We have now come to helieve that the physical, chemi- cal, and biological properties of a molecule are really con- sequences of its characteristic shape. Therefore, if we wish to understand and interpret correctly any molecular pro- cess, we must have a detailed insight into the phenome- non of molecular shapes. Naturally, our attempts to un- derstand molecular shapes have to proceed through the construction of models. But, before we proceed to design a model it would he wise to take stock of the problems which we face. An ideal model for molecular geometry should satisfy the following requirements

B. M. Deb lndion Institute of Technology

Bombay 400076,. India

Flgure 1 Eectron coudr, transverse forces. and term na n-cear mo- t ons Badlng 10 bent or m a r A 0 2 molec. er Tns atom A and tne A-h oond engln are kept loxed durmg r.ch mol on*

HOMO in

1) It should exnlain how molecular shaves are hroueht ahout in r-mmponent x-component of of net force x-component of ndcleor 9.ecbon-nlc eor on n.cle~r p r8putslve forces on p otlroct ve fWCes on p

three-dirne&ioml space. 2) It should make the following quolitatiue predictions

a) Shaoes in emhd. excited. and ionized states. bj change ofshape dn addition reactions, e.g.

BF, '+ NY, = BF, t Sl& Plamar Pyfamidd Pyiadidal Fbhmidkl

c) Change of bond angle and bond length amongst horizon- tally and vertically homologous molecules. Change of force constants amongst such molecules.

d) Distortions in geometry due to static Jahn-Teller effect. e) Transferability of shapes fmm smaller to larger molecules,

e.g., the CH2 fragment in CHI is hent, Like CHz itself. This enables one to predict shapes of larger molecules based an those of their fragments.

fJ Phenomenon of internal motions, such as inversion and rotation. Variation in harrier height from molecule to mole- cule.

3) It should hack up all these qualitative predictions by quanti- tatiue'ones.

X-coordin~te volume integration position of nucleus p in 3-0 spoce ~oordinotes

in space measured from p-th ""Cl9"S

Figure 2. Electrostatic Hellmann-Feynman theorem. These are long-standing problems and serious attempts to tackle them began in the 50's and 60's. Such attempts are expected to continue a t an increased pace in the 70's. Although we have achieved a great deal (1-6) over the years, we do not yet have a model that deals with all the reauirements of aualitative re dictions alone. not to sneak

from the equation (Fig. 2) that if we know the three-dim- ensional electron density, or the wave function, of the molecule, we can calculate the electron-nuclear force without much difficulty. This simple equation is an ex- tremely important theorem: it shows that the net force on any nucleus in a molecule is merely a balance (within the Born-Oppenheimer approximation) between the classical nuclear repulsive forces and the electron-nuclear attrac- tive forces, provided we employ for the latter the quantum-mechanical electron density. The aboue theorem embles us to visllalize chemical processes in three-dim- ensional space. This was not possible before, and so this theorem has been extensively employed in the last 15 years to investieate different chemical nhenomena (see

ofthe quantitative predictions. In this article we would like to Dresent. a simnle account

of a model for molecular shapes which satisfies the re- quirements (1) and (2) above but does not yet deal with requirement (3) (7).

The physical concept with which the model begins is simple. Let us imagine we have an AB2 molecule in three-dimensional space. If its electron cloud is disposed in such a manner that most of the charge is thrown inside the molecular triangle then the electron cloud would at- tract the two B nuclei inward (see arrows in Fig. la) . The molecule would then have a tendency to he hent. On the other hand, if most of the molecular electron density lies outside the molecular triangle (Fig. lh ) then obviously the molecule would tend to he linear. During these nuclear motions the A nucleus and the A-B bond length are kept fixed.

We thus see that electron-nuclear attractive forces in

ref. (8) for a reXew on this). Similar equations hold'for they- and z- components of the net force on a nucleus.

Let us now focus our attention on the electron-nuclear attractive force. Let us express the molecular wave func- tion in terms of LCAO-MO's such that every occupied molecular orbital has either one or two electrons.' Then,

three-dimensional space play a crucial role in deciding molecular shapes. Now, how do we obtain such forces?

