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arXiv:2610.41782

Chiral Torsional Double Wells: Tunneling and Polarizability

Shah Ishmam Mohtashim, Manas Sajjan, Sabre Kais

quant-phSubmitted Oct 30, 2026• Announced Nov 2, 2026
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Abstract

Quantum tunneling between mirror-related wells of a torsional potential sets the timescale for coherent interconversion of chiral conformers. We ask whether slower tunneling also weakens the transition polarizability in hydrogen peroxide, hydrogen disulfide, and hydrogen diselenide. We use relaxed CCSD(T)//MP2 potentials and a kinetic model that removes overall molecular rotation to obtain the torsional states, then use static polarizabilities to calculate transition matrix elements between them. In the molecular frame used here, symmetry makes the matrix element of the even dipole component vanish between opposite-parity torsional states, while the odd polarizability component has a nonzero matrix element. The two quantities probe different parts of the torsional landscape: tunneling depends on penetration through the barriers, while the polarizability matrix element forms mainly within the wells. Consequently, tunneling slows sharply across the molecular series even as the matrix element grows. Higher torsional doublets and deuterium substitution further show that large changes in tunneling time can occur with little change in the transition polarizability matrix element. Calculated torsional energies for H$_2$O$_2$ and D$_2$O$_2$ agree with experimental values. Thus, slowing coherent interconversion need not weaken the intrinsic transition polarizability.

AI Summary

Does suppressing tunneling between chiral conformers also suppress their transition polarizability? This paper compares torsional doublets in H$_2$O$_2$, H$_2$S$_2$, and H$_2$Se$_2$ using relaxed CCSD(T)//MP2 potentials, a one-dimensional kinetic metric with overall rotation removed, Fourier-DVR eigenstates, and static PBE0 polarizabilities. In the chosen molecular frame, reflection symmetry forbids the even dipole component from connecting opposite-parity torsional states but permits the odd polarizability component $\alpha_{yz}$. Across O/S/Se substitution, the ground-doublet splitting falls from $10.984$ to $1.743\times10^{-6}\,\mathrm{cm}^{-1}$, a factor of $6.3\times10^6$, while $|\langle+|\alpha_{yz}|-\rangle|$ increases from $1.57$ to $2.49$ atomic units. Corresponding coherent well-to-well transfer times increase from $1.518$ ps to $9.57\,\mu$s. Comparisons holding the electronic surfaces fixed show similar separation: deuteration reduces the peroxide splitting by $6.75$ while changing the matrix element by $1.01\%$; three selenium doublets span transfer times of $2.07$ ns–$9.57\,\mu$s while retaining at least $93\%$ of the ground-doublet squared coupling. Contribution densities locate the polarizability integral mainly inside the wells, unlike barrier-sensitive tunneling. Calculated peroxide and deuterated-peroxide torsional origins have RMS deviations of $1.51$ and $3.95\,\mathrm{cm}^{-1}$ from spectroscopic values. The model omits other vibrational dynamics, torsion-dependent zero-point energies, and explicit rotation–torsion coupling. Polarizabilities lack correlated-response or optical-frequency benchmarks. Transfer times exclude decoherence and thermal transitions, and the intrinsic matrix elements alone do not determine observable Raman intensities or enantiomer discrimination.

