The angular overlap model as a unified bonding model for main group

A Version Suitable for Undergraduate Inorganic Students. D. E. Richardson ... duced as a unified bondinn model in the undermaduate curriculum at the ...
0 downloads 0 Views 7MB Size
Download PDF
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 electmnic structure of polyatomic molecules in current textbooks (21on 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. I n 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 alternative 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

372

Journal of Chemical Education

many other physical properties of both metal and nonmetal compounds UndergraduateApplications

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 diatomi 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. I n 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 eomplexes. 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 AH2and AH3 main group hydrides. Rather detailed descriptions of the application of the AOM to prediction of molecular shapes have been pubwho has extensively demonstrated lished by Burdett (7,8),

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. However, 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 miscellaneous 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 h a d a n 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

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

AB, where A is a main group element or

ML 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-shelld 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 Figure 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 NH3can 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 inappropriate 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

o ligand orbitals

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.

n ligand orbitals

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

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.

S Ais 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 interactVolume 70 Number 5 May 1993

373

M-L

A

A-B

M

B

M-L

L

.Figure2. 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, Sabineq 2 is given by Sflab

where S, is the "basic" quantity of a overlap at some fixed bond length, and the relevant angle-dependent wntribution 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, a s 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 S2is the relevant quantity for the AOM calculations. In the case of a n interaction, Sabis given by Sxab

where Sx= S, and O,b = Pab 374

Journal of Chemical Education

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.

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.

Orbital Stabilization and Destabilization In MO theory, two inequivalent atomic orbitals, yr. and (the basis functions), with finite o overlap (Sabt 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 IP' ., or deiitahilize ~ . 'i'the bond when occupied by electrons. yrb

.

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 =

Energy Difference: Ligand us. Central-Atom Orbitals I n 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 electronegativity 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 ~ ~ ~and ~ b here ( o ) 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

Fig~re3. Defmtlons of angLlar overlap parameters for vanous central.atom to igand interacttons. For F = 1 , (a) central-atom p to oliqand 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. Volume 70 Number 5 May 1993

375

Z?' Parameters e, and f, are proportional to Sz and S,: re-

spectively, 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, le, l > If, I , so ~ ~ ~ will~be(negative 0 ) (denoting 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 = 1for the ideal overlap shown in Figure 3a, and from eq 3, we get

In the case of central-atom d orbitals, = 1is found for the d2 orbital interacting with a ligand o orbital as shown in Figure 3b. 'The "angularoverlapfactor" Fhas a more restricted meaning here than given to F by Burdett (ref7), who assigns it the same meaning as Bin this article. The Dresent author has introduced Fand P as angular overlap factorstodivide aand ncontributions in the notation. Table 1.

-

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.

Values

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 P2angular overlap factors. For centralatom p orbitals and d orbitals, the maximum overlaps for which P = 1are shown in Figures 3c and 3d, respectively. Tabulated values for F and P2for 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.

Notes for Table 1: Some special angles. 1

sin 30 = -0s 30 = 2

6 sin 60 = 2

$

@ sin 45 = 2 COS

@ CoS 45 = 2

1 60 = 2

For C3v molecules with tetrahedral angles, 0 = sin"(al3) = 70.53', sin e = a13. cos e = 113. 376

Journal of Chemical Education

fl 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.

pyramidal at tetrahedral angles; L = 13, 14, 15; R = 70.Y ..

' T-shape; L = 3,5,6 m 4

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 necess a w to include ir: interactions for main erouD moleculesto mLke predictions concerning molecular shape, so these contributions to orbital energies are not included. Examples 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 orbjtnl is taken as nonbondmg (a severe approxlmatlon'~A haw-orh~talenergy dlngram ~sthen constructed showng the numbers and relative energies of the basis orbitals, in

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 1and 6 for the fluorines. (3) For each central-atomp 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. 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).

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

-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

.

m 3

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

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

-2e,+ 4f,

NB Be 2p

NB Fo

41 Be

-F-Be-FY

'I

z

'

2F

2e0- 4fa

Bes

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

F~gure6. Correlat~onof AOM aerived energy levels with traditonal representations ol BeF, mo ecLlar orbia s. These orbilals assame no

s-p mixmg, which wo~lomlx 'i',,(sj

w tn YNB(o),

Volume 70 Number 5 May 1993

377

I filled

i.

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

Any of then ligand orbital combinations not stabilized remain nonbonding. Example. For linear BeF2,one LGO is stabilized by %hb=2~7-~fo

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~tnlofthe central atom and .U0's ~n order of inerrsnngenerm Then sum the indtwdual anerm 3rhlfls ofthe result~ngwcup~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 partlcular 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 o r s Orbitals That Are Higher in Energy Than Central-Atom p Orbitals I n this case, E(o) > E(P) > E(s) The stabilizations of t h e central-atom p orbitals a r e first calculated using eq 8. T h e destabilizations of t h e c o r n sponding LGO's a r e derived analogously using e q 7.

378

Journal of Chemical Education

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 basisorbital energy diagram (Fig. 7) because N is the more electronegative element. From the in Table 1and 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 t h e f, contributions to orbital energies, h u t n interactions a r e important i n many complexes and must be included via the e, term. (The f, is also ignored.) Two examples of the application of t h e 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 (N31and any other atbmic anions have two n levels per ligaid for a total of 2n. Diatomic ligands, such as CO, CN; Nz, and other isovalent 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.

The dnrbital derived MO's are filled with the valence elretrons 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 sta-

bilization or destab~liratrunof certain d orbitnls. in some cswer, d orbitals wdl he shifted hy both o and n interartions. 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 (PZvalues, Table 2).

F~gure9 AOM MO d~agrammnstrmea for Cr(CO)G.whch has ex tenswe n Wndmg The octahedral d-orbtal spllnmg parameter A 1s shown Note tne s~mplltcat 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).

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 t o o 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

(4) Far each central metal d orbital, its total "o-only" destabi-

lization (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 ( ~ f g8). . ( 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 remains nonbonding.

Example. In square planar lCutNH.,~,l'", the two eorrespondmg l~gandcombmat~onsare 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 enntributed to the MOSE by these electrons. The total MOSE (aonly) is then given fir the d9 complex by eq 12, where the sum is over the q electrons of the central atom.

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 t h e basis-orbital energies a r e placed as in Figure 9. First, o destabilizations of t h e C r 3d orbitals a r e and counted via Table 1and eq 10. It i s seen that d,, L, d, a r e nonbonding with respect to t h e ligand o interactions, whereas dzs.,P a n d ds*a r e raised by -3e,. To account for n bonding, t h e P values (Table 2) a r e applied i n e q 15, and i t is found t h a t d,, d,, a n d d, a r e stabilized by 4e, (because the C r d orbitals a r e lower t h a n t h e CO n* levels). After calmlatine t h e C r d orbital e n e r w shifts. t h e corresponding l i g a n i orbitals m u s t b e shiEed as shown i n Figure 8, with many of t h e ligand orbital combinations remaining nonbonding. The ligand field splitting parameter ("A") is indicated in Figure 9.2To fill i n the electrons, t h e 6 valence electrons for CdO) a r e entered into the shifted d orbitals, and t h e levels lower t h a n t h e dashed line a r e 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 , forthe (xy,xr, yz) orbitals. Volume 70

Number 5 May 1993

379

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

theorv. Masterv of the s i m ~ lAOM e a ~ u m a c in h this article 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 orbit, ~ mixing between s and D orbitals in main als ~ b t a i n e dand group molecules can be