From:	"R.G.A. Bone" <rgab -x- at -x- purisima.molres.org>
Date:	Wed, 27 Apr 94 23:27:47 -0700
Subject:	Symmetry: Definitive References

In view of the current comments on the philosophy of the use of symmetry
in the computation of molecular potential energy surfaces, I thought it
would be useful to put out a compendium of 'primary' references in this
field.  I revisited my (1992) PhD thesis and culled the following.

It is worth restating, though, that there is no theorem which allows you
claim that a minimum-energy structure (in fact, any stationary point) must
have any point group symmetry at all, in the absence of other information,
e.g., a force law.  (The converse is open to serious discussion, nevertheless:
- does a structure of given symmetry have to be a stationary point?)

Symmetry can be used somewhat heuristically, however, with caution.
And, as has also, already been stated, once one has located a stationary point
(confirmed by checking all gradient components), the only sure way of testing
whether or not it is a local minimum, is to calculate the Hessian.


A general discussion of all basic features of PESs can be found in;
  ------------------
P.G.Mezey, Studies in Physical and Theoretical Chemistry, 53, "Potential
Energy Hypersurfaces", Elsevier, (1987).


Symmetry at Stationary Points
-----------------------------
If one is looking for a 'proof' that the energy must be stationary w.r.t.
all non-totally symmetric distortions (i.e., the oft-desired result that
"symmetric structures are (frequently) stationary points") then there are
two papers which address the issue (for a non-degenerate electronic state):

D. J. Wales, J. Am. Chem. Soc., 112, 7908, (1990)
 - a paper which considers the balance of forces on the atoms at various points
in  a given structure.

H. Metiu, J. Ross, R. Silbey and T. F. George, J. Chem. Phys.,61, 3200, (1974)
This paper observes that the discussion is linked closely with the notion of
expanding the local potential surface in normal coordinates (and their
derivatives, etc.).  [Because, of course, normal coordinates are guaranteed
to transform as irreducible representations of the point group.]

This, in turn, leads on to discussions of reaction paths, etc. and an
understanding of how the gradient vector transforms under symmetry operations,
a subject which has been addressed by many authors.


Symmetry of the gradient.
-------------------------
The first of these papers contains a straightforward and rigorous formal proof.
J. W. McIver, Jr. and A. Komornicki, Chem. Phys. Lett., 10, 303, (1971)

The papers by Pearson describe a perturbation-theory approach.
R. G. Pearson, J. Am. Chem. Soc., 91, 1252, (1969)
R. G. Pearson, J. Am. Chem. Soc., 91, 4947, (1969)
R. G. Pearson, Theor. Chim. Acta., 16, 107, (1970)
R. G. Pearson, J. Chem. Phys., 52, 2167, (1970)
R. G. Pearson, Pure Appl. Chem., 27, 45, (1971)
R. G. Pearson, Acc. Chem. Res., 4, 152, (1971)
R. G. Pearson, Symmetry Rules for Chemical Reactions, Wiley Interscience, (1976)

Rodger and Schipper have a formalism based upon normal-mode expansions.
A. Rodger and P. E. Schipper, Chem. Phys., 107, 329, (1986)

A *landmark* paper by Pechukas describes the properties of steepest-descent
paths and leads to cornerstone statements of transition state symmetries.
P. Pechukas, J. Chem. Phys., 64, 1516, (1976)


Transition State Symmetry Rules
-------------------------------
The believed limitations on the symmetries of transition states have lead to
many analyses, beginning with Stanton and McIver:

J. W. McIver, Jr. and R. E. Stanton, J. Am. Chem. Soc., 94, 8618, (1972)
R. E. Stanton and J. W. McIver, Jr., J. Am. Chem. Soc., 97, 3632, (1975)
J. W. McIver, Jr., Acc. Chem. Res., 7, 72, (1974)
A. Rodger and P. E. Schipper, Chem. Phys., 107, 329, (1986)
A. Rodger and P. E. Schipper, Inorg. Chem., 27, 458, (1988)
A. Rodger and P. E. Schipper, J. Phys. Chem., 91, 189, (1987)
R. G. A. Bone, Chem. Phys. Lett., 193, 557, (1992)


