Vibrational Wave Functions Research Articles

Interpretation of the vibrational energy level structure of the astructural molecular ion H5 (+) and all of its deuterated isotopomers.

Variational nuclear motion computations, employing an exact kinetic energy operator and two different potential energy surfaces, are performed to study the first 60 vibrational states of the molecular ion H5 (+)≡ [H2-H-H2](+) and all of its deuterated isotopologues and isotopomers, altogether 12 species. Detailed investigation of the vibrational wavefunctions mostly results in physically intuitive labels not only for the fundamentals but also for the overtone and combination states computed. The torsional motion associated with the left and right diatomics appears to be well separated from the other vibrational degrees of freedom for all species. The unusual structure of the higher-lying bending states and the heavy mixing of the internal motions is partly due to the astructural character of all these molecular ions. The existence of distinct isotopomers in the H5-nDn (+), n = 1-4 cases, in the energy range studied, is confirmed. Two rules determine the stability order of the isotopomers: first, when possible, H prefers to stay in the middle of the ions rather than at the sides, and, second, the isotopomer with a homonuclear diatomic at the side is always lower in energy. The large number of precise vibrational energies of the present study, as well as the detailed assignment of the states, should serve as benchmarks for future studies by more approximate nuclear-motion treatments, such as diffusion Monte Carlo and multiconfiguration time-dependent Hartree.

Presence of C2 molecular Lines in Sunspot Umbral Spectra

The C2 molecule is well known for its astrophysical importance. The radiative transition parameters that include Franck-Condon (FC) factor, r-centroid, electronic transition moment, Einstein coefficient, absorption band oscillator strength, effective temperatures and radiative life time have been estimated for the Swan band (d3Πg−a3Πu) system of C2 molecule for experimentally observed vibrational levels using RKR (Rydberg–Klein–Rees) potential energy curve. The lifetime for the d3Πg state of C2 molecule was found to be 82.36ns for the v′=0 level. A reliable numerical integration method has been used to solve the radial Schrödinger equation for the vibrational wave functions of upper and lower electronic states based on the latest available spectroscopic data and known wavelengths. The estimated radiative transition parameters are tabulated. The effective vibrational temperature of Swan band system of C2 molecule is found agreed with the effective rotational temperature from photosphere spectrum. Hence, the radiative transition parameters and effective temperatures help us to ascertain the presence of C2 molecule in the interstellar medium, photosphere and sunspots.

Proton Quantization and Vibrational Relaxation in Nonadiabatic Dynamics of Photoinduced Proton-Coupled Electron Transfer in a Solvated Phenol-Amine Complex.

Nonadiabatic dynamics simulations of photoinduced proton-coupled electron transfer (PCET) in a phenol-amine complex in solution were performed. The electronic potential energy surfaces were generated on-the-fly with a hybrid quantum mechanical/molecular mechanical approach that described the solute with a multiconfigurational method in a bath of explicit solvent molecules. The transferring hydrogen nucleus was represented as a quantum mechanical wave function calculated with grid-based methods, and surface hopping trajectories were propagated on the adiabatic electron-proton vibronic surfaces. Following photoexcitation to the excited S1 electronic state, the overall decay to the ground vibronic state was found to be comprised of relatively fast decay from a lower proton vibrational state of S1 to a highly excited proton vibrational state of the ground S0 electronic state, followed by vibrational relaxation within the S0 state. Proton transfer can occur either on the highly excited proton vibrational states of S0 due to small environmental fluctuations that shift the delocalized vibrational wave functions or on the low-energy proton vibrational states of S1 due to solvent reorganization that alters the asymmetry of the proton potential and reduces the proton transfer barrier. The isotope effect arising from replacing the transferring hydrogen with deuterium is predicted to be negligible because hydrogen and deuterium behave similarly in both types of proton transfer processes. Although an isotope effect could be observed for other systems, in general the absence of an isotope effect does not imply the absence of proton transfer in photoinduced PCET systems. This computational approach is applicable to a wide range of other photoinduced PCET processes.

Symmetry of extremely floppy molecules: Molecular states beyond rotation-vibration separation.

