Pericyclic chemistry orbital mechanisms and stereochemistry

PERICYCLIC
CHEMISTRY


PERICYCLIC
CHEMISTRY
Orbital Mechanisms and
Stereochemistry
DIPAK K. MANDAL
Formerly of Presidency College/University
Kolkata, India


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Dedicated to the memory of my parents.


PREFACE
It is about 50 years since the appearance of the pioneering work of
R. B. Woodward and R. Hoffmann on the theory of conservation of orbital
symmetry in concerted reactions. The word pericyclic was introduced in 1969

and the application of the concept of orbital symmetry to pericyclic reactions
proved to be a major turning point in understanding organic reaction mechanisms. The 1981 Nobel Prize in Chemistry was awarded to K. Fukui and
R. Hoffmann for developing theories of pericyclic reactions (Woodward died
in 1979 at the age of 62 and could not share this prize; however, he won the
Nobel Prize in 1965 for his work on organic synthesis). Pericyclic reactions
have a remarkable quality of being manifold, extremely elegant, and highly
useful; they reveal stereochemical intricacies and idiosyncrasies and remain
as an integral part of chemistry teaching and research.
Pericyclic chemistry is covered in every graduate course and in advanced
undergraduate courses in organic chemistry. Ergo, this book is addressed
principally to an audience of graduate and advanced undergraduate students.
The purpose of writing this book is entirely pedagogic, keeping in view
that our students crave understanding, not factual knowledge alone. The
book evolves from a series of lecture notes and students’ feedback during
my teaching this course to graduate students for more than 20 years. The
mechanistic descriptions and the stereochemistry resulting from orbital
mechanisms are at the heart of this book; the synthesis of specific target
molecules has been generally given short shrift.
The book contains eleven chapters. An introduction to molecular orbital
theory (Chapter 1) and relevant stereochemical concepts (Chapter 2) have
been provided as a background aid to follow the chapters on pericyclic
chemistry. In the introductory chapter (Chapter 3), I have introduced all
four classes of pericyclic reactions involving three mechanistic approaches
linked through orbital picture representation. This unifying and integrated
style would help enhance the pedagogy of this text. The qualitative perturbation molecular orbital theory has been incorporated as the most accessible
and useful approach to understanding many aspects of reactivity and selectivity. Three chapters (Chapters 4–6) have been devoted to cycloadditions,
the most versatile class, one to electrocyclic reactions (Chapter 7), two to
sigmatropic rearrangements (Chapters 8 and 9), and one to group transfer
reactions (Chapter 10). A separate chapter (Chapter 11) is included to

xv


xvi

Preface

illustrate the construction of correlation diagrams in a practical,
‘how-to-do-it’ manner.
Besides the unifying approach of mechanistic discussion, the most
important difference between this book and others is the emphasis on stereochemistry, specifically how to delineate the stereochemistry of products.
I have found that students are not often quite comfortable to work stereochemistry for themselves. After all, reaction stereochemistry is not easy!
Students need some more help. To address their concerns, I have always
been looking for innovative approaches to stereochemical issues. My efforts
have resulted in formulating simple stereochemical rules/guidelines, some of
which have been published in the Journal of Chemical Education. These published (also some unpublished) rules/mnemonics have been used extensively
in the relevant chapters as an aid to write quickly and correctly the product
stereochemistry in pericyclic reactions.
Usually, the problem sets are given at the end of chapters without or with
answer keys. One pedagogical decision I have made with respect to problem
sets is that more than 130 problems are inserted within the chapters with
detailed worked solutions, reinforcing the main themes in the text. It is
hoped that students could test their learning immediately while reading
through the chapters. These problem sets should be considered an integral
part of the course. A list of selective references to primary and review literature is included at the end of each chapter. These references (about 550)
would enable the students at the advanced levels to supplement the materials
covered in the chapters.
The approach presented in this book is distinct and class-tested. I hope
this book will be of value and interest to the students, teachers, and
researchers of organic chemistry. I encourage the readers to contact me

() with comments, corrections, and with suggestions that might be appropriate for future editions.
I would like to thank the reviewers for helpful suggestions. Special thanks
are due to my undergraduate, graduate, and research students for their loving
insistence, help, and encouragement in writing this book. I am grateful to the
editorial members Emily M. McCloskey and Billie Jean Fernandez, production manager, and other people at Elsevier for their excellent support and
cooperation. Finally, I thank my family, in particular my daughter Sudipta,
for her continuous support and my son Tirtha for his active help in referencing.
Dipak K. Mandal
Kolkata, India


CHAPTER 1

Molecular Orbitals
A basic and pictorial knowledge of molecular orbitals (MOs) is essential for a
mechanistic description of pericyclic reactions. In this context, a simplified
and nonmathematical description of MO theory1–4 is presented in this chapter. We shall deal with three kinds of MOs—σ, π and ω with major emphasis
on π MOs, and discuss their properties with reference to orbital symmetry,
energy and coefficient.

