angular overlap paper

The Angular Overlap Model as a Unified Bonding Model for Main Group and Transition Metal Compounds A Version Suitable for Undergraduate Inorganic Students

D. E. Richardson University of Florida, Gainesville, FL 32611 -2046

This article presents a simple version of the angular overlap model (AOM) that has been successfully introduced as a unified bondinn model in the undermaduate curriculum at the university ofFlorida at the jun&/senior level. The fundamentals of the AOM are introduced with specific instructions for calculation of molecular orbital energies for both main group and transition metal complexes.

Worked examples are given to illustrate the procedures for main group and transition metal compounds. The calculation of molecular orbital stabilization energies is also described.

Molecular Orbital Theory

Molecular orbital (MO) thwry ( I ) has become the most ~onular. but not the onlv. model for describine the elec- . . . . tmnic structure of polyatomic molecules in current textbooks (21 on inorganic chemistry. In addition to MO theory, all inorganic texts also describe a number of other bonding models (e.a.. valence bond theow and lieand field t h w w ~ . each of k&h is optimal or triditionai for rationaliz& various chemical observations.

In principle, MO theory can be used to provide a unified theoretical framework for organizing and predicting the chemistry of all the elements without recourse to alterna- tive models. Unfortunately, i t also has a rather high threshold for quantitative *first principles" application by students to problems in chemical bonding. For example, the preferred geometry of a main group AB, molecule is more directly predicted by the VSEPR method (3, than by doing a geometry-optimized molecular orbital calculation on a computer.

On the other hand, MO theory produces one-electron energy-level diagrams that resemble atomic energy-level diagrams. This approach allows students to predict

the electronic o"gins of spectral transitions the orbital locations of unpaired electrons the valenceelectxon structure of transition metal complexes

many other physical properties of both metal and nonmetal compounds

Undergraduate Applications

The apparent complexity of generating appropriate MO diaerams for laree molecules has traditionallv limited the lev:) of theoretl'cal sophistication expected"in students studying the chemistry of the elements. At the undergraduate level, MO diagrams are typically presented for various types of molecules (diatomic, chains of atoms, polyatomics AB,, metal complexes ML,, etc.) along with nonquantitative discussions concerning orbital overlap.

Students do not routinely obtain the tools of group theory or computation necessary to generate their own MO energy diagrams for molecules more complex than diatom- i c ~ . Thus, MO diagrams for polyatomic molecules are usually presented without providing a method for constructing them systematically and are, in effect, rationalized after the fact.

Introduction of the Angular Overlap Model There are clear instructional and conceptual advantages

in using a unified MO bonding model for main group and metal compounds. In this article I present a version of the angular overlap model (4) (AOM) that has been successfully used in this way in the undergraduate curriculum a t the University of Florida. As will be seen, the AOM allows students to quickly generate fairly sophisticated MO diagrams for both main group and transition metal eom- plexes.

Larsen and LaMar (5) have reviewed the application of AOM to metal complexes, and presentations for transition metal complexes have appeared in undergraduate inorganic textbooks (2a, b). Moore (6) has described the use of the AOM to simplify the use of MO theory for AH2 and AH3 main group hydrides.

Rather detailed descriptions of the application of the AOM to prediction of molecular shapes have been pub- lished by Burdett (7,8), who has extensively demonstrated

372 Journal of Chemical Education

its utility for main group compounds. Although lucid, these two publications are not generally suitable for introduction as supplementary material at the undergraduate level.

An Undergraduate Presentation

This article provides a self-contained presentation of the AOM that can be incoruorated readilv into the undermaduate curriculum, most\sefully in a j"unior1senior levil inorganic course. This presentation is also appropriate for use as supplementary material.

The material is not intended as a replacement for the VSEPR approach, which provides the most direct method for predicting the shapes of main group polyatomics. How- ever, it has been found that the AOM approach supplants presentation of other electronic structure models beyond the VSEPR for largely covalent molecules.

The fundamentals of the AOM will be introduced with specific instructions for calculating molecular orbital energies for main group and transition metal complexes. Worked examples are then given to illustrate the procedures. A future article will describe the derivation and application of structural preference energies as well as mis- cellaneous applicatiom to other subjects typically covered - -

in inorganic courses.