Traditionally, molecules are theoretically described as near-static structures rotating in space. Vibrational motion causing small structural deformations induces a perturbative treatment of the rotation-vibration interaction, which fails in highly fluxional molecules, where all vibrational motions have amplitudes comparable in size to the linear dimensions of the molecule. An example is protonated methane (CH5(+)) [P. Kumar and D. Marx, Phys. Chem. Chem. Phys. 8, 573 (2006); Z. Jin et al., J. Phys. Chem. A 110, 1569 (2006); and A. S. Petit et al., J. Phys. Chem. A 118, 7206 (2014)]. For these molecules, customary theory fails to simulate reliably even the low-energy spectrum [T. Oka, Science 347, 1313-1314 (2015) and O. Asvany et al., Science 347, 1346-1349 (2015)]. Within the traditional view of rotation and vibration being near-separable, rotational and vibrational wavefunctions can be symmetry classified separately in the molecular symmetry (MS) group [P. Bunker and P. Jensen, Molecular Symmetry and Spectroscopy, NRC Monograph Publishing Program (NRC Research Press, 2006)]. In this article, we discuss a fundamental group theoretical approach to the problem of determining the symmetries of molecular rotation-vibration states. We will show that all MS groups discussed so far are isomorphic to subgroups of the special orthogonal group in three dimensions SO(3). This leads to a group theoretical foundation of the technique of equivalent rotations [H. Longuet-Higgins, Mol. Phys. 6, 445 (1963)]. The group G240 (the MS group of protonated methane) represents, to the best of our knowledge, the first example of a MS group which is not isomorphic to a subgroup of SO(3) (nor of O(3) or of SU(2)). Because of this, a separate symmetry classification of vibrational and rotational wavefunctions becomes impossible in this MS group, consistent with the fact that a decoupling of vibrational and rotational motion is impossible. We discuss here the consequences of this. In conclusion, we show that the prototypical, extremely floppy molecule CH5(+) represents a new class of molecules, where customary group theoretical methods for determining selection rules and spectral assignments fail so that new methods have to be developed.

Anharmonic Franck-Condon Factors for the X̃(2)B1 ← X̃(1)A1 Photoionization of Ketene.

The X̃(2)B1 ← X̃(1)A1 photoelectron spectra of ketene and its doubly deuterated isotopologue have been computed from correlated vibrational wave functions as determined from vibrational configuration interaction theory relying on multidimensional Born-Oppenheimer potential energy surfaces being obtained from explicitly correlated coupled-cluster calculations. Duschinsky effects were accounted for in all cases. Excellent agreement with available experimental data was achieved.

Calculating vibrational spectra without determining excited eigenstates: Solving the complex linear equations of damped response theory for vibrational configuration interaction and vibrational coupled cluster states.

It is demonstrated how vibrational IR and Raman spectra can be calculated from damped response functions using anharmonic vibrational wave function calculations, without determining the potentially very many eigenstates of the system. We present an implementation for vibrational configuration interaction and vibrational coupled cluster, and describe how the complex equations can be solved using iterative techniques employing only real trial vectors and real matrix-vector transformations. Using this algorithm, arbitrary frequency intervals can be scanned independent of the number of excited states. Sample calculations are presented for the IR-spectrum of water, Raman spectra of pyridine and a pyridine-silver complex, as well as for the infra-red spectrum of oxazole, and vibrational corrections to the polarizability of formaldehyde.

Giant natural circular dichroism of vibronic transitions in HoAl3(BO3)4

The giant natural optical activity of vibronic replicas of the f–f5I8 → 5F2 transition in HoAl3(BO3)4 is observed. It exceeds the optical activity related to purely electronic transitions by two orders of magnitude and corresponds to the almost complete circular polarization of the transitions. This effect is explained by the local distortions of the crystal in the excited electronic state, which lead to the mixing of electron and vibrational wavefunctions and, as a consequence, to the delocalization of the vibronic wavefunction and the enhancement of spatial dispersion, which is responsible for the optical activity.

