We investigate entirely electronic torsional vibrational modes in linear cumulene chains. The carbon nuclei of a cumulene are positioned along the primary axis so that they can participate only in the transverse and longitudinal motions. However, the interatomic electronic clouds behave as a torsion spring with remarkable torsional stiffness. The collective dynamics of these clouds can be described in terms of electronic vibrational quanta, which we name *torsitons*. It is shown that the group velocity of the wavepacket of *torsitons* is much higher than the typical speed of sound, because of the small mass of participating electrons compared to the atomic mass. For the same reason, the maximum energy of the *torsitons* in cumulenes is as high as a few electronvolts, while the minimum possible energy is evaluated as a few hundred wavenumbers and this minimum is associated with asymmetry of zero point atomic vibrations. Theory predictions are consistent with the time-dependent density functional theory calculations. Molecular systems for experimental evaluation of the predictions are proposed.

## I. INTRODUCTION

Highly efficient and fast vibrational energy transport on a molecular scale has been a subject of theoretical and experimental investigations in recent decades. The possible applications in biochemistry, organic chemistry, and nanotechnology include the development of efficient cooling in microscopic and nanoscopic molecular systems, such as nanowires^{1} and optical limiters, designing efficient energy transport schematics for energy signaling,^{2} as well as optimizing and even promoting chemical reactions by concentrating the excess energy at the reaction center.^{3,4} It is suggested that the quantum vibrational excitations can be manipulated similarly to electrons and photons, thus enabling controlled heat transport. Moreover, delocalized excitations (phonons) can be used to carry and process quantum information.^{5–7} The highest transport speed was found in alkanes (1.44 km/s).^{8}

Possible candidates capable to maintain fast and efficient energy transport are oligomers because of their periodic structure.^{9} In such systems, vibrational states can be substantially delocalized because of the strong interaction of equivalent site states, so that ballistic energy transport takes place as a free-propagating wavepacket. The ballistic constant-speed transport has been observed in bridged azulene-anthracene compounds,^{10} polyethylene glycol oligomers,^{2} alkanes,^{8,9,11} and perfluoroalkanes,^{12,13} and the theory describing this transport and its possible breakdown due to decoherence has been suggested.^{1,14–16}

Phonon wavepackets in carbon based polymer chains can propagate with the group velocity as high as 10 km/s because of a high strength of covalent bonds.^{17,18} Yet the maximum energy of singly excited vibrations does not exceed ca. 3000 cm^{−1}, as the motion is associated with displacements of rather heavy nuclei. Thus, the ballistically transferred energy is much smaller than a typical bond energy exceeding 1 eV (∼10^{5} cm^{−1}). Involvement of multi-phonon transport to increase the amount of transferred energy is expected to enhance the energy relaxation/dissipation. Much larger energy can be carried by excitons, delocalized electronic states.^{19} However molecular excitons are usually strongly coupled to the environment resulting in incoherent energy transport (see, e.g., exciton transport in DNA^{20,21}).

Here we propose to exploit the special vibrational modes of entirely electronic nature capable of efficient delivering energies in the eV range. Such modes can exist in molecules having all atoms aligned along the single axis (see Figure 1) and they are formed by propagating torsional oscillations of electronic nature. Nuclei do not participate in these oscillations because their rotation about the axis they located on is degenerate.

Considering a linear molecule as an elastic rod, four gapless phonon branches are expected based on symmetry, including longitudinal, two transverse, and one torsional modes.^{22,23} The longitudinal and torsional oscillations of frequency *ω* and wavevector *q* are characterized by an acoustic spectrum *ω* = *cq* with a relatively high speed of sound, *c*. As oppose to an elastic rod, for a molecular chain with all atoms located on the same axis, there is no nuclear contribution to the torsional vibrations, since the chain is completely linear. Nevertheless, the system can possess a remarkable torsional stiffness due to anisotropic arrangement of its electronic clouds. Such a situation is found in cumulenes, featuring a chain of carbon atoms coupled to each other by double bonds,^{24} where the anisotropy results from the *π*-bond anisotropy between carbon atoms (see Figures 1(a) and 2). Similar conditions can be realized in transition metals where atoms can form chain bridges between junctions.^{25–27}

The torsional sound should exist in such systems and we expect it to be of a purely electronic nature because nuclei are positioned along the primary axis and cannot participate in the torsional motion. Since electrons are much lighter than atoms, it is natural to expect the speed and a single quantum energy to be much higher than those for nuclei vibrations.

