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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.

We can go ahead and add the "Mulliken charges arising from

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

_______________________________________________

AMBER mailing list

AMBER.ambermd.org

http://lists.ambermd.org/mailman/listinfo/amber

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

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.

We can go ahead and add the "Mulliken charges arising from

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

_______________________________________________

AMBER mailing list

AMBER.ambermd.org

http://lists.ambermd.org/mailman/listinfo/amber

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

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