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Recent developments in the quantum theory of organic polymers
127
Journal of Molecular Structure (Theo&em), 230 (1991) 127-142 Elsevier Science Publishers B.V., Amsterdam
RECENT DEVELOPMENTS ORGANIC POLYMERS
IN THE QUANTUM THEORY OF
JANOS J. LADIK Chair of Theoretical Chemistry, Friedrich-Alexander-University,
Erlangen-Niirnberg
(FRG)
(First received 8 August 1990; in final form 12 October 1990)
ABSTRACT A new theoretical method and its applications will be presented to calculate the interface between two different periodic polymers. Further theoretical developments in extrapolation methods to find out the band structure of an infinite chain from its oligomers, as well as the treatment of correlation in disordered chains, will be discussed. Finally, theoretical methods to treat the interaction of a stationary and non-stationary magnetic and electric field (including a laser pulse) will be shown.
INTRODUCTION
The basic quantum theory of polymers treated as solids is summarized in a book which the present author published in 1988 [ 11. In this monograph the Hartree-Fock crystal orbital (CO) theory is described in detail, both in its ab initio form and in its different semi-empirical forms, together with applications for a larger number of polymers. Theoretical methods for the treatment of disorder, with applications, as well as the correction of the band structure for correlation effects are also described. The interaction between polymer chains and the effect of the environment (which is especially important for biopolymers) on the band structure are then discussed. In the second part of the monograph, the applications of the basic theory for different properties, such as U.V. (using the intermediate exciton theory, starting from correlated band structures), vibrational and photoelectron spectra, transport (both Bloch type and hopping conduction) and mechanical properties are summarized. In one of the last chapters the effects of a static magnetic field on the band structure and the approximations usually applied for polarizabilities and hyperpolarizabilities are described. The aim of the present short review paper is to summarize the recent theoretical developments achieved since the appearance of the above mentioned monograph and not to present basically new results (with the exception of the case of the interaction of a periodic polymer with a laser pulse, where the effect
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Elsevier Science Publishers B.V.
128
of the overtones of the electromagnetic field is also taken into account). First of all a new method to describe the interfaces between two different polymeric chains [ 21 and their applications will be presented. The next step is the application of a modified Romberg algorithm [ 31 to extrapolate from the electronic structure of oligomers to the band structure of an infinite chain (which converges faster than other extrapolation methods, for instance Pad6 approximants [3]). An important new theoretical development is the treatment of correlation in the case of disordered polymers [ 41. The efficiency of this new method will be illustrated by several applications. Essential progress has been achieved in treating the effects of static electric fields [5] and time-dependent magnetic fields [6] on the band structure of polymers. Finally, a general theory for the interaction of a laser pulse with a periodic polymer [ 71 will be sketched. This new theory can provide not only static polarizabilities and hyperpolarizabilities, as can the method described in ref. 5, but also frequency and time-dependent ones [ 71. It should be mentioned that all the methods described here are formulated for quasi-one-dimensional (1D) chains, but their generalization to 2D and 3D systems is obviously straightforward. THE INTERFACE BETWEEN TWO DIFFERENT POLYMER CHAINS
To treat the interface between two polymer chains it is necessary first to solve the Hartree-Fock problem F,(b)di(k),
(la)
=ci(k)iSi(k)di(k)i
(lb)
F,(k)di(k)P=&i(k)zs,(k)di(k), where Fj(k) =
C eikqFj(q)
0’=1,2)
q= -co
(2)
Here the matrix block Fj(q) describes the interactions of the zeroth (reference) cell with the qth one in the jth chain (for more details see Chapter I of ref. 1 or ref. 8). Knowing these band structures the Green matrices of the unperturbed chains can be written as
G;(z) =-& I-
o’= 1,2)
(3)
J
where z =E + iv with q> 0. The elements of Gj”(z) are defined as
=z
Gjo(drp;sv
d:,i(k)jd,,i(k)jei~‘~-~’ -n/t&j
z--Ei(k)j
dlz
(j= 1,2)
(4)
129
Here oj is the elementary translation and n? the number of filled bands in the jth chain, r and s are A0 orbital indices and p and Y cell indices, respectively. With the help of the matrices Gjo one can write down two Dyson equations for the two chains Gj=GjO+G,OVjGj
(j= 1,2)
(5)
where V1 =z[S1 -S,]-
[F, -F,]
=zAS-AF
(6)
and at the zeroth-order iteration V4”1= _V[Ol 1
(7)
The interaction matrices Vj(i= 1,2) given in eqns. (6) and (7) give the effect of chain 1 on chain 2, and vice versa, through the interface. The differences between matrices F1 and F, and between S1 and S2, respectively, determine for how many cells from the interface they have a non-zero block (in other words, the larger the chemical difference between the two interacting chains, the more non-zero blocks will the matrices Vj possess). One can start an iterative procedure using eqn. (4) and the matrices Sj and Fj of the unperturbed chains to determine the matrices Gj” and Vjol. After solving the Dyson eqns. (5) for both chains, one can compute new chargebond order matrices in the basis of the well-known relation EF~
PJ['l
=
_ai
s 2x - n/llj
ImGjol (E)dE
o’= 1,2)
(8)
Here GJ”] stands for the perturbed Green’s matrix of chain j in the zeroth iteration. Continuing this procedure until self-consistency is reached in both chains (mutually consistent field), one can obtain the interface states (both of which fall into the unperturbed bands of the two chains) as well as some other states which lie outside the bulk bands. (For further details and description of the numerical procedure see refs. 2 and 9. It is obvious that this procedure can be easily generalized for interfaces between two 2D or 3D periodic systems. The method outlined was applied first as a test calculation for the interface of an (H,), and (Li2)r chain [9], then between that of truns-polyacetylene (PA) and polytetrafluoroethylene (PTFE), as well as between polyethylene (PE) and PTFE chains [lo], between truns-PA and poly (vinylidene fluoride) ( PVDF ) , and between PE and PVDF [ 111. These pairs of organic chains were chosen to investigate the interface states between similar (PE and PTFE) and rather different chains (PA and PTFE, or PA and PVDF). In all these calculations (which converge in about 30 steps) the geometries of the individual
130
chains were kept constant until the interface (which of course is an approximation used for these first model calculations) and the two different chains were connected by a covalent chemical bond. For instance in the PA-PTFE case (which will be discussed in some detail below) we have found the interface to be
As an example let us consider the interface states PA and PTFE. Tables 1 and 2 give the band structures and interface states, respectively. TABLE 1 Lower and upper edges (in eV) of the energy bands of the alternating trans-polyacetylene and ~l~trafluor~thylene chains Band order 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Polytetrafluoroethylene
Polyacetylene Lower
Upper
Lower
Upper
- 307.099 - 307.052 - 32.473 - 25.806 - 19.548 - 17.460 - 14.487 - 2.423
- 307.056 - 307.016 - 26.467 -21.256 - 17.631 - 14.497 -4.171F 4.813
- 716.060 - 716.060 - 716.054 - 716.050 -318.664 - 318.656 - 50.561 - 50.045 -43.122 -42.911 -34.137 - 30.542 - 25.205 -25.013 -23.118 - 23.065 -21.900 - 20.486 - 19.699 - 19.393 - 19,181 - 19.180 - 14.767 - 14.717 0.730
-
716.060 716.058 716.050 710.050 318.652 318.643 - 49.753 - 49.705 - 42.543 -42.540 - 30.645 - 25.497 - 24.599 - 23.459 - 22.937 - 22.036 - 20.489 - 19.740 - 19.239 - 17.388 - 17.359 - 15.131 - 14.585 - 14.585F 4.915
F denotes the Fermi level of each chain. (In this model calculation an STO-3G basis was used and for the geometries applied see ref. IO).