In the present study, we performed a first principle investigation of the electronic torsional waves in cumulene chains. We found that the speed of sound in cumulenes is to be as high as 1000 km/s and a maximum energy of the quantum as high as 10 eV.

Because this type of motion is not related to the nucleus vibrations, the corresponding quantum quasi-particle is not a usual phonon. As a collective motion of electrons, it can be associated with a plasmon with a specific symmetry and a gapless spectrum, though its symmetry properties differ drastically from Langmuir waves in plasma and typical translational plasmons in conjugated systems. To avoid confusion and emphasize the symmetry properties, we call a quantum of torsional electronic oscillations a *torsiton*. The axial symmetry is exact in case of classical treatment of nuclei, while the quantum nuclei vibrations violate this symmetry. These vibrations result in a small *torsiton* spectral gap of the order of 0.02 eV. The possible ways to observe *torsitons* experimentally are discussed.

We consider the torsional oscillations of a cumulene in a dielectric environment, so the electronic excitations can be neglected. Although the metallic behavior of cumulene was predicted theoretically,^{24,28} it is not confirmed experimentally,^{29,30} so the nature of electronic excitations remains unclear. Here we ignore electronic excitations assuming that there is the significant spectral gap (cf. Ref. 29).

## II. THE SYSTEM

A linear cumulene chain is a compound containing a sequence of *n* carbon atoms with (*n* − 1) double bonds between them R = C = (C = )_{n−2}C = R.^{31} Quantum chemistry calculations were performed for the simplest termination of cumulene chain by two hydrogen atoms on each side; an example of cumulene molecule H_{2}C = (C = )_{3}CH_{2} is shown schematically in Figure 1(a). One can see that orthogonal *π*-bonds between carbon atoms can provide rigidity with respect to twisting with remarkable torsional stiffness, while much smaller stiffness is expected in another carbyne modification, polyyne, which is a chain of carbon atoms with alternating single and triple bonds between them (see Figure 1(b)). For cumulene molecules, the shortened notation H_{2}C_{n}H_{2} (without bond type specification) will be used.

## III. ELECTRONIC TORSIONAL MODE

To estimate the speed of sound for electronic torsional wave, we consider a model of elastic rod (torsion spring) which can be described by the Lagrangian

where dynamical variable *θ*(*z*, *t*) is a twisting angle of the rod along z-axis as a function of coordinate along prime axis and time; two neighboring cross sections at points *z* and *z* + *dz* will rotate with respect to each other with a relative angle $d\theta =\u2202\theta /\u2202zdz$.^{32} Here *z*_{l,r} = ∓*L*/2, *L*—molecule length, *κ* stands for torsional stiffness and *j _{e}* is an average linear density of electronic moment of inertia with respect to z-axis.

We estimate parameters of interest as *j _{e}* = 1.73

*m*Å (0.95 ⋅ 10

_{e}^{−3}u Å) and torsional stiffness

*κ*= 10.6 eV Å as described in Secs. IV and V. Our estimate for the torsional stiffness is consistent with the previous estimate of 10.3 eV Å reported in Ref. 24.

With the angle *θ*(*z*, *t*) and the related angular velocity *dθ*/*dt* considered as dynamical variables, Eq. (1) leads to the Euler equation

which is a wave equation with the dispersion relation $\omega (q)=q\kappa /je$ and the speed of *torsiton* wave

using *j _{e}* evaluated below. This velocity exceeds the typical phonon propagation velocity in polymers by two or three orders of magnitude. Next we also estimate the maximum energy transferred by the electronic torsional mode.

The dispersion relation for longitudinal vibration in a uniform chain with nearest neighbor coupling and the lattice period *a* has the standard form *ω*(*q*) = *ω*_{∗}sin(*aq*/2).^{33}

It should be a good approximation for the torsional mode under consideration because the interaction responsible for the torsional stiffness is due to short range covalent bonding. In the long wavelength limit *q*⟶0, we estimate maximum energy of the *torsiton* as

(the lattice period in cumulenes is given by *a* = 1.28 Å).^{34} This value corresponds to the typical electronic excitations energy range.

Though *torsiton* spectrum is gapless in the infinite chain limit if atomic vibrations are neglected, *ω*(0) = 0, the energy of the first excited *torsiton* mode in the cumulene of finite chain length, *L*, corresponds to the minimum wavevector *q _{min}* =

*π*/

*L*. For sufficiently long molecules, the

*torsiton*spectrum acquires the gap due to zero-point atomic vibrations Δε ∼ 0.02 eV, which is much smaller than typical electronic excitation energies (see also Sec. VII).