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Quantum tunneling between mirror-related wells of a torsional potential sets the timescale for coherent interconversion of chiral conformers. We ask whether slower tunneling also weakens the transition polarizability in hydrogen peroxide, hydrogen disulfide, and hydrogen diselenide. We use relaxed CCSD(T)//MP2 potentials and a kinetic model that removes overall molecular rotation to obtain the torsional states, then use static polarizabilities to calculate transition matrix elements between them. In the molecular frame used here, symmetry makes the matrix element of the even dipole component vanish between opposite-parity torsional states, while the odd polarizability component has a nonzero matrix element. The two quantities probe different parts of the torsional landscape: tunneling depends on penetration through the barriers, while the polarizability matrix element forms mainly within the wells. Consequently, tunneling slows sharply across the molecular series even as the matrix element grows. Higher torsional doublets and deuterium substitution further show that large changes in tunneling time can occur with little change in the transition polarizability matrix element. Calculated torsional energies for H2_22​O2_22​ and D2_22​O2_22​ agree with experimental values. Thus, slowing coherent interconversion need not weaken the intrinsic transition polarizability.
Does suppressing tunneling between chiral conformers also suppress their transition polarizability? This paper compares torsional doublets in H2_22​O2_22​, H2_22​S2_22​, and H2_22​Se2_22​ using relaxed CCSD(T)//MP2 potentials, a one-dimensional kinetic metric with overall rotation removed, Fourier-DVR eigenstates, and static PBE0 polarizabilities. In the chosen molecular frame, reflection symmetry forbids the even dipole component from connecting opposite-parity torsional states but permits the odd polarizability component αyz\alpha_{yz}αyz​.
Across O/S/Se substitution, the ground-doublet splitting falls from 10.98410.98410.984 to 1.743×10−6 cm−11.743\times10^{-6}\,\mathrm{cm}^{-1}1.743×10−6cm−1, a factor of 6.3×1066.3\times10^66.3×106, while ∣⟨+∣αyz∣−⟩∣|\langle+|\alpha_{yz}|-\rangle|∣⟨+∣αyz​∣−⟩∣ increases from 1.571.571.57 to 2.492.492.49 atomic units. Corresponding coherent well-to-well transfer times increase from 1.5181.5181.518 ps to 9.57 μ9.57\,\mu9.57μs. Comparisons holding the electronic surfaces fixed show similar separation: deuteration reduces the peroxide splitting by 6.756.756.75 while changing the matrix element by 1.01%1.01\%1.01%; three selenium doublets span transfer times of 2.072.072.07 ns–9.57 μ9.57\,\mu9.57μs while retaining at least 93%93\%93% of the ground-doublet squared coupling. Contribution densities locate the polarizability integral mainly inside the wells, unlike barrier-sensitive tunneling.
Calculated peroxide and deuterated-peroxide torsional origins have RMS deviations of 1.511.511.51 and 3.95 cm−13.95\,\mathrm{cm}^{-1}3.95cm−1 from spectroscopic values. The model omits other vibrational dynamics, torsion-dependent zero-point energies, and explicit rotation–torsion coupling. Polarizabilities lack correlated-response or optical-frequency benchmarks. Transfer times exclude decoherence and thermal transitions, and the intrinsic matrix elements alone do not determine observable Raman intensities or enantiomer discrimination.
Abstract Previous results for nuclear fluxes during coherent tunnelling of molecules with symmetric double well potentials are extended to fluxes in asymmetric double well potentials. The theory is derived using the two-state approximation (TSA). The symmetric system serves as a reference. As an example, we consider the one-dimensional model of the tunnelling inversion of oriented ammonia, with semiclassical dipole coupling to an electric field. The tunnelling splitting increases with the dipole coupling by a factor The tunnelling time decreases by The nuclear density appears as the sum of two parts: The tunnelling part decreases as times the density of the symmetric reference, whereas the non-tunnelling part is the initial density times Likewise, the nuclear flux decreases by with essentially the same shape as for the symmetric reference, with maximum value at the potential barrier. Coherent nuclear tunnellings starting from the upper or lower wells of the asymmetric potential are equivalent. The results are universal, in the frame of the TSA, hence they allow straightforward extrapolations from one system to others. This is demonstrated by the prediction of isotope effects for five isotopomers of ammonia.
We investigate coherent and incoherent tunneling phenomena in conditions of crossing diabatic potentials. We consider a model of two crossing parabolic diabatic potentials with an independent of coordinates constant adiabatic coupling. As a result of the coupling and level crossing avoiding, we get the asymmetric double-well lower adiabatic potential with a variable shape. We show that the doublet structure of levels (generic for double-well potentials) is remained valid as long as the transition matrix element is smaller than characteristic inter-level spacings (the latter ones in own turn decrease upon increasing of a difference between the diabatic potential minima). In the adiabatic limit the tunneling transitions do not depend on the upper potential. In the over-barrier energy region there is an oscillating behavior due to the resonances between the states in the lower and in the upper adiabatic potentials. A new phenomenom emanated from this oscillating dependence namely, multiple coherent - incoherent regime transitions for the upper adiabatic potential state evolution, is our main concern in this paper. The problem is not only of intellectual interest but also of relevance to various molecular systems undergoing conversion of electronic states or isomerization reactions. Our model exhausts all cases practically relevant for spectroscopy of non-rigid molecules, and can capture many of the features exhibited by experiment.
