When finite becomes infinite: convergence properties of vibrational spectra of oligomer chains
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Abstract
We present a computational study of convergence properties of vibrational IR and Raman spectra for a series of increasingly long units of polyethylene, cis and transpolyacetylenes, and polyynes. Convergent behavior to the spectra of infinitely long polymers was observed in all cases when chains reached lengths of approximately 60 carbon atoms, both with respect to the positions of the bands and to their intensities. The vibrational spectra of longer chains are practically indistinguishable. The convergence rate depends on the degree of the π conjugation in a studied system: Vibrational spectra for oligoethylenes converge noticeably faster than the spectra for the conjugated systems. The slowest convergence is observed for skeletal motions of the oligomer chains, which may require more than a hundred carbon atoms in the chain to show deviations smaller than 1 cm^{−1} to the corresponding solidstate calculations. The results suggest that the boundary between the properties of finite and infinite molecular systems fades away for a surprisingly small number of atoms.
Keywords
Simulations of IR and Raman spectra of polymers Convergence in size evolution SCCDFTBIntroduction
Polymers and molecular wires, molecular sheets, and covalent crystals consist of molecularsized monomer units with regular bonding patterns in one, two, and three dimensions, respectively. Carbon can form all three types of these extended structures: carbon chains and nanotubes (1D), graphene flakes (2D), and nanodiamonds (3D). Their electronic and vibrational spectra converge toward bulk limits that are independent of the system size. It is therefore natural and interesting to ask a fundamental question: What is the smallest possible size—either expressed in units of length or in the number of atoms—of a finite extended structure, for which the effects of its finiteness can no longer be observed? An answer to this question is generally difficult to obtain. A few systematic theoretical attempts to answer this question have employed ab initio and density functional theory (DFT) for the simulation of vibrational spectra of series of homologues compounds with increasing system size. An early work by Raghavachari and coworkers [1] focused on the vibrational frequencies of nalkanes. Karpfen and coworkers [2] reported the size evolution of vibrational spectra of polyenes, and Kertesz and coworkers [3, 4, 5] applied a similar approach to polyynes and linear carbon chains. Naturally, the maximum molecular size in these studies correlates with available computer power at the time of the study: While C_{5}H_{12} was the largest compound treated in the 1986 HartreeFock study [1], by 2007 DFT frequency calculations of C_{72}H_{2} had become possible. Nevertheless, prediction of vibrational spectra based on ab initio and DFT approaches remains difficult for systems with more than a hundred atoms due to the costly and tedious solution to the coupled perturbed KohnSham (CPKS) or HartreeFock (CPHF) equations [6] that elude efficient parallelization. Only recently [7], the auxiliary density perturbation theory [8] developed by Koster and collaborators has been able to constitute a viable way of replacing the CPHF equations and by massive parallelization [7] has enabled spectra calculations for larger nanostructures.
In a recent article [9], we applied the computationally more economical selfconsistentcharge densityfunctional tightbinding (SCCDFTB) [10] quantum chemical method to study the size evolution of Raman spectra of hydrogenterminated nanodiamond models starting from the smallest conceivable model of adamantane (C_{10}H_{16}) up to molecules as large as 3 nm, containing approximately 1000 carbon atoms. Although the simulated Raman spectra displayed convergent characteristics with increasing system size, the convergence toward the single crystal spectrum was rather slow. Thus, in the present study, we simulate the vibrational spectra of carbon aggregates with reduced dimensionality, i.e., onedimensional finite chains consisting of carbon atoms. In analogy with the simulations of the Raman spectra of nanodiamonds, we terminate the chains using hydrogen atoms. In order to make the study as general as possible, we simulate both infrared (IR) and Raman spectra for three families of chains consisting of the sp, sp^{2}, and sp^{3} carbon atoms. The dimensionality reduction allows us to study the convergence rate up to almost a perfect agreement with the solidstate calculations for the corresponding infinite chains: polyyne, cis and transpolyacetylene, and polyethylene. The polyethylene results are used for assessing the convergence rate of the vibrational spectra of crystalline nanodiamond models.
