Journal of Molecular Structure (Theochem), 206 (1990) 39-48 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

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THE EFFECT OF ELECTRIC FIELD ON INFINITE QUASI-ONE- DIMENSIONAL POLYMERS: APPROXIMATE DETERMINATION OF HYPERPOLARIZABILITIES*

J. LADIK

Friedrich-Alexander- University Erlangen-Niirnberg, D-8520 Erlangen, Egerlandstr. 3 (F.R.G.)

(Received 7 June 1989)

ABSTRACT

For the treatment of the effect of a homogeneous electric field on the electronic structure of a quasi-one-dimensional infinite polymer, it is shown that in the SCF LCAO crystal orbital ap- proximation (so-called ab initio Hartree-Fock crystal orbitals) all the matrix elements that occur are finite. In the case where the electric-field strength has a non-zero component along the chain which introduces non-periodicity, a promising simple approximation is proposed. The self-con- sistent wavefunctions were determined in the presence of the field, and expressions for calculating the induced dipole moment and elements of the polarizability and different hyperpolarizability tensors per unit cell are given. A brief outline of the procedure for an inhomogeneous electric field which does or does not destroy the periodicity of the chain is given. Special attention is given to the problem of the interaction of laser light with a periodic chain for which a Green’s matrix formalism (the use of Dyson’s equation) is proposed.

INTRODUCTION

The treatment of the effect of an electric field on an infinite linear polymer seemed to be a formidable problem, because (i) the potential ] e 1 rE corre- sponding to a homogeneous electric field E = constant is unbounded in an in- finite system, and (ii) if the field has a non-zero component in the direction of the chain (say the z axis ) the term ] e 1 zE, occurring in the potential destroys the periodic symmetry of the chain [ 11. These circumstances blocked direct crystal-orbital (CO) calculations on infinite periodic chains, although they would be very necessary to determine the elements of the polarizability tensors and hyperpolarizability tensors. As is well known, these quantities play an important role in electro-optics and non-linear optics and, therefore, knowl- edge of these parameters is of great practical importance for new materials in order to be able to show the desired properties.

To overcome these difficulties, Bodart et al. [ 21 and Delhalle et al. [ 31 cal-

*In memory of Professor Oskar E. Polansky (15 January 1989).

0166-1280/90/$03.50 0 1990 Elsevier Science Publishers B.V.

40

culated the dipole moment induced by a homogeneous electric field and the polarizability and hyperpolarizability tensor elements of different series of similar unsaturated molecules. Using an STO-3G basis set they treated the ethylene and acetylene series [ 21 and the allene series [ 31, taking the first four or three members of the series, respectively. In order to improve the minimal basis results, the above authors scaled their results on the basis of the ay,, values obtained theoretically with the help of a double-zeta basis, or according to the experimental values, respectively, in the case of the first two series [ 21. They then used their combined scaling factor 1.57 [2] for the other tensor elements [ 21 and for other series [ 31.

In the present paper it is shown that (i) despite the fact that rE tends to infinity if r does, the matrix elements of rE are finite; and (ii) the aperiodicity caused by the zE, term can be treated in an approximate way. In this way an approximate CO procedure can be developed for the treatment of the effect of an electric field on the electronic structure of a quasi-one-dimensional system. The results of such calculations can also be applied in the determination of the polarizability and hyperpolarizability tensor elements. Finally, some com- ments are given on the treatment of electric-field effects in the electronic struc- ture of linear polymers in the case of inhomogeneous electric-field strength,

E(r).

FORMULATION OF THE METHOD

Homogeneous electric field

Finite matrix elements of an unbounded operator According to the corresponding Maxwell equation, the electric-field strength

E depends on the scalar electric potential, $, and the magnetic-vector poten- tial, A

1 aA E= -grade-cat

In the case where A is time-independent (time-independent magnetic field strength H) and the field is homogeneous, E = constant, the potential energy is equal to

V= - lel$= lel 1 Edr= (e(Er= lel (E,x+E,y+E,z) (2)

Let us assume further that we have a linear polymer with z being the polymer axis. According to the ab initio SCF LCAO CO (Hartree-Fock) theory of pe- riodic chains with Born-von Karman boundary conditions [ 4,5], the following generalized matrix eigenvalue equation must be solved

41

(3)

(4)

F(k)c,(k)=Ei(k)S(k)Ci(k)

with

S(k)= f eikQ”S(q); [S(q) lT,s = (xi? Ix: > Q= -N

where k is the crystal momentum, 2N+ 1 the number of unit cells in the chain (in the case of an infinite chain N-+ co ) and a is the elementary translation. xy stands for the sth basis function centered on an atom in the qth cell char- acterized by the vector R,

X; =Xs(r-R,-RB) =Ne-a(r-RgB)2(x~)‘(y~)m(z~)~

RqB =R, +RB, stB (5)

if one uses Cartesian Gaussian functions and the orbital s is centered on atom B. Furthermore, the Fock matrix F(k) is equal to

(6)

where AN is the one-electron operator and the Coulomb operators Jj and ex- change operators Rj form the two-electron part of the Fock operator P and IZ* is the number of filled bands. (In an actual ab initio crystal orbital calculation Jj and .& are also expressed in an LCAO form.) In this case, similarly to the molecular case [ 61, sums of multicentre two-electron integrals and generalized charge-bond order matrix elements are obtained which, however, in this case depend also on the cell index q. (For further details see refs. 4 and 5.)

