How to specify ECP parameters in mol files¶

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

Following the usual practice the point nucleus model is used by default for ECP calculations.

Theory¶

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

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

ECP parameters from the library¶

See the simple example file fro the ecp test (HI_ecplib.mol) :

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).

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

Explicitly typed ECP¶

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, the ECP keyword is used. Three numbers after the ECP keyword denote the following:

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

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 at the Stuttgart RECP webpage. One can use it directly without the modification. There are two kinds of SO factors in the RECPs of Christiansen and coworkers. The SO factor in DIRAC is the same as the ones defined for Columbus program.