Stochastic Classical Molecular Dynamics Coupled To

Home , Boltzmann machine

Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 199

Sto chastic Classical Molecular Dynamics Coupled

to Functional Density Theory: Applications

to Large Molecular Systems

K. C. Mundim

Institute of Physics, Federal University of Bahia,

Salvador, Ba, Brazil

and

D. E. Ellis

Dept. of Chemistry and Materials Research Center

Northwestern University, Evanston IL 60208, USA

Received 07 Decemb er, 1998

Ahybrid approach is describ ed, which combines sto chastic classical molecular dynamics and

rst principles DensityFunctional theory to mo del the atomic and electronic structure of

large molecular and solid-state systems. The sto chastic molecular dynamics using Gener-

alized Simulated Annealing GSA is based on the nonextensive statistical mechanics and

thermo dynamics. Examples are given of applications in linear-chain p olymers, structural

ceramics, impurities in metals, and pharmacological molecule-protein interactions.

scrib e each of the comp onent pro cedures. I Intro duction

In complex materials and biophysics problems the num-

b er of degrees of freedom in nuclear and electronic co or-

dinates is currently to o large for e ective treatmentby

purely rst principles computation. Alternative tech-

niques whichinvolvea hybrid mix of classical and quan-

tum metho dologies can provide a p owerful to ol for anal-

ysis of structure, b onding, mechanical, electrical, and

sp ectroscopic prop erties. In this rep ort we describ e an

implementation which has b een evolved to deal sp ecif-

ically with protein folding, pharmacological molecule

do cking, impurities, defects, interfaces, grain b ound-

aries in crystals and related problems of 'real' solids.

As in anyevolving scheme, there is still muchroom

for improvement; however the guiding principles are

simple: to obtain a lo cal, chemically intuitive descrip-

tion of complex systems, which can b e extended in

a systematic way to the nanometer size scale. The

computational scheme is illustrated in Fig. 1, show-

Figure 1. Flowchart describing the hybrid MD+DV+GSA

ing classical Molecular Dynamics MD, Monte Carlo

approach.

MC-GSA sto chastic sampling, and Discrete Varia-

tional DV DensityFunctional DF quantum mechan-

ics pro cedures coupled together. We next brie y de-

200 K.C. Mundim and D. E. Ellis

II Classical Dynamics, Sto chas- metho ds.

- In MM, it is very simple to intro duce time evolu-

tic Metho ds, and Quantum

tion;

Clusters

- In MM, it is p ossible to intro duce the temp erature

as an external p erturbation, and trace its e ects.

Two imp ortant questions arise which are unfavor-

I I.1 Classical Metho dology: Molecular Dynam-

able to the MM metho ds: First, there do not exist well-

ics

de ned rules to evaluate the force constants; and sec-

ond, in order to cho ose the b est force eld is necessary

The idea of molecular mo deling is an attempt to de-

to have a priori some considerable knowledge ab out the

scrib e the quantum chemical b onds in terms of a clas-

molecular system. In molecular mechanics the atom is

sical force eld in Newton's equations. In a Molecular

represented by a spherical b o dy with a particle mass

Mechanics MM mo del the atomic b onds are repre-

equal, in general, to the resp ective atomic mass. In

sented by springs joining the atoms; i.e., the molecule

to day's molecular mechanics, several force eld mo dels

is assumed to b e a collection of masses and springs.

have b een prop osed. In general the molecular energy

In this case is necessary nd a set of parameters, for

p otential function related with the classical force eld

this force eld, that t the quantum atomic interac-

can b e expressed by the following sum [3],[4];

tions with sucient accuracy. Ideally one hop es that

such re nements will eventually lead toward a uni ed

computational mo del that can successfully mimic ob-

V = V + V + V + V + V + V 1

H b tor i tor p C vdW

served molecular prop erties. Academic research e orts

and the pharmaceutical industry interest in developing

new comp ounds in biological molecular systems have

K r r V =

Hn n o H

stimulated the app earance of di erent computational

co des based on classical force elds.

The rst molecular dynamics applications was p er-

K V = formed by Alder and Wainwright, [1] who used a p er-

n n o b

fectly elastic hard sphere mo del to represent the atomic

interactions. One of the most widely used force eld

ones is called MM1 Molecular Mechanics, prop osed

V = K

tor i n n o

by Allinger [2]. At the moment there are an uncount-

able numb er of MM computational co des using a classi-

cal force eld. Each one of these uses a particular force

eld to describ e di erent molecular prop erties and to

K [1 + cosm ] V =

n n n tor p

t some exp erimental results.

Reasons which justify the increasing use of MM can

b e listed, as for example:

1 q q

i j

Vc =

-Possibly the chief reason is the relatively short

4 r

computational time, which for MM metho ds increases

as M , where M is the numb er of atoms in the molecule.

In contrast, the use of the ab-initio quantum metho ds

C ij C ij

12 6

V =

vdW

12 6

in such molecular systems is computationally impracti-

r r

ij ij

cable b ecause the computer time to evaluate the inter-

electronic repulsion integrals increases as N or more

Where, in summary, the rst term of eq. 1 V

rapidly, when correlation is taken into account, where

represent the energy necessary to stretch or compress

N is the numb er of basis orbitals. Normally, there are

the atomic b ond; Vb is the p otential contribution that

at least several basis functions p er atomic orbital shell-

represents the b ond-angle b ending interaction; V ,

tor i

1s, 2s, 2p, etc..

V are the torsional contributions that represent

tor p

the harmonic dihedral b ending interaction and the si- - The MM metho d and its results are conceptually

nusoidal dihedral torsion interactions. The last terms easier to understand than quantum mechanical QM

Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 201

V and V represent the non-b onded interactions the time t + t. In the same pro cedure the forces and

C vdW

such as Coulombic repulsion, hydrogen b onding and the new velo cities are used to calculate the new atomic

van der Waals interactions. Each of these p otential en- p osition at time t + t, and so on.

ergy functions represents a molecular deformation from

Once the force on each atom has b een calculated for

a reference geometry. The interactions given in equa-

time t, the p osition of each atom at some later time t

tion 1 are available in our Molecular Mechanics co de,

+ t is then given by

and are typical of other currently available co des. To

! !

study molecular prop erties in solid systems, additional

r r

+ v t+ a t 3 =

t t t t+t

p otentials have b een intro duced, including Born-typ e

where = /m . Here t = n t, and t 0.001 pico

exp onential repulsion and Morse p otentials.