Fundamentals of the Angular Overlap Model The detailed origins of the AOM will not be described

here. Interested readers can consult several sources, most notably a useful monograph by Burdett (7), for many aspects of the model not covered. It is assumed that the students have hadan introduction to the elementary quantum theory, atomic orbitals, and MO theory for homonuclear and heteronuclear diatomic molecules at a level typical for textbooks on inorganic chemistry.

Basic Concepts and Defhitions

The AOM suggests that the strength of a bond formed using atomic orbitals on two atoms is related to the magnitude of the overlap of the two orbitals. The model provides a specific mathematical approach to calculating relative energies of molecular orbitals that result from the overlap of various types of orbitals on two or more atomic centers.

Central-Atom and Ligand Orbitals In the model, a central atom is surrounded by ligands,

and the overlaps of one or more ligand orbitals with the

o ligand orbitals

various central-atom orbitals are considered. The general molecular formula is

AB,

where A is a main group element or

M L

where M is a transition element, and A or M represent the central atom surrounded by n ligands at various locations, all at the same distance.

In practice, ligands can be atomic centers, such as

H, F, and 0

or polyatomic, such as

CO and NH3

Central-atom orbitals are the valence-shell s and p orbitals (for A, a main mu^ element) or valence-shell d orbitals - . (for M, a transition element). Ligand orbitals are usually of either the o or x type. This is determined by the orientation of the ligand orbital with respect to the A-B or M-L bond axis. When vieweddown the bond axis, a o orbital is unchanged by rotation about the axis, whereas a x orbital has a nodal plane along the axis and is reversed in phase by a 180" rotation about the axis. Examples are shown in Fig- ure 1.

Use of o-Symmetry Orbitals

The ligand "o" orbital shown in Figure 1 sometimes causes some confusion because mixing of molecular orbital theory with atomic orbital hybridization (valence bond theory) seems to be implied. In essence, it is convenient to use this type of orbital to represent the relevant ligand molecular orbitals that have o symmetry.

For example, the lone pair of NH3 can be associated with the highest occupied MO, which has an admixture of N s and p orbitals with some H 1s character. Rather than deal with the complex mathematical form of this MO, we simply indicate the highest MO of NH3 with a o-symmetry orbital when using ammonia as a ligand. Similarly, s and p mixing in atomic ligands, such as flourine, makes it inap- propriate to use only the p orbitals of the ligand to form bonds in the AOM, so a o orbital is used.

Orbital Overlap and Overlap Integrals

The overlap integral S provides the fundamental quantity that describes the "magnitude of overlap" between two atomic orbitals. It is formally calculated by integrating the product of the two atomic wave functions y, and yb over aU space, as in eq 1.

The atomic orbitals yi are in turn described quantum mechanically as a product of an angular function Y and a radial function R. With the exception of two interacting s orbitals, which have no angular dependence, the relative orientation of the two orbitals will affect the value of S* via the angular functions Y, and Yb. The value of the overlap integral (eq 1) can be expressed as

n ligand orbitals

Figure 1. Illustration of a-symmetry (a, b) and n-symmetry (c, d) ligand orbitals. The o obitals are unchanged by rotation about the bond axes, whereas the n orbitals are reversed in phase by a 180' rotation.

SA is an angle-independent contribution that depends on the distance r between the atoms, the quantum numbers n and 1 for both orbitals, and the identity of the two atoms. The subscript h refers to the type of bond and is normally either o or n. Oab is the angle-dependent contribution that depends entirely on the forms of Y, and Yb, which are independent of r and the type of atoms involved. Expressions for 0,b can be determined for the various types of interact-

Volume 70 Number 5 May 1993 373

A A-B B

M-L

M M-L L .Figure 2. Molecular orbital diagrams for the interaction of two basis orbitals. (a) One p odital on A interacts with a o orbital on B to form bonding and antibonding molecular orbitals. The atomic orbitals are oriented to yield the maximum overlap. (b) As in (a) except the oorbital is at an angle to the axis of the p orbital. The reduced overlap leads to reduced stabilization and destabilization energies. (c) AR interaction between a d,orbital on M and a digand orbital oriented to vield maximum overlw. (d) As in (c) except the orbitals are oriented with net zero overlap, thereby forming two nonbon'ding molecular orbitals. .

ing orbitals, and a list of angle-dependent functions for S* can be obtained (7).