Based on an invited lecture delivered at the Golden Jubilee cel- ebrations of the Indian Chemical Society. Calcutta, December

We obtain such electron-nuclear forces through what is known as the electrostatic Hellmann-Feyman theorem in quantum mechanics. This is shown in Figure 2. We see

314 / Jaurml of Chemical Education

. . 1973.

'This means we are excluding configuration interaction, which amounts to fractional electmn occupancy of the MO's.

occupation number i - th MOkeaI) ( = 0 , 1 , 2 )

x-component of electron-nuclear forces on nucleus p

Figure 3. Exampie of calculation of the total three-dimensional electron density as a sum of individual MO densities.

we can express the total three-dimensional electron densi- ty as a sum of individual MO densities (see Fig. 3). Thus, if we know an LCAO-MO we can obtain the electron-nuc- lear force generated by the electron density in this MO; the MO density is obtained by squaring the MO and so contains both square and overlap terms. This gives us very detailed insight, in terms of the occupied MO's, into a chemical phenomenon, e.g. molecular shapes. But, for the purposes of model-building, this also brings us some trouble. For instance, if we wish to predict qualitatively the equilibrium shape of a molecule, then do we have to consider the behavior of all the individual MO forces? Fortunately, the answer is no. We need consider only one such MO force, namely, that from the highest occupied molecular orbital (HOMO).

We postulate that the gross equilibrium geometry of a molecule is determined primarily by the behavior of the HOMO, when non-degenerate, with respect to the bond angle or length examined. In case the HOMO is insensi- tive with regard to a valence angle, the angular behavior of the next lower MO, if sensitive, will determine the shape. If this MO is insensitive t w , then the next lower MO is to be examined and so on. However, there might he two cases where we would not be able to distinguish a unique HOMO: (a) The molecular electmnic s t a te& orbi- tallv deeenerate (static Jahn-Teller effect). In this case, the'orbi'tal to be filled last, if it could be specified unam- biguously, will decide the shape. (b) The two highest oc- cupied MO's have opposing influences on a valence angle

Figure 4. Schematic valence MO's far AH* moiecuies (sp-basis)

and their energies cross each other at or near the midpoint of the range of valence angle studied. In such a case the molecular shape will be determined by the net influence of these two opposing MO's. Now, let us see how well this postulate does work.

We first consider a simple molecule class, namely, AH2 molecules. Their four low-lying valence MO's (using s and p atomic orbitals only) are shown schematically in Figure 4, together witb their energy order. We see that the elec- tron density in the lowest bonding MO(rb1) is concentrat- ed more in the molecular triangle. This will tend to pull the two protons inward, thereby favoring a bent molecule (see Fig. 1). The next bonding MO, $2, throws most of its electron density outside the molecular-triangle; this favors a linear molecule. The "lone pair" orbital 4s has the bigger lobe of the sp hybrid A 0 on A outside the molecu- lar triangle. But, if we remember that in obtaining the transverse force on a proton we have to consider both square and overlap terms in the MO density, then we can see that the linearizing effect of the outside lobe will be more than offset by the other terms which tend to pull the two protons inward. The net effect would be that $3

would also favor a bent molecule. The other "lone pair" orbital, b4, which is perpendicular to the plane of the molecule will obviously not exert any transverse force on a roto on.^

These conclusions are summarized in Figure 5 which,

, BENT POSITIVE FORCE

ZERO FORCE

NEGATIVE FORCE

AH2 MO ENERGY ORDER : a,< a2 < (

Figure 5. Shape diagram for AH2 molecules

for obvious reasons, we call the shape diagram for AH2 molecules. Orbitals which lie above the zero force line are in an area which favors a bent molecule while those lying below this line favor a linear molecule. The orbitals lying on the zero force line have no influence on the valence ande. The enerev order of the MO's is also indicated in " -. this diagram.

From the shape diaaram we see that AH2 molecules with 1, 2, 5, and 6 valence electrons will be bent in their ground states, while those witb 3 and 4 valence electrons will be linear. The 7th and the 8th electron go into the in- different 44 orbital. Since the orbital just below 44, name- ly favors a bent molecule, 7 and 8 valence electron molecules will also be bent. All such shape predictions for AH2 molecules are summarized in Table 1. The only ex- ception to these predictions seems to be BeHz+; approxi- mate valence bond calculations (9) indicate this molecule to have a bent configuration slightly more stable than the linear form. Later calculations (lo), however, have shown that the "linear" form is really bent.