may b e taken seconds. The initial co ordinates

t=0

Molecular dynamics MD calculations consist of

from exp erimental data or may b e randomized; the

analyzing the evolution of the molecular system with

are derived from the Maxwell- initial velo cities

t=0

time. In this case the atoms are continuously moving,

Boltzmann distribution, in general, and temp erature T

the b onds are vibrating, the angles are b ending and

is controlled by a dynamic scaling pro cedure. A co oling

the whole molecule is rotating. In MD, successive con-

or annealing pro cedure of mo difying T provides one way

gurations of the system are generated byintegrating

to approach stationary p oints, and p ossibly the equi-

Newton's equations of motion. The result is a tra jec-

librium state of the system. Once the p otentials are

tory that sp eci es how the p ositions and velo cities of

calculated for each new geometry, the new forces and

the atoms vary with time. The tra jectory is obtained

p ositions of each atom can again found. This pro cedure

by solving the following di erential equations;

is rep eated until convergence in the energy is reached.

There are di erent algorithms or numerical metho ds

d r

i i

2 =

available to solve di erential equation 2. All meth-

dt m

o ds are based on nite di erences and solve the equa-

where is the force over atom i along the co ordinate

tion step by step in time. Often the step size is taken

and m is the atomic mass.

constant, but adaptive metho ds have sometimes b een

Equation 2 is solved by a di erence nite tech-

found useful.

nique with continuous p otential mo dels 1, which are

The simplest and most straightforward but unfor-

generally assumed to b e pairwise additive. The essen-

tunately not suciently accurate approachistouse

tial idea is that the integration is broken down into

the Taylor expansion based on equation 2.

many small stages, each separated in time by a xed

increment dt. For each step the total force on each

! !

r r

= + v t+ a t 4

t+t t t t

atom is calculated as the vector sum of the interactions

with some or all atoms b elonging to the molecular sys-

! !

tem. The force is calculated by taking the gradientof

v v

= + a t

t+t t t

the p otential function represented by eq. 1. Using

Another metho d sometimes used but computation- the forces and accelerations, all atomic p ositions and

ally very exp ensive is the Taylor predictor; velo cities are determined using Newton's equation at

! ! !

2 3

r r r

! ! d 1 d d 1

2 3 4

r r

= + t + t + t + O r 5

t+t t

2 2

dt 2 dt 6 dt

! ! !

2 3

r r r

1 d 1 d ! ! d

2 3 4

r r

t + t t + O r =

tt t

2 2

dt 2 dt 6 dt

202 K.C. Mundim and D. E. Ellis

This algorithm contains no explicit velo cities, and This metho dology, though p edagogically correct, al-

we see that also the third derivative terms can b e made lows the system to get \trapp ed" in lo cal energy min-

to cancel. In this case the velo cities can b e approxi- ima. There is no guarantee that this geometry corre-

mated by sp onds to the global minimum energy. So the usual

MD pro cedure may not b e the b est way to search for

! ! !

r r v

=2t 6 = the global minimum p oint on the complex energy hy-

t1 t+1 t

p ersurface.

Another very well known algorithm is the Verlet

In this context, if the molecular hyp ersurface en-

leap-frog scheme given as;

ergy is non-convex, sto chastic molecular dynamics of-

! !

fers a more ecientway of nding b oth lo cal minimaas

v v

= +F t 7

t+t=2 tt=2

well as the global one. Thus we next discuss the more

! !

imp ortant concepts needed to implement a sto chastic

v r

=r + t

t+t t

t+t=2

molecular dynamics pro cedure.

One consideration for molecular dynamics that im-

mediately invalidates a numb er of these metho ds is the

I I.2 Sto chastic Dynamics: Generalized Simu-

cost of calculation of the force, or gradient of the p oten-

lated Annealing

tial. Computation of the force is extremely lab orious

compared to any manipulationinvolved in up dating the

variables to take one step forward in time. This means

Simulated Annealing [5][6] metho ds have b een ap-

that any metho d that involves more than one force eval-

plied successfully in the description of a variety of global

uation p er step cannot b e a go o d choice. This rules out

extremization problems. Simulated Annealing metho ds

the well-known Runge-Kutta metho d and its variants,

have attracted signi cant attention due to their suit-

that require up to four force evaluations p er step, so

ability for large scale optimization problems, esp ecially

the simple Verlet scheme is generally preferred.

for those in which a desired global minimum is hidden

The MD pro cedure can b e brie y summarized in the

among many lo cal minima. The basic asp ect of the Sim-

following steps:

ulated Annealing metho d is that it is analogous to ther-

i- Guess the initial molecular geometry, the initial

mo dynamics, esp ecially concerning the way that liquids

atomic velo cities and environment temp erature.

freeze and crystallize, or that metals co ol and anneal.

ii- Calculate the force, as the gradient of the p oten-

At high temp eratures, the molecules move freely with

tial 1.

resp ect to one another. If the system is co oled slowly,

iii- Calculate the new atomic p osition or molecu-

thermal mobility is lost. The atoms are often able to

lar geometry using one of the algorithms to integrate

line themselves up and assume a molecular geometry

Newton's equations.

that is in general a lo cal equilibrium state. The simu-

iv- Rede ne the atomic velo cities using the virial

lated annealing pro cedure is actually more complicated

theorem

than the combinatory one, since the familiar problem of

+ *

3 1

2 long, narrow p otential valleys again asserts itself. Sim-

nk T = m v

B i

2 2

ulated annealing, as we will see, tries random steps, but

cy cle

in a long, narrowvalley, almost all random steps are up-

hill. The amazing fact is that, for a slowly co oled sys-

T v

o new

v T tem, nature is able to nd this minimum energy state.

old

So the essence of the pro cess is slow co oling, allowing

Where k is the Boltzmann's constant, T is the ini-

B o

ample time for redistribution of the atoms as they lose

tial or \bath" temp erature and T is the lo cal temp era-

their mobility. This is the technical de nition of an-

ture. v and v are the velo cities of two consecutive

old new

nealing, and it is essential for ensuring that the lowest

steps in the MD pro cedure.

energy state will b e achieved.

v- Saveintermediate information for statistics and

go back to step i until the stabilization of the molec- The rst nontrivial solution along this line was pro-

ular energy is reached. At the end of the MD pro ce- vided by Kirkpatrick et al. [5],[6] for classical systems,

dure the total energy kinetic and p otential contribu- and also extended by Cep erley et al. [7] to quan-

tion must b e constant. tum systems. It strictly follows the quasi-equilibrium

Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 203

Boltzmann-Gibbs statistics using a Gaussian visiting sition.

distribution, and is sometimes referred to as Classical

To generate the random vector x the present GSA

Simulated Annealing CSA or the Boltzmann machine.

routine uses an extension of the pro cedure given in Ref.

