Point Charge Approximations to a Spherical Charge A Random Walk to High Symmetry Jeffrey B. Weinrach', Kay L. Carter, and Dennls W. Bennett2 University of Wisconsin-Milwaukee, Milwaukee. WI 53201 H. Kenh McDowell Los Alamos National Laboratories, Los Alamos, NM 87544
In 1958 Watson demonstrated that spatial correlation problems inherent in the calculation of the electronic structures of anions could be substantially overcome by surrounding them with a sphere of uniformly distributed positive charge (I).While the method works well for HartreeFockcalculations, it does not lend itself to density functional methods, which require the evaluation of quantities on a nonlinear numerical grid (2). This difficulty can be resolved if the chareed is approximated by a polyhedron with .. sphere . an equal fractional ~har~clocalized at each vertex. Unfortunatelv, .. only . coordinates for regular polyhedra are ohtainable from point symmetry generators, and there is no guarantee that any of the regular polyhedra are suitable for a given computational problem. In order to determine the geometries of a general n-point polyhedron, we took a pragmatic approach, based on the same principal as the well-known valence shell electron pair repulsion model (VSEPR) (3),which operates to minimize average repulsion between like charges. For Coulombic forces, the repulsion between each pair of points is proportional to d-I, where d is the distance between the points, and the total energy is the sum of these terms for each pair of points. Intuition implies that minimizing this energy should also provide the most uniform charge distribution. Monte Carlo methods are especially useful for solving problems that do not have analytical solutions. For the spherical charee auestion, a particular configuration has - - - - ~ ~distribution he a global minimum only been mathematicall; proven for 24 points and for 12 or fewer points, and then usually for minimizing forces of the form d-" for large m. Solutions for m = 1are not alwavs the same as those for large m (4). Since thereis noanalytick method for thedeterminationofgener. al minimal potential conformations, a random Monte Carlo walk may he the best means of arriving a t such a minimum (5). Monte Carlo methods have seen wide application in physicochemical modeling, and the calculation of point charge approximation to a uniformly charged sphere provides an especially straightforward example of their utility. The resulting polyhedra exhibit widely varying, often rare, and seemingly unpredictable point symmetries that provide elegant examples for the teaching of point-group symmetry3. ~
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' Current address: Los Alamos National Laboratories, NM.
Author to whom correspondence should be addressed. We have foundonly one referenceto the application of unrestricted Monte Carlo to this question (5). This stated that configurations witholn symmetry were obtained. Coordinates were published for 13 points that defined a slight distortion of our Csvstructure.The energy for these coordinates is 58.8535. compared to 58.8532 for our coordinates, and it would appear that the calculation was terminated before the symmetry became apparent.
The Monte Carlo Method In its simplest form, Monte Carlo minimization operates by allowing parameters to change sequentially in small random increments, with the results of each change evaluated to determine whether the results are closer or further away from some established criterion; quite often the criterion is a minimization of potential energy. For such a case a decrease in potential energy signifies that the change has aided in the approach toward aminimum, while an increase indicates the opposite. Changes that result in decreases are accepted, establishine a new lower energy configuration, wbile those resulting'm increases are simply rejected. The result is a random walk toward a minimum, which is considered to be reached when a selected number of changes in each parameter produces no potential energy change greater than a predetermined value (e.g., 1 X 10-9. In the determination of minimal potential energy configurations for point charges on the surface of a sphere, the electrostatic potential of n equally charged points is minimized by varying the angles between n unit vectors emanatine from the center of the sphere and terminating with a p&t charge. The process begins by placingall of the charges at one point, then allowing a random walk with large steps, replacing a previous configuration with a new one if the net repulsion between the points decreases. The size of a given step is determined by generating a random number between 1and -1 and multiplying i t by amaximum step size, S. This results in an incremental change in the range -S < 0 < S. The charged vectors are rotated individually and sequentialIv until a eiven number of steps (ex., 100) results in no iecrease i;energy, indicating that the difference between the current conformational potential energy and that a t the minimum is significantly less than the maximum step size. When no further changes are noted, the maximum step size is decreased (usually halved), and the process is repeated. The initial large steps allow the points to approach a rough minimum potential conformation quickly, wbile subsequent smaller steps "fine tune" the process. The search for aminimum continues with the maximum step size repeatedly decreasing until it becomes less than some predetermined convereence value, usually a number on the order of the desireiprecision of the specific calculation. At this point the conformation is considered to be a t a local minimum, and further work is required to test for a globalminimum. In any event the results are not rigorous, since the absolute determination of the conformation yielding the true global minimum would require an infinite number of random steps. Every possible local minimum would have to be identified and compared. I t is a t this stage that a little "chemical intuition" becomes invaluable, i.e., if it looks right, it probably is. However, as demonstrated below, such an approach must be undertaken with reasonable caution.
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Polnt-Charge to a Charged Sphere . Approxlmailons .. .