I t is convenient to divide orbital interactions into two groups depending on the type of ligand orbital involved: those with a character and those with n character. In the former case, Sabin eq 2 is given by

S f l a b

where S, is the "basic" quantity of a overlap at some fixed bond length, and the relevant angle-dependent wntribu- tion F)*b has been assigned the notation Fab.

As an example, consider the o overlap of an s orbital with a p orbital on a central atom, as would occur in a molecule such as HzO:

The shaded lobe of the p orbital is arbitrarily taken as the positive lobe.

When the angle 9 = 90°, the maximum (+I+) overlap possible for a given internuclear distance accurs, so

FSp = 1 and Sap = Sm

'When 9 = On, the positive overlap (+I+) is exactly canceled by the negative contribution (+I-), so

Fa, = 0 and Ssp = 0

Finally, at any angle 9 at the same internuclear distance, it can be shown that F = sin 9. From eq 1, we get

S, = Sosin 0

A plot of the s-p overlap as a function of 0 is therefore a sine function. When 180°< 0 < 360°, the value ofS, is negative, but, as will be seen, the sign of S is immaterial since S2 is the relevant quantity for the AOM calculations.

In the case of a n interaction, Sab is given by

S x a b

where

Sx = S, and O,b = Pab


for this type of interaction. As an example, the overlap of a central atom d orbital on

a metal with a ligand x-symmetry p orbital is shown.

Orbital Stabilization and Destabilization

In MO theory, two inequivalent atomic orbitals, yr. and yrb (the basis functions), with finite o overlap (Sab t 0) are combined to form two molecular orbitals. One MO (Yo) has a lower energy than the atomic orbital (bonding), and the other (Yo.) has a higher energy (antibonding) (Fig. 2a).

Because the bonding MO is primarily composed of the lower-energy atomic orbital (yr.1, one could say that yr. has been stabilized by interaction with yrb. Likewise, yrb has been destabilized by its interaction with yr, to form the antibonding MO. Thus, although molecular orbitals are described more accurately by mathematical combinations of the basis functions, the &raction of two atomic orbitals can be thought of as producing "shifted" orbital energy levels that either stabilize I'P., or deiitahilize ~'i'..) the . . bond when occupied by electrons.

Energy Difference: Ligand us. Central-Atom Orbitals

In the AOM, it is assumed that the central-atom valence orbitals are substantiallv different in enerw from the ligand orbitals. Thus, staglization or destabilyiation of an orbital on the central atom depends on whether the ligand orbitals are lower or higher in energy. If the ligand orbitals are lower. the central-atom orbitals can be destabilized to form antibonding molecular orbitals. If the ligand orbitals are higher, the central-atom orbitals can lead to bonding molecular orbitals.

To assess which case applies, the Pauling electronegativ- ity can be used: The higher the eledronegativity, the lower the valence orbital energies. Thus, in NFs the N 2p orbitals are higher in energy than the F o ligand orbitals. In HzO, the 0 2p orbitals are lower in energy than the H 1s ligand orbitals. In typical transition metal complexes, the metal d valence orbitals will always he higher than occupied ligand orbitals.

In the case shown in Figure 2a, the central-atom p orbital yr, bas been stabilized to form a o bonding MO that is lower than the y~, orbital energy by a quantity given as

The o ligand orbital yrb is correspondingly destabilized by ~ d ~ ~ ~ ~ b ( o ) . and here it is assumed for simplicity that the stabilization and destabilization are equal in magnitude.

Orientation and Overlap

The orientation of the orbitals in Figure 2a provides the maximum overlap possible for a o interaction (F = 1) and thus the maximum stabilization of va. If the ligand orbital makes an angle @ to the central p orbital, the stabilization of yfa is lower due to decreased overlap (Fig. 2b), and the energy of the antibonding MO would be correspondingly less above the energy of yrb. At $I = 90. the net overlap is zero, and no stabilization or destabilization occurs.

