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Quantum Mechanical Simulations of Polymers for Molecular Electronics and Photonics Supercomputer Research in Chemistry and Chemical Engineering Downloaded from pubs.acs.org by UNIV OF CALIFORNIA SAN DIEGO on 08/30/15. For personal use only.

M. Dupuis, H. O. Villar, and E. Clementi Scientific and Engineering Computations, Department 48B-MS428, IBM Corporation, Kingston, NY 12401 Ab initio quantum mechanical studies can play an important role in obtaining a detailed understanding of the electronic structure of existing materials, and in predicting the properties of new ones. In this article we give a general outline of our research activity in two areas dealing with new materials, specifically, conducting polymers and polymers with non-linear optical properties. We present the strategy followed for the study of these molecular systems, and an overview of our findings concerning the structure of the prototypical conducting polymer, i.e. pure and doped polyacetylene (PA). We focused our attention on vibrational spectra and infrared and Raman intensities. The results of self-consistent-field (SCF) calculations on charged soliton-like molecules are consistent with experimental observation. In particular we show that the theoretically established accidental mutual exclusion of infrared and Raman bands invalidates the requirement formulated on the basis of the interpretation of experimental data, that defects in PA must have local C2h symmetry. These conclusions are derived from extensive calculations for which supercomputer performance was imperative and carried out on the parallel supercomputer assembled at IBM-Kingston as a loosely coupled array of processors (LCAP). We briefly describe this computer system which has proven to be ideally suited to the methods of ab initio quantum chemistry. "Advanced materials, specialty polymers, ceramics are the absolute core to advanced technologies o f the future". This one statement taken from a W a l l Street Journal article entitled 'Frontiers o f Science'^ indicates clearly that materials research is at the forefront o f science. In fact the same feeling emerges when considering several recently published overview articles on developments in the field of new materials. The October 1986 issue of Scientific A m e r i c a n ^ was fully devoted to this technological area under the heading 'Materials for Economic Growth.' A few weeks earlier the feature article in Chemical & Engineering News presented T h e Organic Solid State' , specifically dealing with organic super and semi-conductors. Finally, the National Science Foundation's report on "Opportunities in Chemistry"^ published in 1985 underlined the key role to be played by chemists in understanding materials and designing new ones. F r o m these articles the reader gets the idea that materials research requires a multidisciplinary effort (1)

