A Novel Pictorial Approach to Teaching Molecular Motions in Polyatomic Molecules John G. Verkade Iowa State University, Ames. IA 5001 1 With the increasing pressures being experienced by our "constant volume" chemistry curricula,' pedagogical short cuts t o teaching complicated concepts are very welcome, provided they are effective and nonsimplistic. Some years ago we proposed such a short cut for teaching molecular orbital honding concepts in polyatomic molecules.2,3 The approach is a highly pictorial and nonmathematical one that is based on atomic orbital symmetries. The shapes of atomic orbitals are easily learned hy students, and they use them to generate molecular orbitals hv means of simole eeometrical relationships. No formal use bf group theor; ismade. This so-called "eenerntor orhitill" noon,ach is a funrlamentnl unifying concept for localized a s bell as delocalized bonding pictures. Moreover, the "generator orbital" (GO) approach is easily applied to the wide array of structural and honding types encountered in inorganic and organic chemistry courses a t the undergraduate level. When appropriately integrated into the curriculum, the GO method substantially reduces the time necessary to teach honding concepts in lower and/or upper level inorganic, organic, and physical chemistry courses while a t the same time teaching these concepts more effectively. In this article a orocedure is described in which the GO approach can he uhized to teach students how to generate the vibrational. rotational. and translational modes of molecules in a compfetely manner.4 Although only a few simple molecular shapes can he treated in this article, the more complex geometries of NH3, CHI, PFe-, and benzene, for example, are also amendable to the GO approach.4 Generator Orbltals and Molecular Motlons The conceot of molecular motions is convenientlv introduced by visualizing a molecule in space, say H20, as initially havina no bonds. Each of the three atoms has three deerees of freedom all of which must be translational. ( ~ o t a t i o of k atoms are not meaningful.5) Thus, there are nine degrees of freedom altogether. If we now allow bonds to form, only the moleculeas a whole can translate along the Cartesian coordinates, and so the other six degrees of Geedom must be of the rotational and vihrational types. Subtracting the three rota-
'
"Symposium on the 50th Anniversary of the Committee on Professional Training-CPT 1936-1986. What next?"; 191st National Meeting of the ACS, New York. 1986. 2Hoffman.D. K.; Ruedenberg, K.; Verkade. J. G. J. Chem. Educ. 1977. 54. 590. Hoffman. D.K.; Ruedenberg. K.; Verkade, J. G. Strucfure Bonding 1977, 33. 59.
'
Verkade.J. G. A PictorialApproach to Molecular Bonding: Springer Verlag: New York, 1986. Electronic motions are so much faster than nuclear movements that the former are not normally considered in descrbing molecular motions Moreover, nucler are orders of magntude smaller than aloms and are treated as point masses. Since points have no volume. the rotation of a point is meaningless. "otation around the internuclear zaxis is meaningless. T h e one vibrational mode follows from the 3n - 5 rule for linear molecules.
