Perturbative Instanton Approach Research Articles

Tunneling splittings in vibrational spectra of non-rigid molecules. X. Reaction path Hamiltonian as zero-order approximation

This paper completes the development of the perturbative instanton approach (PIA), presented in previous papers of this series, as a rigorous semiclassical method for the calculation of the tunneling dynamics in multidimensional potential energy surfaces. The precision of the PIA is established here by calculating upper as well as lower bounds for quantities characterizing the tunneling. Two perturbative solutions of the integral equation of instanton motion are derived using the straight-line path (SLP) and minimum energy path (MEP) as zero-order approximations. For arbitrary coupling strengths between the tunneling coordinate, ( X), and small-amplitude coordinates, { Y k }, the multidimensional extreme tunneling trajectory (ETT) is shown lie inside a strip, the boundaries of which are the trajectories ( X 1 (0), { Y k ( X 1 (0))}) and ( X 2 (0), { Y k ( X 2 (0))}). X 1 (0)( t) and X 2 (0)( t)= X 1 (0)( αt), correspond to 1D tunneling motions along the SLP in the bare potential and along the MEP in the adiabatic potential, modified by taking into account that the effective mass is X-dependent. The scaling coefficient, α, is defined by the spectral density of transverse vibrations. Along the boundary trajectories, the X 1 (0)( t) and X 2 (0)( t) motions induce transverse displacements, Y k , proportional to the coupling constants. Since the boundary trajectories are related by a scaling transformation in time, both perturbative series differ only by a renormalization of the transverse frequencies. The method of two perturbative series is also exploited for solving the semiclassical equations. The contributions of each transverse coordinate to the Euclidean action along the ETT, the width of tunneling channel, and the prefactors of the semiclassical wave functions are shown to be limited from below and from above by the corresponding parameters calculated for the boundary trajectories. These “conditions of the strip” establish the accuracy of tunneling splittings calculated within the PIA. The thus completed PIA is well adapted for the study of non-rigid molecules with strong XY-coupling. As an example, the tunneling dynamics of malonaldehyde is recalculated, and lower and upper boundaries for the proton transfer barrier are determined to equal 4.30 and 4.39 kcal/mol.

Multidimensional tunneling dynamics of proton transfer in malonaldehyde molecule and its isotopomers

The twenty-one-dimensional Hamiltonian of malonaldehyde molecule and a number of its isotopomers (H/D, 13C/12C) was reconstructed in the low-energy region (<3000 cm–1). Parameters of the Hamiltonian were obtained from quantum-chemical calculations of the energies, equilibrium geometries, and eigenvectors and eigenfrequencies of normal vibrations at the stationary points corresponding to the ground state and transition state. Despite substantial variation of the barrier height calculated using different quantum-chemical methods (from 2.8 to 10.3 kcal mol–1), the corresponding potential energy surfaces can be matched with high accuracy by scaling only one parameter (the semiclassical parameter γ, which defines the scales of potential, energy, and action). Scaling invariance allows optimization of the Hamiltonian in such a way that the calculated ground-state tunneling splitting coincides with the experimental value. The corresponding potential barrier height is estimated at 4.34±0.4 kcal mol–1. The quantum dynamics problem was solved using the perturbative instanton approach without reducing the number of degrees of freedom. The role of all transverse vibrations in proton tunneling is characterized. Vibration-tunneling spectrum is calculated for the ground state and low-lying excited states and mode-specific isotope effects are predicted.

Six-dimensional tunneling dynamics of internal rotation in hydrogen peroxide molecule and its isotopomers

The six-dimensional torsion-vibration Hamiltonian of the H2O2 molecule and its H/D- and 18O/16O-isotopomers is derived. The Hamiltonian includes the kinetic energy operator, which depends on the tunneling coordinate, and the potential energy surface represented as a quartic polynomial with respect to the small-amplitude transverse coordinates. Parameters of the Hamiltonian were obtained from DFT calculations of the equilibrium geometries, eigenvectors, and eigenfrequencies of normal vibrations at the stationary points corresponding to the ground state and both the cis- and trans-transition states, carried out with the B3LYP density functional and 6-311+G(2d,p) basis set. The quantum dynamics problem is solved using the perturbative instanton approach generalized for the excited states situated above the barrier top. Vibration-tunneling spectra are calculated for the ground state and low-lying excited states with energies below 2000 cm–1. Strong kinematic and squeezed potential couplings between the large-amplitude torsional motion and bending modes are shown to be responsible for the vibration-assisted tunneling and for the dependence of tunneling splittings on the quantum numbers of small-amplitude transverse vibrations. Mode-specific isotope effects are predicted.