1.1 ATOMIC ORBITALS
An atomic orbital (AO) is described by a wave function ϕ, where ϕ2 denotes
the probability of finding an electron at any point in a three-dimensional
space. The algebraic sign of ϕ may be positive or negative, which indicates
the phase of the orbital (cf. the peaks and troughs of a transverse wave). An
orbital can have nodes where ϕ ¼ 0. On opposite sides of a node, ϕ has
opposite signs. An AO as a graphical description of ϕ shows lobes with
a + or a À sign (the opposite signs of two lobes are also indicated by unshaded
and shaded lobes). On the other hand, ϕ2 is always positive whether ϕ is
positive or negative. As such, the representation of AO in terms of ϕ2 is

made by drawing lobes without a phase sign. This drawing refers to the
probability distribution of AOs, and is indicated in this text as simply orbital
picture.

1.1.1 s, p and Hybrid Orbitals
1s orbital is spherically symmetrical about the nucleus and has a single sign of
ϕ. It is represented as a circle, being one cross-section of the spherical contour. The 2s orbital is also spherically symmetrical but possesses a spherical
node. The node is close to the nucleus and hence the inner sphere is not
important for bonding overlap. The 2s orbital is usually drawn as a single
circle with a single sign omitting the inner sphere.
Unlike an s orbital, the p orbitals are directional, and oriented along the
x-, y- and z-axis. Each p orbital has two lobes with opposite signs and one
node (nodal plane).
Pericyclic Chemistry
/>
© 2018 Elsevier Inc.
All rights reserved.

1


2

Pericyclic Chemistry

Carbon has four AOs (2s, 2px, 2py and 2pz) that are available for bonding.
Though this model of one s and three p orbitals is very useful, there is an
alternative model of four AOs of carbon, based on Pauling’s idea of hybridization. The hybridization involves mixing of 2s and 2p orbitals in various
proportions to produce a new set of AOs. Mathematically, the mixing is
taken to be the linear combination of atomic orbitals (LCAOs). Such

LCAOs on the same atom are called hybrid orbitals. The combination of
2s with one, two or three p orbitals can be used in different ways to produce
different sets of hybrid orbitals, designated as spn where n may be a whole
number or a fraction. For example, a combination of 2s and three 2p orbitals
can be used to generate four equivalent hybrid orbitals called sp3 hybrids.
Each sp3 hybrid orbital has two lobes with opposite signs, but unlike a p
orbital, the two lobes of a hybrid orbital have different sizes.
The schematic representations of s, p and sp3 hybrid orbitals are shown in
Fig. 1.1. In Fig. 1.1A, the orbitals are drawn as graphical description of wave
function (ϕ) showing a phase sign while Fig. 1.1B shows the orbital picture
in terms of ϕ2 with no phase sign.

Fig. 1.1 (A) Sketch of atomic orbitals in terms of ϕ with a phase sign; (B) orbital picture in
terms of ϕ2 without a phase sign.

The unequal size of two lobes of a hybrid orbital, say sp, arises from
the mixing of s orbital with a p orbital on the same atom (Fig. 1.2). The
two lobes of p orbital have the same size, but opposite signs (unshaded
and shaded), and the s orbital has a single sign (unshaded). The combination gives in-phase (same sign) mixing on one side of the nucleus
and out-of-phase (different signs) mixing on the other side, leading to
a large lobe on the left side and a small lobe on the right side of the hybrid
orbital.

Fig. 1.2 Unequal size of two lobes of a hybrid orbital.


3

Molecular Orbitals

Table 1.1 Energies of s and p atomic orbitals
Atomic orbital

H

1s
2s
2p

À13.6

Orbital energy (eV)

C

N

O

À19.4
À10.7

À25.6
À12.9

À32.4
À15.9

1.1.2 Atomic Orbitals of Nitrogen and Oxygen
Nitrogen and oxygen have similar sets of s, p and hybrid orbitals as for carbon. However, the energies are different. The relative energies of an AO on

different atoms follow their pattern of electronegativity. An orbital on a
more electronegative atom will have lower energy (Table 1.1).5

