Rovibrational Averaging Research Articles

Enantiomer Superpositions from Matter-Wave Interference of Chiral Molecules

Molecular matter-wave interferometry enables novel strategies for manipulating the internal mechanical motion of complex molecules. Here, we show how chiral molecules can be prepared in a quantum superposition of two enantiomers by far-field matter-wave diffraction and how the resulting tunneling dynamics can be observed. We determine the impact of rovibrational phase averaging and propose a setup for sensing enantiomer-dependent forces, parity-violating weak interactions, and environment-induced superselection of handedness, as suggested to resolve Hund’s paradox. Using ab initio tunneling calculations, we identify [4]-helicene derivatives as promising candidates to implement the proposal with state-of-the-art techniques. This work opens the door for quantum sensing with chiral molecules.Received 20 January 2021Revised 4 June 2021Accepted 14 July 2021DOI:https://doi.org/10.1103/PhysRevX.11.031056Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasChiralityCold atoms & matter wavesQuantum opticsQuantum sensingPhysical SystemsAtomic & molecular beamsMoleculesTechniquesAb initio calculationsDensity functional theoryInterferometryQuantum chemistry methodsQuantum master equationAtomic, Molecular & OpticalQuantum InformationCondensed Matter, Materials & Applied Physics

Calculating nuclear magnetic resonance chemical shifts in solvated systems

Calculated NMR chemical shifts can be a valuable resource for peak assignments as well as structure determination. Chemical shifts are extremely sensitive to the chemical environment surrounding the nucleus of interest and can be used to deduce precise intramolecular and intermolecular structural information. However, the chemical shift is also very sensitive to solvent effects, conformational and rovibrational averaging, and relativistic effects. All of these effects must be taken into account when calculating magnetic shielding and chemical shift. Here, we review the current state of NMR chemical shift and shielding calculations for solvated systems. Dr. Leah B. Casabianca received her PhD in Physical Chemistry in 2008 working with Prof. Angel C. de Dios at Georgetown University. She then did a postdoctoral fellowship at the University of Illinois at Chicago with Prof. Yoshitaka Ishii working on solid-state NMR of materials and one at the Weizmann Institute of Science with Prof. Lucio Frydman working on Dynamic Nuclear Polarization of rare nuclei. She joined the faculty at Clemson University in 2014. Her work involves developing NMR methods for characterization of molecules interacting with the surface of nanoparticles. Dr. Casabianca was the recipient of an NSF CAREER Award in 2018.

More evidence for bent ro-vibrationally averaged structures of linear triatomic molecules: The non-zero b axis component of the dipole moment and the [formula omitted] structure in electron diffraction

We have previously demonstrated theoretically, by computing expectation values over the bending wavefunction, and experimentally, by re-interpreting observed values of the rotational constant B0, that a linear triatomic molecule will necessarily be observed as being bent on ro-vibrational average. We present here further evidence for this assertion: (1) the non-zero b axis component of the electric dipole moment 〈μb〉 for the linear molecules CO2, HCN, and HCO+, and (2) the internuclear distances rg in electron diffraction (ED) for CO2 and HCN when re-interpreted without an a priori assumption of the linear ro-vibrationally-averaged structure. In either case, the theoretical values calculated as the expectation values over the DVR3D wavefunctions show good agreement with the corresponding experimental values. If, as in a conventional paradigm, the r0 structure of a linear molecule were observed as linear, the 〈μb〉 would be of zero-value and the shrinkage effects in ED (claimed in the 1960s for the observation that the distance between terminal atoms was shorter than the sum of the constituent bond lengths) would become inevitable. Hence, the non-zero value of 〈μb〉 and no shrinkage effect in the reasonable, re-interpreted terminal-atom distance in ED, definitely show the bent ro-vibrationally averaged structure on observation even for the ro-vibrational ground state. Thus, we have additional pieces of indisputable evidence supporting the assertion that the ro-vibrationally averaged structure of a linear molecule is “bent” on observation.

Linear and bent triatomic molecules are not qualitatively different!

I, and other authors, have discussed in several recent publications that “linear” triatomic molecules (defined as having linear equilibrium structures) are necessarily observed as being bent on ro-vibrational average. We have demonstrated this theoretically by calculations of the rotation–vibration expectation values, [Formula: see text], where [Formula: see text] is the bond angle supplement, [Formula: see text] being the instantaneous value of the bond angle of the triatomic molecule A–B–C. Direct experimental evidence of bent average structures has been obtained by other authors in Coulomb explosion imaging experiments, and indirect evidence from re-interpretation of experimentally derived rotational constant values. In spite of a rather significant amount of evidence in support of the bent average structures, the idea has been heavily criticized. In the present work I discuss in more detail some of the arguments for the bent average structures put forward in previous papers, and I hope to correct and clarify some of the misunderstandings leading to the criticisms. Part of the criticism originates in a widespread, but fallacious, belief among spectroscopists that linear and bent chain molecules have qualitatively different energy-level and spectral intensity patterns. This is not true. One can view the linear-molecule energy level and spectral patterns as limiting cases of the bent-molecule ones.