In the presence of a homogeneous electric field, E, the additional term ) e ) rE will occur in operator RN in eqn. (6). In order to investigate the matrix ele- ments of this term, let us first assume that there is only one atom per unit cell and only a 1s function is centered on it. (The generalization of the results which we give for the case of more orbitals per atom and more atoms in the unit cell is straightforward and very simple. ) For this case we have

lel <xYS IKJ+E,Y+E~~Ix’L) (7) =lel ~~~~~I~,~+~,YIxP~~+~x~~I~~~Ix~~~~

If the origin is taken as the zeroth atom (reference cell), it can immediately be seen that the first integral in eqn. (7) vanishes, because x and y change sign when they are reflected at the origin. (This is, however, not the case for an odd basis function in x or y, respectively, or if there is more than one atom in the unit cell. ) Thus

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<XL IJ%x+EyY 1x7, > =o and

@a)

<x: Im+J%YIx:) 20 (8b)

if xr or xs is an odd function of x or y, respectively, or reA and seB #A. On the other hand, even the matrix elements given in eqn. (8b) are finite

because x: and xSQ are Gaussian functions [see eqn. (5) ] and, therefore, they tend to zero much more quickly than x or y tend to infinity.

Let us now investigate the matrix element

(xys 1 I&z Ixt ) =I$, (e -aW+yz+zz) 12 I e --0((r-Rq)2) (9)

=E,(e- cxW+y2+2*) I z I e- a[x2+y*+z* -Z(xX,+yX,+zZ,) +(x:+Yij+z;)l

where x, y and z are measured from the chosen origin (the atom in the zeroth cell).

Since the integrand of the matrix element

(e-olz~~Z~e-““~e+2~zZ~e-“Z :> (IO)

goes to zero if z--too in all cases (2, is finite or tends to infinity also). In this way it can be shown that despite the fact that V= I e I rE is unbounded, all its matrix elements remain finite if a Gaussian basis is used.

Let us now investigate the sum

eikqa(x& I&4xP.J q= -N

(11)

which occurs in eqn. (6). From elementary symmetry considerations it follows that the matrix elements are positive for q> 0 (they are zero for q= 0 in this special case, but they are non-zero and positive also for q=O if there is more than one atom in the unit cell) and they are negative if q < 0. Furthermore, it is easy to see that

<XL IZIXL) = -<XL l~lX2)

Substituting eqn. (12) into eqn. (11) gives

14Ez E eikq”(x~sIZIxPs)=lelEZ[? eikq”(x~s14x4s) q= -N q=l

(12)

+q=I~Neikqa~x~~l~lxL)l=lelE~q~~ (e’kqa-e-ikq”)(x~slZIxP,)

= lej 2 E,2i sin kqa(xys IzlxP,) #O q=l

(13)

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if k # nx. The same conclusion also holds if there is more than one orbital and atom in the unit cell