In Figure 2c, a x ligand orbital is shown interacting with a d orbital on a central metal atom. The optimal overlap shown produces shifts of magnitude E&&) and edestab(n) from the original atomic orbital energies. It can be seen that the overlap integral is zero when 9 = 45' in the case

shown (Fig. 2d), and the atomic orbitals each become nonbonding molecular orbitals (indicated as YINB).

Calculafing Orbital Ener ies Usmg Angular Overlap $onsiderations

The general discussion above suggests that the shiRs of central-atom orbital energies upon interaction with one or more ligands is strongly related to the degree of overlap between the orbitals. Indeed, certain forms of approximate molecular orbital theory lead to the conclusion that the energy shifts are approximate functions of the square of the overlap integral.

The o Interaction

At the level of approximation to be used here, the a interaction of a ligand orbital with a central-atom orbital leads to a stabilization of the lower-energy orbital:

'atah = Ye, - (@l2f, (3)

The destabilization of the higher-energy orbital is given by

These equations are derived in some detail elsewhere (6, 7). The o nature of the interaction is noted by setting 0 =

Fig~re 3. Defm tlons of angLlar overlap parameters for vanous central.atom to igand interacttons. For F = 1 , (a) central-atom p to o- liqand orbitals and ib) central metal atom d? to a hand orbitals. For . . P = 1, (c) central-atom p to p-ligand orbitais, (d) central metal atom d, to p-ligand orbital.


Z?' Parameters e, and f, are proportional to Sz and S:, respectively, and represent "basic" units of orbital interaction. These parameters are also related to the energy difference between the two orbitals and the natures of the two atoms.

For any molecule AB., the actual values of e, and f, will depend on the identity of A and B as well as the A-B bond distance, but both will always be negative and independent of the relative angular orientation of the two orbitals. Furthermore, l e, l > If, I , so ~ ~ ~ ~ ( 0 ) will be negative (de- noting a lowering of energy), and ~ a , ~ b ( a ) will be positive.

From eqs 3 and 4, it is seen that the shifts in orbital energies that oemr as ligands are moved to various positions with respect to the central atom are only related to the angular overlap factors FO because e, and f, are independent of angular orientation. When considering central-atom p orbitals, FO = 1 for the ideal overlap shown in Figure 3a, and from eq 3, we get

In the case of central-atom d orbitals, = 1 is found for the d2 orbital interacting with a ligand o orbital as shown in Figure 3b.

'The "angular overlap factor" Fhas a more restricted meaning here than given to F by Burdett (ref 7), who assigns it the same meaning as Bin this article. The Dresent author has introduced Fand P as angular overlap factors todivide aand ncontributions in the notation.

Table 1. Values

Notes for Table 1 : Some special angles.

1 @ sin 30 = -0s 30 = $ sin 45 = -

@ 2 2

CoS 45 = - 2

6 sin 60 = - 1

2 COS 60 = -

2

For C3v molecules with tetrahedral angles, 0 = sin"(al3) = 70.53', sin e = a13. cos e = 113.

AB. linear L = 1.6 - ~~

bent ~=16:17 AB, trigonal plane L = 2,7,8

pyramidal L = 13,14,15 T-shaoe L = 3.5.6

~ ~>~ ~ -

AB, tetrahedral L =9;10,11,12 square plane L = 2,3,4,5

AB, trigonal bipyramidal L = 1,2,6,7,8 square pyramidal L = 1,2,3,4,5

AB6 octahedral L = 1,2,3,4,5,6

Figure 4. Ligand positions for various AB, geometries.

The n Interaction

For interactions between the central-atom and p-ligand orbitals, the corresponding energy changes are given by eqs 5 and 6.

The factors en and f, are analogous to e, and f, and are both negative, and 0 has been set to P to indicate the n interaction. The energy shifts due to n interactions are then related to the P2 angular overlap factors. For central- atom p orbitals and d orbitals, the maximum overlaps for which P = 1 are shown in Figures 3c and 3d, respectively.

Tabulated values for F and P2 for various ligand positions around a central atom (Fig. 4) are given in Tables 1 and 2. As described below, these tables can be used with

Table 2. f l Values


equations derived from eqs 3-6 to calculate the AOM shifts from basis-orbital enereies in molecules with various numbers and orientations ofligands about a central atom.

