Parity-violating Energy Difference Research Articles

The Interplay between Tunneling and Parity Violation in Chiral Molecules

In this review, the concepts of quantum tunneling and parity violation are introduced in the context of chiral molecules. A particle moving in a double well potential provides a good model to study the behavior of chiral molecules, where the left well and right well represent the L and R enantiomers, respectively. If the model considers the quantum behavior of matter, the concept of quantum tunneling emerges, giving place to stereomutation dynamics between left- and right-handed chiral molecules. Parity-violating interactions, like the electroweak one, can be also considered, making possible the existence of an energy difference between the L and R enantiomers, the so-called parity-violating energy difference (PVED). Here we provide a brief account of some theoretical methods usually employed to calculate this PVED, also commenting on relevant experiments devoted to experimentally detect the aforementioned PVED in chiral molecules. Finally, we comment on some ways of solving the so-called Hund’s paradox, with emphasis on mean-field theory and decoherence.

Enhancement of Parity-Violating Energy Difference of CHFClBr, CHFClI, and CHFBrI by Breaking the Cancellation among Valence Orbital Contributions

Enhancement of Parity-Violating Energy Difference of CHFClBr, CHFClI, and CHFBrI by Breaking the Cancellation among Valence Orbital Contributions

The Role of the CuCl Active Complex in the Stereoselectivity of the Salt-Induced Peptide Formation Reaction: Insights from Density Functional Theory Calculations.

The salt-induced peptide formation (SIPF) reaction is a prebiotically plausible mechanism for the spontaneous polymerization of amino acids into peptides on early Earth. Experimental investigations of the SIPF reaction have found that in certain conditions, the l enantiomer is more reactive than the d enantiomer, indicating its potential role in the rise of biohomochirality. Previous work hypothesized that the distortion of the CuCl active complex toward a tetrahedral-like structure increases the central chirality on the Cu ion, which amplifies the inherent parity-violating energy differences between l- and d-amino acid enantiomers, leading to stereoselectivity. Computational evaluations of this theory have been limited to the protonated-neutral l + l forms of the CuCl active complex. Here, density functional theory methods were used to compare the energies and geometries of the homochiral (l + l and d + d) and heterochiral (l + d) CuCl-amino acid complexes for both the positive-neutral and neutral-neutral forms for alanine, valine, and proline. Significant energy differences were not observed between different chiral active complexes (i.e., d + d, l + l vs. l + d), and the distortions of active complexes between stereoselective systems and non-selective systems were not consistent, indicating that the geometry of the active complex is not the primary driver of the observed stereoselectivity of the SIPF reaction.

Magnetic Circular Dichroism in Archean Stratospheric Oxygen: Enantiomeric Excess of Amino Acids Produced in Volcanic Plumes.

While there is consensus that Archean atmosphere was anoxic with O2 pressure, p(O2) <10-6 PAL (present atmospheric level) at sea-level, evidence suggests that p(O2) at stratospheric altitudes of 10-50 km was orders of magnitude higher, a result of photodissociation of CO2 by UVC sunlight and incomplete mixing of O2 with other gases. Molecular O2 is paramagnetic due to triplet ground state. Magnetic circular dichroism (MCD) by stratospheric O2 is examined in earth's magnetic field and shows maximum circular polarization │(I+ - I-)│ at altitude of 15-30 km (I+/I- is intensity of left/right circularly polarized light). While (I+ - I-)/(I+ + I-) is small (~10-10), it is an unexplored source of enantiomeric excess (EE) by asymmetric photolysis of amino acid precursors produced in volcanic eruptions. The precursors reside in stratosphere for periods of over a year due to relative absence of vertical transport. Due to negligible thermal gradient across equator, they are trapped in the hemisphere where they are produced, with interhemispheric exchange time of over a year. The precursors diffuse through altitudes of maximum circular polarization before getting hydrolyzed on ground to amino acids. Enantiomeric excess of ~10-12 is calculated for precursors and amino acids. While small, this EE is orders of magnitude higher than predicted (~10-18) by parity violating energy differences (PVED) and could be the seed for growth of biological homochirality. Preferential crystallization (PC) is described as a plausible mechanism for amplification of solution EE of some amino acids from 10-12 to 10-2, for period of several days.

Parity Violation Energy Difference Calculation of Atropisomers.

