Carbó-Dorca, Ramon

Revistes Catalanes amb Accés Obert

ARTICLE SCIENTIA gerundensis, 13:177-184 (1987) 
FUZZY SET THEORY AND QUANTUM CHEMISTRY 
R. Carbó 
Divisió de Cikncies. Col.legi Universitari de Girona (UAB). Hospital, 6.17071-Girona. 
RESUM 
Es discuteixen breument algunes consideracions sobre l'aplicació de la Teoria dels 
Conjunts difusos a la Química quintica. Es demostra aqui que molts conceptes químics asso- 
ciats a la teoria són adequats per ésser connectats amb I'estructura dels Conjunts difusos. 
També s'explica com algunes descripcions teoriques dels observables quantics es 
potencien tractant-les amb les eines associades als esmentats Conjunts difusos. La funció 
densitat es pren com a exemple de l'ds de distribucions de possibilitat al mateix temps que 
les distribucions de probabilitat quantiques. 
Se discuten brevemente algunas consideraciones sobre la aplicación de la teoria de 10s 
Conjuntos difusos a la Química cuantica. Se demuestra aqui que algunos conceptos quími- 
cos asociados a la teoria son adecuados para conectarse con la estructura de 10s Conjuntos 
difusos. También se explica de qué modo algunas descripciones teóricas de 10s observa- 
bles cuanticos pueden potenciarse mediante la teoria de 10s Conjuntos difusos. La función 
densidad se toma como ejemplo del empleo de distribuciones de posibilidad al mismo 
tiempo que las distribuciones de probabilidad cuánticas. 
A BSTRACT 
Some considerations on the application of Fuzzy Set Theory in Quantum Chemistry are 
briefly discussed. It is shown that chemical concepts are well suited to be connected with 
fuzzy set structure. 
It is also explained how some theoretical descriptions of quantum chemical observa- 
bles can be successfully enhanced throughout Fuzzy Set Theory. 
The density function is taken to provide examples of the use of possibility distributions 
at the same time as the classical probability ones in Quantum Chemistry. 
Key words: Quantum Chemistry, Fuzzy Set Theory, Density Function, Possibility Distributions. 
INTRODUCTION 
Chemistry has its own set of terms and concepts. Among these, those whose 
origin can be found in the realm of Quantum Chemistry constitute an impor- 
tant to01 to help understanding molecular structure, making available to the 
whole chemical community a picture of the molecular building blocks and 
their transformations. 
Many times, however, the abstract picture Quantum Chemistry gives of 
the molecules has' not the satisfactory generalizable and simple conceptual 
meaning as to be sufficiently pruned in order to become a chemical everyday 
word or concept. 
It is evident that any effort directed towards the connection between theo- 
retical thinking and chemical pictures should be worthwhile, albeit obviously 
difficult. 
In many aspects of the links between mathematical results coming from 
Quantum Theory and simplified pictures associated to experimental reaso- 
ning, the major difficulty lies in the fuzziness of the concepts which chemists 
work with. 
Take, as an example, the fundamental idea of the chemical bond. Every- 
one knows that the line or lines which connect atoms in chemical formulae can 
have a theoretical support which needs, at this moment, a whole encyclopae- 
dia in order to contain all the information gathered on the nature of such a 
simple picture. 
There are sophisticated theoretical tools whose results are not prone to be 
put in such a way, say, as to be learnt in a freshman's course, consequently lac- 
king a fundamental desideratum, which must be recognized to belong to 
Quantum Chemistry: The theoretical results usefulness to understand chemi- 
cal phenomena and thus solve chemical problems better. 
In this sense, nat only Quantum Chemistry must be reflected in its own re- 
sults, but it also has an urgent need to relink the actual brilliant findings and 
outstanding computational development with the overgrown experimental 
pool. This paper can be viewed as a modest goa1 to this rather ambitious state- 
ment. 
FUZZY SETS 
Some twenty years ago, Zadeh (1) developed an interesting mathematical 
idea: The Fuzzy Set Theory, which has found widespread use, as can be dedu- 
ced from a recent review (2). 
In fact, a fuzzy set is nothing but a simple structure associated to a conven- 
tional set. Let us define a set X = {x), attached to a function, the member- 
ship or characteristic function, whose arguments are the elements of set X and 
their values lie on the [0,1] interval. That is: 
1) V x E X-+ 0 (x)  E [0,1] 
Around this definition, the usual operations of set theory can be discovered 
again in terms of the membership function notion. For example, if A and B 
FUZZY SET THEORY 179 
are fuzzy sets with membership functions and QB respectively, then the set 
is a fuzzy set, whose rnembership function is defined by means of: 
3) gc (x )  = max { 0, (x), 0, (x) 1 ; V x E X. 
Or, one can say that A is a subset of B if and only if S @*, that is: 
and so on. 