tional degrees of freedom of the molecule leaves three vihrational modes in conformance with the 3n - 6 rule where n is the number of nuclei. Generator orbitals (GO's) can be used as a device to help students visualize each of the translational, rotational, and vihrational motions. A set of atomic orbitals we will employ as GO'S is shown in Figure 1. The simple systematic rules that enable the student to sketch the nodes and lobes of the members of such an atomic orbital set (and larger ones) are described in detail elsewhere.3.4 Each of the three types of degrees of freedom has motional components along one or more of the Cartesian coordinates. Further, each of these Cartesian components can be thought of as a vector representing a component of an atomic motion taking place in either the plus or minus direction along a Cartesian coordinate. We will refer to these motions as atomic vectors (AV's). As is now demonstrated in several examples, the motions involved in all of the degrees of freedom are linear combinations of the atomic vectors (LCAV's) generated by GO's. Diatomic Z2 As an example of this class of molecules, consider the N2 molecule with the coordinate system shown in Figure 2. We note that there are six degrees of freedom: three translational, two rotationa1,G and one vibrational mode.7 Although the visualization of these motions in a diatomic molecule k trivial, we will apply the GO method to such a system in order to set the stage for more complicated molecules. GO's are placed a t the center of mass of the molecule. Starting with s, we work our way through p, etc., until six different LCAV's are generated. The appearance of each LCAV is obtained by observing how avector is "called in" by the GO. In Figure 3a we see that the s GO with a positive wave function calls in the positive ends (arrowheads) of the nitrogen motional vectors (AV's) along the z axes. We now forget about the GO since it is only a device anyway, and we observe that the motions it generates on the atoms correspond to the compression stage of the nitrogen symmetric vibration. Here, both nitrogens move toward the center of the molecule a t the same rate. We will address the justification of this "calline in" orocess at the end of this article. A negative sign in thes coswould call in the negative (arruwtails, of the nitrogen A V ' i alone. I . Aaain both nitrocen .4V's would be called with equal contrl;butions. Such motion of the molecule represents the elongation phase of the svmmetric vibration and except for the reversal of signs,-the same linear combinations of the two AV's is involved as in the compression. In cuntrast to translations, a vihrational mode (as well as a rotational motion) involves no movement of the molecular center of mass. So far we have accounted for one of the degrees of freedom. Before moving to the p set of GO's. notice in Fieure 3h and c that the svmmetrv of the s GO is not compa~blewith calling in ~ V ' s d i r e c t e halong y and x . This is because the AV's have nodal olanes at their centers where the vector sign changes from plus to minus. An s orbital has no such node. By employing a pz GO, we see from Figure 4 that the
a
Volume 64
Number 5
May 1987
411
Figure 2. Axis system fw the atomic vectors of the atoms in N2.
Figure 3. Atomic vector (AV) combinations in N, that are permlned (a) and not permined [(b),(c)] by an s W.
NA
GO
Ns
I -
Figure 4. AV combination in N. generated by a p r GO. in ihis acd future drawings, the GO center is denoted by a triangle to distinguish such a location from atom centers, whlch are represented by dots.
Figure 5. AV combinations in N. generated by the (a) py, (b)px. ( c )dyz, and (d) dxrGO.
hl'l
l5Wi
Figure 1. Drawings of lobes and nodes of atomic orbitals. The somewhat Unusual alternate ndation far these orbitals is explained in the references cited in footnotes 3 and 4.
LCAV generated by this GO represents a translation along z of the mass center. In that fieure. - . the DZ GO is re~resented by an arrow in which the arrowhead and tail denote its positive and negative wave functions, respectively. We 412
Journal of Chemical Education
could, of course, have used a drawing of a pz orbital such as that shown in Figure I, but students sometimes find the arrow notation easier to sketch. By reversing the "direction" of the pz GO, the translation in the opposite direction along z of the mass center is generated. The four GO'S shown in Figure 5 are seen to generate the remaining two translational motions (Figure 5a,h) and the two rotations (Figure 5c,d). Note in the latter two figures that the dyz and dxz GO'S are each denoted by a doubleheaded and a double-tailed arrow. It should also be recognized that no comhination of AV's can be devised that represents rotation of the molecule around its internuclear axis. Were we to use higher-order GO'S, either an LCAV already generated would be regenerated, or no LCAV would he generated. Thus as soon as the 3n different LCAV's of a molecule have been generated, the process is terminated. Recall that each GO functions only as a device, and it is discarded as soon as a symmetry motion is generated.
Table 1. Generator Table for N, GO'S
AV
Equivalence Sets
s
pz
px
py
dxz
dyz Figure 7. Molecular motions for linear YZY generated by the (a) s, (b) dxz, and (c)dyz Ws. The W s are not shown.