Enantiomers have a different energy due to the parity violation effects. Up to now, these effects are difficult to calculate and their final effect on the choice of one enantiomer in the homochirality issue is still a matter of debate. Nevertheless, many scientists support the role of this tiny energy difference in the triggering of homochirality. In this work, we studied the energy difference in atropisomers, a class of stereoisomers in which the chirality is given by the block of rotation around one bond. Atropisomers might have a low energy barrier for the interconversion and this is interesting for the equilibration of the two enantiomers and the choice of the most stable enantiomer. Moreover, structures might be extended like in the case of polymers or crystals having helical framework and thus giving an additive effect on the parity violation energy of the whole structure. The parity violation energy difference here is discussed with the correlation on the general structure of the final molecule giving a qualitative model to predict the sign of local contributions of atoms.

Chiral selectivity vs. noise in spontaneous mirror symmetry breaking.

Mirror symmetry breaking bifurcations, that occur in nonlinear chemical systems leading to final chiral states with very large enantiomeric excess, can be exploited as an efficient chiral signal selector for even the smallest chiral polarizations. This effect of the chiral polarization requires the system's capacity for overcoming thermal noise, which is manifested as fluctuating reaction rate constants. Therefore, we investigate the chiral selectivity across a range of tiny parity-violating energy differences (PVED) in the presence of inevitable non-equilibrium temperature fluctuations. We use a stochastic differential equation simulation methodology (Ito process) that serves as a valuable tool in open systems for identifying the thresholds at which the chiral force induces chiral selectivity in the presence of non-equilibrium temperature fluctuations. This approach enables us to include and analyze chiral selectivity in the presence of other types of fluctuations, such as perturbations in the rate of fluid flow into and out of the reactor and in the clamped input concentrations. These concepts may be of practical interest (i.e., spontaneous deracemizations) but are also useful for a better understanding of the general principles governing the emergence of biological homochirality.

Resilience of parity-violation-induced chiral selectivity to nonequilibrium temperature fluctuations in open systems

We consider the sensitivity of chemical reactions, able to undergo spontaneous mirror symmetry breaking, to chiral bias in the presence of nonequilibrium temperature fluctuations and derive a selectivity criterion. For this, we estimate the magnitude of fluctuations $\ensuremath{\delta}k$ in chemical reaction rate constants $k$ arising from the nonequilibrium temperature fluctuations $\ensuremath{\delta}T$ about a mean value $T$. To leading order, the relative rate constant fluctuations $\ensuremath{\delta}{k}_{i}/{k}_{i}$ for each reaction $i$ are given by the product of the activation enthalpy $\mathrm{\ensuremath{\Delta}}{H}_{i}^{\ifmmode\ddagger\else\textdaggerdbl\fi{}}/RT$ for the $i\mathrm{th}$ reaction multiplied by the relative rms temperature fluctuations $\ensuremath{\delta}{T}_{\mathrm{rms}}/T$. The latter are determined by the system's specific heat at constant volume: ${C}_{V}$. We test this criterion with simulations carried out for an open-flow fully reversible Frank model, and for a range of parity-violating energy differences (PVED) within the theoretically estimated upper and lower bounds. Depending on the relative magnitudes of the deterministic PVED bias and the temperature fluctuations, the PVED bias can either (i) select the final stable chiral outcome deterministically or (ii) select one of two possible stable chiral outcomes with an asymmetric statistical weighting. For larger temperature fluctuations, the PVED bias loses its selectivity, and the final stable chiral outcomes are stochastic and equally probable. This paper points towards the possible design of small volume chemical flow reactors capable of detecting the elusive PVED bias in bulk systems, provided other sources of fluctuations can be sufficiently controlled and attenuated.

Assessing MP2 frozen natural orbitals in relativistic correlated electronic structure calculations.

The high computational scaling with the basis set size and the number of correlated electrons is a bottleneck limiting applications of coupled cluster algorithms, in particular for calculations based on two- or four-component relativistic Hamiltonians, which often employ uncontracted basis sets. This problem may be alleviated by replacing canonical Hartree-Fock virtual orbitals by natural orbitals (NOs). In this paper, we describe the implementation of a module for generating NOs for correlated wavefunctions and, in particular, second order Møller-Plesset perturbation frozen natural orbitals (MP2FNOs) as a component of our novel implementation of relativistic coupled cluster theory for massively parallel architectures [Pototschnig et al. J. Chem. Theory Comput. 17, 5509, (2021)]. Our implementation can manipulate complex or quaternion density matrices, thus allowing for the generation of both Kramers-restricted and Kramers-unrestricted MP2FNOs. Furthermore, NOs are re-expressed in the parent atomic orbital (AO) basis, allowing for generating coupled cluster singles and doubles NOs in the AO basis for further analysis. By investigating the truncation errors of MP2FNOs for both the correlation energy and molecular properties-electric field gradients at the nuclei, electric dipole and quadrupole moments for hydrogen halides HX (X = F-Ts), and parity-violating energy differences for H2Z2 (Z = O-Se)-we find MP2FNOs accelerate the convergence of the correlation energy in a roughly uniform manner across the Periodic Table. It is possible to obtain reliable estimates for both energies and the molecular properties considered with virtual molecular orbital spaces truncated to about half the size of the full spaces.

