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We consider longitudinal nonlinear atomic vibrations in uniformly strained carbon chains with the cumulene structure ($=C=C=)_{n}$. With the aid of ab initio simulations, based on the density functional theory, we have revealed the phenomenon of the $pi$-mode softening in a certain range of its amplitude for the strain above the critical value $eta_{c}approx 11,{%}$. Condensation of this soft mode induces the structural transformation of the carbon chain with doubling of its unit cell. This is the Peierls phase transition in the strained cumulene, which was previously revealed in [Nano Lett. 14, 4224 (2014)]. The Peierls transition leads to appearance of the energy gap in the electron spectrum of the strained carbyne, and this material transforms from the conducting state to semiconducting or insulating states. The authors of the above paper emphasize that such phenomenon can be used for construction of various nanodevices. The $pi$-mode softening occurs because the old equilibrium positions (EQPs), around which carbon atoms vibrate at small strains, lose their stability and these atoms begin to vibrate in the new potential wells located near old EQPs. We study the stability of the new EQPs, as well as stability of vibrations in their vicinity. In previous paper [Physica D 203, 121(2005)], we proved that only three symmetry-determined Rosenberg nonlinear normal modes can exist in monoatomic chains with arbitrary interparticle interactions. They are the above-discussed $pi$-mode and two other modes, which we call $sigma$-mode and $tau$-mode. These modes correspond to the multiplication of the unit cell of the vibrational state by two, three or four times compared to that of the equilibrium state. We study properties of these modes in the chain model with arbitrary pair potential of interparticle interactions.
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Nonlinear vibrations in strained monoatomic carbon chains are studied with the aid of ab initio methods based on the density functional theory. An unexpected phenomenon of structural transformation at the atomic level above a certain value of the strain was revealed in cumulene chain (carbyne-{beta}). This phenomenon is a consequence of stability loss of the old equilibrium atomic positions that occur at small strain, and appearance of two new stable equilibrium positions near each of them. The aforementioned restructuring gives rise to a softening of {pi}-mode whose frequency tends to zero in a certain region of amplitudes when carbon atoms begin to vibrate near new equilibrium positions. This resembles the concept of soft mode whose freezing is postulated in the theory of phase transitions in crystals to explain the transitions of displacement type. The dynamical modeling of mass point chains whose particles interact via Lennard-Jones potential can approximate our ab initio results well enough. In particular, this study demonstrates an essential role of dipole-dipole interactions between carbon atoms in formation of their new equilibrium positions in the cumulene chain. We believe that computer studying of Lennard-Jones chains enables to predict properties of various dynamical objects in carbon chains (different nonlinear normal modes and their bushes, discrete breathers etc.) which then can be verified by ab initio methods.
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