Crystal Field Theory and the Angular Overlap Model Applied to Hydrides of Main Group Elements E. A. Moore The Open Unlversit~.Walton Hall, Milton Keynes, MK7 6AA. England Many students have difficulties understanding molecular orbital theorv. even of molecules an simple as H?O. In transition metal chdmistry a t the elementary level, we use orbital desmintions that focus on the orbitals of the central atom. The descriptions have the advantage that many properties of molecules can he explained without the full, often fearsome, MO diagram. rrithis paper we explore how crystal field theory and the angular overlap model can he applied to very simple molecules, the di- and trihydrides of main group elements, which can then be used to introduce such concepts as bonding orbitals, MO diagrams and Walsh diagrams. In addition, application of these methods to the main group hydrides can serve as an introduction to the same methods applied to transition metal complexes. Crystal Field Theory What we shall refer to as crystal field theory is a model in which ligands or atoms surrounding a central atom (or ion) are regarded as point charges. We do not use the mathematical formulation in terms of perturbation theory that is used in (for example) analyzing transition metal spectra, however, but estimate qualitatively how these charges affect the central atom orbitals. Our crystal field model for the hydrides of elements on the right of the periodic table regards the central atom as an anion (e.g., 02-) surrounded by protons. The charge on the central atom is determined by subtraction of the protonic charge--thus NH2 is regarded as N2- plus two protons and NH2- as N3- plus two protons. Note the contrast with tran-
sition metal chemistry in which we consider a metal cation surrounded by negatively charged ligands (1, 2). Both the hydrides and the transition metal complexes are regarded as ionic and held together by electrostatic attraction. In the familiar transition metal crystal field theory, the effect of the surrounding ligands is to raise the energy of the central atom d orbitals and cause them to split into several energy levels depending on the symmetry of the complex. For the hydrides, because the protons attract (rather than repel) the electrons on the central atom, the p orhitals will be lowered in energy. Like the d orbitals, though, the p orbitals will split into several levels determined by molecular symmetry. We can thus illustrate the lowering of the p orbital energy uponforminga molecule and the formation of several molecular energy levels from the single, triply degenerate, atomic level. We show below the production of very simple molecular orbital diagrams for di- and trihydrides using our crystal field theory. Dihydrides, AH2
AH2 molecules may be bent or linear. Taking the neutral molecules CH2 and Hz0 as examples, we can discuss the effect of the protons on the 2p orbital energies. The interactions of the 2p orbitals with the protons (considered as point positive charges) are shown in Figure 1.For a bent molecule, the 2p, orbital is out of plane and hence little affected by the protons; it is essentially nonbonding. The 2p, and 2p, orbitals are lowered in energy; these are bonding orbitals. For a slightly bent molecule, a 2p, electron is on average closer to
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VSEPR (5),the electron pair picture would predict a triplet witba bondanglegreater than the tetrahedral angle (103.5") or a singlet with a bond angle less than 120°. Although 140' is greater than 109.5', the deviation is rather large, even allowing for single electrons requiring less space than electron pairs, and the experimental ground state would be better described as close to linear than as a distorted tetrahedron. Similarly 102' is a lot less than 120". A hybrid orbital description of the singlet would again give an angle close to 120° as sp2hybrids would be used. For the triplet we could obtain a more accurate description by assuming that the CH bonds were formed by sp hybrids and that the odd electrons occupied nonhybridized p orbitals. Can we do better using crystal field theory?
Figure 1. Interaction of 2p orbitals w l h protons.
Figure 3. Energy level diagram for C H A a ) linear, (b) bent.
Figure 2. Crystal field energy levels for AHt-(a) bent. (d) extensive bending.