1.2 MOLECULAR ORBITALS
An MO is also described by a wave function ψ which can be expressed as an
LCAOs. The set of AOs chosen for the linear combination is called the basis
set. The total number of MOs will be equal to the total number of AOs combined. The calculation of MOs using all AOs of a molecule presents massive
computational problems. However, the essential qualitative features of
bonding can be understood if the basis set is restricted just to those AOs that
are involved in a particular type of bonding. MOs are designated by the symbols σ, π and ω reflecting the type of bonding that occurs.
Now consider the linear combination of two AOs ϕ1 and ϕ2 on atoms 1
and 2. (Note that the linear combination uses only the first power of wave
function; cf. equation of a straight line.) Two MOs ψ 1and ψ 2 are produced
which are expressed as
ψ 1 ¼ c1 ϕ1 + c2 ϕ2
ψ 2 ¼ c1 ϕ1 À c2 ϕ2
where c1 and c2 are the mixing coefficients which denote the relative contributions of the AOs ϕ1 and ϕ2 to an MO. The coefficients may be positive,
negative or even zero.
The geometry of approach of the two AOs leads to different types of
MOs. This is illustrated below taking, for example, the overlap of two p
AOs centred on two identical atoms (homonuclear), when c1 ¼ c2.
End-on approach: End-on overlap of two p orbitals gives two MOs (ψ 1 and
ψ 2) that are cylindrically symmetrical about the internuclear axis (Fig. 1.3).


4

Pericyclic Chemistry

These are called σ MOs. Here, ‘+’ combination signifies in-phase (same

sign) overlap when ψ 1 has no node. In contrast, ‘À’ combination denotes
out-of-phase (opposite sign) overlap leading to a node (nodal plane) in ψ 2.
The MO ψ 1 has lower energy than p AO and is called bonding σ orbital
(symbolized σ), while ψ 2 has higher energy and is called an antibonding σ
orbital (σ*).

Fig. 1.3 End-on overlap of two p orbitals to produce σ MOs.

Side-on approach: Side-on (lateral) overlap of two p orbitals produces two
MOs (ψ 1 and ψ 2) that are not cylindrically symmetrical about the internuclear axis, and are called π MOs (Fig. 1.4). ψ 1 has lower energy with no node
and is a bonding MO (π), while ψ 2 with one node (nodal plane) is of higher
energy and is an antibonding MO (π*).

Fig. 1.4 Side-on (lateral) overlap of two p orbitals to produce π MOs.

Orthogonal approach: For orthogonal (perpendicular) approach of two p
orbitals, bonding overlap of the same sign is cancelled by an antibonding
overlap of the opposite sign (Fig. 1.5). The net interaction is therefore nonbonding. ψ 1 and ψ 2 have the same energy and are equivalent to individual p
orbitals. These nonbonding MOs are called ω MOs.

Fig. 1.5 Orthogonal approach of two p orbitals to produce ω MOs.

It may be mentioned here that besides nonbonding ω MOs, there are also
nonbonding π MOs that can arise in conjugated π systems (see later).


Molecular Orbitals

5

1.2.1 Energy Diagram
In general, the interaction of two AOs leads to a pair of bonding and antibonding MOs. Consider the formation of σ and σ* MOs for hydrogen molecule. From quantum mechanical calculation, the energy (E) of an electron
in σ and σ* orbital is expressed in terms of three integrals (, and S) as
+
(1.1)
1+S

E ị ẳ
(1.2)
1S
where is the Coulomb integral which denotes the energy of an electron in
an isolated AO; β is the resonance integral which represents the energy of
interaction between two AOs; and S is the overlap integral which indicates
the extent of overlap of the AOs.
Now we focus on the numerator and denominator terms in Eqs (1.1),
(1.2). α and β are negative quantities. Therefore, (α + β) < α (indicating lowering of energy of σ) and (α À β) > α (raising of energy of σ*). The overlap
integral S is a function of internuclear distance and the value of S ranges from
0 to 1. For two interacting orbitals, S > 0. Hence, the denominator (1 + S) >
(1 À S). Thus, the energy increase associated with antibonding σ* orbital is
slightly greater than the energy decrease for bonding σ orbital. These results
are presented qualitatively in the MO energy diagram (Fig. 1.6).
E ị ẳ

Fig. 1.6 MO energy diagram for a two-orbital interaction. x and y indicate, respectively,
the energy decrease for the bonding orbital and energy increase for the antibonding
orbital.

