# Computational methods to obtain relative energies of electronic configurations of atoms

When one learns the Aufbau principle to “predict” electronic configurations, and the $(n + \ell)$ ordering rule (or Madelung rule), one also learns of the exceptions to the rules… some of which (Cr, Cu, Mo, Ag, Au) can be half-explained by adding another “rule”, some of which (Nb, Ru, Rh, …) cannot. We have on this site our fair share of questions dealing with these: Nb, Pt, Pd, Cu, Th, etc.

My question here is not about rules or theories to rationalize and understand these complicated electronic configurations, but rather about computational tools to predict them. What computational methods (and software) would one use to compute the different energies of several electronic configurations of an atom?

For example, take niobium. How would I answer, with computational quantum chemistry methods, the following question:

what is the energy difference between the 5s1 4d4 and 5s2 4d3 electronic configurations of niobium?

And would that method also give some interpretation in terms of what these energy differences arise from?

• For atoms, Full CI should do the job. Otherwise DMC.
– user23061
Commented Aug 12, 2016 at 14:56

The different occupations you suggest correspond to different spin states. Therefore, they are quite simple to distinguish. On the other hand, the ground states are often degenerate (in the terms of 5 equivalent d orbitals available). Therefore the lowest (and sufficient) level of theory is the CAS-SCF with inclusion of the s- and d- orbitals. I tried several calculations using ORCA in def2-svp basis and the results correspond to what one would expect, the time to perform the calculations is in order of seconds.

Zr (should be $4d^2$ $5s^2$, ie. MULT=3)

    CAS-SCF STATES FOR BLOCK  1 MULT= 5 NROOTS= 1
ROOT   0:  E=     -46.4247189198 Eh
0.62280 [     0]: 211110
0.37690 [     2]: 211011

CAS-SCF STATES FOR BLOCK  2 MULT= 3 NROOTS= 1
ROOT   0:  E=     -46.4308999710 Eh
0.49082 [     0]: 221100
0.47988 [     2]: 221001
0.01125 [    18]: 210111
0.00276 [    27]: 201120


Cr (should be $3d^5$ $4s^1$, ie. MULT=7)

    CAS-SCF STATES FOR BLOCK  1 MULT= 7 NROOTS= 1
ROOT   0:  E=   -1043.1949498785 Eh
1.00000 [     0]: 111111

CAS-SCF STATES FOR BLOCK  2 MULT= 5 NROOTS= 1
ROOT   0:  E=   -1043.1559672632 Eh
0.40000 [    15]: 111111
0.30000 [     0]: 211110
0.30000 [    30]: 011112


Nb (should be $4d^4$ $5s^1$, ie. MULT=6)

    CAS-SCF STATES FOR BLOCK  1 MULT= 6 NROOTS= 1
ROOT   0:  E=     -56.3066169626 Eh
0.96220 [     0]: 111110
0.03771 [     2]: 111011

CAS-SCF STATES FOR BLOCK  2 MULT= 4 NROOTS= 1
ROOT   0:  E=     -56.2843641219 Eh
0.89474 [     0]: 211100
0.04255 [     2]: 211001


The lower the energy the better.

From the numbers I'd conclude that it is possible to obtain the ground state configuration by reasonably simple quantum chemical calculations. Nevertheless, I'd expect also less elaborate theories to predict such a fundamental properties.

Edit: I added the multiplicity to the expected ground state configuration

• How do you then distinguish electronic configurations that have the same spin state? Excited states, for example?
– F'x
Commented Aug 15, 2016 at 9:25
• @F'x: As this was not a part of the original question, I took the simplest approach. Nevertheless, it is possible to calculate the excited states as well. But to check whether the ground state has one or two electrons in the s-orbital, the described procedure is enough, as they have to differ in total spin. Commented Aug 15, 2016 at 9:44

In addition to Riccardo's answer, I would like to mention software like Gaussian and GAMESS which can be used for electronic configuration studies (computational approach).

• How would you use typically quantum chem software to compute atomic electron configurations?
– F'x
Commented Aug 12, 2016 at 11:51
• You can input different spin multiplicity values (Molecule specification) and check which one is giving you minimum energy. This is helpful in finding stable electronic state (using high level of theory). Commented Aug 12, 2016 at 12:31
• See the @ssavec's answer below. Here, different multiplicity values [MULT = 5, 3 etc] are giving different energy values and hence helpful in predicting electronic state. Commented Aug 12, 2016 at 13:11

To have such detailed information you need to use a highly correlated method (CI) adding spin-orbit coupling. Dirac should do the job. Of course you cannot "force" the electrons on orbitals, but check the different spin states.