history of mathematical chemistryconf.uni-obuda.hu/ieeeday2017/ivangutman_presentation.pdf ·...
Post on 14-Jun-2020
11 Views
Preview:
TRANSCRIPT
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
top related