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2 Amplitude of Orbitals: Interactions of Different Orbitals

2 Amplitude of Orbitals: Interactions of Different Orbitals

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Elements of a Chemical Orbital Theory



5



Scheme 5  Interaction between different

orbitals



Scheme 6 illustrates the orbitals of the polar s bond in methane resulting from the

interaction between the 1s atomic orbitals of a hydrogen atom and a sp3 hybrid orbital

of the carbon atom. The energy (−13.6 eV) of the 1s orbital is higher than that (−13.9

eV) of the hybrid orbital. The major component of the bonding orbital is the hybrid

orbital on the carbon. This can be compared to the polarized C–H bond with slightly

negatively charged carbon atom and positively charged hydrogen atom. The antibonding

orbital is polarized in the reverse direction with 1s as the major component.



1s

sp3



δ+



δ−



H



C



Scheme 6  Orbitals of a polar s bond in CH4



The interaction of the p-orbitals in the carbonyl C=O group is illustrated in

Scheme 7. The major component of the bonding orbital is the p-orbital of the oxygen

atom lower (−17.8 eV) in energy than that (−11.4 eV) of the carbon atom. The

carbonyl p bond is polar. The oxygen atom is negatively charged and the carbon

atom is positively charged. The antibonding orbital is polarized in the reverse direction.

The p-orbital of the carbon atom is the major component. The relative energies of

atomic orbitals can be guessed from the electronegativity. The energy decreases

with the electronegativity.



6



S. Inagaki



C O



C



δ+



O



C



O



δ−



C O



Scheme 7  Orbitals of a polar p bond



The bond orbitals of sC–H and pC=O relate to the other property of waves apart

from the phase, that is, the amplitude. The bonding orbitals have large amplitudes

on the low-lying atomic orbitals, i.e., on C of sC–H and on O of pC=O (Scheme 8).

The antibonding orbitals have large amplitudes on the high-lying atomic orbitals.



C



O



C



O



Scheme 8  Amplitudes of orbitals



1.3  Strength of Orbital Interactions

The orbital interactions are controlled by the overlap integrals (Scheme 9) and the

energy gap between the orbitals (Scheme 10):

1. The orbital overlap strenthens the interaction

2. The energy gap weakens the interaction



Scheme 9  Overlap strengthens the

interaction



strong



weak



Elements of a Chemical Orbital Theory



7



Scheme 10  Orbital energy gap De weakens

the interaction

∆ε



∆ε



strong



weak



As the interaction is strong, the in-phase combined orbital is stabilized and the outof-phase combined orbital is destabilized. The energy splitting increases between

the in-phase and out-of-phase combined orbitals.

The ionization energies of ethylene and acetylene (Scheme 11) give experimental

evidence of the effects of the orbital overlap on the interaction (Scheme 9). The p

bonding orbitals results from the interaction of the carbon p orbitals. There is no difference in the energy gap. The strength of the interaction is determined by the overlap.

The atomic distance is shorter in acetylene. The p orbitals have greater overlap with

each other. The interaction is stronger. It follows that the bonding orbital lies lower in

energy and that the ionization energy is higher. This is in agreement with the observed

high ionization energy of 11.40 eV for acetylene relative to 10.51 eV for ethylene.



−10.51eV

−11.40eV



H

C

H



H

C

H



H C



C H



1.20Å



1.34Å



Scheme 11  Experimental evidence of the relation between the overlap and the interaction: the

ionization energies of ethylene and acetylene



Substituent effects on the rate constants of SN1 reactions give experimental

evidence of the relation between the energy gap and the interaction. Alkyl substitutions on the carbon atoms bonded to the leaving group X accelerate the reaction.

Alkoxy substitutions accelerate it further. The transition state is late. The geometry

is close to the that of the reaction intermediate carbocation. The rate is qualitatively

estimated by the stability of the carbocation. The carbocations are generally planar.

There is a vacant p orbital on the carbon atom. The sCH bonds interact with the ionic

center. According to the rule for the interaction of different orbitals, the bonding

orbitals of the sCH bonds interact and mix with vacant p orbital in phase to be lowered in energy (Scheme 12). The sCH bonds and therefore the carbocation are stabilized. This is the stabilization by the hyperconjugation. In the RO-substituted



8



S. Inagaki



pC

∆ε



∆ε



C



H

H



nO



σCH



CH2CH3



O



CH3



CH2OCH3



Scheme 12  Experimental evidence of the relation between the energy gap (De) and the interaction:

the substituent effects on the stabilities of the carbocations



carbocations, a lone pair interacts with the cation center. The lone pair orbitals lie

higher in energy than sCH. The ionization energies of the oxygen lone pairs (10.94,

10.64, 10.04, 9.61 eV for CH3OH, C2H5OH, (CH3)2O, and (C2H5)2O, respectively)

are lower than those of the alkanes (13.6, 11.99, 11.51 eV for CH4, C2H6, and C3H6,

respectively). The oxygen lone pairs are closer in energy to the vacant p-orbital.