Simulations of the firstorder Raman and IR vibrational spectra of extended systems are usually done in the framework of solid state physics. The solidstate techniques transform the infinitelydimensional massweighed Hessian matrix into a blockdiagonal form. The resulting blocks are finite with the size 3N × 3N, where N denotes the number of atoms inside the unit cell. Each finite block corresponds to a certain irreducible representation of the discrete translational symmetry group labeled by a triple of indices: (k_{x}, k_{y}, k_{z}). Diagonalization of each block yields a set of phonon frequencies at (k_{x}, k_{y}, k_{z}). The collection of the phonon frequencies for (k_{x}, k_{y}, k_{z}) probing uniformly the first Brillouin zone is usually referred to as the phonon dispersion relations. Since the magnitude of the incident light wave vector is minuscule in comparison to typical phonon wave vectors, onephonon IR and Raman techniques are capable of probing only the proximate vicinity of the zone center. Thus, the positions of the bands in IR and Raman spectra correspond to the phonon frequencies at the Γ point, (k_{x}, k_{y}, k_{z}) = (0, 0, 0). While the positions of the bands are quite readily determined in the solidstate framework, computing the band intensities constitutes a much more difficult problem. The intensities are directly related to the changes of the electrical dipole moment and polarizability during the vibrations—defined as appropriate derivatives of the total energy with respect to the components of external electric field—which leads to serious conceptual problems in the solidstate framework. Therefore, the intensities are usually determined either using quite advanced techniques or by semiempirical procedures based on the concept of effective bond polarizabilities.
A natural alternative available for determination of the firstorder Raman and infrared (IR) vibrational spectra of extended systems are the techniques of quantum chemistry. In this framework, sometimes referred to as the oligomer approach, one usually starts with a small molecular model of a given extended system and elongates/enlarges it until the convergence of a simulated property is observed. The vibrational spectra are obtained directly from the molecular vibrational frequencies and the associated IR intensities and Raman activities. In contrast to the solidstate framework, the derivatives with respect to the components of external electric field are readily available, greatly simplifying the process of simulating the spectra. On the other hand, the terminal atoms or groups used to saturate the extended system surface may introduce spurious bands in the simulated spectra that fade away only in the limit of large N making the calculations computationally expensive.
Although both the procedures described above are quite standard and usually lead to quite accurate IR and Raman spectra, it is appropriate to briefly signalize here their possible shortcomings and limitations. First of all, the solidstate techniques assume that the studied structure is infinite and perfectly periodic. Both of these assumptions are not correct, at least not in the exact sense. Another issue concerns disorder in the crystal structure, which can be mild (structural defects) or strong (amorphous solids). Modeling these types of disordered phases in the solidstate framework usually requires large supercells allowing for spatial separation of defects or achieving quasirandom distribution of atoms that mimics an amorphous solid. It is unknown how big the supercell should be to allow effective decoupling of neighboring defects. This issue can be quite important in practice. Imagine, for example, a simulation of defect dynamics in a crystal. In the supercell approach, one in fact studies dynamics of a coupled network of defects, which—if not completely decoupled—may behave very differently. Another important situation to be mentioned in this context is studying a chemical reaction on the crystal surface. In the supercell approach, one is studying in fact not one but infinitely many reactions spatially separated by some distance d. One usually wants to keep d small to reduce the computational cost. On the other hand, it is not immediately clear how large value of d should be that the reactions do not influence each other; too small a value of d may distort the reaction energetics to a considerable degree. Note that all these problems are completely alleviated in the quantum chemical approach using a finite molecular model, which seems to be a natural choice for studying defective or amorphous systems. Clearly, the quantum chemical techniques also have their own limitations. Some of them—relatively slow convergence, the presence of the finite size effects, and the closely related spurious signals from the terminal atoms and groups—have already been signalized. Another obstacle can be the quite substantial computational cost associated with large molecular models.