Enhancement of the parity-violating energy difference of H2X2 molecules by electronic excitation

The parity-violating energy difference (PVED) between two enantiomers of a chiral molecule is caused by the weak interaction. Because of the smallness of the PVED, nonzero PVED is yet to be discovered in experimental searches. To detect the PVED, the search for molecules with large PVED values is important. Previously, one of the authors proposed that the PVED may be significantly enhanced in ionized or excited states. The significant enhancement of the PVED in some electronic excited states is proven in this study using H$_2$X$_2$ (X=O, S, Se, Te) molecules as examples. The maximum enhancement was an about 360-fold increase for H$_2$Se$_2$. For the PVED calculation, we employ the finite-field perturbation theory (FFPT) within the equation-of-motion coupled-cluster theory based on the exact two-component molecular-mean field Hamiltonian. The relation between the enhancement of the PVED and the contribution to the PVED from the highest occupied molecular orbital is also examined. The effects of computational elements, such as parameters related to the electron correlation and FFPT on PVED values in excited states of H$_2$X$_2$ molecules are studied.

Chirality and Relativistic Effects in Os3(CO)12

The energy and structural parameters were obtained for all forms of the carbonyl complex of osmium Os3(CO)12 with D3h and D3 symmetries using density functional theory (DFT) methods. The calculations took into account various levels of relativistic effects, including those associated with nonconservation of spatial parity. It was shown that the ground state of Os3(CO)12 corresponds to the D3 symmetry and thus may be characterized either as left-twisted (D3S) or right-twisted (D3R). The D3S↔D3R transitions occur through the D3h transition state with an activation barrier of ~10–14 kJ/mol. Parity violation energy difference (PVED) between D3S and D3R states equals to ~5 × 10−10 kJ/mol. An unusual three-center exchange interaction was found inside the {Os3} fragment. It was found that the cooperative effects of the mutual influence of osmium atoms suppress the chirality of the electron system in the cluster.

Contributions of a modified electrodynamics to the molecular biochirality.

In this paper, we investigate the physical basis behind the molecular biochirality from the computation of a parity violation energy difference (PVED) in enantiomers of organic molecules (e.g., amino acids, which occur as levogyrous-type in nature), by considering the influence of fundamental interactions beyond the standard model of elementary particles and interactions. Particularly, we study the role of a 4-D Chern-Simons theory at the origin of this PVED, the Carroll-Field-Jackiw electrodynamics, which violates both Lorentz and parity symmetries. Then, we consider terrestrial and Jovian scenarios where the influence of a modified (effective) magnetic field generated by the planets on the molecules is taken into account in the calculation of PVED. Besides this quantity, we also calculate the relative quantity excess of an enantiomer over the other in a thermal bath. Finally, we compare the obtained results with those ones from other models based on fundamental interactions.

Nuclear spin symmetry conservation studied by cavity ring-down spectroscopy of ammonia in a seeded supersonic jet from a pulsed slit nozzle

We report high-resolution cw laser spectra of the symmetric stretching fundamental in ammonia NH measured with a cavity ring-down setup for mid-IR transitions (around ). In a supersonic jet expansion ( seeded in Ar), we have obtained rotational temperatures below . For the gas expansion, a pulsed slit nozzle with length and 70 µm width has been used. The measured relative line intensities demonstrate that the A and E nuclear spin symmetry isomers do not interconvert in the supersonic jet expansion under these conditions (nuclear spin symmetry conservation). The lateral velocity of the gas expansion has been probed by a Doppler profile measurement, showing a preference for an expansion with an angular distribution following Lambert's cosine law perpendicular to the slit direction and an approximate molecular velocity of about . The experimental setup is described in detail, also in view of high-resolution spectroscopy of chiral molecules as candidates to measure the parity violating energy difference between enantiomers with a barrier to interconversion around .

