Arvi Rauk- Simple Huckel Molecular Orbital Theory

8/3/2019 Arvi Rauk- Simple Huckel Molecular Orbital Theory

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CHAPTER 5

SIMPLE HU CKEL MOLECULAR

ORBITAL THEORY

In this chapter, simple Hu ckel molecular orbital (SHMO) theory is developed. The ref-erence energy, , and the energy scale in units of are introduced.

SIMPLE HU CKEL ASSUMPTIONS

The SHMO theory was originally developed to describe planar hydrocarbons withconjugated p bonds. Each center is sp2 hybridized and has one unhybridized p orbitalperpendicular to the trigonal sp2 hybrid orbitals. The sp2 hybrid orbitals form a rigidunpolarizable framework of equal CC bonds. Hydrogen atoms are part of the frame-work and are not counted. The Hu ckel equations (3.3) described in the rst part of

Chapter 3 apply, namely,

F1ehe1 he1fa1 eafa1 EIE A 2Ma1

ea 5X1

Each MO is expanded in terms of the unhybridized p orbitals, one per center. The over-lap integral between two parallel p orbitals is small and is approximated to be exactlyzero. Thus,

f1 NN

A1 cAwA1

wA1wB1 dt1 dAB 5

X

2

where NN is the number of carbon atoms which is the same as the number of orbitals.Equation (5.2) is just a generalization of equation (3.4). The subsequent steps are pre-cisely those which were followed in Chapter 3. The energy is expressed as an expectationvalue of the MO [equation (5.2)] with the eective hamiltonian

86

Orbital Interaction Theory of Organic Chemistry, Second Edition. Arvi RaukCopyright( 2001 John Wiley & Sons, Inc.

ISBNs: 0-471-35833-9 (Hardback); 0-471-22041-8 (Electronic)


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e f1h 1f1 dt1jf1j2 dt1

NNA1

NNB1 cAcB

wA1h wB1 dt1NN

A1NN

B1 cAcBwA1wB1 dt1

NN

A1 c2AhAA

NNA1

NNBHA cAcBhABNN

A1 c2A

NNA1

NNBHA cAcBSAB

5X3

NN

A1c2AhAA NN

BHA cAcBhABNNA1 c

2A

ND

5X4

e

e

and the variational method applied. Dierentiating equation (5.4) with respect to each ofthe coecients, cA,

q

qcAND1 qN

qcAD

1 ND2 qDqcA

0

qN

qcA e qD

qcA 0

hAA ecA NN

BHA hBAcB 0 for each A f1Y F F F YNNg 5X5

The condition that the NN linear equations have a solution is that the determinant ofcoecients (of the c's) be equal to zero:

h11 e h12 h13 h1NNh21 h22 e h23 h2NNh31 h32 h33 e h3NN

FF

F

FF

F

FF

F

FF

F

hNN1 hNN2 hNN3 hNNNN e

0 5X6

Within the SHMO approximations, all of the diagonal hamiltonian matrix elements,hAA, are equal and are designated . The Hu ckel is the energy of an electron in a 2porbital of a trigonally (sp2) hybridized carbon atom. The o-diagonal matrix elements,hAB, are all equal if the atoms involved are bonded together (since all bond distances areassumed to be equal) and these are designated . The Hu ckel is the energy of interac-tion of two 2p orbitals of a trigonally (sp2) hybridized carbon atoms which are attached

to each other by a sbond. If the two atoms are not nearest neighbors, then hAB is setequal to zero. In summary,

hAA hAB if centers A and B are bondedhAB 0 if centers A and B are not bonded

SIMPLE HU CKEL ASSUMPTIONS 87


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Thus, all of the diagonal elements are e. The o-diagonal elements are if the twoatoms involved are bonded and zero if they are not. It is usual to divide each rowof the determinant by . This corresponds to a change of energy units and leaves either1's or 0's in the o-diagonal positions which encode the connectivity of the molecule.The diagonal elements become ea, which is usually represented by x. While the de-terminant was expanded in Chapter 3 to yield the secular equation, it is more conve-nient in general to diagonalize the determinant using a computer. An interactive computerprogram, SHMO, has been written to accompany this book [102].