atom, (b) linear. (c)slightly
the protons than is a 2py electron. As the molecule bends more the protons become closer to a 2py electron and the 2py level drops below the 2p,. The energy level diagrams are shown in Figure 2, in which the qualitative resemblance to the conventional MO diagrams of the dihydrides is apparent. For a linear molecule, the 2p, electron is strongly attracted by the protons, but the 2p, and 2py electrons are hardly affected; they are nonbonding (Fig. 1). The crystal field energy level diagram for linear AH2 thus shows one bonding and two nonbonding levels as illustrated in Figure 2. HzO has enough electrons to fill all the levels shown in Figure 2. The pattern of levels for the bent molecule fits very nicely with the observed photoelectron spectrum (3),which shows an almost nonhonding peak and two bonding peaks of different energy in the region where we expect to find peaks for orbitals containing 02p. The nonbonding peak can be associated with 2p, in Figure 2 and the bonding peaks with 2py and 2p,. Contrast this with the usual elementary picture of electron pair bonds (Lewis structures) or hybrid orbitals, which eive two eauivalent bond and two eauivalent lone pairs. These descriptions would imply two ddubly degenerate levels eivine onlv two peaks in the P E spectrum. CH2 is parti&lariy interesting and is & example of the special advantage central atom methods have for open shell molecules. CH2 causes problems for simple theories such as electron-pair bonds or hybridization, and some form of molecular orbital theory is necessary for a satisfactory description of this molecule. In terms of electron pair bonds, two CH bonds can be formed, but this leaves two nonbonding electrons in the valence shell of carbon. These two electrons can be mired. eivine a singlet. or un~aired.formine a triplet ( 4 ) ;he Rround stateof CH? &, a triplet state. l?xperi&taiy with a wlde bond anele (about 140") and the first excited state is a singlet with a smaller bond angle of 102'. Using 858
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CH2 hasonly four 2pelectrons. From Figure 2 these would he exoected to eo into 20, and 2 ~for , a bent molecule leaving the nbnbonding 2p, emGy ( ~ i3).i For linear CH2, however, there is onlv one bonding orbital, and two electrons have to go in nonbo"dingorbitak According to Hund's rule, the two nonbonding electrons have to have parallel spins as shown in Figure 3. We thus have to introduce another factor, the spin state, into our crystal field theory when discussing CH2 because the linear and bent formshave different spin multiplicity, the analogy with transition metal chemistry now becomes more apparent. Let us start with the energy level diagram for bent CH2 with a bond angle close t o 180". The effect of reducing the bond angle from 180' is to remove the degeneracy of 2p, and 20.., as in Fieure 3. If the reduction in angle is only small, however, t h i n the energy difference betwe& 2p, andzp, will be small. The mound state of CH? is determined by the interplay of theorbital and the spinkxchange energies. The areument can be quantified if we define two crystal field s$itting parameters 61 (the 2p, - 2py energy difference) and 69 (the 2 ~ , 213, enerwdifference) and aspin pairing energy that represe&ts the-difference in exchange energy between paired and parallel spins. The orbital energies of the triplet and singlet can be compared using the crystal field stabilization energy of the p orbitals. The crystal field stabilization enerw (CFSE) of an electron in a D orbital is the energy by w h i z i t is stabilized relative to the center of gravity of the set of three n orbitals. The total CFSE of the singlet state is greater th& that of the triplet by al. ~ o w e v e r , t h esinglet has one more air of s d n s than the triplet, and so its exchange e n e r k i s large; by an amount P. The total energy difference between singlet and triplet is thus (P- 61). If the spin pairing energy, P, is larger than 61, then the triplet state will be morestable; if 61- is larger, then the singlet state will be the more stable. The ground state bond angle, 140°, is still close enough to 180" for 61 to be small. The exchange energy is then the dominant term. In the first excited state, however, the bond angle is reduced to 102'. 61 is now much larger and the
.
~
trast to the pyramidal form of NH3. BH3 has two less electrons than NH?, and so the hiehest nondeaenerate level in form &d nonbonding Figure 5, bondkg for the for the olanar form, is unoccupied. Looking a t Figure 5, it can be seen that in t h e planar-form the doubly degenerate level has agreater stabilization energy than in the pyramidal form. The more stable form of BH3 will thus he the trigonal planar form. No T-shaped hydrides are known, and molecules in which this shape might be important (e.g., the hypothetical FH3 or the oossihle mass-s~ectrometerion LiHq+) need extensions of okr crystal field Aodel either, becawithe central atom is electro~ositiveand hence should be reearded as Mf surround& by hydride ions or because there are more than six electrons to fill the 2p levels. T-shaped hydrides can be handled very simply, however, using the angular overlap model, which we discuss next.
Figwe 4. Axes for planar AHl mlecuie.
