MATCH Communications in Mathematica/ and in Computer Chemislry
MATCH Commun. Math. Computo Chem. 56 (2006) 509-518 ISSN 0340
- 6253
AssigningWave Functionsto Graphs: A Way to Introduce Novel TopologicalIndices
Jorge
Gálvez.a,
Ramón
García-Domenecha,
J.Vicente
de Julián-Orti~a.b
.Unidad de Disefio de Fánnacos y Conectividad Molecular, Departamento de Química Física, b Red de Investigación de Enfermedades Tropicales. Dept. de Biología Celular y Parasitología, a,bFacultat de Farmacia, Universitat de Valencia, Av. V. Andrés Estellés, O, 46100 Burjassot, Valencia, Spain. . Corresponding author:
[email protected]
= (2/a)~ sin(n 7tx/a)
(1)
where"a" is fue bond length and "n" is an integernumberfrom one on up. Let us consider fue one-edgehydrogen depleted graph, that is, fue ethane graph. Its connectivity term Cij, is Cll = (101)-1/2= 1.000 It may be realized that this value is equal to the maximun value for n= 1 and a=2 in equation(1). If we do the samecalculation for a 1-2 edge, as for instanceeither of fue two edgesin propane'sgraph, then its connectivity term is C12= (102)-1/2 = 0.7071.Likewise, this value is fue maximum wave function value for n=2 , a=4. The samestandsfor fue CI3 and C14terms, for which fue equivalenciesare n= 3, a= 6 and n= 4, a= 8, respectively.The connectivity terms are so equivalent to the maximum probability values for particles(electrons)placed at different quantumlevels within boxesof different lengths. The valuesfor eachedgewave functions are: cl>ll
= (2/2) y, sin (7t x/2)
cl>12 = (2/4) ~ sin (7t x/2) cl>13 = (2/6) y, sin (7t x/2) cl>14 = (2/8) ~ sin (7t x/2)
An altemative way to reach fue sameresult is taking into accountfue topological valence valuesfor eachedge,while maintainingconstantfue box length, as follows: cl>lj= 1/(~lj) ~ sin (~Ij 7tx 12)
-512That expressionis pledgedbecausethe probability of locating an electronwithin a given 1j edge(or bond into a chemicaltranslation)is inverseto Oljo[15] The wavefunctionsassociatedto the (1,1) (1,2) (1,3) and (1,4) edgesare: «/111 = (1/1) ~ sin (7t x/2) «/112 = (1/2) ~ sin (7t x) «/113 = (1/3) ~ sin (37t x/2) «/114 = (1/4) ~ sin (27t x)
The correspondingwavelengths(in arbitrary units) are 4, 2, 1.333 and 1.000respectively and the correspondingtopologicallevels as defined for their valence values are 1, 2, 3, 4, respectively. It is easyto demonstratethat any other possible edge into a graph has an associatewave function that is derived from the four edges outlined, which can be denominated as fundamentaledges. From theseresults, it is noteworthy that topologicallevel values are the sameas quantum leveis in the particle-in-a-box approach.This is a one anotherlink betweenboth formalisms, i.e. betweenmoleculartopology and quantumchemistry. Furthermore, it. is easy to realize that the maximum of the edges-wavefunctions (correspondingto sinusvalues=1) is equalto the connectivity termsCijo Since any graph is composedof the four fundamentaledges(1,1), (1,2) (1,3) and (1,4), what are the edgesassociatedto ethane,propane,isobutaneand neopentane,respectively,the discrete characterof the connectivity indices can be associatedto different wave function quantum levels. This algo agreeswith the Fourier theorem, which stands that any wave equation,despiteits complexity, mar be expressedasa sum of sinusoidalfunctions. Thus, if we considerisopentane,its wave function would be:
=
'P¡sopentane «/113.«/112+2 «/113+
= 0.408sin (37tx/2) sin (7tx) + 1.1547si.n(37tx/2)+
«/112
+0.707sin (7tx). It is interesting to realize that, under this formalism, the Randié index, or first arder connectivityterm, X , is just the maximumvalue o( the graph-wavefunction. Table 1 showsthe correspondingwave function equationsfor a set of alkanes,as well as the values of the differences between maxima and minima for ~ach graph wave function (maximum-minimumgap, MMG).
