Np atom¶
In this tutorial we will try to get the \([Rn] 5f^{4} 6d^{1} 7s^{2}\) configuration of Np. This is a tough cookie. First we prepare the mol file:
DIRAC
C 1
93. 1
Np 0.0000000000 0.0000000000 0.0000000000
LARGE BASIS dyall.v2z
FINISH
The next step will be to remove the valence electrons and to compute the [Rn] configuration and to keep the DFCOEF file:
**DIRAC
.WAVE FUNCTION
.ANALYZE
**HAMILTONIAN
.X2C
**INTEGRALS
*READIN
.UNCONTRACT
**WAVE FUNCTION
.SCF
*SCF
.CLOSED SHELL
42 44
**ANALYZE
.MULPOP
*MULPOP
.VECPOP
1..oo
1..oo
*END OF
This converges nicely. Now we could be tempted to restart from DFCOEF, reintroduce the \(7s^{2}\) and the open shells \(5f^{4}\) and \(6d^{1}\) and impose overlap selection like this:
**DIRAC
.WAVE FUNCTION
.ANALYZE
**HAMILTONIAN
.X2C
**INTEGRALS
*READIN
.UNCONTRACT
**WAVE FUNCTION
.SCF
*SCF
.CLOSED SHELL
44 44
.OPEN SHELL
2
1/10,0
4/0,14
.OVLSEL
.NODYNSEL
**ANALYZE
.MULPOP
*MULPOP
.VECPOP
1..oo
1..oo
*END OF
However this does not converge (please try). Why not? To understand this have a look at the Mulliken population analysis of the [Rn] configuration (first calculation). In there you will see that gerade orbitals 22–26 are d, and 27 is an s. It is important to know that DIRAC assumes that open-shell orbitals come at the end, after closed-shell orbitals. In this case DIRAC would occupy the d orbital 22 fully and then distribute one electron over 4 d and 1 s orbital. But this is not what we want. What we will do is to restart from [Rn] DFCOEF and reorder the virtual gerade orbitals. We move the empty d orbitals after the empty s orbital:
**DIRAC
.WAVE FUNCTION
.ANALYZE
**HAMILTONIAN
.X2C
**INTEGRALS
*READIN
.UNCONTRACT
**WAVE FUNCTION
.SCF
.REORDER
1..21,27,22,23,24,25,26
*SCF
.CLOSED SHELL
44 44
.OPEN SHELL
2
1/10,0
4/0,14
.OVLSEL
.NODYNSEL
**ANALYZE
.MULPOP
*MULPOP
.VECPOP
1..oo
1..oo
*END OF
So in the above input we ask to keep gerade orbitals 1–21 as they are, then to move 22-26 after 27. The empty line that is following means that ungerade orbitals should be kept as they are.
Alright, we run this input, we keep the new DFCOEF and in a final run we restart again, we remove reordering (because orbitals are already reordered) and we relax the overlap selection:
**DIRAC
.WAVE FUNCTION
.ANALYZE
**HAMILTONIAN
.X2C
**INTEGRALS
*READIN
.UNCONTRACT
**WAVE FUNCTION
.SCF
*SCF
.CLOSED SHELL
44 44
.OPEN SHELL
2
1/10,0
4/0,14
**ANALYZE
.MULPOP
*MULPOP
.VECPOP
1..oo
1..oo
*END OF
This converges nicely. We can verify that the final valence orbitals are as they should be. First the gerade:
* Electronic eigenvalue no. 22: -0.2053405574542 (Occupation : f = 1.0000) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Ag Np s
--------------------------------------
alpha 1.0000 | 1.0000
beta 0.0000 | 0.0000
* Electronic eigenvalue no. 23: -0.1931234363014 (Occupation : f = 0.1000) m_j= -3/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Ag Np dxx L Ag Np dyy L B1gNp dxy L B2gNp dxz L B3gNp 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. 24: -0.1931234361060 (Occupation : f = 0.1000) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Ag Np dxx L Ag Np dyy L Ag Np dzz L B2gNp dxz L B3gNp 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. 25: -0.1829590945173 (Occupation : f = 0.1000) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Ag Np dxx L Ag Np dyy L Ag Np dzz L B2gNp dxz L B3gNp 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. 26: -0.1829590945233 (Occupation : f = 0.1000) m_j= 5/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Ag Np dxx L Ag Np dyy L B1gNp dxy
--------------------------------------------------------------------
alpha 1.0000 | 0.2500 0.2500 0.5000
beta 0.0000 | 0.0000 0.0000 0.0000
* Electronic eigenvalue no. 27: -0.1829590945048 (Occupation : f = 0.1000) m_j= -3/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L Ag Np dxx L Ag Np dyy L B1gNp dxy L B2gNp dxz L B3gNp 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
And here is the 5f4 shell:
* Electronic eigenvalue no. 23: -0.3986217183147 (Occupation : f = 0.2857) m_j= 5/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L B3uNp fxxx L B3uNp fxyy L B2uNp fxxy L B2uNp fyyy L B1uNp fxxz L B1uNp fyyz L Au Np fxyz
--------------------------------------------------------------------------------------------------------------------------------
alpha 0.1429 | 0.0000 0.0000 0.0000 0.0000 0.0357 0.0357 0.0714
beta 0.8571 | 0.1071 0.3214 0.3214 0.1071 0.0000 0.0000 0.0000
* Electronic eigenvalue no. 24: -0.3986217184250 (Occupation : f = 0.2857) m_j= -3/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L B3uNp fxxx L B3uNp fxyy L B3uNp fxzz L B2uNp fxxy L B2uNp fyyy L B2uNp fyzz L B1uNp fxxz
--------------------------------------------------------------------------------------------------------------------------------
alpha 0.7143 | 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1786
beta 0.2857 | 0.0214 0.0071 0.1143 0.0071 0.0214 0.1143 0.0000
Gross | L B1uNp fyyz L Au Np fxyz