Before leaving the diatomic molecule, let us record in abbreviated form in Table 1 the pictorial results we have obtained. Entered in the left column of this table are the equiualence sets of AV's. From the symmetry of Nz, it is clear that vectors along the z axes on NAand NBare symmetrv oartners, and therefore these ~ a r t n e r should s be treated simultaneoisly. The same is true-for the pairs NA(x),N&) and NA(Y), Ndv). However. these two nartner sets. althoueh identical in chemical environment to one another, are notin the same environment as the NA(z), NB(z) equivalence AV set. The "m" entries in Table 1denote those motions of the AV equivalence sets that are compatible with the symmetries of the specific GO'S heading the GO column. As we will see, "generator tables" such as Table 1will be particularly useful in treating more complicated molecules.
T,bd
ylD4
la1
Ibl
Y~IPVI lkl
Figure 8. Motions for llnear YZY obtained by linear combination of (a) p b generated,(b) px-generated, and (c)py-generated symmetry motions.
Llnear YZY
In a linear triatomic tnulecule I'ZY, the twoend aromsare chcmicnlls diffrrent from the middle one (as in CY,, even if all the at& in the molecule are identical (as in il-j. hereforr: wr divide rhe mulecule into a munoaromic Z component and a Yz diatomic moiety as implied in Figure 6. he AV's generated on the Z atom by GO'S at the mass center of this atom (which is also the center of the molecule) will be those generated by px, py, and pz. The AV's generated are, of course, the three translational degrees of freedom of Z, which in fact are all the degrees of freedom this atom bas. Before generating the LCAV's in the Yz portion of YZY, let us record in abbreviated form in Table 2 the pictorial results we have obtained so far. In the AV equivalence set column of Table 2, an AV alongz belonging to Z (i.e., Z(z)) is easilv seen from Fieure 6 to form a uuioue eauivalence set in the symmetry of aiinear YZY molecuie. o n t h e other hand Z(x) is equivalent to Z(Y),and therefore these two AV's form a partne; set in this geometry. The "m" entries for the Z atom in Table 2 correspond to its translational symmetry motions. The symmetry motions for the Y2 portion of the
molecule are identical to those we have already generated pictorially for Nz, and so the completion of generator Table 2 is simple. Having generated the symmetry motions of the Z and Y2 parts separately, we now must determine what happens to these motions when we bind the Yz moiety to the Z atom. This is done by recognizing from generator Table 2 that some columns have only one "m" entry and that others have two. The symmetry motions from the former type of column are depicted in Figure 7, and they give rise to one vibration and two rotations of the molecule. The pz, px, and py columns of generator Table 2 each contain two symmetry motions. Because each member of one of these pairs has the same symmetry (i.e., each is generated by the same GO) we must take linear combinations of both of these members (e.g., Z(z) - YA(z) YB(z) = translation along z = Tz, and -Z(z) - YA(z)+ YB(z)= asymmetric vibration = "2). All the linear combinations of the pz, px, py-generated pairs are shown pictorially in Figure 8. In each case, a translation and a vibration is produced. We have now pictorially generated both rotations, all three translations and the four vibrational modes of linear YZY. The 3n - 5 rule for the number of vibrations is also seen to be verified. Although we generated these vibrations in the order given by their subscripts in Figures 7 and 8, it should be noted that by convention spectroscopists refer to the symmetric stretch in triatomic molecules (Fig. l a ) as vz and the asymmetric stretch (Fig. 8a) as
+
u3.
Figure 6. Atomic vector axis system fara linear YZY molecule. The central dot also SeNeS as the GO center.
Table 2. Generator Table for Llnear YZY GO'S
AV
Equivalence Sets
s
YAWYdx);Ydfi.YdYI
px
py
m
m
m
m
dxz
dyz
m
m
m
Z(4 Z(x). Z(v) Yd4. Ye(=)
pz
m
X
m
Figure 9. Axis system far NO2-. Volume 64
Number 5
May 1987
413
Table 3. Generator Table for Bent NO.-
v21andvt.TzI
Rz
tcl
Id)
Figure 10. Symmehyand molecular motions for NO