Tunneling splittings in vibrational spectra of non-rigid molecules: VIII. Six-dimensional tunneling dynamics of hydrogen peroxide and its isotopomers

The six-dimensional torsion–vibration Hamiltonian of H 2O 2 is derived. This Hamiltonian includes a tunneling coordinate dependent kinetic energy operator and a potential energy surface, quartic in the small-amplitude transverse coordinates. The parameters of this Hamiltonian are derived from the equilibrium geometries and the eigenvectors and eigenfrequencies of the normal modes in the ground and the two ( cis/ trans) transition states. The quantum dynamical problem is solved within the perturbative instanton approach developed in previous papers of this series, which is generalized here for excited states situated above the barrier and for anharmonic transverse vibrations. Globally uniform semiclassical wave functions and tunneling splittings in the ground and low-lying excited states are calculated. The strong kinematic and squeezed potential couplings between the wide-amplitude torsional motion and bending vibrations are shown to be responsible for vibrationally assisted tunneling and for the significant dependence of the tunneling splittings on the quantum numbers of transverse vibrations. Mode specific isotope effects of tunneling splittings are predicted. The dependence on the quantum numbers of overall rotation is found to be insignificant.

Tunneling splittings in vibrational spectra of non-rigid molecules: IX. Malonaldehyde and its isotopomers as a test case for fully coupled multidimensional tunneling dynamics

Twenty one dimensional potential energy surfaces (PES) and the tunneling coordinate dependent kinematic matrices of malonaldehyde and of several of its isotopomers (D, 13C) are constructed in the low-energy region (<3000 cm −1) using quantum-chemical data. Even though the barrier heights calculated with different methods differ strongly (from 2.8 to 10.3 kcal mol −1), all PES become near identical after scaling, fitting only one semiclassical parameter γ, which defines the scales of potential, energy and action. The results of dynamical calculations carried out in dimensionless variables are therefore quite insensitive to the choice of the quantum-chemical method used for the construction of the Hamiltonian. Choosing the value of γ such that the calculated tunneling splitting in the ground state coincides with the experimental value, the corresponding barrier height is determined as 4.30 kcal mol −1 and the vibrational spectrum of the transition state is obtained. The perturbative instanton approach developed in the previous papers of this series is used to solve the dynamical problem without reducing the number of degrees of freedom. The role of all 20 transverse vibrations in proton tunneling is characterized. The tunneling path and globally uniform semiclassical wave functions are evaluated from the fourth-order Hamilton–Jacobi equation and the second-order transport equation. Tunneling splittings in the ground and low-lying excited states are calculated and isotope effects of H/D and 13C/12C substitutions are predicted.

Tunneling splittings in vibrational spectra of non-rigid molecules: VII. Globally uniform semiclassical wave functions within the instanton approach

The perturbative instanton approach developed in the previous papers of this series (V.A. Benderskii et al., Chem. Phys. 219 (1977) 119, 143; 234 (1998) 153, 173; 244 (1999) 273, 299) is shown to reproduce tunneling splittings in low-lying excited states of multiwell 1D potentials with the same accuracy as second-order perturbation theory for anharmonic oscillators. Instanton wave functions of highly excited states in these potentials are derived using the asymptotically smooth matching of semiclassical wave functions with solutions of the Schrödinger equation for the regions near the external and internal turning points. Multidimensional, globally uniform, semiclassical wave functions of both high and low energy states are expanded over a basis set including the above 1D functions and localized functions of small-amplitude motions coupled with the tunneling coordinate. The numerical procedure for the solution of multidimensional, deep tunneling problems is discussed.

Tunneling dynamics of internal rotation in the nitric acid molecule

The Hamiltonian of internal rotation about theC2 axis in the HNO3 molecule and its H/D-, O18/O16-, and N15/N14-isotopomers was reconstructed using the results of quantumchemical calculations. The Fermi resonance between the torsional (2v9) and ONO bending (v5) vibrations is a characteristic feature of the molecule. Tunneling splittings in the ground and excited states were calculated using the perturbative instanton approach. Abnormally large changes in the splittings upon isotope substitution of heavy atoms are predicted.