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Table l. Wave function equations and the values of fue differences between maxima and minima into each graph wave functioD for a set of alkanes. Alkane
Sinusoidal wave function,,;,.c,
=
= sin (n x/2)~!!,";,f,¡;I,,~}i;¿f!}7i'iJj};f'é¡
MMGa
2.00
Ethane
'Petane
ProPa:ne
'Ppropane = 211)12= 1.4142 sin(n x)
2.83
'Pisobutane = 311)13 = 1.7320sin(3n x /2)
3.46
'Pbutane =211)12+ 11)1211)12= 1.4142sin(n x )+ 0.5sin1nx )
2.91
Isobutane
Butane
11)11
Neopentane 'Pneopentane =411)14 = 2 sin( 2n x)
2M4
4.00
'P2M4=11)1311)12 + 211)13 + 11)12 = 0.4082sin(3n x /2)sin(nx)+
3.72
1.1547sin(3nx /2) + 0.7071sin(nx) n-pentane 'Pn-pentane =211)12+ 211)1211)12 = 1.4142sin(n x)+ 1.0sin1nx) 22MM4
23MM4
2.91
'P22M4=11)1411)12+ 311)14 + 11)12 = 0.3535sin(2nx)sin(nx)
4.41
+1.5sin(2nx)+0.7071sin(nx)
-
'P23MM4 = 11)1311)12+
4.61
211)13+ 11)12 =
0.4082sin(3nx /2)sin(nx) +
1.1547 sin(3n x /2) + 0.7071sin(7tX) 2M5
'P2MS=211)13+11)1311)12+11)1211)12+11)12 = 1.1547sin(3nx/2)+
3.72
0.4082sin(3n x /2)sin(nx) + 0.5 sin1n x ) + 0.7071sin(nx) 3M5
'P3MS= 211)12 + 211)1311)12+ 11)13 = 1.4142sin(n x) +
4.00
0.8 164sin(3n x /2)sin(nx) + 0.5773sin(3n x /2) n-hexane
'P n-hexane = 211)12 + 311)1211)12 = 1.4142sin(nx )+ 1.5 sin1n x )
3.25
23MM5
'P23MMS =311)13+ 11)1311)12+ 11)1311)13+ 11)12 = 1.7320sin(3nx /2) +
4.88
0.4082sin(3nx /2)sin(nx) + 0.3333sin13n x/2 ) + 0.7071sin(nx) aMMG= M~imun-mimimum gap.
Other algorithms allowing manipulation of wave functions are possible as well. For examplefue (1,1), (1,2) (1,3) and (1,4) edgescan be describedas:
=
11)11 sin (n x/2 +7t/2);
11)11 (0)= 1.0000 = XII
11)12 = sin (n x/2 +7t/4)
11)12 ( 0)= 0.7071 = XI2
11)13 = 2/3 sin (n x/2 +n/3)
11)13 ( 0)= 0.5773 = XI3
11)14 = sin (n x/2 +7t/6)
11)14 (0)= 0.5000 = Xl4
-514As in the fonner description, any other graph edge may be derived from these four fundamentaledges.In this case,fue differencesin fue associatedwave functionsare not in the amplitudebut in the phaseangle(exceptfor fue (1,3) edgethat includesboth). In this case,the connectivity index X is just the interceptfor eachgraphequation(X = \}/ (O)).
The polynomial approach Many previous attempts are reported on fue graph characterization through their characteristicpolynomials. [16, 17] However, the use of fue single polynomials leads to asymptoticcurves,whosecharacteristicpoints arenot very representativeas graphdescriptors. It is interestingto realize that the eigenvaluesfor fue four fundamentalgraphsabovedefined (correspondingto ethane,propane,isobutaneand neopentane)are identical to fue X valuesfor eachone. 2. '1'
W
Figure 1.- Comparisonbetweenthe singlepolynomial and the Hennite-like function for fue nbutanegraph.Observethat fue AUC makesfinite for fue exponentialquasi-wavefunction. A possible altemative is combining the polynomial expression with the squared exponential in a similar way as fue Hennite polynomials are introduced into the wave function ofthe harmonicoscillator. Thus, we define the quasi-wavefunction for a graphas:
-515-
'I'=P(x) being P(x) the graph characteristic
e(-x2/2)
polynomial.
The difference
between the two
representations is displayed in Figure l. The most significant difference is that the afea under fue curve becomes finite in the exponential approach, which makes sense to the concept of quantum probability as the overall integral between O and oo. lt is noteworthy fue appearing of a maximum at about x=2.4. That maximum also stands for fue rest of alkanes in fue range between x=1 and x=5. Because of fue similarity to fue Hermite approach for harmonic oscillator wave functions we call our approach as Hermitelike wave functions. Table 2 illustrates fue characteristic polynomial,
as well as the X connectivity index,
maximum eigenvalues, ordinates ofthe maxima placed between x=1 and x=5 ('I'max 1-5),afea under the curve and boiling temperatures for a set of alkanes.