The narrow energy gap leads to stronger interaction and more stabilization of the

in-phase combined orbital as stated above as a rule of the orbital interaction

(Scheme 10). This is the stabilization by the resonance.



1.4  Electron Delocalization

Delocalization of electrons is important in chemistry. Electron delocalization is a

major factor of the stabilities and the reactivities of molecules. The delocalization

occurs through the interaction of an occupied orbital with a vacant orbital (Scheme

13). The two electrons occupy the stabilized orbital. There are no electrons in the

destabilized orbital. The stabilization results from the interactions between the

occupied and unoccupied orbitals.



stabilized

(a) stabilization



Scheme 13  Electron delocalization and stabilization by

the interaction between the occupied and unoccupied

orbitals



(b) electron delocalization



Elements of a Chemical Orbital Theory



9



The electrons occupy the in-phase combined orbital after the interaction. They

are distributed not only in the orbital occupied prior to the interaction, but also in

the overlap region and the orbital vacant prior to the interaction. The electrons

localized in the occupied orbital before the interaction delocalize to the overlap

region and the vacant orbital after the interaction (Scheme 13).

Electron delocalization occurs through the interaction between the occupied and

unoccupied orbitals and leads to the stabilization.



1.5  Exchange Repulsion

The interaction between the occupied orbitals leads to the destabilization (Scheme

14). The two electrons in the stabilized orbital lead to stabilization, but there are

two more electrons, which occupy the destabilized orbitals. The destabilization

overcomes the stabilization, and net destabilization results.



destabilized



stabilized

(a) destabilization



Scheme 14  Exchange repulsion and destabilization

by the interaction between the occupied orbitals



(b) exchange repulsion



Two electrons occupy the in-phase combined orbital. The probability density

increases in the overlap region. Two more electrons occupy the out-of-phase combined

orbital and reduce the density there. The decrease is greater than the increase.

The electrons are expelled from the overlap region.

The destabilization is caused by the exchange of electrons between the occupied

orbitals through the orbital overlap. The force is then termed exchange repulsion or

overlap repulsion. The exchange repulsion is a major cause of the steric repulsion.

There are many occupied orbitals in the sterically crowded space.



1.6  Stabilization and Number of Electrons

In the interaction of a pair of atomic orbitals, two electrons form a bond and four

electrons form no bond (Sect. 1.1). The substituted carbocations are stabilized by

the electron delocalization (hyperconjugation and resonance) through the interaction

of the doubly occupied orbitals on the substituents with the vacant p-orbital on the

cation center. The exchange repulsion (Sect. 1.5) is caused by four electrons. Now



10



S. Inagaki



we see that two-electron interaction leads to the stabilization and four-electron

interaction leads to the destabilization. The stabilization/destabilization by the

orbital interaction is determined by the number of electrons.

Radicals and excited states have an orbital occupied by one electron. The interaction

of the singly occupied orbital with a vacant orbital (Scheme 15) and with a singly

occupied orbital (Scheme 16) leads to the stabilization. The stabilized orbitals

occupy one and two electrons, respectively. There are no electrons in the destabilized

orbital. For the interaction with a doubly occupied orbital there are two electrons in

the stabilized orbital and one electron in the destabilized orbital (Scheme 17).

Although the destabilization of the out-of-phase combined orbital is greater than

the stabilization of the in-phase combination, there is one more electron in the stabilized

orbital. Net stabilization is then expected.

The particpation of one through three electrons in the orbital interaction gives

rise to stabilization. The destabilization occurs when four electrons participate.



stabilized



Scheme 15  The stabilization by the interaction

between a singly occupied orbital and a vacant orbital



stabilization



stabilized

stabilized



Scheme 16  The stabilization by the interaction between singly occupied orbitals



stabilization



destabilized



Scheme 17  The stabilization by the interaction between singly and doubly occupied

orbitals



stabilized

stabilization



Elements of a Chemical Orbital Theory



11



2 Applications to Molecular Properties: Interactions

of Bond Orbitals

Chemists have developed, established, and advanced an idea of chemical bonds

which localize between a pair of atoms. The idea is useful for understanding and

designing molecules and chemical reactions. Chemists will never give up the idea

of chemical bonds.