Proton transfer in guanine-cytosine base pair analogues studied by NMR spectroscopy and PIMD simulations.

It has been hypothesised that proton tunnelling between paired nucleobases significantly enhances the formation of rare tautomeric forms and hence leads to errors in DNA replication. Here, we study nuclear quantum effects (NQEs) using deuterium isotope-induced changes of nitrogen NMR chemical shifts in a model base pair consisting of two tautomers of isocytosine, which form hydrogen-bonded dimers in the same way as the guanine-cytosine base pair. Isotope effects in NMR are consequences of NQEs, because ro-vibrational averaging of different isotopologues gives rise to different magnetic shielding of the nuclei. The experimental deuterium-induced chemical shift changes are compared with those calculated by a combination of path integral molecular dynamics (PIMD) simulations with DFT calculations of nuclear shielding. These calculations can directly link the observable isotope-induced shifts with NQEs. A comparison of the deuterium-induced changes of 15N chemical shifts with those predicted by PIMD simulations shows that inter-base proton transfer reactions do not take place in this system. We demonstrate, however, that NMR isotope shifts provide a unique possibility to study NQEs and to evaluate the accuracy of the computational methods used for modelling quantum effects in molecules. Calculations based on the PBE functional from the general-gradient-approximation family provided significantly worse predictions of deuterium isotope shifts than those with the hybrid B3LYP functional.

Computational molecular spectroscopy of [formula omitted] NCS: Electronic properties and ro-vibrationally averaged structure

For NCS in the X̃2Π electronic ground state, three-dimensional potential energy surfaces (3D PESs) have been calculated ab initio at the core-valence, full-valence MR-SDCI+Q/[aug-cc-pCVQZ (N, C, S)] level of theory. The ab initio 3D PESs are employed in second-order-perturbation-theory and DVR3D calculations to obtain various molecular constants and ro-vibrationally averaged structures. The 3D PESs show that the X̃2Π NCS has its potential minimum at a linear configuration, and hence it is a “linear molecule.” The equilibrium structure has re(N–C) = 1.1778 Å, re(C–S) = 1.6335 Å, and ∠e(N–C–S) = 180°. The ro-vibrationally averaged structure, determined as expectation values over DVR3D wavefunctions, has 〈r(N–C)〉0 = 1.1836 Å, 〈r(C–S)〉0 = 1.6356 Å, and 〈∠(N–C–S)〉0 = 172.5°. Using these expectation values as the initial guess, a bent r0 structure having an 〈∠(N–C–S)〉0 of 172.2° is deduced from the experimentally reported B0 values for NC32S and NC34S. Our previous prediction that a linear molecule, in any ro-vibrational state including the ro-vibrational ground state, is to be “observed” as being bent on ro-vibrational average, has been confirmed here theoretically through the expectation value for the bond-angle deviation from linearity, 〈ρ¯〉, and experimentally through the interpretation of the experimentally derived rotational-constant values.

Bending wavefunctions for linear molecules

The bending motion of a linear triatomic molecule has two unique characteristics: the bending mode is doubly degenerate and only positive values of the bending angle, expressed by the bond angle supplement ρ¯, can be observed. The double degeneracy requires the wavefunction to be described as a two-dimensional oscillator. In the present work, we first review the conventional expressions based on two, symmetrically equivalent normal coordinates. Then we discuss an alternative expression for the bending wavefunction in terms of two geometrical coordinates, the bond angle supplement ρ¯ (=π-τ⩾0, where τ is the bond angle) and the rotation angle χ (0⩽χ<2π) describing rotation of the molecule around the molecular axis. In this formalism, defined for the (ρ¯,χ) polar-coordinate space with volume element ρ¯dρ¯dχ, the one-dimensional wavefunction resulted through re-normalization for χ has zero amplitude at ρ¯=0, and the ro-vibrational average of the bending angle, i.e., the expectation value 〈ρ¯〉, attains a non-zero, positive value for any ro-vibrational state including the vibrational ground state. This conclusion appears to cause some controversy since much conventional spectroscopic wisdom insists on 〈ρ¯〉 having the value zero.