Table 2.- Connectivity index X; characteristic polynomial; maximum eigenvalues; maximum '1' between x=1 and x=5; afea under the curve AUC and boiling temperature, Tb, for a set of alkanes. Alkane
X
Characteristic polinomial
M,aximum
'l'max,exp.
AUC
Tb(K)
elgenvalues x=Oandx=S
ethane propane isobutane butane neopentane 2M4 n-pentane 22MM4 23MM4 2M5
1.0000 1.4142 1.7320 1.9142 2.0000 2.2700 2.4142 2.5610 2.6430 2.7700
x;t- 1 X3- 2x X4- 3x2 X4- 3x2+1 XS- 4X3 XS_4X3+2x XS_4X3+3x X6- 5X4+3x2 X6- 5X4+4x2 x6 - 5x4 +5x2
1.000 1.414 1.732 1.618 2.000 1.845 1.732 2.074 2.000 1.902
0.4463 0.5592 0.8962 0.9486 1.6922 1.8379 1.9188 3.9767 4.1095 4.2497
0.6065 0.7358 1.1594 1.2765 2.1630 2.4772 2.6775 5.3290 5.6415 5.9898
184.4 230.9 261.3 272.5 282.5 300.9 309.1 322.7 331.0 333.0
3M5 n-hexane 23MM5
2.8080 2.9140 3.1807
X6- 5X4+5x2 -1 x6-5x4+6x2-1 x7-6xs+8x3_2x
1.932 1.802 2.053
4.2318 4.3784 10.4415
.5.9877 6.3219 14.894
336.6 341.7 362.8
Results and discussion The formalism we introduce here, i.e. assigning wave or quasi-wave functions to molecular graphs, has a widespread potential applicability.
One of fue most interesting usefulness is
opening a new way to introduce novel topological índices, as specific points of such functions.
---
-516-
A detailed description on how this approach can be achieved is out of fue goal of this paper and it will be hopefully dealt in fue near future, but let's illustrate, just as an example, a simple case: The good fitting between the maximum amplitude (MA)of
the sinusoidal wave
functions and the boiling points for groups of alkane isomers. The same happens with the ordinate values ofthe maxima as well as with AUC for fue Hermite-like approach. The following regression equations were the best (highest F-ratio), obtained correlating boiling points with a set of topological indices including the new enes introduced by first time here, as well as the connectivity up to fue fourth order: BP
= 59.80 X +
N=58
167.92
R2= 0.9591 SE=8.23
F = 128.4
BP = 1.444 (MA) + 54.23 X + 179.66 N=58
R2 = 0.9688
BP= - 4.93('1' max\-5) + ;N=58
R2 = 0.9878
SE= 6.78
F =191.2
87.28 X + 112.85 SE= 4.36
(2)
F =366.4
Thus, although the X index cpntinues to be fue best single descriptor for boiling points prediction for the selected set of alkanes, the inclusion of some of the novel topological indices led to a significant improvement in fue prediction as compared with fue Test of connectivity indices. . In closing, we may conclude that fue use of graph wave functions allows, within fue framework of fue independent and fundamental nature of topological indices, to consider molecular energies (at least electronic and vibrational)
as particular cases of topological
indices. In fact, whilst energy is the eigenvalue ofthe Hamiltonian operator over a given wave function, topological indices would result on applying different operators over the selected wave functions. Under such a view,' energy would be just a particular case of topological index, what could explain why electronic and vibrational energies [7, 15] corre late so well with some topological indices. This is interesting because topology, in the sense we use this word here, would be an altemative and independent approach to energy, or at least, it may suggest new ways of dealing with the well-known concepts of molecular energies and wave functions.
,;¡ ",;;,
-517Conclusions The possibility of describinggraph by continuousfunctions is a very useful approachfor severalreasons:
-
First, it makes possible introducing new topological indices as specific points of continuousfunctionsdescribingeachgraph.
-
Second,it allows setting up a parallelism betweenfue quantizationof energy and the discretecharacterof topological descriptors.
-
Finally, this formalism involves a straightforward relationship between molecular energy and topological descriptors.Under this view, molecular energies,particularly electronic and vibrational ones, would be particular cases of topological indexes derived from the application of fue Hamiltonian operator, just one of the many possibleoperatorsto apply ayer the wave functions.
Acknowledgments The authors acknowledge financial support from fue SpanishRed de Investigación de Centrosde EnfermedadesTropicales,RICEJ: (CO3/04), the project FIS-2005 PI052128 of fue Fondo de Investigación Sanitaria, Instituto de Salud Carlos 1Il, Ministerio de Sanidad, Spain and the grant number BQU2003-07420-C05of fue CorroerMinisterio de Ciencia y Tecnologíawithin fue SpanishPlan Nacional I+D.
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