7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
1/34
Dr. Ottorino ORIDr. Ottorino ORIActinium Chemical ResearchActinium Chemical Research
RomeRome
[email protected]@gmail.com
WEST UNIVERSITY OF TIMIWEST UNIVERSITY OF TIMIOARAOARAFACULTY OF CHEMISTRY, BIOLOGY, GEOGRAPHY
DEPARTMENT OF BIOLOGYDEPARTMENT OF BIOLOGY--CHEMISTRYCHEMISTRY
LAB. OF STRUCTURAL AND COMPUTAT IONAL PHYSICALLAB. OF STRUCTURAL AND COMPUTAT IONAL PHYSICAL--CHEMISTRY FORCHEMISTRY FORNANOSCIENCES & QSARNANOSCIENCES & QSAR(CFSCNQ)(CFSCNQ)
Prof. dr. dr. habil. Mihai V. PUTZProf. dr. dr. habil. Mihai V. PUTZ
Director CFSCNQDirector CFSCNQ
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
2/34
Chemical Graph Theory is an important branch of
Theoretical Chemistry.
Its basic element is the so-called molecular
graph G. Vertices V(G) of this graph denote
molecular atoms and edges E(G) are the bonds of
the molecule.
Normally, G corresponds the carbon atomskeleton of the molecule. This is the hydrogen-
depleted representation of the molecular
graphs. Atomic valence is the vertex degree r.
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
3/34
Example Lets consider a connected graph G
like this one with n nodes and m edges:
n =10; m =11
Chemically, G represents Naphtalene molecule
with n carbon atoms and m bonds.
Vertex degree r=2 or 3 corresponds to atomic
valence. For carbon, silicon compounds r 4.
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
4/34
Graph Theory properties are transferred to
molecular structures like the following TheFirst Theorem of Graph Theory:
Any graph contains an even number ofvertices of odd degree r
which becomes: Every molecule contains aneven number of atoms of odd valence
Then C61 fullerene and others with odd
carbon atoms are forbidden also by topology.
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
5/34
In Chemical Graph Theory gr aph invar i ant s or
t opologi cal i ndices are used to correlate theproperties of molecules to their chemical
structure.
A t opologi cal i ndex is defined by a map from a
graph G into a number or a finite string ofnumbers:
Topologica l indicesshould be graph invar i ant,
i.e. isomorphic graphs lead to the same
topological index.
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
6/34
Isomor phic gr aphs
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
7/34
Isomor phic gr aphs
EXERCISE are these graphs isomorphic ?
See: Lecture Notes on GRAPH THEORY
Tero Harju (2007), Department of Mathematics
University of Turku, FIN-20014 Turku, Finland
e-mail: [email protected]
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
8/34
Isomor phic gr aphs
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
9/34
An impor t ant t opological gr aph invar i ant t he Gr aph Ener gy
See : Ivan Gutman, Bo Zhou and Boris Furtula
The Laplacianenergy Like Invariant is an Energy Like Invariant
MATCH Commun. Math. Comput. Chem. 64 (2010) 85-96
ISSN 0340 - 6253 AND RELATED REFERENCES
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
10/34
Pr opert i es of t he Gr aph Ener gy
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
11/34
Then the total -electron energy of fullerenes and nanotubes
is determined solely by their carbon atom content and depends
very little on other structural features.
Pr oper t i es of t he Gr aph Ener gy f or r egular gr aphs,
i .e. whit vert ices wi t h t he same number of bonds (degr ee) r
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
12/34
Given two vertices u and v in V(G), we
denote by d(u,v) or du,v the distancebetween u and v, i.e. the length of the
shortest path connecting u and vby traveling
along the graph edges (or molecular bonds).
This distance belongs to the invariants of the
graphs conveying crucial information about
the structure.