In the horizontally homologous series BHz-, CHz, NHz,

2The force on a proton due to 4. acts only along the A-H bond.

Volume 52, Number 5, May 1975 / 315

and Hz0 the bond angle should remain more or less the same since the last two molecules differ from the former two only in the occupancy of the indifferent $4 orbital. The actual angles are 102" (calcd), 104" (singlet carbene), 104", and 104.5", respectively. In the series BHz, CHz, NHz, and HzO, the A-H hond length will decrease he- cause the "lone pair" orhitals @, and d r are progressively filled up; these will tend to pull the terminal protons towards the A atom. The hond lengths in these molecules are 1.18A, 1.11A, 1.02A, and 0.958A, respectively. If we substitute the oxygen atom in HzO by a sulfur atom then this would increase the bond length because there would now he less concentration of charge in the A-H binding region, since the sulfur atom's s and p AO's are more dif- fuse than those of the oxygen atom. However, this in- creased diffuseness means that the tail of the central atom sp hybrid in $3 (see Fig. 4) can overlap more effectively with the hydrogenic orhitals. Therefore, the hond angle in HzS would be less3 than that in H20. This is exemplified in the vertically homologous series Hz0 (0.958& 104.5"), HzS (1.334A, 92.Z0), HzSe (1.47A, 91% and HzTe (1.7.k 89.5"). We can also say that the stretching and bending force constants in Hz0 (7.76, 0.69 mdyn/A) would be greater than those in NHz and HzS (4.14,0.45 mdyn/A).

We may now proceed to deal with other molecules in a similar manner. Figure 6 depicts the schematic valence MO's for an AH3 molecule. The three orbitals 61, @z,, and dzb are bonding orbitals; the last two form a degenerate pair. $3 is a "lone pair" orbital on the central atom while the orbital 64 is antibonding between the central and the terminal atoms. If we remember what we did with the AH2 MO's then it is easy to see that the orhitals $1 and $3 would favor a pyramidal molecule; for, their electron densities tend to pull the protons-inward. The orhitals $2,

and 4zb will favor a planar molecule although they lead to different consequences as described later. A consideration of the overlap forces arising from $4 shows that this too favors a planar molecule. The shape diagram for AH3 molecules is shown in Figure 7 and the resulting shape predictions are listed in Table 2. We see again that there are two apparent exceptions to our predictions, namely, the seven valence electron molecules CH3 and NH3+. The

3 This can he understood in the fallowing way. If the A 0 on the A atom were concentrated as a point charge on A then it would exert no transverse force an the terminal protons. As the tail (positive part) of the sp hybrid on A becomes more and more dif- fuse it can overlap more and more with the hydrogenic orhitals. This overlap density tries to pull the protons inward.

CH, radical is planar or nearly planar (11) and the NH3+ ion has been calculated (12) to he planar.

In the horizontally homologous series LiH3+, BeH3, and BH3 the bonding orhitals $2. and dab are progressively filled up. Therefore, the out-of-plane bending force con- stant will increase from LiH3+ to BHa. Again, because of incrgased occupancy of the "lone pair" orbital 63 the bond angle in NH3 (106.6') and H30+ (110.4") is smaller than

AH3 MO ENERGY ORDER 4, < & , OEb< o3 < '$4

Figure 6. Schematic valence MO's far AH3 molecules (sp-basis).

PYRAMIDAL IPoSITNE FORCE

AH3M0 ENERGY ORDER a,< @20 ,aEb< t$ < f14 Figure 7. Shape diagram for AH, molecules.

Table 1. S h a ~ e Predictions For AH. Molecules

Number of valence Ground state Eroited state Apparent

electrons geometry Examples geometry Examples ereeptiona

1, 2 Bent Ha*. LiHzt, HeHz'+ ~inesr (HOMO 0 3 3, 4 Linear BeHlt, BeH.. HeH1*, BHz+ Bent (HOMO 08) BeHl BeH9+ 5-8 Bent CHI, NHI, HaO, BHI-, BH,, Bent (HOMO 04) CH,, NH?

