# Which one, Mulliken charge distribution and NBO, is more reliable?

Sometimes the Mulliken and NBO turn out to be so different that I can't decide which one I can trust. I've heard that Mulliken is inaccurate, but is NBO always accurate? And should I use Gaussian or other software to calculate charge distribution? Are there methods more accurate than Mulliken or NBO?

And charge distribution doesn't represent charge density, right?

• There is no proper, strict and agreed definition of atom border, that's the real problem. Gaussian includes option to fit charges on atoms (assumed for the sake of the procedure as points) so they produced electrostatic field as accurately fitting once calculated from the electronic density distribution as possible, and maybe (I haven't used Gaussian for a long time, I'm not sure) dipoles too. – permeakra Aug 24 '14 at 12:16
• NBO is more accurate than Mulliken in very many ways. However, NBO is not always accurate. Generally speaking, you would never want to use Mulliken charges for any sort of production level work unless its been calibrated. – LordStryker Aug 24 '14 at 19:33

## 2 Answers

As I do not want to drag this out in the comments, I decided to give a brief answer. Let me first cite the very helpful comments to keep the for (relative) eternity and add my thoughts at the end.

1. There is no proper, strict and agreed definition of atom border, that's the real problem. Gaussian includes option to fit charges on atoms (assumed for the sake of the procedure as points) so they produced electrostatic field as accurately fitting once calculated from the electronic density distribution as possible, and maybe (I haven't used Gaussian for a long time, I'm not sure) dipoles too. – permeakra

2. NBO is more accurate than Mulliken in very many ways. However, NBO is not always accurate. Generally speaking, you would never want to use Mulliken charges for any sort of production level work unless its been calibrated. – LordStryker

Mulliken is of course the cheapest and fastest way to compute charges. However, this method tends to give qualitative results at best. The reason for this is obvious. It will divide the canonical orbitals equally amongst the participating atoms. There is no polarisation whatsoever. Another problem is, that they are very basis set dependent. And that being said, the worst part is, that the description becomes worse by increasing the basis set.

NBO charges are much more reliable, since the operate on the electron density instead. Localised natural atomic orbitals can be used to describe the computed electron density. Polarisation of bonds is therefore considered. This method obviously only works very well for systems, that can actually be (quite well) described by a Lewis structure. However, they usually give a nice qualitative picture and are quite robust when it comes to increasing the basis set.

There are a couple of more population analysis tools, that can be used and should be considered when it is necessary to derive some conclusions from charges.

Quite common is also the Hirshfeld charges approach, which divides the electron density into spherical basins around the nucleus. Maybe interesting to read is "Are the Hirshfeld and Mulliken population analysis schemes consistent with chemical intuition?" (I can't tell, I do not have access.)

There are also several schemes to obtain partial charges from the electrostatic potential.

Probably the most rigorous approach to atomic partial charges is the Quantum Theory of Atoms in Molecules (QTAIM). Here the Laplacian of the electron density is used to describe the partitioning in electronic basins. The best feature of this analysis is, that it can be also used for experimentally obtained electron densities. However, as everything, this has also major drawbacks. One of the most important is the computational effort. Unfortunately the integration scheme is not always completely robust when it comes to some particular molecules. However, it is almost completely basis set independent and results could be cross referenced with experimental data.

Since permeakra's initial statement holds true, especially for the agreed part, the best way to deal with partial charges is to be careful and maybe check with different methods. In gaussian there are several schemes implemented, most commonly, Mulliken, Hirshfeld and NBO (3.1) and some others, see the population keyword for more detail.

• I've read your answer and the links you provided. Now I see that Mulliken is expendable. Thanks. However, after I look at NBO result, I suddenly feel it's no use. Usually, charge distribution is used to know the dipole moment. But we can know the dipole moment directly from the Gaussian output. I can't think of other uses of charge distribution. – OhLook Aug 26 '14 at 17:49
• @Ath There are certainly uses for partial charges and they often provide more insight into chemical bonding. The dipole moment is just a vector, but it does not give you the polarity of the molecule. Atomic charges may become important when working on chemical reactions - it can be as useful as oxidation states might have been before that. The NBO concept is a very important one as it provides a lot of chemical insight. – Martin - マーチン Aug 26 '14 at 18:45

The Mulliken analysis is the most common population analysis method, it is also one of the worst and is used only because it is one of the oldest and simplest. Table (picture) shows an example of the variation in partial charge with respect to basis set for the Mulliken analysis. Note that the basis sets which would be best for energy calculations are some of the worst for population analysis. This is why the Mulliken analysis is not used, it is highly basis set dependent.