The next interesting step in this sub ject was Szu's pro-

[13]. Wehave calculated x =g ! using a numer-

p osal [8] to use a Cauchy-Lorentz visiting distribution, ical integration of the visiting distribution probability

instead of a Gaussian distribution. This algorithm is g x. Where ! is a random vector [0,1] obtained from

an equiprobability distribution and g is the inverse referred to as the Fast Simulated Annealing FSA or

of the integral of g x given by Cauchy machine.

On the other hand, a Generalized Simulated An-

nealing GSA approach which closely follows the

g ! =inverse g xdx 9

recent Tsallis statistics [9],[10] has b een prop osed

Mathematical details of the structure of the distri-

[11],[12],[13],[14]; GSA includes b oth the FSA and CSA

bution function g and its inverse g-1 are given in the

pro cedures as sp ecial cases. Wehave implemented the

reference [13]. In summary, the complete algorithm for

GSA algorithm as a metho d to calculate the mini-

mapping and searching for the global minimum of the

mum energies of conformational geometries for di er-

energy is:

ent molecular structures. This technique can b e ap-

i Fix the parameters q ;q . We note that

A v

plied in either quantum [12] or classical [13] metho ds.

q ;q = 1;1 and 1;2 resp ectively corresp ond to the

A v

The GSA metho d is based on the correlation b etween

Boltzmann and Cauchy machines. Start at t = 1, with

the minimization of a cost function conformational en-

arbitrary internal co ordinates and high enough value

ergy and the geometries randomly obtained through a

for visiting temp erature T 1 and co ol as follows:

slow co oling. In this technique, an arti cial temp era-

ture is intro duced and the system is gradually co oled in

q 1

2 1

complete analogy with the well known annealing tech-

T t=T 1 10

qv qv

q 1

1 t 1

nique, frequently used in metallurgy when a molten

where t is the discrete time corresp onding to steps of

metal reaches its crystalline state global minimum of

computer iteration.

the thermo dynamic energy. In our case the temp er-

ii Next, randomly generate the new atomic co or-

ature is intended as an external noise, which acts as

dinate x from x as given by the visiting distribution

t+1 t

a convenient sto chastic source for eventual detrapping

probability gqv as follows:

from lo cal minima. Near the end of the pro cess the sys-

tem hop efully is within the attractive basin of the global

x = x + g ! 11

t+1 t

minimum. The challenge is to co ol the system as fast

as p ossible and still have the guarantee that no irre-

For suciently long time simulations this pro cedure

versible trapping at any lo cal minimum has o ccurred.

assures that the system can b oth escap e from any lo cal

More precisely,we search for the quickest annealing ap-

minimum and can explore the entire energy hyp ersur-

proaching a quenching which maintains the probability

face. This equation is used in the GSA routine and

of nishing within the global minimum equal to one.

di ers from the general prop osal given in [11]; instead

The pro cedure to search the minima global and

we build a minimization vector using 8.

lo cal or to map the energy hyp ersurface consists in

iii Then calculate the conformational energy

comparing the conformational energy for two consecu-

Ex from the new molecular geometry using the

t+1

tive random geometries x and x obtained from the

t+1 t

classical force eld [3]. The new energy value will b e

GSA routine. x is a N- dimensional vector that con-

accepted according to the following rules:

tains all atomic co ordinates N to b e optimized. The

if Ex Ex , replace x byx;if

t+1 t t+1 t

geometries, for two consecutive steps, are related by

Ex > Ex , run a random number r 2[0,1];

t+1 t

if r > P acceptanceprobability retain

x = x +x 8

t+1 t t

x ; otherwise, replace x byx .

t t t+1

where x is a random p erturbation on the atomic p o- The acceptance probability is given by t

204 K.C. Mundim and D. E. Ellis

if E x E x

t+1 t

P =x ! x = 12

q t t+1

if E x >Ex

q 1

t+1 t

1+ [1+q 1E x E x E x =T t]

A t+1 t t qv

treating lo calized covalentinteractions where there is iv- Calculate the new temp erature T t using

hybridization of orbitals with neighb oring atoms. The Eq. 12 and go back to ii until the convergence of

emb edded cluster density functional ECDF scheme Ex is reached within the desired precision.

p ermits analysis of wavefunctions and their derived In order to make clear the pro cedure to construct

prop erties in a restricted volume, with interactions to the presently used computational co de, we present the

the extended host included by e ective p otentials and following owchart as Fig. 2:

b oundary conditions. The present implementation of

the ECDF scheme employs 'guard atoms' on the surface

of the cluster to saturate interior b ond structures, and

synthesis of total charge and spin densities including

the environment, to determine e ective cluster p oten-

tials. Scanning algorithms by which the 'view window'

of the cluster is moved over the volume of interest, p er-

mit treatment of extended systems. A key concept in

obtaining self-consistency of the multiple overlapping

views of a complex system like a heterogeneous inter-

face, is that of equilibration of the chemical p otential,

or Fermi energy EF, among the comp onent clusters.

This metho dology has b een develop ed in several forms

over the years, and successfully applied to a varietyof

molecules and extended solids, including metals, semi-

conductors, and insulators; see for example [15].