Angular Overlap Model (AOM)
The angular overlap model has been used extensively in transition metal chemistry in recent years (6-11).I t has the attraction of heine a functional eroun a .. ~ o r o a c hto inoreanic chemistry and has been particuiarly useful for l o w - s y k e trv com~lexes.This model is not confined to transition metalche&istry, but, with the notable exception of Burdett's book (12).it has not been used much for main group molecules. As we are concerned with an elementary introduction to MO theory, we shall only consider a honding here. In the simple a-bonding model, the interaction between a central atom orhital and a ligand a orbital can be approximated as @S2,where @ is a measure of the strength of interaction between the orhitals and S is the overlap integral between them. The combination of central atom orbital and ligand orhitals produces a bonding orbital stabilized by @S2 and an antibondine orhital destabilized bv BS2 relative to the central atom oibital. If there is more than one ligand, then the contributions from each lieand are added. Thus the total stabilization energy of the honding orbital is x,3S,'. If all thesurroundine lieandsare identical. then the radial Dart of S is the same fdr ail ligands and for the rhree 2p orbiials, and theonlv variahlesare theaneular Dartsof the 2~orbital3 of the ligands.- he contribution>^;^ for a and the ligand on the z axis overlapping with 2p, is defined as BS&. Each term @Si2 can then be written in terms of this standard term ,4S2 and the square ofthe 2p angular functions. @S,2is reeardegas an emnirical oarameter to he obtained from spktroscopic or othkr data.*~he stabilization energy due to a lieand with aneular coordinates 8. d is as eiven in Table 1. $owlet ussee cow thismodel can beapplie:to the hydrides. ~
Fleure 5. Crystal tield energy levels tor AHs-(a) dal.
atom. (b) planar, (c) pyrami-
excited state is a singlet. These two states, then, illustrate the interplay of orhital and spin exchange energies in a very simple molecule. The idea can then he transferred, in undergraduate teaching, to transition metal complexes. Trihydrides, AH8
Three possible geometries can he considered for the trihydrides-trigonal planar (D& pyramidal (C3"), and T shaped (Cz,). We consider first a trigonal planar molecule. The axes are defined as in Figure 4 with the z axis out of plane along the C3 axis and the x axis along one A-H hond. An electron in a p, orhital will scarcely be affected by the three protons and so will be nonbonding. The p, and p, orhitals are equally affected by the protons and so have the same stabilization energy. Thus, as shown in Figure 5, the crystal field diagram shows a doubly degenerate bonding level and a nondegenerate nonbonding level. If the protons move out of planeto form a pyramidalmolecule, the formerly nonbonding p, orbital becomes bonding. The p, and p, orhitals remain hondine hut their stabilization enerw decreases. The crystal fie12 energy level diagram for thepyramidal form is c o m ~ a r e dwith that for the trieonal ~ l a n a r and form in Figure 5. ?he most familiar trihydrideis N H ~ 'this is pyramidal. NH3 on our crystal field model will contain N3- and thus have six 2p electrons. These will fill the three levels shown in Figure 5c. As in MO treatments, the highest occupied molecular orhitals (HOMO'S) of NH3 are a nondegenerate weakly bonding orbital and two degenerate bonding orbitals. The peaks in the P E spectrum (3)show that this model is onalitativelv correct. As with Hp0 this can he contrasted Gith the v&nce hond model whilh, for N H ~predicts , a lone pair and three equivalent N-H bonding orhitals. If NH3 is compared with the unstable BHs it can be seen why the latter will adopt a planar trigonal structure in con~~~
~
.
~~~~
Table 1.
Stablllzatlon Energy ol Zp Orbltals due to a Llgand at 8,
A orbital
stabilization enerw18sg sin2 0 cos2 $ sln2 t sinz $ cn.2 R
2P. ~ P Y 20.