Molecular Orbital Diagrams Constructing MO Diagrams for Main Group A& Molecules

A systematic procedure for constructing MO diagrams for main group polyatomic molecules is given here. The focus is on the D orbitals of the central atom and their interactions with\igand o orbitals. It is not generally neces- s a w to include ir: interactions for main erouD moleculesto - &

mLke predictions concerning molecular shape, so these contributions to orbital energies are not included. Exam- ples of the application of the procedures are given to clarify the required manipulations.

Ligand s or a Orbitals That Are Lower in Energy Than Centml-Atomp Orbitals

(1) Each ligand contributes one o orbital (or s orbital for H) for interaction with central-atom p orbital8 in A&, for a total of n lieand orbitals. The central-atom s valenee or- - bjtnl is taken as nonbondmg (a severe approxlmatlon'~ A haw-orh~tal energy dlngram ~s then constructed showng the numbers and relative energies of the basis orbitals, in

Example. For BeF2, the two tluorines contribute a total oftwo o hgand nrhitals. The hasis orhitals are shown in Rgure 5 as the levels on the left and right sldw labelled "Be 2p", "Be Zs", and -2 Fa".

(2) Select ligand positions for the following geometries (see Fig. 4).

-2 linear; L = l , 6 bent 90'; L = 16,17; with 8 = 45' bent at any angle; L = 16, 17; with 8 = 112 of B-A-B bond angle

m3 . trigonal plane; L = 2,7,8 pyramidal any angle; L = 13, 14, 15

Be -F-Be-F- z

'I 2F

Y X

Figure 5. AOM MO diagram constructed for linear BeF2. The bond axes are on the zaxis. Note that ligand p orbitals perpendicular to the

pyramidal at tetrahedral angles; L = 13, 14, 15; R = 70.Y . . ' T-shape; L = 3,5,6 m4

tetrahedral; L = 9, 10, 11, 12 square-planar; L= 2,3,4,5

AB. trigonal bipyramidal; L = l ,2,6,7,8 square pyramidal; L = 1,2,3,4,5

ms octahedral, L = l,2, 3, 4, 5, 6

Example. A linear geometry is chwen for BeF, using positions 1 and 6 for the fluorines.

(3) For each central-atom p orbital, its total e, destabilization (if any) is obtained by summing the values (Table 1) under the orbital for all the ligand positions chosen in step 2. The total destabilization including the f, term is then given by

Example. For linear BeF2, the following sums are obtained from Table 1.

Thus p, and p are nonhonding, and p, is destabilized (Fig. 5) ac- cording to eq {by

(4) The n ligand o orbitals always lead to n combinations (ligand group orbitals (LGO's) (9)), one of which corre- sponds to any central-atom p orbital that is destabilized. The total stabilization of such a corresponding LGO is given by

-2e,+ 4f,

NB Be 2p

NB Fo

41 2e0- 4fa

'' Bes

F~gure 6. Correlat~on of AOM aerived energy levels with traditonal representations ol BeF, mo ecLlar orbia s. These orbilals assame no s-p mixmg, which wo~lo mlx 'i',,(sj w tn YNB(o),


Figure 7. AOM MO diagram constructed for ammonia with H-N-H angles of 109.5'.

Any of then ligand orbital combinations not stabilized re- main nonbonding.

Example. For linear BeF2, one LGO is stabilized by

% h b = 2 ~ 7 - ~ f o

while the other stays nonbonding (Fig. 5).

(5) The total molecular orbital stabilization energy (MOSE) resulting from o interactions (Zo)) for a given geometry and valence-electron count (the number of o bonding and lone pair electrons appearing in the Lewis dot structure; electron pairs contributing to multiple bonds and lone naira on lieands are not counted since the treatment here Is o-only, ircalrulated by doubly filhng the sorh~tnl ofthe central atom and .U0's ~n order of inerrsnngenerm Then sum the indtwdual anerm 3rhlfls ofthe result~ng wcup~ed orbitals. Thus, for an x-valence electron system, we get

Oeeunied nonbondine orbitals and the o orbital contribute - ~~~~ ,~~~~~~~~ ~~~ ~- -~ ~

zero to thin sum. Ua) represents the stability of the par- tlcular wangement relative to the case of 4nfinitrly new arated" A-B bonds (which has a relative energy of zero).