The efficiency of solidstate techniques based on the concept of discrete translational symmetry suggests that the departure from the idealized infinite model in real crystals does not have serious consequences. It is, therefore, natural and interesting to ask a fundamental question what is the smallest possible size—either expressed in units of length or in the number of atoms—of a finite real crystal, for which the effects of its finiteness cannot be observed. The same question can be expressed somewhat differently by asking, what is the smallest possible size of the finite model in the quantum chemical approach that effectively gives the same results like the solidstate framework. In the present work, we shed some light on this problem by modeling IR and Raman spectra of series of finite polyynes, cis and transpolyacetylenes, and polyethylenes and studying the convergence of the computed frequency and intensity patterns with growing N. These systems have been studied quite extensively in the solidstate framework, while the finite quantum chemical investigations were limited to rather small molecular models owing to prohibitive computational costs [11]. The results of our simulations demonstrate that “infinity” may start in the molecular world at "quite finite scales". We believe that our results and discussion will prove helpful for researchers aiming at studying extended systems, for which either the finite size, lack of translational periodicity, and/or the presence of defects play an important role.
Computational details
The simulations presented here have been performed using the selfconsistentcharge densityfunctional tightbinding (SCCDFTB) method [10]. This technique, which can be treated as a careful approximation to density functional theory (DFT), is a modern semiempirical method known to reproduce well molecular geometries, energies, and vibrational frequencies of medium and large molecular systems [12, 13, 14, 15, 16]. Recent reoptimization of the repulsive potentials allowed us to accurately compute the vibrational frequencies with errors comparable to those from DFT [17]. The reoptimized set of the C–C and C–H repulsive SCCDFTB potentials is used in this study. Efficient analytical Hessian code [15] allowed for modeling IR [18] and Raman [19] spectra of large molecules. Numerous tests [9, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27], performed for fullerenes, nanotubes, and other carbon nanostructures showed that SCCDFTB is a viable method of computing the harmonic vibrational frequencies and eigenvectors of organic molecules and carbon clusters.
Hydrogen terminated, finite hydrocarbon chains containing up to a few hundred carbon atoms have been used in this study. The largest considered models correspond to 150 unit cells for polyynes, 75 unit cells for polyacetylene, and 100 unit cells for polyethylene. While it was technically possible to study much larger systems (of approximately 1000 carbon atoms), it has been found that the here employed models are sufficiently large to demonstrate the convergence of simulated vibrational spectra. For each of the finite models, we performed full geometry optimization, followed by the determination of harmonic vibrational frequencies, and computation of IR intensities and Raman activities using doubleharmonic approximation. In the optimization, the force convergence criterion has been set to 10^{−6} a.u. and the charge convergence criterion, to 10^{−12} a.u. The spectra have been simulated assuming a Gaussian spectral envelope with a halfwidth of 5 cm^{−1}. The method of transforming Raman activities to Raman intensities was presented elsewhere [19].
The convergence of the positions for the IR and Raman bands was evaluated with respect to the corresponding infinite limits obtained from solid state SCCDFTB calculations. The phonon dispersion curves were computed using the DFTB+ code [28] and the standard supercell approach. It has been found that for the evaluation of the Hessian matrix, it is sufficient to take two translational replicas of the unit cell in each direction. Larger supercells have yielded virtually the same phonon curves. Some of the acoustic phonon dispersion relation curves show mild numerical instabilities (imaginary frequencies for some segments of the curves), originating most probably from the coupling of the chain vibrations to the rotation and translations. We have not pursued this topic further, as the phonon frequencies at the Γ point—needed to assess the finite oligomers frequencies convergence—are not affected by these instabilities. Moreover, as we mentioned, the instabilities are related only to the acoustic phonons, which are not in the scope of the IR and Raman spectroscopy.
We stress again that the goal of this work is the evolution of vibrational spectra with system size, not the exact prediction of vibrational spectra of particular molecular systems. We therefore avoided arbitrary fitting of the DFTB parameters to experimental spectra that would reduce their transferability, and we did not resort to elaborate scaling techniques that were employed for instance in refs. [2, 3]. The reported vibrational spectra are therefore based on the unscaled, raw harmonic SCCDFTB vibrational frequencies. We also stress that the present calculations correspond only to a very simplified model, in which we analyze a single, gas phase polymer chain at 0 K using harmonic vibrational frequencies and intensities. Bringing this model closer to experimental reality would require including anharmonic effects in the formalism, considering many entangled chains interacting via intermolecular forces, including finite temperature effects, and considering many further methodological amendments.