For π and π* orbitals, a similar pattern follows; however, the value of S
for π overlap is much smaller. If we assume S ẳ 0, we obtain
Eị ẳ + ; E ị ẳ

6

Pericyclic Chemistry

where α is the energy of an electron in an isolated p orbital and β represents
the energy of interaction between two adjacent p orbitals.
1.2.1.1 Remarks
The above MO energy diagram serves qualitatively a general pattern for a
two-orbital interaction that may involve any AOs or MOs. It can be seen
from Fig. 1.6 that if each interacting orbital is completely filled providing
a total of four electrons, both bonding and antibonding orbitals would be
completely filled. The net interaction would thus be repulsive because
the increase in energy in the antibonding combination is greater than the
decrease in energy in a bonding combination. This repulsive filled
orbital/filled orbital interaction is the underlying reason for the steric strain
(repulsion between closed-shell molecules or groups), and is included in the
first term of the Salem–Klopman equation (see Section 4.1).

1.2.2 CdH and CdC σ Bonds
A CdH bond is formed by the interaction of a hybrid orbital (say, sp3) of
carbon with the 1s orbital of hydrogen. This two-orbital interaction leads to
bonding σCH and antibonding σ*CH as shown in Fig. 1.7A. Note that the
energy of sp3C (À16.1 eV) is somewhat lower than that of 1sH (À13.6 eV).
A CdC σ bond is formed by the end-on overlap of two hybrid orbitals, one
from each carbon. Fig. 1.7B shows the formation of a bonding σCC and an
antibonding σ*CC from two sp3 hybrid orbitals. (Note the conventional
MO sketches with the AOs instead of the delocalized sketches of MOs.)
The bonding σ MO has no node between lobes of the same sign, but antibonding σ* MO has one node between lobes of opposite sign (shaded and

unshaded).

Fig. 1.7 Bonding and antibonding MOs for (A) CdH bond and (B) CdC bond.


Molecular Orbitals

7

We shall see later that the more important orbitals in connection with
reactivity are the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). These are called the frontier
MOs. Thus, for the σ component (CdH or CdC), the HOMO is σ
and the LUMO is σ*.
Besides MOs, orbital picture representation (without phase sign) of a
σ-component is used in the mechanistic analysis of pericyclic reactions.
Fig. 1.8 shows the orbital pictures of CdH and CdC σ components.
The σ component is labelled as σ2 (2 is the number of electrons in the
component).

Fig. 1.8 Orbital pictures of CdH and CdC σ components.

€
1.3 HUCKEL
MOLECULAR ORBITAL (HMO) THEORY
FOR ACYCLIC CONJUGATED π SYSTEMS
H€
uckel theory1,6 treats a π system independently of the σ framework (the π
and σ orbitals being orthogonal to each other). The HMO theory assumes
the following:
(1) Each Coulomb integral (α) has the same value.

(2) The resonance integral (β) is same for any two adjacent atoms but zero
for two atoms not directly bonded.
(3) The overlap integral (S) is zero for the interaction between two
p orbitals.
It might be surprising that S is assumed to be zero, while the whole concept
of chemical bonding is based on the overlap of orbitals (!). In fact, overlap is
not really neglected because it is implicitly included in other parameters such
as β which is roughly proportional to S. The assumption that S ¼ 0 greatly
simplifies the calculation.
The π MO wavefunction (ψ j) is described by a linear combination of p
AOs (ϕr) as
Xn
ψj ¼
c ϕ
(1.3)
r¼1 jr r
where n is the total number of p orbitals involved and j ¼ 1, 2, 3,…,n.


8

Pericyclic Chemistry

Here, we shall consider the linear conjugated systems and obtain their π
MOs and energies using Coulson equations7 as follows:
rffiffiffiffiffiffiffiffiffiffi 

2
jrπ
sin

(1.4)
cjr ¼
n+1
n+1


jπ
Ej ¼ α + 2β cos
n+1


(1.5)

1.3.1 Linear Conjugated System With Even Number
of p Orbitals
1.3.1.1 Ethylene
The simplest system is ethylene in which two p orbitals (n ¼ 2) are conjugated to each other in forming a π bond. According to Eq. (1.3), the
MO wavefunctions are
ψ 1 ¼ c11 ϕ1 + c12 ϕ2
ψ 2 ¼ c21 ϕ1 + c22 ϕ2

qffiffi
The coefficients are evaluated using Eq. (1.4). Thus c11 ¼ 23 sin π3 ¼
0.707. Similarly, c12 ¼ c21 ¼ 0.707, c22 ¼ À 0.707. Substituting these values,
we obtain
ψ 1 ¼ 0:707ϕ1 + 0:707ϕ2
ψ 2 ¼ 0:707ϕ1 À 0:707ϕ2
(Note that the signs of coefficients are arising from the calculation using
Eq. 1.4.)
The wavefunctions (π MOs) can now be sketched as

The relative magnitudes of the coefficients are usually indicated by the relative sizes of the lobes. Here the two coefficients have the same size for both
ψ 1 and ψ 2. For ψ 1, the coefficients have the same sign indicating in-phase
(bonding) overlap. For ψ 2, the coefficients have opposite signs indicating
out-of-phase (antibonding) overlap which creates a node (nodal plane).