131 TABLE 2 Interface states (in eV) of polyacetylene and polytetrafluoroethylene solution of the PA-PTFE interface problem Order of state
Polyacetylene
Polytetrafluoroethylene
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
- 714.488 - 705.515 - 684.274 - 657.076 - 620.825 -577.311 - 528.092 - 473.990 - 416.792 -358.011 - 299.243 - 242.008 - 187.988 - 138.645 -95.221 - 59.068 -31.496 - 10.799 - 1.392* 12.090+ 24.599+
- 710.826 -700.517 - 682.842 -653.611 -618.116 -575.011 - 526.551 - 473.398 - 415.972 -357.741 - 299.449 - 243.460 - 188.877 - 140.371 -97.573 -61.791 -34.018 - 15.123 - 5.346’ 4.627+ 14.696+
chains obtained form the
*Indicates states lying above the Fermi level of the considered chain; +denotes new states obtained by using extended energy domains (in a second calculation energy values E which lie far above the Fermi level, were included in eqn. (3) ) .
From Table 2, we deduce the following properties of the interface states of PA-PTFE. The first interface states of PA lie below the lower edge of the first occupied band; states 11-16 lie in the gap between the second and third bands; state 17 occurs inside the third band, state 18 lies inside the valence band, state 19 lies in the conduction band and extra states exist above this band. The first interface state of PTFE falls inside its fourth occupied band: states 2-10 lie between bands 4 and 5, states 11-16 occur between the sixth and seventh bands; state 17 lies inside the eleventh band, state 18 lies between bands 22 and 23; state 19 occurs in the gap between the valence and conduction bands; the first extra state lies inside the conduction band and the second extra state lies far above this band. In Fig. 1 the interface states of PA and PTFE falling in the valence and conduction band regions of both chains are shown. It should be pointed out that if we had examined, instead of two interacting
ca 0
PA
PTFE
I
Fig. 1. The higher energy bands and interface states (discontinuous lines) of PA and PTFE chains.
polymer chains, a real interface between two 3D solids, a large number of extra states would occur at the interface. This would cause a so-called “bending” of the band structures at the interface of the two systems, a phenomenon which is experimentally well known in solid state physics. A MODIFIED ROMBERG ALGORITHM FOR EXTRAPOLATION
FROM OLIGOMERS
TO THE BAND STRUCTURE OF AN INFINITE PERIODIC CHAIN
The extrapolation of the results of finite-chain calculations to the infinitechain limit is an alternative way of directly performing ab initio crystal orbital calculations [8] on such chains. Such a procedure may be of importance in cases where the crystal orbital calculations diverge, but the corresponding finite ring calculations still converge. (In ab initio SCF crystal orbital calculations - according to our experience - insurmountable convergence difficulties occur much more frequently than in MO calculations.) However, even in the crystal orbital calculations there are truncation radii up to which the interaction integrals are calculated explicitly. The extrapolation of crystal orbital results obtained with a small number of interacting neighbouring cells to the exact limit is another application of extrapolation schemes in polymer calculations, which would be useful even in the standard case where the calculation converges. In a recent study [4] the usefulness of a recently-proposed modified Romberg extrapolation algorithm [ 31, applied to both finite ring and infinite-chain ab initio crystal orbital calculations of hydrogen molecules as a simple model system with realistic electron-electron and electron-nucleus interactions [ 1,2], was shown. All Hartree-Fock calculations were executed using the STO-3G minimal
133 TABLE 3 Energy results for alternating hydrogen rings (in a.u. using an STO-3G basis) [ 121
1 2 3 4 5 6 7 8 9
-0.9713517 - 1.0870394 - 1.0897328 - 1.0909935 - 1.0916524 - 1.0920329 - 1.0922712 - 1.0924300 - 1.0925411d
- 1.2027272 - 1.0413158 -1.1121363 - 1.0881830 - 1.0938101 - 1.0929155 - 1.0930176 - 1.0929544
-
1.0914453 1.0928545 1.0929909 1.0929881 1.0929932 1.0929872
4.90 2.19 1.90 1.91 1.88 1.93
(HH~*~H-H~~~H-H~~~,dn_n=l.326a.u., dn...n=2.362a.u.). “Direct cluster calculation. bOriginal Romherg algorithm. ‘Modified Romberg algorithm. dThe result of an infinite alternating Hz ring calculation, using the CO method with 9th neighbours’ interactions, was - 1.0929865.
basis [3] for the hydrogen atoms. The geometry of the alternating hydrogen chain was taken as d = 1.326 au and d = 2.362 au. Two series of calculations have been performed. In the first case the results obtained for finite rings were extrapolated to the infinite alternating hydrogen chain. The geometry of the rings was determined as follows: if n is the number of H-units in the ring then r_
-
{d: +d2, -2d,d,cos[a(n-l)/n]}“2 Bsin(x/n)
(9)
is the radius of the ring [ 131 and a=2arcsin(d,/2r)
(19)
/3=2arcsin(d,/2r)
(11)
are the two possible angles between the position vectors to adjacent hydrogen atoms. As the first test of the new model, calculations were performed for rings with n = 2N+ 1 H-units, N running from 1 to 9. The results for the total energy per H-unit are given in Table 3 under the heading EN. Let Qi,O=Ei+1 be the partial sequence of energies per unit cell with indices i=o, .... N- 1. Then the elements qi,kof the Romberg array can be calculated by the following recursion formula [3,14]
(12) where I
=hJhi_k
with hi = (i+ l)-“.
