The Valence Bond Interpretation of Molecular Geometry
Derek W. Smith School of Science University of Waikato Hamilton. New Zealand
Simple models which explain the shapes of molecules have played an important part in the practice and teaching of chemistry. The popular VSEPR (Valence Shell Electron Pair Repulsion) model ( I ) , is easy to handle and is usually reliable in its uredictions. However. there are a number of well-known exceitions to the VSEPR rules. For example, some alkaline earth dihalide molecules are bent (2).although the VSEPR m d r l suggests that they should be linwr. N'hilr most AH.. species withour lone pairz on the central ntont are trigmal hilpramidal, ns predicted hy the \'SEPR model, some arr square pyramidal. I'crhapa the mcgst seri(ms ohj~ctionto thr \'3EPR model rat least in thr form in whirh it is usunll\." Dre. . sented to students) lies in its concentration on repulsive forces as dominant in determining molecular geometrv, ignoring complctrly the hmding forces (which must I>(. yriuter in n stable molrrulel. A theorv whirh describes the honding in H molecule, while a t the same time explaining its geometry, would be preferable. Simplified molecular orbital (MO)approaches to molecular geometry have been advocated (3-6), but these are more difficult to apply than the VSEPR model. The purpose of this paper is to show that the valence bond (VB) theory not only provides an attractive means of describing the bonding in a molecule but can also explain its geometry. The ideas behind the following discussion are not new; they follow directly from the principles laid down by Pauling and others (7) in the 1930's. The Energetics of Molecule Formation The standard enthalpy change for the process (1)
In the valence state of A (denoted by A*), each of n hybrid orbitals, directed along the hond axes, is singly-occupied; any remaining valence electrnns of A will be in doubly-occupied h e pair orbitals. The standard euthalpy change for the Process (2) AB,(g) A * k ) t nB*(g) (2) is the total intrinsic bond energy of AB,, and exceeds the total thermochemical hond energy by the amount of promotion energy required by the atoms to attain their valence states. The promotion energies for the B atoms are usually disregarded; atoms such as hydrogen, halogen and oxygen can he considered to he in suitable condition for bonding in their ground states. Thus the promotion energy involved in the formation of AB, can he taken to be that of A alone. It then follows that the most stable configuration of A& is that which affords the best compromise between the total intrinsic bond energy and the promotion energy. These two factors must now he considered in more detail.
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Promotion Energies Consider an atom in a valence state, for which we have $ 2 , . . . &, using the constructed a s e t of n hybrid orhitals 91, s,p, and d orbitals in the valence shell of the atom. The orbital 9j can he described as sniphido, where the superscripts indicate the relative amounts of s,p, and d character in the hybrid. The valence state arises from the configuration srp:prd", where x n;a;l(ai + bi + c;) y =
-
AR,,(g) A(g) + nR(g) (1) is the total th~rmochemicalbo'nd energ.y of the molecule AB,. When divided by n , this gives the mean th~rmochemicalbond energy in AB,. Since the standard entropy of AB,(g) is not strongly dependent on its geometry, the most stable configuration of AB, will he that which maximizes the total thermochemical hond energv.
Valence States (VS's), Valence State Conliguratlons (V.S.C.'s) Promotion Energies (P.E.'s) and 29 for Various Values ofO(=~-A-B) Molecule
0 (deal
V.S.
V.S.C.
106 1 Journal of Chemical Education
P.E.
