# [SIESTA-L] Question about Mulliken Population Analysis

```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
```