Example. BeFz has a total of four valence electrons tin s bonds (the other 12 in F lone pairs are not counted). Two electrons occupy the B 2s orbital. The other pair is placed in the lowest energy MO (Fig. 5). The MOSE is calculated fmm eq 9 as

2(2, - 4fJ = 4% - 8fa

Ligand a or s Orbitals That Are Higher in Energy Than Central-Atom p Orbitals

In this case,

E(o) > E(P) > E(s)

The stabilizations of the central-atom p orbitals are first calculated using eq 8. The destabilizations of the corn- sponding LGO's are derived analogously using eq 7.

I filled

i. 4 NH, a

Figure 8. AOM MO diagram constructed for square planar [cu(NH,)$+.

Example: NH3 Ammonia has three H ligands contributing three o 1s orbitals. The pyramidal structure with tetrahedral H-N-H bond angles of 109.5' is used here to illustrate the use of pentries that contain 0 (L= 13,14,15; 9 = 70.5'= sin-' 6/31, The N 2p orbitals are set at a lower energy than the H 1s in the basis- orbital energy diagram (Fig. 7) because N is the more electronega- tive element. From the in Table 1 and eqs 7 and 8, p,, py, and p, are stabilized, and the corresponding H 1s combinations are destabilized, as shown in Figure 7. With eight valence electrons (one lane pair and three o bonding pairs) Iilled in as shown, we get

This MOSE cannot be directly compared to the value for BeF2 derived ahnve because the e, and f, values are different.

Constructing MO Dia rams for Transition Metal Ampiexes ML,

The procedure for transition metal complexes focuses on the d valence orbitals of the central metal atom. It i s not necessary for most purposes to incorporate the f, contributions to orbital energies, hut n interactions are important i n many complexes and must be included via the e, term. (The f, is also ignored.) Two examples of the application of the steps are given.

(1) In all cases, ligand o and n orbitals are lower in energy than metal d orbitals. Ligand n* orbitals (e.g., on CO) are usually higher in energy than the transition metal d orbitals.

(2) Each ligand contributes one o-type orbital. The following considerations apply for enumeratingn and n* orbitals on some common ligands.

Halides (X3. oxide (0% sulfide (5%). nitride (N31 and . .. any other atbmic anions have two n levels per ligaid for a total of 2n. Diatomic ligands, such as CO, CN; Nz, and other isova- lent diatomics. have two n and two n* levels, but then orbitals are usuallv ienored . .. NR, (R = H, alkyl) m i n e ligands, H20, and related ligands have little n bonding with metal centers. They can be considered o-only.


F~gure 9 AOM MO d~agram mnstrmea for Cr(CO)G. whch has ex tenswe n Wndmg The octahedral d-orbtal spllnmg parameter A 1s shown Note tne s~mpllt cat on oy mdlcatmg oasls orblta s in boxes and the filled orbitals by a dashed line.

Example. In [Cu(NH3),12+, the NH3 ligands are o-only, so n interactions are not considered.

(3) Select ligand positions as in the main-group procedure (Fig. 4).

Example. For square planar [CU(NH~) d2+, positions L = 2,3, 4 , 5 are used with Table 1 to calculate shifts in the basis-orbital energies (Fig. 8).

(4) Far each central metal d orbital, its total "o-only" destabilization (if any) is obtained by summing Fa values (Table 1). The net destabilization is given by

Example. The d , 4, and 4, orbitals are nonbonding in square planar copper%( complexes, but &a.,a and &a orbitals are destabilized by +. and -3e,, respectively (~ fg . 8).

(5 , As in the main p u p case, a combination of lizand oorbit- als (an LGOI exists for each d orbital that is destabihred. The stabiliratron of this combmation is p e n by

Any of the n ligand group orbitals not destabilized re- mains nonbonding.

Example. In square planar lCutNH.,~,l'", the two eorre- spondmg l~gand combmat~ons are stabllmd by e. and 3e0, nnd the other two become nnnhondmg moleculnr orb~tals (Rg. HI.