The extrapolated value is E* = qN_ l,N_ I while the exponent a is determined by minimizing the asymptotic error bound 2 ]qN__l,N_ 1- qN_ 1,N_2] [ 141. The results &*, and the corresponding values for a! are given in Table 3. For comparison we give also the results E$ which are obtained with the simple fixed exponent (cy= 1) Romberg algorithm. Similar calculations have been performed [ 31 for an infinite alternating hydrogen chain applying the ab initio HF-CO method with different neighbours’ interactions. It was found in both cases that the modified Romberg algorithm converges very well to the infinite chain values (both for the total energy per unit cell and for the widths and positions of the valence and conduction bands, respectively ) . It should be emphasized that after these first model calculations the method can be extended safely to calculate in an accurate way the band structures of different organic polymers. Note that standard ab initio CO calculations show, usually, a slowly (logarithmically) convergent behaviour of the ~~vidual band energies (a;-(k) curves). Therefore the Romberg algorithm (especially in its modified form) provides a much faster convergence than most other methods, for instance Pade approximants. TREATMENT
OF ELECTRON COR~LATION
IN DrSORDERED POLYMERS
Let us assume we have a binary disordered polymer consisting of units A and B. Here one has to determine again first the ab initio HF band structures of A and B [8] and correct it for correlation using a generalized electron polaron model (quasi-particle (QP) band structures [ 151 in the second order ~oeller-Plesset many-body perturbation theory ] 16 3 ) . One can rewrite the above method in a Green’s function form, where the ionization potentials (IP) and electron affinities (.#A) of the valence electrons are obtained as the poles of the one particle Green’s function rPi =Wi,
EAj = pi
(13)
According to the generalized Koopmans’ theorem, these poles can be identified as the corresponding QP one-electron levels in the valence and conduction bands, respectively. The w values (QP energies) can be obtained from the inverse Dyson equation [ 17 J which we write, for quasi-particle poles and in the diagonal approximation [ 18 1, as
135
&i+Mii-
(14)
(oi)=fBi
where Mii, the irreducible self-energy part (which will be evaluated to secondorder at Q), is Mii=
Vijkl(2 Vijkl-
Vijlk )
C jsocc@i+&j-&k-&~+i~ k,lEOCC
Vijkl(2Kjkl-
+c jeocc~i+&j-&k-&l-i~
Vijlk)
(15)
k&XC
where occ denotes the set of one-particle states occupied in the Hartree-Fock ground state, Vij,, the two-electron integrals and 9 is a positive infinitesimal tending to zero. Equation (14) remains valid for periodic polymers if the index i is replaced by I= (i&i), i.e. a composite index containing band and quasi-momentum. The aperiodic system will be treated as a long, but finite, chain considered as a large molecule. We obtain the quasi-particle energies of the system by solving the following generalized eigenvalue problem: (F+A)U=SUO
(16)
where F is the Fock matrix for the system, U the eigenvector matrix of F+A, S is the overlap matrix, o a diagonal matrix containing the OJivalues and A=SUMU+S
(17)
where M is a diagonal matrix containing the Mii values, i.e. the shifts of the quasi-particle levels. The consistency of eqns. (16) and (17) can be easily checked by multiplying eqn. (16) form the left by U+ (and realizing that U+SU= 1 to restore eqn. (14). The matrix F + A will be subdivided into blocks according to the elementary cells occurring in the aperiodic polymer, e.g. for the sequence A-B-A... the structure of F will be
(18) In eqn. (18) only first-neighbour interaction blocks have been kept in F as a first, and not too bad, approximation, according to our experience with matrixblock negative factor counting (NF’C) calculations. Given the blocks of F and A, a histogram of the density of o values can be calculated by the matrix-block NFC method [ 191,By this method one calculates the sign changes of D(n) as a function of L for a given grid of il values. D (A) is in our case given by D(n) =det(F+A-IS)
(19)
136 TABLE 4 Band edges for periodic and aperiodic poly (Gly,Ala) in the Hartree-Fock (QP) approximation
(HF ) and quasi-particle
VB”
CBb
Gap
Periodic
- 10.83 -9.52
3.72 3.62
14.55 13.14
Aperiodic
- 10.61 -9.36
3.71 3.61
14.32 12.97
“Upper valence band edge. bLower conduction band edge.
By adding to F (eqn. 18)) A defined by eqn. (17) (for the determination of its blocks see ref. 4) one can use the matrix block NFC method to determine the density of states (DOS) of the disordered polymer. Note that the method in its described form does not require an iterative procedure and therefore no convergence problems occur by its application. This method has been successfully applied to compute the DOS of random (Li,),(Hz)% chains, as well as to disordered truns PA-polycarbyne chains [ 201. Further it was applied to a mixed random polyglycine-polyalanine chain [21] in the P-pleated sheet conformation [22], applying Clementi’s doublezeta basis set [ 231. The necessary quasi-particle shifts Mii were taken from ref. 24. According to the results obtained (see Table 4) the introduction of correlation (both in the case of periodic and random binary chains) causes a decrease of the fundamental gap. It should be noted that a relatively smaller effect was observed in these calculations because for computation of the QP shifts, Mii not all virtual bands of polyglycine and polyalanine, respectively, were taken into account. In subsequent, more accurate, calculations the effect of correlation on the gap will be certainly substantially larger. THE EFFECT OF ELECTRIC AND TIME-DEPENDENT A LASER PULSE ON THE BAND STRUCTURE
MAGNETIC FIELDS AND OF
OF A PERIODIC POLYMER
It was usually assumed that the effect of a static electric field 2=&r&. on the band structure of a periodic polymer, if the field strength has such a strong non-zero component in the direction of the polymer axis z, that at least for larger segments of the polymer an average Z= ( z1+.zz) /2 rule is applicable cannot be calculated because the potential V= 1e 1l&z is unbounded [ 25,261. This assumption also causes rather large difficulties in the calculation of the elements of the polarizability and hyperpolarizability tenso:: Though it is certainly true that the potential V= 1e 1Er can be unbounded,
137
in ref. 5 the author was able to show that any matrix element of V (if the periodicity of the polymer is not destroyed) is finite if one uses a Gaussian basis set or any other fast-decaying basis. 0:
I&slZ
Pm
z--r00
if
(26)
Where x: is any kind of Gaussian centred on atom A (re A) in the reference cell and xi is centred on atom B (seB ) in the qth cell. This derivation makes it possible to perform CO calculations in infinite periodic chains in the presence of a static field and to determine the static polarizabilities and hyperpolarizabilities of different order (for details see ref. 5). In another paper [ 61 using a perturbational-variational method, the effect of an oscillatory time-dependent magnetic field with a vector potential A($)
=A’(r3e-‘“t+~(r3ei”tlf=curl
A’
(21)
on the band structure of a periodic polymer was formulated. For this eurppO,e, in the+Hamiltonian, the momentum P has to be substituted by P-P (P(e/c)A ), the many-electron wave function can be put in the form @(r’,t) =e-iWo”An$i(j?,t)
(22)
i
where 3i(ct) =$i(a+A#i
(?)e-iw”+A$F
(r)eiot
(23)
and IV0is the field-free (unperturbed) total energy of the polymer. Computing the stationary value of the functional
(24) (Frenkel’s variational functional [ 271) , one can derive coupled HF equations for the determination of A@ - and A$ + in the form (fn-Ei+W)
]A@? ) +h’]#i)
1 -P,_, r12
+
-I
‘-r:w2 @j>}l@i)=O 49 >+
Here
(26) #i is the MO (CO) of the unperturbed problem and the operator h T is defined as follows