lelEL ? eikYX~ Islx8> q=-N

if k#nx, reA, seB#A. In deriving eqn. (14) we used the hermiticity of the sum

~~~~,~~~l~,,+~~~-~,~,~l~,,=I~l~,~~~~I~I~~~eikq” +tx%Ix:)e-ikq”l

tWGJ+lr,s+ [W-q&)+1,, = [ (xp )zIX,yq)e-ikq”+ (x: IzlxB)eikq”] lelEL

= tFbWl,s+ [W-q&)lr,s (15)

On the basis of eqn. (15) it can be stated that the presence of E does not destroy the hermiticity of the Fock matrix

E&E)+ =F(k,E) (16)

Approximate treatment of aperiodic& caused by the term E,z If the homogeneous electric field E has only x and y components then E, # 0,

EY# 0 and E,= 0, and their presence will not destroy the periodicity of the chain. On the other hand, if E,# 0, then because of the occurrence of the term I e I Ep the chain will become non-periodic. In this case, however, the following procedure can be used if E, is not very large. Instead of calculating the matrix elements (x: I E,z Ix: ) (which, as has been shown, can be done, but for an aperiodic chain the whole formalism given by eqns. (3) to (61 is no longer valid) one tries to subdivide the chain of 2N+ 1 unit cells into 2N+ 1 segments (a< N) , where each segment consists of a certain number of unit cells and it is assumed that in this segment z has an average value 2,. Obviously, 8 must be a function of the value of E,, of the basis set applied, the size of the unit cell and the number of neighbours explicitly taken into account. If a constant num- ber of cells is used for the segmenJ length 1 (which of course is only a first approximation ), then 2N+ 1 = 1(2N+ 1) and the eqn. (14 ) becomes

I+% f eikq”(x~I~lx~)=I~lE,[(x~l~lx~) q= -N

44

+ 5 2i sin kqa q=l

=+$l z,Sr,,(d= [~(kq&Hr,,; q>O (17)

Equation (17) means that if the E, values are not too large and if there is a finite number of neighbour interactions, then instead of treating an aperiodic chain, different periodic chains are treated and the average of their band struc- tures is taken. Finally, for all the Fock-matrix elements in the presence of an electric field

[FUG) I,+ = 2 q= -iv

+E,y)IX~)+lelE,[z;S,,(O)+ : 2isink.w; z Z,Sr,,(q)l (18) q=l m 1

At the first sight the approximation introduced in eqn. (17) seems to be very rough. It should not be forgotten, however, that in actual polymer calculations a finite number (not very large) of neighbour interactions must be taken, which means that for a segment of, say, Z=3 unit cells, fi will be 1,2 or 3 depending first of all on the size of the unit cell. If this is taken into account and if the last two terms in eqn. (18) are not large as compared with the other ones (the opposite would happen only in the special case if 1 E 1 is large and E, >> E, or E,) eqn. (18) will provide a tolerably good approximation of the problem. Of course, after programming eqn. (18) only (which we plan to do) different nu- merical tests can be used to determine the quality of the approximation intro- duced above.

Determination of the polarizability and hyperpolurizability tensor elements per unit cell

If the generalized eigenvalue problem of the matrix F (k,E ) defined by eqn. (18) is solved

F(k,E)ci(k,E) =Ei(k,E)S(k)ci(k,E) (19)

and not of that of F (k), instead of the original ci(k) eigenvectors [defined by eqn. (3) 1, new ci (k,E ) eigenvectors and a new band structure Ei (k,E ) are ob- tained. Using these new eigenvectors, the one-electron wavefunctions (which are linear combinations of Bloch functions) can be written as

h(k,r,E)= -&F-i ,Leikqa 6 ci,,(k,E )XZ E @I(E) , I= i,k (20)

where Ci is the number of basis functions per unit cell. Using the wavefunctions $i(k,E ), the dipole moment per unit cell induced

by an electric field (rind) can be calculated as

n/a

c ind ~2 C (h,(E) Iilk >=2: 1 a z (&(kE) lerl$i(k,E)> (21)

is1 -n/a

On the other hand,

P i”d=aE+; BE’++ yE2E+...= tFl ; e(EY (22)

Equation (22) defines the polarizability tensor a and the hyperpolarizability tensors j3, y, etc. To describe the effect of lasers on solids one must set m = 4 or 5 in eqn. (22). This equation can be written out in detail for the ith component (i=l,2or3) ofpindas

On the basis of eqn. (23)) the tensor elements can be calculated as

a2pd Pijk - aEjaE, E=.

a3&d yijkl=aEjaEkaEI E=O

(23)

(244

Mb)

(24~)

According to the numerical results obtained for the case of finite molecules [2,3], the results of the applied numerical differentiation (the analytical de- pendence of pind on the components of E is usually unknown) are very sensi- tive to the quality of the Hartree-Fock procedure used. Therefore, instead of using the usual 10M4 as the convergence criterion, 10e5 or 10e6 was set for the elements of the charge-bond order matrix and a basis set of at least double- zeta quality is required [ 2,3].

According to the experiences of Bodart et al. [ 21 and Delhalle et al. [ 31, the differentiations occurring in eqns. (24) can be quite well performed by using numerical-difference quotients. For instance, these authors have calculated the value of CY,, as

(25)

with values of EZ of -0.002, -0.001,0.001 and 0.002 au., and have obtained numerically quite stable results.

The same procedure can be applied in the case of Find de~rmined using eqn. (21) for an infinite chain. In this case, eqns. (24a) to (24~) would provide the polarizability and hyperpolarizability tensor elements per unit cell. Again only numerical tests could determine the quality of these tensor elements deter- mined with the help of the above-described procedure.

