Why Do Some Molecules Have Symmetry Different from that Expected? Edgar Heilbronner Universitat Basel, Klingelbergstrasse 80, CH-4056 Easel, Switzerland
Hamm: Jhime les uieilles questions. Ah les uieilles questions, les uieilles rbponses, il n'y a que $a! Samuel Beckett, Fin de Partie Quite apart from the major role that symmetry breaking plays in physics, it has become an important phenomenon in chemistry. Under particular circumstances a previously symmetrical molecule will lose spontaneously some (or all) of its symmetry elements to assume an equilibrium structure of lower symmetry. A typical example is the occurence of a Jahn-Teller distortion toward lower symmetry on ionization or excitation of a higher symmetrical molecule ( I ) . In the following we shall address amuch more modest, but didactically important problem, that for want of a better name we designate as "symmetry avoidance". By this we mean that a given molecular system assumes an equilibrium structure of lower symmetry than expected on the basis of chemical intuition and/or the current meta-theoretical concepts. In contrast t o symmetry breaking, where two obseruable states are being compared, "symmetry avoidance" refers to the situation where we match the observed geometry of a molecule with the one expected on--obviously inadequate-theoretical grounds. The problem of interest is to explain where and why our meta-theory went wrong. Two remarks are necessary. Firstly, modem high-quality ab initio (2) or semi-empirical (3) quantum-chemical procedures will yield the true structure and symmetry of our molecule, in agreement with experimental observation, except in a few pathological cases. Accordingly the "observed geometry" mentioned above, refers also to the one that would be obtained from a sophisticated quantum-chemical treatment. Secondly, it must be emphasized that we use the term "meta-theory" for a naive (qualitative) molecular orbital model, e.g., of Hiickel type, or for the simple resonance formalism, which chemists have learned t o use for the rationalization of experimental and/or theoretical results derived from good quantum-chemical calculations. Although the problem of symmetry avoidance is a rather general one, as intimated by the historical example given below, we shall limit the main argument-mainly for didactic reasons-to the special case of simple s systems. Indeed, this restricted class of molecules is especially well suited to allow a transparent analysis of the factors involved.
Sexangula (4) to his patron, Johann Mathitus Wacker von Wackenfels as a New Year's present, in which he showed that the hexagonal habitus of snowflakes could be explained by the two-dimensional (planar) closest hexagonal packing of its "elementary" particles, in this instance of suhmicroscopic ice droplets. The fundamental assumptions were (1) The "elementary" particles are spherical, and (2) there exists a force F(R) between any two of them, which is
attractive for all interparticle distances R. Extension of the concept t o three dimensions leads to the well-known hexagonal and cubic closest packings of spheres, two tvpes of structure in which manv pure metals and the rare gases crystallize. (It should be mentioned that Thomas Harriot (5) had developed similar ideas in 1599, but, as was his custom, he did not puhlish them.) Obviously, many crystals are of much lower symmetry than that corresponding to hexagonal or cubic closest packing, which means that one or the other or both of the above assumptions 1 and 2 are violated. The nowadavs rather obvious idea that most molecules are far from spherical (assumption I), was put forward by William Hyde Wollaston (1766-1828) in his Bakerian Lecture "On the Elementary Particles of Certain Crystals" (6) in 1812. From our point of view, the much more intriguing suggestion that assumption 2 is a t fault, had heen discussed much earlier bv the Jesuit Ruder Boikovii: (17111778) (7). He proposed that the forces F(R) between two atoms are complicated multimodal functions of the distance R, as shown in Figure 1, taken from his Theoria Philosophiae Naturalis published in 1763 (8).(Note that in contrast convention. attractive forces in Fieure 1 to resent-dav sien u are positive, and repulsive ones negative.) From such a curve one deduces that there are locallv stable eauilihrium ~ o s i tions a t points such as G, L, and P, because &all posit&e or negative changes in R, away from such a point, result in a
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Symmetry Expectation The hypothesis that the properties of matter are conditioned by the symmetry of its constituent particles goes back t o Plato (428-348 B.C.), who assumed that the atoms of the four elements fire, earth, air, and water have the highly symmetrical shapes of a tetrahedron, a cube, an octahedron, and an icosahedron, respectively. However, the first scientist to link an observable macroscopic symmetry to the suhmicroscopic structure of its constituents was Johann Kepler. In 1611 he dedicated a small treatise with the title De Niue Dedicated to Hans Bock, on the occasion of his 6Mh birthday.