The SHMO calculation on ethylene yields the results shown in Figure 5.1a. The p

and p orbitals are precisely one unit above and below . The SHMO results are pre-sented in Figure 5.1b in the form of an interaction diagram. In this case, heL and heUare assigned the same value, namely 1jj, in the spirit of SHMO theory, but we knowthat the eect of proper inclusion of overlap would yield heU b heL.

The SHMO results for the series of ``linear'' p systems allyl, butadiene, and penta-dienyl are shown in Figure 5.2. The molecular species are portrayed with realistic angles(120) and, in the case of the last two, in a specic conformation. Simple Hu ckel MOtheory does not incorporate any specic geometric information since all non-nearest-neighbor interactions are set equal to zero. As a result, the SHMO results (MO energiesand coecients) are independent of whether the conformation of butadiene is s-trans, as

shown in Figure 5.2, or s-cis, as may be required for the DielsAlder reaction. Likewisethe results for the pentadienyl system shown in the ``U'' conformation in Figure 5.2 areidentical to the results for the ``W'' or ``sickle'' conformations. The MOs are displayed aslinear combinations of 2p atomic orbitals seen from the top on each center, with changesof phase designated by shading. The relative contribution of each atomic 2p orbital tothe p MO is given by the magnitude of the coecient of the eigenvector from the solu-tion of the SHMO equations. In the display, the size of the 2p orbital is proportional

h

h

a bFigure 5.1. (a) SHMO results for ethylene. (b) The interaction diagram for ethylene: note that

heL heU because overlap is assumed to be zero in SHMO theory.

88 SIMPLE HU CKEL MOLECULAR ORBITAL THEORY


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to the magnitude of the coecient. In some MOs, such as p2 of allyl, the node passesthrough a nucleus and a 2p orbital is not shown because its coecient is identically zero.The orbitals which are near will be of most interest in various applications. In the caseof allyl, this orbital, p2, is LUMO in the allyl cation, SOMO (singly occupied molecularorbital) in the allyl radical, and HOMO in the allyl carbanion. In the pentadienyl sys-tem, p3 plays the same role. In butadiene, the HOMO is p2. Since the energy of the

HOMO is higher than the energy of the HOMO of ethylene, one might conclude thatbutadiene is more basic than ethylene and more reactive toward electrophilic addition.Caution should be exercised in jumping to this conclusion, however, since the largestcoecient of butadiene's HOMO, 0.60, is smaller than the coecient of the 2p orbitalof the HOMO of ethylene, 0.71. The smaller coecient would imply a weaker intrinsicinteraction (hAB) with Lewis acids and therefore reduced reactivity. Clearly, the energyfactor and the intrinsic interaction (as judged from the coecients) are in opposition,the rst predicting higher reactivity and the second lower. As is often the case in orbitalinteraction theory, one must resort to experimental observations to evaluate the relativeimportance of opposed factors. Since, experimentally, dienes are more susceptible to elec-

trophilic attack than unconjugated alkenes, we can conclude that the energy factor ismore important than the relatively small dierence in coecients.Simple Hu ckel MO results for the series of cyclic p systems cyclopropenyl, cyclo-

butadiene, cyclopentadienyl, and benzene are shown in Figure 5.3. Several points maybe noted. The lowest MO in each case has energy identical to 2jj, a result which canbe proved to be general for any regular polygon. Each ring has degenerate pairs of MOsas a consequence of the three- or higher-fold axis of symmetry. The orbitals of each

Figure 5.2. The SHMO orbitals of allyl, butadiene, and pentadienyl. The vertical scale is energy in

units ofjj, relative to . Coecients not specied may be obtained by symmetry.

SIMPLE HU CKEL ASSUMPTIONS 89


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degenerate pair may appear very dierent. The orientation of the nodal surfaces of thedegenerate MOs is entirely arbitrary. Two equivalent orientations are shown for cyclo-

butadiene. A perturbation at one of the vertices such as by a substituent will rotate thenodes of the degeneate set so that one node passes through that vertex. The orientationsshown are those which should be adopted for the purposes of interaction diagrams involv-ing a single substituent on the ring. The pairs of degenerate MOs which form the HOMOof benzene is at exactly the same energy as the HOMO of ethylene. In orbital interactionterms, we would predict that, even though the HOMOs are of the same energy, benzenewould be less susceptible to electrophilic attack than ethylene for the reason that the

Figure 5.3. SHMO orbitals for cyclopropenyl, cyclobutadiene, cyclopentadienyl, and benzene. Theenergies are in units ofjj relative to . Two alternative but equivalent representations are shown forthe degenerate p orbitals of cyclobutadiene. Sizes of the 2p orbitals are shown proportional to the

magnitudes of the coecients whose numerical values are given. Coecients not specied may beobtained by symmetry.



