  ( 2)     7.39252180916936  ( 2)     7.61998005503501  ( 2)    21.10496588735050  ( 2)
```

In this output, the molecular spinors are generated with the inclusion of spin-orbit coupling in the SCF step. The orbital degeneracies showing up in AREP calculation are broken with SOREP (for comparison, see AREP calculation results below).

Note: Compared to all-electron calculations, only small number of occupied molecular spinors are listed in the above example. Here, only four occupied molecular spinors are shown. This is because only eight (upper) electrons are described - seven electrons on the iodine atom and one electron on the hydrogen atom. Remaining forty-six electrons in iodine are omitted and are substituted by relativistic effective core potentials.

## AREP calculation¶

We set the one-component scalar relativistic effective core potential (AREP) calculation by omitting the SO parameters in SOREP (`HI_arep.mol`).

After the SCF step, the following molecular orbitals are generated (see also the `full output`)

```* Fermion symmetry E1
* Closed shell, f = 1.0000
-0.91858300709907  ( 2)    -0.53332424176610  ( 2)    -0.38664035068657  ( 4)
* Virtual eigenvalues, f = 0.0000
0.03415914639101  ( 2)     0.05878598572499  ( 2)     0.08511594399606  ( 2)     0.08655831374568  ( 4)     0.13616239596081  ( 2)
0.20167198225135  ( 4)     0.23444552560595  ( 4)     0.26659777181880  ( 2)     0.33379422568499  ( 2)     0.34298998830327  ( 4)
0.46036400731232  ( 2)     0.49614437446042  ( 4)     0.51225813066935  ( 2)     0.61491923213700  ( 4)     0.61871900761802  ( 4)
0.63458123096312  ( 4)     0.63671447698655  ( 2)     0.68126113881102  ( 4)     0.75708271906160  ( 4)     0.82156075161775  ( 2)
0.92185115859998  ( 2)     0.98434317399476  ( 4)     1.01698231678790  ( 4)     1.19528790938055  ( 2)     1.32084962430606  ( 4)
1.47297468375170  ( 4)     1.48649707416567  ( 4)     1.52558291015577  ( 2)     1.66381720482403  ( 4)     1.68291895964979  ( 2)
1.99293721990332  ( 4)     2.00580508554321  ( 4)     2.23700102551750  ( 2)     2.96678284298137  ( 2)     3.86917720594751  ( 4)
4.03098540560673  ( 4)     4.26054689341582  ( 4)     4.35208715121292  ( 2)     4.68579647949884  ( 2)     7.21075466518969  ( 4)
7.50571888682965  ( 2)    21.10478953048176  ( 2)
```

Compared to molecular spinors from the SOREP calculation above, the spin-orbit coupling is neglected in AREP-based molecular orbitals. As a result, some degeneracies in molecular orbitals appear in the relativistic scalar (spin-free, AREP) only treatment. Detailed comparison of molecular orbitals can be done using the **ANALYZE functionality.

## Exercises¶

1. Identify and compare few occupied and virtual orbitals (including HOMO and LUMO) obtained without (AREP) and with (SOREP) spin-orbital relativistic effects.

2. Optimize (numerically, see *OPTIMIZE) the internuclear distance with (SOREP) and without (AREP) spin-orbital interaction. What is effect of spin-orbital coupling on the bond distance ? Compare against other published works (cite:Park2012). Help - the input file, `geopt_HF.inp`.

3. Compute the dipole moment, electric polarizability and first hyperpolarizability of the HI molecule with (SOREP) and without (AREP) spin-orbital relativistic effects. How do spin-orbital effects influence this property ? Compare with two-component (X2C) all electron calculations. Input files `HI.acv2z.mol`, `x2c.scf_prop.inp` and `x2c_sf.scf_prop.inp`.

4. At the X2c-level evaluate the effect of the picture change for the dipole moment, electric polarizability and the first hyperpolarizability (input file `x2c.scf_prop_nopctr.inp`). What is the size of the picture change effect affecting the properties ? Apply these findings to the previous ECP property calculations.

5. To evaluate the size of spin-orbital effects on properties, you might use (in addition to the X2C approach of point 3.) the four-component Dirac-Coulomb Hamiltonian.