Subject: Re: CCL:G:How to keep a particular state

From: [email-masked] (Mike Frisch)
Date: Thu, 16 Nov 1995 16:47:01 -0500 (EST)

There have been several questions about SCF and different electronic
states, and enough mis-information has been posted that a coherent
reply is needed.

Several different issues have been mixed together in the responses:

1.  Getting Gaussian to find an SCF solution of a particular symmetry.
2.  Whether SCF always converges to the ground state.
3.  Whether an SCF solution different from the ground state (lowest)
    solution but of the same symmetry is phycially meaning ful.
4.  Whether one always wants the ground state in all calculations, or
    all calculations involving a reaction.

The original question was about either 1 or 3 (it was open to either
interpretation).

First of all, an SCF can converge to a minimum or a saddle point in the
space of possible wavefunctions.  The nature of the stationary point
found can be testing by finding if the orbital rotation Hessian has
any negative eigenvalues (Stable keyword in G94, or Stable=Opt to move
downhill to a local minimum).  Once a minimum is found, it is not
guaranteed to be the global minimum, although it usually is in
practice.  (In fact, it takes some effort to produce an example of
multiple local minima, and this usually involves stretching multiple
bonds.)

There is a rather useless theorem that if pure SCF (without DIIS or
any other extrapolation) converges, then the resulting wavefunction is
a local minimum.  In practice, pure SCF is so slow and unreliable that
everyone uses some extrapolation method.  This greatly reduces the
number of SCF iterations required and greatly increases the likelihood
of finding a solution but does create the possibility of finding a
saddle point instead of a minimum.  The theorem has led some people to
the false assumption that if the SCF converges, even with normal
extrapolation proceedures, then it must be stable.  In practice, the
stability should be tested if there is any doubt or uncertainty about
the solution -- this includes any case in which significant
convergence difficulties are encountered.

If a solution of the SCF equations which has a different symmetry than
the ground state is found, then the variational principle applies and
the energy of this state is an upper bound on the energy of the lowest
excited state of its symmetry.  If the SCF is started with an initial
guess of a different symmetry than the ground state and is constrained
to consider wavefunctions of only this symmetry, then such a solution
can be found.  This is typically done by keeping the number of occupied
orbitals of each symmetry type fixed.  The route command for this in
Gaussian 94 is SCF=Symm.

A solution of the standard SCF equations which is of the same symmetry
as the ground state only provides an upper bound on the ground state
energy.  A solution with the same symmetry as the ground state which
is not the lowest solution is an arbitrary mixture of the ground state
and excited states of the same symmetry.  It does not provide an upper
bound on any energy except the ground state, and is not a good
approximation to the the wavefunction for any excited state, since it
has an unknown amount of ground state mixed in.  These general excited
states must be studied by a method which enforces orthogonality to the
ground state, such as CIS or CASSCF.

While typical reactions occur entirely on the ground state potential
energy surface, as one poster noted, there are many cases of surface
crossings.  These are an important feature of photochemical reactions.
The usual case of ground state reactions can be reliably studied in
Gaussian by using Stable=Opt to ensure that the wavefunctions used
at each structure are minima in wavefunction space.  For details on
studying conical intersections and related photochemistry, see
the appropriate references to Mike Robb's work in the Gaussian manual.

Mike Frisch
[email-masked]