We can also roughly estimate the *torsiton* mean free path. While a precise analysis of dephasing in cumulene is beyond the scope of the present study, for an estimate, we use the reported data for electronic dephasing in several organic systems. The coherence time of electronic excitations in FMO complexes ranges from 300 to 660 fs,^{35,36} and the electron-phonon scattering rate in bilayer graphene,^{37} another carbon allotrope, is saturated at 5 ps^{−1}, which corresponds to a coherence time of 200 fs. Assuming coherence time for the *torsiton* as *T* ≳ 100 fs, one can estimate its mean free path as *l*_{0} = *cT* ≳ 100 nm, which exceeds a length of any realistic cumulene chain. This estimate is valid for *torsiton* energies not matching other electronic excitations in cumulene. Otherwise the strong scattering will be expected.

Thus we found the electronic torsional sound wave velocity and energy unprecedentedly high compared to typical phonon parameters which makes this system very attractive for energy transport applications. The energy transferred by a single quantum is sufficiently large for chemical applications: bond making-bond breaking, energy release, and energy transfer to reaction center.

Below we derive our estimates for electronic moment of inertia and for torsional stiffness, discuss the limitations of our result due to zero point atomic vibrations, and propose the way to observe the ultrafast energy transport due to electronic torsional sound.

## IV. ELECTRONIC MOMENT OF INERTIA

The linear density of electronic moment of inertia is defined as

where *ρ* is the electronic density. We calculated the linear density of electronic moment of inertia for cumulene molecule using density functional theory (DFT) with B3LYP hybrid functional and 6-31(d, p) basis sets, as implemented in a Gaussian 09 software package.^{38} The electron density as a function of coordinates is extracted with the uniform grid of 0.1 Bohr radius (0.0529 Å), and the symmetric limits in X-Y plane (the plane perpendicular to the prime axis) were chosen ±6.5 Bohr radius (±3.44 Å). Either doubling of the limits or decrease of the grid by the same factor changes the result by less than 1%.

To estimate the accuracy of the numerical result, we tested the same approach on the hydrogen atom. The theoretical value of the moment of inertia of hydrogen electron cloud in the ground state can be calculated using the electron wave-function^{39} as $J0=2mea02$, where *m _{e}* is the electron mass and

*a*

_{0}is the Bohr radius. The result of numerical calculations obtained using the same method as for cumulene is $Jnum=1.90mea02$, which is within 5% accuracy.

In Figure 3, we show dependence *j _{e}*(

*z*) obtained from DFT-calculations for H

_{2}C

_{11}H

_{2}. One can see that

*j*(

_{e}*z*) is a smooth function weakly deviating from its average

*J*/

_{e}*L*(∼5%), so for simplicity coordinate dependent moment of inertia density

*j*(

_{e}*z*) can be replaced with the constant

*j*≃

_{e}*J*/

_{e}*L*≃ 1.73

*m*Å (0.95 ⋅ 10

_{e}^{−3}u Å). As shown in Figure 3, we define molecular length

*L*as a distance between the second left and the second right carbon atoms, where

*j*(

_{e}*z*) is still not affected by boundaries.

## V. TORSIONAL STIFFNESS CALCULATION

Combining the proposed model with the first principles calculations of the hydrogen atom torsional vibrational mode associated with the relative torsional oscillations of pair of hydrogen atoms (“whiskers,” see Figure 2), we introduce Lagrangian

where Φ_{l}(*t*), Φ_{r}(*t*) are the angles of the “whiskers” deviation from equilibrium on the left and right side, *J _{l}* =

*J*=

_{r}*J*/2 are moments of inertia of the whiskers,

*J*is the entire atomic moment of inertia along primary axis, defined by 4 hydrogen atoms.

*θ*(

*z*,

*t*) is the same as in Eq. (1) with boundary conditions

*θ*(∓

*L*/2) = Φ

_{l,r}.

In this model, the potential energy is originated from the torsional strain of the electronic spring and the kinetic energy is entirely defined by the motion of hydrogen atoms, so long as the kinetic energy of electrons is neglected. The latter assumption is justified as long as *J _{e}* ≪

*J*(0.028 vs 3.43 u Å for

*n*= 25).

