**1. Introduction/Background History**

Polyacetylene is selected for review because of its relative simplicity; the small periodic repeat permits polyacetylene to be treated by sophisticated computational methods. The path from bond-alternate potential minimum to symmetric bond-equivalent maximum is along a single normal mode (the Peierls mode). The vibrational spectrum of polyacetylene is relatively simple. In particular, the Raman spectrum is quite sparse and the oligopolyenes that lead to polyacetylene show characteristic frequency shifts and relative intensity changes which increase in chain length. On the experimental side, however, polyacetylene is entirely insoluble, reactive with itself, and has not been obtained in crystalline form that yields to single crystal diffraction, a feature that it shares with most polymers. Another reason for this review is that there have been several recent experimental and theoretical studies of polyacetylene and finite chain oligopolyenes of known length that are relevant to studies of the structure of polyacetylene. Furthermore, the evidence for the existence of bond alternation in polyacetylene has never been critically reviewed. This exercise shows that the fully extended chain with an all *trans* geometry has never been made. We outline a new method for the preparation of polyacetylene with these properties.

The first publication on the electrical conductivity of doped polyacetylene in 1977 [1], and in a recent 2017 review [2], it is stated as known that *trans*-polyacetylene exhibits bond alternation as a consequence of Peierls instability. It should be noted that Peierls instability refers to a negative curvature in the potential energy for a one-dimensional lattice. The resulting "dimerization", if it occurs, would mean that the periodic repeat for the electron exchange integral is two CH units, and thus the material has filled bonding and empty antibonding π-orbital bands. Thus, *trans* polyacetylene in its fully extended periodic form would be a semiconductor if it has bond alternation and a conductor

if it does not. A close reading of the literature suggests an alternative interpretation of the argument: "since polyacetylene is not a conductor, it must be a semiconductor, and thus it must exhibit bond alternation". The alternative to this argumen<sup>t</sup> is that "polyacetylene" is really a mixture of finite chains and cross-linked polymer segments.

In this first of many papers [1], it was noted that bond alternation can also be inferred in long linear conjugated polyenes because the allowed optical "band gap" transition converges to a constant value with increase in conjugation length. An alternative view based on strong electron correlation for this limit that does not require bond alternation [3] in order to exhibit a finite limiting electronic excitation energy was mentioned in this initial work [1]. It is argued below that this alternative Mott semiconductor (Hubbard Hamiltonian) picture is substantially correct for very long chains.

Bond alternation is the key ingredient in the interpretation of the observations for polyacetylene in terms of a semiconductor band gap and "doping" enhancement of electrical conduction. In the recent review [2] it is stated that this Peierls barrier is very high on the basis that polyacetylene cannot be made to have equal bond lengths at any reasonable temperature. This proposed double well potential is rarely shown with a quantitative energy scale. The Peierls effect in the case of a π-electron system with a stiff σ-framework might lead to a negligibly small barrier. It is only in classical mechanics that atomic positions are at the bottom of the potential.

The earliest reasonable estimation of the "dimerization energy" (Peierls barrier) is a 1983 work [4], where Hartree Fock and small basis set MP2 (Moller-Plesset second order) methods were used. The MP2 value for the difference in energy at the symmetry point and either minima of the potential is 2 kJ/mol (700 cm<sup>−</sup>1) [4]. A more extensive 1997 treatment [5] used multiple linear conjugated polyenes of increasing length, for which optimized structures were compared with equal bond structures or with structures that had one of several variations of bond alternation with optimized bond lengths at the molecular ends changing to equal bonds in the middle of the chain. The resulting dimerization energy extrapolated to 1/N = 0 at the MP2/6-31G\* level was 0.4 ± 0.1 kJ/mol depending on the method of structure variation used. The low end of this range is 0.3 kJ/mol or 105 cm<sup>−</sup>1. The point that has never before been considered in this context is that the harmonic vibrational motion at the bond alternation geometry has a frequency known since at least 1958 to be in the 1500–1600 cm<sup>−</sup><sup>1</sup> range corresponding to a double bond stretching motion [6–8]. The harmonic zero-point energy is higher than the estimated barrier height.

We have shown, as discussed below, that while polyacetylene must have a potential energy surface with two equivalent minima, it cannot exhibit bond alternation, i.e., a periodic alternation between short and long bonds. The band structure argumen<sup>t</sup> that leads to this conclusion, leads conversely to the conclusion that if polyacetylene did not have bond alternation, it would be metallic. It would appear, however, that polyacetylene is not metallic unless it is "doped" with electron donors or acceptors. The terminology is from elemental semi-conductors where the doping is elemental. Here the "dopant" is usually molecular. Thus, the experimental evidence appears to differ from a conclusion in regard to ground state structure of polyacetylene that seems elementary. The alternative point of view investigated in this paper is that this conflict between our conclusions regarding the necessary lack of bond alternation of polyacetylene and experimental results for so-called "polyacetylene" is that what is called "polyacetyene" is not in fact polyacetylene, but is, rather, a mixture of finite chain polyenes of various lengths. The addition of dopants permits electron transport from chain to chain. These oligopolyenes do exhibit bond alternation and give rise in spectroscopic studies this property but this is due to end effects.