(27) (Assuming that the magnetic field is not very strong, the term =:A 2 has been neglected in the derivation.) Introducing an LCAO ansatz for A& and A@ and taking into account the periodicity of the polymer, one ends up with the matrix equation (system of inhomogeneous equations) A-(k) ( B(k)
B(k) A+(k)
) (F- (k))=($ F+(k)
g)
(28)
Here the matrices and vectors are defined as follows A’ (12)=,=tNA’ B(k)=
5
(q)eik4”
(29a)
B(q)eikq”
(29b)
q=-N
[B(q)li,s,j,t
(SOa)
IB(q)li,~,~,~=(~~(r~)x9(r2)~l-~~~~i(r,)m,(r,))
@Ob)
[A’
(q)
l+,j,t= txk’lb -G~~IxII)+
with P 1-2 being the exchange operator. Further FT =U+CT ,
G’=U+DT
(31)
(the unitary matrix U block-diagonalizes the original cyclic hypermatrices A * and B of the problem (see Chapter I of ref. 1) ). D’ (k) =,=tND’
(q)eikqa
(32)
with elements
[Dr(q)li,,=(xs”I-~Tl~i)
(33)
The solution of the hypermatrix equation (28) provides than the unknown vectors C-i(k) and C z (k), that is the LCAO coefficients in the corrections A@, and A$+. *Finally we can write down now the CO of a polymer in the presence of a time-dependent magnetic field with frequency o as
(34)
139
If one wishes to treat the interaction of a laser pulse of frequency no and Gaussian shape in time, one can apply a very similar formalism [ 71. Only now the band structure will be simultaneously dependent both on $($t)=&(?)exp(--‘t2) (Cr2=4/T2whereZ’/2isthetimeduringwhichthe electric field strength changes from its peak value & (3 to (%&/e) ) and on the magnetic vector potential A (?&CO). One should point out here that in the case of a laser pulse two different times t’ and t, respectively, occur in the equations: t’ occurs in the frequency-dependent factors of eqn. (37) in analogy with the case of the time-dependent magnetic field (see eqn. (34)) while t is characteristic of the Gaussian shape of the laser pulse, i.e. the factor exp ( - a!2t2) in eqn. (37) ) . Further in this case one has to take into account, according to classical electrodynamics, the relation
I?=-grad@-;
1 a&ttl,o) .&
(35)
which leads to the expression for the electric potential ,~lEo(r)+~Im[A(r)e’O”])di
V(r)=j(
(36)
Finally since a laser pulse has a large intensity, the quadratic term in A 2 in the operator (P- (e/c)A)2 cannot be neglected. If one performs the variational calculation applying again Frenkel’s principle [ 271, one ends up with coupled HF equations similar to eqn. 25. All the terms in (25) will remain, only some extra terms are present due to eqn. (36) and to the term A2 (for details see ref. 7). Introducing again an LCAO ansatz and taking into account the periodic symmetry of the polymer, one obtains formally a very similar hypermatrix equation to (28)) only the definitions of A’ (k) and of the vectors D’ (k) will change due to the extra terms and the exp( -u2t2) time dependence of the laser pulse. One can write down again the CO in a similar form to eqn. (34), only the CO & will now be dependent on more variables 1
E 2 eikqa{ci,r(k) @ = (2N+ 1)1’29= --N ,.=I
(37)
+ 5 e-“2t2[c~~,,,,[k,E,(r)]e-i”“t’+cit,,[k,E,(r)]ei”“t’]X:!(r) n=l
In eqn. (37) nob means that, besides the basic frequency, o, the overtones nw have also been taken into 2ccount. Knowing the CO @i(kyEoo,t,t’,T) one can express the induced dipole moment as
140
(38) Using a series expansion in j!Zfor pd
(39) one can finally express the elements of the frequency and time-dependent polarizability and hyperpolarizability tensors in a standard way. For instance
d2pjnd(Eo,o,t’,t) rBi,j,k(Wt)
=
aEjaEk
>E=O
(40)
After the completion of the necessary (by no means simple) program packages, it will be possible to compute the elements of the dynamic different-order polarizability tensors. This will give an important contribution to the understanding of various non-linear optical phenomena which are of great practical importance. Note that until now only second-order perturbation theory was applied to this problem [ 281 which is certainly not satisfactory in the case of strong (laser-pulse) electromagnetic fields. CONCLUSION
One can see from the foregoing discussion of different problems of polymer theory, that the theory is nearly complete (at least for quasi-1D systems) and the next task will be to develop such highly efficient program packages (both fully vectorized and parallelized) which make it possible to compute rapidly the HF and QP band structures both of periodic and disordered polymers, even those containing larger units. These programs also have to take into account interactions between different chains and the effect of the environment. After the development of such highly efficient programs, one will be in the position to calculate quite accurately different ground and excited state properties of polymers simultaneously (of which no parametrized semi-empirical method is capable, nor can it be achieved with the help of any model Hamiltonian). In this way one will be able to design different polymers with prescribed 3-4 properties (taylor-made polymers). Since one can change the electronic structure (and thus the various properties) of an organic polymer more easily and on a much larger scale, than for inorganic crystals, the achievement of this goal will be of great practical importance in different fields (computer technology, communications, space research, non-linear optical devices, new polymer-based construction elements, etc. and in the case of biopolymers a better understanding of their functions).
141 ACKNOWLEDGEMENTS
The author expresses his gratitude to Professor I.G. Csizmadia for inviting him to present a keynote lecture at the II. WATOC World Congress, which has given him the occasion to write this short review. He is further indebted to his former and present colleagues, Professors L. Dalton, F. Martino, P. Otto and M. Seel, Priv.-Doz.-s C.-M. Liegener and S. Suhai, Drs. M.A. Abdel-Raouf, A.K. Bakhshi, R. Day and W. Fijrner, as well as to Dipl.-Phys. A. Sutjianto for their very fruitful cooperation and many highly useful discussions. He is further indebted to the Deutsche Forschungsgemeinschaft, to the National Foundation for Cancer Research, for the Deutsche Akademische Austauschdienst, to Siemens A.G., to the Chemistry Department of the University of Southern California at Los Angeles and to the Fonds der Chemischen Industrie for substantial financial support.
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2 3
9 10 11 12 13 14 15 16 17
J.J. Ladik, Quantum Theory of Polymers as Solids, Plenum Press, New York-London, 1988. J. Ladik, Prog. Surf. Sci. 26 (1987) 135. F. Beleznay, J. Phys., A19 (1986) 551 and references therein; C.-M. Liegener, F. Beleznay and J. Ladik, Phys. Lett., Al23 (1987) 399, for extrapolations with Pad6 approximants see J. Cioskowski, Theor. Chim. Acta, 77 (1990) 253. C.-M. Liegener, Chem. Phys., 133 (1989) 173. J. Ladik, J. Mol. Struct. (Theochem), 206 (1990) 39. J. Ladik, J. Mol. Struct. (Theochem), 199 (1989) 55. J.J. Ladik and L.R. Dalton, J. Mol. Struct. (Theochem), in press. G. Del Re, J. Ladik and G. Biczo, Phys. Rev., 155 (1967) 997; J.-M. Andre, L. Gouverneur and G. Leory, Int. J. Quantum Chem., 1 (1967) 427,451; J. Ladik, in J.-M. Andre and J. Ladik (Eds.) Electronic Structure of Polymers and Molecular Crystals, Plenum Press, New York, 1975, p. 23. M.A. Abdel-Raouf, P. Otto, J. Ladik and M. Seel, Phys. Rev. B, 40 (1989) 1450. M.A. Abdel-Raouf and J. Ladik, Chem. Phys. Lett., 156 (1989) 296. M.A. Abdel-Raouf and J. Ladik, Solid State Commun., 71 (1989) 1019. W.J. Hehre, R.F. Stewart and J.A. Pople, J. Chem. Phys., 51 (1969) 2657. R. Day, J. Ladik and F. Martino, Chem. Phys. Lett., A107, (1989) 79. W. Romberg, Det. K. Nov. Vidensk. Selsk. Fork., 28 (1955) 7. S. Suhai, Phys. Rev. B, 27 (1983) 3506; C.-M. Liegener and J. Ladik, Phys. Rev., B35 (1987) 6403; see also Chapter V of ref. 1. C. Moeller and S. Plesset, Phys. Rev., 46 (1939) 618. L.S. Cederbaum, W. Domcke, J. Schirmer and W. von Niessen, Adv. Chem. Phys., 65 (1986) 115.
