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Graphical Representation of Hückel Molecular Orbitals
Zhenhua Chen*
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sı Supporting Information
*
ABSTRACT: In this paper, we develop a general but very simple mathematical
foundation for the predefined coefficient graphical method of Hückel molecular
orbital theory (HMO). We first present the general solution for the recurrence
relation of the coefficients of Hückel molecular orbitals (MOs). Subsequently, for all
the three unbranched hydrocarbons, i.e., open-chain, cyclic Hückel and Möbius
polyenes, different boundary conditions are explored for obtaining the MOs and their
energy levels. The analytic continuation of the recurrence relation, in which one
extends the domain from integral to real, allows us to analyze the symmetric
properties of Hückel MOs in an elegant fashion without even knowing the actual
expressions. In fact, we can use the symmetric properties to derive the Hückel MOs of
the unbranched hydrocarbons and some branched hydrocarbons such as naphthalene.
Consequently, this work also provides a pedagogical alternative to present the HMO model for students in an advanced physical
chemistry course. Finally, the graphical approach could be a good mnemonic device for students’ comprehension of the HMO
theory.
KEYWORDS: Upper-Division Undergraduate, Graduate Education/Research, Physical Chemistry, Quantum Chemistry,
Theoretical Chemistry, Mnemonics/Rote Learning, Computational Chemistry, MO Theory
ückel molecular orbital theory (HMO),1,2 as the first and
simplest approximation in a hierarchy of semiempirical
methods a posteriori to ab initio quantum chemistry approaches,
still plays an irreplaceable role in chemistry. For two reasons, the
HMO approximation can prolong its impact over the years. On
one hand, it gives a simple interpretation to molecular orbital
(MO) theory, since several systems have closed analytical
solutions. On the other hand, even more importantly, the HMO
model is widely adopted in chemical education. This remarkable
reputation was accumulated via the contributions from some
outstanding scientists as Lennard-Jones,3 Coulson4,5 and
Longuet-Higgins,6 Ruedenberg,7 Hoffman,8 Shaik,9 and Kutzelnigg,10 etc. Even in contemporary chemistry, the Hückel
approach is still very useful for learning chemical concepts on a
sound basis. For example, by using the Hückel mnemonics,
Shaik has proposed a qualitative valence bond (VB) model to
show that VB theory does not fail to reproduce the rules of
aromaticity and antiaromaticity.11 Later in the monograph on
VB theory,12 Shaik and Hiberty use the same mnemonics to
further demonstrate that VB theory reproduces also the
photoelectron spectrum of methane. Recently, based on the
Hückel methodology, Humbel has developed the Hückel Lewis
configuration interaction method13 to recast the concept of
Lewis structures into the Hückel formalism.
For simple unbranched hydrocarbons, the treatments of
Lennard-Jones3 and Coulson4 are straightforward and most
general. In this method, one first expands the secular equation
into a characteristic polynomial. Then, the MO energies are
obtained by using the recurrence relation of the characteristic
H
© XXXX American Chemical Society and
Division of Chemical Education, Inc.
polynomial. Finally, the atomic orbital (AO) expansion
coefficients are determined by substituting back the MO
energies one by one into the secular equation. Later, this
standard method has been adopted by Coulson and Streitwieser
in a dictionary,14 by Heilbronner and Bock in the voluminous
books,15 and by Lowe in a popular quantum chemistry
textbook.16 In another excellent modern textbook,17 Levine
obtained similar results by using properties of some special
determinants, i.e., continuant and circulant.
In order to visualize the Hückel MOs and their energies,
several graphical techniques and mnemonic devices have been
proposed. For polyenes and cyclic polyenes, Frost and
Musulin18 described a simple circle mnemonic for MO energies.
Heilbronner19 first proposed the insightful idea that large-ring
Hückel systems might be twisted to give Möbius systems and has
presented a formula for their MO energies. Zimmerman20 used
the circle mnemonic to produce the MO energy levels of Möbius
systems. For pedagogical consideration, some simple and
elegant plotters have recently been developed as useful
educational tools for the visualization of Hückel MOs, as have
been reported in this Journal.21,22 Very recently, Litofsky and
Viswanathan23 described the modern matrix diagonalization
Received: July 23, 2019
Revised: December 14, 2019
A
https://dx.doi.org/10.1021/acs.jchemed.9b00687
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method for teaching HMO theory in computational chemistry
course using an Excel spreadsheet.
