Download Proof of the Formulae for the Molecular Orbitals and Energy Levels

CROATICA
CHEMICA
CCACAA 59 (3) 533-538
ACTA
(1986)
YU ISSN 0011-1643
UDe 541
Original Scientific Ptiper
CCA-1666
Proof of the Formulae for the Molecular Orbitals and
Energy Levels of Mobius Annulenes, Based on the Theory
of Skew-Circulant Matrices
A. C. Day
Dyson
Perrins
Laboratory,
University
of
Oxford,
United
Kingdom
R. B. MaHion
The King's
School, Canterbury,
United
Kingdom
and
M. J. Rigby
St. Patrick's
House 47, Park
Hill Road, WaHington,
Received September
Surreu,
United
Kingdom
4, 1985
The formulae for the molecular orbitals and corresponding
energy-levels
that arise in a Hiickel molecular-orbital
(HMO)
treatment of Mi:ibius systems are derived by appeal to the theory
of skew-circulant matrices. The approach adopted is analogous to
that previously used to obtain the orbital energiea and HMO's of
'Hiickel' annulenes from the theory of circulant matrices.
INTRODUCTION
In his classic 1964 paper, Heilbronner! stated (without proof) the expressions for the molecular orbitals, and corresponding energy-levels, that arise
in a Hiickel-molecular-orbital
(HMO) treatment of Mčbius systems - results
which very quickly inspired others=? to an intensive theoretical study of
M6bius molecules.? These same formulae were subsequently proved in Zimmerman's book"; this proof is, however, not a short one, making appeal, as
it does, to the ideas of group theory and the use of trial solutions. In this
note, therefore, we show how these resuits may be elegantly obtained, outside
the immediate realms of HMO theory, by judicious exploitation of the theory
of skew-circulant matrices.P'"! The arguments called upon are entirely analogous to those previously invoked by one of US13,14 to derive the orbital
energies and HMO's of 'Hiickel' annulenes from the theory of circulant
matrices (e. g., ref. 13).
A. C. DAY ET AL.
534
THE
The circulant
PROPERTIES
OF
CIRCULANT
MATRICES
matrix
1
/
°2
an
•••••••
(
.
\
02
~
0t..
• • • • ••
°1
k = 1,2, ... , n, and
{Ak'}
has been shown10-15 to have eigenvalues,
vectors, {Vk'} k
1, 2, ... , n, where
=
eigen(1),
and
(2),
v'k
and
(Uk
is an
nth
root of unity -
i. e., one of the n roots of the scalar equation
=
wn
THE
We
POW
PROPERTIES
(3).
1
OF SKEW-CIRCULANT
show that the skew-circulant
B,
···.. .
···...
·. ·...
.... ... ..... ......
···...-
01
-an
B
matrix
MATRICES
O2
03
an
01
02
~-1
01
°r>-2
-on_1
-an
-02
-03,
-04
01
has eigenvalues and eigenvectors formally given by equations
provided that (Uk in those equations is everywhere replaced by
is an nth root of minus one.
(1) and (2),
where ek
@k,
First consider the quantity
(4)
FORMULAE
where @k is one particular
of the scalar equation
Equation
MtlBIUS
535
ANNULENES
root of minus one -
nth
i. e., one of the n roots
19n=-1
(5) has solutions
19k = cos [(2k
FOR
{19k}
k
(5),
= 1,2, ...
, n, which are given by
+ 1) 1r/nJ + i sin [(2k + 1) 1rlnJ =
exp
[(2k
+ 1)i 1rlnJ
(6).
It will be clear from equations (4) and (5) that the following equations
true:
lk = al + a2 19k + a319k2
+ ... + an 19kn-l
19k1k =
=-
19k2lk
This set of equations
al
-an
+ al
an_l -
19k + a2 19k2 +
an 19k +
al
19k! +
implies the following matrix
O
0
3
an
al
O
2
an 1
Qk
-an
al
°n_2
Q2
k
2
..................
.....
-03
which, on the left-hand
the vector*
-04
+ an_l
+ an_2
19kU-l
19kn-l
(7)
equation:
/'\
Qk
=
Ak
al
siđe, contains
are
\
Q2
k
(8)
:~)
Q
k
the skew-circulant
matrix
B. Renee,
(9)
is an eigenvector of B, with eigenvalue Ak given by equation (4). Since @k
is just one of the n roots (equation (6)) of equation (5), it follows that there
will be a Vk (equation (9)) and a Ak (equation (4)) corresponding to every
distinct @k, k = 1,2, ... , n. {vd k = 1,2, ... , n and {Ad k = 1,2, .. , n are
thus the complete set of eigenvectors and corresponding eigenvalues of B.