(6) The molecular orbitals below the d orbitals will generally be filled, and it can he shown that a total of 2ne, m enn- tributed to the MOSE by these electrons. The total MOSE (a only) is then given fir the d9 complex by eq 12, where the sum is over the q electrons of the central atom.

The dnrbital derived MO's are filled with the valence elre- trons drtermined for the formal oxrdation state of the metal. .Fw example, TI(IIII: q = I; FelI111: q = 5 ; PttII,: q

Example. Since coppcr(ll, is a d9 metal center, nmeelectrons are accommodated in the d-type molcculnr orbitals [Fig. 8,. All lower MO levels nre filled. From eq 12, we get

(7, In accuunting for n interactions, the net result can be stabilization or destab~liratrun of certain d orbitnls. in some cswer, d orbitals wdl he shifted hy both o and n interar- tions. Typically,

1e.l > le,l

but their ratio can vary widely depending on the metal and the ligand. Two kinds of n ligands can be considered:

For ligands with two n orbitals E-, 02, N3-, SZ, etc.) the destabilization of a d orbital is calculated from the n-overlap factor table (PZ values, Table 2).

One n LGO is stabilized for each destabilized d orbital by the negative of the quantity in eq 13. Again, the molecular orbitals below the d orbitals are all filled, and then interaction is seen to contribute 4ne, to the MOSE. The total MOSE due too and n interactions for the molecule is then given by eq 14, where the sums on the right side are taken over molecular orbitals that are derived from over the filled molecular orbitals that were derived from d orbitals.

For ligands with two unoccupied n* levels per ligand (e.g., CO), the stabilization of a d orbital is calculated as

One n* LGO is stabilized far each destabilized d orbital by the negative of the quantity in eq 15. The total MOSE is

Example of MO Diagram Construction for a Transition Metal Complex with n Bonding Ligands: Cr(C0)s

The carbonyl ligands contribute six o-ligand orbitals and twelve n* orbitals. The octahedral geometry is used (L = 1, 2,3,4,5,6), and the basis-orbital energies are placed as in Figure 9. First, o destabilizations of the C r 3d orbitals are counted via Table 1 and eq 10. It i s seen that d,, L, and d, are nonbonding with respect to the ligand o interactions, whereas dzs.,P and ds* are raised by -3e,. To account for n bonding, the P values (Table 2) are applied i n eq 15, and i t is found that d,, d,, and d, are stabilized by 4e, (because the C r d orbitals a re lower than the CO n* levels).

After calmlatine the C r d orbital e n e r w shifts. the corresponding l igani orbitals must be shiEed as shown i n Figure 8, with many of the ligand orbital combinations re- maining nonbonding. The ligand field splitting parameter ("A") is indicated in Figure 9.2 To fill i n the electrons, t h e 6 valence electrons for CdO) are entered into the shifted d orbitals, and the levels lower than the dashed line are all filled. ~ r o m eq 16, we get

he large value of A for n-acceptor complexes, such as Cr(C0)6, compared to relatively small A values for halide complexes can be related in part to the sign of the e, contribution. For a MX8 complex (where X is a halide), the n interactions lead to a destabilization of 4 e , for the (xy, xr, yz) orbitals.


theorv. Masterv of the s im~le AOM a~umach in this article

Both a and n terms contribute to the stability of Cr(CO)fi, and the x interactions are usually referred to as x back- bonding.

Checking AOM Diagrams with the Sum Rules

A useful method for checkine the accuracv of derived AOM diagrams is based on the application of &e sum rules of eqs 17 and 18, where n is the number of ligands.

. ,

The first sums are over all bonding molecular orbitals only, and the second sums are simply the coefficients e, and en for those orbitals.

The AOM diagrams derived for the four examples in this article can be checked to verify eqs 17 and 18. for example, for NH3 (Fig. 7) the sum from eq 17 is

(113) + ( 4 3 ) + ( 4/31 = 3 = n

For CrCO)6 (Fig. 9) the sum from eq 18 is

Limitations and Conclusions The AOM is based on rather severe approximations and

is not a t ~ l v auantitative MO method. Neither s-D mixing nor n~c1ea;r~~ulsions are incorporated into the model. 6 is bv no means infallible in its re dictions concerning mo- leklar shapes and cannot model the more subtle aspects of bonding in polyatomic molecules.

