Mollweide Projections Molecular Orbital Symmetries o! : the Spherical Shell C. M. Quinn and J. G. McKiernan Department of Chemistry D. B. Redmond Department of Mathematics St. Patrick's College. Maynooth, Co. Kildare, Ireland
Two recent papers ( I , 2) have initiated interest in the generator-orbital approach to the construction of symmetry-adapted LCAO functions. A difficulty in this approach or any other approach to molecular orbital symmetries is the problem surrounding the presentation of' three-dimensional details in a general, easily visualized form. A very convenient device for this ourDose is nrovided IIV the Mollweide oroiection (3) of cartography &hereby detail on the surfa'ce of a sphere can be presented as a two-dimensional map. We present in this paper a first application' of this device within the generator-orbital aooroach to molecular orbital svmmetries. ~ ~ m m e t r ~ - a d a pfunctions &d are obtained directly and oictoriallv. for -given molecular structures on the basis of the nodal structures of the central harmonic functions for which the Mollweide oroicrtioni are oresented. In this uauer the technique is illustrated for the ekes of ligand groipbrbitals. both o and a t m e in Oh svmmetrv. In the oaner which follows this (4) and eiiewheri(5, 6) we shall show how other interactions can be handled with eanal facilitv and summarize the usefulness of this new approach for receit cluster-orbital theory (7.8). Later (9) we shall show the utilitv of our approach for the presentation of correlation diagrams of the Walsh kind (lo), and the discussion of orbital symmetry conservation in reaction mechanisms (11). Mollwelde Proiectlons of the Soherlcal Harmonics In the generator-orbital approach, symmetry-adapted LCAO functions or localized eauivalent functions (1.2) result from the local matching a t lfgand orbital positions of the phases of harmonic functions or their hvbrid . spherical . equivalents, which are considered to he a t the center of the molecular structure. The validity of this technique, of course, stems from the fact that the spherical harmonics provide a set of basis functions for the irreducible representations of the spherical group RR and, therefore, present an overcomplete set for the reoresentiltion of all the irreducible representations of the molecular point groups, which determine the possible molecular orbital symmetries for a given molecular structure. The principal difficulty is the conceptual one of three dimensions. A knowledge of the phases of the spherical harmonics (or their hybrid equivalents for localized functions) in terms of their Mollweide projections removes this difficulty and in manv cases reduces the eeneration of aonrooriate .. . ligand orbital combinations to an exercise in the superimposition of svrnbols for the lieand orbitals on the basic set of ~ o l l w e i d eprojections. Mollweide's ellintical ~roiection for the surface of a unit . . sphere, i.e., a hollow globe is drawn in Figure 1.Viewed from along the +x axis the hollow globe is imagined to be split from top to bottom along a line passing from the point (OOI), the north pole, to the point (00 - I), the south pole, through the point (-loo), and then opened out and flattened onto a twodimensional plane so that the original equatorial circumference is preserved and divided in equal quarters bounded by the directions @,,@,-@,, and -& In Figure 1 these unit
from along the + x axis vectors identifying the dispositions of the axes of the Cartesian coordinate svstem throueh the unit snhere are marked in spherical polnr coordinatrs that conform to the rules 1) 0 inire:ws down thr p r u p w i u n f n m tr tt, n on paqsing from 8, t1,0.11in i ~ h mr x d ~ n a t e s r,l l -9, rl.i.0,. 