* Or, (ef. Heilbronner's comment-), the appropriate real linear-combination of
Vk and Vk*, to give linear-combination-of-atomic-orbital
(LCAO) molecular-orbitals
(MO's) with real coefficients.
A. C. DAY ET AL.
536
It should be noted that the set of eigenvectors, {vd, is the same for all
skew-circulant
matrices, but that the eigenvalues depend explicitly upon
the elements that comprise any row of the particular skew-circulant matrix
under consideration.
APPLICATION
TO MOBIUS
ANNULENES
As may be seen from earlier work (e. g. refs. 1, 8, 16-19), a suitable
adjacency-matrix for the cyclic graph CnM representing the carbon-atom eonnectivity of a Mobius [n]-annulene would be a skew-circulant matrix with
first row (O, 1, O,... , O, - 1) - that is, a matrix of the form**
o
o
o
A(C~)
o
o
-1
o
o
o
o
o
1
o
o
o
o
-1
o
o ••••
In the terminology of the matrix B, therefore,
are non-zero, and so, from equation (4),
lk
It Iollows from equation
equation (11) as
only a2 (= 1) and
an (= -1)
= ek - ek
(11).
=-
Bk-i, which allows us to write
n 1
-
(5) that Bkn-l
lk
(lO).
= ek + ek-
1
whence, by equation (6), we obtain
}'k
=
2 cos [(2k
+ 1) n/n]
(12).
The eigenvalue list for the matrix A (CnM) is thus {lk}
k = 1,2, ... , n. It
should be observed that the eigenvalues are, of course, entirely real, since
A (CnM) is real-symmetric, even though this is not the case for the general
skew-circulant matrix B.
Finally, since, in the HMO interpretation,
orbital energies (e kM) are
measured in units of a resonance integral {J' and with reference to a Coulomb
integral a, namely,
we may write equation (12) as
€kM
= a + 2 j3' cos [(2k + 1) n/n]
The solution for the corresponding
H
€k
=a
(13).
Hiickel system takes the form
+ 2 j3 cos (2k
n/n)
(14).
** In its quantum-mechanical interpretation, equation (10) implies the choice
of a basic set of atomic orbitals that has onlyasingle < I}) , IH Il})j > - term negative. This choice is always possible for a Mobius ring.l.8.16-1D
FORMULAE
FOR
MdBIUS
ANNULENES
537
Writing P' in equation (13) and P in equation (14) allows for the faet that
the dihedral angles (eu) between adjacent p-orbitals are different for a regular
Mčbius and a regular Hiickel system - namely, n/n (= eu) and zero, respeetively, the resonanee integrals being related by ~' = .~eos (n/n).
Comparison of the eosine term in equation (13), which may be expressed
as eos (2krc/n + n/n), with that in equation (14) shows that the Mčbius and.
the Hiickel solutions for a given n are n/n »out of phase« with eaeh other.
In Frost and Musulin's mnemonie device'" for representing the RMO energy-lev el spectrum of Huckel annulenes, the regular n-gon is oriented with one
vertex »down«. The phase angle of n/n by which equations (13) and (14)
differ eorresponds in this eontext to a rotation of Frost and Musulin's polygon20 through an angle of n/n - i. e., one half of the angle subtended by
an edge of the polygon at its centre. That is why, in the eorresponding
mnemonie deviee introdueed by Zimrnerrnanš-š'" for Mčbius systems, the
regular polygon is oriented, in a circle of radius 2 P', with one edge »down«.
To eomplete the RMO interpretation,
we recall-! that the eomponents
of the eigenveetor Vk of A (enA!) (or appropriate linear-eombinations
of Vk
and Vk* if real eoefficients are required-A'") are the weighting coefficients
of the individual basis atomie-orbitals, {c1\JIl = 1,2, ... , n, of the kth linear-eombination-of-atomie-orbitals
Hiickel MO, Pk, for the M6bius [n]-annulene i. e. (ef. refs. 1, 8):
pk
=
n-1j2
n
~
(19k)rr1 <Pil
p=l.
=
the last step being via equation
n
n~lj2 . ~
p=l
{exp [(2k
+ 1) (tA- -
1) i n/n] } <Pil
(15),.
(6).
In the eourse of diseussion of this paper at the Symposium in Dubrovnik,
Professor D. J. Klein kindly pointed out21 to the author who read the paper
at the Symposium (R. B. M.) that the solution to the Mčbius/Huckel matrix-problem may also be effeeted (a) by use of double-group representation?
and (b) by applieation of a suitable (diagonal) unitary-transformation
to yielđ
an Herrnitian (though not real) circulant-matrix."