Energy threshold for chiral symmetry breaking in molecular self-replication.

The homochirality of biological molecules (right-handed sugars and left-handed amino acids) is a signature of life. Extensive research has been devoted to understanding how enrichment of one enantiomer over the other might have emerged from a prebiotic world. Here, we use experimental data from the model Soai autocatalytic reaction system to evaluate the energy required for symmetry breaking and chiral amplification in molecular self-replication. One postulate for the source of the original imbalance is the tiny difference in energy between enantiomers due to parity violation in the weak force. We discuss the plausibility of parity violation energy difference coupled with asymmetric autocatalysis as a rationalization for absolute asymmetric synthesis and the origin of the homochirality of biological molecules. Our results allow us to identify the magnitude of the energy imbalance that gives rise to directed symmetry breaking and asymmetric amplification in this autocatalytic system.

Parity violating energy difference for mirror conformers of DABCO linker between two M2+ cations (M = Zn, Cd, and Hg).

We have studied the potential energy surface of cations (M = Zn, Cd, and Hg), which are considered as a model for the DABCO linker in metal-organic frameworks, a new prospective class of materials. Relativistic four-component and two-component calculations of parity violating energy difference (PVED) for twisted isomers of cations have been performed. The right-twisted conformers of are more stable than the left-twisted ones. The increase in PVED with the nuclear charge of the transition metal atom M (Z M ) is discussed.

Comments on 'Hydrogen bonds in crystalline d-alanine: diffraction and spectroscopic evidence for differences between enantiomers'.

The recent paper by Belo, Pereira, Freire, Argyriou, Eckert & Bordallo [(2018 ▸), IUCrJ, 5, 6-12] reports observations that may lead one to think of very strong and visible consequences of the parity-violation energy difference between enantiomers of a molecule, namely alanine. If proved, this claim would have an enormous impact for research in structural chemistry. However, alternative, more realistic, explanations of their experiments have not been ruled out by the authors. Moreover, the theoretical calculations carried out to support the hypothesis are unable to differentiate between enantiomers (molecules or crystals). Therefore, the conclusions drawn by Belo et al. (2018 ▸) are deemed inappropriate as the data presented do not contain sufficient information to reach such a conclusion.

An oxorhenium complex bearing a chiral cyclohexane-1-olato-2-thiolato ligand: Synthesis, stereochemistry, and theoretical study of parity violation vibrational frequency shifts.

In our effort towards measuring the parity violation energy difference between two enantiomers, a simple chiral oxorhenium complex 5 bearing enantiopure 2-mercaptocyclohexan-1-ol has been prepared as a potential candidate species. Vibrational circular dichroism revealed a chiral environment surrounding the rhenium atom, even though the rhenium is not a stereogenic center itself, and enabled to assign the (1S,2S)-(-) and (1R,2R)-(+) absolute configuration for 5. For both compound 5 and complex 4, previously studied by us and bearing a propane-2-olato-3-thiolato ligand, relativistic calculations predict parity violating vibrational frequency differences of a few hundreds of millihertz, above the expected sensitivity attainable by a molecular beam Ramsey interferometer that we are constructing.

High-resolution mid-infrared spectroscopy of buffer-gas-cooled methyltrioxorhenium molecules

We demonstrate cryogenic buffer-gas cooling of gas-phase methyltrioxorhenium (MTO). This molecule is closely related to chiral organometallic molecules where the parity-violating energy differences between enantiomers is measurable. The molecules are produced with a rotational temperature of approximately 6 K by laser ablation of an MTO pellet inside a cryogenic helium buffer gas cell. Facilitated by the low temperature, we demonstrate absorption spectroscopy of the 10.2 μm antisymmetric Re=O stretching mode of MTO with a resolution of 8 MHz and a frequency accuracy of 30 MHz. We partially resolve the hyperfine structure and measure the nuclear quadrupole coupling of the excited vibrational state. Our ability to produce dense samples of complex molecules of this type at low temperatures represents a key step towards a precision measurement of parity violation in a chiral species.