For the purpose of solving the Hückel secular equation
efficiently, considerable effort has been devoted to develop
graphical approaches. Most of these approaches target the
characteristic polynomial whose roots represent the HMO
energies. Harary24 and Sachs25 have independently developed
the graphical technique for evaluating the characteristic
polynomial. Alternatively, Heilbronner26 and Schwenk27 have
proposed different approaches. Independently, Graovac et al.28
and Hosoya29 have presented different methods for enumerating the coefficients of the characteristic polynomials for
conjugated hydrocarbons. Mallion et al.30 and Aihara31
extended the graphical method to conjugated systems with
heteroatoms. Also, Tang32 and Kiang33 obtained the characteristic polynomials for Hückel and complex matrices. In the
standard methods listed above, the MO energies are obtained
first, and then, each one is individually substituted back into the
secular equation to obtain the AO coefficients.
Zhang et al.34 first proposed an alternative graphical approach
to obtain both the AO coefficients and the energies of the
Hückel MOs. The graphical approach first predefines the AO
coefficients in terms of trigonometric functions by using the
recurrence relation of the coefficients, and then these
trigonometric functions are explicitly solved to compute AO
coefficients by exploring various boundary conditions for
different molecules. In the past 40 years, we found that the
predefined coefficient graphical method is very suitable for
teaching HMO theory in an elegant manner for undergraduate
students in advanced physical chemistry courses.
In this paper, we begin by describing the HMO model and
deducing the general solution of the recurrence relation of the
AO coefficients of Hückel MOs for unbranched hydrocarbons.
We then present results for the open-chain polyenes and cyclic
polyenes of both Hückel and Möbius systems, along with a
discussion on the symmetric classifications of π-MOs of openchain and cyclic polyenes, and then apply the graphical approach
to derive the π-MOs of naphthalene. After that, we present a
discussion on the pedagogical implementation of the graphical
approach, followed by concluding remarks.
For unbranched (open-chain, cyclic, or Mö bius trip)
hydrocarbons, by appropriate numbering of the atoms, the
adjacency matrix takes a form of
ÄÅ
É
t ÑÑÑÑ
ÅÅÅ 0 1
ÅÅ
ÑÑ
l
0, polyene
o
ÅÅ 1 0 1
ÑÑ
o
o
ÅÅ
ÑÑ
o
ÅÅ
ÑÑ, t = o
1, cyclic polyene
m
ÅÅ 1 ∏ ∏ ÑÑ
o
ÅÅ
ÑÑ
o
o
ÅÅ
Ñ
o
o
∏ 0 1 ÑÑ
system
̈
ÅÅ
o
ÑÑ
n−1, Mobius
ÅÅÅÅ t
ÑÑÑ
1
0
(5)
Ç
Ö
■
GENERAL SOLUTION OF THE RECURRENCE
RELATION OF THE COEFFICIENTS OF HÜ CKEL MOS
The entries in the kth eigenvector {Cjk} are the AO expansion
coefficients of the corresponding Hückel MO. We note that
there is a recurrence relation between these AO coefficients, as
can be seen from the secular equation and the topological
structure of the adjacency matrix. For all the three kinds of
unbranched hydrocarbons, the recurrence relations of the AO
coefficients are the same and given as
Cj − 1, k − λk Cjk + Cj + 1, k = 0
y ′′ − λy′ + y = 0
q 2 − λq + 1 = 0
q = (λ ± λ 2 − 4 )/2 . Since for unbranched hybrocarbons
|λ| ≤ 2, we can change the variable as
(9)
λ = 2 cos θ
Here, we note that this kind of variable change not only
diminishes the square root sign but also allows us to use Euler’s
formula to simplify the root expressions. The final expressions of
the two roots are as simple as q1 = eiθ and q2 = e−iθ. Therefore, we
immediately have that cm = (q1)m = eimθ and cm = (q2)m = e−imθ are
two special solutions of the linear homogeneous recurrence
relation
(1)
c k − 1 − λc k + c k + 1 = 0
(10)
The general solution is written as
cm = ae imθ + be−imθ
(11)
(3)
which satisfies both the recurrence relation and the initial
conditions. In eq 11, the coefficients a and b are two complex
numbers which can be determined by the initial conditions. For
unbranched hydrocarbons, the initial conditions are presented
in the form of restrictions, including the normalization of the
MO wave function, and the boundary conditions on the starting
and terminal atoms.