Dihydrides, AH2
Let us take Hz0 as an example. With the x axis out of plane and the y axis bisecting the HOH angle, the two H atoms are a t angles of 37.75O and 142.25" to the z axis ( 8 ) and a t 90' to the x axis (@)for a bond angle of 104.5O. The bonding orbitals of Hz0 are thus stabilized by 0 (Zp,), (0.375 BS& 0.375 @S&)(2pJ and (0.625 PSi, 0.625 @$,) ( 2 ~ ~ ) . These splittings agree with the crystal field picture and qualitatively with the PES data (3).The agreement with the PES ionization energies is not quantitative (observed values 2p, 12.8 eV, 2p, 14.8 eV and 2p, 18.6 eV) since the simple
+
+
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model neglects, for example, the interaction of the 2s orbital on oxygen, which is of the same symmetry as 2p,. Introducing 2s overlap raises the 2p, honding orbital and brings the ratio of the stabilization energies of Zp, and 2p, closer to the observed values. However, for the general AH2 case at the elementary level it is probably best to stick to the simple 2p a-bonding model. As well as constructing energy-level diagrams for molecules of known or assumed geometry, the student can use this very simple model to investigate the variation of energy levels with geometry as in Walsh diagrams (13).For a bond angle 0'. the bonding orbitals will lie at 0 (Zp,), (2 cos2(8'l 2)pS&) (2pJ and 2 sin2(O'/2),8Sio (2pA The variation of energy with 0' can he plotted as in F~gure6, giving a Walshtype diagram. In the extreme case of a linear molecule, there are two nonbonding orbitals (2p, and 2p,), and 2p, is stabilized by 2pSi,. It is instructive to use Figure 6 to discuss the
For open-shell molecules such as CH3 and BH3, it is instructive to discuss the relative stabilities of planar and pyramidal forms (Fig. 7). The total stabilization energy for the closed shell molecule, NH3, is 6PS& for both the planar and pyramidal forms on this model. CH3 and BH3 have one and two electrons, respectively, less than NH3. In the pyramidal form there must be vacancies in bonding orbitals. The total s t a b ' i t i o n energy of pyramidal CH3 and BH3 is thus leas than 68S$ However, in the planar form the vacancies are in nonbondii orbitals, and the stabilization energy remains at 6pS$ The case of CH3 is particularly interesting because VSEPR predicts a planar shape only if the odd electron is ignored. Finally the possibility of T-shaped AH3 molecules has to be considered. Although 90" and 180° angles between the AH bonds are assumed, other angles can be treated without any problem. There is, however, a factor for T-shaped molecules that need not be taken into account for planar and pyramidal molecules--the three A-H bond lengths are not ~ geometry all identical. The angular overlap model f o this therefore requires two PS& values, say p(Sp,)2 for the two bonds in the cross-piece and p(Sp,)2for the stem A-H bond.
Figure 6. Variation of energy d AH. molecule with angle using simple AOM.
radical: BeH2. For BeH2 there are onlytwo 2p electrons to consider, and these will go into the lowest energy orbital. The maximum stabilization energy for this molecule will occur at the minimum of the curve in Figure 6, that is when 2p, has a stabilization energy of 2,8S&. This corresponds to a linear molecule, as BeH2is. Trlhydries, AH3
As in the case of the hydrides, the angular overlap model results for trihydrides are similar to those for the crystal field model but can be made quantitative. For a planar molecule the protons lie at 0 = 90°, 6 = 0; 0 = go", 6 = 120"; and 0 = 90D,6 = -120°. The stabilization energies of the 2p orbitals are given in Table 2. If the molecule distorts into a pyramidal shape so that the A-H bonds are at an angle Ofrom thex-y plane, then the Zp,, 2p,, and 2p, orbitals are lowered by the amounts given in Table 3. For example NH3 has 0 = 27.5", and hence 2pz is stabilized by 0.648S&, 2p, by 1.18!3Si,, and 2p, by 1.18@,. That is 2p, and 2p, are degenerate and lie below the 2p, honding level in agreement with the PES data.
planar
pyramidal
Flgwe 7. Energy level diagrams fw CH. and BH*. Table 2.
Stablllzatlon Energles for a
&,, AHSMolecule Table 3.
Stablllzallon Energles for an AHo Molecule ol Symmetry
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The stabilization energies of the 2p orbitals are then 0 (p,), l3(S,,j2(pX) and 2P(SpJX(p,). In T-shaped molecules the cross-viece A-H bonds are longer than the 3tem bond, and so the h;drogen 1s orbitals ov&lap less with the central p orbital, and thus S" < The difference in bond length is as to make (S",J2 twice as large as unlikely t o be (S',,)2, and so the p, level will lie below p,. Figure 8 shows theeffect of changing the ratioS"Yg2 between the limits (1) corresponding to three equal 1-$bond lengths and (2) corresponding to p, and p, being degenerate as in the planar trigonal case.
sue%
82.
the Jahn-Teller effect. In the trigonal planar form the molecule would be in a degenerate state; going from the regular shape to a T shape removes the degeneracy. Thus this very s i m ~ l moleculeshows e clearlv that a less reeular shave can that &dents be favored over a more reguiar one, a sometimes find difficult to see with more complex systems such as Cu(I1) salts. Ab initio calculations on LiHsC (14) indicate that the molecule, although planar, will not be trigonal or T-shaped but a Y shape that is best envisaged as an Hz molecule joined perpendicularly to the lithium atom of LiHC. Nonetheless, our much simplified description does indicate that this molecule will not have C8 symmetry. Another teaching point that arises from a study of Tshaped hydrides is that orbital energies depend on bond length as well as bond angle. Figure 6 was a simple illustration of variation of orbital energy with angle; Figure 8 is a simple illustration of how varying bond length (in this case the ratio of stem and cross-piece bonds) can alter the energy.