Tunneling splittings in vibrational spectra of non-rigid molecules: VI. Asymmetric double-well potentials

The perturbative instanton approach (PIA) developed in the previous papers of this series [V.A. Benderskii, E.V. Vetoshkin, S.Yu. Grebenshchikov, L. von Laue, H.P. Trommsdorff, Chem. Phys. 219 (1997) 119; V.A. Benderskii, E.V. Vetoshkin, L. von Laue, H.P. Trommsdorff, Chem. Phys. 219 (1997) 143; V.A. Benderskii, E.V. Vetoshkin, H.P. Trommsdorff, Chem. Phys. 234 (1998) 153; V.A. Benderskii, E.V. Vetoshkin, Chem. Phys. 234 (1998) 173] has been extended to asymmetric double-well potentials, characterized by an energy difference, D, between the two potential wells. For multidimensional double-well potentials with an asymmetry comparable to or smaller than the fundamental frequency, Ω, of the tunneling mode in the wells, the semi-classical wave functions and tunneling splittings in the ground and low-lying excited vibrational states are calculated. The semi-classical equations are written in a form, in which the energy asymmetry, D, and eigenvalues, as terms of first-order of ℏ, are replaced from the Hamilton–Jacobi equation (HJE) into the transport equation. This replacement, symmetrizing the HJE, makes it possible to avoid the construction of a tunneling trajectory between inequivalent wells and to use the extreme classical trajectory in the upside-down symmetric potential. Comparison of the results of the quantum and PIA calculations shows that the latter provide an evaluation of the tunneling matrix element with an accuracy of better than 10%, provided that D/ ℏΩ≤3 and the semi-classical parameter γ>5. A resonant increase of the tunneling splitting is predicted, if one of the transverse frequencies or the difference between longitudinal and transverse frequencies become equal to D. The relation between these resonances and the Flynn–Stoneham and Skinner–Trommsdorff mechanisms of vibrationally assisted incoherent tunneling is discussed.

Tunneling splittings in vibrational spectra of non-rigid molecules: IV. Kinematic couplings

The perturbative instanton approach (PIA) developed previously [Benderskii and co-workers, Chem. Phys. 219 (1997) 119; 219 (1997) 143; 234 (1998) 153] is generalized to multidimensional Hamiltonians in which the potential is non-separable and the kinetic energy operator is tunneling coordinate dependent, i.e. both potential and kinematic couplings between the tunneling and transverse coordinates are considered. The potential couplings uniquely define the symmetry of the kinematic couplings. The first-order extreme tunneling trajectory (ETT) and the semi-classical wave functions are determined, without using the Eckart–Sayvetz conditions, from the solutions of classical equations of motion and the semi-classical Hamilton–Jacobi and transport equations. The second-order effect of kinematic couplings is shown to result in the appearance of the pseudo-potential terms in the above equations. For active transverse vibrations, kinematic couplings contribute to an action proportional to the products of the potential and kinematic coupling constants. Analytical expressions for the ETT, the semi-classical action and the prefactor of the wave function in the vicinity to the ETT are given. For 2D potentials, tunneling splittings evaluated within PIA agree well with the results of quantum calculation.

Tunneling splittings in vibrational spectra of non-rigid molecules:: III. Tunneling coordinate-dependent coupling between small amplitude motions

The perturbative instanton approach (PIA), developed previously to evaluate tunneling in multidimensional potential energy surfaces, is extended to include tunneling coordinate ( X)-dependent interactions between small amplitude vibrations Y k . The form of the coupling terms considered here depends on the symmetry of the two, coupled, transverse vibrations. When both vibrations have the same symmetry, the coupling is ∼ X 2 Y k Y l , while ∼ XY k Y l coupling exists between coordinates of the different symmetry. When the large amplitude motions is described by an angular coordinate φ with period 2π/ n, the corresponding interaction terms have the form Y k Y l ·cos( nφ) and Y k Y l ·sin( nφ), respectively. These couplings cause a rotation of the transverse coordinate system along the extreme tunneling trajectory (ETT), and the semi-classical action depends on the sign of coupling constants. The first-order analytical expressions for the ETT, the semi-classical action and the prefactors of the wave functions of the ground and first excited states are found from perturbative solutions of the equations of motion, the Hamilton–Jacobi and the transport equations, respectively. When the longitudinal frequency in the well approaches the renormalized frequency of one of the transverse vibration, a resonant increase of the tunneling splitting results for the corresponding vibrational level.