Molecular Orbitals

9

The energies of the π MOs are estimated using Eq. (1.5) as
π
E1 ¼ α + 2β cos ¼ α + β
3
2π
E2 ¼ α + 2β cos ¼ α À β:
3
Fig. 1.9 shows the MO energy diagram where ψ 1 is a bonding MO (π) as
it has lower energy than the energy of p orbital (α) and ψ 2 is an antibonding
MO (π*) having an energy higher than α.

Fig. 1.9 MO energy diagram of ethylene.

The ground state π electron configuration of ethylene is π2π*0. Therefore, π is HOMO and π* is LUMO. A thermal pericyclic reaction is a
ground state process whereas a photochemical reaction is a first excited state
process. On photochemical excitation by absorption of light, one electron is
promoted from π (HOMO) to π* (LUMO) with the conservation of spin,
and the resulting excited state is a singlet with singly occupied π (formerly
HOMO) and singly occupied π* (formerly LUMO) (Fig. 1.10).

Fig. 1.10 Frontier orbitals in the ground state and in the excited state of ethylene.


10

Pericyclic Chemistry

A useful convention8 is to designate the singly occupied excited state
orbitals of a molecule by its former ground state HOMO/LUMO labels. This
excited state frontier orbital convention as shown in Fig. 1.10 will be used
while dealing with the frontier orbital analysis of photochemical reactions.
(An alternative excited state nomenclature that specifies π* as HOMO and
π as NHOMO, next lower HOMO, will not be used in this text.)
At the instant of excitation, the nuclei retain the planar ground state
geometry (Frank–Condon principle). The π bond order [½ (no. of bonding
electrons À no. of antibonding electrons)] in the excited state is zero. The
initial planar excited state then quickly relaxes to the minimum energy
geometry in which the two sp2 carbons are twisted by about 90 degrees
when there is no π overlap (Fig. 1.11). This twisted, excited state, sometimes
called the p state, permits the possibility of returning to either E or Z configuration of the ground state alkene.

Fig. 1.11 Orbital pictures of the ground state and excited state of ethylene.

We shall see later that the phase relationship of terminal p orbitals in
HOMO/LUMO is important in the mechanistic analysis of pericyclic reactions. This phase relationship is characterized by orbital symmetry. Fig. 1.12
shows the orbital symmetries of HOMO and LUMO of ethylene. The
HOMO (π) has the same phase at the two ends and possesses a plane of symmetry (symbolized by m) since a lobe (labelled *) reflects to a lobe of the same
sign. Note that the mirror plane (m) bisects the CdC bond and is perpendicular to the plane of the molecule. On the other hand, the LUMO (π*) has
opposite phases at the two ends and is characterized by C2 symmetry as the

C2 operation brings a lobe (labelled *) to a position of a lobe with the same
sign. Note that the C2 axis bisects the CdC bond and is lying in the plane of
the molecule.

Fig. 1.12 Orbital symmetries of HOMO and LUMO of ethylene.


Molecular Orbitals

11

1.3.1.2 Butadiene
The basis set orbitals of butadiene (ϕ1–ϕ4) are four p orbitals (n ¼ 4). The
wavefunctions (ψ 1–ψ 4) representing four π MOs are written as
ψ 1 ¼ c11 ϕ1 + c12 ϕ2 + c13 ϕ3 + c14 ϕ4
ψ 2 ¼ c21 ϕ1 + c22 ϕ2 + c23 ϕ3 + c24 ϕ4
ψ 3 ¼ c31 ϕ1 + c32 ϕ2 + c33 ϕ3 + c34 ϕ4
ψ 4 ¼ c41 ϕ1 + c42 ϕ2 + c43 ϕ3 + c44 ϕ4
qffiffi
qffiffi
Using Eq. (1.4), c11 ¼ 25 sin π5 ¼ 0.371, c12 ¼ 25 sin 2π5 ¼ 0.600. The
other coefficients are calculated similarly. We obtain
ψ 1 ¼ 0:371ϕ1 + 0:600ϕ2 + 0:600ϕ3 + 0:371ϕ4
ψ 2 ¼ 0:600ϕ1 + 0:371ϕ2 À 0:371ϕ3 À 0:600ϕ4
ψ 3 ¼ 0:600ϕ1 À 0:371ϕ2 À 0:371ϕ3 + 0:600ϕ4
ψ 4 ¼ 0:371ϕ1 À 0:600ϕ2 + 0:600ϕ3 À 0:371ϕ4
The wavefunctions (π MOs) of butadiene can be sketched in s-trans or
s-cis conformation. Fig. 1.13 shows the butadiene π MOs in more stable
s-trans conformation. (We shall see later that it is the s-cis form which is
the reactive component in most pericyclic processes.) The relative sizes of