18 19 20 21 22
L.S. Cederbaum, Theor. Chim. Acta, 31 (1973) 239. R.S. Day and F. Martino, Chem. Phys. Lett., 84 (1981) 86; J. Ladik, M. Seel, P. Otto and A.K. Bakhshi, Chem. Phys., 108 (1986) 215. C.-M. Liegener, A.K. Bakhshi, A. Sutjianto and P. Otto, Chem. Phys., 133 (1989) 177. C.-M. Liegener, A. Sutjianto and J. Ladik, Chem. Phys., accepted. L. Pauling and R.B. Corey, Proc. Natl. Acad. Sci., U.S.A. 39 (1953) 253.
142 23 24 25 26 27 28
L. Gianolio and E. Clementi, Gazz. Chim. Ital., 110 (1980) 179. C.-M. Liegener, A.K. Bakhshi, P. Otto and J. Ladik, J. Mol. Struct. (Theochem), 188 (1989) 205. See ref. 1 p. 371 and references therein. V.P. Bodart, J. Delhalle, J.-M. Andre and J. Zyss, Can. J. Chem., 63 (1985) 1631; J. DelhaIIe, V.P. Bodart, M. Dory, J.-M. Andre and J. Zyss, Int. J. Quantum Chem., 519 (1986) 313. J. Frenkel, Wave Mechanics, Advanced General Theory, Dover, New York, 1950. J.F. Ward, Rev. Mod. Phys., 37 (1965) 1.
Journal of Molecular Structure (Theo&em), 230 (1991) 127-142 Elsevier Science Publishers B.V., Amsterdam
RECENT DEVELOPMENTS ORGANIC POLYMERS
IN THE QUANTUM THEORY OF
JANOS J. LADIK Chair of Theoretical Chemistry, Friedrich-Alexander-University,
Erlangen-Niirnberg
(FRG)
(First received 8 August 1990; in final form 12 October 1990)
ABSTRACT A new theoretical method and its applications will be presented to calculate the interface between two different periodic polymers. Further theoretical developments in extrapolation methods to find out the band structure of an infinite chain from its oligomers, as well as the treatment of correlation in disordered chains, will be discussed. Finally, theoretical methods to treat the interaction of a stationary and non-stationary magnetic and electric field (including a laser pulse) will be shown.
INTRODUCTION
The basic quantum theory of polymers treated as solids is summarized in a book which the present author published in 1988 [ 11. In this monograph the Hartree-Fock crystal orbital (CO) theory is described in detail, both in its ab initio form and in its different semi-empirical forms, together with applications for a larger number of polymers. Theoretical methods for the treatment of disorder, with applications, as well as the correction of the band structure for correlation effects are also described. The interaction between polymer chains and the effect of the environment (which is especially important for biopolymers) on the band structure are then discussed. In the second part of the monograph, the applications of the basic theory for different properties, such as U.V. (using the intermediate exciton theory, starting from correlated band structures), vibrational and photoelectron spectra, transport (both Bloch type and hopping conduction) and mechanical properties are summarized. In one of the last chapters the effects of a static magnetic field on the band structure and the approximations usually applied for polarizabilities and hyperpolarizabilities are described. The aim of the present short review paper is to summarize the recent theoretical developments achieved since the appearance of the above mentioned monograph and not to present basically new results (with the exception of the case of the interaction of a periodic polymer with a laser pulse, where the effect
0166-1280/91/$03.50
0 1991-
Elsevier Science Publishers B.V.
128
of the overtones of the electromagnetic field is also taken into account). First of all a new method to describe the interfaces between two different polymeric chains [ 21 and their applications will be presented. The next step is the application of a modified Romberg algorithm [ 31 to extrapolate from the electronic structure of oligomers to the band structure of an infinite chain (which converges faster than other extrapolation methods, for instance Pad6 approximants [3]). An important new theoretical development is the treatment of correlation in the case of disordered polymers [ 41. The efficiency of this new method will be illustrated by several applications. Essential progress has been achieved in treating the effects of static electric fields [5] and time-dependent magnetic fields [6] on the band structure of polymers. Finally, a general theory for the interaction of a laser pulse with a periodic polymer [ 71 will be sketched. This new theory can provide not only static polarizabilities and hyperpolarizabilities, as can the method described in ref. 5, but also frequency and time-dependent ones [ 71. It should be mentioned that all the methods described here are formulated for quasi-one-dimensional (1D) chains, but their generalization to 2D and 3D systems is obviously straightforward. THE INTERFACE BETWEEN TWO DIFFERENT POLYMER CHAINS
To treat the interface between two polymer chains it is necessary first to solve the Hartree-Fock problem F,(b)di(k),
(la)
=ci(k)iSi(k)di(k)i
(lb)
F,(k)di(k)P=&i(k)zs,(k)di(k), where Fj(k) =
C eikqFj(q)
0’=1,2)
q= -co
(2)
Here the matrix block Fj(q) describes the interactions of the zeroth (reference) cell with the qth one in the jth chain (for more details see Chapter I of ref. 1 or ref. 8). Knowing these band structures the Green matrices of the unperturbed chains can be written as
G;(z) =-& I-
o’= 1,2)
(3)
J
where z =E + iv with q> 0. The elements of Gj”(z) are defined as
=z
Gjo(drp;sv
d:,i(k)jd,,i(k)jei~‘~-~’ -n/t&j
z--Ei(k)j
dlz
(j= 1,2)
(4)
129
Here oj is the elementary translation and n? the number of filled bands in the jth chain, r and s are A0 orbital indices and p and Y cell indices, respectively. With the help of the matrices Gjo one can write down two Dyson equations for the two chains Gj=GjO+G,OVjGj
(j= 1,2)
(5)
where V1 =z[S1 -S,]-
[F, -F,]
=zAS-AF
(6)
and at the zeroth-order iteration V4”1= _V[Ol 1
(7)
The interaction matrices Vj(i= 1,2) given in eqns. (6) and (7) give the effect of chain 1 on chain 2, and vice versa, through the interface. The differences between matrices F1 and F, and between S1 and S2, respectively, determine for how many cells from the interface they have a non-zero block (in other words, the larger the chemical difference between the two interacting chains, the more non-zero blocks will the matrices Vj possess). One can start an iterative procedure using eqn. (4) and the matrices Sj and Fj of the unperturbed chains to determine the matrices Gj” and Vjol. After solving the Dyson eqns. (5) for both chains, one can compute new chargebond order matrices in the basis of the well-known relation EF~
PJ['l
=
_ai
s 2x - n/llj
ImGjol (E)dE
o’= 1,2)
(8)
Here GJ”] stands for the perturbed Green’s matrix of chain j in the zeroth iteration. Continuing this procedure until self-consistency is reached in both chains (mutually consistent field), one can obtain the interface states (both of which fall into the unperturbed bands of the two chains) as well as some other states which lie outside the bulk bands. (For further details and description of the numerical procedure see refs. 2 and 9. It is obvious that this procedure can be easily generalized for interfaces between two 2D or 3D periodic systems. The method outlined was applied first as a test calculation for the interface of an (H,), and (Li2)r chain [9], then between that of truns-polyacetylene (PA) and polytetrafluoroethylene (PTFE), as well as between polyethylene (PE) and PTFE chains [lo], between truns-PA and poly (vinylidene fluoride) ( PVDF ) , and between PE and PVDF [ 111. These pairs of organic chains were chosen to investigate the interface states between similar (PE and PTFE) and rather different chains (PA and PTFE, or PA and PVDF). In all these calculations (which converge in about 30 steps) the geometries of the individual
