84

Geometry Optimization Using Symmetry Coordinates

H. Kondo, H. H. J&6,* H. Y. Lee, and William J. Welsh Department of Chemisty, Universio of Cincinnati, Cincinnati, Ohio 45221 Received 25 April 1983; accepted 26 August 1983

The use of symmetry coordinates (SC) in geometry optimization is discussed. A computer program incorporating the use of SC, together with analytical calculation of the gradient and quadratic acceleration, is described. Also reported are careful test results on a series of small molecules and typical results with a long series of molecules up to quite large size (4&60 atoms).

INTRODUCTION

For the last several years we have been inter- ested in the use of molecular symmetry in molec- ular orbital calculations.' The program we have developed, CNDO/S,~ makes extensive use of point-group symmetry throughout.

In this article we would like to report on the simplifications afforded by the use of symmetry in the process of geometry optimization. The Hamiltonian operator, and consequently the gradient of the potential energy, transform of necessity in the totally symmetric irreducible representations (irrep) of the applicable point group (PG). As a result, any derivatives of the gradient with respect to coordinates that are not totally symmetric (or do not contain a totally symmetric component) vanish identically. There- fore when the energy is expressed in terms of symmetry coordinates (sc), derivatives (gradient elements) are required only with respect to those coordinates that transform in the totally sym- metric irrep.

A molecule consisting of N atoms has 3N - 6 (3N - 5 if it is linear) internal degrees of free- dom (DF); without use of symmetry, all the inter- nal DF must be optimized. Moreover, while it is generally easy to separate out the 3 translational DF, separation of the 3 (or 2) rotational ones is not trivial. On the other hand, even molecules with only relatively low symmetry have signifi- cantly fewer than 3N totally symmetric DF. Thus in formaldehyde, H,CO, transforming in the PG C,,, there are 6 internal DF, but only 4 totally symmetric SC. In acetone, also C,,, there are 24

*To whom all correspondence should be addressed.

internal coordinates, but only 9 totally symmet- ric sc. The larger the molecule and the higher its symmetry, the greater the simplification achieved by the use of the sc.

Optimization using sc of any molecule requires an initial test structure which transforms in some PG, say G. The converged structure will trans- form in the same group, or possibly in a PG to which G is a subgroup; however, the final struc- ture cannot be of lower symmetry than the test structure. Special provisions have to be made in case the test structure is too symmetric (cf. be- low).

The model used in this work is the semi- empirical C N D O / ~ m e t h ~ d , ~ which has permitted relatively simple analytic expressions for the gradient elements.4 The program we have devel- oped is fully automatic, using an initial test structure to get started, and proceeds via a steepest descent method with a quadratic accel- e r a t i~n .~?

SYMMETRY COORDINATES

We have chosen the most ndive method of defining the sc, l. Different definitions were required for asymmetric and symmetric top molecules, i.e. molecules having no axis, proper or improper, of order greater than two, and molecules having just one such axis, respectively. Spherical tops having several noncoincident axes of order greater than two are treated in a sub group of lower symmetry.

For the asymmetric tops, the { are simply linear combinations of the Cartesian displace- ment coordinates of sets of symmetrically equiv- alent atoms. Projection of the x, y, and z

Journal of Computational Chemistry, Vol. 5, No. 1, 84-88 (1984) 0 1984 by John Wiley & Sons, Inc. CCC 0192-8651/84/010084-05$04.00

Geometry Optimization Using Symmetry Coordinates 85

Table I. The totally symmetric symmetry coordinate for acetone.

ii'

X

t6 = 2 + 2 5, = 41 -5 -6

c 7 = x t x - x '2 = 12 -7 -8 -9 -XI0

$ = Z3'5 c 9 = -7 2 t Z -8 C Z -9'ZlO

c 5 = x - x -5 4

coordinates of one atom in each set by a projec- tion operator in the totally symmetric irrep gen- erates all the needed sc for that set; at most one totally symmetric sc is generated from each Cartesian coordinate of any atom. Table I shows the sc thus generated for acetone.

For symmetric top molecules, the Cartesian coordinates are first transformed into cylindric polar coordinates. The scs are then the ap- propriate linear combinations of the latter. Thus, for ethane, = z1 - z2, J2 = z3 + z4 + z5 - z6 - 2 7 - 28,

and l3 = p3 + p4 + p5 + & -t p7 + p8. Note that with 18 internal coordinates for 4h or D3d, only 3 are totally symmetric.

Provisions have been made within the pro- gram to freeze certain sc or to allow optimiza- tion of only selected sc.

The use of the totally symmetric sc autumati- cally removes from consideration most of the redundant (rotational and translational) coordi- nates, provided the molecule has a minimum of

symmetry. Actually, in our experience carrying along one or more redundant coordinates has not slowed convergence.