At a first glance, fuzzy set definition looks somewhat like a probability dis- 
tribution, but it is not. The membership function can be considered as apossi- 
bility distribution, quite independently of Probability Theory. 
FUZZY SETS AND QUANTUM CHEMISTRY 
Zadeh's Fuzzy Set Theory may be connected with some theoretical concepts, 
which in turn can be attached to simple chemical practice. 
Connection between probability distributions, as defined in Quantum 
Theory in terms of the density function, according to the first postulate, and 
possibility distributions of chemical interest is the task which will be sketched 
here. 
One can think of the density function as a formal chemical system observa- 
ble representation containing all the information on the system one can have. 
Thus, entangled with the probability interpretation of Quantum Mechanics, 
density functions must have information on the possibility distributions of the 
attached molecular system too. 
At this stage, the problem can be stated as follows: 
Given a quantum mechanical probability distribution, it is feasible to construct a 
possibility distribution? 
The problem of obtaining a probability distribution from a fuzzy set struc- 
ture has already been discussed in other contexts (3), and although it is also 
interesting it will not be dealt with here. 
An almost trivial way to answer the question above starts from the heuris- 
tic statement that if a possibility distribution exists as a derivation of a proba- 
bility one, it will not be unique. To see this, let us consider a molecular density 
function ~ ( r ) ,  one can build a new function up, such as: 
integrating over a finite volume V in the position space, where the system is 
defined. Equation 5, according to the usual Quantum Theory concepts, can be 
interpreted as the probability of finding the system inside the volume V. But it 
can also be taken as the degree or strength of localization of the system in V. 
In fact, if V is taken to be spherical in shape and centered at some point given 
apriori, then n (V) is simply a function of the radius of the sphere defining V, 
thus x (V) = x (R). 
A sequence of radii: {Ri } E [O, + m], will give necessarily a set of function 
values: n(Ri) E [0,1], and if adequately chosen, will fulfill the inequality: 
Thus, x (R) as defined has not the structure of a probability distribution, but 
nicely outlines the definition of a fuzzy set membership function. 
SOME EXAMPLES 
As a first application example, let us take a paradigmatic function used in 
LCAO Theory, a Slater type orbital, which can be written: 
7 )  X {n,l ,m) (r, 0 ,  rp) = N (n,a)  rn-' exp (-ar) Y{l,m) ( O ,  cp),  
where N (n, a )  is a normalization factor and Y {l,m} a normalized spherical 
harmonic function. The density function defined as Q = I x l 2  can be easily inte- 
grated over a sphere of radius R,  yielding: 
8) n(R) = J: Jn e (r,  0 .  'P) r2 dr d n  
2n 
= 1 - exp (-2aR) Z (2aR)P 1 p! 
p=o 
Taking constant values of the parameters n and a ,  but allowing the parameter 
R to vary over a discrete set of values, one can obtain a set of numbers 
{n(Ri)), which although being in the interval [0,1] do not correspond to any 
probability distribution. 
The same results may be obtained choosing in equation 5 two volume sets 
centered at different points A and B. Two fuzzy sets will be generated with 
membership function n, and n ~ .  It is straightforward to see how Zadeh's fra- 
mework can be applied to such sets. Other volume shapes can be chosen: an 
ellipsoid, for instance, then x (a, b, c), with {a, b,  c} being the ellipsoid's prin- 
cipal axes, will behave in the same manner. A more general way to define 
x(V) can be related to Daudel's loge theory (4) or to the density function 
analysis of Bader et al. (5). 
ENTROPY AND LOCALIZATION 
The possibility distribution, defined in equation 5, and the example in the in- 
tegral 8, are typical functions of cumulative probability. As such they will 
FUZZY SET THEORY 181 1 
behave in a general manner very much like an hyperbolic tangent function. 
A possible transformation may be described, such as to confer to every 
function, obtained through a procedure similar to equation 5 ,  a more conve- 
nient shape. 
Enl'ropy, within the ideas of Information Theory, can be used for this pur- 
pose. In fact, this can be achieved defining the entropy function through natu- 
ral logarithms as: S(x) = -xlog(x) with X E  [0,1], and using the convention 
S(O)=O in order to prevent the singularity at the origin. 
Then S(x) has value on the interval [O,l]. If scaled by the number e ,  it has a 
maximum at x= l le  and is a convex function, which can be used as member- 
ship function in a fuzzy set definition based on the integral n(R). 
Taking definition 5 ,  then perhaps, a better membership function defined 
as an entropy: 
9) S(x(R)) = - e n(R) log (n(R)) 
can be used. This seems very convenient, because the interpretation of the 
membership function in the case of the n(R) functions has some difficulties 
coming from its shape. 
Returning again to the Slater function 8, one can see that as n(R) increases 
the values of S(n(R)) increase from zero unti1 a maximum is reached, and then 
decreases to nullity as R goes to infinite. Then the entropy S measures the de- 
gree of localization and acquires a maximal value when n(R) assumes the 
value lle. 