Conclusion Crystal field and angular overlap models can be used a t an elementam level to introduce basic ideas of molecular orbital theory-using very simple main group compounds as examples. The results agree qualitatively with spectroscopic data and are particularly useful for gaining insights into open-shell molecules. The ideas gained by such studies can be transferred to more rigorous MO approaches to main group compounds and to crystal field and angular overlap models of transition metal complexes. Literature Clled Figure 8. Variation of 2p energy levels In a T-shaped molecule.
Although there are no known T-shaped trihydrides, we can discuss hypothetical molecules that would be expected to adopt this shape. For our example we choose LiH3+ (a molecule that might well be formed in a mass spectrometer). This has only one 2p electron, and so planar shapes are more stable than a pyramidal shape. In a trigonal planar molecule the electron would have to be in one ofthe deeenerate levels. In the T-shaped molecule the degeneracy orthese levels is removed.. and the electron eoes into the lower of the two degenerate levels that replace them. I t is therefore advantaeeous for LiH%+to adopt a T shave. An interesting teaching point is that this is an example of
1. Bethe, H. Ann.Phy& 1923.3.133-206. 2. Majt undergraduate inorganic chemistry tertbwks, e.g., Cotton, F.A,; Wilkinson, G. AduoneedInorgonie Chemistry, 5th ed.;lnteneience: New York. 1988: PP 652-656. Huheey, J. E. Inorganic Chemistry, 2nd ed.:Harper and Row: New York, 1978: pp 34b389. Jolley, W . L. Modarnlnorgonic Chemlatly,IntemationelStudentEdition; McGraw-Hill: 1935; pp 361-365. Cotton, F. A,: Wiikinaon, G. Basic Inorganic Chemi~try;Wiley: New Yark. 1976;pp 354459. 3. Bsllard, R. E. Pholoelectmn Speelroacopy and Makcvlor Orbital Theory; Hilgcr (now o w e d by the Institute of Physic$): Brktol, U.K., 1978: p 32. 4. Hertlberg,G.;Johns, J. W . C . J.Chem.Phw. 1971.54.22762278. 5. Gillespie. R. J. Molecular Geometry; VanNaatrmd-Reinhold:London, 1972. 6. Schaffer. C. E. Pure Applied Cham. 1970,24,361-392. 7. Larsen. E.: La Mar. G. N. J.Chom. Educ. 1974,51,63340. 8. Purcdl, K. F.; Kotz J. C.Inorganir Chemlatry: Saunden: Phila&lphi, 1977:p 613. 9. Huheey. J.E.Inorg~nieChrmisfry,3rdd.:Harperandnow:NewYark,1933:pp42110. De Koek. a. L.; Gray. H. B. Chemieol Struclum and Bonding: BenjaminICumming~: MenloPark, CA, 198a 11. Burdett. J. K. Struetura Bandim 197L 31.67. A"*
7"".
The Agricultural and Food Chemistry Division (AGFDj ir pleared to announce and call for applications far the first AGW Graduate Candidate Symposium to be held during the Ni6t ACS National Meeting in Atlanta, GA, April 14-19, 1991. Participation in this symposium, designed hy ACFI) to ihoacase graduate degree candidate research talents to prospective employers (industrial, academic, governmental), is open to all year-to-be-degreed graduate students at all certified US. universities doing advanced degree research in areas of food and agricultural chemistry. Owing to fund limitations,the participating graduate candidates, whose travel expenses up to $1WO will be covered by AGFD, M'llbe chosen by an AGFD Advisory Board. In order to apply, qualifying graduate candidates should submit a cover letter, with resume, complete college/ graduate school transcripts, a 2-page summary of their degree program reseal&, letters of recommendation from two major degree committee professors (one preferably the graduate advisor), and a 150-word abstract of the proposed oral presentation the graduate candidate would make at the Atlanta ACS Meeting. This completed applicationshould be received by October 1,1990, by Dr. Charles J. Brine, FPP Division, FMC Corporation, Box 8, Princeton, NJ 08543; Tel. (609)520-3681.For further information contact the above.
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