(4)
OPEN-CHAIN POLYENE
For open-chain polyenes, the starting atom can be chosen as the
first atom. The boundary condition at the starting atom is
(2)
It can be verified that H and M matrices have the same
eigenvectors {Ck}
■
where the eigenvalues are related by
εk = α + βλk
(8)
This equation has a pair of roots, namely,
where M is the adjacency matrix. Its matrix elements are either 1
or 0 depending on whether the two atoms connect with each
other or not
HCk = εk Ck and MCk = λk Ck
(7)
where y is a function of variable x, we seek solutions of this
equation by using the basic exponential functions y = eqx. It is
obvious that y = eqx is a solution of eq 7 if and only if q is a root of
the characteristic quadratic equation
HMO MODEL
Within the HMO model, the molecular orbitals and their
energies are the eigenvectors and eigenvalues of the Hückel
matrix, which can be written as
l
o1, if μ , ν are indices of two connected atoms
Mμν = o
m
o
o 0, otherwise
n
(6)
The recurrence equation recalls us a method for solving linear
homogeneous differential equation with constant coefficients.
Considering the differential equation
■
H = αI + β M
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−2 cos θc1 + c 2 = 0
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(12)
ϕk =
Substituting c1 = aeiθ + be−iθ, c2 = aei2θ + be−i2θ, and 2 cos θ =
eiθ + e−iθ into eq 12, we have
where
(13)
a+b=0
(14)
Graphically, these AO coefficients are shown in Figure 1.
Figure 1. Graphical representation of Hückel MO coefficients for npolyenes.
Substituting eq 14 for the last two terms of coefficients into
the boundary conditions on the terminal atom, one has
sin(n − 1)θ − 2 cos θ sin nθ = 0
(15)
ϕk =
By using trigonometric identity
sin(A + B) + sin(A − B) = 2 cos A sin B
(21)
is a normalization factor.
2
(sin θkχ1 + sin 2θkχ2 + sin 3θkχ3 + sin 4θkχ4 )
5
(22)
Substituting sine values for all θ’s, we have
sin(n + 1)θ = 0
l
ϕ1 = 0.3717χ1 + 0.6015χ2
o
o
o
o
o
o
o
o
o ϕ2 = 0.6015χ1 + 0.3717χ2
m
o
o
o
o ϕ3 = 0.6015χ1 − 0.3717χ2
o
o
o
o
o
ϕ = 0.3717χ1 − 0.6015χ2
o
n 4
(17)
Visually, eq 17 gives us an interpretation that the AO
coefficient on the n + 1 carbon vanishes, since n-polyene does
not have the n + 1 carbon. The solutions of eq 17 are
kπ
, k = 1, 2, ..., n
n+1
(18)
εk = α + βλk = α + 2β cos
2kπ
, k = 1, 2, ..., n
n+1
− 0.3717χ3 − 0.6015χ4
− 0.3717χ3 + 0.6015χ4
+ 0.6015χ3 − 0.3717χ4
(23)
■
CYCLIC POLYENE: HÜ CKEL SYSTEM
For cyclic polyenes, every carbon atom can be selected as the
starting one. Then the secular equation on the first site gives the
boundary condition that
cn − 2 cos θc1 + c 2 = 0
(19)
(24)
By using 2 cos θ = eiθ + e−iθ and the general term expression,
eq 11, for the AO coefficients, after some simply deduction, we
have
where we have used the eigenvalues of the topological matrix
deduced from eqs 9 and 18
2kπ
n+1
+ 0.6015χ3 + 0.3717χ4
Equation 23 shows clearly that the Hückel MOs are either
symmetric or antisymmetric with respect to the mirror
perpendicular to the polyene molecular plane and bisecting
the molecule. It is interesting to note that all odd number MOs
are symmetric, and all even MOs are antisymmetric for arbitrary
n-polyenes. Furthermore, we will see in the later text that this
symmetric property can be used to deduce the Hückel MOs in
an alternative but somewhat a slightly simpler way, rather than to
clarify MOs only.