This maximum is displaced over R in terms of the parameters n and a ,  in 
such a way as localization is found nearest to origin as any of the two condi- 
tions hold: a) n is decreased and b) a is increased, in good accord with the well 
known properties of one-electron basis functions, Thus diffuse Slater orbitals 
will present entropy maximal at R values far away from R=O and core orbitals 
near the same point. Two Slater orbitals can be compared numerically in this 
manner, and from the conventional fuzzy set point of view can be treated as 
two convex fuzzy sets described by entropy membership functions. 
LOCALIZATION AND MOLECULES 
Using this mathematical picture, a fuzzy localization procedure on a set of 
density functions P= { ei)  may be described as follows: 
a) A region R in the position space is chosen. 
b) To each density function of the set P, a function ni(R) can be computed 
with equation 5 for any region of the set. 
c)  An entropy membership function S(ni(R)) defines a fuzzy set Fi on the 
whole set of possible regions. 
The order of localization of the density functions set P can be obtained 
using a fuzzy localization procedure with the additional condition: 
d) The density function ei can be said more localized in the region R than 
the density function ~j if the following inequality holds: S(n,(R)) > S(nj(R)). 
The above conditions can be used, in turn, to describe a molecule. If some 
density function is known and a localization procedure is performed on this 
function, it is only necessary to use the definitions a, b and c adding the follo- 
wing ones: 
e) There is some finite region M for which S(n(M)) > p is true, being y E 
[0,1] a constant cut achieving certain entropy value. 
f) The space where the molecule is embedded can be decomposed into a 
bonding volume defined as: 
and the remaining regions N,, where: 
VMke N,, then S(x(M,)) < F. 
In this context the Bonding volume is the fuzzy representation of the molecule 
and the complementary set Ny plays the role of the molecular nonbonding 
regions. 
FUZZY DENSITY FUNCTIONS SETS 
The previous discussion has been essentially associated to a unique density 
function. But it is possible to obtain some kind of relationship between desinty 
pairs, which can be related to the same system: MO pairs, for instance, or two 
densities attached to different states or molecular structures. In this last 
choice, an attempt to answer the question, which has a daily importance in 
chemical research. ccHow similar is a molecule to another?)), was made some 
time ago (6). Answering this new question has implications, perhaps, in the 
scene of Structure - Activity relationships (7). 
But the interesting fact appears when given two density functions, 
{ eA,  eB),  and O being a positive definite operator, one computes the correla- 
tion coefficient: 
FUZZY SET THEORY 183 
in order to find some relationship between the density pair { e,, eB). Then, as 
by construction the following property holds: 
The measure of the correlation coefficient generates a membership function 
in the cartesian product of the available densities, which plays the role of a 
fuzzy set, being easy to show that the property 11 fullfills Zadeh's fuzzy set 
description. As a consequence, one has no need at all to attach any statistical 
significance to every correlation coefficient, but the power to create an order 
within the density function set, which can be related to the ordering structure 
of molecular properties present into the systems described by the density func- 
tions. 
FINAL REMARKS 
Correlation coefficients and euclidean distances are related measures when 
computed over a numerable set of objects. On the other hand, the concept of 
localization is in the same way related to the fuzzy idea of chemical bond, the 
sharing of electrons between atoms to form molecular structures. Localization 
and energy minimization are strongly related computational features (10). 
When solving Schrodinger's equation in the LCAO-MO Theory frame- 
work by means of a direct minimization procedure using elementary Jacobi 
rotations, as described in reference (8), the rotations must be applied to MO's 
belonging to different molecular shells, including virtual or empty orbitals. It 
is well known that orbital mixing within the same shell leaves the energy inva- 
riant. This property allows us to perform Jacobi rotations, in order to obtain 
not only optima1 orbitals throughout intershell mixing, but localized ones by 
means of intrashell mixing: adopting Edminston-Ruedenberg's localization 
algorithm (9), for instance, which consists of maximization of the sum of Cou- 
lomb selfrepulsion of the set of MO's within a given shell. 
This procedure can be easily related to maximal variation of coulombic eu- 
clidean distances between pairs of MO intrashell density functions. In this 
sense, localization procedures, which are not unique, on the contrary as 
energy minimization, may be considered as allowed manipulations of the 
quantum mechanical tools, due to the fuzzy set structure underlying the defi- 
nition of localized densities. 
COROLLARY 
Perhaps, as a conclusion, one can say that it is possible to add, for chemical 
purposes, a simple corollary to the first postulate: 
Every set of density functions generates a collection of fuzzy sets. 
This corolary is nothing more than the statement of the geometrical character 
supporting Quantum Theory applied to Chemistry. How far this fuzzy con- 
nection can go into the increment of chemical lore can only be seen with fur- 
ther understanding and practice. 
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4. DAUDEL, R. (1956). Les Fondements de la Chimie Théorique. Gautier-Villars. 
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