In outline, the following main steps are p erformed

Figure 2. Schematic diagram of Generalized Simulated An-

in ECDF calculations:

nealing pro cess.

i Charge and spin densities of the extended sys-

tem are constructed by summing comp onent atom-like

I I.3 Quantum Metho dology: DensityFunctional

densities for the cluster and host:

Emb edded Cluster Scheme

host

The DensityFunctional DF theory is a standard

= + 13

cluster

to ol for describing electronic structures, which is rst-

principles in concept and exible in execution. DF the-

ii LCAO-typ e basis functions are generated bynu-

ory has b ecome a ma jor to ol for analyzing electronic

merical solution of DF equations for atoms/ions in

structure of molecules and solids, with high intrinsic

a p otential well, determined from the previous self-

accuracy and reasonable computational e ort. For ex-

consistent- eld iterations; the basis is thus iterated as

tended systems with no p erio dicity to exploit, a clus-

part of the overall optimization pro cedure.

ter approach o ers many advantages, providing that

iii Form matrix elements of the e ective Hamilto-

a reasonable emb edding scheme is used. De nition of

nian

'reasonable emb edding' dep ends up on the nature of the

system- while a p ointcharge environment is accept-

h = t + V + V 14 able for a highly ionic comp ound, it is quite p o or in

C xc;

Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 205

the capability of feeding information b etween the com- and overlap op erators over the AO basis , p os-

p onents provides us with a p owerful new to ol for study- sibly transformed into symmetry orbitals, and orthog-

ing materials prop erties. An area of intense develop- onalized against a frozen core. Here t, V , and V

C xC

ment in our lab oratories is design and construction of a are kinetic energy,nuclear and electronic Coulombpo-

graphical user interface which will p ermit the use of this tential, and exchange-correlation p otential resp ectively.

system of programs by non sp ecialists, and particularly The index refers to spin orientation. The DF p oten-

by exp erimentalists wishing to analyze new materials. tial is formed from the densities, and b oundary condi-

In the remainder of this rep ort, we give four illustra- tions are applied to force lo calization of solutions to the

tive examples related to current problems in materials cluster.

design and optimization.

iv Solve the Schrodinger nonrelativistic or Dirac

Beyond the general pro cedures outlined ab ove, suc- relativistic equation, to obtain a variational expansion

cessful applications dep end up on some degree of exp e-

of the one-electron wavefunctions as

rience and 'art'. For example; when is it most ecient

to use MD and when is it b etter to use MC/GSA to de-

~r = C 15

n j j;n

termine the minimum-energy con guration of a system

as complex as a ceramic grain b oundary containing im-

v Pro ject the cluster densities onto a lo calized 'ef-

purities? In practical terms, we observe an initial rapid

fective atom' expansion as

convergence with the MC/GSA approach, followed bya

long 'tail' of sampling in which only small incremental X X

= f j j = 16

cluster n n

improvements are found. Thus an optimal scheme in-

volves switching from GSA to MD-gradient based pro-

Here f are Fermi-Dirac o ccupation numb ers and

cedures at a rather well de ned p oint in the simulation,

enumerates atoms within the cluster.

when an accurate energy minimum is desired. Further

vi Analyze the interior `seed atom' region of each

details will b e given elsewhere; here we only wish to

cluster to extract comp onent atomic densities. New

comment brie y on another imp ortant practical asp ect:

atomic densities for the next iteration are obtained

the use of b oundary conditions and tight constraints to

after equilibration of the system and application of

p ermit ecient sampling of regions of interest. In tradi-

constraints such as electroneutrality of crystalline unit

tional MD/MC one often tries to work with the largest

cells. Equilibrate the various clusters selected to span

p ossible simulation volume, to minimize b oundary ef-

the region of interest, either by matching Fermi energies

fects, and distortions due to imp osed p erio dicity. Here,

or sp ectral features, and iterate steps i-vi until adequate

we delib erately cho ose quite restricted simulation vol-

convergence of charge and spin distributions, sp ectral

umes, with tight b oundary conditions, in order to fo cus

features, and total energy is obtained.

up on system resp onse to p erturbations related to de-

vii Extract densities of states, optical and X-ray

fects, surfaces, and interfaces. Of course, to o tight con-

absorption co ecients, electron energy loss sp ectra

trol completely prejudices the outcome of simulation,

EELS, cohesive energies, and other desired prop erties.

so convergence and stability tests have to b e made as

Compare charge distribution and cohesive energies with

in any lo cal scheme.

data assumed in construction of MD/GSA p otentials.

During the dynamic pro cess the MD/MC pro ce-

Up date MD/GSA p otentials as required.

dures may call the SCF-DV pro cedure to rede ne the

atomic charges. In case of ionic interactions, e.g.

molecules do cking with ionic lattices, this o ers the

III Examples of Applications

opp ortunity to determine p olarization and dynamic

charges 'on the y' as a function of geometry. In gen- Each of the comp onents of the hybrid scheme: MD,

eral, the DV quantum results pro duced can b e used to MC/GSA, and ECDF have b een develop ed indep en-

adjust the classical force eld See owchart in Fig.1; dently and has a considerable history of successful ap-

however, the interaction among various p otential pa- plications. The linkage b etween these comp onents, with

206 K.C. Mundim and D. E. Ellis

rameters has to b e considered. For example, in a typ-

ical Born-typ e ionic interaction with p oint-charge long

range terms, exp onential repulsion, and inverse p ower

short range attraction, mo di cation of p ointcharge val-

ues also implies some mo di cation of short-range pa-

rameters to maintain correct b ond lengths and cohesive

energies.

I I I.1 Applications Using a Coupled DVM-GSA

Metho d

I I I.1.1 A Stacked Linear Porphyrin Chain

The chemical b onding of metal p orphyrins, p or-

phyrazines, and phthalo cyanines is of imp ortance in

Figure 3. Electrostatic p otential in a 3D isosurface.

understanding biophysical and catalytic pro cesses. The

crystalline materials like copp er phthalo cyanine CuPc

form stacked chains, and when partially oxidized by io-

dine CuPcI for example, b ecome interesting 'molec-

ular wire' conductors. Dop ed systems like Cu1-

xNixPcI show quasi-one dimensional magnetic inter-

actions of considerable theoretical interest. Comp onent

molecules monomers like CuPc are quasiplanar and

small enough to b e treated by standard quantum chem-

ical approaches; however, understanding the electronic

coupling b etween monomers requires a DF approach,

and understanding the dynamics of coupling requires

a MD/MC classical treatment. An ECDF/MD/MC

study of a CuPc stack has b een recently carried out

[16]; here we wish to give a few highlights of the re-

Figure 3a. Linear stacking of a Porphyrin Chain.

sults.