More strictly speaking, we may indeed have other solutions
with other integer k values that are either negative or larger than
n in the right-hand side of eq 18. Nevertheless, all these
periodical duplications may just be discarded, since sin θ is a
periodical function, which repeats over an interval of 2π. We can
see from eq 14 that these solutions give no new MOs but only
duplications. Moreover, the AOs of n-polyene can superpose to
produce exactly n MOs, inasmuch as MOs are written as linear
combination of AOs. In addition, we can assign the number k as
an index to address the corresponding MO.
Subsequently, with the θ solutions at hand, we have the MO
energy levels
λk = 2 cos θk = 2 cos
m=1
(16)
we have that
θk =
2
n+1
n
∑ sin mθkχm
In a short summary, the AO coefficients of Hückel MOs in
open-chain polyenes can be determined by using the graphical
representation as follows:
(1) Draw out the skeleton of n-polyene using the cycle
(denotes AO) and the linking line (represents the β
integral) between two adjacent cycles.
(2) Assign sin θ for the first AO coefficient, and then assign sin
mθ for the mth AO coefficient by using the recursive
relation, for m = 2, ..., n.
(3) Deduce the boundary condition equation, eq 17, by using
the secular equation of the terminal atom and the assigned
AO coefficients in step 2.
(4) Solve the boundary condition equation to get θ values, eq
18.
(5) Obtain the energy levels by eq 19 and the MOs by eq 21.
Here, we use the 1,3-butadiene as an illustrative example. It
has 4 carbon atoms. So, the boundary condition is sin 5θ = 0,
kπ
which implies that θk = 5 , for k = 1, 2, 3, 4. Therefore, we have
the energy levels that ε1 = α + 1.6180β, ε2 = α + 0.6180β, ε3 = α
− 0.6180β, and ε4 = α − 1.6180β. By using eq 21, we have the
general expression of the Hückel MOs of 1,3-butadiene
Therefore, the general coefficient expression reduces to cm =
a(eimθ − e−imθ) = i2a sin mθ. Because the total MO wave function
can always be normalized for an arbitrary nonvanishing
coefficient vector, parameter a is free to set. Therefore, after
ignoring the normalization factor, we have a general term for the
AO coefficients of the mth atom as simple as
cm = sin mθ
2
n+1
Article
cn = ae inθ + be−inθ = a + b
(20)
Then by using eq 14, we have the normalized MOs
(25)
Meanwhile, the boundary condition on the last site
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cn − 1 − 2 cos θcn + c1 = 0
where k = 1, 2, ..., [(n − 1)/2]. For convenience, we have
rearranged the ordering of the real MOs according to their
energy levels.
(26)
implies that
c1 = ae iθ + be−iθ = ae i(n + 1)θ + be−i(n + 1)θ
■
(27)
CYCLIC POLYENE: MÖ BIUS SYSTEM
Similar to the case of cyclic polyenes, the boundary conditions
for Möbius system at the first and the last carbon atoms can be
deduced as
Here, we have two equations, eqs 25 and 27, for the three
parameters a, b, and θ. To fulfill these two boundary conditions,
in the Supporting Information we prove that one only requires θ
to satisfy
e inθ = 1
(28)
Meanwhile, very interestingly, a and b can take any values, so
long as they are not both zero. From eq 28, we have n solutions
θk =
2kπ
, k = 0, 1, 2, ..., (n − 1)
n
cn = ae inθ + be−inθ = −(a + b)
(38)
c1 = ae iθ + be−iθ = −[ae i(n + 1)θ + be−i(n + 1)θ ]
(39)
To fulfill these two boundary conditions, here θ only needs to
satisfy
(29)
e inθ = −1
Here we label the energy levels starting from zero. We will see
later that this labeling is more convenient as compared to that
starting from one.
Since a and b can take arbitrary values, except both cannot be
zero, we have an infinite number of solutions for the wave
function. Obviously, the simplest one is to set a = 1 and b = 0.
Then the general term of the coefficient is cm = eimθ.