0097-6156/87/0353-0146$06.00/0 © 1987 American Chemical Society

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involving experimental and theoretical chemists, physicists, material scientists, and electrical engineers. In this article we are concerned with the possible role and impact o f quantum mechanical simulations on the study o f new materials. A s the result o f new methodological developments and growing access to supercomputers, theoretical physicists and chemists are i n a position to con tribute extensively to this research. The present state o f methods o f theoretical physics is such that it is possible to go beyond idealized models and to examine real materials^. These studies provide a detailed understanding o f structural and electronic properties o f solids and may lead to the design o f new materials with specific properties. The methods o f theoretical chemistry have also evolved tre mendously i n recent years. Molecular self-consistent-field ( S C F ) methods are routinely applied to systems with 50 to 100 atoms***, sometimes more. The infor mation available is no longer restricted to energies. Forces acting on atoms i n molecules are readily calculated, leading to optimized molecular structures and vibrational s p e c t r a ^ . The study presented below takes full advantage o f these capabilities. The ability to treat very large systems at this level o f theory opens new avenues o f theoretical research on new materials. We must remark, however, that the key to accurate quantitative prediction resides i n new developments aimed at treating the electron-electron correlation effects i n large systems. This problem has only started to be addressed and, without any doubt, will require supercomputers. Finally we note that extensions o f molecular methods o f chem istry to describe I D - and 3D-periodical systems*!^ have been available for some time, but their computational requirements are such that supercomputer perform ance is required to extend their domain o f application. We have initiated a research program to study the electronic structure o f conducting polymers and materials with non-linear optical properties. A s such, these studies o f molecular electronics and photonics are only the first step i n a global simulation o f these materials in the spirit o f the approach described by Lie and Clementi (Lie, G . C ; Clementi, E . ; this volume.). The two prototype molec ular systems often considered i n these two areas are polyacetylene ( P A ) and polydiacetylene (PDA) ^=^>. Polyacetylene is the simplest polymer insulator in its pure state and conductor i n a doped state. The electronic structure o f the defects in the doped state and o f the charge carriers is o f great importance and the subject o f considerable research*^!*). Findings related to this prototype system are critical and should provide a detailed understanding o f conductivity i n this recently discovered class o f polymers. In the following sections we will give an overview o f the results obtained so far in our research. Polydiacetylenes are molecular systems which display the strongest non linear optical properties^. These molecules are known to undergo a conforma tional (rod to coil) transition which is responsible for a change i n non-linear response related to the second-order hyperpolarizability γ. O u r contribution to this field o f research relies on our ability to calculate such electronic properties as polarizability and first and second hyperpolarizabilities which are responsible for the non-linear response o f the molecules to incident light. The computational task is much more complex than getting the energy alone, since the first- and second-order response o f the electronic wavefunction to the perturbation (here an incident radiation) must be determined* *. A likely technical implication o f deter mining these properties accurately may be that a larger 'basis set' than otherwise used will be needed. The difficulty is also compounded by the size o f the molec ular systems for which these properties are o f interest. It is very noticeable that the systems with non-linear optical characteristics are most often large, even very large by today's computational standards. The calculation o f these properties is therefore a challenge i n itself for which access to a supercomputer is imperative. Programs for the calculation o f these properties are under development in our laboratory. In section II o f this paper we outline the computational strategy that we have used successfully to study tjie electronic, structure . o f defects i n doped P A , (

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and which has general applicability for the study of electronic structure and reac tivity of molecular clusters, in catalysis for example. In section III we describe the physical problem associated with doped P A , mainly the identification of the electronic structure of defects. The combined use of equilibrium structure deter mination and vibrational spectra calculation, including infrared and Raman absorption intensities, has given a quantitative basis to some experimental obser vations. We interpreted these results as providing strong evidence for the iden tification of charged solitons as defects in P A . In section IV we turn our attention briefly to what distinguishes this work from most other research, beyond the content of the scientific problem discussed: the calculations were per formed on the experimental parallel supercomputer ( L C A P ) assembled in our laboratory which has proven very powerful for many scientific and engineering applications. COMPUTATIONAL METHODOLOGY The starting point for all o f our research in electronic structure theory is the Hartree-Fock ( H F ) or self-consistent-field ( S C F ) method for the solution of the molecular electronic Schrodinger equation* **. The total wavefunction of the molecular system is expressed as an antisymmetrized product (Slater determinant) of molecular orbitals, each orbital being a linear combination of basis functions (also called atomic orbitals) assigned to the atomic centers. The total energy of the molecule is a function of the position of the nuclei R , E ( R ) . The molecular orbitals are determined by solving the H F equations, which take into account explicitly the electrostatic effects among the electrons, including the electronelectron repulsion. The molecular energy depends on the basis set used to expand the molecular orbitals. The more extended the basis set, the more accurate the description of the molecular system, but the more time consuming the calcu lation. Note that in this formalism each electron feels the average potential created by the other electrons. Electron correlation is what makes up a more realistic picture of the electron interaction where each electron feels an instanta neous potential created by the other electrons. Electron correlation is a key ingre dient for quantitatively accurate molecular calculations. A s mentioned earlier methods to treat electron correlation in large molecular systems are seldom used owing to their very high computer time requirements. In recent years, methods to calculate efficiently, the forces acting on atoms in molecules have been developed: the calculation o f energy gradients dE/dR and second derivatives d E/dR are routinely carried out even for large systems* *. Sometimes the second derivative matrix is obtained by finite difference of the energy gradients. The second derivative matrix is just what is needed for a vibrational analysis carried out in the harmonic approximation. F r o m first princi ples then, without empirical parameters, it is possible to obtain the complete set of vibrational frequencies and normal modes. It is now well established that the vibrational frequencies obtained at the S C F level o f theory are 10 to 20% too high compared to the observed frequencies*?^*. W i t h the availability of the normal modes expressed in the cartesian coordinate space or in the internal coor dinate space, it is easy to move further toward a more complete representation of the vibrational spectra by calculating the infrared and Raman intensities of the vibrational modes. These quantities are obtained by calculating the derivatives of the molecular dipole moment and polarizability with respect to the cartesian coordinates followed by a transformation to derivatives with respect to the normal modes. A n elegant and inexpensive method has been proposed by Komornicki and Mclver* * which involves calculating the energy gradients in the presence of a small uniform field. It is based on the simple relationship δμ/dR = d('dE/ô¥)/dR = -d(dE/dR)/d¥ where F represents the electric field. The importance of getting a complete semi-quantitative vibrational spectrum can not be over emphasized. In the study presented below we made use of 1

