:orphan:
Nb atom
=======
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].4d\ :sup:`4`\ .5s\ :sup:`1`\ 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 Nb\ :sup:`5+`\ 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
Nb\ :sup:`5+`\ 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