Manipulation and visualization of two-dimensional phase distribution of vibrational wave functions in solid parahydrogen crystal

Solid parahydrogen, which is known to have an exceptionally long vibrational coherence lifetime as a molecular solid, offers an ideal testbed to perform coherent control experiments in the condensed phase. Here we demonstrate the spatial manipulation and visualization of the relative phase of vibrational wave functions in solid parahydrogen. Spatial distribution of vibrational excitation is generated by femtosecond impulsive Raman excitation. It is shown that the imprinted initial phase can be manipulated by wave-front modulation of the excitation laser pulses with a spatial light modulator. An interferometric measurement is used to convert the spatial phase distribution of the vibrational wave functions to the amplitude distribution. We have confirmed that the spatial profile of the scattered anti-Stokes pulse reveals the spatial phase distribution of the wave functions. The read-and-write scheme demonstrated in this experiment is applicable to a broad range of Raman memory systems accessible by $\mathrm{\ensuremath{\Lambda}}$-type transitions.

Anharmonic Calculation of Structural and Spectroscopic Parameters of the cis-DONO Molecule.

The equilibrium nuclear configuration of the cis-DONO molecule is calculated in the MP2/aug-cc-pVTZ approximation. The main objective of this work is to calculate the frequencies and absolute intensities for fundamental transitions of cis-DONO, to examine the influence of H/D substitution on the form of vibrational modes and structural and spectral parameters, and to formulate a feasible and reliable approach for calculating such parameters of H-bonded complexes formed by cis-DONO. Vibrational wave functions and energy values of cis-DONO are derived from variational solutions of anharmonic equations in one to four dimensions with the potential energy surfaces calculated by the MP2/aug-cc-pVTZ method. The equilibrium geometry and one-dimensional anharmonic frequencies and intensities of cis-DONO are also computed using the ccsd(t)/aug-cc-pVQZ approximation. All the calculated results are compared with the experimental data, the most accurate results of other authors, and the values obtained earlier for trans-DONO and cis- and trans-HONO using the MP2/aug-cc-pVTZ approximation.

Settling time of a vibrational wavepacket in ionization.

The vibrational wavepacket of a diatomic molecular ion at the time of ionization is usually considered to be generated on the basis of the Franck–Condon principle. According to this principle, the amplitude of each vibrational wavefunction in the wavepacket is given by the overlap integral between each vibrational wavefunction and the ground vibrational wavefunction in the neutral molecule, and hence, the amplitude should be a real number, or equivalently, a complex number the phase of which is equal to zero. Here we report the observation of a non-trivial phase modulation of the amplitudes of vibrational wavefunctions in a wavepacket generated in the ground electronic state of a molecular ion at the time of ionization. The phase modulation results in a group delay of the specific vibrational states of order 1 fs, which can be regarded as the settling time required to compose the initial vibrational wavepacket.

State to State Photodissociation Dynamics of Vibrationally Excited D2O in B Band

The state-to-state photodissociassion dynamics for the B band of D2O have been explored from quantum dynamical calculations including the electronic ̃X and ̃B states. The calculations were carried out using a Chebyshev real wave packet method. The calculated absorption spectra, product state distributions, and branching ratios from different initial vibrational states show different dynamic features, due to the different shapes of the vibrational wavefunctions. The initial bending mode (0,1,0) generates two lobes with a shallow minimum on the absorption spectrum and a slight inverted vibrational population of OD(̃X) product at high total energies. The rotational state distributions of OD(̃X, v=0) product are highly inverted and depend weakly on the initial state and total energy. On the other hand, the ro-vibrational distributions of OD(̃A) product strongly oscillate with the total energy, which are dominated by the long-living resonances and depend sensitively on the potential surfaces. The antisymmetric stretching mode (0,0,1) has large OD(̃A)/OD(̃X) branching ratios at high total energies, which indicates that the B band dissociation proceeds mainly via the adiabatic pathway in some cases.

Rotation/Torsion Coupling in H5(+), D5(+), H4D(+), and HD4(+) Using Diffusion Monte Carlo.

