1 3
Theor Chem Acc (2015) 134:114
DOI 10.1007/s00214-015-1709-4
REGULAR ARTICLE
Hydrocarbon chains and rings: bond length alternation in fi nite
molecules
Jen o ˝ Kürti
1 · János Koltai
1 · Bálint Gyimesi
1 · Viktor Zólyomi
2,3
Received: 19 June 2015 / Accepted: 28 July 2015 / Published online: 9 September 2015
© Springer-Verlag Berlin Heidelberg 2015
more recent publications about the BLA in carbon chains,
see, e.g. Refs. [ 3 – 6 ] and references therein. Polyyne in particular remains a hot topic experimentally and theoretically,
in part due to the diffi culty in determining the band gap of
this material [ 6 ]. One reason why it is diffi cult to correctly
calculate the band gap in Peierls distorted systems is that
the bond length alternation directly determines the band
gap, i.e. if the bond length alternation is underestimated (as
is often the case in, e.g. density functional theory), the band
gap will be underestimated as well. As such, the matter of
correctly describing bond length alternation is a current and
important topic.
In addition, Peierls distortion brings with it some interesting fundamental questions, such as whether the bond
length alternation appears in carbon rings . It is known that
in long linear chains, as we increase the length of the chain,
we approach the infi nite polymer where the Peierls distortion is realized. However, the effect cannot depend on the
boundary conditions, and hence, a hydrogen terminated
fi nite chain and a cyclic carbon ring must converge to the
same Peierls distorted geometry in the infi nite limit. All that
can be different is the speed of convergence. In this work,
we discuss the question of Peierls distortion in carbon rings
and at what number of carbon atoms the alternation appears.
The BLA in conjugated polymers can be very well
described by an improved Hückel-type method, the semiempirical Longuet-Higgins–Salem (LHS) model [ 7 , 8 ].
1
The corresponding solid-state physics approximation is the
Su–Schrieffer–Heeger model [ 9 , 10 ]. The relationship
between the LHS and the SSH models was pointed out in
Ref. [ 11 ].
1 Developing the model and the corresponding HUGEH code and
applying it to various conjugated polymers was a very fruitful collaboration between P.R. Surján and J. Kürti.
Abstract We present a theoretical study of Peierls distortion in carbon rings. We demonstrate using the LonguetHiggins–Salem model that the appearance of bond alternation in conjugated carbon polymers is independent of the
boundary conditions and does in fact appear in carbon rings
just as in carbon chains. We use the Hartree–Fock approximation and density functional theory to show that this
behaviour is retained at the fi rst principles level.
Keywords Peierls distortion · Conjugated polymers ·
Annulenes · Longuet-Higgins–Salem model · Density
functional theory
1 Introduction
Peierls distortion [ 1 ] in one dimension (1D) is an inherent trait of linear conjugated polymers, leading to a bond
length alternation (BLA). The usual chemical nomenclature for it is conjugation. Examples include polyacetylene
and polyyne, two materials which have been studied by
the physics and chemistry community for a very long time
(see, e.g. the book [ 2 ] and the papers cited therein). For
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Jenő Kürti
kurti@virag.elte.hu
1
Department of Biological Physics , Eötvös University
Budapest , Budapest , P.O.B. 32 , 1518 , Hungary
2
Physics Department , Lancaster University , Lancaster LA1
4YB , UK
3
Wigner Research Institute , Hungarian Academy of Sciences ,
Budapest , P. O. B. 49 , 1525 , Hungary
81
Reprinted from the journal
Theor Chem Acc (2015) 134:114
DOI 10.1007/s00214-015-1709-4
REGULAR ARTICLE
Hydrocarbon chains and rings: bond length alternation in fi nite
molecules
Jen o ˝ Kürti
1 · János Koltai
1 · Bálint Gyimesi
1 · Viktor Zólyomi
2,3
Received: 19 June 2015 / Accepted: 28 July 2015 / Published online: 9 September 2015
© Springer-Verlag Berlin Heidelberg 2015
more recent publications about the BLA in carbon chains,
see, e.g. Refs. [ 3 – 6 ] and references therein. Polyyne in particular remains a hot topic experimentally and theoretically,
in part due to the diffi culty in determining the band gap of
this material [ 6 ]. One reason why it is diffi cult to correctly
calculate the band gap in Peierls distorted systems is that
the bond length alternation directly determines the band
gap, i.e. if the bond length alternation is underestimated (as
is often the case in, e.g. density functional theory), the band
gap will be underestimated as well. As such, the matter of
correctly describing bond length alternation is a current and
important topic.
In addition, Peierls distortion brings with it some interesting fundamental questions, such as whether the bond
length alternation appears in carbon rings . It is known that
in long linear chains, as we increase the length of the chain,
we approach the infi nite polymer where the Peierls distortion is realized. However, the effect cannot depend on the
boundary conditions, and hence, a hydrogen terminated
fi nite chain and a cyclic carbon ring must converge to the
same Peierls distorted geometry in the infi nite limit. All that
can be different is the speed of convergence. In this work,
we discuss the question of Peierls distortion in carbon rings
and at what number of carbon atoms the alternation appears.
The BLA in conjugated polymers can be very well
described by an improved Hückel-type method, the semiempirical Longuet-Higgins–Salem (LHS) model [ 7 , 8 ].
1
The corresponding solid-state physics approximation is the
Su–Schrieffer–Heeger model [ 9 , 10 ]. The relationship
between the LHS and the SSH models was pointed out in
Ref. [ 11 ].
1 Developing the model and the corresponding HUGEH code and
applying it to various conjugated polymers was a very fruitful collaboration between P.R. Surján and J. Kürti.
Abstract We present a theoretical study of Peierls distortion in carbon rings. We demonstrate using the LonguetHiggins–Salem model that the appearance of bond alternation in conjugated carbon polymers is independent of the
boundary conditions and does in fact appear in carbon rings
just as in carbon chains. We use the Hartree–Fock approximation and density functional theory to show that this
behaviour is retained at the fi rst principles level.
Keywords Peierls distortion · Conjugated polymers ·
Annulenes · Longuet-Higgins–Salem model · Density
functional theory
1 Introduction
Peierls distortion [ 1 ] in one dimension (1D) is an inherent trait of linear conjugated polymers, leading to a bond
length alternation (BLA). The usual chemical nomenclature for it is conjugation. Examples include polyacetylene
and polyyne, two materials which have been studied by
the physics and chemistry community for a very long time
(see, e.g. the book [ 2 ] and the papers cited therein). For
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Jenő Kürti
kurti@virag.elte.hu
1
Department of Biological Physics , Eötvös University
Budapest , Budapest , P.O.B. 32 , 1518 , Hungary
2
Physics Department , Lancaster University , Lancaster LA1
4YB , UK
3
Wigner Research Institute , Hungarian Academy of Sciences ,
Budapest , P. O. B. 49 , 1525 , Hungary
81
Reprinted from the journal