Subsequently, the normalized MOs is expressed as
∑ eimθ χm
θk =
(30)
m=1
λk = 2 cos
where the normalization factor can be easily verified, as
n
∑ e−imθ eimθ
k
k
(2k − 1)π
n
(42)
n
(31)
ϕk = n−1/2
The MO energies can then be written as
2kπ
εk = α + 2β cos
, k = 0, 1, 2, ..., (n − 1)
n
∑ χm
k
(43)
The energy levels for Möbius systems can be represented
graphically be using Zimmerman’s mnemonic,20 which is an
extension of the Frost and Musulin diagram for Hückel aromatic
systems. The energies are double degenerate
(32)
λn − k + 1 = λk for k = 1, 2, ..., [(n + 1)/2]
(33)
Meanwhile, when n is an even number, the highest energy
orbital is also nondegenerate, with an energy of εn−1 = α − 2β. Its
wave function is written as
ϕ′2k − 1 =
1
(ϕ + ϕn − k + 1) =
2 k
2
n
i
(ϕ − ϕn − k + 1) =
2 k
2
n
n /2
ϕn − 1 = (n)−1/2
∑ (χ2m− 1 − χ2m )
ϕ′2k = −
(34)
m=1
The other orbitals are doubly degenerate
εn − k = εk , k = 1, 2, ···, [(n − 1)/2]
ϕ′2k =
i
(ϕ − ϕn − k ) =
2 k
1
(ϕ + ϕn − k ) =
2 k
2
n
2
n
m=1
∑ cos mθkχm
(45)
n
∑ sin mθkχm
m=1
(46)
SYMMETRY CLASSIFICATION OF HÜ CKEL MO
Now, we first assume that all atoms in an n-polyene are in sites
on a line; the position of the atoms can be labeled by natural
numbers starting from one, as we have just done. However, we
have other choices. Actually, the position of the origin is free for
us to select without changing the solutions of the Hückel MOs,
since the number line has no physical influence on the polyene.
As we know, the MOs of an n-polyenes are either symmetric or
antisymmetric. Therefore, a proper choice of the origin may
display the symmetry properties of these MOs in an obvious and
(36)
n
m=1
∑ cos mθkχm
m=1
■
n
∑ sin mθkχm
n
for k = 1, 2, ..., [(n + 1)/2].
(35)
where θk + θn−k = 2π and cos(2π − θ) = cos θ have been applied.
The real form of Hückel MOs can be obtained from the
superposition of the two corresponding degenerate MOs, as
ϕ′2k − 1 = −
(44)
where we have used eq 42, θn‑k+1 + θk = 2π, and cos(2π − θ) =
cos θ. The real form of the MOs can be obtained by
superposition of the two corresponding degenerate complex
ones, as
n
m=1
∑ eimθ χm
m=1
These energy levels can be represented graphically by using
the mnemonic diagram of Frost and Musulin18 for Hückel
aromatic systems. The lowest energy orbital is nondegenerate,
with a value of ε0 = α + 2β. Its wave function is written as
ϕ0 = (n)
(41)
and the wave function in its simple form is given as
=1
m=1
−1/2
(2k − 1)π
, k = 1, 2, ..., n
n
and
k
⟨ϕk |ϕk ⟩ = n−1
(40)
Equation 40 implies that a 360° rotation converts each AO
into its negative sign in a Möbius system, while in a Hückel
system eq 28 implies that the AO keeps unchanging as the
original one after the rotation.
Therefore, we have that
n
ϕk = n−1/2
Article
(37)
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second one passes through two opposite carbon atoms. Using
the first symmetry plane, benzene can be skeletonized from
hexatriene by drawing an extra line for the two terminal carbons,
see Figure 4a. Therefore, graphically the symmetric and
antisymmetric MOs of benzene are very similar to those in
linear hexatriene, see Figure 4b,c. The only difference here is that
the two terminal atoms are linked in benzene.
From Figure 4b, we have that the boundary condition for the
symmetric MO is written as
automatic way. To this end, let us perform the analytic
continuation for the linear recurrence relation. The domain of
the variable in linear recurrence relation can be extended from a
natural number to a real one, namely
f (x − 1) − 2 cos θf (x) + f (x + 1) = 0, x ∈ R
(47)
where we have replaced the natural number k in eq 10 by x to get
eq 47. The general term expression takes a similar replacement
f (x) = ae ixθ + be−ixθ
(48)
cos
where a and b are complex numbers which satisfy the initial
conditions. Since the general term expression is deduced only
from the recurrence relation, it is valid for any value of real x. For
a symmetric MO, one requires that
f (x ) = f ( − x )
Article
5
5
3
θ − 2 cos θ cos θ + cos θ = 0
2
2
2
(50)
After simplification, it reduces to
cos
(49)
7
5
θ = cos θ
2
2
(51)
Visually, it seems that the symmetric MO coefficients of the
terminal atom can be obtained in two ways, either clockwise or
anticlockwise, which apparently should be equal.