Two methods for studying the rotation/torsion coupling in H5(+) are described. The first involves a fixed-node treatment in which the nodal surfaces are obtained from a reduced dimensional calculation in which only the rotations of the outer H2 groups are considered. In the second, the torsion and rotation dependence of the wave function is described in state space, and the other internal coordinates are described in configuration space. Such treatments are necessary for molecules, like H5(+), where there is a very low-energy barrier to internal rotation. The results of the two approaches are found to be in good agreement with previously reported energies for J = 0. The diffusion Monte Carlo treatment allows us to extend the calculations to low J, and results are reported for the three lowest energy torsion excited states with J ≤ 3. For the level of rotational and vibrational excitation investigated, only modest changes in the vibrational wave functions are found. The effects of deuteration are also investigated, focusing on D5(+) and the symmetric variants of H4D(+) and HD4(+).

Full-Dimensional Quantum Dynamics of Vibrational Mediated Photodissociation of HOD in Its B Band.

The photodissociation dynamics for the ground and three fundamental vibrational states of HOD were explored from quantum dynamical calculations including the electronic X̃, Ã, and B̃ states. The calculations were based on a Chebyshev real wave packet method. Due to the different shapes of the initial vibrational wave functions and isotopic effect, the calculated absorption spectra, product state distributions, and branching ratios show different dynamic features. The initial bending excited vibtaional state (0, 1, 0) generates a bimodal behavior on the absorption spectrum and an inverted vibrational population of OD(X̃) fragment at some total energies. The rotational state distributions from four vibrational states have two different behaviors. One has a single broad peak, whereas the other one has a bimodal structure. Large OD(Ã)/OD(X̃) ratios are found for photodissociation from four vibrational states at high total energies, which indicate that the H atom dissociates mainly via the adiabatic pathway. We also calculated the OD/OH isotopic branching ratios from four vibrational states and found that the OD + H production channel is dominant over the OH + D channel in the energy range considered. The calculated results are consistent with the available observed ones.

Stimulated Raman signals at conical intersections: Ab initio surface hopping simulation protocol with direct propagation of the nuclear wave function.

Femtosecond Stimulated Raman Spectroscopy (FSRS) signals that monitor the excited state conical intersections dynamics of acrolein are simulated. An effective time dependent Hamiltonian for two C-H vibrational marker bands is constructed on the fly using a local mode expansion combined with a semi-classical surface hopping simulation protocol. The signals are obtained by a direct forward and backward propagation of the vibrational wave function on a numerical grid. Earlier work is extended to fully incorporate the anharmonicities and intermode couplings.

Fourier Grid Hamiltonian Method for calculating the Einstein coefficients, Franck–Condon factors, r-Centroids, average internuclear separations and radiative lifetimes for N2 and CO molecules

The Einstein coefficients and radiative lifetimes for diatomic molecules such as N2 and CO have been computed using the Fourier Grid Hamiltonian (FGH) method and a comparison with other methods has been made. The Hamiltonian matrix is constructed using the Hulburt and Hirschfelder (HH) potential model and the Rydberg-Klein-Rees (RKR) potential model for diatomic molecules. The FGH method works well for any model of the potential including the tabulated forms of experimental potential energy data. The eigenvalues and vibrational wave-functions for the ground and the exited state of N2 and CO molecules have been computed by diagonalizing the Hamiltonian matrix. The vibrational wave-functions and eigenvalues for the ground and the excited states are used to calculate the radiative transition parameters such as Einstein coefficients (Av″v′), radiative lifetimes, electronic transition dipole moments (Re(x)), Franck–Condon factors (FCFs), r-Centroids and average internuclear separations for both the models. FGH method work well for both the models. There are good agreements between our computed values of radiative transition parameters for both the models and the values estimated by other methods.

Towards a global model of spin-orbit coupling in the halocarbenes.