TECHNICALLY AND COMPUTATIONALLY THIS
IS ONE OF SPECIALTIES OF Prof.PUTZs LAB
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
13/34
1 5 9 7 3
2 6 10 8 4
After an arbitrarilyarbitrarily numbering of the nodes
we calculate the length dij of the shortestpath connecting pair wise the n atoms.
n=10; m=11
Some cases: d11=0 d12=d21=1
d16=d19=2 .. Etc..
dmax=d14=5
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
14/34
composing - with all dij entries - the Matrix
D of the chemical distances of G:
1 5 9 7 3
2 6 10 8 4
D 1 2 3 4 5 6 7 8 9 10
1 0 1 4 5 1 2 3 4 2 3
2 1 0 5 4 2 1 4 3 3 2
3 4 5 0 1 3 4 1 2 2 3
4 5 4 1 0 4 3 2 1 3 2
5 1 2 3 4 0 3 2 3 1 2
6 2 1 4 3 3 0 3 2 2 1
7 3 4 1 2 2 3 0 3 1 28 4 3 2 1 3 2 3 0 2 1
9 2 3 2 3 1 2 1 2 0 1
10 3 2 3 2 2 1 2 1 1 0
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
15/34
D contains interesting structural information.
Lets zoom in it first by sorting its rows:1 5 9 7 3
2 6 10 8 4
D wi
1 0 1 1 2 2 3 3 4 4 5 13
2 0 1 1 2 2 3 3 4 4 5 13
3 0 1 1 2 2 3 3 4 4 5 134 0 1 1 2 2 3 3 4 4 5 13
5 0 1 1 2 2 2 3 3 3 4 11
6 0 1 1 2 2 2 3 3 3 4 11
7 0 1 1 2 2 2 3 3 3 4 118 0 1 1 2 2 2 3 3 3 4 11
9 0 1 1 1 2 2 2 2 3 3 8,5
10 0 1 1 1 2 2 2 2 3 3 8,5
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
16/34
the action of D in possibly grouping
structurally-equivalent atoms emerges:
1 5 9 7 3
2 6 10 8 4
D wi
1 0 1 1 2 2 3 3 4 4 5 13
2 0 1 1 2 2 3 3 4 4 5 13
3 0 1 1 2 2 3 3 4 4 5 134 0 1 1 2 2 3 3 4 4 5 13
5 0 1 1 2 2 2 3 3 3 4 11
6 0 1 1 2 2 2 3 3 3 4 11
7 0 1 1 2 2 2 3 3 3 4 118 0 1 1 2 2 2 3 3 3 4 11
9 0 1 1 1 2 2 2 2 3 3 8,5
10 0 1 1 1 2 2 2 2 3 3 8,5
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
17/34
Row half-sum wi invariant provides atom-i
contribution to Wiener index Wof G:
W = ij dijW = wiw1 = 13, w5 = 11
W = 109 w9= 8,5We call wi the Wiener weight of atom-i; lets
zoom again in D by sorting by wi ..
1 5 9 7 3
2 6 10 8 4
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
18/34
the action of wi in suggesting possible
most-stable atoms emerges:
1 5 9 7 3
2 6 10 8 4
D wi
9 0 1 1 1 2 2 2 2 3 3 8,5
10 0 1 1 1 2 2 2 2 3 3 8,5
5 0 1 1 2 2 2 3 3 3 4 11
6 0 1 1 2 2 2 3 3 3 4 11
7 0 1 1 2 2 2 3 3 3 4 11
8 0 1 1 2 2 2 3 3 3 4 11
1 0 1 1 2 2 3 3 4 4 5 13
2 0 1 1 2 2 3 3 4 4 5 13
3 0 1 1 2 2 3 3 4 4 5 13
4 0 1 1 2 2 3 3 4 4 5 13
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
19/34
Exercise (HOMEWORK): for any (or all) of the
below PAH structures:
Write the ADJACENCY MATRIX
Write the DISTANCE (BONDING) MATRIX
Calculate the Wiener indexby making use of molecular symmetry
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
20/34
Much attention is devoted on FULLERENES.
Luckily they admit a representation withplanar molecular graphs the Schlegel
d iagramBuckminsterfullerene C60
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
21/34
Planar connected graphs fulfil Eulers theorem
C20nv nf ne 2
nv=20;ne=20*3/2=30
nf=12whereisthe12th
face?20+1230=2W=500=20*w1
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
22/34
Wiener index comutation of C20 fullerene
C20 W=500=20*w1w1=25
Shells Atoms k s*k1 3 3
2 6 12
3 6 18
4 3 12
5 1 550
wi 25W=n*wi 500
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
23/34
Ri-prediction of
the magic LEM:
C28H4
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
24/34
Topological prediction of saturn like-fullerane C50H10 from C50D5h 10 equatorial reactive nodes (maximal wi ):
NOT BAD AT ALL FOR A PURE TOPOLOGICAL WIENER BASEDNOT BAD AT ALL FOR A PURE TOPOLOGICAL WIENER BASED
CALCULATIONS OF A MILLISECOND!CALCULATIONS OF A MILLISECOND!