Like any predictor, the value of the AOM is that it usually leads to correct predictions. As with predictions made using other simple molecular orbital models, AOM predictions are more reliable when the electronegativities of the central atom and the ligands are not too different and when a mostly covalent description is appropriate. (A second article will describe the application of the orbital diagrams and MOSE values to the prediction of molecular shapes and other physical properties.)

Another advantage of the AOM is that it serves as a rather straightforward introduction to molecular orbital

3Most molecular omital symmetry labels can be oetermined eas. y by referring to the character table tor tne molecule's pomt group. The xy y, and 2 representations are attached to the p orbitals and their corresoondina LG0's. The remainina LGO svmmetries (if anv) must be delermin& bv the usual ~rocedure of determinino the reducible representation oitne n oasis'orbrals followed by deciinposrt~on Into IrredLCde representations.

will Geld MO diagrams t l k contain; significant amount of information that will not change markedly for more sophisticated types of treatments (e.g., ab initio calculations). A modest amount of group theory (10) will aUow the student to attach symmetry labels to the molecular orbitals ~bta ined,~ and mixing between s and D orbitals in main group molecules can be <ncorporated (llj.

At first glance, the procedures described here may seem complicated and difficult to commit to memory. However, their application is simple when presented by way of exam~le. and the a ~ ~ m a c h is easv to remember. At the UN- ve;sit; of Florid; ;ndergradua"tes and graduate students wnerallv mastered the method in a short time (after about - 2 3 lect&es and a homework set), allowing the instructor to concentrate on applications of MO theory in under- standing chemistry. In teaching the AOM concepts and ~rocedures, a meat amount of basic MO theorv is Dre- sented thatis n&nally covered anyway; therefore, the in- clusion of the AOM does not require additional lectures in an already cmwded outline. The abstract, nonmathemati- cal MO theorypf general chemistry then becomes a model that can be aftive& manipulated by the student to address the central concept of chemical bonding

Acknowledgment The author is grateful to R. Drago and D. Talham for

their helpful suggestions during the wmpletion of this article. Helpful discussions with D. Talham and J. Boncella regarding their classroom experiences with the AOM are also acknowledged.

Literature Clted 1. Afew recent monographs: (alAlbrigh2 T. A; Burden, J. K, Wangbo, M.-H. Orbital

Infermfio~ in Chemistry; Wiley: New Yo& 1985. (b) DeKoek, R. L.; Gray, H. B. Chemical S l m t u m ~d Bondiw, Benjamin Cummings: Menlo Park, 1980. (c) Webster, B. Chomiml Bonding Thmry; Blsckrvell Scientific: Oxford, 1990; (d) Hehre, W J.; Radom, L; Rhleyer, P Y. R.;Pople, J.AAb Initio MalPeulnr Orbital Theory; Wiley: NewYork 1986.

2. (a)PureeU, K; Kotz, J.InowanieChpmis~;Saundera:NewYork, 1971. mIMiesaler, c L: Tan, D. A. 1nogonlc chemiahy; RenticeHall: E n g l e d Chi,%, 1991. (4 Shrive., D. E.;Atkins,P W.: langfmd, C. H. Inaganic Chemislry: W H. Freeman: NewYork, 1990. ldlCotton,FA.;wlk~son,G.;Gaua,P.L.BmieIieg~nieChem- isfry, 2nd ed.; W~ley: New Yark, 1987. (el Huheey, J. E. Inorganic Chemutry, 3rd ed.; Harper and Row: New Yo*, 1983. (0 Bufleq I. S.; Harrad, J. E Inorganic Chemist?; BenjamidCvmmings: Redwood City, 1989. (gl Douglas, B.: McDa?iel, D. H.;Alerander, J. J.ConceptsondMdela oflnownicChpmkfry.2nd ed.: Wdey: NewYork, 1983.

3. Gilleaoie. R. VSEPR M&lofMolPculalaCoMVff: Prprtiie-Hall: NeuYark. 1990. 4. seh&rc. E. ~ u ~ ~ ~ ~ ~ ~ i ~ d c h . l w n , 24.36i.' 5. Laraen. E.: LaMar. 0. N. J. Chem Edue 1974.51.611.


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