The torsional energy has a minimum at constant torsional angle gradient (∂*θ*/∂*z*) = (Φ_{r} − Φ_{l})/*L* suggesting that electrons adiabatically follow atomic motion. For the only hydrogen torsional oscillator mode, one can assume antisymmetric condition −Φ_{l} = Φ = Φ_{r}. Then the Euler equation for Lagrangian (6) is

This equation describes the harmonic oscillator with the frequency defined as

Using the same DFT calculation, from harmonic vibrational analysis, one can find *ω*_{τ} of H_{2}C_{n}H_{2} for different *n*. In Figure 4, we presented the related frequency *ω*_{τ} for *n* = 5, 6, 8, 10, 12, 16, 24, 25. Since the choice of length *L* includes some arbitrariness (our choice is illustrated in Figure 3), the correct fit should include some length parameter *B* ∼ *a*, so that $\omega \tau 2=A/(B+L)$. Using optimum fitting analysis, we found *B* = 4.22 Å and the torsional stiffness is given by *κ* = *AJ*/4 ≃ 2.89 ⋅ 10^{6} cm^{−2} u Å^{3} ≃ 10.6 eV Å (u stands for the atomic mass unit), while atomic moment of inertia *J* is defined by end groups only and does not depend on *n*. These estimates were used to evaluate the speed of *torsitons*. Below we discuss the quantum approach to the problem and analyze the effect of zero-point atomic vibrations on the *torsiton* spectrum.

## VI. QUANTUM MECHANICAL APPROACH

In Sec. III, we treated cumulene electronic torsional oscillations classically. Here we consider the quantum mechanical aspects of the problem.

Using periodicity of the system, one can arrange all the electrons into “blocks” or unit cells. In cylindrical coordinates, position of each electron in the *n*th block is described by sets {*θ _{i}*,

*r*,

_{i}*z*},

_{i}*i*= 1…

*N*, where

*N*is the number of electrons in the block. One can introduce new coordinates using the average angles

*ϕ*for description of the torsional angle of the whole block (double bond). Then we suggest to represent the kinetic energy related to these collective variables

_{n}*ϕ*as

_{n} where $\xi \u02c6n=\u2212i\u0127\u2202\u2202\varphi n$ is the operator of the *z*-axis projection of the electron angular momentum in the *n*-th block, and *J _{n}* is a corresponding electronic moment of inertia of the block. Such an expression leads to the right behavior for the overall rotation corresponding to the

*q*= 0 mode. Indeed, one can rewrite Eq. (9) as

where $M\u02c6q=\u2211peiqp\xi \u02c6p$ is defined in the manner that $M\u02c60$ corresponds to the total angular momentum and *J _{n}N* is the total moment of inertia of the molecule with respect to the

*z*-axis. For the long wavelength modes

*qa*≪ 1 we expect that one can use the limit of

*q*⟶0.

The potential energy can be derived solving the corresponding Born-Oppenheimer problem. The expected form would be a sum like

satisfying the axial symmetry requirements, where {*U _{nk}*} are the functions of the previously introduced angles {

*ϕ*}; more complicated constructions than binary interactions can also be included, but they will not affect our further consideration. It is assumed that all high energy (“fast”) degrees of freedom are excluded, while the low energy (“slow”) degrees of freedom left, which are related to the long wavelength

_{n}*torsitons*possessing a small energy compared to the other electronic excitations. Leaving only the nearest neighbor interactions and making expansion near the minimum, we come up with the quadratic Hamiltonian (cf. Eq. (1)),

where the moment of inertia on the *n*th block can be taken as *J _{n}* ≃

*j*⋅

_{e}*a*, where

*a*is a distance between two adjacent atoms and

*κ*/

*a*is a torsional stiffness per unit cell (see also Sec. IV). This Hamiltonian has a spectrum

^{40}

This prediction is consistent with the results of time-dependent density functional theory (TD DFT) calculations of the first excited electronic state (see Fig. 5). Indeed, the energy of this state scales as the inverse molecular length, in agreement with Eq. (13) for *q* = *π*/*L*. The obtained energy is also consistent with the estimate based on the electronic stiffness. We do not expect that this mode is associated with the electron delocalization since in that case a 1/*L*^{2} dependence for the minimum energy is expected, similarly to the particle in the box problem.^{39}

In Fig. 5 we present the results of TD DFT calculations with B3LYP hybrid functional and 6-31(d, p) basis sets, as implemented in a Gaussian 09 software package^{38} (see also Sec. IV) for the first excited singlet electronic state in cumulene (red diamonds) and polyyne (green squares) with their fit based on the theoretical predictions (Eq. (13), Ref. 40). The determined speed of sound is very close to that obtained with Eq. (3). The result deviates from the nearest neighbor model Eq. (12) at small molecular length, which is probably a consequence of the next neighbor interactions. We also did the calculations for polyyne and found the saturation of minimum energy at about 1.27 eV, which estimates the gap in the electronic spectrum there.