16O/18O oxygen isotope exchange kinetics in MnO 4 − as probed by 55Mn NMR

The effect of the 16O ↔ 18O substitution in the coordination sphere of permanganate anion MnO4− on the chemical shift of 55Mn nuclei have been studied by 17O and 55Mn NMR. Time constants τn,k of oxygen exchange in the water–permanganate anion system have been estimated. In nearly neutral solutions (pH ≈ 6.8–7.2), the oxygen exchange time is on the order of tens of hours. Bubbling gaseous HCl through this solution for a few seconds leads to the equilibrium distribution of oxygen isotopes in the manganese coordination sphere. The observed temperature dependences of isotope-induced 55Mn NMR shifts in Mn16 O44-n18On– (n = 0–4) have been treated as a result of rovibrational averaging of Mn–O bond lengths. The change in the Mn—O bond length in caused by the 16O → 18O isotope substitution is on the order of 10–4 A.

Ro-vibrational averaging of the isotropic hyperfine coupling constant for the methyl radical.

We present the first variational calculation of the isotropic hyperfine coupling constant of the carbon-13 atom in the CH3 radical for temperatures T = 0, 96, and 300 K. It is based on a newly calculated high level ab initio potential energy surface and hyperfine coupling constant surface of CH3 in the ground electronic state. The ro-vibrational energy levels, expectation values for the coupling constant, and its temperature dependence were calculated variationally by using the methods implemented in the computer program TROVE. Vibrational energies and vibrational and temperature effects for coupling constant are found to be in very good agreement with the available experimental data. We found, in agreement with previous studies, that the vibrational effects constitute about 44% of the constant's equilibrium value, originating mainly from the large amplitude out-of-plane bending motion and that the temperature effects play a minor role.

Recent Advances in Theoretical and Physical Aspects of NMR Chemical Shifts

In the first part of this review, theoretical aspects of nuclear magnetic shielding include (a) general theory, for example, newly developed approaches in relativistic theory of nuclear shielding, the relation between the spin-rotation tensor and shielding in relativistic theory, ab initio methods for treating open shell systems and a complete theory of chemical shifts in paramagnetic systems, the link between the definitions of the elusive concepts aromaticity and anti-aromaticity and the magnetic properties: the magnetizability tensor and the nuclear magnetic shielding tensor via delocalized electron currents and electron current maps, (b) ab initio and DFT calculations, both relativistic and non-relativistic, for various nuclei in various molecular systems using various levels of theoretical treatment. Physical aspects include (a) anisotropy of the shielding tensor, usually from solid state measurements, and calculations to support these, (b) shielding surfaces and rovibrational averaging, paying special attention to the sensitive relationship between shielding and bond angles or torsion angles that makes shielding such a powerful tool for structural/conformational determination in macromolecules, (c) chemical shifts that arise from isotopic substitution of NMR nucleus or neighboring nuclei, (d) intermolecular effects on nuclear shielding, and (e) absolute shielding scales.

Ro-vibrational properties of FeCO in the [formula omitted] and [formula omitted] electronic states: A computational molecular spectroscopy study

The present work complements our previous study of the geometry and electronic structure in the ground and low-lying electronic states of FeCO. Here, we report three-dimensional potential energy surfaces (PESs) for the 3Σ- electronic ground state and its high-spin counterpart, the excited state ã5Σ-, calculated ab initio at the MR-SDCI+Q_DK3/[5ZP ANO-RCC (Fe, C, O)] level of theory. These PESs are employed in 2nd-order-perturbation-theory and DVR3D calculations of the rotation–vibration energies and ro-vibrationally averaged structures. The equilibrium structures determined from the 3D PESs have re(Fe–C)=1.7247Å, re(C–O)=1.1587Å, and ∠e(Fe–C–O)=180°for the X̃3Σ- state, and re(Fe–C)=1.8429Å, re(C–O)=1.1522Å, and ∠e(Fe–C–O)=180° for the ã5Σ- state. The ro-vibrationally averaged structures, determined as expectation values over DVR3D wavefunctions, have 〈r(Fe–C)〉0=1.7303Å, 〈r(C–O)〉0=1.1631Å, and 〈∠(Fe–C–O)〉0=172.6°for the X̃3Σ- state, and 〈r(Fe–C)〉0=1.8471Å, 〈r(C–O)〉0=1.1568Å, and 〈∠(Fe–C–O)〉0=171.4° for the ã5Σ- state. The coordinate-covalent Fe–C bond in the X̃3Σ- state, which elongates significantly as the molecule bends, is shown to exhibit normal large amplitude bending motion with strong coupling (manifested by the large value of the relevant third order force constant) between bending and Fe–C stretching modes. The ionic Fe–C bond in the ã5Σ- state shows anormal bending behavior due to a severe Fermi resonance which also gives rise to a large coupling between the bending and the Fe–C stretching motions, even though the corresponding third order force constant is small. The Yamada–Winnewisser quasi-linearity parameter γ0 is calculated to be −1.00 and −0.90, values characteristic for a linear molecule, for the X̃3Σ- and ã5Σ- states, respectively. The ro-vibrationally averaged structures of the X̃3Σ- state are discussed in detail and it is concluded that any triatomic, linear molecule will be observed as being bent on ro-vibrational average.