The term "finite polyene" here means that the molecule exhibits bond alternation with the terminal carbon–carbon bond length being shorter than the average of the values near the center of the molecule by an amount of 0.003 Å or more; barely measurable but otherwise arbitrary. We do not know and cannot easily compute how long a polyene needs to be such that it has two minima in its bond alternation potential and further how long it must be to have two equal energy minima. Whatever that is, it defines "approaching infinite". Minima of exact energy equality energy define "infinity" for the chain length.

To reiterate, our conclusion is not that polyacetylene has a single minimum potential energy for the bond alternation atomic displacement mode, but rather that the zero-point level is above the barrier that separates two minima. It is the probability distribution of the zero-point barrier that determines the structure, not the minima of the potential. Both minima are equally populated in the zero-point probability distribution. Simulations with reasonable parameters predict, in fact, that the maximum of the probability distribution is at the symmetry point where the potential is a maximum.

When referring to studies of the preparations, we use the notation "polyacetylene". We restrict the terms *cis-* or *trans-*polyacetylene to hypothetical infinite chains in their fully extended conformation. We are primarily concerned here with *trans*-polyacetylene. An infinite transationally symmetric chain is the starting point for the standard treatment of polymer vibrations in general as developed by Born and von Karman. As in all periodic problems, the description of the nuclear motion is the product of a local function times a function with translational symmetry along the chain propagation direction. It is the vibrational levels of the periodic repeat unit that count in establishing the zero-point level energy and the "optical" vibrational excitations in infrared and Raman spectra.

The relevant internal degree of freedom in this case is the bond alternation or Peierls distortion mode, a mixture of double bond expansion and single bond contraction. The potential energy in which the nuclei move for this degree of freedom necessarily has two minima that are exactly equivalent in energy only for the infinite chain. The equivalence of the minima derives from the fact that the energy difference between the two patterns of bond alternation becomes negligibly small per repeat unit as the chain length becomes very long. This argumen<sup>t</sup> makes no statement as to the height of any barrier between the two minima, if there is one.

Assuming that there is a barrier, then the issue of how to treat the nuclear motion arises. In this regard, there are two relevant observations. (1) The harmonic vibrational frequency that corresponds to motion along the Peierls degree of freedom is computed to have a value in the harmonic approximation of ca. 1500 cm<sup>−</sup>1. This corresponds to the strongest Raman active mode at a similar wavenumber. (2) The best estimate of the height of the Peierls barrier via extrapolations discussed above is 100–300 cm<sup>−</sup><sup>1</sup> [5]. The harmonic zero point level is thus 2–7 times larger than this barrier height. Use of the harmonic approximation is clearly not justified.

### **2. Summary of This Review**

We first review in Section 3 the double minimum problem in general, for two molecular cases, and then in Section 4 the specific case of polyacetylene. This is followed in Section 5 by a survey of experimental observations on polyacetylene in the literature that are relevant to bond alternation. It is found that X-ray diffraction, solid state NMR (Nuclear Magnetic Resonance), and polarized IR studies are compromised by ambiguities internal to the studies or to the presence of the finite chains, or both. The electronic spectroscopy of finite conjugated polyenes is then discussed in Section 6. The conclusion of these spectroscopic studies is that a low-lying doubly excited "diradical" state with the same symmetry as the ground electronic state is the lowest energy electronic excitation. This conclusion for the best studied case of octatetraene has recently received theoretical treatment, whose results are in excellent agreemen<sup>t</sup> with experiment. Admixture of this 21Ag excited state with the ground 11Ag state is the origin of the double minimum barrier for polyacetylene in its ground electronic state. It is also the basis of the difficulty in dealing theoretically with the ground state of polyacetylene with current periodic quantum chemical computational methods, since it requires inclusion of at least all doubly excited configurations at a non-perturbative level. This is followed by a brief review in Section 7 of the experimental electronic and vibrational Raman spectra of finite linear polyenes facilitated by the recent availability of such materials in homologous series. In Section 8, it is shown how these Raman spectra are relevant to our ongoing experimental Raman and vibrational inelastic neutron scattering studies of a molecular crystal for which photochemical elimination polymerization has been demonstrated

to occur that leads to polyacetylene constrained to be fully extended in parallel channels formed by an inert lattice that also prevents cross-linking reactions. The iodine atoms that are photochemically cleaved are able to leave the host crystal as iodine vapor. In Section 9, the salient features gleaned from the literature are reviewed and an outlook is presented.

