CHEMISTRY and

MATHEMATICS

Ivan Gutman Faculty of Science

University of Kragujevac

Kragujevac, Serbia

MATHEMATICS science studying

abstract objects:

quantities (numbers), structures, spaces

and their changes

and

relations between these

natural numbers: 1, 2, 3, 4, 5, …

rules for addition, subtraction, multiplication, division

prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, …

Fermat’s (small) theorem:

If n is not divisible by a prime p, then

1 1(mod )pn p

20 1 2 0a a x a x equation soluble in radicals

21 1 0 21

0

22 1 0 21

0

14

2

14

2

x a a a aa

x a a a aa

20 1 2 0a a x a x

2 30 1 2 3 0a a x a x a x

2 3 40 1 2 3 4 0a a x a x a x a x

2 3 4 50 1 2 3 4 5 0a a x a x a x a x a x

equation soluble in radicals

equation

equation

equation

soluble in radicals

soluble in radicals

NOT soluble in radicals

MATHEMATICS

in

CHEMISTRY

atoms/elements: C, O, H, N, …

rules for mutual connection

valency: 1, 2, 3, 4, …

4C O H

3C O H

NOT POSSIBLE

2 6C O H

6 14C H

2 2n nC H Clifford, 1878

Arthur Cayley (1821-1895)

Chemical Graph Theory

~ 1860 STRUCTURAL FORMULAS

(Kekulé, Couper, Butlerov, …)

1874 STRUCTURAL FORMULA = GRAPH

(Arthur Cayley)

1874 ENUMERATION OF ALKANES

A. Cayley “On the Mathematical

Theory of Isomers”, Phil. Mag.

6 14C H has 5 isomers

HOW MANY ISOMERS

with n carbon atoms ?

n=1 one

n=2 one

n=3 one

n=4 two

n=5 three

n=6 five

n >6 difficult

Enumeration of Alkanes

1874 Cayley: unsuccessful

1932 Henry Henze & Richard Blair

1935 György Pólya

György Pólya (1887-1985)

Number of isomeric alkans n number of isomers 1 1

2 1

3 1

4 2

5 3

6 5

7 9

8 18

9 35

10 75

15 4347

20 366319

30 4111846763

40 62481801147341

50 1117743651746953270

80 10564476906946675106953415600016

MATHEMATICS in

CHEMISTRY

is much older

JABIR ibn HAYYAN ~ (721 – 815)

JABIR ibn HAYYAN (= Geber) ~ (721 – 815)

CLAIMS THAT THE NUMBERS

1

3

5

8

17 28 = 4 7

ARE OF GREAT IMPORTANCE

EVERYTHING IN THE WORLD

IS DETERMINED BY THE

NUMBER 17

e. g. METALS HAVE 17 POWERS

(PROPERTIES, QUALITIES)

4 9 2

3 5 7

8 1 6

4 9 2

3 5 7

8 1 6 17

28

MAKE A CIRCLE OF MALE AND

FEMALE, THEN A SQUARE, FROM

THAT A TRIANGLE; MAKE A

CIRCLE, AND THOU SHALT HAVE

THE PHILOSOPHER’S STONE

MICHAEL MAIER, 1618

let’s return to

contemporary chemistry

Hückel Molecular Orbital Theory

Erich Hückel (1896-1980)

Hückel Molecular Orbital Theory

1931 Hückel: molecular orbital theory

1931 Hückel: π-electron configuration of benzene

1956 Günthard & Primas:

HMO ~ Spectral Graph Theory

Hückel Molecular Orbital Theory

HΨ = EΨ

H = Hamiltonian operator

H = αI + βA

A = adjacency matrix

of molecular graph

biphenylene molecular graph

of biphenylene

H = αI + βA

n ...21 eigenvalues of molecular graph

Energy of i-th -electron:

i iE

Hückel Molecular Orbital Theory

Hückel Molecular Orbital Theory

1964 Sachs: Sachs theorem

1972 Graovac-Gutman-Trinajstić-

Živković: application of Sachs theorem

Hückel Molecular Orbital Theory

1972 Graovac-Gutman-Trinajstić-

Živković: application of Sachs theorem

1972 extensive chemical application of

spectral graph theory

~103 papers

A Graph Theoretical

THEORY of

CYCLIC CONJUGATION

Contribution of a cycle (ring) C

to total π-electron energy

0

2 ( , )ln

( , ) 2 ( , )

G ixef dx

G ix G C ix

Gutman

+0.1586 +0.0534 +0.1586

+0.0198 +0.0275 +0.0275

Total -electron energy

/2

1

( 1)/2

( 1)/21

2

2

n

ii

n

n ii

n

E

n

Hückel

1 '( , )

( , )

ix G ixE n dx

G ix

Coulson

ENERGY of GRAPH

n

i

iGEE1

||)(

Gutman (1978)

1 2( , ) ( , )k b G k b G k

2

20

1( ) ln ( , ) k

k

dxE G b G k x

x

1 2( ) ( )E G E G

Gutman

NUMEROUS APPLICATIONS

for finding graphs with extremal

(maximal or minimal) energy

PAPERS ON GRAPH ENERGY

IN THE XXI CENTURY

(2001 )

Nov 2017: >806

Dec 2015: >630

Dec 2016: >710

806:17=47.4 per year

47.4:12=3.95 per month

Thank you