# Re: [SIESTA-L] Question about Mulliken Population Analysis

```Hi Ricardo,

I'm by no means an expert on the subject, but from what I understand
from Mulliken populations, (see, for example, Jensen's Computational
Chemistry,) the diagonal terms are the number of electrons in a
particular orbital.  The off-diagonal entries are equal to half the
number of electrons shared between two different atomic orbitals.  To
get the number of electrons associated with an atom, one sums all
contributions from atomic orbitals for that atom.  You will notice
that you have some off-diagonal entries which are negative, which is
physically impossible, and is one of the problems associated with
Mulliken analysis.```
```
Hope this helps,
Chris Rowan

2009/6/9 Ricardo Wagner Nunes <rwnu...@fisica.ufmg.br>:
> Dear all,
>
> We have done some calculations using the option:
>
>  WriteMullikenPop       3
>
> I would like to know what is precisely the meaning of the off-diagonal terms
> that are printed in the output (see example with relevant input and output
> below).
>
> I understand that the diagonal terms give the Mulliken charges on the atoms
> from the diagonal terms of the rho*S, where rho = density matrix and S is
> the overlap of basis orbitals. If that is not correct, I would appreciate if
> somebody could enlighten me.
>
> But how about the off-diagonal? I considered the possibility that they would
> give the off-diagonal terms of the rho(k)*S(k) matrix, i.e., in the basis of
> Bloch sums, but they are not printed per k-point, so this does not seem
> right.
>
> To put it another way, since atom i and j can be neighbors more than once,
> considering the periodic images, what would the off-diagonal terms in the
> matrix below represent?
>
> Thanks a lot in advance for any help on this.
>
> Ricardo Nunes
> Departamento de Fisica - UFMG
>
>
> ================
> Input:
> ================
> WriteMullikenPop       3
> # Species and atoms
> NumberOfSpecies        1
> NumberOfAtoms          2
> %block ChemicalSpeciesLabel
>  1  79  Au
> %endblock ChemicalSpeciesLabel
>
> LatticeConstant     2.673694  Ang
> %block LatticeVectors
>    9.350359    0.000000    0.000000
>    0.000000    9.350359    0.000000
>    0.000000    0.000000    1.000000
> %endblock LatticeVectors
>
> # Atomic coordinates
> AtomicCoordinatesFormat     Ang
> %block AtomicCoordinatesAndAtomicSpecies
>    0.04618072    0.04352611   -0.00000014   1  Au        1
>    0.04608645    2.47934396    1.33684744   1  Au        2
> %endblock AtomicCoordinatesAndAtomicSpecies
>
> ================
> Output:
> ================
> mulliken: Overlap Populations between Atoms
>
>                 1       2
>           1  10.488   0.100
>           2   0.100  10.489
>
> mulliken: Overlap Populations between Orbitals
>
>                   1        2        3        4        5        6        7
> 8
>                  6s       6s       5dxy     5dyz     5dz2     5dxz
> 5dx2-y2  5dxy
>   1 6s          0.780   -0.018    0.000    0.000    0.000    0.000    0.000
> 0.000
>   2 6s         -0.018    0.138    0.000    0.000    0.000    0.000    0.000
> 0.000
>   3 5dxy        0.000    0.000    1.954    0.000    0.000    0.000    0.000
> 0.008
>   4 5dyz        0.000    0.000    0.000    1.853    0.000    0.000    0.000
> 0.000
>   5 5dz2        0.000    0.000    0.000    0.000    1.795    0.000    0.000
> 0.000
>   6 5dxz        0.000    0.000    0.000    0.000    0.000    1.923    0.000
> 0.000
>   7 5dx2-y2     0.000    0.000    0.000    0.000    0.000    0.000    1.898
> 0.000
>   8 5dxy        0.000    0.000    0.008    0.000    0.000    0.000    0.000
> 0.003
>   9 5dyz        0.000    0.000    0.000    0.014    0.000    0.000    0.000
> 0.000
>  10 5dz2        0.000    0.000    0.000    0.000    0.001    0.000    0.000
> 0.000
>  11 5dxz        0.000    0.000    0.000    0.000    0.000    0.010    0.000
> 0.000
>  12 5dx2-y2     0.000    0.000    0.000    0.000    0.000    0.000    0.014
> 0.000
>  13 6Ppy        0.000    0.000    0.000    0.000    0.000    0.000    0.000
> 0.000
>  14 6Ppz        0.000    0.000    0.000    0.000    0.000    0.000    0.000
> 0.000
>  15 6Ppx        0.000    0.000    0.000    0.000    0.000    0.000    0.000
> 0.000
>  16 6s         -0.024    0.012    0.000    0.003    0.000    0.000    0.000
> 0.000
>  17 6s          0.012    0.008    0.000    0.003    0.000    0.000    0.004
> 0.000
>  18 5dxy        0.000    0.000    0.001    0.000    0.000    0.000    0.000
> 0.000
>  19 5dyz        0.003    0.003    0.000   -0.002   -0.001    0.000   -0.002
> 0.000
>  20 5dz2        0.000    0.000    0.000   -0.001   -0.001    0.000    0.000
> 0.000
>  21 5dxz        0.000    0.000    0.000    0.000    0.000    0.000    0.000
> 0.000
>  22 5dx2-y2     0.000    0.004    0.000   -0.002    0.000    0.000    0.000
> 0.000
>  23 5dxy        0.000    0.000    0.000    0.000    0.000    0.000    0.000
> 0.000
>  24 5dyz        0.000    0.000    0.000    0.000    0.000    0.000    0.000
> 0.000
>  25 5dz2        0.000    0.000    0.000    0.000    0.000    0.000    0.000
> 0.000
>  26 5dxz        0.000    0.000    0.000    0.000    0.000    0.000    0.000
> 0.000
>  27 5dx2-y2     0.000    0.000    0.000    0.000    0.000    0.000    0.000
> 0.000
>  28 6Ppy        0.007    0.005    0.000    0.008    0.000    0.000    0.003
> 0.000
>  29 6Ppz        0.005    0.002    0.000    0.000    0.002    0.000    0.004
> 0.000
>  30 6Ppx        0.000    0.000    0.005    0.000    0.000    0.001    0.000
> 0.000
>
```