I have a question about the details of the Transition State
decomposition procedure in ADF. I've read a few of the papers and I
feel like I understand the general ideas, but every so often I run
into something that stumps me about the details.

Today, in an effort to deepen our understanding, we played around with
a very simple molecule (H2) and printed out all kinds of information.
All of the following results are from a calculation using two H atoms
at a distance of 0.6 Angstroms (non-physical, but it makes some
things more visible) and basis set I (to make everything as simple as
possible). It's all LDA (VWN paramerization). I've broken the
"discussion" down into parts corresponding to the pieces of the
bonding energy. All energies are in hartrees.

------------------------------------------------------------
   -> Electrostatic Energy (-0.0036)
I think this is straightforward. It's calculated just from the total
electrostatic interaction:
 electrons(A) -- nucleus(B)
 electrons(B) -- nucleus(A)
 electrons(A) -- electrons(B)
   nucleus(A) -- nucleus(B)

It's pretty small in this case because the atoms are really too
close together.

------------------------------------------------------------
   -> Pauli Repulsion (0.6311)

I think (but I'm not sure) that I have a handle on this as well. One
takes the fragment orbitals, orthonormalizes them, and evaluates the
Pauli repulsion in some magical way using the resulting density. Along
the way one gets a set of orthonormal Lowdin orbitals. From our
simple H2 calculation, these are:

SIGMA.g:
 ====== Lowdin Orbitals expressed in primitive basis functions

  column 1 2
  row
    1 5.343245243556E-01 5.343245243556E-01

SIGMA.u:
 ====== Lowdin Orbitals expressed in primitive basis functions

  column 1 2
  row
    1 1.417896195411E+00 -1.417896195411E+00

There is one electron in each of these Lowdin orbitals (at least
that's what I assume the bit about Occupied CSFOs in the output file
means). Assuming that is true, the electron density of this system
is:
 Rho(Lowdin) = 1 * Psi(Sigma.g)^2 + 1 * Psi(Sigma.u)^2

Now, I'm not clear on what the energies of the Lowdin orbitals are.
It seems like they are:
Sigma.g: 0.6270
Sigma.u:-1.7674
Those are the "Gross Orbital Energies in Transition Field". These
don't make a whole heck of a lot of sense to me, but I can live with it.

Regardless, the orthonormalization procedure causes the energy of the
system to go up. I'm happy with that.

------------------------------------------------------------
   -> Orbital Interactions (-0.9507)
Sigma.g: -.4074
Sigma.u: -.5433

At first this seemed strange. The Sigma.u orbitals, which are
completely unoccupied, are contributing to the Total Bonding Energy?!?!?
How can that be? Then I realized that Psi(Sigma.u) is contributing to
the Lowdin electron density, but not to the SCF electron density,
which is:
Rho(SCF) = 2 * Psi(Sigma.g)^2
Due to the symmetry of the situation and the lack of flexibility in
the basis set, Psi(Sigma.g) here is *exactly* the same as Psi(Sigma.g)
in the construction of the Lowdin orbitals.

If the orbital interaction energy is calculated according to the
equation on page 19 of the ADF Users Guide (v2.3 from May 1997), then
it involves a Delta(Rho) which is defined as:
Delta(Rho) = Rho(SCF) - Rho(Lowdin) = Psi(Sigma.g)^2 - Psi(Sigma.u)^2
Consequently there are contributions to the orbital interaction from
electrons being transferred from Psi(Sigma.u) to Psi(Sigma.g).

--------------------------------------------------------------------
Once we reached this point, we started to get a little overconfident.
Except for the energies of the Lowdin orbitals (are those the right
numbers above?), everything seemed clear and understandable... mostly
at least (more on the confusing bits below).

Thinking about the construction of the Orbital Interactions Energy, we
figured we could test our understanding by doing a calculation on H2-
(oh yes, this is totally unrealistic, but it's just to understand what
the program is doing).

The Lowdin orbitals and MOs of H2- are exactly the same as those of H2
(in basis set I), so that's good. Based upon the thoughts above, it
seems like things should work out like this:

Rho(Lowdin) = 2 * Psi(Sigma.g)^2 + 1 * Psi(Sigma.u)^2

because Psi(Sigma.g) ought to be doubly occupied whilst Psi(Sigma.u)
is singly occupied.

Rho(SCF) = 2 * Psi(Sigma.g)^2 + 1 * Psi(Sigma.u)^2

therefore:

Delta(Rho) = Rho(SCF) - Rho(Lowdin) = 0

and the orbital interaction energy should be zero.

This is, unfortunately, not the case.

The orbital interaction energy from the Sigma.u orbitals is zero, as
expected. But the Sigma.g contribution is -0.0930.

Now I feel like I have no idea whatsoever what is going on. Can
someone explain this?

Accompanying this series of questions is one about the relationship
between the orbital interaction energy and the energies of the
Kohn-Sham orbitals, but I'll save that for later.

Thanks in advance for any suggestions/pointers to articles/answers.

-greg

-- 
---------------------
Dr. Greg Landrum  (landrum_at_foreman.ac.rwth-aachen.de)
Institute of Inorganic Chemistry
Aachen University of Technology
Prof.-Pirlet-Str. 1, D-52074 Aachen, Germany
Phone: 049-241-80-7004
Fax: 049-241-8888-288