Professor Klein also made
the intriguing observation'" that the eigenvalue problem pertaining to the
Mčbius/Huckel type of matrix may be generalised in other ways: one generalisation is to a matrix in which the two elements that, in the conversion
of a 'Htiekel' matrix into a 'Mčbius' one, are ehanged from + 1 to -1 are
replaced by ei<l> and e-i<l> (eP being real). (A 'Huckel' matrix therefore corresponds to eP = 2krc, k an integer, while the 'Mčbius' matrix is reprodueed
when eP = (2k + 1) rc). The eigenvalue solution to this matrix may then be
obtained'" via method (b), above, and constitutes a simple generalisation of
the Frost-Musulin'"
'shadow diagram'š": their polygon is simply rotated by
eP/n. A seeond generalisation is in terms of a Pariser-Parr-Pople
type of
model that includes eleetron-eleetron interaction=; in this case, simple, elegant solutions do not seem to be known=, though some general properties
of the ground-state eigen-solution have recently been established."
538
A. C. DAY ET AL.
REFERENCE S
1.
2.
3.
4.
5.
6.
7.
8.
E. He i 1b r o n ner,
Tetrahedron
Letters (1964) 1923.
S. F. Ma s o n, Nature 205 (1965) 495.
H. E. Zi m m e r man,
J. Am. Chem. Soc. 88 (1966) 1564.
M. J. S. De war,
Tetrahedron
Supplement
8 Part I (1966) 75.
H. E. Zi m m e r man,
Angew. Chem. Int. Ed. Eng!. 8 (1969) 1.
H. E. Zi m m e r man,
Accounts
Chem. Res. 4 (1971) 272.
H. E. Zi m m e r man,
Tetrahedron
38 (1982) 753.
H. E. Zi m m e r man,
Quantum
Mechanics For Organic Chemists, Academic
Press, London, United Kingdom, 1975, pp. 113-116.
9. A. G rao vac
and N. Tri naj s t i ć, Croat. Chem. Acta 47 (1975) 95.
10. G. Ko w ale w s k i, Determinantentheorie,
Third Edition, Chelsea Publishing
Co., New York, United States of America, 1943, p. 26.
11. W. Ka u z man n, Quantum
Chemistry:
An Introduction,
Academic Press,
New York, United States of America, 1957, pp. 48-50.
12. P. J. Da v i s, Circulant
Matrices,
John Wiley & Sons, New York, United
States of America, 1979.
13. R. B. Mall i o n, BuU. Soc. Chim. France (1974) 2799.
14. C. A. C o u 1s o n, B. O' Lea r s, and R. B. Mall i o n, Hilckel Theory For
Organic Chemists, Academic Press, London, United Kingdom,
1978, Appendix
C, pp, 156-158.
15. W. L. Fer r a r, Algebra, Second Edition, Oxford University Press, Oxford,
United Kirigdom, 1957, pp. 23-24.
16. A. G rao vac
and N. Tri naj s t i ć, J. Molec. Structure
30 (1976) 416.
17. E. C. Hass
and P. J. Plath,
Z. Chem. 22 (1982) 14.
18. A. C. Day, R. B. Mall i o n, and M. J. R i g b y, On The Use of Riemannian
Surfaces
In
The Graph-Theoretical
Representation
of Mobius
Systems, in
Proceedings
of the Symposium on Chemical
AppLications
of Topology and
Graph Theory, held at the University
of Georgia, Athens, United States of
America, April 18th April 22n", 1983 (Editor: R. B r u ceK
i n g), Elsevier
Science Publishing Co., New York, United States of America, 1983, pp. 272-284.
19. N. Tri naj s t i ć, Chemical Graph Theory, Vol. I, C. R. C. Press, Inc., Boca
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21. D. J. K 1e i n, Personal Communication
to R. B. M., at Dubrovnik,
September, 1985.
22. R. A. Ha r r i s, Chem. Phys. Letters 95 (1983) 256.
23. D. J. K 1e i n and N. Tri naj s t i ć, J. Am. Chem. Soc. 106 (1984) 8050.
SAŽETAK
Dokaz formula
za molekularne orbitale i energijske razine Mobiusovih anulena
zasnovan na teoriji specijalnih cikličkih matrica
A. C. Day, R. B. MaHion
i M. J. Rigby
Izvedene su formule za molekularne orbitale i energijske razine, koje se javljaju
u okvirima Hiickelove MO teorije, i to za Mčbiusove anulene, s pomoću teorije specijalnih cikličkih matrica.