the lobes indicate qualitatively the relative values of the coefficients. The
number of nodes (indicated by dashed line) in ψ 1, ψ 2, ψ 3 and ψ 4 is 0, 1,
2 and 3, respectively. This indicates that an orbital ψ j has ( j À 1) nodes.

Fig. 1.13 Sketches of π MOs of butadiene in s-trans conformation.

As the number of nodes increases, the energy of the orbital increases in
the order: ψ 1 < ψ 2 < ψ 3 < ψ 4. In another fashion, if we count the number of
bonding/antibonding interactions between the adjacent p orbitals, it is seen
that ψ 1 with three bonding overlaps and ψ 2 with two bonding and one antibonding interactions become bonding MOs whereas ψ 3 with two antibonding and one bonding interactions and ψ 4 with three antibonding interactions


12

Pericyclic Chemistry

become antibonding MOs. The MO energy therefore increases in the same
order as shown above.
The energies of the π MOs can however be estimated using Eq. (1.5).
For example, E1 ¼ α + 2β cos π5 ¼ α + 1:618β. The estimated energies of all
π MOs are shown in the MO energy diagram (Fig. 1.14). Note that the more
positive or less negative β values imply a decrease in energy; less positive or
more negative β values indicate an increase in energy.

Fig. 1.14 MO energy diagram of butadiene.

The bonding/antibonding classification of the π MOs is now clearly evident. ψ 1 and ψ 2 have energies lower than the energy (α) of a p orbital and are
therefore bonding MOs whereas ψ 3 and ψ 4 are antibonding MOs as their
energies are higher than α.
(Since the decrease or increase in energy of an MO is considered relative

to the energy of the AO, α can be arbitrarily assumed to be zero and the MO
energy can be expressed in only β terms; however in this text MO energy is
expressed in both α and β terms, as obtained from the energy expression.)
The ground state π electron configuration of butadiene is ψ 21ψ 22. Thus, ψ 2
is HOMO and ψ 3 is LUMO. In the first excited state, one electron is promoted from ψ 2 (HOMO) to ψ 3 (LUMO). As per the frontier orbital convention used in this text (see p. 10), the singly occupied ψ 2 and ψ 3 also
represent HOMO and LUMO in the first excited state.
The HOMO/LUMO energies in s-trans and s-cis conformations of butadiene are not the same. In s-cis conformation, the HOMO energy is raised
and the LUMO energy is lowered relative to those in the s-trans form
(Fig. 1.15). Unlike the s-trans form, the s-cis conformation has a possible
interaction between the two terminal p orbitals. An antibonding interaction
raises the HOMO energy, while a bonding interaction lowers the LUMO
energy in s-cis conformation.


Molecular Orbitals

13

Fig. 1.15 Relative energies of frontier orbitals in s-trans and s-cis conformations of
butadiene.

In frontier orbital analysis, the higher energy HOMO and lower energy
LUMO would make the s-cis-butadiene more reactive than s-transbutadiene in pericyclic reactions. It may be mentioned here that the lower
HOMO/LUMO energy gap in s-cis conformation leads to UV absorption at
a longer wavelength (253 nm) for a homoannular diene locked in s-cis conformation compared with a λmax of 215 nm for an acyclic or a heteroannular
diene existing predominantly or exclusively in s-trans conformation.
The orbital symmetry of frontier orbitals is important in the context of
mechanism of pericyclic reactions. Fig. 1.16 shows that the HOMO has the
opposite phase relationship at the two termini and is characterized by C2
symmetry while the LUMO with the same phase relationship at the two

ends exhibits m symmetry.

Fig. 1.16 Orbital symmetries of ΗΟΜΟ and LUMO of butadiene.