I I I.1.2 Carb on Interstitials in Copp er

Fig.3 represents the electrostatic p otential in a 3D-

mapp ed isosurface of a p orphyrin chain. The molecular

In metallic conductors like copp er, small particle

equilibrium conformation was obtained using a GSA

precipitates are found to improve hardness and wear

technique, coupled to a classical force eld. Wehave

resistance of contacts. For high temp erature applica-

carried out semi- empirical Hamiltonian PM3 calcu-

tions, matrix-emb edded carb on b ers have also b een

lations to draw the electrostatic p otential isosurface.

considered, to provide sti ness and prevent ductile fail-

One of the signi cant conclusions drawn from this work

ure, and may b e useful to reduce sliding friction [17].

is that standard force elds are capable of repro duc-

An understanding of C di usion in Cu, and its b ehav-

ing b oth monomer and intermolecular interactions with

ior at the Cu/C interface is fundamental to developing

considerable accuracy.Thus, while the underlying elec-

successful strategies for improving b er-matrix comp os-

tronic interactions p orphyrinic-ring pi versus metal-d;

ite p erformance. Wettability-enhancing e orts based

sigma b onding in-plane versus pi-interaction b etween

up on alloying elements b ene t by connection b etween

molecules are complex and require detailed analysis,

the fundamentals of the wetting of solids by liquid met-

a simple classical parametrization captures the main

als and the practice of the preparation of metal-matrix

structural results. This observation helps to justify and

comp osites MMC [18]. In [19]itwas shown that

motivate the mixed classical/quantum metho dology. the interfacial energy in a metal-carb on b er system

Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 207

and emb edded cluster DF schemes. is mo di ed by the interfacial adsorption of the alloying

elements which are added to the metal matrix.

Our MD simulations [20] are here delib erately con-

High-resolution scanning electron microscopy

strained to small displacements of the Cu host atoms,

HRSEM shows the existence of a 50nm solid so-

for which the framework of elastic Ho oke's- law Cu-Cu

lution zone at the copp er-carb on interface, after an-

interactions is quite adequate. As follows from sim-

nealing. After heat treatment of one hour at 1000 C

ple pseudop otential calculations, the Cu-C interaction

the interface faded and we can see a copp er-carb on

is repulsive at least to several co ordination shells sur-

solid solution region of width 50 nm. Formation

rounding the carb on atom. The repulsive part of in-

of a very dilute Cu-C interstitial solid solution at the

teraction may b e tted byany functional form, and we

interface may b e di usion- controlled. To predict the

normally cho ose Lennard-Jones LJ parameters to re-

di usion b ehavior of carb on atoms wehave to study the

pro duce the exp erimentally measured prop erties. Un-

structure and interatomic interactions in such alloys.

fortunately, exp erimental data on dilute Cu-C solid so-

Phenomenological studies of these phenomena have

lutions are absent, so we use the results of the previ-

b een augmented by quasi-continuum mo dels [20] which

ously mentioned pseudop otential studies.

re ect some asp ects of the underlying electronic struc-

The expression used for the classical system energy ture. Now the analysis is extended to the atomic scale,

is thus using a combination of GSA/MD atomistic simulations

X X

1 C i; j C i; j

6 12

E r = K r R + + E

ij ij 0 0

12 6

2 r r

ij ij

i;j i

The last term, E , represents the volume-dep endent

part of the energy and includes information ab out elec-

tronic density redistribution, which is gathered from

the emb edded cluster density functional scheme, or

may b e obtained from other electronic structure cal-

culations. The parameter values are: R =2.475A,

2 6

K=6.920 eV/A ,E=-0.875 eV, C6=41.548 eV/A ,

C12= 2989.105 eV/A .

Among the ma jor conclusions of this work we nd

the following:

i. Carb on strongly prefers the o ctahedral intersti-

tial site over tetrahedral sites, as predicted from earlier

works. Wehave determined the relative site energies

and the rather long range relaxation of the Cu host

around the impurity.

Figure 4. Two carb on atoms in a fcc-cell of Cu.One carb on

atom is lo cated in a substitutiona l p osition and the other

ii. Surface sites incomplete o ctahedral interstitials

one b elongs to a tetrahedral interstitial p osition.

are at lower energy than volume sites, consistent with

the extremely low solubility of C in Cu.

I I I.1.3 Scheelite, a Candidate Host for Matrix-

iii. Carb on dimers and hints of clustering app ear at

Fib er Comp osites

higher impurity densities.

Calcium tungstate, CaWO , also known as scheel-

iv. The graphite-Cu interface region app ears to b e

ite, is a prototyp e for a large class of naturally o ccurring disordered to a depth of several Cu atomic layers.

208 K.C. Mundim and D. E. Ellis

minerals of the same structure [22]. The WO tungstate terial. These are precisely the characteristics which

group exhibits strong b onding of mixed ionic and cova- makescheelite interesting as a host matrix for ceramic

lentcharacter of great rigidity; as a result the scheel- matrix- b er comp osites CFC. A useful CFC should

ite lattice has a rather anisotropic resp onse to pres- b e air stable to T > 1300 C, and under stress, crack-

sure, with compression along the c-axis see Fig. 5 ing should take place in a controlled manner along the

b eing 1.2 times that in the a-b plane. The W-O b ond matrix- b er interface, allowing the b ers to pull out

length is 1.75A ; in contrast, the Ca-O b ond length is and minimize structural damage. A typical b er can-

2.41A and the Ca environmentmay b e describ ed b est didate would b e -Al O and and so the surface struc-

2 3

as an eight-fold co ordinated cage, formed from oxygens tures of scheelite and alumina, and their interfaces are

of eight di erent surrounding WO tetrahedra. of primary imp ortance. Various crystallographic sur-

faces of alumina have b een characterized by exp eriment

and theory; however, very little is known ab out the low

energy surface structures of CaWO . Simple consider-

ations suggest that oxygen-terminated surfaces which

preserve the WO units should b e of relatively low en-

ergy; thus wehave carried out GSA mo delling of the

001 oxygen terminated scheelite surface. A p ortion of

the hemispherical 001 surface volume mo del is shown

in Fig. 6; rigid emb edding atoms included in the force

eld are not shown.

Figure 6. Hemispherical dynamical volume representing

scheelite 001 oxygen terminated surface in two di erent

views.

Figure 5. Lattice structure of scheelite.