2) $increases to the right from +&I Thus, along the equator ( B = 9/27 the other fundamental directions are &, ((l,?r/2,0) or (l,lr/Z,Zn)), PY(l,a/2,a/2), -&,(l,a/Z,a occurring a t both right and left equatorial extremes), and -&(l,n/2,3~/2). This scheme is somewhat different from the original cartographic form (31, in which equal areas are emphasized and which requires different scalings . of B and 4. In Figures 2-4 the Mollweide projections of the s to R spherit.aI harmonics are prrsented. In the figures, arras on the unit sphere that correspund to regions of positive amplitude d t h r functions are shaded. ' h e "kublc harmonic" 1131 , , forms ~-~~~~ are used for the f and g projections since these are required for the representation of the irreducible symmetries of the is an important example. An cubic point groups, of which 01, examination of Fieure 3. usine for comoarison earlier mesentations (13-15)"of tbe'compiicated f-iarmonic functibns, demonstrates the usefulness of the nroiection techniaue for . . the visualiz:ition of these three-dimensional distributions, and this i i further em~haaizedin Figure 4 wherein they k u b ~ c harmonics are for the-first time.2 Note that the Mollweide projections for theg(x4 y 4 z4) andg(2z' - y 4 - 2 4 ) functions have been idealized. There are slight distortions from circular in the nodal lines for these projections but these have been ignored in the drawing of Figure 4. ~~~
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The aulhors are aware of only one previous use of these projections in the chemical literature. J. W. Linneti presenled the spherical hamwnic
solutions for the "particle-on-the-sphere" problem as Mollweide projections in his monograph on "Wave Mechanics and Valence Thewy" 1< 9.9) .-,.
The squared functionsin cross-section for the general subsets of
g and &spherical harmonics are to be found in various publications
Volume 61
Number 7 Julv 1984
569
d".
Figure 4. Mollweide projectionsfor Me *cubic harmonics on the unit sphere identified using the Muiiiken symbols for 4 irreducible representations and Bethe's notation. Note that in both Figures 3 and 4 there are repetitions of the ineducible representationsof Oh(T," and Tagfor example) f w which lower-order hamwnics (pard 4provide basis hmctions. In such casesue f-and ghamnnics are onhagonaiired to the lower order functions ( 12) and this is manifest, for example, in the f$ funnion by the conical nodes about the xaxes distinguishing this function from the p, harmonic.
Figure 2. Mollweide projections of the s-, p. and d-spherical harmonics on the unit sphere classifiedalso as basis functions for the irreducible symmetries of the point group 4.
Figure 5. (a) Ligand geometry (Xs) exhibiting Oh Symmetry in usual 3-D perspective. (b) Mollweide projection for this arrangement of six ligsnd atoms. The perspectiveham above the page is along the - x axis towards the coordinate origin; thus atom 4 appears in the center of the projection at the position -& IC) Suitable symbols for s- or powbitais at ligand positions. For sorbitals filied Circles correspond to positme orbital amplitude on the unit sphere about iigand positions: Open circles correspond to the contribution -1 times the sorbital amplitude about ligand positions. For pa-orbitals the symbol used denotes the plobe visible radially ham outside the sphere. Id) Suitable symbols for the plr-vertical set (a:) of ligand pwbitals: filied circles-positive amplitude: open circle-negative amplitude. (e) Suitable symbols for the plr hariroml set (sf) of ligand porbitals. Figure 3. Maliweide projectionsof the f-kubic harmonics ( 12)and their ciassification~for the irreducible symmetries of 4. The general and cubic sets of f-harmonics are identical in the cases f,s, fa+-pl and ,,,f and it is a useful exercise to compare these Projections with the earlier presentations of these general functions ( 13. 14). The remainingspecificallycubic linear combinations also are available for comparison in reference ( 15).B e W s ( 12)original notation is given in brackets.The projections are arbitrarily changed in sign to agree with referenceI15)where necessary, since it is customary to display the probability amplitude functions as positive about a posltive axis when this is convenient.
570
Journal of Chemical Education
Llgand Group Orbitals in Oh Symmetry The generation of LCAO approximations of molecular orbitals requires components of comparable energies to match in symmetry; the resultant molecular orbital energies can then he assessed in terms of both overlap considerations and the non-crossing rule for limited hnsis-set expansions. The symmetry-matrhing requirement then becomes straightforward
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