Mulliken charges, Wrong Prediction by ATK

[Shan]

Dear QW team,
Wishes of the Day.

I like to report a problem that I encountered several times. ATK could not predict the mulliken charges correctly. Most of the times it gives completely wrong mulliken results.
For example, the mulliken charges extracted using ATK-DFT for NO2 molecule are
N  -0.123e  (gained)
O  +0.062e (lost)
O  +0.062e (lost)

The same using Gaussian09 DFT package gives,
N  +0.479e (lost)
O  -0.241e   (gained)
O  -0.238e   (gained)

This type of complete contradictory results in case of mulliken population were witnessed several times by me.

Awaiting your reply.
Shan 

[Jess Wellendorff]

1) I think you got the sign convention in the ATK Mulliken analysis wrong. The electron charge on the O atoms correspond to 6.62 electrons, i.e. 0.062 electrons gained.
2) It is well known in literature that the numbers obtained from Mulliken analysis depend a lot on level of theory (HF vs. DFT-LDA), basis set, and software. For example, DZP basis sets appear to consistently give small oxygen Mulliken charges than larger and more advanced basis sets.

[Anders Blom]

Which in turn doesn't mean the DFT results in ATK are wrong - it's just that Mulliken populations are not a particularly good way of measuring e.g. electron transfer since the numbers depend strongly on the basis set even if the DFT solutions (and by that I mean the electron density) are very similar.

[Shan]

Dear Dr. Jeff  and Dr. Anders

I was not wrong in interpreting the signs of mulliken population. below are the original mulliken population shown by atk-dft on NO2,
N   5.123
O   5.939
O   5.938
As per my knowledge these mulliken populations indicate charge gaining of nitrogen and losing of oxygens. aren't they?
(Theory used, ATK-DFT, GGA-RPBE, DZDP)

Whereas the same calculation of NO2 through gaussian (Theory used: DFT, B3LYP, LANL2DZ) as mentioned earlier
N   0.479e (lost)
O  -0.241e   (gained)
O  -0.238e   (gained)

similarly, the Mulliken populations extracted for NH3 using ATK-DFT are,
N   4.842
H   1.053
H   1.053
H   1.053
This shows, charge gained by hydrogens and lost by nitrogen.

However the same using gaussian shows,
N -1.167 (gained)
H  0.389 (lost)
H  0.389 (lost)
H  0.389 (lost)

As I mentioned earlier, this type of contradictory results are observed through ATK several times by me.
There may be variations in quantity of charge due to level of theory used. But, variation in sign (charge transfer direction) is not acceptable.

awaiting your reply,
shan

 

[Anders Blom]

I don't know Gaussian well enough to compare. However, since we are talking about signs, it's important to be ultra-specific.

N   5.123
O   5.939
O   5.938

(from ATK) means 0.123 additional electrons are allocated to the N orbitals. It would important for this discussion to show raw log file results, not processed tables with possible sign interpretation. The words "lost" and "gained" are added by you - if I read just "O  -0.241e" from Gaussian, to me this seems to indicate electrons lost, just like in ATK (unless e is negative, which we don't know).

[Shan]

Dear Dr. Anders,
Here are the raw log data from both ATK and Gaussian. (Gaussian DFT chemical package is based full potentials and is well recommended package for molecular analysis)

From ATK:

