Extremely branched alkanes

Zoran Markovic, Vesna Ivanov-Petrovic, Ivan Gutman*

Faculty of Science, University of Kragujevac, P.O. Box 60, Kragujevac, YU-34000, Serbia and Montenegro

Received 21 January 2003; revised 3 March 2003; accepted 6 March 2003

Abstract

Chemical trees are the graph representations of alkanes. Recently the extremely branched chemical trees have been

determined [Z. Naturforsch. 57a (2002) 49], namely the species that among all chemical trees with a given number n of vertices

have the smallest connectivity and Wiener indices, and the greatest first ordinary and first Laplacian eigenvalues. Because of

steric reasons, for larger values of n these extremal molecular graphs do not correspond to chemically sound molecules. By

means of the PM3 semiempirical MO method we now determine the value of n until which the ‘extremely branched alkanes’ are

stable molecules: (n around 16), as well as the value of n beyond which these cannot exist (n around 20).

q 2003 Elsevier B.V. All rights reserved.

Keywords: Branching; Branched alkanes; Molecular graph; Chemical tree; PM3

1. Introduction

A lot of work has been done to characterize the

‘extent of branching’ of the carbon-atom skeleton of

organic molecules [1–18]. These researches resulted

in a number of graph-based molecular-structure-

descriptors [19,20] that provide numerical measures

of branching. Consequently, molecular graphs pos-

sessing extremal (minimal or maximal) values of

these structure-descriptors may be viewed as pertain-

ing to extremely branched molecules.

In what follows we shall be concerned only with

alkanes—the simplest (from a structural point of

view) acyclic systems—and their molecular graphs

called ‘chemical trees’. (A chemical tree [21,22] is a

connected acyclic graph in which no vertex has degree

greater than 4.) The number of carbon atoms of an

alkane and the number of vertices of the respective

chemical tree will be denoted by n:

Finding the extremal values of a structure-descrip-

tor in the class of all n-vertex chemical trees proved to

be a difficult task. Especially difficult is the character-

ization of chemical graphs representing the maximally

branched alkanes. A major breakthrough in this

direction was achieved by Fischermann et al. [23,24],

who established the structure of the trees having the

smallest Wiener index [23]. Subsequent studies

revealed that the very same chemical trees possess

the greatest first graph eigenvalue [24] as well as the

greatest first Laplacian eigenvalue [25]. These trees

have also the smallest connectivity index [26], but with

regard to this structure-descriptor they are not unique.

Anyway, all the results obtained [23–26] point

0166-1280/03/$ - see front matter q 2003 Elsevier B.V. All rights reserved.

doi:10.1016/S0166-1280(03)00196-9

Journal of Molecular Structure (Theochem) 629 (2003) 303–306

www.elsevier.com/locate/theochem

* Corresponding author. Tel.: þ381-34-331-876; fax: þ381-34-

335-040.

E-mail addresses: gutman@knez.uis.kg.ac.yu (I. Gutman),

mark@knez.uis.kg.ac.yu (Z. Markovic).

towards the conclusion that these have to be considered

as the maximally branched chemical trees.

A detailed procedure for the construction of the

maximally branched chemical trees (which is quite

complicated) is outlined elsewhere [24]. In Fig. 1. are

depicted the respective trees for n up to 26.

By inspecting Fig. 1, a skeptical person may

immediately ask if the corresponding alkanes do

really exist or-at least-have a reasonable chance to be

synthesized. In what follows we offer arguments in

favor of the conclusion that for n up to 16 the answer

is affirmative whereas for n ¼ 20 and beyond it is

negative.

Highly branched chemical trees have, of course,

attracted the attention of theoretical chemists also in

the past. Such chemical trees were investigated by

Diudea [27,28] and others [29]; in the works [27–29]

references to the extensive experimental research on

the underlying chemical compounds, called ‘dendri-

mers‘, can also be found.

For the present study an approach to maximally

branched chemical trees (and to the existence of the

corresponding alkanes), put forward by Klein [30], is of

particular importance. Klein realized that some highly

branched species cannot be self-avoidingly embedded

in a tetrahedral lattice. In his opinion, the respective

molecules cannot exist because their atoms are more

voluminous than the available space. Klein’s calcu-

lations [30] show that there are no such ‘forbidden’

chemical trees with n ¼ 22 or fewer vertices, and that

there are 1, 6, and 23 “forbidden” chemical trees with

23, 24, and 25 vertices, respectively.

From the results outlined in the present paper, it is

seen that Klein’s restrictions are insufficient to infer

Fig. 1. The molecular graphs ( ¼ the carbon-atoms skeletons) of the extremely branched alkane species CnH2nþ2 up to n ¼ 26: The general

procedure for constructing such molecular graphs is described in Refs. [23,24].)

Z. Markovic et al. / Journal of Molecular Structure (Theochem) 629 (2003) 303–306304

about the existence of highly branched alkanes.

Whereas ability to be embedded in a tetrahedral

lattice is certainly a necessary condition for existence,

it is far from being sufficient. Our calculations show

that sterically forbidden alkanes are encountered at

much lower values of n : for sure at n ¼ 20; and

possibly already at n $ 17:

2. The method

The structures of the branched alkanes were

generated by means of Chem3D; version 6.0 [31].