Conformational surfaces for 2-8 points generated by the described method are the polyhedra expected from a VSEPR-like treatment and can provide the student with a hands-on verification of the VSEPR model. It is of interest to note that the square antiprism results from the calculation of the Bpoint conformation, even when the starting point is acube (6).Thelower energy 20-point polyhedron isa D s h structure rather than the expected dodecahedron. Aeain. this structure is obtained even if the random walk startskith a dodecahedra1configuration.Several other similar results indicate that the energy barriers between conformations are low and that the method is likely to produce a global minimum. However, this is not always the case. Suggested minima have been published for up through 16points (4),and our results coincide except for 14 and 16 points. For configuration was earlier believed to he 14 points, our the minimum, but a lower energy D 4 h configuration has been identified. For 16 points, our tetrahedral arrangement was not among those considered. We have generated polyhedra containing up to 50 points to date; only one exhibits a trivial point group symmetry (49 points), and many have point symmetries that are rarely observed in molecular studies. The point groups for 2 to 50 point polyhedra generated by the described method are listed in the table. No obvious pattern emerges from the list, and, although the highest symmetries resulted from even numbers of points, even numbers also generated some of the lowest symmetry conformations,while many polyhedra with an odd number of vertices exhibited relatively high symmetries. Arrangements of low symmetry were expected to be increasinalv likelv as the number of ~ o i n t sincreased. but this was %t always the case, with l&h symmetries' still arising for 40 or more points. Although all of the polyhedra provide interesting point group examples, several are of particular interest, and we will focus on those. The 22-point polyhedron is shown in Figure 1.Figure l a i s a view down a twofold rotational axis, while Figure I b is a view down a threefold axis. The structure has f;ll tetrahedral symmetry, Td.For clarity, the points lying at the corners of the tetrahedron in the structure are filled in. The 28-
A
./ a
b
Figure 1.22-point polyhedron. Td with points on thevertices of the tetrahedron marked, viewed down (a) a twofold axis. (b) a threefold axis.
Figure 2.28-point polyhedron. T, wnh points on the vertices of me tetrahedron marked, viewed down (a) a twofold axis, (b) a threefold axis.
Polnt Symmetry Groups of the Polyhedra RewHlnp from Monte Carlo Mlnlmlratlon Number of Points
Point Qoup
Number of Points
Point Cmup
b
c
Figure 3.24-poim polyhedron. Oviewed down (a)afourfoldaxls. (b)athreefold axis. (c)a twofold axis.
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Journal of Chemical Education
point polyhedron is shown in Figure 2. Figure 2a is a view down a twofold axis, while Figure 2h is a view down a threefold axis. However, unlike the 22-point structure, the 28point polyhedron does not contain mirror planes bisecting the angle between the twofold axes. Therefore, the 28-point polyhedron belongs to the Tpoint group, the pure rotational subgroup of the Td group. The 16-point polyhedron also has T symmetry. The 24-point polyhedron in Figure 3 is another example of a rarely obsewed point group, 0, the pure rotational subgroup of the octahedralgroup Oh. View (a) is down a fourfold axis. Note that the vertices of the octahedron lie at the centers of the square faces, and that the twisted orientation of the square faces precludes the existence of the mirror planes required for Oh symmetry. The 44-point configuration with full Oh symmetry is shown in Figure 4 for comparison. Twenty-four points define the vertices of one of the Archimedian figures whose vertices are surrounded by similar arrangements of faces composed of two or more regular polygons. This polyhedron is called the snub cube, and it is enantiomorphic. This configuration has been shown to he the elohal enerw minimum-for forces of the form d-"' with m large (7),but i t is clearly not unique, since the mirror imane of this oolvhedron would be another snuh cube with the same enerR),.- he 48-point polyhedron has a similar chiral 0 eeometrs. The 16- and 28-ooint confieuratiuns, belonging i o group T, are also chiral. Figure Sa illuntrates a 45-point polyhedron belonging to the U, ~ o i n erouo t viewed down the threefold .~~~~~ axis. The 23~-- . and ?9;pointpo&hedra also h a v e symmetry. ~ ~ The 36point solid has D7svmmetrv as shown in Fieure 5h. viewed down a twofold a& with tl;e other twofoldaxes hirizontal and vertical. The 50-point structure shown in Figure 6 contains a sixfold axis and dihedral mirror planes, giving it Ded symmetry. Figure 7 illustrates the I-, 17-, 27-, and 42-point polyhedra, each viewed down afivefold axis. All four polyhedra belong to the D5h point group.
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Figure 4. 46polnl polyhedron, Oh viewed down (a) a fourfold axis. (b) a threefold axis. (c)a twofold axis.
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Fiewe 5. Dihedral groups. (a) 45-point polyhedron. h, viewed down M e Mreefold axis. One twofold axis is horizontal. (b) 36point polyhedron. D2. viewed downahvdoldaxis. meothertwofoldaxasare horizontal and vmlcal.