2

2

12

2


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all of these computational techniques. The availability o f the infrared and Raman intensities was the key to suggesting that charged solitons and not bipolarons are a better description of defects in doped P A . The S C F method for molecules has been extended into the Crystal Orbital (CO) method for systems with I D - or 3D- translational periodicity*lMI>. The C O method is in fact the band theory method of solid state theory applied in the spirit of molecular orbital methods. It is used to obtain the band structure as a means to explain the conductivity in these materials, and we have done so in our study of polyacetylene. There are however some difficulties associated with the use of the C O method to describe impurities or defects in polymers. The periodicity assumed in the C O formalism implies that impurities have the same periodicity. Thus the unit cell on which the translational periodicity is applied must be chosen carefully in such a way that the repeating impurities do not interact. In general this requirement implies that the unit cell be very large, a feature which results in extremely demanding computations and thus hinders the use of the C O method for the study of impurities. In the limit of very large clusters of repeating units in one dimension, a molecular system will behave as an infinite chain. The number of units needed to reach this limiting behavior will vary from system to system depending on the short or long range character of the interaction between units. When simulating infinite systems with finite clusters, it is necessary to check the convergence of the properties of interest as a function of the size of the cluster i n order to establish some degree of confidence. We have used this approach in our investigation of polyacetylene, where we have compared the band structure calculated for increasingly large clusters with the band structure of an infinite polymer. Other parameters used to gauge the convergence behavior in finite cluster calculations are geometry, atomic charges for example. T o study defects or impurities we increase the size of the molecular cluster in which the defect is simulated and again select some criteria to gauge the convergence of the description of the defect. These criteria may be geometrical parameters i f we deal with optimized geometries for each cluster size, and/or any properties related to the electron density. In our study of polyacetylene we have used geometrical parameters, atomic charges, and bond indices to characterize charged soliton and bipolaron models by comparison with isolated P A clusters. E L E C T R O N I C S T R U C T U R E OF DEFECTS I N PA Frequent attention has been given to polyacetylene P A for the study of charge carriers in conducting polymers* **. Because of its simplicity P A is amenable to both experimental and theoretical research. It is now well accepted that in its undoped state trans P A displays C C bond length alternation o f 0.08 Â (the C - C single bond length is 1.44 Â, the C = C double bond length is 1.36 À). In a doped state P A conducts electricity. The charge carrier models most often discussed are neutral solitons, positively and negatively charged solitons, polarons and bipolarons. The charged soliton picture led apparently to a successful modelling of the experimental vibrational spectrum of doped PA* ^. Recently on the basis of the work o f Campbell et al. ^ and of Boudreaux et al.*^* the idea of bipolarons has gained acceptance* *. One key element in the situation is the experimentally deduced mutual exclusion of the infrared active and Raman active absorption lines. Such a mutual exclusion implies a defect with local symmetry belonging to the C h point group, to which bipolaron models belong, thus favoring the bipolaron picture of defects in PA* *. Previous theoretical studies of isolated and doped P A have mostly dealt with geometrical structures and charge and spin waves*^*. A few have dealt with the vibrational spectrum* ^) using force fields derived from fits to experimental infrared and Raman spectra of smaller polyenes or from scaled force constants obtained from semi-empirical S C F calculations on these systems. Very little the1