They were optimized using MOPAC 2000, version 1.3

[32], which is an integrated part of Chem3D: In order to

provide high reliability for investigating the properties

of the molecules and radicals, the MNDO-PM3

method was used [33 –35]. We determined the

optimized ground-state geometries and standard

enthalpies of formation of the respective alkanes (in

their most stable conformations) as well as of the

radicalic open-shell species obtained by infinitely

extending the carbon–carbon bond connecting the

two most branched carbon atoms. In the case when this

‘bond-extension’ process is exothermic, we also found

its transition state and determined the corresponding

activation enthalpy. The transition states for the

compounds C15–C22 were found using the appropriate

MOPAC facilities (TS, SADDLE). Such a transition

structure could not be established only for the C23

alkane. Eventually, the geometries of the transition

states were refined by means of Bartel’s method (non-

linear least squares gradient minimization routine),

and then checked by means of vibrational analysis for

the existence of a single ‘negative’ vibration. Further

details and results of our calculations are available

from the authors (from Markovic), upon request.

The main results obtained are collected in Table 1.

3. Discussion

From Table 1 we see that—according to expec-

tations—the length of the bond between the two most

branched carbon atoms monotonically increases with

n whereas the respective bond order monotonically

decreases. A remarkably large jump is observed

between n ¼ 19 and n ¼ 20: At n ¼ 24 there is no

energy minimum corresponding to the alkane species,

indicating that there is no chemical bond between the

respective two carbon atoms.

The dissociation enthalpies decrease monotoni-

cally, but jumpless, becoming negative (exothermic)

at n ¼ 15:

The activation enthalpies, computed only for n $

15; monotonically decrease, apparently becoming

equal to zero at n ¼ 24: The activation enthalpy for

n ¼ 23 was not found, so that one can only suppose

that its value is close to zero.

From the data given in Table 1 we see that the

breaking of the critical carbon–carbon bond becomes

exothermic at n ¼ 15: Therefore, for n # 14 the

extremely branched alkanes CnH2nþ2 (whose molecu-

lar graphs are depicted in Fig. 1) may safely be

predicted as stable compounds, capable of existence.

Starting with n ¼ 15 the breaking of the critical

carbon–carbon bond becomes energetically favorable

and thus (according to our calculations) the respective

compounds are thermodynamically unstable. How-

ever, for n ¼ 15 and the next few n values, the

activation enthalpy for such a bond dissociation is so

Table 1

The parameters characterizing the stability and the possibility of

existence of the most branched alkanes CnH2nþ2 (cf. Fig. 1):

BL ¼ distance (in pm) between the two most branched carbon

atoms, in the (geometry optimized) most stable ground-state

conformation of the respective molecule; BO ¼ the bond order of

the same carbon–carbon bond; DHdisoc ¼ change of enthalpy (in

kJ/mol) in the (virtual) process when the distance between the two

most branched carbon atoms is extended from its optimal value

( ¼ BL) to infinity; DHactiv ¼ activation enthalpy (in kJ/mol) of the

same process; for details see text.

n BL BO DHdisoc DHactiv

5 152.7 0.984

8 156.7 0.950 þ121.34

11 158.2 0.948 þ92.75

14 161.2 0.943 þ9.95

15 161.3 0.941 219.80 þ106.20

16 163.7 0.940 257.47 þ83.18

17 164.5 0.938 282.29 þ71.25

18 165.5 0.936 294.75 þ66.19

19 169.9 0.927 2126.23 þ45.89

20 176.0 0.908 2186.21 þ27.94

21 183.5 0.897 2222.12 þ20.95

22 194.7 0.874 2301.24 þ8.21

23 208.3 0.847 2416.06 <0

24 – 0 0

Z. Markovic et al. / Journal of Molecular Structure (Theochem) 629 (2003) 303–306 305

large that there seems to be a realistic chance that—

once synthesized and kept at a low temperature—

these compounds will not immediately decompose.

Anyway, for n exceeding 19 the activation enthalpy

seems to be not large enough to prevent bond

dissociation. This is paralleled by the sudden decrease

of the bond order and increase of the bond length.

Consequently, we predict that for larger values of n; the

earlier designed [24–26] ‘maximally branched n-

vertex chemical trees’ (see Fig. 1) do not correspond to

alkane molecules that can exist in reality. According to

our calculations, this is true beyond any doubt for n $

24; and most probably already for n $ 20:

4. Concluding remarks

The present study indicates that the ‘maximally

branched alkanes’ CnH2nþ2, as determined by means of

the methods of chemical graph theory [24–26], are

reasonably stable compounds, capable of being

synthesized, only up to (around) n ¼ 16: For n

exceeding 19 we predict that the respective compounds

cannot exist, because the bond between the two most

branched carbon atoms will spontaneously break (in

order to reduce the enormously large steric strain).

The cases n ¼ 17; n ¼ 18; and n ¼ 19 are border-

line. We doubt, but are not certain, that these alkane

species can exist in reality.

Acknowledgements

The present investigations were partially supported

by the Ministry of Sciences, Technologies and Devel-

opmentofSerbia,within theProjectsno.1389and1448.

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