For the antisymmetric MOs, we have a similar boundary
condition such that
Substituting eq 48 into eq 49, we have a = b. In this case, the
general term expression can be written as f S(x) = cos xθ, where S
denotes symmetry, and we have again ignored the normalization
factor. Similarly, for an antisymmetric MO, we have fA(x) = sin
xθ, where A denotes antisymmetry.
For an odd number n, we can put the origin on the (n + 1)/2
atom. We can then depict the graphical representations for the
symmetric and antisymmetric MOs, respectively, in Figure 2a,b.
sin
7
5
θ = −sin θ
2
2
(52)
We can get six possible θ values from eqs 51 and 52, namely,
1
2
1
2
that θ S = 0, 3 π , 3 π , and θ A = 3 π , 3 π , π . Therefore, we have
the following MO energy levels
ε0 = α + 2β , ε1 = ε2 = α + β , ε3 = ε4 = α − β ,
and ε5 = α − 2β
l
o
o
ϕ0 =
o
o
o
o
o
o
o
o
o
o
o
ϕ1 =
o
o
o
o
o
o
o
o
o
o
o
ϕ2 =
o
o
o
o
m
o
o
o
o
ϕ3 =
o
o
o
o
o
o
o
o
o
o
o
o ϕ4 =
o
o
o
o
o
o
o
o
o
ϕ5 =
o
o
o
n
(53)
The corresponding MOs are written as
Figure 2. Graphical representations of the (a) symmetric and (b)
antisymmetric MOs for n-polyenes (n = odd).
The boundary condition on the terminal atoms for the
1
symmetric MO is cos 2 (n + 1)θ = 0. Meanwhile, for the
1
antisymmetric MO, it is sin 2 (n + 1)θ = 0. Combining these
two boundary conditions, we have eq 17. Therefore, we can get
exactly the same n solutions for θ as in eq 18.
Nevertheless, when n is an even number, the origin also sits on
the middle of the chain, and the graphical representations for
MOs are depicted in Figure 3. Again, the boundary conditions
1
1
are cos 2 (n + 1)θ = 0 and sin 2 (n + 1)θ = 0, respectively, for
the symmetric and antisymmetric MOs.
For cyclic polyene, here we take benzene as an illustrative
example. There are two types of symmetric planes that can be
used to classify the MOs. The first symmetry plane passes
through the middle point of two opposite CC bonds, and the
1
(χ + χ2 + χ3 + χ4 + χ5 + χ6 )
6 1
1
(2χ + χ2 − χ3 − 2χ4 − χ5 + χ6 )
12 1
1
(χ + χ3 − χ5 − χ6 )
2 2
1
(χ − χ3 + χ5 − χ6 )
2 2
1
(2χ − χ2 − χ3 + 2χ4 − χ5 − χ6 )
12 1
1
(χ − χ2 + χ3 − χ4 + χ5 − χ6 )
6 1
(54)
Meanwhile, the symmetric properties of the MOs can also be
classified by using the second type of symmetric plane that
passes through two opposite carbons. This time, the benzene
Figure 3. Graphical representations of the (a) symmetric and (b) antisymmetric MOs for polyenes (n = even).
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Article
Figure 4. (a) Graphical correspondence of hexatriene and benzene, and the graphical representations of the (b) symmetric and (c) antisymmetric
MOs for benzene.
Alternatively, we can use both σv symmetric planes to classify
the MOs. In fact, for the purpose of classifying benzene’s MOs, it
is more convenient to use the C2v subgroup rather than its
highest symmetric D6h point group. The first type of MO
belongs to the a1 irreducible representation, which is symmetric
under the operations of both σv(xz) and σv(yz) planes. The
graphic representation of the a1 MO of benzene is very simple,
see Figure 6. Since the σv(xz) passes through the first atom on
ring can be derived from 7-polyene by overlapping the two
terminal atoms, see Figure 5a.