Figure 1. Dspendmce of me force between two atoms on melr Interatom c distance. according to R. Boixovlt ( 1 763)
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restorina force that moves the atom back to its oriainal position; This is the fimt example of a complex interatomic potential function V(R),albeit (up tosign) in the formof the corresponding force F(R) = -dV(R)IdK. A Hlstorlcal Example for "Symmetry Avoidance": The Water Molecule We now turn to an instructive historical examnle that illustrates the interplay of symmetry expectation a i d symmetry avoidance and that involves at one stage a "BoBkoviC function". Both Dalton in 1808 (9) and Wollaston in 1813 (6) assumed that in a molecule AB2 the two soherical atoms B should occupy "for reasons of s b m e t r y 7 'bpposite poles of the central, equally spherical atom A, to yield a linear system BAB. (This, however, did not refer to the water molecule, which at the time was helieved to be diatomic, i.e., HO!) In 1916 Kossel nublished his theorv of the formation of chemical bonds ( l o ) ,which involves a two-step mechanism: First, theatoms. from which the eiven molecule iseoine to be built. exchange 'electrons until each has reached-a r t k g a s elec: tronic configuration. Secondly, the ions thus formed will cluster under the influence of purely coulombic forces to reach a minimum energy structure, under the limiting condition that each ion is assigned a fixed radius r, which cannot be penetrated by the other ions. Applying these rules to the water molecule HzO, Kossel obtained the result shown in Figure 2, where the two protons, of negligible radius, occupy ion of radius r, opposite positions on the surface of the 0%yielding an H 2 0 molecule of D-h symmetry, as would have been expected "on grounds of symmetry" by Dalton and Wollaston, had they known that it helongs to the AB2 type. In the meantime, Dehye had started to investigate the dipole moments of molecules (10, and Jona, one of his coworkers, obtained for the gas-phase dipole-moment of the Hz0 molecule fi(H2O) = 1.87 D (12). This result proved that the Hz0 molecule somehow avoids the highly symmetrical D.h structure. In 1924 an explanation was provided by none
less than Werner Heisenbere- .(13). .. who investieated the dependence of the force acting on one of the p;otons in the Kossel model. if it is moved awav from the remainine HOmoiety along thez axis, taking inio account the polariahilitv n of the 02-ion. He ohtained the "BoSkoviE curve" F(R) shown schematically in Figure 3 (now with the usual sign convention). The force F(R) vanishes at points B and C, but only the latter corresponds to a locally stable position, i.e., to a local minimum in the potential energy curve V(R).Accordingly, Heisenberg proposed that the HzO molecule has two OH bonds of unequal length, so that its symmetry is only C,,. Whereas the length of the short bond is equal to the radius r of the 02-ion, the length of the long bond depends on the polarizability a,which means that the experimental dipole moment fi(Hz0) can be reproduced by an ad hoc choice of a. This curious model was rather short-lived. A year later Friedrich Hund, of molecular orbital and Hund's rules fame, showed (14) that it was unstahle with regard to displacements of the protons away from the common z axis, as indicated in Fieure 4. Denendine on a.the two orotons come to rest on the &face of ihe oz-:on a t an a n g l k ~so , that the H10 molecule assumes a structure with two OH bonds of equal length, hut of Cz, symmetry. As Eucken had shown that the rotational constants of HsO are comnatible with either 0 = 64' or 9 = 110' (15),it wairather unf&tunate that Hund's theoretical value r r ~ d d 6 4 ' ) = 1.34 D for the d i ~ o l e moment was in much betteragreement with the experimen0~) tal one (fi(Hz0) = 1.87 D (12)), than was f i ~ ~ d ~ 1 ( 1=1 4.32 D. Therefore Hund's conclusion was that the Hz0 molecule should be acute-angled, which it is not. We mention in nassine that a Kimhall-tvne model (16). using the uniformiy charged sphere approzmation yields the result shown in Fieure 5. The anele 0 = 102' is ohtained independently of thcassumed value of the 02-radius r, which can therefore be scaled to reproduce the experimental dipole moment p(Hz0) = 1.87 D exactly. Such a calibration yields the valuer = 80 pm ROH= length of the OH bond, which is very reasonable considering the crudeness of the model.