Nuclear magnetic shielding and quadrupole coupling tensors in liquid water: a combined molecular dynamics simulation and quantum chemical study.

Nuclear magnetic resonance (NMR) shielding tensors for the oxygen and hydrogen nuclei, as well as nuclear quadrupole coupling tensors for the oxygen and deuterium nuclei of water in the liquid and gaseous state, are calculated using Hartree-Fock and density functional theory methods, for snapshots sampled from Car-Parrinello molecular dynamics trajectories. Clusters representing local liquid structures and instantaneous configurations of a single molecule representing low-density gas are fed into a quantum chemical program for the calculation of the NMR tensors. The average isotropic and anisotropic tensorial properties of 400 samples in both states, averaged using a common Eckart coordinate frame, are calculated from the data. We report results for the gas-to-liquid chemical shifts of (17)O and (1)H nuclei, as well as the corresponding change in the nuclear quadrupole couplings of (17)O and (2)H. Full thermally averaged shielding and quadrupole coupling tensors are reported for the gaseous and liquid-state water, for the first time in the case of liquid. Electron correlation effects, the difference of classical vs quantum mechanical rovibrational averaging, and different methods of averaging anisotropic properties are discussed.

Size-consistent ab initio calculation of the electric quadrupole moment of Cl 2

The molecular electric quadrupole moment ( Θ) of Cl 2 has been calculated using SDCI, and (SC) 2-SDCI wave functions as well as CCSD, CCSD(T), and CC3 methods. All these correlation methods are single reference. All of them, but SDCI, are free of the size-extensivity error. The variation of Θ from the separated atoms to the equilibrium region is reported. The present results leads to an estimated value of 2.3520 a.u. (10.55 × 10 −40 Cm 2) corresponding to a CC(3) calculation at the CBS approach and including the ro-vibrational and thermal averaging corrections. This value is compatible with two experimental values and points to one of them as slightly more reliable.

Conformational dependence of deuterium-induced isotope effects on the olefinic one-bond13C–1H and three-bond1H–2H coupling constants in cis- and trans-stilbene

Secondary deuterium isotope effects on the olefinic one-bond 13C, 1H and three-bond 1H, 2H spin–spin coupling constant were determined in a series of specifically deuteriated cis- and trans-stilbene isotopomers. Inverse (1H detected) heteronuclear multiple quantum correlation (HMQC) NMR pulse sequence was used to measure the coupling constants in these π-conjugated systems of a moderate size. Complete spectral analyses were performed in cases where higher order perturbations took place. All secondary isotope effects on 1J(13Cα, 1H)/Hz are negative, i.e. the reduced values from –0.06 to –0.02 were observed. In contrast, the effects on 3J(1Hα, 2Hα′)/HZ are all positive from 0.04 to 0.13, thus being on average relatively higher than those on 1J(13Cα, 1H). Only two primary effects on 1J(13Cα, 1H) were determined to see whether the same trend holds as for secondary effects. The real values of olefinic coupling constants are given with a high accuracy. In order to relate isotope effects with structural parameters, AM1 semiempirical calculations were performed. Subtle conformational changes due to the deuteriation are discussed in terms of rovibrational averaging and overall geometry.

Nuclear magnetic shielding of nitrogen in ammonia

The nitrogen shielding surface in ammonia is calculated using the localized orbital-local origin (LORG) method of Hansen and Bouman, in terms of the symmetry coordinates for the molecule. With respect to the inversion coordinate, the N shielding surface has a shape similar to the potential surface. Rovibrational averaging of the N shielding in NH3 and ND3 molecules is carried out using numerical wave functions which are solutions to the inversion potential which best fits the spectra of all isotopomers. The other coordinates are vibrationally averaged in the usual way, assuming small amplitude motions. The calculated temperature dependence of the N shielding due to inversion is in the opposite sense to that observed for a large number of molecules, and is nearly canceling the contributions from all the other coordinates. The temperature dependence of the nitrogen shielding in ammonia has been measured in the range 300–400 K in samples with densities in a hundredfold range (0.37–33 amagat). When the temperature-dependent intermolecular effects are separated out, the remaining temperature dependence is small and is consistent with the calculations. The inversion contribution to the deuterium-induced isotope shift is of opposite sign to the contributions from all other coordinates. The agreement with the experimental isotope shift in the liquid phase is satisfactory.

