Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantifying the Chirality of Vibrational Modes in Helical Molecular Chains

Ethan Abraham, Abraham Nitzan

TL;DR

The study tackles the ambiguity in defining chirality for molecular vibrational modes by introducing two quantitative measures: the Continuous Chirality Measure (CCM) applied to both molecular structure and individual normal modes, and a momentum-based pseudoscalar spectrum (MPS) that captures handedness in vibrational motion. By analyzing a two-stranded polyethylene double helix across varying twist, the authors show that structural twist drives the normal-mode CCM to higher values, and that MPS reveals band-dependent handedness, with twist-induced shifts in low- and high-frequency bands. They furthermore define a thermal chirality $\xi_z(T)$ that behaves quantum-mechanically (nonzero at low $T$) and discuss parity properties, showing CCM is parity-even while MPS is parity-odd. These findings provide a concrete framework linking molecular structure chirality to the chirality of vibrational excitations and suggest potential implications for chiral phonon phenomena and related thermal effects in twisted molecular systems.

Abstract

Chiral phonons have been proposed to be involved in various physical phenomena, yet the chirality of molecular normal modes has not been well defined mathematically. Here we examine two approaches for assigning and quantifying the chirality of molecular normal modes in double-helical molecular wires with various levels of twist. First, associating with each normal mode a structure obtained by imposing the corresponding motion on a common origin, we apply the Continuous Chirality Measure (CCM) to quantitatively assess the relationship between the chirality-weighted normal mode spectrum and the chirality of the underlying molecular structure. We find that increasing the amount of twist in the double helix shifts the mean normal mode CCM to drastically higher values, implying that the chirality of molecular normal modes is strongly correlated with that of the underlying molecular structure. Second, we assign to each normal mode a pseudoscalar defined as the product of atomic linear and angular momentum summed over all atoms, and we analyze the handedness of the normal mode spectrum with respect to this quantity. We find that twisting the double-chain structure introduces asymmetry between right and left-handed normal modes so that in twisted structures different frequency bands are characterized by distinct handedness. This may give rise to global phenomena such as thermal chirality.

Quantifying the Chirality of Vibrational Modes in Helical Molecular Chains

TL;DR

The study tackles the ambiguity in defining chirality for molecular vibrational modes by introducing two quantitative measures: the Continuous Chirality Measure (CCM) applied to both molecular structure and individual normal modes, and a momentum-based pseudoscalar spectrum (MPS) that captures handedness in vibrational motion. By analyzing a two-stranded polyethylene double helix across varying twist, the authors show that structural twist drives the normal-mode CCM to higher values, and that MPS reveals band-dependent handedness, with twist-induced shifts in low- and high-frequency bands. They furthermore define a thermal chirality that behaves quantum-mechanically (nonzero at low ) and discuss parity properties, showing CCM is parity-even while MPS is parity-odd. These findings provide a concrete framework linking molecular structure chirality to the chirality of vibrational excitations and suggest potential implications for chiral phonon phenomena and related thermal effects in twisted molecular systems.

Abstract

Chiral phonons have been proposed to be involved in various physical phenomena, yet the chirality of molecular normal modes has not been well defined mathematically. Here we examine two approaches for assigning and quantifying the chirality of molecular normal modes in double-helical molecular wires with various levels of twist. First, associating with each normal mode a structure obtained by imposing the corresponding motion on a common origin, we apply the Continuous Chirality Measure (CCM) to quantitatively assess the relationship between the chirality-weighted normal mode spectrum and the chirality of the underlying molecular structure. We find that increasing the amount of twist in the double helix shifts the mean normal mode CCM to drastically higher values, implying that the chirality of molecular normal modes is strongly correlated with that of the underlying molecular structure. Second, we assign to each normal mode a pseudoscalar defined as the product of atomic linear and angular momentum summed over all atoms, and we analyze the handedness of the normal mode spectrum with respect to this quantity. We find that twisting the double-chain structure introduces asymmetry between right and left-handed normal modes so that in twisted structures different frequency bands are characterized by distinct handedness. This may give rise to global phenomena such as thermal chirality.
Paper Structure (7 sections, 18 equations, 12 figures)

This paper contains 7 sections, 18 equations, 12 figures.

Table of Contents

  1. Computational Details
  2. Molecular Models
  3. CCM Normal Mode Spectra
  4. Phonon Angular Momentum
  5. Momentum Pseudoscalar Spectra
  6. Behavior of the Normal Mode Chirality Measures Under Parity
  7. Inter-Chain Distance

Figures (12)

  • Figure 1: (a) The model system for this study: a two-stranded $N=98$ polyethylene wire containing various levels of (left-handed) twist (shown from top to bottom: 0, 4, 8, 12 twists). The $z$-axis is taken to be the axis of the chain. Terminal atoms shown in red. (b) The CCM (Eq. (\ref{['ccm_mol']})) of the wire as a function of the number of twists. Results are shown when the structure $Q$ is taken to be a segment of the wire corresponding to (pink) one helical turn, (green, red, gold) 5, 10, or 20 monomers, and (blue) the entire structure. (c) Histograms of the normal mode distribution binned by CCM for the structures shown in the top panel.
  • Figure 2: (a) Frequency polygons showing the spectral densities of the two-stranded $N=98$ polyethylene wire when the wire is (red) untwisted, or (blue, purple, green) containing 2, 4, and 12 twists. All twists are left-handed. (b) Frequency polygons showing the density of modes binned by an axial momentum pseudoscalar (MPS) score (based on Eq. (\ref{['PzLz']})) for the same structures as the top panel. The left panel plots only the modes from the low frequency band, and the right from the high frequency band. (c) Spectrum of the MPS as a function of frequency obtained by averaging $\Sigma_i p_{k,i}^zL_{k,i}^z$ for modes within bin width $\Delta(\omega).$
  • Figure 3: Thermal chirality plotted as a function of the number of (left-hended) twists for the two-stranded $N=98$ polyethylene wire. Results are shown for (red) $T = 0$ K, (green) $T = 300$ K, (blue) $T = 1000$ K and (purple) classical limit.
  • Figure S1: (Top and middle) Histograms of the normal mode distribution binned by CCM for two stranded polyethelene wires of length $N=98$ for various levels of twist. The number above each plot denotes the number of twists. Results are shown using two configurations of the same molecular structure sampled from two arbitrary timepoints throughout MD simulations. (Bottom) Same as above but for one trial of a polymers of length $N=36.$
  • Figure S2: The mean of the normal mode CCM distributions shown in trial 1 of \ref{['FIG1']} as a function of the Twist Fraction given by $N_T/N_T^{\text{max}}(N)$. Results are shown for lengths (red) $N=36$ and (blue) $N=98.$ Error bars show the standard deviation of the distributions.
  • ...and 7 more figures