1.3.1.3 Hexatriene
1,3,5-Hexatriene is a conjugated π system of six p orbitals (n ¼ 6). Using Eqs
(1.3), (1.4), the wavefunctions (ψ 1 – ψ 6) for six π MOs are obtained as follows:
ψ 1 ¼ 0:232ϕ1 + 0:418ϕ2 + 0:521ϕ3 + 0:521ϕ4 + 0:418ϕ3 + 0:232ϕ4
ψ 2 ¼ 0:418ϕ1 + 0:521ϕ2 + 0:232ϕ3 À 0:232ϕ4 À 0:521ϕ3 À 0:418ϕ4
ψ 3 ¼ 0:521ϕ1 + 0:232ϕ2 À 0:418ϕ3 À 0:418ϕ4 + 0:232ϕ3 + 0:521ϕ4
ψ 4 ¼ 0:521ϕ1 À 0:232ϕ2 À 0:418ϕ3 + 0:418ϕ4 + 0:232ϕ3 À 0:521ϕ4
ψ 5 ¼ 0:418ϕ1 À 0:521ϕ2 + 0:232ϕ3 + 0:232ϕ4 À 0:521ϕ3 + 0:418ϕ4
ψ 6 ¼ 0:232ϕ1 À 0:418ϕ2 + 0:521ϕ3 À 0:521ϕ4 + 0:418ϕ3 À 0:232ϕ4


14

Pericyclic Chemistry

Note the pattern in the size (not sign) of coefficients: the first three and
last three values in each MO hold a mirror image relationship. The energies
of the π MOs are estimated using Eq. (1.5). For instance,
E3 ¼ α + 2β cos 3π
7 ¼ α + 0:445β. The sketches of six π MOs, their nodal
properties and energies are shown in Fig. 1.17. It is evident that the six π
MOs constitute three bonding and three antibonding MOs.
In the ground state, the π electron configuration of hexatriene is ψ 21ψ 22ψ 23.
Thus, ψ 3 is HOMO and ψ 4 is LUMO. In the first excited state, ψ 3 and ψ 4
also represent HOMO and LUMO when each is half-filled. (Only HOMO
and LUMO are labelled with coefficient values in the figure.)

Fig. 1.17 MO energy diagram and sketches of the π MOs of 1,3,5-hexatriene.

In pericyclic reactions, the two termini of the hexatriene component
ought to be generally close to each other. This is achieved when the middle
double bond is Z and the molecule adopts s-cis conformation. The symmetry
of the frontier orbitals of hexatriene is shown in Fig. 1.18. The HOMO has
m symmetry while the LUMO is characterized by C2 symmetry.

Fig. 1.18 Orbital symmetries of ΗΟΜΟ and LUMO of 1,3,5-hexatriene.


Molecular Orbitals

15

1.3.2 Linear Conjugated System With Odd Number
of p Orbitals
We have seen that the π MOs of a conjugated system (n ¼ even) comprise
equal number of bonding and antibonding MOs. When n ¼ odd, the conjugated π system is a reactive intermediate (carbocation, carbanion or carbon
radical) when the π MOs will contain a nonbonding MO besides bonding
and antibonding orbitals as described below.
1.3.2.1 Allyl System
The allyl system (cation, radical or anion)
represents a conjugated
system of three p orbitals (n ¼ 3). The wavefunctions (ψ 1–ψ 3) for three
π MOs are, according to Eq. (1.3), given by
ψ 1 ¼ c11 ϕ1 + c12 ϕ2 + c13 ϕ3
ψ 2 ¼ c21 ϕ1 + c22 ϕ2 + c23 ϕ3
ψ 3 ¼ c31 ϕ1 + c32 ϕ2 + c33 ϕ3

The coefficients are evaluated using Eq. (1.4). For instance,
qffiffi
qffiffi
c21 ¼ 24 sin 2π4 ¼ 0.707; c22 ¼ 24 sin 4π4 ¼ 0. With the calculated values of
the coefficients, the wavefunctions are
ψ 1 ¼ 0:500ϕ1 + 0:707ϕ2 + 0:500ϕ3
ψ 2 ¼ 0:707ϕ1 À 0:707ϕ3
ψ 3 ¼ 0:500ϕ1 À 0:707ϕ2 + 0:500ϕ3
In ψ 2, the coefficient of ϕ2 is zero which indicates that there is no contribution of ϕ2 to ψ 2. Physically, this implies that ϕ2 is orthogonal to ϕ1 and
ϕ3 in ψ 2. The energies of the π MOs can be estimated using Eq. (1.5). The
sketches of the MOs, their nodal properties and the estimated energies are
shown in Fig. 1.19. The nodal properties indicate that ψ j has again ( j – 1)
nodes, as observed for the system with even number of p orbitals. In case
of ψ 2, a node passes through the middle carbon C-2. The energy of ψ 2 is
α, which is same as the energy of a p orbital, and hence ψ 2 is a nonbonding
MO. It is seen that ψ 1 is bonding and ψ 3 is antibonding. Thus, the three π
MOs of an allyl system comprise a bonding, a nonbonding and an
antibonding MO.
The frontier orbitals of an allyl system depend on whether it is a cation, a
radical or an anion. The number of π electrons in allyl cation, radical and
anion is 2, 3 and 4, respectively. The frontier orbitals are given below:


