Effective core potential (ECP) is efficient method in many quantum mechanical calculations. This reduces basis set demands in heavy elements by replacing core electrons with effective potential. For the account of spin-orbit and other relativistic effects, many ECP parameters developed recently are based on atomic Dirac-Fock calculations, and sometimes called the relativistic effective core potential (RECP).

Originally, the spin-orbit term is included in the RECP (SOREP) as following [Lee1977],

\[U^{SOREP} = U_{LJ}^{SOREP}(r) + \sum_{l=0}^{L-1}\sum_{j=|l-1/2|}^{l+1/2}\sum_{m=-j}^{j}
[U_{ij}^{SOREP}(r)- U_{LJ}^{SOREP}(r)]|ljm\rangle \langle ljm|\]

The SOREP is divided into two types of potential - averaged relativistic effective core potential (AREP) and spin-orbit potential. (\(U^{SOREP} = U^{AREP} + U^{SO}\)) [Ermler1981]

\[U^{AREP} = U_{L}^{AREP}(r) + \sum_{l=0}^{L-1}\sum_{m=-l}^{l}
[U_{l}^{AREP}(r)- U_{L}^{AREP}(r)]|lm\rangle \langle lm|\]

where

\[U_{l}^{AREP} = \frac{1}{2l+1}[l\cdot U_{l,l-1/2}^{SOREP}(r)
+ (l+1)\cdot U_{l,l+1/2}^{SOREP}(r)]\]

The spin-orbit potential (\(U^{SO}\)) is defined as,

\[U^{SO} = s \cdot \sum_{l=1}^{L} \frac{2}{2l+1}\Delta U_{l}^{SOREP}(r)
\sum_{m=-l}^{l} \sum_{m^{\prime}=-l}^{l}|lm\rangle
\langle lm|l|lm^{\prime}\rangle \langle lm^{\prime}|\]

with

\[\Delta U_{l}^{SOREP(r)} = U_{l,l+1/2}^{SOREP(r)} - U_{l,l-1/2}^{SOREP(r)}\]

```
INTGRL
HI MOLECULE
I:Christiansen RECP, H:Aug-cc-pVTZ
C 2 2 Y Z A
53. 1
I 1.59900000 0.00000000 0.00000000
LARGE BASIS ECPCE46_TZ
ECPLIB ECPCE46SO
1. 1
H 0.00000000 0.00000000 0.00000000
LARGE BASIS aug-cc-pVTZ
FINISH
```

1. Basis set library for ECP (line 7): The format is same as the large component basis set. (see large component basis set) However, one should use the basis set parameters corresponding to the RECP.

2. ECP parameters from the library (line 8): ECPLIB keyword should be used for the use of RECP parameters from the library.

Note: In DIRAC program, some ECP libraries are provided in basis_ecp directory for your conveniences. However, it is safe to check ECP parameters from authors’ webpages. The name of ECP parameters used here is following, ECP(XX)(YY)(ZZ)(SO/SF).

- XX: representative authors, (CE=Christiansen, Ermler, and coworkers), (DS=Dolg, Stoll, and coworkers), (HW=Hay, Wadt, and coworkers), (SB=Stevens, Basch, and coworkers)
- YY: number of core electrons replaced by potential
- ZZ: further information (if exist),
- (SO/SF) : spin-orbit(SOREP) or spin-free(AREP), The difference of SO and SF is the existance of SO parameters.

For example, ECPDS60MDFSO indicates Stuttgart RECP from Dolg, Stoll, and coworkers with 60 electrons replaced by potential which is fitted from relativistic calculation (spin-orbit potential is included).

Further correlation-consistent basis sets by Kirk Petereson and co-workers that can be used with the Stuttgart-Cologne ECPs are available here .

```
INTGRL
HI MOLECULE
I:Christiansen RECP, H:Aug-cc-pVTZ
C 2 2 Y Z A
53. 1
I 1.59900000 0.00000000 0.00000000
LARGE BASIS ECPCE46_TZ
ECP 46 4 3
# AREP
# f
4
2 .922500 -1.447005
2 2.569100 -14.188832
2 7.908600 -43.263306
1 25.061100 -27.740856
# s-f
6
2 1.503600 -124.037542
2 1.874600 235.545458
2 2.682800 -261.475363
2 3.446600 144.184313
1 1.124200 31.003269
0 11.458300 6.512373
# p-f
6
2 1.245400 -95.368794
2 1.582600 188.963297
2 2.242900 -221.153567
2 2.930100 104.680452
1 .950300 30.809252
0 12.782000 5.414640
# d-f
6
2 .668500 -50.340552
2 .828000 102.276429
2 1.115700 -133.296812
2 1.419000 75.010821
1 .503000 19.150171
0 4.557600 8.099959
# spin-orbit
# p-f
6
2 1.245400 5.416752
2 1.582600 3.711418
2 2.242900 -25.208180
2 2.930100 27.780892
1 .950300 -2.699622
0 12.782000 .351446
# d-f
6
2 .668500 -.406356
2 .828000 2.040806
2 1.115700 -3.283400
2 1.419000 1.842046
1 .503000 -.126998
0 4.557600 .008442
# f
4
2 .922500 -.019085
2 2.569100 .035451
2 7.908600 -.007995
1 25.061100 .096830
1. 1
H 0.00000000 0.00000000 0.00000000
LARGE BASIS aug-cc-pVTZ
FINISH
```

For the explicitly typed ECP, ECP keyword is used. Three numbers after the ECP keyword denote the following.

1.Number of core electrons: This number of core electrons is substituted by RECP. In this case, 46 core electrons in iodine atom are omitted and 7 electrons in the valence are described.

2.Number of AREP blocks: The numbers of AREP blocks to be read.

3.Number of SO blocks: The numbers of SOREP blocks to be read. In the case of AREP (spin-free) calculation, the number of SOREP blocks is set to 0.

After reading block numbers, AREP (and SOREP) blocks are read.

1.AREP blocks: Three parameters in each line in AREP blocks are \(n_{li}\), \(\alpha_{li}\), and \(C_{li}\) respectively in the following equation.

\[U_{l} = \sum_{l}C_{li}r^{n_{li}-2}e^{-\alpha_{li}r^{2}}\]

2.SO blocks: Three parameters in each line in SO blocks are same as in AREP blocks.

Note: coefficients in the spin-orbit block are sometimes defined differently by RECP developing groups. Spin-orbit potential in DIRAC is defined as [Park2012],

\[U_{l}^{SO,DIRAC} = \frac{2}{2l+1}\Delta U_{l}^{SOREP}\]

For example, factors (2/2l+1) are already multiplied in SO parameters in [Stuttgart RECP webpage]. One can use it directly without the modification. There are two kinds of SO factors in [RECPs of Christiansen and coworkers]. SO factor in DIRAC is same as the ones defined for Columbus program.