We have learned about bond orbitals which represent chemical bonds. In this

section, we learn how interactions of bonds determine molecular properties.

Interactions of bond orbitals give molecular orbitals, which show behaviors of the

electrons in molecules.



2.1  From Bond Orbitals to Molecular Orbitals

Butadiene has two p bonds. The interaction between the two p bonds is one of the

simplest models to derive molecular orbitals from bond orbitals. A p bond in butadiene is similar to that in ethylene. The p bond is represented by the bonding and

antibonding orbitals. The interactions occur between the p bonds in butadiene. The

bond interactions are represented by the bond orbital interactions.

The bonding orbitals pa and pb of ethylenes are combined in phase to be the lowest

p molecular orbitals (p1) of butadiene (Scheme 18). The out-of-phase combined orbital

(p2) is the highest occupied molecular orbital (HOMO). The in-phase combination of

out of phase



p a*



p b*



in phase

out of phase



pa



pb



in phase



Scheme 18  The p molecular

orbitals of butadiene from the

bond orbitals



H

H



C



a



C



H



H



H



H



H

C C

H

C C

H

H



H

H



C



b



C



H

H



12



S. Inagaki



the antibonding orbitals (p3) gives the lowest unoccupied molecular orbital (LUMO)

of butadiene. The out-of-phase combination gives the highest molecular orbital (p4).

There is energy gaps between the bonding and the antibonding orbitals (between

pa and pb*, between pa* and pb), but no gaps between the bonding orbitals pa and

pb). and between the antibonding orbitals (pa* and pb*). The pa–pb* and pa*–pb

interactions are weak relative to the pa–pb and pa*–pb*interactions (Sect. 1.3), and

can thus be neglected here.



2.2  Energy, Phase, and Amplitude of Orbitals

The energies, the phases and the amplitudes of the p molecular orbitals of butadiene

are shown in Scheme 19. The p1, p2 , p3, and p4 orbitals corresponds to half, one,



Scheme 19  The energies, the phases

and the amplitudes of the p molecular

orbitals of butadiene



Elements of a Chemical Orbital Theory



13



one and a half, and two waves, respectively. The energies of the p orbitals increase

with the number of waves or with the number of out-of-phase combined neighboring

pairs of the atomic orbitals. The amplitudes at the inner and terminal p-orbitals in

Scheme 18 are identical to each other because the bond orbitals of ethylene are

combined. The actual p molecular orbitals have larger amplitudes at the inner

p-orbitals in p2 and p3, and at the terminal p-orbitals in p1 and p4. The difference in

the amplitudes cannot be reproduced until the interactions between p and p* of

ethylene are taken into consideration (Chapter “Orbital Mixing Rules”).

The energy, the phase, and the amplitude characterize salient features of orbitals.

This can be seen in atomic orbitals and bond orbitals (Sect. 1).



2.3  Ionization Energies

The energy splitting by the orbital interaction is confirmed by the ionization energies

of ethylene and butadiene. The ionization energy of ethylene is 10.51 eV. The first

and second ionizations are observed at 9.09 and 11.55 eV for butadiene. One is lower

than that of ethylene, the other being higher. This is in agreement with the orbital

energy ordering: the p1 and p2 orbitals of butadiene lie lower and higher than p of

ethylenes, respectively. The difference 1.42 eV of p2 from p is greater than that (1.04

eV) of p1 from p. This is in agreement with the rule that the out-of-phase orbital (p2)

is destabilized more than the in-phase combined orbital (p1) is stabilized.



2.4  Electronic Spectra

The p orbitals of butadiene (Scheme 18) qualitatively obtained from the orbitals of

ethylenes are also supported by the electronic spectra of polyenes. The HOMO of

butadiene is higher that the HOMO of ethylene since the former is the out-of-phase

combination of the latter. The LUMO of butadiene is the in-phase combination of

the LUMOs of ethylene and lies lower than the LUMO of ethylene. The energy gap

between the HOMO and the LUMO is smaller in butadiene. In fact, the wavelength

(lmax) is longer for butadiene (217 nm) than for ethylene (165 nm). The wavelength

increases with the chain length of the polyenes.



3 Applications to Chemical Reactions: Interactions

of Frontier Orbitals

Atomic orbitals interact with each other to give bond orbitals (Sect. 1), which mutually interact to give molecular orbitals (Sect. 2). Here we will examine interactions

of molecular orbitals, especially those of frontier orbitals important for chemical

reactions.



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