To explain the qualitative difference in the cumulene and polyyne spectra, we matched our model to the well investigated model of the Josephson junction array,^{41}

The kinetics energy in this model is associated with the Coulomb repulsion of superconducting granules, while the cosine term describes the Josephson coupling between granules. The parameters of the matching model consistent with our model for cumulene are *E _{C}* =

*ħ*

^{2}/(

*j*) = 3.45 eV,

_{e}a*E*=

_{J}*κ*/

*a*= 8.28 eV. This model shows a quantum phase transition between superconducting state with a weakly oscillating phase difference between granules and the insulating state with weakly correlated phases of granules at

*G*=

_{c}*E*/

_{J}*E*= 1.23. In our case this is the phase transition between an ordered ground state (cumulene) having gapless torsional mode (

_{C}*torsiton*) and an uncorrelated state with the gap in the spectrum realized in polyyne. Indeed, in the first case corresponding to cumulene (

*E*/

_{J}*E*= 2.40 >

_{C}*G*), the system has a continuous spectrum of gapless excitations (

_{c}*torsitons*in our case) while in the second case, there is the gap ε

_{p}= 1.27 eV in the spectrum as we found in polyyne (see Fig. 5). In polyyne the interaction between bond orientations is much weaker than that in cumulene because the triple and single bonds are almost axially symmetric. Therefore this system is likely on the “insulating” gapped side of the quantum phase transition. Quantum phase transition in one dimensional systems takes place at zero temperature only. Since our system is finite and the thermal energy

*k*is small compared to electronic interaction energies the zero temperature consideration should be still applicable.

_{B}TIn spite of our quantum mechanical derivation is not rigorous we believe that being considered together with TD DFT simulation results and classical arguments of Sec. III it supports our expectations of the existence of the electronic sound spectral branch, i.e., *torsitons*.

## VII. EFFECT OF ZERO-POINT ATOMIC VIBRATIONS

In our description of the electronic torsional mode, we implicitly used Born-Oppenheimer approximation, considering electronic motion in an axially symmetric field of motionless nuclei, positioned along the z-axis. This axial symmetry is reflected by the symmetry of the Lagrangian in Eq. (1) with respect to the change of the function *θ*(*z*) by arbitrarily constant.

In reality, the nuclei participate in zero-point vibrations in the ground state, which does not possess an axial symmetry because this ground state is adjusted to the electronic ground state where this symmetry is broken (see Figure 2). Indeed, to find this ground state, one needs to consider interacting nuclei in the field of electronic cloud with already calculated anisotropic electronic density.

Thus, the potential energy depends on the angle *θ* even in the absence of torsion and the energy minimum is realized at some angle *θ*_{0} which we can set to zero. The potential energy can be expanded over the small displacement from this minimum as *αθ*^{2}/2. This term incorporated to the Lagrangian in Eq. (1) as −*αθ*^{2}/(2*L*) leads to the gap in the spectrum of torsional waves. Correcting Eq. (2) by −*αθ*/(*j _{e}L*) term in the right hand side, we obtain a new dispersion relation

Since *ω*(0) ≠ 0 the mode is not exactly acoustic due to the gap $\Delta \omega =\alpha /jeL$.

To estimate the parameter *α* consider the change of classical energy $\delta E(\theta )=">H\u02c6(\theta )\u2212H\u02c6(0)g$, where $H\u02c6$ is the atomic chain quantum Hamiltonian and $">\cdots g$ is an average over the ground state of carbon atoms considering their zero-point vibrations.

All the normal modes of carbon atoms in the molecule, which do not include motion of hydrogen atoms with respect to the adjacent carbon atoms, can be either longitudinal or transverse. For *D*_{2d} symmetry point group with coordinate system defined above, there are only two possible second order invariants: *z*^{2} and *x*^{2} + *y*^{2}, so the transverse modes of the harmonic Hamiltonian of the atomic chain are expected to be double-degenerate and the corresponding eigenfunctions possess axial symmetry.^{42} Practically, this degeneracy is observed in DFT-calculated IR spectra of H_{2}C_{n}H_{2} for odd *n*, while for even *n* all energy levels are split, because such molecules belong to *D*_{2h} symmetry group. Indeed, the splitting is entirely an effect of sides, because in even *n* molecules the side CH_{2} groups lie in the same plane (while in odd *n* they are orthogonal), so X-Z and Y-Z planes become distinguishable, while for an infinite chain this effect would disappear.