130
chains were kept constant until the interface (which of course is an approximation used for these first model calculations) and the two different chains were connected by a covalent chemical bond. For instance in the PA-PTFE case (which will be discussed in some detail below) we have found the interface to be
As an example let us consider the interface states PA and PTFE. Tables 1 and 2 give the band structures and interface states, respectively. TABLE 1 Lower and upper edges (in eV) of the energy bands of the alternating trans-polyacetylene and ~l~trafluor~thylene chains Band order 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Polytetrafluoroethylene
Polyacetylene Lower
Upper
Lower
Upper
- 307.099 - 307.052 - 32.473 - 25.806 - 19.548 - 17.460 - 14.487 - 2.423
- 307.056 - 307.016 - 26.467 -21.256 - 17.631 - 14.497 -4.171F 4.813
- 716.060 - 716.060 - 716.054 - 716.050 -318.664 - 318.656 - 50.561 - 50.045 -43.122 -42.911 -34.137 - 30.542 - 25.205 -25.013 -23.118 - 23.065 -21.900 - 20.486 - 19.699 - 19.393 - 19,181 - 19.180 - 14.767 - 14.717 0.730
-
716.060 716.058 716.050 710.050 318.652 318.643 - 49.753 - 49.705 - 42.543 -42.540 - 30.645 - 25.497 - 24.599 - 23.459 - 22.937 - 22.036 - 20.489 - 19.740 - 19.239 - 17.388 - 17.359 - 15.131 - 14.585 - 14.585F 4.915
F denotes the Fermi level of each chain. (In this model calculation an STO-3G basis was used and for the geometries applied see ref. IO).
131 TABLE 2 Interface states (in eV) of polyacetylene and polytetrafluoroethylene solution of the PA-PTFE interface problem Order of state
Polyacetylene
Polytetrafluoroethylene
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
- 714.488 - 705.515 - 684.274 - 657.076 - 620.825 -577.311 - 528.092 - 473.990 - 416.792 -358.011 - 299.243 - 242.008 - 187.988 - 138.645 -95.221 - 59.068 -31.496 - 10.799 - 1.392* 12.090+ 24.599+
- 710.826 -700.517 - 682.842 -653.611 -618.116 -575.011 - 526.551 - 473.398 - 415.972 -357.741 - 299.449 - 243.460 - 188.877 - 140.371 -97.573 -61.791 -34.018 - 15.123 - 5.346’ 4.627+ 14.696+
chains obtained form the
*Indicates states lying above the Fermi level of the considered chain; +denotes new states obtained by using extended energy domains (in a second calculation energy values E which lie far above the Fermi level, were included in eqn. (3) ) .
From Table 2, we deduce the following properties of the interface states of PA-PTFE. The first interface states of PA lie below the lower edge of the first occupied band; states 11-16 lie in the gap between the second and third bands; state 17 occurs inside the third band, state 18 lies inside the valence band, state 19 lies in the conduction band and extra states exist above this band. The first interface state of PTFE falls inside its fourth occupied band: states 2-10 lie between bands 4 and 5, states 11-16 occur between the sixth and seventh bands; state 17 lies inside the eleventh band, state 18 lies between bands 22 and 23; state 19 occurs in the gap between the valence and conduction bands; the first extra state lies inside the conduction band and the second extra state lies far above this band. In Fig. 1 the interface states of PA and PTFE falling in the valence and conduction band regions of both chains are shown. It should be pointed out that if we had examined, instead of two interacting
ca 0
PA
PTFE
I
Fig. 1. The higher energy bands and interface states (discontinuous lines) of PA and PTFE chains.
polymer chains, a real interface between two 3D solids, a large number of extra states would occur at the interface. This would cause a so-called “bending” of the band structures at the interface of the two systems, a phenomenon which is experimentally well known in solid state physics. A MODIFIED ROMBERG ALGORITHM FOR EXTRAPOLATION
FROM OLIGOMERS
TO THE BAND STRUCTURE OF AN INFINITE PERIODIC CHAIN
The extrapolation of the results of finite-chain calculations to the infinitechain limit is an alternative way of directly performing ab initio crystal orbital calculations [8] on such chains. Such a procedure may be of importance in cases where the crystal orbital calculations diverge, but the corresponding finite ring calculations still converge. (In ab initio SCF crystal orbital calculations - according to our experience - insurmountable convergence difficulties occur much more frequently than in MO calculations.) However, even in the crystal orbital calculations there are truncation radii up to which the interaction integrals are calculated explicitly. The extrapolation of crystal orbital results obtained with a small number of interacting neighbouring cells to the exact limit is another application of extrapolation schemes in polymer calculations, which would be useful even in the standard case where the calculation converges. In a recent study [4] the usefulness of a recently-proposed modified Romberg extrapolation algorithm [ 31, applied to both finite ring and infinite-chain ab initio crystal orbital calculations of hydrogen molecules as a simple model system with realistic electron-electron and electron-nucleus interactions [ 1,2], was shown. All Hartree-Fock calculations were executed using the STO-3G minimal
133 TABLE 3 Energy results for alternating hydrogen rings (in a.u. using an STO-3G basis) [ 121
1 2 3 4 5 6 7 8 9
-0.9713517 - 1.0870394 - 1.0897328 - 1.0909935 - 1.0916524 - 1.0920329 - 1.0922712 - 1.0924300 - 1.0925411d
- 1.2027272 - 1.0413158 -1.1121363 - 1.0881830 - 1.0938101 - 1.0929155 - 1.0930176 - 1.0929544
-
1.0914453 1.0928545 1.0929909 1.0929881 1.0929932 1.0929872
4.90 2.19 1.90 1.91 1.88 1.93
(HH~*~H-H~~~H-H~~~,dn_n=l.326a.u., dn...n=2.362a.u.). “Direct cluster calculation. bOriginal Romherg algorithm. ‘Modified Romberg algorithm. dThe result of an infinite alternating Hz ring calculation, using the CO method with 9th neighbours’ interactions, was - 1.0929865.
basis [3] for the hydrogen atoms. The geometry of the alternating hydrogen chain was taken as d = 1.326 au and d = 2.362 au. Two series of calculations have been performed. In the first case the results obtained for finite rings were extrapolated to the infinite alternating hydrogen chain. The geometry of the rings was determined as follows: if n is the number of H-units in the ring then r_
-
{d: +d2, -2d,d,cos[a(n-l)/n]}“2 Bsin(x/n)
(9)
is the radius of the ring [ 131 and a=2arcsin(d,/2r)
(19)
/3=2arcsin(d,/2r)
(11)
are the two possible angles between the position vectors to adjacent hydrogen atoms. As the first test of the new model, calculations were performed for rings with n = 2N+ 1 H-units, N running from 1 to 9. The results for the total energy per H-unit are given in Table 3 under the heading EN. Let Qi,O=Ei+1 be the partial sequence of energies per unit cell with indices i=o, .... N- 1. Then the elements qi,kof the Romberg array can be calculated by the following recursion formula [3,14]
(12) where I
=hJhi_k
with hi = (i+ l)-“.