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largest available coecient, 0.58 in p3, is smaller than 0.71, the coecient in the HOMOof ethylene.

Conjugated p systems which do not contain any odd-membered rings are called

alternant, and provided all the atoms are the same, alternant systems have a symmetricaldistribution of orbital energies about the mean ( for C). The coecients also repeat inmagnitude in MOs which are equidistant from the mean. These features are readilyapparent in the orbitals portrayed in Figures 5.15.3.

Of the four cyclic conjugated p systems shown in Figure 5.3, only benzene, with sixelectrons in the p orbitals, is stable kinetically and thermodynamically. Neutral cyclo-propenyl and cyclopentadienyl, with three and ve p electrons, respectively, are free radi-cals. The cyclopropenyl cation, with two p electrons, and the cyclopentadienyl anion,with six, have lled shells and constitute aromatic systems in that they exhibit unusualstability, compared to other carbocations and carbanions, respectively. Cyclobutadiene

is a special case. With two electrons in the degenerate HOMOs, one would expect thatthe electrons would separate and that the ground state would be a triplet. However,a distortion of the geometry from square to rectangular would eliminate the degeneracyand permit a singlet ground state. The ground state of cyclobutadiene has been shownexperimentally to be singlet with a barrier for the rectangular-to-square deformation inthe range 7hHz 42 kJ/mol [103]. Theoretical computations suggest that the highervalue may be correct [104].

CHARGE AND BOND ORDER IN SHMO THEORY: (SABF0, ONE ORBITAL

PER ATOM)

It is of interest to enquire how the electrons are redistributed during an interaction andhow a bond is aected. We use a simplied Mulliken population analysis [see AppendixA, equations (A.77)(A.79)]. The simplication consists of dropping all terms involvingthe overlap of atomic orbitals and assuming that, in any given MO, there is only oneatomic orbital on any given center). Thus, we may assume that the following relationshold:

fa n

A1wAcAa SAB dAB

n

A1c2Aa 1 5X7

where n is the number of atomic orbitals (which equals the number of nuclear centers,since there is one orbital per center). A measure of the electron population on each centeris easily obtained as below.

Electron Population and Net Charge of Center A

The electron population of center A is dened as

PA na1

nac2

Aa 5X8

where na 0Y 1Y 2 is the number of electrons in the ath MO and the sum runs over theMOs (there are as many MOs as there are AOs and atomic centers). The net charge of

CHARGE AND BOND ORDER IN SHMO THEORY 91


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center A, CA, depends on the number of electrons, nA, which are required for a neutralatomic center A. Thus

CA nA PA 5X9

Notice that CA is negative if the population of A, PA, exceeds the number of electronsrequired to give a neutral center. It is easily veried that

nA1

PA NenA1

CA net charge on molecule 5X10

Exercise 3.1. Verify equations (5.10).

Exercise 3.2. Verify that the net charge at each carbon atom of each of the neutral ringsystems shown in Figure 5.3 is zero (to two signicant gures).

Bond Order between Centers A and B

Strictly speaking, there should be no electron population between pairs of atoms inSHMO since orbitals are assumed not to overlap. However, it is conventional to set alloverlap integrals to unity for the purpose of dening a ``bond order.'' The bond order,BAB, between centers A and B is dened as

BAB na1

nacAacBa 5X11

where the quantities are dened as above. A positive value indicates bonding. Smallnegative values of BAB may result. These are indications of antibonding or repulsiveinteraction between the centers concerned.

Exercise 3.3. Show that the maximum value for the bond order due to a single bond is

1 and that it occurs when na 2 and cA cB 1a 2p

.

Exercise 3.4. Show that the bond order for benzene is 0.67 from the data in Figure 5.3.