GRADIENT ALGORITHM

The total electronic energy of a molecule can be broken up into two parts: a part containing terms each of which depends only on a single atom, and another part which depends on pairs (and larger groups) of atoms. Within the CND0/2 (and CNDO/S) approximation:

E = E E A + C E A B (1) A A < B

where A and B refer to atoms. The first set of terms in eq. (1) is geometry independent, and hence need not concern us further. The second set of terms, E A B , in the CNDO methods can be broken up into three parts:

E A B ( R A B , s p a , Y A B ) = E i A s p a ) + EA'B(YAB)

+ EAh ( R A B )

Here, R A B is the interatomic distance, Spa is the overlap integral between atomic orbitals p on A and u on B, and yAB is the semi-empirically averaged electron repulsion integral of CNDO. Spa and YAB, of course, themselves are functions of R A B . With these definitions, the gradient ele- ments become

Here the qmn are the three coordinates of the two atoms A and B. All partial derivatives in eq. (2) can be readily evaluated analytically.

STEEPEST DESCENT AND QUADRATIC ACCELERATION

The gradient is a vector pointing in the direc- tion of the steepest increase of the energy. Thus in the nth iteration a new structure is de-

86 Kondo et al.

termined by moving in the negative direction of the gradient by an arbitrarily chosen step a":

where Ig") is the gradient. Whenever an optimization is carried out by p

steepest descent method on a function that is quadratic or nearly quadratic in each of the independent variables, quadratic acceleration methods improve the rate of convergence. We have chosen the method of Fletcher and Powell7 to accelerate convergence. The improved coordi- nates qn+' at the ( n + 1)th step are generated by the recursion relation:

where at the nth iteration the Hessian matrix H" is an approximation to the inverse of the matrix G of the 2nd-order partial derivatives

The initial matrix Ho is taken as the unit matrix, and successive approximations are gener- ated by the recursion relation

with (a"l = -a"lHnlgn) and ly") = Ign+l) -

The use of the quadratic acceleration method assumes the potential energy function to be quadratic in each of the SC, li. One frequently finds in the literature statements. that this method insures convergence in a calculable num- ber of iterations. However, the assumption of quadratic dependence of the energy on geometry variables is, at best, an approximation, and hence no statement about the rate of convergence can be made in general. Furthermore, we have en- countered a number of cases in which the ap- proximation to the Hessian matrix is inadequate. Such behavior has been particularly frequent when torsional angles were involved. In such cases optimization sometimes converged more rapidly by a steepest descent without quadratic acceleration. Occasionally, the Hessian becomes non-(positive definite); in that case, quadratic acceleration is restarted from a unit matrix for H.

En)*

Table 11. The treatment of step size.

'The iteration n is rejected. bExcept in the first iteration after an (Y

reduction.

CONVERGENCE

A crucial, but unfortunately difficult, parame- ter in obtaining good convergence is the step size, a". The initial step size ao must be held quite small, otherwise problems in the self-consistent- field (SCF) convergence are frequently encoun- tered. However, convergence of the optimization is greatly accelerated by continually increasing the step size. Although no truly optimal pattern could be found, we have chosen the pattern shown in Table 11.

It may appear natural and desirable to use the magnitude of the change in the sc between successive iterations as the criterion when con- vergence has been achieved. However, there is no guarantee that at a given precision of calculation a preset degree of precision of sc can be achieved. Therefore we have chosen the absolute magni- tude of the maximum gradient element g,, as the convergence criterion. Although this magni- tude is an adjustable parameter, we have used 0.1 eV/A for most calculations.

REDUCED SYMMETRY

As indicated above, optimization using sc can- not reduce the symmetry below that of the ini- tial test structure. This must be true because any sc that distorts a structure to one of lower sym- metry is necessarily nontotally symmetric, and hence the corresponding gradient element van- ishes. This is illustrated in Figure 1.

In order to cope with this problem, we have included a special program step which permits distortion of the converged structure in any desired manner by a small but finite amount. Such distortions must be made according to chemical insight into likely structures of the molecule. Distortion in the direction of each non- totally symmetric sc is insufficient, at least in

Geometry Optimization Using Symmetry Coordinates 87

Figure 1. Saddle point and minimum.

the absence of normal coordinates. If the energy is decreased by such a distortion, indicating that the geometry obtained is a saddle point rather than a (local) minimum, the optimal geometry has lower symmetry, and optimization is reini- tiated automatically in the distorted symmetry.

RESULTS

The program has been tested on a large variety of molecules. It is found that the final converged geometry is independent of the initial test struc- ture. With a convergence criterion of 0.1 eV/A, in a series of 6 molecules containing 4 to 10 atoms, bond lengths never deviated from the average by more than 0.001 A, and bond angles had an average variation of & 0.15 O . In some 45 runs, convergence required an average of 12.7 iterations reducing the energy and 1-2 iterations which were rejected because the energy in- creased. The number of iterations appears to be, because of the quadratic acceleration and the dynamic variation of step size, nearly indepen- dent of starting geometry. Typical data are shown in the Appendix? The total energy of the converged structures also was independent of starting geometry, in each case agreeing to better than 0.001 eV.