Synchrotron-Based Highest Resolution Terahertz Spectroscopy of the ν24 Band System of 1,2-Dithiine (C4H4S2): A Candidate for Measuring the Parity Violating Energy Difference between Enantiomers of Chiral Molecules

The chiral C2 symmetric molecule 1,2-dithiine (1,2-dithia-3,5-hexadiene, C4H4S2) has been identified as a possible candidate for measuring the parity violating energy difference between enantiomers. We report here the observation and analysis of the low-frequency fundamental ν24 using highest resolution synchrotron-based interferometric Fourier transform infrared (FTIR) spectroscopy in the terahertz range with a band center of ν0 = 6.95375559 THz (ν̃0 = 231.952319 (10) cm-1) and two related hot bands, the (ν13 + ν24) ← ν13 band at ν0 = 6.97256882 THz (ν̃0 = 232.579861 (33) cm-1) and the 2ν24 ← ν24 band at ν0 = 7.01400434 THz (ν̃0 = 233.962001 (14) cm-1). This success in the difficult analyses of the THz spectrum of a complex chiral molecule of importance for fundamental tests on molecular parity violation is enabled by the ideal setup of an FTIR experiment of currently unique resolution with the very stable and bright synchrotron radiation at the Swiss Light Source (SLS).

High resolution GHz and THz (FTIR) spectroscopy and theory of parity violation and tunneling for 1,2-dithiine (C4H4S2) as a candidate for measuring the parity violating energy difference between enantiomers of chiral molecules.

We report high resolution spectroscopic results of 1,2-dithiine-(1,2-dithia-3,5-cyclohexadiene, C4H4S2) in the gigahertz and terahertz spectroscopic ranges and exploratory theoretical calculations of parity violation and tunneling processes in view of a possible experimental determination of the parity violating energy difference ΔpvE in this chiral molecule. Theory predicts that the parity violating energy difference between the enantiomers in their ground state (ΔpvE ≃ 1.1 × 10(-11)(hc) cm(-1)) is in principle measurable as it is much larger than the calculated tunneling splitting for the symmetrical potential ΔE± < 10(-24) (hc) cm(-1). With a planar transition state for stereomutation at about 2500 cm(-1) tunneling splitting becomes appreciable above 2300 cm(-1). This makes levels of well-defined parity accessible to parity selection by the available powerful infrared lasers and thus useful for one of the existing experimental approaches towards molecular parity violation. The new GHz spectroscopy leads to greatly improved ground state rotational parameters for 1,2-dithiine. These are used as starting points for the first successful analyses of high resolution interferometric Fourier transform infrared (FTIR, THz) spectra of the fundamentals ν17 (1308.873 cm(-1) or 39.23903 THz), ν22 (623.094 cm(-1) or 18.67989 THz) and ν3 (1544.900 cm(-1) or 46.314937 THz) for which highly accurate spectroscopic parameters are reported. The results are discussed in relation to current efforts to measure ΔpvE.

Infrared laser induced population transfer and parity selection in (14)NH3: A proof of principle experiment towards detecting parity violation in chiral molecules.

We have set up an experiment for the efficient population transfer by a sequential two photon-absorption and stimulated emission-process in a molecular beam to prepare quantum states of well defined parity and their subsequent sensitive detection. This provides a proof of principle for an experiment which would allow for parity selection and measurement of the time evolution of parity in chiral molecules, resulting in a measurement of the parity violating energy difference ΔpvE between enantiomers of chiral molecules. Here, we present first results on a simple achiral molecule demonstrating efficient population transfer (about 80% on the average for each step) and unperturbed persistence of a selected excited parity level over flight times of about 1.3 ms in the beam. In agreement with model calculations with and without including nuclear hyperfine structure, efficient population transfer can be achieved by a rather simple implementation of the rapid adiabatic passage method of Reuss and coworkers and considering also the stimulated Raman adiabatic passage technique of Bergmann and coworkers as an alternative. The preparation step uses two powerful single mode continuous wave optical parametric oscillators of high frequency stability and accuracy. The detection uses a sensitive resonantly enhanced multiphoton ionization method after free flight lengths of up to 0.8 m in the molecular beam. Using this technique, we were able to also resolve the nuclear hyperfine structure in the rovibrational levels of the ν1 and ν3 fundamentals as well as the 2ν4 overtone of (14)NH3, for which no previous data with hyperfine resolution were available. We present our new results on the quadrupole coupling constants for the ν1, ν3, and 2ν4 levels in the context of previously known data for ν2 and its overtone, as well as ν4, and the ground state. Thus, now, (14)N quadrupole coupling constants for all fundamentals and some overtones of (14)NH3 are known and can be used for further theoretical analysis.