The break of axial symmetry takes place in the third order anharmonic interaction. To express potential energy in normal modes, representation introduces notations *u*_{xi} and *u*_{yi} for transverse modes with energy *ħ ω*_{xi} = *ħ ω*_{yi} = *ħ ω _{i}* and

*u*

_{zk}for

*k*th longitudinal mode. Thus the third order anharmonic energy is expressed by

where it is assumed that *V*_{xixjz} = *V*_{yiyjz} for any *i*, *j*.

With $V\u02c63$ as a perturbation, a meaningful correction to the ground state in the first order of perturbation theory is given by

The rotation of electronic cloud about the z-axis by an angle *θ* changes the energy in diabatic approximation by

Assuming for the small displacement from the minimum sin*θ* ≃ *θ*, one can find $\delta E(\theta )=\psi 0\delta H\u02c6(\theta )\psi 0=\alpha \theta 2/2$, with *α* given by

Using anharmonic frequency analysis of H_{2}C_{5}H_{2}, we calculated third-order anharmonicity constants.^{38,43} To exclude effect of hydrogen atoms, we considered the only transverse and longitudinal normal modes with the nearest integer of reduced mass greater than or equal to 2 atomic units. Applying Eq. (19) we found *α* = 3.2 cm^{−1}, so that the energy gap can be estimated using Eq. (15) as Δε ≃ 130 cm^{−1}.

To answer a question how crucial is the described effect for the acoustic mode, one can find the length *L* of a cumulene chain where this energy becomes comparable to the minimum *torsiton* energy, which can be estimated as

Thus the length required to make the gap value of the same order as *ħ ω _{min}* is

*L*

_{∗}≃

*ħ πc*/Δε ≃ 128 nm, that is, much larger than the real molecule length.

^{44}

Axial symmetry can be violated also by the fourth-order anharmonic interaction, however its contribution into the energy gap does not change qualitatively the presented estimate.

## VIII. EXPERIMENT SUGGESTIONS

As shown above, the electronic torsional mode features an unprecedented speed of 1000 km/s = 1 nm/fs and can transfer energy up to 10 eV, which is comparable to the energies of the strongest chemical bonds (C=C, N ≡ N, etc.). Such a high transferred energy brings an opportunity of performing chemistry at distances, including chemical bond breaking reactions. Figure 6 shows a schematic of the compound suitable for the proof of principle experiment on remote chemistry initiation. The compound features two surface-anchored end-groups connected by a cumulene chain. Laser initiated bond breaking at the initiation (left) end-group can result in generation of a strong torque at the chain which will propagate as a wave-packet along the chain and can result in bond breaking at another end group, the target. The energy released by the initiation end-group can be tuned by selecting convenient functional groups. Spectroscopic observation of the transported energy can aim at detecting the formation of the products at the target or detection of the excess energy at the target. In the latter case, a longer cumulene chain is required as for the chain length of 50 carbon atoms the transport time is only ca. 5 fs. Compounds with such long chains have been synthesized for polyynes^{45} and we hope that this should be possible for cumulenes as well.

## IX. CONCLUSION

In this article, we considered electronic torsional waves in cumulene chains (*torsitons*) which are torsional sound waves of entirely electronic nature. We evaluated the speed of *torsiton* propagation as high as 1000 km/s. A single *torsiton* can carry energy from almost 0 to 10 eV. Similar waves should exist in other atomic chain with anisotropic bonds including recently discovered transition metal linear chains. While the largest band energy computed for cumulenes at 10 eV, the computations neglected electronic excitation, which will likely be contributing at such high energies. It will be interesting to see how the ground electronic state *torsitons* are perturbed by electronic excitations at higher *torsiton* band energies and how the quasi-particles of two types, *torsitons* and excitons, interact. Nevertheless, the presented band calculations are expected to be free of electronic excitation effects at smaller energies. Importantly, the transport speed supported by the lower half of the *torsiton* band is similar to that of the full band (with small corrections due to the *torsiton*-vibron coupling).

## Acknowledgments

We thank Marina Matherne for the suggestion of cumulene chain to consider, Dmitry Polyanski for the discussion of bond breaking reaction at distance, and Noa Marom for suggestions on electronic density calculation. This work was supported by the National Science Foundation (Grant No. CHE-1462075) program.