FACTORS GOVERNING ENERGIES OF MOs: SHMO THEORY

Reference Energy and Energy Scale

Most organic molecules are made up of the elements C, H, N, and O, with lesser amounts

of S, P, and X (Cl, Br, I). Molecular orbitals are built up by the interaction of the atomicorbitals of these elements held together at bonding separations. It is convenient at thispoint to adopt an energy scale derived from SHMO theory, in which the ``Coulomb inte-gral'' C is the reference point on the energy scale,

C 2pC1he12pC1 dt1 5X12



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and the absolute value of the ``resonance integral'' jj jCCj is the unit of energy,

CC 2pC1Ahe

12pC1B dt1 5X13

Thus, C is the core energy of an electron localized to the 2p atomic orbital of acarbon atom, and CC is the energy associated with the interaction of two carbon2p orbitals overlapping in a p (parallel) fashion at the CC separation of benzene (orethylene).

HETEROATOMS IN SHMO THEORY

In SHMO, the core energies of heteroatoms, X, are specied in terms of and , andthe interaction matrix elements for p orbitals overlapping in a p fashion on any pair ofatoms, X and Y, are specied in terms of. Thus,

X hXjj 5X14XY kXYjj 5X15

In SHMO theory the energy ofp bond formation in ethylene is 2 (since the strength ofa CC p bond is about 280 kJ/mol, one may consider

j

jto be about 140 kJ/mol). The

energies of electrons in 2p orbitals of N and O, normally found in p bonding environ-ments (i.e., as dicoordinated N and monocoordinated O) are given by hN2 0X51 andhO1 0X98. These are suitable values for a pyridine N and a carbonyl O. The p-typeinteraction matrix elements of most pairs, except those involving F, are approximatelygiven by kXY 1. The energies of electrons in 2p orbitals of N and O in normal (satu-rated) bonding environments (i.e., as tricoordinated N and dicoordinated O) are givenby hN3X 1X37 and hO2 2X05. Thus a tricoordinated N is more electronegative thana dicoordinated N, and similarly for dicoordinated versus monocoordinated O. The p-type interaction matrix elements of these kinds of orbitals with the normal C 2p orbitalsare approximately given by kCN

0X8 and kCO

0X67, reecting the smaller size of

the orbitals. The value for the CF interaction, kCF 0X5, is small for the same rea-sons. The change in eective electronegativity of the 2p orbitals is a consequence of theincrease in the number of atoms to which they are coordinated (see below), although inusual SHMO usage, the distinction is made in terms of the number of electrons whichthe atom contributes to the p system. A more complete list of hX and kXY parameters,derived on the basis of Pariser-Parr-Pople (PPP) calculations by Van Catledge [105], isgiven in Table 5.1.

Effect of Coordination Number on and

A decrease in the coordination number at N from three to two reduces the eectiveelectronegativity of the remaining nonbonded p orbitals at the center. The change in hNis 0X86jj. A decrease in the coordination number at O from 2 to 1 similarly reduces theeective electronegativity of the remaining nonbonded p orbitals at the O center. Thechange in hO is larger, 1X12jj. Within the same molecule, the coordination number ofN or O may readily be changed by the process of protonation or deprotonation, as

HETEROATOMS IN SHMO THEORY 93


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TABLE5.1.