2 SZ
z
n,b;l(a; + bi
= 13 n;c;l(ai + b;
+ c;)
+ c;)
and ni is the occupation number ( O , l , or 2) of $; in the valence state. If the ground state of the atom arises from the configuration s2pm,the promotion energy P is given by (3),
+ ( r n - >)P,d (3) are respectively the energies required to
P = ( 2 -xIP"d
where
P,d
and
P,,d
where P,, is the energy required to promote an electron from s to p . Thus for any valence state, we can write down the promotion energy in terms of P,,, PSd and Pdp. However, the numerical evaluation of these auantities is not easv. Promotion energies are usually obtained from atomic spectra (8),hut ~ r e c i s ecalculations are not straightforward (9).and values quoted in the literature are best regarded as rough estimates (10). Intrinsic Bond Energies We need to determine the relative magnitudes of the total intrinsic bond energies for the possible configurations of a molecule. This can he done by invoking the concept of orbital "strength," introduced by Pauling (71, The strength S of an orbital is the magnitude of the wave function along a specified axis, relative to that of an s orbital having the same principal quantum number. The intrinsic hond strength obtainable from an orbital is proportional to S2.Thus S for an s orbilnl is, by definition, equal to unity. For a p orbital directed along the bond axis, S is 3'12. For a d z 2orbital, S is 5'12 along the
z-axis and 51/2/2 in the xy plane. For a d,z-,z orbital, S is 1 5 ~ / ~along / 2 the x - and y-axes. These values are strictly applicable only to hydrogen-like orbitals. A hybrid orbital of the typesp" takes the form (1 n)-'l' (S n1/2p). The strength S for such a hybrid is given by (5). S = (1+ n)-'12[1+ ( 3 r 1 ) ' ~ I (5) Similar formulae can be derived for hvbrids involving d orbitals, depending on which d orbital(;) is or are beingwed, and on the orientation of the honds with resoect to the d orb i t a l \ ~ )The . tut:d ~ntrinsicbond energy tor a-givm gromrrry w~llhe oroourtion to 2.5". the s ~ l m m a t i mbeing perlmned over alith' hybrid bond orbitals. We can now determine how the promotion energy and the total intrinsic hond energy for any molecule depend on its geometry, and hence deduce the geometry which will maximize the total thermochemical bond enerev. "" We shall first consider AB.E, molecules, where A is a main group atom with the around state electronic confiauration s2p"+2m-2. B is a univaient atom or radical, and E represents a lone pair. For n rn < 4, only the s and p valence orbitals of A are considered. For n rn > 4, d orbital participation is invoked. We then discuss briefly the extension of the model to molecules containing multiple bonds. Arguments relating to molecules are deemed to he also valid for isoelectronic ions.
+
+
+
+
AB2 Molecules
The valence state for an AB2 molecule is ( ~ p " ) ~ ( s p " ) l , where n = -sec 0 , 0 being the B-A-B angle. Promotion energies and ZS2for various values of 0 are given in the table. The minimum orornotion enerw is obtained with fl = 180' and n = 1. As O H i lowered from 1800, the intrinsic bond energy increases sliahtlv. - .. reachina a maximum a t 109.5", while the promotion energy increases steadily. If the total intrinsic bond energy is sufficiently large compared with the promotion energy, such bending might be energetically favorable. This would explain the observed geometries of the alkaline earth dihalide molecules. The value of P, falls down the alkaline earth group atoms from about 300 kJ mol-I for Be to about 170 kJ mol-I for Ba, while the total thermochemical hond energies range from 1200 kJ mol-I for MF2 to about 700 kJ mol-1 for MI? (11). All the barium dihalide molecules are Iwnt. nc >ireSrF?, SrCI,, CaF2, ;and MgF,, ahile the rest are h e m . Parricioation bv (n - lld ort1i1.d~ as well as nz and n p orbitals would also f a k r bending; the best s d hyhrids are at rieht angles, while the best spd hybrids make an angle of 133" the observed geometries can be kxplained qualitatively with d orbitals. For Zn, Cd and Hg, P, is higher than for the alkaline earths, about 400-500 kJ mol-l; these atoms form linear dihalide molecules.
(3.ow ever,
AB. and AB,E Molecules
The dependrncc of the prnmotim energy and total intrinsic on the H - A - Bangle a is shown bond energit,; in A H , molc~c~~ls. in the tahlv. A deviation from plnnarity to\riird a n\.ramidal configuration with 0 = 109.5' w31 result in a negligil& increase in the total intrinsic bond energy, a t the expense of a considerable increase in promotion energy. Clearly, AB:, molecules should he planar, as found experimentally. For AB2E molecules (in their sinelet states). the valence state is (sor')' ( ~ p " ) ' ( s p ~with ) ~ , rn = 2/(n - 1). The promotion energy falls and the total intrinsic bond energy increases as the molecule is bent toward the tetrahedral angle of 109.5°. If the molecule is further bent toward a bond angle of 90°, the promotion energy continues to fall while the intrinsic hond energy hegins to decrease. The best balance between these two will be struck somewhere hetween 90 and 109.5"; the exact value may depend on other factors, such as nonhunded atom interactions. According to the VSEPR model, the interbond angle s h d d he somewhat less than 120°; experimentally, molecules such as CH:, CF2 and SiF2 have hond angles in the range 100105".
AB3E and AB,E, Molecules - . For an ;\H, mnleculi~,the tt.tr;lhedml sp.. h ~ h r i d cmsrirute s the hrsr p~swhlcset; ncmrq~~i\,alent .