The contrasting b ond structures are the underly-

The p otential parameters used are given in Table 1.

ing reason for the apparently contradictory high melt-

Note that the strong W-O `b ond spring' and angular

ing temp erature and low fracture strength of the ma- spring constants are chosen to repro duce the observed

Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 209

rigidity of the WO structural unit. The exp ected re- ated with changes in co ordination and b ond lengths.

sult is that the greatest resp onse to crystal cleavage will A preliminary rep ort on this work has b een presented

b e some rearrangement of the Ca ions with resp ect to [24]; a detailed analysis of cohesion and b onding e ects

the tungstate network. The surface relaxation energy is will b e given elsewhere. A general conclusion that can

calculated to b e 17 meV/A with resp ect to the cleaved b e drawn from the surface and interface studies is that

unrelaxed crystal, with tungstate angles held xed, and reduction in cation co ordination and concomitant re-

duction in average metal-oxygen b ond lengths leads to 20 meV/A when exure is p ermitted.

reduced ionic charges at solid-vacuum and oxide-oxide

The 001 oxygen-terminated cleavage plane, with

interfaces. In the case of scheelite and related tetrahe-

its intact WO groups, app ears as the most obvious low

drally b onded materials like monazite, LaPO , the ro-

4 energy surface. There are several other cleavage planes

bust tungstate phosphate groups resist deformation

parallel to the c-axis whichmay b e of relatively low

and reduction pro cesses, leading to sp ecial fracture b e-

energy.Interestingly, simulations of these surfaces by

havior and high temp erature chemical stability.

MD techniques suggest that cleavage along such places

Finally,we mention results of recent simulations

will result in sp ontaneous rupture along the 001 plane

on the scheelite 100: -Al O 0001 interface [23].

[23]. 2 3

Rather large unit cells were used in slab geometry with

Wehave carried out ECDF calculations on the bulk

two-dimensional p erio dicity, using a gradient-search

and surface environments to determine ma jor features

MD scheme. It app ears that relaxation is essentially

of the electronic structure. The simplest analysis of

lo calized to a few atomic layers on either side of the

the results can b e made in terms of Mulliken atomic

interface, due to the considerable rigidity of the Al-O

orbital p opulations and densities of states DOS dia-

and W-O b onds. The main relaxation mechanism is

grams. In Table 2 we give the self-consistent orbital

the rotation of the WO groups to accommo date inter-

p opulations and charges for bulk and surface regions,

a+2 +6 2

face strain. ECDF calculations have b een undertaken

noting that the formal valencies are C ,w ,O .

to verify the lo cal electronic structure and to reconcile

Consistent with our qualitative discussion ab ove, we see

this structure with the force eld mo del.

that calcium is near its nominal ionic value, indicating

the relatively weak and long-range nature of its inter-

I I I.2 Molecular Mo deling in New Drug Devel-

actions. In contrast, the e ectivecharge of +3 on

opment

tungsten emphasizes the covalentcharge sharing in the

WO - group. As is generally found in metal oxides,

4 The development of a new pharmaceutical is a long

the oxygen 2p band, while formally fully o ccupied, has

and exp ensive pro cess. A new comp ound must not only

in fact 0.5e vacancies p er atom, due to charge transfer

pro duce the desired resp onse with minimal side- e ects,

from the metals. The b onding-antib onding gap in en-

but must b e demonstrably b etter than existing thera-

ergy levels of the crystal-emb edded WO - complex

pies, and pro duced at a reasonable cost. Finding novel

is found to b e 5.0eV, indicative of its great stabil-

comp ounds with desirable prop erties can b e a dicult

ity. The o ccupied valence band region shows W 5d, 6sp

problem, and one where molecular mo deling has much

structures spanning 6eV and forming two distinct

to o er. The development of molecular mo deling tech-

sub-bands with the O 2p, which can b e qualitatively

niques is contributing to the understanding of biological

lab eled as - and -typ e or equivalently denoted as e-

pro cesses at the atomic-molecular level and to the pro-

and t- symmetry of the T lo cal p oint group.

d p osal of new molecular structures with high biological

In order to b egin to explore the oxide-oxide in- eciency. The action of many drugs, hormones and

terfaces critical for understanding matrix- b er inter- membrane proteins is dep endent on the spatial con-

actions, wehave generated a preliminary mo del for formation adopted by the molecules in the inhomoge-

CaWO 001: MgO001. To induce steps and irreg- neous environment formed by the cell membrane and

ularities, the two surfaces have b een made to intersect its neighborhood.

at an angle of 25 degress. The volume b etween the The mo deling of biological molecules has led to

two p erfect bulk crystals was 'regrown' using the GSA great progress in the design of new molecular struc-

scheme with variable atomic comp osition and p ositions. tures with desired sp eci c prop erties. Such pro ducts

ECDF studies were made of selected interface regions could b e obtained synthetically or by genetic engineer-

to examine rearrangement of electron densities asso ci- ing pro cedures. Molecular mo deling can lead to the

210 K.C. Mundim and D. E. Ellis

understanding of the dynamic circumstances compati- pro duce the observed b ehavior and therefore to infer

ble with exp erimental observations, p ermitting a formal patterns of b ehavior for situations of interest.

de nition of the conditions that would b e exp ected to

Bond W-O Bond Ca-O L-J W-Ca

R 1:750A R 2:440A C 25:152eV A

o o 6

2 2 12

K 26:019eV =A K 0:300eV =A C 710:646eV A

ij ij 12

Bond I-J-K K 0:300eV =A deg r ees

O-W-O 5.204 103.80

W-O-Ca 5.204 125.17

O-Ca-O 4.336 138.52

Bond X-I-J-Y K eV =r ad deg r ees

X-O-Ca-Y 0.039 138.5

X-O-W-Y 0.039 103.8

O-X-Y-O 3.469 35.0

Table 1 - Ho ok's law, Lennard-Jones, Covalent b ond angle, prop er and improp er dihedral parameters of GSA

simulations for CaWO :

Bulk Surface

interface will b e renormalized by the in uence of the di-

W5d 2.51 3.10

electric discontinuity. The p olarization eld pro duced

6s 0.08 0.06

at the surface of discontinuitybyapointcharge can b e

6p 0.11 0.17

calculated by the metho d of images. A ctitious charge

net charge 3.35 2.72

is placed in the opp osite phase: the distance and the

value of the image charge are xed by taking the ap-

Ca 3d 0.03 0.02

4s 0.01 0.01

propriate electrical b oundary conditions at the surface.