SHMO

ValuesforHeteroatoms:X

hXjj,XY

kXYjja

X

hX

kXY

byYAtoms

Numberof

Electrons

Cb

Bb

N2c

N3b

O1d

O2c

Fd

Sib

P2

c

P3b

S1d

S2c

Cld

Cb

1

0.0

0

1.0

0

Bb

0

0.4

5

0.7

3

0.8

7

N2c

1

0.5

1

1.0

2

0.6

6

1.0

9

N3b

2

1.3

7

0.8

9

0.5

3

0.9

9

0.9

8

O1d

1

0.9

7

1.0

6

0.6

0

1.1

4

1.1

3

1.2

6

O2c

2

2.0

9

0.6

6

0.3

5

0.8

0

0.8

9

1.0

2

0.9

5

Fd

2

2.7

1

0.5

2

0.2

6

0.6

5

0.7

7

0.9

2

0.9

4

1.0

4

Sib

1

0.0

0

0.7

5

0.5

7

0.7

2

0.4

3

0.6

5

0.2

4

0.1

7

0.6

4

P2c

1

0.1

9

0.7

7

0.5

3

0.7

8

0.5

5

0.7

5

0.3

1

0.2

1

0.6

2

0.63

P3b

2

0.7

5

0.7

6

0.5

4

0.8

1

0.6

4

0.8

2

0.3

9

0.2

2

0.5

2

0.58

0.6

3

S1d

1

0.4

6

0.8

1

0.5

1

0.8

3

0.6

8

0.8

4

0.4

3

0.2

8

0.6

1

0.65

0.6

5

0.6

8

S2c

2

1.1

1

0.6

9

0.4

4

0.7

8

0.7

3

0.8

5

0.5

4

0.3

2

0.4

0

0.48

0.6

0

0.5

8

0.63

Cld

2

1.4

8

0.6

2

0.4

1

0.7

7

0.8

0

0.8

8

0.7

0

0.5

1

0.3

4

0.35

0.5

5

0.5

2

0.59

0.6

8

a

Ref.

105.

b

Tricoordinated,plana

rgeometry.

cDicoordinated.

d

Monocoordinated.



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a consequence of a change of pH, for instance. The results of SHMO calculations on theenolate anion and on enol are shown in Figure 5.4. In the enolate case, hO1 0X97,kCO 1X06, while for enol, the values hO2 2X09, kCO 0X66 were used accordingto Table 5.1. The contributions of the individual 2p orbitals of C and O are displayedwith sizes proportional to the magnitudes of the coecients. The eect of the proto-nation can be seen as a lowering of the energy and changed polarization of all of the

MOs, including the HOMO.Normally, no distinction is made between the kind of atom or group which is co-ordinated to the center of interest, but this may be a gross oversimplication in extremecases. It is not reasonable to expect that the 2p orbital of a methyl group will haveapproximately the same energy as the 2p orbital of a triuoromethyl group (assumingit were planar). Because of the strong inductive eect of the electronegative uorineatoms acting in the s framework, the carbon atom of triuoromethyl would be signi-cantly denuded of electrons. The 2p orbital is in eect more electronegative and fallsbelow .

Can one deduce reasonable values for the eective electronegativity of the p orbitals

of C upon reduction of the coordination number from 3 to 2 (i.e., C2), as in alkynes,allenes, nitriles (RCN), or carbenes, or even to 1, as in CO, isonitriles (RNC), oracetylides? A linear extrapolation from dicoordinated O O2 2X09jj and di-coordinated N N2 0X51jj to dicoordinated C yields an estimate of the energy ofthe 2p orbital as 0X86jj. This value is probably too high. It places the energy of the2p orbital of a dicoordinated carbon above that of the 2p of a tricoordinated boron, butthe same is not true in the case of a dicoordinated N and a tricoordinated C, and the

a bFigure 5.4. SHMO results for enolate (a, using O1 parameters) and enol (b, with O2 parameters).

HETEROATOMS IN SHMO THEORY 95


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electronegativity dierences in the series B, N, C (Table 5.1) are very similar. A reason-able compromise is to place C2 midway between B and C,

C2 0X23jj 5X16Hybridization at C in Terms of and

The 2s to 2p promotion energy of atomic C is about 800 kJ/mol. In a molecular envi-ronment, this value is expected to be somewhat less where the presence of other nucleimay stabilize p orbitals relative to s. The coordination number of the carbon atom has adirect eect on the orbital energies, just as it had on the energies of heteroatom orbitalsdiscussed in the previous section. Mullay has estimated the group electronegativities ofCH3, CHCH2, and CCH to be 2.32, 2.56, and 3.10, respectively [106]. The last value issimilar to his estimate for NH2. Boyd and Edgecombe have placed all three values near2.6 [107]. Reed and Allen, using their bond polarity index, have assigned values of 0.000,0.027, and 0.050, respectively (compared to H 0X032 and F 0.189) [108]. Withoutattempting to be too quantitative, convenient values of the core energies of ``hybrid''atomic orbitals, in jj units, recognizing that changes in coordination number also occur,are approximately

sp 0X50jj coordination number 1 5X17sp2 0X33jj coordination number 2 5X18sp3

0X25

j

j coordination number 3

5X19

The interaction energies of the pairs of hybridized orbitals interacting in a s fashionwould be strongly distance dependent. At typical single-bond distances, one may adoptks 1X5kpjj for all of them, but the value rises steeply as the separation is decreased.When the separation is that of a double bond, a value of ks 2X0kpjj is more appro-priate. These values are suggested only to help place sbonds or s orbitals more or lesscorrectly relative to p bonds and p orbitals when both may have similar energies.