1

(1

12

2

11

1

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(

oretical work has been done on I R and Raman intensities o f isolated PA ^> and, to our knowledge, no intensity studies on models for doped P A have been pub lished. This work is the first to combine structure and vibrational spectrum deter mination, including calculation o f I R and Raman intensities, for relatively large polyenes at the ab initio level of theory. This approach has led to a consistent picture of these molecular systems, and has made it possible to provide a theore tical quantitative basis to the question o f mutual exclusion o f I R and Raman lines in doped P A . We found that the cations (with C symmetry) used as models for charged solitons show an accidental mutual exclusion of I R and Raman bands. This result would indicate that the requirement that doped P A have local C symmetry in the region of the defect may not be necessary. Further the calculated spectra for the various models show better overall agree ment with experiment in the case of cations as models for charged solitons. We have used the systems C H with η = 2,4,...,22, C „ H with n = 3,5,...,21, and C „ H with n = 4,6,...,22 to represent pure P A , positively charged solitons, and bipolarons respectively. S C F wavefunctions were calculated with a double-zeta quality basis set (denoted 6-3 l G ) ^ and optimized geometries for all these systems were determined. In addition for the molecules with η up to 11 or 12 we calculated the vibrational spectrum, including infrared and Raman intensities. 2 v

2 h

+

n

n + 2

n + 2

+ 2

n + 2

ISOLATED P A M O D E L . The calculated structures o f the neutral clusters with η = 22 display bond alternation. The central unit in C H is characterized by R ( C - C ) = 1.452 A , R ( C = C ) = 1.337 A , the C - C = C angle is 124.2 degrees, and the H - C = C angle 119.2 degrees. The agreement with the experimental bond lengths is very good. The convergence of the π orbital energies with the cluster size can be seen in Figure 1 where the π band edges are shown. The right hand side points are the result of a C O calculation on an infinite polymer using the same basis set and indicate convergence o f the electronic properties o f the polyenes with increasing chain length. The band gap deduced from the C O calcu lation is 6.8 eV which is rather far from the accepted experimental value of 1.5 eV. Most of the difference can be attributed to the electron-electron correlation effects which are not accounted for in the S C F formalism. This result under scores the need to go beyond the present level o f theory for quantitative predic tion. Experimentally three I R active and four Raman active vibrations have been observed in undoped P A . F o r the neutral clusters up to η = 12 we have identified the vibrational modes which best correspond to these bands. The frequencies cal culated for C , H , 4 are presented in Table I. 2 2

2 4

2

Table I.

Experimental frequencies of PA and corresponding calculated frequencies of the C H , cluster l2

Expt. (CH)n 3013 1474 1292 1291 1080 1016 1015

Expt. (CD)n (m) (s) (vw) (vvw) (s) (w) (vs)

2231 1357 916 1200 855 745 752

Calc. (C H ) I 2

I 4

3331 1873 1466 1465 1322 1081 1175

4

Calc. (C D ) I 2

Activity

Description

IR Raman IR Raman Raman Raman IR

C - H stretch C = C stretch C H bend C H bend C - C stretch out of plane bend out of plane bend

14

2477 1822 991 1003 1261 873 857


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Quantum Mechanical

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151

χ 6

"

-

π *

LUMO^ "

