Nb atomΒΆ
DIRAC presently does not perform an open-shell Hartree–Fock in a strict sense, rather an average-of-configuration calculation, which amounts to the optimization of the average energy of a set of configurations (or determinants) generated from the specification of a given number of open shells and their electron occupations. As an example of such a calculation, showing in particular tricks to help convergence, we consider the Nb atom.
The ground state configuration of Nb is [Kr].4d4.5s1 and so a reasonable input for an average Hartree–Fock calculation would be:
**DIRAC
.WAVE FUNCTION
.ANALYZE
**HAMILTONIAN
.X2C
**INTEGRALS
*READIN
.UNCONTRACT
**WAVE FUNCTION
.SCF
*SCF
.CLOSED SHELL
18 18
.OPEN SHELL
2
4/10,0
1/2,0
**ANALYZE
.MULPOP
*MULPOP
.VECPOP
1..oo
1..oo
*END OF
Here we create separate open shells for the four 4d electrons and the single 5s electron to reduce the number of states in the average.
We use the following mol file:
DIRAC
C 1
41. 1
Nb 0.0000000000 0.0000000000 0.0000000000
LARGE BASIS dyall.v2z
FINISH
Unfortunately this calculation does not converge:
ERROR, SCF not converged, Dirac stops!
--- SEVERE ERROR, PROGRAM WILL BE ABORTED ---
The reason is that the open-shell 4d and 5s orbitals mix. We can see this in the Mulliken population analysis:
* Electronic eigenvalue no. 10: -0.2308050163326 (Occupation : f = 0.4000) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Ag Nb s L Ag Nb dxx L Ag Nb dyy L Ag Nb dzz
-----------------------------------------------------------------------------------
alpha 1.0000 | 0.8647 -0.0540 -0.0540 0.2432
beta 0.0000 | 0.0000 0.0000 0.0000 0.0000
* Electronic eigenvalue no. 11: -0.2230586181555 (Occupation : f = 0.4000) m_j= -3/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Ag Nb dxx L Ag Nb dyy L B1gNb dxy L B2gNb dxz L B3gNb dyz
--------------------------------------------------------------------------------------------------
alpha 0.2650 | 0.0663 0.0663 0.1325 0.0000 0.0000
beta 0.7350 | 0.0000 0.0000 0.0000 0.3675 0.3675
* Electronic eigenvalue no. 12: -0.2205588496606 (Occupation : f = 0.4000) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Ag Nb s L Ag Nb dxx L Ag Nb dyy L Ag Nb dzz L B2gNb dxz L B3gNb dyz
-----------------------------------------------------------------------------------------------------------------
alpha 0.2178 | 0.0285 0.0481 0.0481 0.0931 0.0000 0.0000
beta 0.7822 | 0.0000 0.0000 0.0000 0.0000 0.3911 0.3911
* Electronic eigenvalue no. 13: -0.2186062869816 (Occupation : f = 0.4000) m_j= -3/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Ag Nb dxx L Ag Nb dyy L B1gNb dxy L B2gNb dxz L B3gNb dyz
--------------------------------------------------------------------------------------------------
alpha 0.7350 | 0.1837 0.1837 0.3675 0.0000 0.0000
beta 0.2650 | 0.0000 0.0000 0.0000 0.1325 0.1325
* Electronic eigenvalue no. 14: -0.2178694489641 (Occupation : f = 0.4000) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Ag Nb s L Ag Nb dxx L Ag Nb dyy L Ag Nb dzz L B2gNb dxz L B3gNb dyz
-----------------------------------------------------------------------------------------------------------------
alpha 0.7822 | 0.1068 0.1728 0.1728 0.3298 0.0000 0.0000
beta 0.2178 | 0.0000 0.0000 0.0000 0.0000 0.1089 0.1089
* Electronic eigenvalue no. 15: -0.2161776019252 (Occupation : f = 0.5000) m_j= 5/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Ag Nb dxx L Ag Nb dyy L B1gNb dxy
--------------------------------------------------------------------
alpha 1.0000 | 0.2500 0.2500 0.5000
beta 0.0000 | 0.0000 0.0000 0.0000
We see significant mixing of 4d and 5s orbitals. This mixing is forbidden from atomic symmetry. The problem is that we run the atom only in linear supersymmetry and so the mixing is introduced by numerical noise due to the energetic proximity of these orbitals.