GROSS CLASSIFICATION OF MOLECULES ON THE BASIS OF MO ENERGIES

Frontier orbital energy is not the only criterion which governs the chemical character-istics of a compound. For example, the magnitudes of the atomic orbital coecientson any given atom may be responsible for the reduced basicity of benzene relative toethylene, both of which have the same HOMO energy. Nevertheless, the energy criterionmay be applied to deduce gross features. In Figure 5.5 are shown four extreme casesinto which molecules can be categorized on the basis of their frontier orbital energies.Compounds which have a large HOMOLUMO gap b1X5jj will be stable against self-reaction, for example, dimerization, polymerization, and intramolecular rearrangements.If the HOMO is low in an absolute sense (` 1jj, the HOMO of ethylene), the com-pound will be chemically resistant to reaction with Lewis acids. If the LUMO is high inan absolute sense (b 1jj, the LUMO of ethylene), the compound will be chemicallyresistant to reaction with Lewis bases:

. Compounds with a high LUMO and a low HOMO (Figure 5.5a) will be chemi-cally inert. Saturated hydrocarbons, uorocarbons, and to some extent ethers fallin this category.



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. Compounds with a low HOMO and LUMO (Figure 5.5b) tend to be stable to self-reaction but are chemically reactive as Lewis acids and electrophiles. The lower theLUMO, the more reactive. Carbocations, with LUMO near , are the most power-

ful acids and electrophiles, followed by boranes and some metal cations. Where theLUMO is the s of an HX bond, the compound will be a LowryBronsted acid(proton donor). A LowryBronsted acid is a special case of a Lewis acid. Wherethe LUMO is the s of a CX bond, the compound will tend to be subject tonucleophilic substitution. Alkyl halides and other carbon compounds with ``goodleaving groups'' are examples of this group. Where the LUMO is the p of a CXbond, the compound will tend to be subject to nucleophilic addition. Carbonyls,imines, and nitriles exemplify this group.

. Compounds with a high HOMO and LUMO (Figure 5.5c) tend to be stable to self-reaction but are chemically reactive as Lewis bases and nucleophiles. The higherthe HOMO, the more reactive. Carbanions, with HOMO near , are the mostpowerful bases and nucleophiles, followed by amides and alkoxides. The neutralnitrogen (amines, heteroaromatics) and oxygen bases (water, alcohols, ethers, andcarbonyls) will only react with relatively strong Lewis acids. Extensive tabulationsof gas-phase basicities or proton anities (i.e., hG of protonation) exist [109,110]. These will be discussed in subsequent chapters.

. Compounds with a narrow HOMOLUMO gap (Figure 5.5d) are kinetically re-active and subject to dimerization (e.g., cyclopentadiene) or reaction with Lewisacids or bases. Polyenes are the dominant organic examples of this group. The

diculty in isolation of cyclobutadiene lies not with any intrinsic instability of themolecule but with the self-reactivity which arises from an extremely narrowHOMOLUMO gap. A second class of compounds also falls in this category,coordinatively unsaturated transition metal complexes. In transition metals, theatomic n d orbital set may be partially occupied and/or nearly degenerate with thepartially occupied n 1 sp n set. Such a conguration permits exceptional reactivity,even toward CH and CC bonds. These systems are treated separately inChapter 13.

a b c dFigure 5.5. (a) High LUMO, low HOMO, large HOMOLUMO gap; thermodynamically stable

and chemically inert. (b) Low LUMO, low HOMO, large HOMOLUMO gap; thermodynamically

stable and chemically reactive as Lewis acid. (c) High LUMO, high HOMO, large HOMOLUMOgap; thermodynamically stable and chemically reactive as Lewis base. (d) Low LUMO, high HOMO,

small HOMOLUMO gap; may be thermodynamically stable but chemically amphoteric and self-

reactive.

GROSS CLASSIFICATION OF MOLECULES ON THE BASIS OF MO ENERGIES 97

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Arvi Rauk- Simple Huckel Molecular Orbital Theory

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