4p 0.06 0.04

When twocharges i and j are on the same side of

net charge 1.92 1.94

the interface, the p otential on i due to j will havea

contribution due to the image of j i.e., j', so that the

O2s 1.96 1.98

Coulomb p otential on i will b e expressed by

2p 5.33 4.88

net charge -1.29 -0.86

1 1 1

Table 2 - Self-consistent Mulliken atomic orbital p op-

1 2

V = q +

ulations and net atomic charges for di erent sites of

4 r + r

1 ij 1 2

001 oxygen terminated scheelite.

where

I I I.2.1 Conventional Molecular Dynamics Pro-

1=2

2 2 2

cedure Applied to Biomolecular Systems

r = x x +y y +z z

ij i j i j i j

Our computational co de, based on the classical force

eld presented in section 2A, has an additional ca-

1=2

0 2 2 2

r = x x 2x +y y +z z

i j s i j i j

pability of mo deling molecules of biological interest in

aqueous and ap olar media, as well as at their inter-

and where r and r are the distance b etween i and

ij ij

face. The software establishes two media with di er-

j, and i and j' resp ectively.X is the p osition of the

ent continuous dielectric constants, separated by a cy-

surface b etween the two media. When twocharges are

toplasm/membrane environment. The approach used

at di erent sides of the interface the metho d of image

to simulate the membrane interface is expressed bya

gives the following result:

discontinuity in the dielectric constant, taking into ac-

count the di erent electrical p olarizability of the aque- Wehave applied this MD approach to study to

ous and the hydro carb on phases. Then, the electro- study a new mutant sequence of E. Coli. , mo deling

static interaction b etween non b onded atoms at this a 21 amino acid p eptide, a mutant sequence from E.

Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 211

Coli maltoporian [3],[4]. An imp ortant asp ect of p ep- 21 residues and shows a deletion of four residues in the

tide simulation is the great numb er of degrees of free- hydrophobic region relative to the wild sequence. Such

dom for linear conformations, which can lead to sev- a deletion should ab olish its capacity for helix formation

eral conformations with equivalentvalues of minimum and therefore its functionality. These e ects were con-

energy. In this investigation wehave studied a mutant rmed exp erimentally.However, 50 was re-established

signal sequence 78r of the LamB gene pro duct, also by replacing a Gly residue with a Cys residue at residue

called receptor or maltop orin, a protein of the exter- 13 of the mutant p eptide. The relevant sequences are

nal membrane of E. coli. The mutant sequence contains

Wild typ e: MMITLRKLPLAVAVAAGVMSAQAMA

78r1 : MMITLRKLP|||VAAGVMSAQAMA

steps of t 1 femtosecond, due to the intrinsic vibra- Polarization e ects on the p eptide conformations

tional frequencies. This means that ab out 6x10 MD have b een investigated through the electrostatic charge

steps would b e needed to simulate the protein folding image metho d as describ ed ab ove. A similar technique

pro cess, whichisbeyond computational p ossibility. to simulate p olarization e ects in p eptide conforma-

tions has b een prop osed by some of the authors in Ref.

[25]. These authors carried out a classical force eld

simulation containing electrostatic image charge cal-

culations to investigate the conformations of p eptides

characterized by di erenthydrophobicities at a water-

membrane interface mo del. The interface is represented

by a surface of discontinuitybetween two media with

di erent dielectric constants, taking into account the

di erence b etween the p olarizabilities of the aqueous

medium and the hydro carb on one.

I I I.2.2 Sto chastic Molecular Dynamics Pro ce-

dure Applied to Biomolecular Systems

Figure 7. Ico-alanine p olyp eptid e in a linear- and helix-

con guration.

The biological function of a protein or p eptide is

often intimately dep endent up on the conformations

An alternative solution to this problem is to use that the molecule can b e adopt. X-ray crystallogra-

sto chastic simulation pro cedures such as GSA. Re- phy and NMR are two metho ds used to provide de-

cently, Moret [26] has discussed the eciency of tailed information ab out protein structures. Unfortu-

sto chastic simulation to study protein folding by the nately, the rate at which new protein sequences are de-

GSA metho d. The author simulated -helix con gura- termined exp erimentally is much greater than that for

tion of the ico-alanine p olyp eptide Fig. 7 and found determination of structures. There is thus consider-

the necessary conditions for GSA parameters to ob- able interest in theoretical metho ds for predicting the

tain adequate computational eciency in protein fold- three-dimensional structure of a protein from its amino

acid sequence; this is called the protein folding problem. ing simulation. The GSA pro cedure with visiting index

Prediction of protein folding is a case where the usual q =1,05 was shown to b e more ecient than the usual

MD is not a go o d theoretical metho d, since the real simulated annealing metho d of Kirkpatrickq =2

physical time of a folding pro cess may b e on the scale and the fast simulated annealing metho d of Szu q =

of minutes. To solve the time evolution of Newton's 1 for calculation of an -helix conformation equilib-

equation in MD it is necessary to discretize the time in rium structure.

212 K.C. Mundim and D. E. Ellis

sical/quantum pro cedure. Although GSA/MD/ECDF IV Conclusions:

is already a p owerful to ol for the study of complex sys-

A nearly optimum pro cedure to search for minimum-

tems, it is not optimized for pro cesses which are essen-

energy structures is to initially scan using the MC/GSA

tially dynamic, such as di usion of atoms and molecules

scheme. As the slow-convergence region is attained, we

along interfaces and surfaces. It will b e imp ortantto

switch to the MD/annealing scheme. The algorithms

develop the capability for treating the longer time scales

are coupled together in suchaway as to p ermit this

implied by di usion pro cesses; it is encouraging that the

strategy, in a uni ed and convenient manner.

protein folding problem which has a long time scale

To extract electronic structure information in 'in-

has already b een shown to b e amenable to the GSA

teresting' nuclear con gurations, and not only in the

approach.

state of lowest p otential energy,wehave the capacity

Graphical interfaces are essential now for controling

to activate the Emb edded Cluster DensityFunctional

and monitoring the mo dels of complex systems. On the

comp onents of the co des from within the running dy-

'output side' wehavevarious to ols, which use commer-

namics pro cedures. This p ermits generation of 'snap-

cial programs to visualize the results. Currently,we are

shots' of electronic states during a pro cess of molec-

working in the direction of increasing the `input side'

ular interaction or do cking of a molecule with a sur-

to help the user de ne and set up the mo del system.

face, wall of a molecular sieve, or macromolecule. An

imp ortant application which helps in dynamically re-

App endix

parametrizing the interatomic p otentials, is the deter-

mination of atomic charges 'on the y'.