Proton dynamics and the tautomerization potential in benzoic acid crystals

AbstractTwo‐proton exchange along the two hydrogen bonds mediates the tautomerization in benzoic acid (BA) dimers. Optical spectroscopy and quasi‐elastic neutron scattering (QENS) have been employed to characterize the proton dynamics in doped and pure BA crystals. The proton motion in BA is governed by a multidimensional potential energy surface (PES), and recent theoretical methods, based on a perturbative instanton approach, to describe tunneling in such PES are presented. This PES is also modulated by the interaction with the solid state environment as manifest by the energy difference between the otherwise equivalent tautomers. The value of this energy difference in pure crystals is an important parameter in the data analysis of NMR and QENS. Both methods give mutually consistent values that differ significantly from earlier determinations via infrared and 13C NMR as well as a recent evaluation using neutron diffraction data. The energy difference between tautomers is altered for dimers in the vicinity of impurity molecules. This is the basis for the optical spectroscopic methods, which enable a direct and accurate determination of the level structure and tautomerization dynamics of these coupled dimers in the limit of very low temperatures, where coherent tunneling is also observed in some cases. Measurements with new impurity molecules make it possible to monitor simultaneously at least 10 different tautomer configurations and prove that the influence of the probe molecules on the proton dynamics is small. The transition to thermally activated barrier crossing at higher temperatures is accessed via the width of the QENS line that is determined by the inverse of the proton correlation time. The quantitative data analysis of the scattered intensity as a function of temperature and scattering angle yields the energy difference, A, between the two wells (A/kB=90+20 K), the length (0.686 Å), and direction of the proton jump vector. These measurements complement NMR investigations presented in the preceding paper.

Hydrogen bond dynamics in benzoic acid crystals

We report quasi-elastic neutron scattering (QENS) and optical spectroscopic measurements of the proton motion in the hydrogen bond of benzoic acid crystals, a system in which proton transfer occurs in a near-symmetric double-well potential and both coherent and incoherent proton tunneling have been measured. Inter-well relaxation is characterized by a correlation time that determines the width of the QENS line. Our new measurements cover the transition from the classical to the quantum regime. The optical spectroscopic data, using new dopant molecules, extend measurements of the proton dynamics to very low temperatures, where coherent tunneling is observed. Theoretical methods, based on a perturbative instanton approach, have been developed to describe tunneling in multidimensional potential energy surfaces and are used to model our observations.

Tunneling splitting in vibrational spectra of non-rigid molecules. I. Perturbative instanton approach

A perturbative approach to calculating tunneling splittings in multidimensional potential energy surfaces (PES) is developed. In two-dimensional (2D) model PES, represented by a quartic X 4 potential and a harmonic oscillator Y with frequency ω, both coupled by linear ( CXY) or gated ( CX 2 Y) terms, the extreme tunneling trajectories (ETT) of zero energy are determined by solving the classical equations of motion in the inverted potentia, − V( X, Y), in the form of rapidly converging Taylor series of C/ ω < 1. The series for X( t) and Y( t) contain only even or odd powers of C/ ω, respectively. The semiclassical action on the ETT expands into a series of ( C/ ω 2). When C/ ω < 0.5), second order action reproduces the exact value with an accuracy of better than 5%. On multidimensional PES with one saddle point, the contributions to the action of mutually uncoupled transverse vibrations are additive, which enables us to introduce their spectral density, characterizing the tunneling dynamics. The semiclassical wave function of the ground state is found within the approximation of small fluctuation about the ETT. From this wave function, tunneling splittings are calculated, using the Lifshitz-Herring formula. The values obtained are in satisfactory agreement with the results of numerical diagonalization of the Hamiltonian matrix. Hydrogen transfers in malonaldehyde and in formic acid dimers are treated as examples for the application of this approach.

Tunneling splitting in vibrational spectra of non-rigid molecules. II. Excited states

The perturbative instanton approach (PIA), developed previously [1], is used to evaluate tunneling splittings of excited vibrational levels in multidimensional double well potential surfaces. Solutions of the two-dimensional semi-classical equations of motion are expanded into a perturbative series of the parameter C/ ω (< 1), where C is the coupling constant between the tunneling coordinate and the coordinate of a transverse mode with frequency ω. Semi-classical action, found from the Hamilton-Jacobi equation, is shown to have a minimum on the zero energy extreme tunneling trajectory, defined by the classical equations of motion in the inverted potential. Prefactors of semi-classical wave functions of excited vibrational states are derived from the first order solution of the transport equation. Coupling between longitudinal and transverse vibrations leads to a mixing of their excited states. Even if the coupling parameter is as small as ≈10 −2, resonant mixing increassesthe tunneling splitting in the transverse ladder by several orders of magnitude as compared to the ground state. The generalized Lifshitz-Herring formula, taking into account this mixing, is presented. At Fermi resonances, the tunneling splittings become equal for both states. Possibilites of experimental observations of these resonances in Raman spectra of matrix isolated non-rigid molecules are discussed briefly.