Results
Oligoynes
The simulated IR spectra of finite polyynes [29] consist of two strong bands located at approximately 566 and 3254 cm^{−1} and three less distinct bands at 424, 1970, and 2041 cm^{−1}. The intensive bands correspond to the C≡C−H bending and the C−H stretching, respectively. The small satellite band at 424 cm^{−1} is associated with bending of the terminal C−C≡C−H group. All these spectral features originate from the hydrogen termination of the finite models and clearly are not present in the infinite chain spectrum. Their positions converge very fast to constant values; for chains consisting of more than ten carbon atoms (five unit cells of the corresponding polymer), no further change in their vibrational frequencies is observed. The remaining spectral features, the lowintensity broad bands with maxima approximately at 1970 and 2041 cm^{−1} are associated with the C≡C stretching. Their spectral composition displays an interesting and quite complex patterns. The convergence of the band maxima toward constant positions is considerably slower than for the modes involving hydrogens. Intensities of all five computed bands converge toward constant values, which has quite significant consequences as can be seen by the following reasoning. The total IR intensity of a sample is proportional to its volume V = nv, where v is the volume of a single molecule and n is the number of molecules in the sample. Clearly, for linear chains, the molecular volume v grows linearly with the number N of carbon atoms in the molecule. Consequently, the number of molecules n in a constantvolume sample will scale inversely proportional to N. Thus, since the IR signal from a single molecule converges to a constant, the total IR intensity of the studied constantvolume sample will scale like 1/N, vanishing completely in the limit of a perfect crystal (N → ∞). This conclusion is consistent with the corresponding solidstate picture. The longitudinal optical (LO) phonon of polyyne (located by SCCDFTB at 1951 cm^{−1} at the Γ point) is of even parity and thus does not contribute to the IR spectrum. Note that the constant intensity pattern is unique for the IR spectrum of polyyne; all other simulated spectra show linear scaling of the band intensities for large N, which leads to a nonvanishing, constant signal from a given volume of the sample.
The calculated Raman spectra of finite polyynes are particularly simple, consisting of a single strong band. Its position converges rapidly to the LO phonon frequency at Γ (1951 cm^{−1}) with deviation of +5.3 cm^{−1} for N = 60, +1.7 cm^{−1} for N = 100, and + 0.8 cm^{−1} for N = 140. We note that SCCDFTB predicts the absolute value of this frequency too high: scaled DFT predictions and experimental observations place this mode around 1850–1860 cm^{−1} [3]. The spectral structure of this band is quite complex with approximately 90% of its intensity carried by a single peak and the rest by a comb of less and less intensive, blueshifted peaks giving rise to a characteristic asymmetric shape of the band. Again, the same conclusion can be drawn from the solidstate phonon dispersion curves, which show characteristic LO phonon softening around the Γ point. Remembering that the Raman scattering probes only the vicinity of the zone center, one may expect the same asymmetric band shape. The intensity of the band in our simulations displays a simple linear relationship to the number of carbon atoms for chains longer than 60 atoms. Following the earlier discussion, we see that the Raman intensity per constant volume (or alternatively: per unit cell of a polyyne 1D crystal) converges to a constant. Further discussion of these phenomena is given in ref. [29].
Cis and transoligoacetylenes
The IR spectra of finite cisoligoacetylenes computed with the SCCDFTB method consist of four intensive bands located at approximately 473, 715, 1334, and 3019 cm^{−1}. These bands correspond to the following vibrational modes of the infinite cispolyacetylene chain: inplane CCC deformation (ν_{11}), outofplane CH deformation (ν_{20}), inplane CH deformation (ν_{10}), and CH stretch (ν_{12}) [30, 31]. The experimentally observed positions of these bands are 448, 740, 1328, and 3057 cm^{−1} [30]. The calculated SCCDFTB Raman spectra of cisoligoacetylenes display three intensive bands located at approximately 860, 1091, and 1447 cm^{−1}. These bands correspond predominantly to the C−C stretching mode (ν_{4}), the C=C stretching mode (ν_{3}), and the C−H bending mode (ν_{2}), respectively. Experiment locates those bands at 910, 1250, and 1540 cm^{−1} [31, 32]. Note that the vibrational eigendecomposition of these three bands computed by Mulazzi et al. and quoted by Lichtmann et al. [32, 33, 34] is quite different than ours, with all modes having substantial contribution from all three vibrations located by us.