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Figure 2. Model of the H 2 0 molecule according lo W. Kossel(1916).
Figure 4. Model of the H.0 molecule accadlng to F. Hvnd (1925)
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Figure 3. Model of the HtO molecule accwding to W. Heisenberg (1924).
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Figure 5. Klmbal model of the H P molecule using the unifmly charged spherical cloud approxlmatlon.
Although the above, rather naive electrostatic models are no longer relevant for the rationalization of the geometry of molecules, e.g., of the HzO molecule, they provide an important take-home lesson. The four Hz0 models presented above involve successive refinements that lead to the symmetry descent D,h C,. CZ.The question is: What is the crucial reason for the departure from the highly symmetrical D,h structure, i.e., from the structure originally expected on the basis of the Kossel model? If RI = RoH(I),RZ = RoH(z),and R3 = RH(I)H(z) are the distances between the respective atoms, then the total potential energy V T of~ the ~ Kossel model is the sum of three coulombic terms,
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V,,
= Vo,(R1)
+ Von(R2)+ V&RJ
each of which depends only on a single interatomic distance. However.. Vr* is not simolv . "the sum of inde~endent pairwise interactions, because a hond-length change ARM for a eiven bond A-B will also lead to chanees in the Dotential energy contribution of another bond, say the bond GD. In other words, V ~ will D depend not only on RCD,but also on Rm, and in fact on all other interatomic distances within the molecule. In a first and usually sufficient a~proximatiou, this interdependence can be taken care of by&luding sec: ond-order cross-terms for each pair of interatomic distances,
In our above example this cross-linking is provided by the polarizability n of the 0 2 - ion. Only if a is assumed to be zero, will the electrostatic model of minimum potential energy lead to the Dm,,Kossel structure. For n Z 0, the model avoids this high symmetry and reduces it to Cz,. This leads us to expect that naive "meta-theoretical" models that do not take such second-order cross-terms explicitly into account, may-under certain conditions-predict too high a molecular symmetry. (Of course, such cross-terms are implicitly included in more sophisticated treatments.) In the following section we demonstrate the importance of the explicit inclusion of second-order cross-terms for even qualitative discussions of molecular geometry (symmetry), using-for the reasons mentioned above-the naive Htickel model of s systems as an example. However, the conclusions reached are, cum grano salis, of more general validity (cf. concluding remarks below). HMO r Energy and Double-Bond Flxatlon
I t is well known that certain ?r systems, especially those that do not obey Hiickel's rule (17), exhibit lower symmetry than expected on the basis of simple resonance and/or molecular orhital arguments. A classical example is pentalene, C8H6 (18). which assumes a C2h structure 1A (or 1B) with peripheral bond alternation, i.e., that of a Kekul6 structure, rather than the Dzh structure corresponding to the resonance hybrid 1A t.1B (18,191. The theory of double-bond fixation, which explains this observation, has a long and welldocumented history (20). In particular it has been shown (21) that the deoarture from aDeh structure suaaested bv the superposition
