CCL: Dipole moment calculation from non-zero charge distribution

 Sent to CCL by: "David F. Green" [dfgreen{=}]
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.
 David F. Green                                  <dfgreen||>
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  Applied Mathematics and Statistics
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 Marc Baaden baaden;; wrote:
 Sent to CCL by: "Marc  Baaden" [baaden^^^]
 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 ?