The simulated IR spectra of the finite transoligoacetylenes consist of three intensive bands located at approximately 995, 1177, and 2957 cm^{−1}. These bands correspond to the outofplane CH deformation (ν_{7}), the inplane CH deformation (ν_{6}), and the CH stretching mode of the infinite transpolyacetylene chain, observed in experiment at 1012, 1170, and 3013 cm^{−1}, respectively. The calculated Raman spectra of the finite transoligoacetylenes display only two intensive bands located at approximately 901 and 1407 cm^{−1}. The higherfrequency mode corresponds to the inplane CH deformation (ν_{2}) of the infinite transpolyacetylene chain and is observed in experiment at 1457 cm^{−1}. The lower frequency mode corresponds in our calculations to the CC stretch (ν_{4}) and is observed in experiment at 1066 cm^{−1}.
Oligoethylenes
The simulated IR spectra of finite oligoethylenes consist of four distinct bands located at approximately 1237, 1466, 2891, and 2995 cm^{−1}. The two highfrequency modes correspond to the symmetric (2891 cm^{−1}) and asymmetric (2995 cm^{−1}) CH stretch of the methylene groups. The equivalent vibrations of polyethylene, denoted as ν_{1}(π) and ν_{6}(π), are observed in experiment [36, 37, 38, 39, 40] at 2851 and 2919 cm^{−1}, respectively, while the scaled (0.966) B3LYP/631G* calculations locate them at 2927 and 2976 cm^{−1} [40]. The SCCDFTB band at 1466 cm^{−1} corresponds to the scissoring motion of the methylene groups. Its polyethylene counterpart ν_{2}(π) is observed experimentally at 1468 cm^{−1}. The remaining band at 1237 cm^{−1} corresponds to the methylene wagging ν_{3}(0), which is observed in experiment at 1176 cm^{−1}. The B3LYP/631G* calculations locate ν_{2}(π) and ν_{3}(0) at 1492 and 1174 cm^{−1}, respectively.
It is appropriate to explain here the nomenclature used for the classification of the optically active vibrational modes of polyethylene. The symmetry of the vibrational modes of an infinite polyethylene chain is traditionally described using a purely translational unit cell containing two methylene units. One could in principle use the smallest possible unit cell, containing only a single methylene fragment, with two additional symmetry elements: a screw axis and a glide plane. In this case, one would obtain nine different phonon dispersion curves, two acoustic (ν_{5} and ν_{9}) and the remaining ones optical. These modes can be IR or Ramanactive either at the Γ point (k_{x}, k_{y}, k_{z}) = (0, 0, 0) or in the X point (k_{x}, k_{y}, k_{z}) = (π, π, π). Enlarging the unit cells to two methylene units folds the Brillouin zone (BZ) in half, mapping the X point of the smaller unit cell onto the Γ point of the larger unit cell and giving 18 phonon frequencies at the new Γ point. These new frequencies are labeled as ν_{1}–ν_{9} with additional index (0) or (π), referring to the position of the point in the twice larger BZ. Note that the additional index can also be interpreted as the phase difference between the adjacent methylene fragments for a given vibrational mode. Note also that for all the IR and Ramanactive vibrational modes in our SCCDFTB simulations of finite oligoethylenes, the adjacent twomethylene unit cells vibrate in phase, in a close analogy to the translationally invariant vibrations of infinite polyyne chain. The important difference, however, concerns the amplitude of the vibration. In an infinite chain, the amplitude is identical for each translationequivalent unit cell, while for the finite oligoethylenes studied here, the distribution of the amplitudes reproduces the shape of the fundamental harmonic vibration of a finite string.