AlH2. N&+, NHz-, FHz+ or Linear (HOMO $3 BHr

Table 2. Shape Predictions For AH8 Molecules

Number of Ground state Excited state Apparent valence ektrons geometry Examplea geometry Examplea exceptions

1, 2 Pyramidal Planar (HOMO dr., drs)

3 6 Plansr L i i a + , HeH,+, GsH*, BeHz, Pyramidal CHa*. BeHs. BHs (HOMO 0d

7, 8 Pyramidal CHa(?), NHI. HatO, P b r CHI: SbH& NHatC), SiHs (HOMO h) NHz CHI. NHI+

9. 10 Planar

316 / Journal of ChernicalEducatbn

that in CH3 (120°?). In the series LiH3+, BeH3, BHs (1.2211. calcd). CH. (1.0811). and NH3 (1.0211) the bond length'will decrease because here bonding and "lone pair" orbitals are filled up more and more. But, in the vertically homologous series NH3 (1.0211, 106.6'), PH3 (1.42A, 93.5"). and AsH3 (1.5211, 91.8") the bond length will in- crease and the bond angle will decrease because the cen- tral atom AO's hecome more diffuse (cf. AH2 molecules). For the same reason the inversion barrier in NHa (-8 kcal/mole) will be higher than that in CHa- (1.22 kcal- /mole calcd). 4 3

Flgure8. Doubly degenerate MO'sfor AH1 molecules

A glance a t the @?, and @a* orbitals (Fig. 8) reveals how we can predict the nature of the static Jahn-Teller dis- tortion (see ref. (8)) in a molecule like LiH3+. Let us com- pare LiH3+ with the hypothetical pyramidal molecule LiH32+. When we feed an electron into LiH32+ this can go into either mzn or &,, resulting in a planar molecule. If the @zn is occupied then the Li-HI bond length will he shorter compared with the other two, because the LCAO coefficient of the HI atomic orbital in @la is twice that of either Hz or H3 atomic orbital. If the $20 orbital is occu- pied instead, then this would result in a T shape in which the Li-Hz and Li-H3 bonds will be shortened compared with the third. Thus, either occupancy leads to a Czu structure for the LiH3+ molecule.

Figure 9. Change of shape of BF, on being added to NH..

Figure 10. Transferability of AB2 shapes10 AB4 and ABr molecules.

Let ,u now consider the addition reaction between the planar BFJ molecule4 and the pyramidal NH3 molecule. This reaction may be regarded as one involving the trans- fer of one electron from the doubly occupied "lone pair" orbital m3 in NH3 to BF3. This means that in the addition compound BF3 - NH3 the BF3 fragment will be pyram- idal whereas in the NH3 fragment the bond angle is ex- pected to be larger than that in the reactant NH3 mole- cule. However, a t least two theoretical calculations have assumed that the geometry of NH3 does not change as a result of this reaction (13) (see Fig. 9).

We may a h try to predict the shapes of larger mole- cules based on the shapes of their fragments. Consider a molecule like CIF3. This will be planar.4 If we imagine that the CI atom and one F atom in the molecular plane remain fixed.then the overall shaoe will be determined bv the linear or bent shape of the remaining ClFz fragment. This mav he reearded as a 22 valence electron fragment, taking one electron from the remaining Cl-F bond. The ClFz fraement is thus linear.' Therefore, the C1F3 mole- culehas> T shape. In a similar manner, one can predict the shapes of many AB, and AB5 molecules. In the former case we make a choice between a tetrahedral a n d a square planar configuration. Figure 10a depicts the nuclear mo- tions which take the tetrahedron into the square planar form. We see that the motions of the nuclei 1 and 2 will

According to the present model AB, molecules with 24 and 28 valence electrons will he planar, whereas those with 25 valence electrons will he pyramidal (see ref. ( I ) ) ; AB2 molecules having 21 or 22 valence electrons are linear, whereas those with 20 elec- trons are bent.