The extrapolated value is E* = qN_ l,N_ I while the exponent a is determined by minimizing the asymptotic error bound 2 ]qN__l,N_ 1- qN_ 1,N_2] [ 141. The results &*, and the corresponding values for a! are given in Table 3. For comparison we give also the results E$ which are obtained with the simple fixed exponent (cy= 1) Romberg algorithm. Similar calculations have been performed [ 31 for an infinite alternating hydrogen chain applying the ab initio HF-CO method with different neighbours’ interactions. It was found in both cases that the modified Romberg algorithm converges very well to the infinite chain values (both for the total energy per unit cell and for the widths and positions of the valence and conduction bands, respectively ) . It should be emphasized that after these first model calculations the method can be extended safely to calculate in an accurate way the band structures of different organic polymers. Note that standard ab initio CO calculations show, usually, a slowly (logarithmically) convergent behaviour of the ~~vidual band energies (a;-(k) curves). Therefore the Romberg algorithm (especially in its modified form) provides a much faster convergence than most other methods, for instance Pade approximants. TREATMENT
OF ELECTRON COR~LATION
IN DrSORDERED POLYMERS
Let us assume we have a binary disordered polymer consisting of units A and B. Here one has to determine again first the ab initio HF band structures of A and B [8] and correct it for correlation using a generalized electron polaron model (quasi-particle (QP) band structures [ 151 in the second order ~oeller-Plesset many-body perturbation theory ] 16 3 ) . One can rewrite the above method in a Green’s function form, where the ionization potentials (IP) and electron affinities (.#A) of the valence electrons are obtained as the poles of the one particle Green’s function rPi =Wi,
EAj = pi
(13)
According to the generalized Koopmans’ theorem, these poles can be identified as the corresponding QP one-electron levels in the valence and conduction bands, respectively. The w values (QP energies) can be obtained from the inverse Dyson equation [ 17 J which we write, for quasi-particle poles and in the diagonal approximation [ 18 1, as
135
&i+Mii-
(14)
(oi)=fBi
where Mii, the irreducible self-energy part (which will be evaluated to secondorder at Q), is Mii=
Vijkl(2 Vijkl-
Vijlk )
C jsocc@i+&j-&k-&~+i~ k,lEOCC
Vijkl(2Kjkl-
+c jeocc~i+&j-&k-&l-i~
Vijlk)
(15)
k&XC
where occ denotes the set of one-particle states occupied in the Hartree-Fock ground state, Vij,, the two-electron integrals and 9 is a positive infinitesimal tending to zero. Equation (14) remains valid for periodic polymers if the index i is replaced by I= (i&i), i.e. a composite index containing band and quasi-momentum. The aperiodic system will be treated as a long, but finite, chain considered as a large molecule. We obtain the quasi-particle energies of the system by solving the following generalized eigenvalue problem: (F+A)U=SUO
(16)
where F is the Fock matrix for the system, U the eigenvector matrix of F+A, S is the overlap matrix, o a diagonal matrix containing the OJivalues and A=SUMU+S
(17)
where M is a diagonal matrix containing the Mii values, i.e. the shifts of the quasi-particle levels. The consistency of eqns. (16) and (17) can be easily checked by multiplying eqn. (16) form the left by U+ (and realizing that U+SU= 1 to restore eqn. (14). The matrix F + A will be subdivided into blocks according to the elementary cells occurring in the aperiodic polymer, e.g. for the sequence A-B-A... the structure of F will be
(18) In eqn. (18) only first-neighbour interaction blocks have been kept in F as a first, and not too bad, approximation, according to our experience with matrixblock negative factor counting (NF’C) calculations. Given the blocks of F and A, a histogram of the density of o values can be calculated by the matrix-block NFC method [ 191,By this method one calculates the sign changes of D(n) as a function of L for a given grid of il values. D (A) is in our case given by D(n) =det(F+A-IS)
(19)
136 TABLE 4 Band edges for periodic and aperiodic poly (Gly,Ala) in the Hartree-Fock (QP) approximation
(HF ) and quasi-particle
VB”
CBb
Gap
Periodic
- 10.83 -9.52
3.72 3.62
14.55 13.14
Aperiodic
- 10.61 -9.36
3.71 3.61
14.32 12.97
“Upper valence band edge. bLower conduction band edge.
By adding to F (eqn. 18)) A defined by eqn. (17) (for the determination of its blocks see ref. 4) one can use the matrix block NFC method to determine the density of states (DOS) of the disordered polymer. Note that the method in its described form does not require an iterative procedure and therefore no convergence problems occur by its application. This method has been successfully applied to compute the DOS of random (Li,),(Hz)% chains, as well as to disordered truns PA-polycarbyne chains [ 201. Further it was applied to a mixed random polyglycine-polyalanine chain [21] in the P-pleated sheet conformation [22], applying Clementi’s doublezeta basis set [ 231. The necessary quasi-particle shifts Mii were taken from ref. 24. According to the results obtained (see Table 4) the introduction of correlation (both in the case of periodic and random binary chains) causes a decrease of the fundamental gap. It should be noted that a relatively smaller effect was observed in these calculations because for computation of the QP shifts, Mii not all virtual bands of polyglycine and polyalanine, respectively, were taken into account. In subsequent, more accurate, calculations the effect of correlation on the gap will be certainly substantially larger. THE EFFECT OF ELECTRIC AND TIME-DEPENDENT A LASER PULSE ON THE BAND STRUCTURE
MAGNETIC FIELDS AND OF
OF A PERIODIC POLYMER
It was usually assumed that the effect of a static electric field 2=&r&. on the band structure of a periodic polymer, if the field strength has such a strong non-zero component in the direction of the polymer axis z, that at least for larger segments of the polymer an average Z= ( z1+.zz) /2 rule is applicable cannot be calculated because the potential V= 1e 1l&z is unbounded [ 25,261. This assumption also causes rather large difficulties in the calculation of the elements of the polarizability and hyperpolarizability tenso:: Though it is certainly true that the potential V= 1e 1Er can be unbounded,
137
in ref. 5 the author was able to show that any matrix element of V (if the periodicity of the polymer is not destroyed) is finite if one uses a Gaussian basis set or any other fast-decaying basis. 0:
I&slZ
Pm
z--r00
if
(26)
Where x: is any kind of Gaussian centred on atom A (re A) in the reference cell and xi is centred on atom B (seB ) in the qth cell. This derivation makes it possible to perform CO calculations in infinite periodic chains in the presence of a static field and to determine the static polarizabilities and hyperpolarizabilities of different order (for details see ref. 5). In another paper [ 61 using a perturbational-variational method, the effect of an oscillatory time-dependent magnetic field with a vector potential A($)