Exceptions and Pitfalls
-----------------------
The application of symmetry and many assumptions about the form of a PES can
lead to problems if unusual features are present.  The oft-cited example is
that of a (putative) monkey-saddle (3rd order 'degenerate' critical point),
but except where symmetry itself may impose severe restrictions on the form
of the potential function, it is safe to assume that there will always be
minor perturbations which lift the degeneracy.  Near 4th-order critical points
can arise on exceptionally flat surfaces and when symmetry demands that
3rd order terms in the potential must vanish.  Far more frequent are bifur-
cations.  Strictly a "bifurcation" is a feature of a path.  Valtazanos and
Ruedenberg demonstrated their association with "valley-ridge inflection points"
on a PES:
P. Valtazanos and K. Ruedenberg, Theor. Chim. Acta., 69, 281, (1986)

Bifurcations are legion.  For description of one on a very simple and surpris-
ing surface, see:
R. G. A. Bone,  Chem. Phys., 178, 255, (1993).

Other more exotic examples, include:
J. F. Stanton, J. Gauss, R. J. Bartlett, T. Helgaker, P. Jorgensen,
H. J. Aa. Jensen and P. R. Taylor, J. Chem. Phys., 97, 1211, (1992).
T. Taketsugu and T. Hirano, J. Chem. Phys., 99, 9806, (1993)

The main significance of bifurcations in the context oif symmetry is that they
are points at which, symmetry may be 'broken'.

For a qualitative discussion of the plethora of forms of PES and the influence
of point group symmetry, see:
R. G. A. Bone, T. W. Rowlands, N. C. Handy and A. J. Stone,
Molec. Phys., 72, 33, (1991)

And, a paper which describes concepts useful for characterisation the
local symmetry of potential energy surfaces in Jahn-Teller systems.
A. Ceulemans, D. Beyens and L. G. Vanquickenbourne,
J. Am. Chem. Soc., 106,  5824, (1984)


Symmetry-breaking in Open-Shell Systems
---------------------------------------
This subject has a long history, so I will be selective here:

General considerations/review:
E. R. Davidson,  J. Phys. Chem., 87, 4783, (1983)

UHF calculations:
L. Farnell, J. A. Pople and L. Radom, J. Phys. Chem.,  87, 79, (1983)
D. B. Cook, Int. J. Quant. Chem., 43, 197, (1992)
D. B. Cook, in "The Self-Consistent Field Approximation", Ed. R. Carbo
D. B. Cook,  J. Chem. Soc. Far. Trans. 2, 82, 187, (1986)
P. O. Lowdin, Rev. Mod. Phys., 35, 496, (1963)(
W. D. Allen and H. F. Schaefer, III,  Chem. Phys., 133, 11, (1989)

Calculations with MC-SCF/CAS-SCF
P. Pulay and T. P. Hamilton, J. Chem. Phys., 88, 4926, (1988)
J. M. Bofill and P. Pulay, J. Chem. Phys., 90, 3637, (1989)

Calculations with a Coupled-Cluster Wavefunction
J. D. Watts and R. J. Bartlett,  J. Chem. Phys., 95, (1991)


Final Comments:

There is no definitive answer to the question of what role symmetry should
play in a given molecular problem.  There will always be a compromise between
(computational) resources available and thoroughness of analysis.  Sometimes
it makes sense to start in high symmetry, sometimes not.  Sometimes it is only
possible to use symmetry - because of the difference it makes to the size of
the calculation.  In calculating a reaction path, it's probably a good idea
to use as little symmetry as possible.  When optimizing a geometry, symmetry
will help you get to a stationary point (probably) but only a frequency
calculation can tell you what it is.  When searching for saddle-points, the
theorems tell you that you'll only increase in symmetry (at the T.S.) if you're
finding a T.S. for a 'degenerate' rearrangement.  Otherwise, your transition
state cannot be searched for by minimizing in a symmetry-constraint.  This
means that the majority of T.S.'s don't have any useful distinguishing
symmetries.   Sad, but true.

Despite the zeal with which chemists have pursued those symmetric 'beauties',
the Fullerenes, Nature more often displays low-symmetry than high-symmetry,
but a bit of thought shows that two low-symmetry structures are often
related to one another by a symmetry operation in a higher-symmetry 'midpoint'
so symmetry of the PES is always conserved.

Richard Bone

================================================================================

R. G. A. Bone.
Molecular Research Institute,
845 Page Mill Road,
Palo Alto,
CA 94304-1011,
U.S.A.

Tel. +1 (415) 424 9924 x110
FAX  +1 (415) 424 9501

E-mail  rgab - at - purisima.molres.org


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