# CCL: Dipole moment calculation from non-zero charge distribution

*From*: "David F. Green"
<dfgreen-,-ams.sunysb.edu>
*Subject*: CCL: Dipole moment calculation from non-zero charge
distribution
*Date*: Fri, 04 Nov 2005 06:41:42 -0500

Sent to CCL by: "David F. Green" [dfgreen{=}ams.sunysb.edu]
Marc,

`The standard definition of the dipole moment holds regardless of the
net
``charge. However, the problem is that the dipole moment is
``translationally dependent; you will obtain a different dipole moment
``depending on what centre of expansion you choose. In general, only the
``leading term of any multipole expansion is invariant in this manner.
If
``you are simply looking for a natural point of expansion, there are a
``view choices:
`

`1. The centre of charge (this is the same as your suggestion). In
this
``case, the dipole moment VANISHES, making the calculation pretty easy :)
`` However, higher order terms will be non-zero and need to be
``calculated. An interesting pair of papers to read about this (and also
``how to define an analogous centre of dipole for neutral molecules) are:
` Platt, D. E. and Sliverman, B. D. J. Comput. Chem. 17:358-366 (1996).
Silverman, B. D. and Platt, D. E. J. Med. Chem. 39:2129-2140 (1996)

`2. The geometric centre of the molecule. There is no formal reason
why
``this is a good choice, but it is a common one. Certainly, it is better
``to expand around a point within the boundary of the set of atoms. For
a
``good example of why, thinking about what the dipole moment of a charged
``molecule would be if you expand around a point far away from the
molecule.
`

`3. If you are working with a set of related molecules, you may choose
an
``expansion point defined by the common region. For example, choose the
``centre of the phenyl ring for a set of modified benzenes. If you are
``working a set of molecules with pre-defined positions and orientations
``(such as a set of bound ligand structures in a binding site) you may
``also use this information to define a common centre of expansion.
`

`I should point out though, that due to the translational variance
``problem, you should be very careful about what you use the multipole
``expansion for. The expansion is great for reducing the complexity of
``the system, and giving a few variables describing the majority of the
``electrostatic interactions. The choice of the centre of expansion may
``affect the convergence properties, so a reasonable choice is important.
`` However, the individual values (with the exception of the leading
``term) are pretty much meaningless. Trying to read too much into the
``value of non-leading terms is a commonly made mistake; only the full
``expansion (all terms) up to some cutoff is relevant.
` I hope this helps.
Cheers,
David.
========================================================================
David F. Green <dfgreen||ams.sunysb.edu>
Assistant Professor http://www.ams.sunysb.edu/~dfgreen/
Applied Mathematics and Statistics
Stony Brook University Office: +1-631-632-9344
Math Tower, Room 1-117 Mobile: +1-617-953-3922
Stony Brook, NY 11794-3600 Fax: +1-631-632-8490
========================================================================
Marc Baaden baaden;;smplinux.de wrote:

Sent to CCL by: "Marc Baaden" [baaden^^^smplinux.de]
Dear CCL readers,
I am looking for a formular/recipe to calculate the dipole moment
for a charged molecule. In that case my guess is that you first
have to "factor out"/remove the monopole, but I couldn't find a
precise formula for this case.
In one textbook the dipole moment is simply given as the integral
over r*p(r) where r is the position of the charge and p(r) the charge
density at r. But for a charge distribution with net charge, this does not
correspond to a separation of equal amounts of positive and negative
charge ...
.. as a naive suggestion, I could imagine calculating the geometrical
centre of all negative charge and of all positive charge, remove the
monopole charge from those and then calculate the dipole moment for this.
But I would like confirmation (maybe even a reference or textbook) that
explicitly handles this case.
Thanks in advance,
Marc Baaden
NB: maybe I should add that this is for a classic (molecular mechanics)
model of a protein with fixed point charges. The protein is charged
due to the protonation states of its ionizable residues.
Also in this context, is there a software package that can take a
charge distribution (ideally a PDB file with charges in the last
column) and calculate monopole + dipole + octapole + hexadecapole
moments for this ?