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

!
! This test calculates the total energy 
! and the dipole moment of
! employing valence-only Hamitonian
! for relativistic effective-core-potentials.
!
**DIRAC
.TITLE
HI molecule with RECP
.WAVE F
.ANALYZE
.PROPERTIES
**HAMILTONIAN 
.ECP
**WAVE FUNCTIONS
.SCF
*SCF
.CLOSED SHELL
8
**ANALYZE
.MULPOP
*MULPOP
.VECPOP
1..oo
**PROPERTIES
.DIPOLE
*END OF DIRAC

together with the molecular input (HI_sorep.mol)

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    4    1    1    1    1
F   7    4    0
9655217 -0.0054740  0.0000000  0.0000000  0.0000000
5610628  0.0250170  0.0000000  0.0000000  0.0000000
8188811 -0.8900750  0.0000000  0.0000000  0.0000000
4925532  1.3195220  0.0000000  0.0000000  0.0000000
2875149  0.0000000  1.0000000  0.0000000  0.0000000
0824299  0.0000000  0.0000000  1.0000000  0.0000000
0237922  0.0000000  0.0000000  0.0000000  1.0000000
F   7    4    0
8262036  0.0122770  0.0000000  0.0000000  0.0000000
2639587 -0.0457030  0.0000000  0.0000000  0.0000000
1561771  0.2251760  0.0000000  0.0000000  0.0000000
5506509 -0.2514460  0.0000000  0.0000000  0.0000000
2444091  0.0000000  1.0000000  0.0000000  0.0000000
0908574  0.0000000  0.0000000  1.0000000  0.0000000
0299745  0.0000000  0.0000000  0.0000000  1.0000000
F   3    3    0
4320000  1.0000000  0.0000000  0.0000000
2200000  0.0000000  1.0000000  0.0000000
1000000  0.0000000  0.0000000  1.0000000
F   2    2    0
4265000  1.0000000  0.0000000
1700000  0.0000000  1.0000000
ECP 46 4 3
4
      .922500       -1.447005
     2.569100      -14.188832
     7.908600      -43.263306
    25.061100      -27.740856
6
     1.503600     -124.037542
     1.874600      235.545458
     2.682800     -261.475363
     3.446600      144.184313
     1.124200       31.003269
    11.458300        6.512373
6
     1.245400      -95.368794
     1.582600      188.963297
     2.242900     -221.153567
     2.930100      104.680452
      .950300       30.809252
    12.782000        5.414640
6
      .668500      -50.340552
      .828000      102.276429
     1.115700     -133.296812
     1.419000       75.010821
      .503000       19.150171
     4.557600        8.099959
#spin-orbit
6
     1.245400        5.416752
     1.582600        3.711418
     2.242900      -25.208180
     2.930100       27.780892
      .950300       -2.699622
    12.782000         .351446
6
      .668500        -.406356
      .828000        2.040806
     1.115700       -3.283400
     1.419000        1.842046
      .503000        -.126998
     4.557600         .008442
4
      .922500        -.019085
     2.569100         .035451
     7.908600        -.007995
    25.061100         .096830
        1.    1
H     0.00000000           0.00000000        0.00000000
LARGE BASIS aug-cc-pVTZ 
FINISH

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
03426904787073  ( 2)     0.05875502261229  ( 2)     0.08417741978544  ( 2)     0.08661765422042  ( 2)     0.08729608075284  ( 2)
13645335183505  ( 2)     0.20146339739788  ( 2)     0.20209264550882  ( 2)     0.23363818798538  ( 2)     0.23500980856689  ( 2)
26681617711053  ( 2)     0.33365537629186  ( 2)     0.34280937791647  ( 2)     0.34322959805083  ( 2)     0.45833702543722  ( 2)
48328741534790  ( 2)     0.50649128877099  ( 2)     0.51736684713002  ( 2)     0.61447310303433  ( 2)     0.61538818610103  ( 2)
61831033924315  ( 2)     0.61894855200797  ( 2)     0.63389639932323  ( 2)     0.63480881100894  ( 2)     0.63698525034339  ( 2)
67886357619745  ( 2)     0.68325244000339  ( 2)     0.75638614817881  ( 2)     0.75745722311601  ( 2)     0.82161646959771  ( 2)
92201376113948  ( 2)     0.98455089264787  ( 2)     0.98468688973278  ( 2)     1.01721143371678  ( 2)     1.01778243972275  ( 2)
19559448686781  ( 2)     1.32033135104656  ( 2)     1.32175612476556  ( 2)     1.47246800149637  ( 2)     1.47312441361493  ( 2)
48573327678144  ( 2)     1.48673373305796  ( 2)     1.52583503659726  ( 2)     1.66340941889701  ( 2)     1.66434543208265  ( 2)
68305757276943  ( 2)     1.99128262601425  ( 2)     1.99405394098767  ( 2)     2.00507423191293  ( 2)     2.00643391158823  ( 2)
23720017057984  ( 2)     2.96761200756551  ( 2)     3.86994566085372  ( 2)     3.87002312971009  ( 2)     4.03165192324828  ( 2)
03197238122657  ( 2)     4.26133522183310  ( 2)     4.26149442194385  ( 2)     4.35237329819399  ( 2)     4.68664685801378  ( 2)
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).

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    4    1    1    1    1
F   7    4    0
 8.9655217 -0.0054740  0.0000000  0.0000000  0.0000000
 4.5610628  0.0250170  0.0000000  0.0000000  0.0000000
 1.8188811 -0.8900750  0.0000000  0.0000000  0.0000000
 1.4925532  1.3195220  0.0000000  0.0000000  0.0000000
 0.2875149  0.0000000  1.0000000  0.0000000  0.0000000
 0.0824299  0.0000000  0.0000000  1.0000000  0.0000000
 0.0237922  0.0000000  0.0000000  0.0000000  1.0000000
F   7    4    0
 4.8262036  0.0122770  0.0000000  0.0000000  0.0000000
 2.2639587 -0.0457030  0.0000000  0.0000000  0.0000000
 1.1561771  0.2251760  0.0000000  0.0000000  0.0000000
 0.5506509 -0.2514460  0.0000000  0.0000000  0.0000000
 0.2444091  0.0000000  1.0000000  0.0000000  0.0000000
 0.0908574  0.0000000  0.0000000  1.0000000  0.0000000
 0.0299745  0.0000000  0.0000000  0.0000000  1.0000000
F   3    3    0
 0.4320000  1.0000000  0.0000000  0.0000000
 0.2200000  0.0000000  1.0000000  0.0000000
 0.1000000  0.0000000  0.0000000  1.0000000
F   2    2    0
 0.4265000  1.0000000  0.0000000
 0.1700000  0.0000000  1.0000000
ECP 46 4 0
4
  2        .922500       -1.447005
  2       2.569100      -14.188832
  2       7.908600      -43.263306
  1      25.061100      -27.740856
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
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
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
        1.    1
H     0.00000000           0.00000000        0.00000000
LARGE BASIS aug-cc-pVTZ
FINISH

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¶

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