Contents: Table of Contents   Polarizabilities  Chapter 6  Distributed Multipole Analysis

Distributed Multipoles by Numerical Integration

The multipole analysis discussed in the previous section is Stones's technique which distributes the multipoles by partitioning the contributions from pairs of gaussian basis functions amongst the verious centres. This is a simple and quick way to arrived at a Distributed Multipole Analysis. Its one disadvantage is that the individual distributed multipoles are sensitive to the basis set used, especially if diffuse basis functions are being used. Objectively this should not really matter, unless one is hoping to identify lower order multipoles as characteristic of certain atoms or groups and transfer them between molecules.

A more stable distribution is obtained if one partitions the actual physical charge density into regions. A complicated way to do this is the Atoms-in-Molecules technique due to Bader. A simpler way is the following. The integration grids used in DFT automatically divide a molecule into atom centred regions (Voronoi polydehra). If one integrates the moments in each polyhedron one gets atom centred charges, dipoles, quadrupoles etc.

For example,

TITLE
Calculation of Properties
SYMMETRY
CNV 2
END
VARIABLES
OH 0.956 A
HOH 52.25 D
END
BASIS DZP
ATOMS
OXYGEN 8.0 0.0 0.0 0.0
HYDROGEN 1.0 POL OH HOH 0.0
END
START

NUMPROP
START
FINISH

The NUMPROP preforms this integration and produces multipole moments up to hexadeacpole on each atom. The multipole moments are given in spherical harmonic form.

The output would look like this

 Atomic Charges
    OXYGEN      -0.20411
    HYDROGEN     0.10203
    HYDROGEN     0.10203
 Atomic Dipoles     x        y         z
    OXYGEN       0.00000   0.00000   0.62744
    HYDROGEN    -0.02376   0.00000   0.01348
    HYDROGEN     0.02376   0.00000   0.01348
 Atomic Quadrupoles  xx        xy        yy        xz        yz        zz
    OXYGEN       1.03638   0.00000  -1.10059   0.00000   0.00000   0.06421
    HYDROGEN     0.13475   0.00000  -0.15695  -0.14047   0.00000   0.02220
    HYDROGEN     0.13475   0.00000  -0.15695   0.14047   0.00000   0.02220
 Atomic Spherical Harmonic Quadrupoles
    OXYGEN       0.06421   0.00000   0.00000   1.23378   0.00000
    HYDROGEN     0.02220  -0.16221   0.00000   0.16841   0.00000
    HYDROGEN     0.02220   0.16221   0.00000   0.16841   0.00000

etc.

The moments produced will not be the same as with the DMA technique, but are just as valid, and more stable to changes in the basis set.