Figure 0. SC-polnt polyhedron, W, viewed down me sixfold axis
Figure 7. Ds, polyhedra viewed down the fivefold axes: (a) 7 points. (b) 17 points, (c) 27 points. (d) 42 points.
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Figure 8. 31-point polyhedron. CZ.. viewed down ihethreefold axis.
Figure 10.
Figure 9. 46point polyhedron. CS,viewed down the twofold axis.
C, polyhedra. mirror planes vertical: (a) 3 3 points. (b) 43 points.
Cyclic groups were also ohtained. Figure 8 shows the 31point Cau configuration looking down the rotational axis. Figure 9 is the 46-point polyhedron belonging to point group C2.Its onlv svmmetrv element is one twofold rotational axis.
a
'--F illustrated by the 33- and 43-point polyhedra shown in ~ i ~ : ure 10. The five Platonic solids, polyhedra with faces consisting of a single regular polygon, are the tetrahedron, octahedron, cuhe, icosahedron, and dodecahedron. The vertices of the Figure 11. 32-point polyhedron, b, viewed down (a) a fivefold axis. (b) a tetrahedron, octahedron, and icosahedron result from the threefold axis (c)a twofold axis. The points marked lie on the vertices of an equal repulsion of 4, 6, and 12 points, respectively, hut, as icosahedron. mentioned previously, the cuhe and dodecahedron are not generally from the mutual repulsion of 8 or 20 points. However, the dodecahedron does show itself most elegantly in the 32-point polyhedron, providing an excellent example of both the icosahedron and the dodecahedron belong to the Ih the concept of duals. For every polyhedron, there is another point group, and since the vertices of each lie on the centers polyhedron that is called its dual. The dual may be conof faces of the other, the entire 32-point polyhedron must structed by locating a vertex for the dual a t the center of also belong to the Ih point group. each face of the original polyhedron. If the faces meet alone I t has been ~reviouslvnoted that aleorithms that create ~ ~ an edge, the corresponding vertices of the dual are connectei polyhedra by minimizing d-'" tend to favor triangular faces by edges. Thus the dual of apolyhedron has as many vertices since they are rigid and can not he distorted (8).All of the as the polyhedron has faces, and the dual has as many faces polyhedra generated with the Monte Carlo algorithm clearly as the original polyhedron has vertices. The dual of the demonstrate this tendencv. Indeed.. onlv .the nolvhedra with tetrahedron is another tetrahedron. The cuhe is the dual of 8,17,33,35,44,47, and 48;ertices appear to Eoniain quadrithe octahedron, and vice versa. The icosahedron and dolateral faces in addition to trianeular faces. decahedron are duals of one another. Dual polyhedra belong Thealgorithm used for thecomputarionsdescrihed herein to the same point symmetry group, since any symmetry has heen coded in both FORTR.AI< and microsofr QuickHAoperation will interchange the vertices of one polyhedron in SIC. I t is designed to generate polyhedra for any n"mher of the same way that it interchanges the faces of the other. The points, hut larger numbers may take extremely long compu32-point polyhedron is shown in Figure 11, viewed down tational times. We are in the process of adding a simplex fivefold, threefold, and twofold axes. The 12 points marked optimization routine to the program in order to speed up lie on the vertices of an icosahedron, while the remaining 20 convergence once an approximate structure is obtained from points lie on the vertices of its dual, a dodecahedron. Thus the random walk (9).
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Journal of Chemical Education
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The method can be easily modified for other applications of Monte Carlo minimization. We are currently working toward parameterization of the model to take into account differing s mimic lone pairs and vari-- - - .- ---e~charges on the ~ o i n t to ~ U substituents S in orde;toprovide an even cioser parallel to the VSEPR model.
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Acknowledgment
DWB and JBW acknowledae Virdnia chemicals, 1% for providing funding for sulfur-oxy anion research that required the generation of the spherical charge approximations. We thank S. C. Johnson & Son, Inc., for the generous
use of computational facilities. KLC gratefully acknowledges the support of a Graduate Fellowship from the National Science Foundation. Literature Clted I, wauon, R. E. m a .
[email protected], ~ m e - ~ L O . 2. Dun1ap.B. I.:Connolly,J.W. D.; Sabin, J. R. J. Chem.Phy8. 1979,71,33963402. ~ Moieevlor , Geometry; van Nostrand-Reinhold: New York, 1972; land 3. ~ i l h ~R.i J. references therein). 4. M ~ I D Y ~ .W T .. ; K ~ Oo.;srni~, ~, w R . C ~ " J. . them. 1977.55.1745-1761. 6. Webb, S. Nature 1986,323, 20. cotton.^. A,: W ~ I L ~ ~ ~ PdO ~~ .~G ~ . ~ ~ d ~ ~ ~W ~~I E Y~: N ~~W Yi O ~~L . ~ 1986 PP 15-16.
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