2

—

—

—

χ

— -2

-β

-10 -

π

LOV\T^

χ

-14 ι 2

ι

ι

ι

6

ι

ι

ι

10

Number

14

of

Carbon

Figure 1. π band edges for neutral C H spond to an infinite polymer (CH) . n

n

n + 2

I

I 18

I

.1 22

I

Atoms

. The right hand side points corre

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The frequencies and relative intensities for the clusters n = 6 to η =12 are given in Table II. A s mentioned earlier, the calculated frequencies are 10 to 20 % too high. The corresponding vibrational modes have the character of P A vibra tions in so far as the atoms which vibrate most are in the central region o f the cluster.

Table II.

IR active frequencies and relative intensities

Description

C H 6

C H 2 a s y m . stretch C H stretch C H stretch C H out-of-plane CH2out-of-plane

3415 3334 3327 1177 1124 1087

C H

8

(0.83) (0.71) (0.66) (1.00) (2.40) (0.63)

8

3415 3333 3329 1176 1135 1097

1 0

(0.51) (0.75) (0.54) (1.00) (1.12) (0.79)

C H l 0

3415 3333 3330 1176 1145 1105

C Hi

1 2

(0.38) (0.88) (0.42) (1.00) (0.67) (0.71)

I 2

3415 3333 3331 1175 1151 1113

(0.30) (0.65) (0.72) (1.00) (0.47) (0.54)

For each molecular cluster we have chosen the mode with frequency near 1175 cm-' ( C H out-of-plane bend) as the reference mode for I R intensity analysis. The mode with highest frequency shown in Table II is the antisymmetric stretch of the terminal C H groups. With increasing cluster length its intensity dimin ishes. The nearly degenerate modes with frequencies close to 3330 c m do not show a well defined pattern but appear to maintain a significant intensity. The mode with frequency 3333 cm-' in C i H involves terminal C H stretches, the mode with frequency 3331 cm-' involves the bulk C H stretches. This latter mode arise from in phase C H stretches shown in Figure 2 which create a molecular dipole responsible for the absorption. The mode with frequency near 1175 c m , also displayed in Figure 2 corresponds to in phase out-of-plane C H bends. A l l the Η atoms are displaced out of the molecular plane on the same side. The molec ular dipole thus created gives rise to the strongest absorption. The trends in intensity demonstrated by these calculations are in accord with experimental observation. The intensities of the bands have been analyzed by Zannoni and Z e r b i ^ in terms of charge flux. These authors point out that the C H stretching mode absorbs parallel polarized light as well as perpendicular polarized light, while the C H out-of-plane bend shows a strong preference for perpendicular polarized light. F o r the mode at 3331 c m we have obtained: 2

1

2

I 4

1

1

1

2

in units o f ( D / Â ) a m u / . experimental observation.

4

These results lend quantitative significance to the

D O P E D P A M O D E L S . We selected two criteria to characterize the structure of the mono- and di-cations. The wavefunctions o f the cations at their respective optimized geometries were used to determine Mayer's bond indices which reflect the strength of the interatomic bonds. The differences in the cations and also the neutral molecule emerge very clearly from Table III.

Figure 2. Selected infrared and Raman active vibrational modes of C i H .

2

1 4


154

SUPERCOMPUTER RESEARCH Table III. Interatomic C C distances (in angstroms) and Bond Indices for C H 4 , C , H , C H 2 . CI labels the center carbon atom, C l 1 the terminal carbon atom +

22

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22

2

;H

distance ci c r C l C2 C2 C3 C3 C 4 C4 C5 C5 C6 C6 C 7 C 7 C8 C8 C 9 C9 CIO C10C11