The simulated SCCDFTB Raman spectra of oligoethylenes consist of two families of bands. The highfrequency bands are composed of two peaks, at 2863 and 2959 cm^{−1}, corresponding to the symmetric and asymmetric CH stretch of the methylene groups. The analogous vibrations of infinite polyethylene chain, ν_{1}(0) and ν_{6}(0), are observed in experiment at 2848 and 2883 cm^{−1} [36, 37, 38, 39, 40]. The second family of bands is located between 900 and 1450 cm^{−1} and displays a quite complex shape. A careful analysis allows for differentiating six main peaks, located at 938, 1069, 1204, 1305, 1393, and 1443 cm^{−1}. The inspection of the associated vibrational eigenvectors shows that those peaks display onetoone correspondence with the experimentally observed Ramanactive peaks of the infinite polyyne chain, which are listed here in the analogous order: CC stretch (ν_{4}(π) at 1061 cm^{−1}), skeletal angle bend (ν_{4}(0) at 1131 cm^{−1}), the rocking motion of the methylene groups (ν_{7}(0) at 1168 cm^{−1}), methylene twist (ν_{7}(π) at 1295 cm^{−1}), methylene wagging (ν_{3}(π) at 1370 cm^{−1}), and the scissoring motion of the methylene groups (ν_{2}(0) at 1440 cm^{−1}) [36, 37, 38, 39, 40, 41, 42].
The analysis of the simulated IR and Raman spectra of finite oligoethylenes suggests that the convergence toward the infinite systems is already achieved for much shorter chains than for oligoynes and oligoacetylenes. The positions of the bands, shown in detail in Fig. 3, are practically identical for chains longer than 30 methylene units with the exception of the ν_{3}(0) and ν_{4}(0) modes, which require a chain of approximately twice longer length for a similar degree of consistence with the solidstate calculations. Note that the intensity convergence is achieved even faster; for chains longer than C_{20}, the band intensities scale approximately linearly with the number of carbon atoms, suggesting constant IR and Raman signals for a sample of a constant volume.
Discussion
The simulated convergence rates for particular vibrations of the finite carbon chains can be used for estimating the convergence rates for higherdimensional carbon structures. In particular, the presented results for oligoethylene chains can be used for assessing the convergence rate of nanocrystalline diamond models, which could not be accessed from direct simulations [9]. As we mentioned above, the intensity convergence is achieved very fast, for chains containing approximately 20 carbon atoms. The associated length of such a chain is equal to 24.7 Å. It is thus reasonable to expect that analogous 3D structures with the volume of (24.7 Å)^{3} would display a similar degree of the intensity convergence rate. Equation (1) from ref. 1 allows us to relate these quantities to the number of carbon atoms in a given nanodiamond model, which are 626 for the octahedral symmetry and 573 for the tetrahedral symmetry. Since the studied nanodiamond models were larger than the number determined above, we observe that the intensity convergence can be analyzed only if the evolution pattern of the associated vibrational eigenvectors has been sufficiently converged. As signalized earlier, the convergence rate for the positions of the bands is considerably slower. It is observed for chains with approximately 30 carbon atoms with the exception of the skeletal modes, which need a model approximately twice the size to show a similar rate of convergence to the solidstate model. The two modes of polyethylene, which correspond to the Ramanactive T2 mode of diamond, are ν_{4}(π) and ν_{4}(0), both with clear skeletal character. Their frequency convergence is illustrated in Fig. 3. One needs at least 22 carbon atoms to get the positions of the bands with an error smaller than 10 cm^{−1}, 31 atoms for an error smaller than 5 cm^{−1}, and 64 atoms for an error smaller than 1 cm^{−1}. These quantities correspond to 815, 2145, and 17,169 of carbon atoms, respectively, in the octahedral nanodiamond models and to 730, 1800, and 12,761 of carbon atoms, respectively, in the tetrahedral nanodiamond models. The obtained estimates suggest that it should be possible, in principle, to observe in simulation a clear and smooth evolution pattern between the calculated Raman spectra of nanodiamonds and the experimentally recorded Raman spectrum of a single crystal.