Ri-prediction of
the SATURN:
C50H10
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
25/34
Lets focus on the minimal vertices in green givingthe minimal contribution w to wiener index W
w=8.5
Minimal vertices are distinguished atoms in a graph,they are in fact the most compactly embedded in the
structure.
Now a question naturally arises: how to measurethe ability of the remaining atoms in contributing tothe overall compactness of graph G?
1 5 9 7 3
2 6 10 8 4
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
26/34
A possible solution consists in computing thefollowing invariant :
= W/Nw 1
= 1.282
Index , called topological efficiency indextopological efficiency index, firstappeared in Ori Ottorino, Cataldo Franco and GraovacAnte (2009) Topological Ranking of C28 FullerenesReactivity, Fullerenes, Nanotubes and CarbonNanostructures, 17:3,308 323.
1 5 9 7 3
2 6 10 8 4
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
27/34
It is very interesting noticing that for the the
buckminsterfullerene C60 Ih we have:
= 1 C60 is in fact a very round hollow cage showing,
for any of the N=60 nodes, the same {bim}={3 6 8
10 10 10 8 3 1} coordination shells with M=9;wi=139=w and therefore W=N*w=8340. In this
stable structure any node works as minimalminimal
nodenode.
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
28/34
Assuming as a measure of sphericity it gives
interesting (and sometime unexpected) results: Benzene is spherical =1;
Closed square lattice (torus) is spherical
Closed graphene lattice is spherical
Sodalite closed zeolitic lattice
is spherical =1 too and so onand so on..
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
29/34
Assuming to determine sphericity produces
interesting (and sometime unexpected) results:
C60 is spherical =1 as we said;
C20
polyhedron is also spherical =1
Other fullerenes Cn have >1 so less
spherical than C20but surely more stable
Warning: better we try correlating and
relative stability only among isomers of the
same Cn fullerene.
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
30/34
is then the topological sphericityof a chemical graph
The lower is the more spherical
and stable the structure will be. So we adopted for invariant the
new alias topological sphericitytopological sphericity
indexindex
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
31/34
The minimum principle works BETTER on topological efficiency indextopological efficiency index
1 as it has been successfully tested on C66 fullerene 4478 isomers, see
Topological efficiency of C66 fullerene, D. Vukicevic, F. Cataldo , O. Ori,A. Graovac, Chemical Physics Letters 501 (2011) 442445.
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
32/34
Int. J. Mol. Sci. 2011, 12(11), 7934-7949;
doi:10.3390/ijms12117934Topological Anisotropy of Stone-Wales Waves in Graphenic
FragmentsOttorino Ori, Franco Cataldo and Mihai V. Putz
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
33/34
Index has been put in FULLERENE generator
program by Prof. Peter Schwerdtfeger
http://ctcp.massey.ac.nz/index.php?group=&page
=fullerenes&menu=fullerenes
free DOWNLAOD PLS USE IT
Indices W andIndices W and tool quickly sieve interesting candidatestool quickly sieve interesting candidatesfor stable chemical structures (molecules, 1for stable chemical structures (molecules, 1--22--3D3D
lattices, fractalslattices, fractals))
WEST UNIVERSITY OF TIMIWEST UNIVERSITY OF TIMIOARAOARA
7/27/2019 TOPO Chemistry NANO Chemistry Lecture 01
34/34
Questions ?
Ottorino ORIActinium Chemical Research
Rome
WEST UNIVERSITY OF TIMIWEST UNIVERSITY OF TIMIOARAOARAFACULTY OF CHEMISTRY, BIOLOGY, GEOGRAPHY
DEPARTMENT OF BIOLOGYDEPARTMENT OF BIOLOGY--CHEMISTRYCHEMISTRYLAB. OF STRUCTURAL AND COMPUTAT IONAL PHYSICALLAB. OF STRUCTURAL AND COMPUTAT IONAL PHYSICAL--CHEMISTRY FORCHEMISTRY FOR
NANOSCIENCES & QSARNANOSCIENCES & QSAR(CFSCNQ)(CFSCNQ)
Prof. dr. dr. habil. Mihai V. PUTZProf. dr. dr. habil. Mihai V. PUTZ
Director CFSCNQDirector CFSCNQ