The ground-state energy splitting due to tunneling in two-dimensional double wells of the form $V(x,y)=\frac{{({x}^{2}\ensuremath{-}{R}^{2})}^{2}}{8{R}^{2}}+\frac{{x}^{2}\ensuremath{-}{R}^{2}}{{R}^{2}}\ensuremath{\gamma}y+\frac{{\ensuremath{\omega}}^{2}}{2}{y}^{2}$ is calculated. Several results are reported. First, we give a systematic WKB expansion of the splitting in series in powers of ${R}^{\ensuremath{-}2}$, each term of the series being a finite polynomial in ${\ensuremath{\gamma}}^{2}$. We find an ascending sequence of the values of the parameter $\ensuremath{\gamma}$ characterizing the curvature of the classical path, for which the successive corrections to the leading order vanish. This effect arises because curvature of the path and quantum nature of motion cancel each other; it does not appear for one-dimensional double wells. Second, we find that for large curvatures, such as for those describing the proton transfer in a malonaldehyde and hydroxalate anion, this expansion is of no practical use. Thus, the WKB expansion is reordered to a strong coupling form, each term of the series in powers of ${R}^{\ensuremath{-}2}$ being an infinite series in powers of ${\overline{\ensuremath{\gamma}}}^{2}$, $\overline{\ensuremath{\gamma}}=\ensuremath{\gamma}/R$. Third, we find that the radius of convergence of the series is determined by the singularity at ${\overline{\ensuremath{\gamma}}}_{s}=\ensuremath{\omega}/2$. At the singularity the system changes its character from being a double well to become a single well. Close to this singularity the classical action and its first quantum correction are found to be nonanalytic functions of $\overline{\ensuremath{\gamma}}$, most likely of the form ${[1\ensuremath{-}{(\overline{\ensuremath{\gamma}}/{\overline{\ensuremath{\gamma}}}_{s})}^{2}]}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}=1/2$ and $\ensuremath{\alpha}=\ensuremath{-}1/2$ for the classical action and its first quantum correction, respectively. Since in the semiclassical regime of large RRR the splitting is exponentially dependent on the value of the classical action and its first quantum correction, close to the singularity we establish strong sensitivity of the splitting on slight variations of the parameter $\overline{\ensuremath{\gamma}}$ entering the Hamiltonian linearly.
We discuss the effect of molecular symmetry on coherent tunneling in symmetric double-well potentials whose two molecular equilibrium configurations are interconverted by nuclear permutations. This is illustrated with vibrational tunneling in ammonia molecules, electronic tunneling in the dihydrogen cation, and laser-induced rotational tunneling of homonuclear diatomics. In this contribution, we reexamine the textbook picture of coherent tunneling in such potentials, which is depicted with a wavepacket shuttling back and forth between the two potential-wells. We show that the common application of this picture to the aforementioned molecules contravenes the principle of the indistinguishability of identical particles. This conflict originates from the sole consideration of the dynamics of the tunneling-mode, connecting the double-well energy minima, and complete omission of all the remaining molecular degrees of freedom. This gives rise to double-well wavepackets that are nonsymmetric under nuclear permutations. To obey quantum statistics, we show that the double-well eigenstates composing these wavepackets must be entangled with the wavefunctions that describe all the omitted molecular modes. These wavefunctions have compensating and opposite nuclear permutation symmetry. This in turn leads to complete quenching of interference effects behind localization in one potential-well or another. Indeed, we demonstrate that the reduced density of probability of the symmetrized molecular wavefunction, where all the molecular coordinates but the tunneling-mode are integrated out, is symmetrically distributed over the two potential-wells, at all times. This applies to any multilevel wavepacket of isotropic or fully aligned symmetric double-well achiral molecules. However, in the case of coherent electronic or vibrational tunneling, fully aligned molecules may exhibit dynamical localization in the space-fixed frame, where the tunneling-mode density shuttles between the opposite directions of the alignment axis. This dynamical spatial-localization results from linear combinations of molecular states that have opposite parity. In summary, this study shows that dynamical localization of the tunneling-mode density on either of the two indistinguishable molecular equilibrium configurations of symmetric double-well achiral molecules is forbidden by quantum statistics, whereas its dynamical localization in the space-fixed frame is allowed by parity. The subtle distinction between these two types of localization has far-reaching implications in the interpretation of many ultrafast molecular dynamics experiments.
Superpositions of handed molecular states give rise to achiral eigenstates, delocalized across a double-well potential via tunneling. A coherent superposition of these energy eigenstates could dynamically relocalize the molecules into chiral states, which has only been addressed theoretically. Here, we present a microwave pump-probe study to create and probe coherent chiral tunneling dynamics in a given rotational state. Through a time-resolved scheme, we uncover the periodic time evolution of the induced chiral wave packet under field-free conditions. Moreover, we demonstrate precise phase control of this coherence via phase modulation during pump excitation.
The movement of a proton which moves in a one-dimensional effective double-minimum potential V(x) and can pass by means of the tunnel effect from one well into the other was investigated. The lingering times of the proton in the potential wells (a) and (b) were calculated for more than 150 potential models as functions of the height Vmax of the potential barrier separating the two wells, the energy difference ΔV, and separation d of the minima. In the determination of the lingering times, it was assumed that the proton at time t = 0 is to be found with certainty in one of the wells and can be represented by means of a statistical operator related to the eigenstates of a one-well Hamiltonian H(a)orH(b), respectively. The times determined on this basis were compared with those which result with a quasiclassical method from the classical frequency of oscillation of a particle in a well and from the penetration coefficient of the potential barrier. The deviation between the values determined with both methods is largest if an eigenvalue of H(a) becomes approximately equal to an eigenvalue of H(b). For the potential model V(x) a polynomial of fourth degree in the position coordinate x was chosen. The eigenfunctions of H(a)andH(b) as well as those of the effective Hamiltonian H corresponding to V(x) were represented as linear combinations of eigenfunctions of the harmonic oscillator with angular frequency ω. ω was chosen so that the mean square deviations of the approximated eigenvalues of H became as small as possible for the states of lowest energy.