An efficient way to handle such a case is to strip off the open-shell electrons and simply calculate first Nb5+ with the electron configuration of Kr and to save the converged DFCOEF file:
**DIRAC
.WAVE FUNCTION
.ANALYZE
**HAMILTONIAN
.X2C
**INTEGRALS
*READIN
.UNCONTRACT
**WAVE FUNCTION
.SCF
*SCF
.CLOSED SHELL
18 18
#.OPEN SHELL
# 2
# 4/10,0
# 1/2,0
**ANALYZE
.MULPOP
*MULPOP
.VECPOP
1..oo
1..oo
*END OF
This calculation converges smoothly and now the Mulliken population analysis tells us that the gerade orbitals 10–14 are purely d, whereas gerade orbital 15 is pure s. Please verify this.
We will now restart from the converged DFCOEF and reinsert the open-shell electrons and impose vector selection by overlap:
**DIRAC
.WAVE FUNCTION
.ANALYZE
**HAMILTONIAN
.X2C
**INTEGRALS
*READIN
.UNCONTRACT
**WAVE FUNCTION
.SCF
*SCF
.CLOSED SHELL
18 18
.OPEN SHELL
2
4/10,0
1/2,0
.OVLSEL
.NODYNSEL
**ANALYZE
.MULPOP
*MULPOP
.VECPOP
1..oo
1..oo
*END OF
This makes sure that the orbitals obtained in the calculation on Nb5+ stay in place. Again, we keep the DFCOEF file for further restart.
The final step is again to restart from DFCOEF but this time we relax the SCF by removing overlap selection:
**DIRAC
.WAVE FUNCTION
.ANALYZE
**HAMILTONIAN
.X2C
**INTEGRALS
*READIN
.UNCONTRACT
**WAVE FUNCTION
.SCF
*SCF
.CLOSED SHELL
18 18
.OPEN SHELL
2
4/10,0
1/2,0
#.OVLSEL
#.NODYNSEL
**ANALYZE
.MULPOP
*MULPOP
.VECPOP
1..oo
1..oo
*END OF
We obtain convergence and our open-shell orbitals are pure d and s. This is what we wanted:
* Electronic eigenvalue no. 10: -0.2557501903224 (Occupation : f = 0.4000) m_j= -3/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Ag Nb dxx L Ag Nb dyy L B1gNb dxy L B2gNb dxz L B3gNb dyz
--------------------------------------------------------------------------------------------------
alpha 0.8000 | 0.2000 0.2000 0.4000 0.0000 0.0000
beta 0.2000 | 0.0000 0.0000 0.0000 0.1000 0.1000
* Electronic eigenvalue no. 11: -0.2557501876526 (Occupation : f = 0.4000) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Ag Nb dxx L Ag Nb dyy L Ag Nb dzz L B2gNb dxz L B3gNb dyz
--------------------------------------------------------------------------------------------------
alpha 0.4000 | 0.0667 0.0667 0.2667 0.0000 0.0000
beta 0.6000 | 0.0000 0.0000 0.0000 0.3000 0.3000
* Electronic eigenvalue no. 12: -0.2505426815019 (Occupation : f = 0.4000) m_j= 5/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Ag Nb dxx L Ag Nb dyy L B1gNb dxy
--------------------------------------------------------------------
alpha 1.0000 | 0.2500 0.2500 0.5000
beta 0.0000 | 0.0000 0.0000 0.0000
* Electronic eigenvalue no. 13: -0.2505426814916 (Occupation : f = 0.4000) m_j= -3/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Ag Nb dxx L Ag Nb dyy L B1gNb dxy L B2gNb dxz L B3gNb dyz
--------------------------------------------------------------------------------------------------
alpha 0.2000 | 0.0500 0.0500 0.1000 0.0000 0.0000
beta 0.8000 | 0.0000 0.0000 0.0000 0.4000 0.4000
* Electronic eigenvalue no. 14: -0.2505426791823 (Occupation : f = 0.4000) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Ag Nb dxx L Ag Nb dyy L Ag Nb dzz L B2gNb dxz L B3gNb dyz
--------------------------------------------------------------------------------------------------
alpha 0.6000 | 0.1000 0.1000 0.4000 0.0000 0.0000
beta 0.4000 | 0.0000 0.0000 0.0000 0.2000 0.2000
* Electronic eigenvalue no. 15: -0.2044678705701 (Occupation : f = 0.5000) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Ag Nb s
--------------------------------------
alpha 1.0000 | 1.0000
beta 0.0000 | 0.0000
