B. M. Deb lndion Institute of Technology Bombay 400076,. India
HOMO in
We have now come to helieve that the physical, chemical, and biological properties of a molecule are really consequences of its characteristic shape. Therefore, if we wish to understand and interpret correctly any molecular process, we must have a detailed insight into the phenomenon of molecular shapes. Naturally, our attempts to understand 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
Flgure 1 Eectron coudr, transverse forces. and term na n-cear moBadlng 10 bent or m a r A 0 2 molec. er Tns atom A and tne A-h engln are kept loxed durmg r.ch mol on*
t ons oond
1) It should exnlain how molecular shaves are hroueht ahout in
three-dirne&ioml space.
2) It should make the following quolitatiue predictions a)
Shaoes in emhd. excited. and ionized states.
x-component of
r-mmponent
of net force on n.cle~r p
x-component of ndcleor r8putslve forces on p
9.ecbon-nlceor otlroct ve fWCes on p
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 horizontally and vertically homologous molecules. Change of force constants amongst such molecules. d) Distortions in geometry due to static Jahn-Tellereffect. 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 molecule. 3) It should hack up all these qualitative predictions by quantitatiue'ones.
X-coordin~te of nucleus p
volume integration in 3-0 spoce
position ~oordinotes in space measured from p-th ""Cl9"S
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 ofthe quantitative predictions. In this article we would like to Dresent. a simnle account of a model for molecular shapes which satisfies the requirements (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 attract 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. l h ) 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 three-dimensional space play a crucial role in deciding molecular shapes. Now, how do we obtain such forces? 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
Figure 2. Electrostatic Hellmann-Feynman theorem. from the equation (Fig. 2) that if we know the three-dimensional electron density, or the wave function, of the molecule, we can calculate the electron-nuclear force without much difficulty. This simple equation is a n extremely 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 attractive forces, provided we employ for the latter the quantum-mechanical electron density. The aboue theorem embles us to visllalize chemical processes in three-dimensional 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 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 function in terms of LCAO-MO's such that every occupied molecular orbital has either one or two electrons.' Then, Based on an invited lecture delivered at the Golden Jubilee celebrations of the Indian Chemical Society. .. Calcutta, December 1973. 'This means we are excluding configuration interaction, which amounts to fractional electmn occupancy of the MO's.
occupation number (=0,1,2)
i - t h MOkeaI)
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 density as a sum of individual MO densities (see Fig. 3). Thus, if we know an LCAO-MO we can obtain the electron-nuclear 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 insensitive 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 t e & orbitallv deeenerate (static Jahn-Teller effect). In this case, the'orbi'tal to be filled last, if it could be specified unambiguously, will decide the shape. (b) The two highest occupied MO's have opposing influences on a valence angle
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 electron density in the lowest bonding MO(rb1) is concentrated 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 molecular 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