Figure 5. (a) Graphical correspondence of 7-polyene and benzene, and
the graphical representations of the (b) symmetric and (c)
antisymmetric MOs for benzene.
Figure 6. Graphic representation of a1 MO of benzene. Here and after,
(k) and [l] in the graph denote cos mθ and sin lθ, respectively.
The graphic representations of the symmetric MOs are
labeled in Figure 5b. The boundary condition is written as
cos 2θ − 2 cos θ cos 3θ + cos 2θ = 0
the left, its coefficient can be predefined as cos 0. Then, the atom
next to it on the top has a coefficient of cos θ, according to the
recurrence relation. Then, by using the reflection operations of
both σv planes, symmetrically we have the coefficients of the
other atoms. Now, the boundary condition of the terminal atom,
which can be any one of the four atoms lying on the top or the
bottom, can be written as
(55)
It can be simplified as
cos 4θ = cos 2θ
(56)
which means that the clockwise and anticlockwise approaches
produce equal coefficients for the atom nearest to the terminal
one. Meanwhile, the antisymmetric MO is depicted diagrammatically in Figure 5c, and the boundary condition is simply
written as
sin 3θ = −sin 3θ
cos θ = cos 2θ
(58)
where we have used the symmetric property of the a1 MO in the
left-hand side, and applied the recurrence relation in the righthand side for the coefficients of the terminal atom. Solving eq 57,
we get the two a1 π-MOs of benzene, that is to say ϕ0 and ϕ4 in
eq 47. The other MOs of different symmetries are graphically
presented in the Supporting Information. Interested readers can
deduce the AO coefficients of these MOs very easily by using the
graphs therein.
(57)
From the boundary conditions, we get six θ solutions, i.e.,
1
2
1
2
θ = 0, 3 π , 3 π , π for symmertic MOs, and θ A = 3 π , 3 π for
antisymmertic MOs.
S
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Figure 7. Graphical representations of π-MOs for naphthalene. The symbols a1, a2, b1, and b2 are irreducible representations of the C2v point group. Sx,
Sy (Ax, Ay) denote that the MO is symmetry (antisymmetry) under the reflection of σv(xz) and σv(yz) planes, respectively.
■
π-MOS OF NAPHTHALENE
For naphthalene, we can use again the C2v subgroup for the
purpose of classifying its MOs. Figure 7 plots the graphical
representation of all four types of π-MOs. As another illustrating
example, let us now deduce the b1 orbitals of naphthalene in
Figure 7c. These orbitals are symmetric with respect to the
reflection of the σx(xz) mirror which bisects the molecule
horizontally. Therefore, the first atom on the far left takes a value
1
of cos 2 θ as the coefficient. Then, the atom next to it on the top
ϕ1b1 = 0.263(χ1 + χ4 − χ5 − χ8 ) + 0.425(χ2 + χ3 − χ6
1
− χ7 ), λ1b1 = (1 + 5 )
(61)
2
ϕ2b1 = 0.425(χ1 + χ4 − χ5 − χ8 ) − 0.263(χ2 + χ3 − χ6
1
− χ7 ), λ 2b1 = (1 − 5 )
(62)
2
Using the graphical representations of π-MOs for naphthalene
in Figure 7, we can derive all the other MOs. We have
summarized the boundary conditions and eigenvalues of MOs in
the Supporting Information and leave the deductions as
excersies for students.
3
has a coefficient of cos 2 θ . Then, by using the σx(xz) symmetric
plane, we have the coefficients for the lower two atoms on the
left, and then, the coefficients of the right four atoms can be
obtained by taking corresponding values on the right atoms but
with opposite sign, since the b1 orbital is antisymmetric with
respect to the reflection of the σv(yz) plane. Moreover, the
bridge atoms take vanishing coefficients. Therefore, the b1
orbital can be expressed as
3
θ(χ + χ4 − χ5 − χ8 )
2 1
1
+ cos θ(χ2 + χ3 − χ6 − χ7 )
2
■
PEDAGOGICAL IMPLEMENTATION
The graphical approach presented in this article can be
implemented for classroom use in an advanced physical
chemistry lecture. The graphical approach originally proposed
by Zhang et al.34 has been carried out for classroom application
for about 40 years.