Final structures of the test molecules trans- formed in various PG, including C,, C,,, Czh, and Dzh. In some cases, optimization was initiated

from a structure transforming in a subgroup of the PG of the final structure. In all such cases, the final structure was the same as when optimi- zation was initiated in the proper PC. Even the number of iterations required for convergence was not significantly affected in this way, al- though, of course, the computing time was increased because of the larger number of sc required.

In the test molecules, converged geometries agreed with experimental values within an aver- age of kO.04 A for bond lengths and + 3 O for bond angles. This level of agreement, of course, depends only on the model (CNDO/~) chosen, not on the optimization method.

Further, less extensive tests have been made on a large number of molecules in symmetries as low as C, and as high as DSh.’ Convergence in cyclic molecules tends to be somewhat slower, but the final geometries also are independent of starting ones, and energy constancy was ob- served whenever tests were made.

For large molecules, 40-60 atoms, at times convergence is slow, particularly where, in the potential surface, narrow troughs exist. Some- times as many as 50-100 iterations are required for convergence in these cases.” A practical mea- sure that seems to improve efficiency is to alter- nate between using and not using quadratic acceleration every so many (5-10) iterations. This device reduces the extent of oscillation within the potential energy trough and so speeds up convergence.

One of the many useful features that our C N D O / ~ program incorporates is an option allow- ing for rotation of groups of atoms about desig- nated bonds prior to geometry optimization. This has proved particularly useful in the conforma- tional analysis of several large molecules where C N D ~ / ~ energies have been calculated as a func- tion of some rotational angle, E ( + ) vs. 9. Here the number of iterations needed for optimization can be reduced significantly for each successive rotational angle by a judicious choice of initial geometry. Specifically, since in general geometri- cal parameters (bond lengths and bond angles) vary smoothly with changing rotational angle, it is advantageous to use the previously converged (optimized) structural geometry corresponding to a nearby angle as the initial geometry for a successive calculation a t a new + value, rather than to repeatedly reinitialize the geometry at some arbitrary starting point. In this way, need- less computational redundancy is obviated.

88 Kondo et al.

Figure 2. Some irrational structures and the corre- sponding converged geometries.

The power of the method is also indicated by experiences with the optimization of some irra- tional structures which had been generated ex- clusively to test certain symmetry routines. In each case, the optimization converged on a chem- ically reasonable structure, even if i t was neces- sary to break up the array of atoms into two or more molecules. Figure 2 illustrates some rather astounding examples of such cases. The most amazing finding, perhaps, is the fact than an SCF calculation with a single Slater determinant con- verges, both for the ridiculous test structures and for the converged structures involving several separated molecules in the same determinant.

The improvement of energy in the first iteration in such cases sometimes exceeded lo00 eV.

The C N D O / ~ method has frequently been criti- cized for giving poor optimized structures." Our program, which uses this method, obviously can- not provide results better than the model. How- ever, in our hands the method has produced excellent results with a great economy in com- puting time.

References

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H. H. Jaffe' and R. L. Ellis, J. Comput. Phys., l0 ,20 (1971). J. Del Bene and H. H. Jaffe', J. Chem. Phys., 43, 1807 (1968). J. A. Pople, D. P. Santry, and G. A. Segal, J. Chem. Phys., 43, 5129 (1965); J; A. Pople and G. A. Segal, J. Chem. Phys., 43, 5136 (1965). Cf., e.g., P. Pulay and F. Torok, Mol. Phys., 26, 1153 (1973). F. S. Acton, Numerical Methods that Work, Harper & Row, New York, 1970, Chap. 17. Cf. also J. W. McIver and A. Komornicki, Chem. Phys. Lett., 10, 303 (1971). R. Fletcher and M. J. D. Powell, Comput. J., 6, 163 (1963). Copies of the appendix are available from H. H. J. upon request. In view of the relative scarcity of molecules trans- forming as spherical tops, and of the special problems involved with the absence of preferred direction, the program can deal with spherical tops only in sym- metric or asymmetric top subgroups. W. J. Welsh, H. H. Jaffe', J. E. Mark, and N. Kondo, Makromol. Chem., 183,801 (1982); W. J, Welsh and J. E. Mark, J. Muter. Sci., in press; W. J. Welsh and J. E. Mark, Polym. Bull., 8, 21 (1982); W. J. Welsh, V. Cody, J. E. Mark, and S. F. Zakrzewski, Cancer Bwchem. Bwphys., in press. D. Perahia and A. Pullman, Chem. Phys. Lett., 19, 173 (1973).