In general, the convergence of the simulated IR and Raman spectra is very smooth, both with respect to the positions of the bands and to their intensities. The only irregularities detected during the analysis of Fig. 3 are those for the mode ν_{4} of transoligoacetylenes and the mode ν_{11} of cisoligoacetylenes. Since these irregularities have a quite interesting interpretation, we discuss them here in detail. The detailed shape of the frequency convergence for the two modes in question is shown in Fig. 4. For the mode ν_{11}, the plot consists of five disjointed segments and for the mode ν_{4}, two disjointed segments. In both cases the discontinuities are caused by strong mixing of these modes with other molecular vibrational modes of quasidegenerate frequency. The other modes carry very small IR and Raman intensity, but the strong mixing results in an intensity borrowing leading to two, almost degenerated signals in the vibrational spectra. Note that the magnitude of the resulting splitting is rather small and these effects are practically undetectable from the analysis of Figs. 1 and 2. The convergence irregularity is particularly intriguing for ν_{11}, for which five nearly parallel modes are responsible for the observed effect. These modes correspond to the lowenergy overtone deformations of the chain and in the limit of infinitely many atoms would converge to the acoustic phonons of the polymer.
Comparison of the simulated and experimentally observed frequencies of the IRactive vibrational modes for short oligoyne chains
Molecule  Experiment^{a}  SCCDFTB 

C_{4}H_{2}  628  576 
2006^{b}  2019  
3332^{b}  3260  
C_{6}H_{2}  622  569 
2121  2086  
3327  3256  
C_{8}H_{2}  622  567 
2023  2026  
3329^{b}  3254 
The comparison of the simulated IR spectra of finite cisoligoacetylenes, trans oligoacetylenes, and oligoethylenes with the experimental [46] IR spectra of cispolyacetylene, transpolyacetylene, and polyethylene is performed in Fig. 6a. For cispolyacetylene, the correspondence between the calculations and the experiment is reasonable, even if the intensity pattern of some of the peaks is distorted. For the remaining two systems, the correspondence is rather poor with most of the spectral features either reproduced with too small or too large of an intensity. Note that the differences are so distinct that for these two systems it is even difficult to state positively that the simulated and experimental spectra refer to the same molecular system. Probably, the largest obstacle in reproducing the experimentally recorded spectrum lies in the fact that we try to reproduce the solid system of many entangled oligoacetylene and oligoethylene fibers using single, noninteracting fibers in its gasphase equilibrium geometry. It is clear that in this way we completely neglect the chainchain interactions, nonequilibrium geometry effects, and possible sample imperfections (e.g., chain branching). These effects can be the explanation of the fact that the quite substantial IR intensity at 1237 cm^{−1} associated with the inphase wagging motion of the CH_{2} groups in the simulated oligoethylene spectra is almost completely depreciated by the experimental conditions, which allow neither for undisturbed hydrogen wagging nor for the collective motion of the whole system required for producing sizable dipole moment change. On the other hand, the experimental spectrum shows bands located at around 725 cm^{−1} and corresponding to the CH_{2} rocking. It would be definitely very interesting to investigate why in the SCCDFTB calculation this mode is almost invisible due to its low intensity. We feel that the next natural step toward reproducing the experimentally observed IR spectrum of these systems should be based on probing the dipole moment of the solidlike cluster of oligoacetylenes or oligoethylenes during the finitetemperature SCCDFTB molecular trajectory and obtaining the IR signal from Fourier transform of the corresponding autocorrelation function like for example in ref. [50].
The comparison of the simulated Raman spectra of analogous systems, given in Fig. 6b, gives much closer resemblance between the experimental and theoretical findings. For cispolyacetylene, the experimental spectrum [43] displays three intensive peaks in addition to other, less distinct spectral features. Our simulated Raman spectrum reproduces this pattern even if the most intensive peak is shifted toward lower energies by some 160 cm^{−1}. For transpolyacetylene, the correspondence between the experiment [44] and theory is even better with two intensive peaks reproduced by simulations with too small frequencies (approximately 170 cm^{−1} for the more intensive band). The analogous comparison for polyethylene shows that the calculated and experimental [45] Raman spectra have a quite similar pattern of bands. In the CH stretch region, positions of both bands are quite well reproduced despite their too small relative intensity. For the spectral window between 800 cm^{−1} and 1500 cm^{−1}, not only the peak pattern but also the intensity in SCCDFTB calculation are quite similar.