The lecture should be carried out next to a lecture in which the
HMO model has been taught. In the lecture, the instructor may
first introduce that solving a higher-order secular equation or
characteristic polynomial is very tedious, when we apply the
conventional HMO theory to slightly complex open-chain and
cyclic polyenes, such as hexatriene and benzene, because it
requires advanced knowledge of group theory. Then, the
students will learn that, for unbranched hydrocarbons, the AO
coefficients of Hückel MOs exhibit some types of regularities,
which are due to the linear recurrence relation of the coefficients.
Then, the instructor may introduce the general method in
solving the linear homogeneous recurrence relation, and then
discuss its connection to the method in solving linear
ϕb1 = cos
(59)
The boundary condition for the b1 orbital can be written as
5
cos θ = 0
2
(60)
where we have used the symmetric property on the right-hand
side and applied the recurrence relation on the left-hand side for
the terminal bridge atom. Equation 60 has two solutions,
1
3
θ = 5 π , 5 π . Thus, after normalization, we have the wave
function of the two b1 orbitals,
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Notes
homogeneous differential equation, which may have been
presented in first-year calculus course. In what follows, the
graphic approach can be applied for open-chain n-polyenes by
using n = 4, 5, and/or 6 as examples. In the classroom, the
instructor derives the π-MOs and their energies for butadiene by
following the procedure given in the text. Then, students
perform calculations for the remaining two examples to gain a
deeper understanding of the π-MOs of open-chain n-polyenes.
For cyclic Hückel and Möbius types of polyenes, the learning
processes can be accomplished similarly, but with an emphasis
on the degeneracy of energy levels for these two systems. The
Frost and Musulin diagram, and Zimmerman’s extension for the
Möbius system, may then be taught as mnemonics for the
purpose of gaining a pleasing learning experience for students.
Then, students can be encouraged to classify the symmetry of πMOs of all three systems one by one, and subsequently to use the
symmetric properties to deduce AO expansion coefficients by
the graphical approach. The deduction of the π-MOs of
naphthalene can be assigned as an advanced example to educe
students’ enthusiasm for applying the graphical approach.
Consecutively, one or two lectures on the conservation of MO
symmetry,35 the frontier MO theory,36 Hückel’s 4n + 2 rule, and
Möbius aromaticity can be more effectively presented after the
students have learned and appreciated fully the symmetry
properties and energy levels of MOs of open-chain and cyclic
polyenes derived in this project.
The author declares no competing financial interest.
■
ACKNOWLEDGMENTS
This project is supported by the Natural Science Foundation of
China (21673186, J1310024). The author is very grateful to
Prof. Menghai Lin and Prof. Qianer Zhang for their helpful
discussion. The author would also wish to thank the associate
editor Prof. Halpern and three anonymous referees for their
valuable comments and the suggested references (11, 12, and
14−17), which substantially improve the quality of the paper.
■
CONCLUDING REMARKS
In summary, we reformulate the predefined coefficient graphical
method to solve Hückel MOs for unbranched hydrocarbons and
some simple derivations. The graphical approach gives every
vivid picture for students to capture the periodical trends in the
AO coefficients of Hückel MOs via the examples discussed here.
Moreover, the students could gain improved comprehension of
the symmetric properties of the MOs of open-chain and cyclic
polyenes by using them to derive the AO coefficients, rather than
knowing first the exact form of the MOs and then finding the
symmetric properties among them. By applying the graphical
representation of the HMO theory to more challenging
molecules, students would stimulate their appetite for learning
more theoretical chemistry. We found that the graphical
approach is a good mnemonic device for students’ comprehension about the HMO theory. Moreover, the strict
mathematical deduction of the graphical method can provide
inspiration for our future study of more complex systems.
ASSOCIATED CONTENT
sı Supporting Information
*
The Supporting Information is available at https://pubs.acs.org/doi/10.1021/acs.jchemed.9b00687.
■
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Article
Student assignments and additional instructional resources (PDF)
AUTHOR INFORMATION
Corresponding Author
Zhenhua Chen − Xiamen University, Xiamen, China;
orcid.org/0000-0001-5545-4462; Email: zhhchen@
xmu.edu.cn
Complete contact information is available at:
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