### **3. Double Minimum Potential Vibrational Energy Levels: Ammonia and [18]-Annulene**

The mathematical technology for determination of the vibrational energy levels of arbitrary one-dimensional potential is now straightforward. These methods were developed to treat numerous molecular potentials [9–15] that have two equivalent minima. The most famous of these is ammonia, where the tunneling splitting is ca. 0.8 cm<sup>−</sup>1. A potential that fits the precise vibrational data is shown in Figure 1 [9–13]. A potential that has the form V(x) = C2x<sup>2</sup> + C4x<sup>4</sup> with C2 = −9000 cm<sup>−</sup><sup>1</sup> A−<sup>2</sup> and C4 = 10,000 cm<sup>−</sup><sup>1</sup> A−<sup>4</sup> and a reduced mass of 1.008 amu has a tunneling splitting of 0.45 cm<sup>−</sup><sup>1</sup> (vs. 0.79 cm<sup>−</sup><sup>1</sup> of Figure 1). The 0 to 1 transitions of 932.5 and 968.3 cm<sup>−</sup><sup>1</sup> are computed to be at 940.3 and 969.8 cm<sup>−</sup>1. The barrier height of 2031 cm<sup>−</sup><sup>1</sup> is 2025 cm<sup>−</sup><sup>1</sup> in this simple treatment using the efficient FGH (Fourier Grid Hamiltonian) method [14,15]. The reduced mass for ammonia varies along the out-of-plane umbrella coordinate. For the equilibrium pyramidal geometry, the value is 1.18 amu, while at the trigonal D3h maximum it is 1.20 amu. This increase relative to the mass of H reflects the small geometry-dependent contribution of the N atom to the inversion normal mode. The zero-point level tunneling splitting of ammonia corresponds to an inversion time for the pyramidal superposition state of about 11 ps. This follows from the tunneling splitting 0.45 cm<sup>−</sup><sup>1</sup> for NH3 in the simplest model treatment. This same model gives 792 ps for the tunneling splitting for ND3.

**Figure 1.** Umbrella mode potential for NH3 with transition and level splittings indicated [9,10].

**Figure 2.** One of the D3h Kekule structures of [18]-annulene.

Another molecular example of more relevance to polyacetylene is [18]-annulene, Figure 2 [16–18]. This simple cyclic C18H18 compound is the 4*n* + 2 analog of benzene (*n* = 1) with *n* = 4, and is thus

expected to be aromatic. To make a complicated story short, this conclusion is consistent with the observation of six-fold equivalent bonds in the X-ray diffraction structure but not with the computed NMR spectrum (for which the inside and outside protons are not shifted in opposite directions by the same amount as is the case for the D6h symmetry). It has been proposed that [18]-annulene has a D3h bond-alternate structure. A method of computation is found that results in a D3h bond-alternate structure that results in agreemen<sup>t</sup> with the NMR spectrum [16]. This proposed geometry is either one of the structures corresponding to the minima of the potential in Figure 3. The zero-point level and probability distribution are shown. This proposed geometry is either one of the structures corresponding to the minima of the potential in Figure 3. The zero-point level and probability distribution are shown. A vibrational normal mode analysis at the symmetry point maximum and also at the minima gives in each case a reduced mass of 9.315 amu. The proton NMR spectrum computed for 200 points along bond order displacement coordinate weighted by the probability of Figure 3 gives a value in reasonable agreemen<sup>t</sup> with experiment. Other details of this density functional theory (DFT) and FGH treatment for [18]-annulene are in [17]. A classical MD (Molecular Dynamics) treatment for NMR averaging that includes this case is in [18]. An important factor for this case is that one of the normal modes of this molecule converts the structure from the maximum of the potential to either one of the minima and back. This example provides a demonstration that zero-point heavy atom averaging is expected in such cases because of the very stiff nature of the bonds prohibits localization into one of the minimum energy wells. The general point to keep in mind is that even with heavy atom motion, it is impossible to localize a carbon-based structure into a localized bond-alternate structure for a period of time that is significant on an experimental time scale. Benzene is the obvious example.

**Figure 3.** Computed potential energy as a function of displacement from 6-fold symmetry for [18]-annulene (black line) showing the two lowest vibrational energy levels (red) and the probability distribution for the ground state (blue) [17].

### **4. Double Minimum Potential Vibrational Energy Levels: Polyacetylene**

For cases like ammonia, where the double minimum potential represents displacement of the three H atoms out of the molecular plane, and this case of a cyclic hydrocarbon, the potential must necessarily contain only even terms. The potential energy variation for polyacetylene must also necessarily be symmetric due to translational symmetry.