Comparing the simulated IR and Raman spectra with experiment, one probably cannot miss the impression that the Raman spectra are reproduced in a much better degree than the IR spectra. We do not fully understand this phenomenon and can only speculate here on its origin. One of the possible explanations may come from a weaker dependence of the invariants of the molecular polarizability on the environment. Another possible explanation can come from the way SCCDFTB assesses the molecular dipole moments and polarizabilities needed for computing the IR and Raman intensities. While the first ones are computed directly from Mulliken induced charges, the second ones are computed as derivatives of the total energy with respect to the external electric field.
Conclusions

Convergence of the band intensities in the simulated IR (see Fig. 1) and Raman spectra (see Figs. 1 and 2, respectively) is achieved for chains containing approximately 60 carbon atoms. The convergence is noticeably faster for chains without πconjugation.

The convergence of band intensities manifests itself in two possible ways: a) convergence toward a constant value and b) convergence toward linear scaling with the number of carbon atoms in the chain. The convergence of type a is associated with the vibrational activity of the terminal fragments of a given chain; it signifies that such a band will be optically inactive in an infinitely long polymer. The convergence of type b is associated with vibrational modes resembling the fundamental harmonic vibrations of a finite string, i.e., modes, in which the adjacent monomer fragments vibrate in phase.

The convergence of band locations to their solidstate positions is achieved with a few exceptions for relatively short chains containing approximately 30–50 monomer units. The exceptions usually concern the skeletal vibrational modes of the chains. For these modes, the rate of convergence seems to be related to the degree of the π conjugation in a given chain. The skeletal vibrations in oligoethylenes (systems without π conjugation) converge for approximately 60 carbon atoms, while for the conjugated systems (oligoynes and oligoacetylenes) no perfect (within 1 cm^{−1}) correspondence to the solidstate calculations is observed even for chains containing 150 carbon atoms.

It is remarkable that the observed convergence patterns, for both intensities and the locations of the bands, are very smooth. The only (practically negligible) irregularities correspond to the bifurcations observed in the frequency convergence of a single mode of cisoligoacetylenes and a single mode of transoligoacetylenes, which originate from very local mixing of these modes with other, quasidegenerate molecular vibrations. For details, see Figs. 3 and 4.

The simulated Raman spectra of long oligomers show quite reasonable correspondence to the available experimental data. The simulated IR spectra display much worse agreement with experiment. Possible explanations for this behavior are discussed.
The simulated convergence rate for the oligoethylene chains can be used to estimate the minimal number of atoms necessary to observe the convergence in the evolution of Raman spectra of nanocrystalline diamonds. Our calculations suggest that models containing approximately 2000 carbon atoms should reproduce the positions of the IR and Raman active bands with errors smaller than 5 cm^{−1} and models containing approximately 15,000 carbon atoms, with errors smaller than 1 cm^{−1}. As the observed intensity pattern usually requires a smaller number of atoms to observe the evolution convergence, we expect that quantum chemical calculations of Raman spectra of nanodiamonds up to 20,000 carbon atoms should answer all the possible ambiguities concerning the observed and simulated Ramanactive bands in diamondcontaining materials.
We believe that our results demonstrate clearly that “infinity” can manifest itself in molecular systems at a surprisingly small scale. We hope that our results (see also refs. [26] and [29]) will be useful and will give valuable motivation to all those who study various aspects of finiteness in extended systems: lack of translational periodicity, presence of defects, effects of finite size, convergence of various physical and chemical properties, etc.
Notes
Acknowledgments
Ministry of Science and Technology, Taiwan (MOST 1052113M009018MY3) and the Center for Emergent Functional Matter Science of National Chiao Tung University from the Featured Areas Research Center Program within the framework of the Higher Education Sprout Project funded by the Ministry of Education, Taiwan. We are grateful to the National Center for Highperformance Computing, Taiwan for computer time and facilities.
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