To generate the random vector x the present GSA With the concept of self-consistentemb edding, we

routine use an extension of the pro cedure used in ref- have gained the ability to resolve large-size structures,

erence [12]. Wehave calculated the x = g ! us- containing thousands of atoms, and a size scale of at

ing a numerical integration of the visiting distribution least 10 nm, as wehave illustrated in the examples given

probability g x. Where ! is a random vector [0,1] here. Studies in progress on oxide surfaces, molecule-

obtained from an equi-probability distribution and g zeolite interactions, and grain b oundaries in metals will

is the inverse of the integral of g x given by help to determine future evolution of the hybrid clas-

0 1

@ A

g ! =inv er se g xdx = inv er se! ; 17

1 0

1 q 1

1 v

2 Z

1=2

[T t]

q 1

C B

q v

dx ; g ! =inv er se

A @

1 1

q 1 2

f1+ q 1 g

q 1 2

V v

t] [T

where q q is the acceptance index visiting index, and

A V

q 1 1

1=2

[T t]

q 1 q 1

: 18 g 4x =

q t

1 1

q 1 2

f1+q 1 g

v 2

q 1 2

[T t]

The integral of g x has an analytical solution only if q ;q = 1; 1 or q ;q = 1; 2. For the general

q A V A V

case it is necessary to makea numerical integration. In this pap er, g has b een calculated using a series integration

and taking the inverse of a p olynomial series, whose expansion is cut o at the 17 order.

The integral in equation 17 can b e written as

q 1 1

2 Z

1=2

q 1

q 1 a

! = ! x= dx ; 19

2 c

1 1

1 + bx

q 1 2

1 v

Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 213

q 1

1 1

. , b = where a =[T t] , c =

q 1 2 v

[T t]

Solving equation 19 using p ower series, wehave;

1 1 1 1 1

3 2 2 2 5

! = + ax abcx + ac b + acb x + 20

2 3 5 2 2

3 5 n

! x=! + A x + A x + A x + = ! + A x x 21

o 1 2 5 o n o

n=1

We lo ok for an inverse function g = x = x! , such that x! x =0.From Eq.21, we see that the inverse

function may b e expressed in terms of a p ower series,

x! =x + B ! ! : 22

o n o

n=1

This is guaranteed by a theorem: Let ! x be analytic at x = x , and ! x 6=0,then the inverse of ! x

o o

. exists and is analytic in a suciently smal l region about ! x and its derivative is

d o

! x

The co ecients B may b e expressed in terms of A .However, it is p ossible to derive the co ecients more

n n

elegantly by employing Cauchy's formula. In this case, B takes the general form

1 A A nn +1n 1+ + + +

2 3

+ + +

B = 23 1

nA ! ! A A

1 1

; ;

The rst few B co ecients can b e expressed as

; B =

1 3 4 A A 3

2 3

B = ;

3A 2! A 1! A

1 1

4 5 6 4 5 4 1 A A A A

2 2 3 4

+ B = : 24

4A 3! A 1!1! A A 1! A

1 1 1 1

This last inversion was intro duced in the GSA package, while the original prop osition [11] uses a L evy-Flight

distribution as the visiting distribution.

527 1994. References

[5] S. Kirkpatrick, J. Stat. Phys. 34 , 975 1984.

[1] B.J. Alder and T.E. Wainwright, J. Chem. Phys. 27,

[6] S. Kirkpatrick, C.D. Gelat and M.P.Vecchi, Science

1208 1957; Phys. Rev. 1, 18 1970.

220, 671 1983.

[2] N.L. Allinger and U. Burkert, \Molecular Mechanics",

[7] D. Cep erly, C. Alder, Science 231, 555 1986.

ACS-Monograph, 177 1982.

[8] H. Szu and R. Hartley,Phys. Lett. A 122, 157 1987.

[3] E.P.G. Areas, P.G. Pascutti, S. Schreier, K.C. Mundim,

P.M. Bisch, J. Phys. Chem. 99, 14882 1995.

[9] C. Tsallis, J. Stat. Phys. 52, 479 1988.

[10] E.M.F. Curado and C. Tsallis, J. Phys A 24, L69 [4] E.P.G. Areas, P.G.Pascutti, S. Schreier, K.C.

1991; Corrigenda: J. Phys. A 24, 3187 1991. Mundim, P.M. Bisch, Brazilian J. Mol. Biol. Res. 27,

214 K.C. Mundim and D. E. Ellis

[19] U. Gangopadhyay and P. Wynblatt, Journ. of Mater. [11] C. Tsallis and D.A. Stariolo, Phys. A 233 , 395 1996.

Sci. 30, 94 1995; S. Hara, K. Nogi and K. Ogino, in

[12] K.C. Mundim, C. Tsallis, Int. J. Quantum Chem. 58,

Pro ceedings of Int. Conf. \High-temp erature capillar-

373 1996.

ity".

[13] M.G. Moret, P.G. Pascutti, P.M. Bish and K.C.

[20] S. Dorfman and D. Fuks, Comp osites 27A, 697 1996.

Mundim, J. Comp. Chem. 19, 647 1998.

[21] D.E. Ellis, K.C. Mundim, D. Fuks, S. Dorfman and A.

[14] K.C. Mundim, T.J. Lemaire and A. Amin Bassrei,

Berner, Mat. Res. So c. Symp., Pro c. 527, 69 1998.

Physica A 252, 405 1998.

[22] H.W. Wycko , Acta Cryst. 11, 199 1958.

[15] D.E. Ellis and J. Guo, in Electronic Density Functional

Theory of Molecules, Clusters, and Solids, ed. D.E. El-

[23] C-K. Guo, D.E. Ellis, O. Warschkow, unpublish ed.

lis, Kluwer, Dordrecht, 1995 p263.

[24] D.E. Ellis and K.C. Mundim, in Proc. of 9th CIMTEC-

[16] L. Guo, D.E. Ellis, K.C. Mundim and B.M. Ho -

World Ceramics Congress Forum on New Materials,P.

man, Journal of Porphyrins and Phthalo cyanin es 2,

Vincenzine, Ed., Techna, Florence, 1998, to b e pub-

1, 1998

lished.

[17] S. Dorfman and D. Fuks, Comp osite Interfaces 3, 431

[25] P.G. Pascutti, K.C. Mundim, A. S. Ito and P.M. Bisch,

1996 and references therein.

J. Comp. Chem. to b e published 1999.

[18] F. Dellanay,L.Froyen and A. Deruyttere, Journ. of

[26] M.A. Moret, M. Sc. Thesis IF-UFBa 04/1996.

Mater. Sci. 22, 1 1987.