2

+

2 3

2 2

2 4

c

2 4

2I

H

C H 24

+

22

2 3

bond index

distance

bond index

distance

1.762 1.082 1.762 1.082 1.764 1.080 1.768 1.077 1.784 1.066 1.856

1.396 1.379 1.413 1.364 1.428 1.353 1.439 1.344 1.452 1.332

1.332 1.442 1.238 1.552 1.166 1.646 1.117 1.726 1.078 1.832

1.449 1.342 1.443 1.351 1.426 1.370 1.398 1.498 1.369 1.435 1.338

1.338 1.450 1.338 1.450 1.338 1.451 1.337 1.453 1.337 1.458 1.329

+

2

bond index 1.095 1.719 1.113 1.638 1.178 1.488 1.305 1.307 1.493 1.144 1.725

The C C bond lengths follow very closely the variations in the bond indices. C , H , the charged soliton-like molecule, displays a single defect site at the center of the molecule, and gradually recovers the bond alternation pattern seen in the neutral species. C H , the bipolaron-like molecule, has a split defect with bond alternation observed in the middle. These results are in qualita tive agreement with the semi-empirical calculations of Boudreaux et a l ^ \ A small difference is obtained i n the extend of the defect which extends over 15 carbon atoms in our work, but only 11 carbon atoms in their work. Our results ought to be considered more reliable on the basis of the higher level o f theory used here. A t the S C F level o f theory we found that for η > 6 the band gap o f the cation is smaller than for the dication, which in turn is smaller than for the neutral. The calculated band gap for C i H is 5.8 eV, for C H it is 6.4 eV to be com pared with 7.2 eV for the neutral system C H . However, we note that the largest dication cluster may be too small to predict accurately the behavior of bipolarons. The experimental spectrum o f doped P A (p-doped and η-doped) shows two I R active modes at 1397 and 888 c n r which are also active in deuterated doped P A at 1120 and 780 c m . The active modes for the mono-cation and d i cation series are given in Tables I V and V . The active modes o f C , , H with frequency o f 1656 and 1271 cm- are shown in Figure 3. They are in plane vibra tional modes involving C C stretches mixed with C H bends. F o r the deuterated cations, one intense mode appears at 1518 c m , and a second mode at 990 cm* seems to gain in intensity as the cluster becomes larger. The spectra o f the dications are more complicated. The most intense mode is associated with the stretching o f the C C bonds near the two charged sites which comprises the defect. The corresponding deuterated spectra show only one band at 1562 cm* The modes o f C , H , are shown in Figure 4. There is no indication of the appear ance of a second band with low frequency. The trends in the spectra o f the dications are not in as satisfactory an agreement with experiment as the trends observed in the cation spectra. We note, however, that C H , may be a better representation o f charged solitons than +

2

2 3

+ 2

2 2

2 4

(

+

2

+ 2

2 3

2 2

2 2

2 4

2 4

1

1

+

I 3

1

1

1

1

+ 2

2

4

+

n

3


DUPUIS ET AL.

J

Figure 3. Selected infrared and Raman active vibrational modes of C n H ofc„H -.

13

Quantum Mechanical Simulations of Polymers

CnHa 1304cm" 1

1 3

+

and


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C HÎ 12

4 2

1699 cm" 1

Figure 4. Selected infrared and Raman active vibrational modes of C H i

1 2

4

+ 2

.

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+ 2

Ci H of bipolarons from consideration o f the cluster size. Further calculations of the spectrum o f larger clusters are needed to substantiate the conclusions derived from the present study. 2

I 4

+

Table I V . Calculated vibrational frequencies and IR intensities for the C H to C u H . The frequencies are given in cm- and the intensities are relative to the most intense mode for each molecule. The types represent the following vibrations : SCI = CH2 in plane bend , CCS = Carbon carbon stretching, C H B = C H in plane bend 5

+

7

1


i 3

C„H

C„D

+ 1 3

+ I 3

MODE

TYPE

FREQ.

INT.

FREQ.

INT.

B; B

CHB CCS CCS

1271 1304 1656

0.84 0.10 1.00

990 1302 1518

0.10 0.01 1.00

2

C D„

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