Table 3. Shape Predictions For Larger Molecules Based on Those For Smaller Molecules

No. of Ground Corresponding No. of Gmund d e n - state parent valence state

Fragment Example electron. geometry molecule electrons geometry Examples

CClz 18. Bent 32 Ts, Dad CCL, 10,; S e O P , TiCL, FeCL; ZnCL-1, AICIA-H~CL.'-CU(CN)I",

A B 1 AB. BF.-, Be%- , XeO., POClr, NF.+, FINO,PO<I-

SF8 20 Bent 34 Cb SFc XeOaF,, IOIFP- XeF. 22 Linear 36 Square XeF,, BrF..

plsns10

-PC12 20' Bent 40 Trigonal PCL, Sb(CIldrBn, bipyramid NbCL, (CHdPF., CI.PFa, SOF.,

Volume 52, Number 5, May 1975 / 317

be little affected by the presence of two electrons, say, in the remaining two A-B bonds because the latter lie in a plane perpendicular to the plane of motions of 1 and 2. Therefore, whether an n valence electron AB4 molecule will be tetrahedral (regular or distorted) or square planar depends on whether the corresponding (n - 2) valence electron ABz fragment is bent or linear, respectively. Likewise, for AB5 molecules we chwse between a trigonal bipyramid and a tetragonal pyramid form. As shown in Figure lob, if we fix the nuclei 3, 4, and 5, then the trigo- nal bipyramid passes into the tetragonal pyramid by mov- ing the nuclei 1 and 2. Since two A-B bonds are perpen- dicular to the plane of nuclear motions the two electrons from them do not affect significantly the motions of the nuclei 1 and 2. Therefore, whether an n valence electron AB5 molecule will be a trigonal bipyramid or a tetragonal pyramid depends on whether the corresponding (n - 2) valence electron ABz fragment is bent or linear, respec- tively. Table 3 lists the resultant shape predictions for a number of AB4 and AB5 molecules. From this table we see that a number of transition metal compounds also fits inta this scheme. In such cases we count the number of valence electrons of the transition metal in the following way: Let the metal have a d' configuration in the com- pound, where i = 0, 5, and 10, and let n be the number of additional electrons the metal has over the preceding inert eas atom. Then (n - il is the number of valence electrons - of the transition metal concerned. This procedure amounts to assumine that the soherical d sub-shell of the transition metal is "indifferent-to molecular shapes. In Table 3 Ti, Os, and Nb are assumed to bave a do configu- ration, Fe a dJ configuration, and Zn, Hg, and Cu a dl0 configuration (see ref. (3), pp. 178, 188).

In the case of 34 valence electron AB4 molecules such as SF4, TeC4, etc. i t is not possible to distinguish qualita- tively between a Td, D z ~ . or CzU configuration. However, since XeOzFz and IOzFz- are expected to have a Cz, structure one may predict that other 34 electron mole- cules such as SF4 and TeCll will also have Cz, symmetry. In the case of SF4 both the SF2 fragments are bent since they are 20 electron AB2 molecules (see footnote 4). In the case of the AOzFz molecules (A = Xe, I-) the 22 electron AF2 fragment will be linear whereas the 20 electron AOz fragment will be bent.

We give below a number of other examples where this idea of transferability of shapes of fragments in larger molecules may be exploited (see also refs. (2) and (14) for the familiar Walsh rules)

1) B(OH)S: This, a 24 valence electron AB3 molecule, will be pla- nar with the three oxygen atoms at the vertices of an equilat- eral triangle. A (HOB < I fraement in this molecule will be ~~ ~~ " . ~. bent smre this ma). he regarded as a 12 valence electron HAB molecule ( 7 , The overall rnolcculsr shape is thus

2) N3H: This is a planar molecule. The (-Nd fragment will be linear since this may be regarded as a 16 electron ABz or ABC molecule 17). The (H-N=N4 fraement will be bent as this . . . " is a 14 electron HAB molecule (7). The overall molecular shape is

/w* H

3) (CH.d(CH2)Li: Taking one electron from each C-H bond this is a 14 electron ABC molecule. Hence, the Li-C-C frame is linear (7).

4) CH; = CHBr: This is an 18 electron ABC molecule. Hence, the Br-C-C frame is bent (7).

5) C2H1: This may be looked upon as a 12 electron HAAH male- cule and so will bave both cis and trans isomen (7). Since one cannot distinguish between the two hydrogens on a carbon atom, this means tbat CzHl is planar.

6) CH2=CH-CH=CHa: The (CH1=CH-CH=) fragment is bent since this is an 18 electron ABC molecule. Since (a) one cannot distinguish between the two terminal carbon atoms, and (b) there are two ways of bending with respect to the fourth carbon atom. the molecule will have both cis and trans isomen.