Remarks on the effect of an inhomogeneous electric field

If E (r ), the simple relation [eqn. (2) ] V= 1 e 1 Er is not valid, and one has to use the more general expression

V= lcl 1 NW= lel [I ~,~~~y,~~~+~,~~~,~~~y+E,~~~,~~~l (261

Substituting eqn. (26) into the expression for the Fock matrix elements [(F(q) lT,s gives

W(dlr,,= <x9 1-F’+ Vlx! > (27)

Using this procedure it is still certain that, independent of the form of the function E (r ) , all the matrix elements remain finite and F (k,E ) remains a Hermitian matrix. On the other hand, it depends strongly on the form of V(x,y,z ) whether the aperiodicity occurring in the cases E, (z ) , E,, (z ) and E, # 0 can be approximated in such a simple way as in the case of the homogeneous field.

In the special case where E, (y,z), Ey (x,z) and E,( z,y ) the simple expression V= I e 1 E,r can still be applied, but besides E,z the terms E,x and E,y will also cause aperiodicity in the polymer chain. If E, (y,z) and EY (x,z) are only slowly varying functions of z and E,(x,y) is not very large compared to them, then besides the average values Z; average functions &,,, (y,&) and EYm (x,&) can also be defined for the mth segment of the linear chain. In this case the matrix elements [F (K) ] r,s still can be written in analogy to eqn. ( 18) as

WW,EWlL,,= 5 eikq”(xy lP+ Jel l- (I= -N

+&AG%)YI Ix:> + If4 [

<xl) IJU~,Y)Z; 1x3

+ f %sinkqaj$ i (xPf&ky)~mix~~ q=l m 1 1 (28)

If the above given conditions are not fulfilled, but the components of E (r ) are only slowly varying functions of z, it is necessary to return to eqn. (26), but an attempt can still be made at an averaging procedure for the three inte- grals in eqn. (26).

It should be pointed out that if E (1:) is rather small in its whole domain, a perturbation theoretical treatment can be used, using as unperturbed wave- functions and energies those of the field-free periodic polymer and V as per- turbation operator. However, in order to obtain somewhat accurate results the calculation must be made at least to second order, but most probably higher order terms will also be needed. Furthermore, there is no experience on the convergence of this perturbation series. All this makes the application of per- turbation theory, even in the case of weak fields, tedious and rather questionable.

Finally, it should be mentioned that if a short wavelength laser beam is in- teracting with a segment of a polymer, E (I) is large and E is a strongly variable function of r, but because the amplitude of the electric field strength wave coupled with the photons is small, the perturbation V(r) can be visualized as a rather localized one. In this case, a Green’s matrix formalism in the form of the Dyson equation must be used

G=G,+GVG, (29)

in which the unperturbed Green’s matrix Go can be determined in a standard way from the solution of the periodic problem and the matrix V can be calcu- lated from its elements

(30)

In eqn. (30) the potential V is again given in the general case by eqn. (26). Having determined the matrix G (z), z = E+ ir,~ the well-known relation [ 71

can be used

Ef

P-; I

I,G + (2Z)d.E --oo

(31)

where

G+(E)= Zim G(z) (32) q-o+

With the help of P piind can also be computed with the aid of eqn. (21) by expressing &(E ) in the LCAO form [ eqn. (20) 1.

CONCLUSION

The theoretical developments given above show that the effect of a homo- geneous or inhomogeneous electric field on the electronic structure of an infi- nite quasi- one-dimensional chain can be treated, in most cases, in a reasonable approximation. In the case of a homogeneous field the method can also be applied to the determination of the induced dipole moment per unit cell and for computing the elements per unit cell of the polarizability and different hyperpolarizability tensors which have a large practical importance in several fields. After writing the appropriate codes, we intend to perform numerical tests to check the reliability of the proposed methods for different cases and different infinite chains.

Finally, it should be pointed out that all the derivations made here for the sake of simplicity for quasi-one-dimensional systems can be easily generalized to two- and three-dimensional infinite periodic systems.

ACKNOWLEDGEMENTS

The author would like to express his gratitude to Professors J. Delhalle and P. Otto and to Dr. W. Fiirner for calling his attention to the problem and for the very fruitful discussions. The financial support of the “Fonds der Chem- ischen Industrie” is gratefully acknowledged.

REFERENCES

See, for example, J.J. Ladik, Quantum Theory of Polymers as Solids, Plenum Press, New York, 1988, p. 369. V.P. Bodart, J. Delhalle, J.-M. Andre and J. Zyss, Can. J. Chem., 63 (1985) 163. J. Delhalle, V.P. Bodart, M. Dary, J.-M. Andre and J. Zyss, Int. J. Quantum Chem., S19 (1986) 313. G. Del Re, J. Ladik and G. Biczd, Phys. Rev., 155 (1967) 997. See also, J.J. Ladik, Quantum Theory of Polymers as Solids, Plenum Press, New York, 1988, pp. 9-20. J.-M. Andre, L. Gouverneur and G. Leroy, Int. J. Quantum Chem., 1 (1967) 427,451. C.C.J. Roothaan, Rev. Mod. Phys., 23 (1951) 69. See, for example, G. Del Re and J. Ladik, Chem. Phys., 49 (1980) 321.