For the case of polyacetylene [19], for which periodic boundary conditions [20] apply, we have followed two independent paths of enquiry in Figures 4 and 5. In Figure 4, we compute the energy of the –CH–CH– periodic repeat using B3LYP/6-311G(2d,2p) with periodic boundary conditions evaluated at 240 points along the potential in one direction. This is then symmetrized by reflection. The barrier height computed by this DFT method is 110 cm<sup>−</sup>1.

**Figure 4.** Computed potential energy of polyacetylene using periodic boundary conditions-density functional theory (PBC-DFT) with B3LYP 6-311G(2d,2p) at 240 points (black points) along one displacement direction with subsequent generation of the symmetric potential shown as blue dotted trace [19]. The horizontal red lines are the two lowest energy levels; the light blue line is the probability distribution.

**Figure 5.** An analytical model potential energy (black curve) for the Peierls bond alternation mode of polyacetylene. The function is a harmonic oscillator plus a Gaussian barrier [19]. The lines are as in Figure 4.

Figure 5 shows the results of an empirical treatment of polyacetylene. The barrier height for polyacetylene has been established by the extrapolation procedure of Guo and Paldus [5] to be less than 200 cm<sup>−</sup>1. In their treatment, a series of computational methods are applied to three structures for each member in a series of a finite polyene chains with an even number of carbon atoms. The structures are (1) the optimized bond-alternate structure, (2) the equal bond length (barrier) structure, and (3) the bond order reversed structure corresponding to the other minimum in the infinite chain case. These energies values relative to the optimized structure are plotted against 1/N, where N is the number of C=C bonds. The values of this energy difference for the bond reversed and bond optimized cases must, of course, extrapolate to zero as 1/N goes to zero. The plot for the barrier height when extrapolated linearly gives a finite value of about 100 cm<sup>−</sup>1. Figure 5 uses a barrier of 200 cm<sup>−</sup>1. This value comes from the observation that for chain lengths that are sufficiently long, the computed values before extrapolation are below that value, so this value is an upper limit. The harmonic force constant for the model of Figure 5 is chosen to match the value of the force constant for C–C single bonds based on harmonic normal mode analyses for simple molecules like ethane. This is the lowest reasonable value. Higher values of this force constant parameter will result in a higher zero-point energy. The reduced mass for both cases is 4.33 amu. This is derived from a Gaussian computation for finite polyenes which uses the Wilson, Decius & Cross prescription [21]. This value depends on the C–C–C bond 120◦ C–C–C angle but is not crucially dependent on this value.

Calculations of the vibrational frequencies of long linear polyenes using MP2 wavefunction methods has been used to compute the force constants needed for harmonic treatment of polyacetylene in the periodic limit. It is found from the resulting vibrational eigenvalues that the vibrational motion that gives rise to the strongest feature in the Raman spectrum is the bond alternation or Peierls motion mode. Figures 4 and 5 show the computed first excited vibrational mode for these potentials are at 1379 cm<sup>−</sup><sup>1</sup> for the DFT computed potential and at 1460 cm<sup>−</sup><sup>1</sup> for the variable parameter treatment. At the time of this work, it was thought that the strongest Raman active mode of what was thought to be polyacetylene was at 1459 cm<sup>−</sup>1. The 1460 cm<sup>−</sup><sup>1</sup> value was chosen as a target value in adjusting the barrier width of this analytic empirical model of Figure 5.

If this modelling procedure is followed with larger values of the barrier height including readjustment of the harmonic force constant so that the same value of the vibrational frequency is obtained, then with a barrier height of 2000 cm<sup>−</sup><sup>1</sup> the ground state zero-point level has a double maximum. Because of the symmetry, the probability of being in one well is the same as being in the other in this and every other state. If the barrier height is raised to 20,000 cm<sup>−</sup>1, then the energy levels occur in pairs with a splitting for the lowest level of 20 cm<sup>−</sup><sup>1</sup> corresponding to femtosecond time scale tunneling. Bond-order alternation states will be exceedingly ephemeral.

### **5. Review of Experimental Observations on Polyacetylene with Emphasis on Bond Alternation**

In the literature, especially in theoretical publications, bond alternation is often used to mean that there are two equivalent minima in the potential energy function. It is our argumen<sup>t</sup> above that this cannot give rise to populated bond-alternate structures with a periodic difference in bond lengths. If the zero-point energy is considered, then both structures are equally populated and, for what is considered to be at least a reasonable approximation to reality, the most probable geometry is at the symmetry point were the two bond lengths are equal. It might be argued that bond alternation in polyacetylene is known to occur on experimental grounds. This is, however, not the case when the experimental studies are viewed critically taking into account the likely presence of finite chains in the sample.

This section on experimental observations is divided into (1) X-ray diffraction, (2) infrared dichroism, (3) solid state NMR spectroscopy, (4) resonance Raman spectroscopy, and (5) a cautionary note on doping.
