# Re: [AMBER] Coulson vs. Mulliken, CM1, etc. in antechamber + mopac

From: Timothy J Giese <timothyjgiese.gmail.com>
Date: Mon, 26 Jul 2010 10:22:50 -0500

Thanks for sending me a link to this dac.

I've looked this over last Friday and after some debate,
here my conclusions.

In brief,
(1) When you are using mopac, the "Coulson" charges are
the Mulliken charges of the SCF density.
(2) When you are using mopac, the "Mulliken" charges are
the Mulliken charges of a density that is NOT the SCF
density.
(3) When you are using sqm, then charges being printed
are the Mulliken charges of the SCF density.

Before going into more detail, here are some definitions...

Mulliken charges
Qa = Za - \sum_{i \in a} (P.S)ii

Coulson charges
Qa = Za - \sum_{i \in a} Pii Sii

Now in more detail,

(1) When you are using mopac, the "Coulson" charges are the
Mulliken charges of the SCF density. In MNDO/NDDO methods, the
overlap matrix is the unit matrix; both forms reduce to the same
Eq., i.e.,
Qa = Za - \sum_{i \in a} Pii

(2) When you are using mopac, the "Mulliken" charges are the
Mulliken charges of a density that is NOT the SCF density.
Although the charges are mulliken charges, the density matrix is
perhaps not what you were expecting. When using the MULLIK keyword,
mopac will construct a new set of orbitals from the SCF converged
orbitals via a transformation into a new basis
Cnew = S^{-1/2} . Cscf
Cnew no longer diagaonlizes the Fock matrix, however, it does
hold the property
Cnew^T . S . Cnew = UnitMatrix
where the matrix S in the above equation is NOT the overlap matrix
used in the eigenvalue problem.
Mopac then constructs a new density matrix
Pnew_{ij} = 2 \sum_{k=1}^{Nocc} Cnew_{i,k} Cnew_{j,k}
It is important to note that this new density matrix is quite fictitious
in that it does not correspond to the density that gives rise to
the electron-electron and electron-nuclear energies. Pnew contains
two-center components of the density, whereas the matrix used in the
SCF calculation does not contribute to two-center density contributions.
Mopac then computes the mulliken charges using this new density matrix,
i.e.,
Qa = Za - \sum_{i \in a} (P.S)_ii

(3) When you are using sqm, then charges being printed are the Mulliken
charges of the SCF density because it evaluates the Coulson charges,
which, as pointed out previously, are the Mulliken charges when the
overlap matrix is the unit matrix.

re-orthogonalized MOs" to the sqm code, but we should pay attention to
what it is called in the printout.

I fear that when people read the mopac manual, they see "Oh, these are
Coulson charges, but I wanted Mulliken charges, therefore, I must want
to use the MULLIK keyword"; however what they probably meant to say is
"I want the Mulliken charges of the SCF density", in which case they
want the "Coulson" charges for reasons previously stated. Taking on
the mopac convention of what these things are called has the benefit
of using a consistent nomenclature with another program, but may also
perpetuate the confusion caused by it because they are indeed both
Mulliken charges, but using different density matrices.

-Tim

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Received on Mon Jul 26 2010 - 08:30:03 PDT
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