=A’(r3e-‘“t+~(r3ei”tlf=curl
A’
(21)
on the band structure of a periodic polymer was formulated. For this eurppO,e, in the+Hamiltonian, the momentum P has to be substituted by P-P (P(e/c)A ), the many-electron wave function can be put in the form @(r’,t) =e-iWo”An$i(j?,t)
(22)
i
where 3i(ct) =$i(a+A#i
(?)e-iw”+A$F
(r)eiot
(23)
and IV0is the field-free (unperturbed) total energy of the polymer. Computing the stationary value of the functional
(24) (Frenkel’s variational functional [ 271) , one can derive coupled HF equations for the determination of A@ - and A$ + in the form (fn-Ei+W)
]A@? ) +h’]#i)
1 -P,_, r12
+
-I
‘-r:w2 @j>}l@i)=O 49 >+
Here
(26) #i is the MO (CO) of the unperturbed problem and the operator h T is defined as follows
(27) (Assuming that the magnetic field is not very strong, the term =:A 2 has been neglected in the derivation.) Introducing an LCAO ansatz for A& and A@ and taking into account the periodicity of the polymer, one ends up with the matrix equation (system of inhomogeneous equations) A-(k) ( B(k)
B(k) A+(k)
) (F- (k))=($ F+(k)
g)
(28)
Here the matrices and vectors are defined as follows A’ (12)=,=tNA’ B(k)=
5
(q)eik4”
(29a)
B(q)eikq”
(29b)
q=-N
[B(q)li,s,j,t
(SOa)
IB(q)li,~,~,~=(~~(r~)x9(r2)~l-~~~~i(r,)m,(r,))
@Ob)
[A’
(q)
l+,j,t= txk’lb -G~~IxII)+
with P 1-2 being the exchange operator. Further FT =U+CT ,
G’=U+DT
(31)
(the unitary matrix U block-diagonalizes the original cyclic hypermatrices A * and B of the problem (see Chapter I of ref. 1) ). D’ (k) =,=tND’
(q)eikqa
(32)
with elements
[Dr(q)li,,=(xs”I-~Tl~i)
(33)
The solution of the hypermatrix equation (28) provides than the unknown vectors C-i(k) and C z (k), that is the LCAO coefficients in the corrections A@, and A$+. *Finally we can write down now the CO of a polymer in the presence of a time-dependent magnetic field with frequency o as
(34)
139
If one wishes to treat the interaction of a laser pulse of frequency no and Gaussian shape in time, one can apply a very similar formalism [ 71. Only now the band structure will be simultaneously dependent both on $($t)=&(?)exp(--‘t2) (Cr2=4/T2whereZ’/2isthetimeduringwhichthe electric field strength changes from its peak value & (3 to (%&/e) ) and on the magnetic vector potential A (?&CO). One should point out here that in the case of a laser pulse two different times t’ and t, respectively, occur in the equations: t’ occurs in the frequency-dependent factors of eqn. (37) in analogy with the case of the time-dependent magnetic field (see eqn. (34)) while t is characteristic of the Gaussian shape of the laser pulse, i.e. the factor exp ( - a!2t2) in eqn. (37) ) . Further in this case one has to take into account, according to classical electrodynamics, the relation
I?=-grad@-;
1 a&ttl,o) .&
(35)
which leads to the expression for the electric potential ,~lEo(r)+~Im[A(r)e’O”])di
V(r)=j(
(36)
Finally since a laser pulse has a large intensity, the quadratic term in A 2 in the operator (P- (e/c)A)2 cannot be neglected. If one performs the variational calculation applying again Frenkel’s principle [ 271, one ends up with coupled HF equations similar to eqn. 25. All the terms in (25) will remain, only some extra terms are present due to eqn. (36) and to the term A2 (for details see ref. 7). Introducing again an LCAO ansatz and taking into account the periodic symmetry of the polymer, one obtains formally a very similar hypermatrix equation to (28)) only the definitions of A’ (k) and of the vectors D’ (k) will change due to the extra terms and the exp( -u2t2) time dependence of the laser pulse. One can write down again the CO in a similar form to eqn. (34), only the CO & will now be dependent on more variables 1
E 2 eikqa{ci,r(k) @ = (2N+ 1)1’29= --N ,.=I
(37)
+ 5 e-“2t2[c~~,,,,[k,E,(r)]e-i”“t’+cit,,[k,E,(r)]ei”“t’]X:!(r) n=l
In eqn. (37) nob means that, besides the basic frequency, o, the overtones nw have also been taken into 2ccount. Knowing the CO @i(kyEoo,t,t’,T) one can express the induced dipole moment as
140
(38) Using a series expansion in j!Zfor pd
(39) one can finally express the elements of the frequency and time-dependent polarizability and hyperpolarizability tensors in a standard way. For instance
d2pjnd(Eo,o,t’,t) rBi,j,k(Wt)
=
aEjaEk
>E=O
(40)
After the completion of the necessary (by no means simple) program packages, it will be possible to compute the elements of the dynamic different-order polarizability tensors. This will give an important contribution to the understanding of various non-linear optical phenomena which are of great practical importance. Note that until now only second-order perturbation theory was applied to this problem [ 281 which is certainly not satisfactory in the case of strong (laser-pulse) electromagnetic fields. CONCLUSION
One can see from the foregoing discussion of different problems of polymer theory, that the theory is nearly complete (at least for quasi-1D systems) and the next task will be to develop such highly efficient program packages (both fully vectorized and parallelized) which make it possible to compute rapidly the HF and QP band structures both of periodic and disordered polymers, even those containing larger units. These programs also have to take into account interactions between different chains and the effect of the environment. After the development of such highly efficient programs, one will be in the position to calculate quite accurately different ground and excited state properties of polymers simultaneously (of which no parametrized semi-empirical method is capable, nor can it be achieved with the help of any model Hamiltonian). In this way one will be able to design different polymers with prescribed 3-4 properties (taylor-made polymers). Since one can change the electronic structure (and thus the various properties) of an organic polymer more easily and on a much larger scale, than for inorganic crystals, the achievement of this goal will be of great practical importance in different fields (computer technology, communications, space research, non-linear optical devices, new polymer-based construction elements, etc. and in the case of biopolymers a better understanding of their functions).
141 ACKNOWLEDGEMENTS
The author expresses his gratitude to Professor I.G. Csizmadia for inviting him to present a keynote lecture at the II. WATOC World Congress, which has given him the occasion to write this short review. He is further indebted to his former and present colleagues, Professors L. Dalton, F. Martino, P. Otto and M. Seel, Priv.-Doz.-s C.-M. Liegener and S. Suhai, Drs. M.A. Abdel-Raouf, A.K. Bakhshi, R. Day and W. Fijrner, as well as to Dipl.-Phys. A. Sutjianto for their very fruitful cooperation and many highly useful discussions. He is further indebted to the Deutsche Forschungsgemeinschaft, to the National Foundation for Cancer Research, for the Deutsche Akademische Austauschdienst, to Siemens A.G., to the Chemistry Department of the University of Southern California at Los Angeles and to the Fonds der Chemischen Industrie for substantial financial support.
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