--------------------------------------
alpha | 0.1786 0.3571
beta | 0.0000 0.0000
* Electronic eigenvalue no. 25: -0.3986217184451 (Occupation : f = 0.2857) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L B3uNp fxxx L B3uNp fxyy L B3uNp fxzz L B2uNp fxxy L B2uNp fyyy L B2uNp fyzz L B1uNp fxxz
--------------------------------------------------------------------------------------------------------------------------------
alpha 0.4286 | 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1286
beta 0.5714 | 0.0429 0.0143 0.2286 0.0143 0.0429 0.2286 0.0000
Gross | L B1uNp fyyz L B1uNp fzzz
--------------------------------------
alpha | 0.1286 0.1714
beta | 0.0000 0.0000
* Electronic eigenvalue no. 26: -0.3652679300791 (Occupation : f = 0.2857) m_j= -7/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L B3uNp fxxx L B3uNp fxyy L B2uNp fxxy L B2uNp fyyy
-----------------------------------------------------------------------------------
alpha 0.0000 | 0.0000 0.0000 0.0000 0.0000
beta 1.0000 | 0.1250 0.3750 0.3750 0.1250
* Electronic eigenvalue no. 27: -0.3652679301320 (Occupation : f = 0.2857) m_j= 1/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L B3uNp fxxx L B3uNp fxyy L B3uNp fxzz L B2uNp fxxy L B2uNp fyyy L B2uNp fyzz L B1uNp fxxz
--------------------------------------------------------------------------------------------------------------------------------
alpha 0.5714 | 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1714
beta 0.4286 | 0.0321 0.0107 0.1714 0.0107 0.0321 0.1714 0.0000
Gross | L B1uNp fyyz L B1uNp fzzz
--------------------------------------
alpha | 0.1714 0.2286
beta | 0.0000 0.0000
* Electronic eigenvalue no. 28: -0.3652679300598 (Occupation : f = 0.2857) m_j= -3/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L B3uNp fxxx L B3uNp fxyy L B3uNp fxzz L B2uNp fxxy L B2uNp fyyy L B2uNp fyzz L B1uNp fxxz
--------------------------------------------------------------------------------------------------------------------------------
alpha 0.2857 | 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0714
beta 0.7143 | 0.0536 0.0179 0.2857 0.0179 0.0536 0.2857 0.0000
Gross | L B1uNp fyyz L Au Np fxyz
--------------------------------------
alpha | 0.0714 0.1429
beta | 0.0000 0.0000
* Electronic eigenvalue no. 29: -0.3652679300957 (Occupation : f = 0.2857) m_j= 5/2
==========================================================================================
* Gross populations greater than 0.00010
Gross Total | L B3uNp fxxx L B3uNp fxyy L B2uNp fxxy L B2uNp fyyy L B1uNp fxxz L B1uNp fyyz L Au Np fxyz
--------------------------------------------------------------------------------------------------------------------------------
alpha 0.8571 | 0.0000 0.0000 0.0000 0.0000 0.2143 0.2143 0.4286
beta 0.1429 | 0.0179 0.0536 0.0536 0.0179 0.0000 0.0000 0.0000
Other similar cases: \(Lu^{+}\), \(Hf^{2+}\), \(Ta^{3+}\) and \(W4+\) with broken orbital degeneracies¶
The \(Lu^{+}\), \(Hf^{2+}\), \(Ta^{3+}\), and \(W4+\) are closed shell cations with the electronic structure of the Yb atom (Z=70), which are not converging in simple SCF procedure.
In the SCF output one observes broken degeneracies of their atomic p,d,f... shells (using four- and two-component Hamiltonians), even with spin-free and nonrelativistic Levy-Leblond Hamiltonians. This is in contrast with the Yb closed shell atom, whose electronic structure is emulated, that comes out with proper spinor/orbital degeneracies.
The reason for this problem is that the start guess for orbitals is based on a bare nucleus or other simple model Hamiltonian. In the first diagonalization this Hamiltonian gives orbitals that do not have the expected order 4f, 6p but rather 6p, 4f. The program then proceeds by occupying the three 6p orbitals, complemented by a remaining four 4f orbitals. Thus resulting density does not have the correct spherical symmetry and offsets the SCF procedure yielding a symmetry-broken solution.
The leading idea behind is to fix the wrong order of orbitals which is causing the degeneracy.
The ‘default’ wrong order of forming molecular orbitals is for s-shell (LUMO) beeing inserted after the first d-shell (HOMO). The user has to apply the \(M_{J}\)-selection for the linear symmetry, or the orbital reordering scheme to have s-orbitals separated from d-orbitals. In this way it is possible to get converged MOs not only for linear, but also for D2h and C2v symmetries. However, for the C2v it takes many more iterations.
One could also experiment with providing a better starting guess of orbitals, e.g. by first calculating the highly-charged cation with a Xenon electronic configuration. This gives a symmetric density that can be used as starting set of MOs (using the DFCOEF file) for the real calculation. Some care is needed with this restart as also the orbitals for the Xe-like ion may have an undesired order. Howver, this may require use of the MO-selection options of the program, as mentioned above.