Magnetic optical rotation in H2 and D2

Results are reported of the first definitive calculation of the Verdet constant for H2 and D2. This constant governs the Faraday effect. A new and compact formalism is introduced and applied with the aid of explicitly electron correlated wave functions. After ro-vibrational and thermal averaging (factors which affect the results by about 10%), our values are in good agreement with the experimental ones, which, at best, are probably only accurate to 1%. Approximations and an appropriate dispersion formula are also discussed. Our results show that for H2 and D2 the exact constant is almost exactly proportional to the so-called normal Verdet constant for the experimentally accessible frequencies. The recommended dispersion formula for H2 is V≂2.0701 (ℏω/Eh)2/[0.2435−(ℏω/Eh)2]2×10−7 rad e a0 ℏ−1.

ChemInform Abstract: Effects of Rovibrational Averaging on the 93Nb Chemical Shift in the (Nb(CO)6)‐ Ion, Based on NMR and Vibrational Spectra.

AbstractThe 93Nb NMR chemical shift changes with temp. and isotopic mass of the ligands.

ChemInform Abstract: Isotope and Temperature Dependence of Transition‐Metal Shielding in Complexes of the Type M(XY)6

The temperature coefficients of the chemical shift of /sup 51/V in carbonyl complexes in solution are reported here. For transition metal nuclei the effects of rovibrational averaging can be obtained from these temperature coefficients without gas-phase studies in the zero-pressure limit. They present a theory for the large temperature coefficients and large isotope shifts observed in transition metal complexes. They use the vibrational analysis of V(CO)/sub 6//sup -/ and Co(CN)/sub 6//sup 3 -/ to provide dynamic averages of the displacements in the V-C, C-O, Co-C, and C-N bonds as typical examples. The temperature and mass dependencies of these dynamic averages successfully account for the signs and magnitudes of the observed temperature coefficients and isotope shifts. They report observations of a correlation between the temperature coefficients and chemical shifts. This correlation is qualitatively consistent with a simple electrostatic model for shielding. The limitations of the electrostatic model for shielding are discussed.

Isotope and temperature dependence of transition-metal shielding in complexes of the type M(XY)6

The temperature coefficients of the chemical shift of /sup 51/V in carbonyl complexes in solution are reported here. For transition metal nuclei the effects of rovibrational averaging can be obtained from these temperature coefficients without gas-phase studies in the zero-pressure limit. They present a theory for the large temperature coefficients and large isotope shifts observed in transition metal complexes. They use the vibrational analysis of V(CO)/sub 6//sup -/ and Co(CN)/sub 6//sup 3 -/ to provide dynamic averages of the displacements in the V-C, C-O, Co-C, and C-N bonds as typical examples. The temperature and mass dependencies of these dynamic averages successfully account for the signs and magnitudes of the observed temperature coefficients and isotope shifts. They report observations of a correlation between the temperature coefficients and chemical shifts. This correlation is qualitatively consistent with a simple electrostatic model for shielding. The limitations of the electrostatic model for shielding are discussed.

Rovibrational averaging of nuclear shielding in MX6-type molecules

Calculations of the mean M–X bond displacements in octahedral MX6 molecules by the L tensor and Bartell methods using anharmonic force fields for SF6, SeF6, TeF6, WF6, PtCl=6, and PtBr=6 are compared with electron diffraction data in SF6, and are used in the interpretation of 19F, 77Se, 125Te, and 195Pt chemical shifts in these molecules. The temperature and mass dependence of M and X chemical shifts can be written in terms of 〈ΔrMX〉, and the two together provide a critical test of anharmonic force fields. The direct proportionality of the isotope shifts to the mass factor (m′−m)/m′ is found to be a direct consequence of the calculated linear dependence of 〈Δr〉−〈Δr〉′ and 〈(Δr)2〉−〈(Δr)2〉′ on this mass factor. The observed isotope shifts and temperature dependent chemical shifts in the zero pressure limit can be used to determine the sensitivity of the nuclear magnetic shielding to bond extension.