7) Excited CzH2: The (HCC-) frame in the ground state is lin- ear since this is a 10 electron HAB molecule (7). On excitation one electron goes inta an orbital favoring a hent molecule. Since one cannot distinguish between the two terminal hydm- gen atoms, excited acetylene should have both cis and trans isomers. The cis form, however, has not been detected.

8) HCONH2: The (H2NC--) fragment in famamide is a 14 elec- tron H2AB molecule. Hence, it is pyramidal (7). Similar con- siderations may be helpful in dealing with conformational problems in polypeptides.

It is also possible to predict the shapes of reactive inter- medi'ates in organic and biochemical reactions. This is helpful for those species which are tw short-lived for ex- perimental observation and for which only gross shapes need he known for understanding reaction mechanisms. One such example is the vinyl cation, CHz=C+H, which has a linear (HCC=) frame because it is a 10 electron HAB molecule. However, the 11 electron vinyl radical, CHz=CH has a hent (HCC=) frame (15).

Let us pause here for a moment and think over what we bave done so far. I t seems quite remarkable that we can extract so much qualitative information out of this model by putting in so little. The simple idea of the HOMO densi- ty being primarily responsible for molecular shapesprovides a consistent uiew of the entire field of molecular geometry in three-dimensional space. In fact, all the requirements of qualitative predictions listed in the beginning of this article have been dealt with. However, we have not made any quantitative predictions. We hope that this will be possible in future.

A question, however, arises a t this point: Why should the HOMO density be of such importance in molecular geometry? We can look a t the problem in the following way: Consider a molecule in its equilibrium configuration. Now let us move the nuclei away from this configuration. Those parts of the molecular electron density which are tightly bound to the nuclei will move with the nuclei rig- idly. Such densities are of little importance in molecular

,shapes. Those parts of the molecular density which are loosely bound to the nuclei will not be able to move along with the nuclei; they will either precede or lag behind the nuclei. Such densities play a significant role in deciding molecular shapes. The density in the HOMO, being most loosely bound to the nuclei, will follow nuclear motions least readily. I t is not surprising, therefore, that one can make such successful shape predictions based on the HOMO postulate. It is also obvious tbat this simple ap- proach to the prediction of molecular shapes has great pedagogic value.

Literature Cited

1972. (4) Pesmon. R. G.. J. Amer Chem Soa.. 91. 4947 (19691; J. C h m . Phyr., 52. 2167

(1970). (51 Schnuelle. G. W., and Pam, R. G . , > . A ~ P ~ C h m . Boe, 94,8974 (1972). (61 Nskatsuji. H.. J. Amer Chem. Sor., 95, 345, 354, 2084, 689411973): 96. 24, 30

(1974). (7) A detailed and formal account of this work may be found in Deb. B. M., J Amer.

Cham. So% 96, 2030 (1974); Dab, B. M., Sen, P. N., and Boa% S. K.. J. Amer. Cham. Soc.. 96.2044 (19741.

(8) Deb. 9. M.. Re". Mod. Phys., 45, 22 (1973): meslsoref. (6). (9) Poshunte,R.D.,Haugen,JiA., and2etik.D.F.. J. Cham.Phya.. SI,3343(19691.

318 / Journal of Chemical Education

(101 Parhusta, R.D.. Liberles, A., and Kllnt, D. W., J. Chrm. Phys., 55,252 (19711. cev. M.. J. Chem. Soe. Forodrody II. 561 (19731: lbl Archibald, R. M., Arm8tmng, (111 Henbcr%,G.,pn'vsCCmmmlmiefion. D. R., nndPerkins, P. G., J. C&m. Soc.I'omdomdy, 11.1793 (1973). (1%) Kari. R. E.. and Csizrnadia, I. G.. In6.J. Qunlvm Chrm.. 6.401 (19721. 1141 Cou1son.C.A.. andDeb,BM..Int. J Quantum Chem.. 5.411 (1971). I131 (a1 Barber. M., Connor. J. A., Gueat. M. F., Hillier, I . H. , Sehwarz, M.. and Sta- I151 Lathan. W. A,, Hehre. W. J.. and Paple, J, A,, J. Amar. Chon. Soc.. 93. 8CB

(19711.

v o l m 52, Number 5. May 1975 / 319