# Title: NO2.nc - gID002
# Type:  MullikenPopulation
+------------------------------------------------------------------------------+
|                                                                              |
| Mulliken Population Report                                                   |
|                                                                              |
| ---------------------------------------------------------------------------- |
|                        |                                                     |
| Element   Total  Shell | Orbitals                                            |
|                        |                                                     |
|                        |      s                                              |
|   0   N   5.123  0.113 |  0.113                                              |
|                        |      y      z      x                                |
|                  0.473 |  0.141  0.147  0.185                                |
|                        |      s                                              |
|                  1.319 |  1.319                                              |
|                        |      y      z      x                                |
|                  2.991 |  1.168  0.897  0.927                                |
|                        |     xy     zy  zz-rr     zx  xx-yy                  |
|                  0.099 |  0.034  0.003  0.009  0.021  0.032                  |
|                        |     xy     zy  zz-rr     zx  xx-yy                  |
|                  0.129 |  0.043  0.006  0.009  0.035  0.035                  |
|                        | --------------------------------------------------- |
|                        |      s                                              |
|   1   O   5.939  0.097 |  0.097                                              |
|                        |      y      z      x                                |
|                  0.538 |  0.182  0.200  0.157                                |
|                        |      s                                              |
|                  1.545 |  1.545                                              |
|                        |      y      z      x                                |
|                  3.723 |  1.294  1.235  1.194                                |
|                        |     xy     zy  zz-rr     zx  xx-yy                  |
|                  0.011 |  0.003  0.001  0.001  0.003  0.003                  |
|                        |     xy     zy  zz-rr     zx  xx-yy                  |
|                  0.025 |  0.008  0.002  0.003  0.006  0.007                  |
|                        | --------------------------------------------------- |
|                        |      s                                              |
|   2   O   5.938  0.097 |  0.097                                              |
|                        |      y      z      x                                |
|                  0.538 |  0.193  0.200  0.145                                |
|                        |      s                                              |
|                  1.545 |  1.545                                              |
|                        |      y      z      x                                |
|                  3.722 |  1.386  1.235  1.101                                |
|                        |     xy     zy  zz-rr     zx  xx-yy                  |
|                  0.011 |  0.007  0.000  0.001  0.004 -0.000                  |
|                        |     xy     zy  zz-rr     zx  xx-yy                  |
|                  0.025 |  0.007  0.000  0.003  0.007  0.008                  |
+------------------------------------------------------------------------------+

From Gaussian,

Summary of Natural Population Analysis:                 
                                                         
                                       Natural Population
                Natural  -----------------------------------------------
    Atom  No    Charge         Core      Valence    Rydberg      Total
 -----------------------------------------------------------------------
      N    1    0.47932      1.99977     4.48234    0.03856     6.52068
      O    2   -0.24115      1.99989     6.23083    0.01043     8.24115
      O    3   -0.23817      1.99989     6.22784    0.01043     8.23817
 =======================================================================
   * Total *    0.00000      5.99956    16.94102    0.05943    23.00000

the same have been attached to this post.

Awaiting your reply,
Shan

[Anders Blom]

It is quite important to notice that a "Natural Population Analysis" is not at all the same as Mulliken populations.

As discussed e.g. here, and as is well known, Mulliken populations are a very blunt instrument with rather little physical relevance, whereas natural population analysis is one of what Wikipedia calls the "modern methods for computing net atomic charges" which work around the problems associated with a simple Mulliken population analysis.

Specifically, we just ran some tests which show that by changing the basis set to SG15 you can reverse the sign of the net Mulliken charges for NO2. This, however, does not mean in any way that the real-space electron density (i.e. the physics, or chemistry if you so prefer) has radically changed; in fact, it's basically identical in the two cases.

Thus, in conclusion, the disagreement comes from comparing two different quantities that cannot be compared. The good news, however, is that one of these (Mulliken) is not particularly relevant or useful. You may want to try the Bader charge analysis in ATK instead, although that method also has some issues when applied to densities obtained with pseudopotential.

[Shan]

well, thanks a lot for such clear reply Dr. Anders,

I was aware that mulliken is not as accurate as Natural population analysis. However, as mentioned earlier, it  was not the quantity of the charge that bothered me, but the direction of charge.

If possible, please fix the issue in the upcoming version.

Have a good day sir.

[Anders Blom]

Fix what? The computation of the Mulliken populations is correctly implemented in ATK as, essentially, the trace of the density matrix. If these numbers are representative of the true charge on the atom or not depends on the basis set - sometimes it is, sometimes (and usually) it's not. The only "wrong" here is assuming that Mulliken charges hold a strong physical relevance.

As another example of how Mulliken populations can be greatly misleading is that you can have negative numbers for some orbitals. How can you have a negative number of electrons on an orbital? Of course you can't, but the mathematical definition of Mulliken populations allows it.

[Anders Blom]

A very enlightening article on this is
http://scitation.aip.org.globalproxy.cvt.dk/content/aip/journal/jcp/83/2/10.1063/1.449486

In particular point 3 in the introduction which states that "Mulliken populations seem to give an unreasonable physical picture of the charge distribution in compounds having significant ionic character ... [and] in some cases leading to charges opposite in sign to those expected from electronegativity differences". Moreover they cite CoIlins and Streitwieser, J. Comput. Chem. 1, 81 (1980), as saying "Mulliken populations involving Li compounds are wholly unreliable and should not be given serious consideration."

Also Table II shows that Mulliken can have a reverse sign compared to NPA, depending on the basis set. As mentioned, we see the same in ATK.