16

Pericyclic Chemistry

Fig. 1.19 The coefficients, nodal properties and energies of the π MOs of an allyl system.
Allyl cation (ψ 21)

Allyl radical (ψ 21ψ 12)

Allyl anion (ψ 21ψ 22)

HOMO

LUMO

SOMO

HOMO

LUMO

ψ1

ψ2

ψ2

ψ2

ψ3

The frontier orbital for an allyl radical is ψ 2 which is a singly occupied
molecular orbital (SOMO). The symmetry properties of the π MOs are
shown in Fig. 1.20.

Fig. 1.20 Orbital symmetries of π MOs of an allyl system.

Problem 1.1

Derive the π MOs of a pentadienyl system using Coulson equations. Sketch
the MOs in s-cis conformation of the molecule showing node(s). Indicate
the frontier orbitals for cation, anion and radical species with symmetry.
Answer
The pentadienyl system (cation, radical or anion) is
The π MOs (ψ 1 – ψ 5) and their energies are derived as
ψ 1 ¼ 0:288ϕ1 + 0:500ϕ2 + 0:576ϕ3 + 0:500ϕ4 + 0:288ϕ5 E1 ¼ α + 1:732β
ψ 2 ¼ 0:500ϕ1 + 0:500ϕ2 À 0:500ϕ4 À 0:500ϕ5 E2 ¼ α + β
ψ 3 ¼ 0:576ϕ1 À 0:576ϕ3 + 0:576ϕ5 E3 ¼ α
ψ 4 ¼ 0:500ϕ1 À 0:500ϕ2 + 0:500ϕ4 À 0:500ϕ5 E4 ¼ α À β
ψ 5 ¼ 0:288ϕ1 À 0:500ϕ2 + 0:576ϕ3 À 0:500ϕ4 + 0:288ϕ5 E5 ¼ α À 1:732β


Molecular Orbitals

17

The energies indicate that ψ 1 and ψ 2 are bonding MOs; ψ 3 is a
nonbonding MO; and ψ 4 and ψ 5 are antibonding MOs.
The sketches of the π MOs are shown below. It is seen that an orbital ψj
has ( j – 1) nodes.

The frontier orbitals for the pentadienyl cation, radical and anion, and
their symmetry are given below:

Symmetry:

Cation (ψ 21ψ 22)

Radical (ψ 21ψ 22ψ 13)

Anion (ψ 21ψ 22ψ 23)

HOMO

LUMO

SOMO

HOMO

LUMO

ψ2
C2

ψ3
m

ψ3
m

ψ3
m

ψ4
C2

1.3.2.2 A Short-Cut Method for Sketching π MOs
The sketches of the π MOs can be simplified by ignoring the difference of
coefficients, where the coefficients are not important in a mechanistic
description of pericyclic reactions. A short-cut method to drawing such a
simplified picture of π MOs is to use the nodal properties which indicate that
an MO ψ j has (j – 1) nodes. The procedure is illustrated for the drawing of
frontier orbitals in Fig. 1.21. Note that the nodes are to be placed in the most
symmetrical manner in the prospective MO.
The symmetry properties of the π MOs can be summarized as follows:
ψ j ( j ¼ odd) ) m symmetry;
ψ j ( j ¼ even) ) C2 symmetry.
Besides the MOs, orbital picture representation (without phase sign) of
the π components is also used in mechanistic analysis of pericyclic reactions.
The orbital pictures of the diene and triene components are sketched as


18

Pericyclic Chemistry

Fig. 1.21 Sketches of frontier orbitals from nodal properties.

1.4 CARBONYL π SYSTEM
The symmetry properties of π MOs of the carbonyl system are similar to
those of alkenes; however, the energy and coefficient patterns differ. The
energy of a p orbital on oxygen (À15.9 eV) is much lower than that on carbon (À10.7 eV) (see Table 1.1). This would lead to lowering of energy of
both πCO and π*CO compared with those for alkene (Fig. 1.22).
For the carbonyl group, the lower energy pO would contribute more to
the lower energy πCO and the higher energy pC would contribute more to
the higher energy π*CO. As a result, there will be a larger coefficient on oxygen in πCO and a larger coefficient on carbon in π*CO as shown in Fig. 1.22.

πCO is HOMO and π*CO is LUMO, the polarization of HOMO and

Fig. 1.22 Energy diagram of π MOs of carbonyl group vis-a-vis alkene π MOs.