We report a global analysis of spin-orbit coupling in the mono-halocarbenes, CH(D)X, where X = Cl, Br, and I. These are model systems for examining carbene singlet-triplet energy gaps and spin-orbit coupling. Over the past decade, rich data sets collected using single vibronic level emission spectroscopy and stimulated emission pumping spectroscopy have yielded much information on the ground vibrational level structure and clearly demonstrated the presence of perturbations involving the low-lying triplet state. To model these interactions globally, we compare two approaches. First, we employ a diabatic treatment of the spin-orbit coupling, where the coupling matrix elements are written in terms of a purely electronic spin-orbit matrix element which is independent of nuclear coordinates, and an integral representing the overlap of the singlet and triplet vibrational wavefunctions. In this way, the structures, harmonic frequencies, and normal mode displacements from ab initio calculations were used to calculate the vibrational overlaps of the singlet and triplet state levels, including the full effects of Duschinsky mixing. These calculations have allowed many new assignments to be made, particularly for CHI, and provided spin-orbit coupling parameters and values for the singlet-triplet gaps. In a second approach, we have computed and fit full geometry dependent spin-orbit coupling surfaces and used them to compute matrix elements without the product form approximation. Those matrix elements were used in similar fits varying the anharmonic constants and singlet-triplet gap to reproduce the experimental levels. The derived spin-orbit parameters for carbenes CHX (X = Cl, Br, and I) show an excellent linear correlation with the atomic spin-orbit constant of the corresponding halogen, indicating that the spin-orbit coupling in the carbenes is consistently around 14% of the atomic value.

Vibrational states of the triplet electronic state of [formula omitted]. The role of non-adiabatic Jahn–Teller coupling

Vibrational energies and wave functions of the triplet state of the H3+ ion have been determined. In the calculations, the ground and first excited triplet electronic states are included as well as the most important part of the non-Born–Oppenheimer coupling between them. The diabatization procedure proposed by Longuet-Higgins is then applied to transform the two adiabatic ab initio potential energy surfaces of the triplet-H3+ state into a 2×2 diabatic matrix. The diabatization takes into account the effect of the geometrical phase due to the conical intersection between the two adiabatic potential surfaces. The results are compared to the calculation involving only the lowest adiabatic potential energy surface of the triplet-H3+ ion and neglecting the geometrical phase. The energy difference between results with and without the non-adiabatic coupling and the geometrical phase is about one wave number for the lowest vibrational levels.

From photoelectron detachment spectra of BrHBr(-), BrDBr(-) and IHI(-), IDI(-) to vibrational bonding of BrMuBr and IMuI.

Photoelectron detachment XLX(-)(00(0)0) + hν → XLX(vib) + e(-) + KER (X = Br or I, L = H or D) at sufficiently low temperatures photoionizes linear dihalogen anions XLX(-) in the vibrational ground state (v1v2 (l)v3 = 00(0)0) and prepares the neutral radicals XLX(vib) in vibrational states (vib). At the same time, part of the photon energy (hν) is converted into kinetic energy release (KER) of the electron [R. B. Metz, S. E. Bradforth, and D. M. Neumark, Adv. Chem. Phys. 81, 1 (1992)]. The process may be described approximately in terms of a Franck-Condon type transfer of the vibrational wavefunction representing XLX(-)(00(0)0) from the domain close to the minimum of its potential energy surface (PES) to the domain close to the linear transition state of the PES of the neutral XLX. As a consequence, prominent peaks of the photoelectron detachment spectra (pds) correlate with the vibrational energies EXLX,vib of states XLX(vib) which are centered at linear transition state. The corresponding vibrational quantum numbers may be labeled vib = (v1v2 (l)v3) = (00(0)v3). Accordingly, the related most prominent peaks in the pds are labeled v3. We construct a model PES which mimics the "true" PES in the domain of transition state such that it supports vibrational states with energies EXLX,pds,00(0)v3 close to the peaks of the pds labeled v3 = 0, 2, and 4. Subsequently, the same model PES is also used to calculate approximate values of the energies EXMuX,00(0)0 of the isotopomers XMuX(00(0)0). For the heavy isotopomers XHX and XDX, it turns out that all energies EXLX,00(0)v3 are above the threshold for dissociation, which means that all heavy XLX(00(0)v3) with wavefunctions centered at the transition state are unstable resonances with finite lifetimes. Turning the table, bound states of the heavy XLX are van der Waals (vdW) bonded. In contrast, the energies EXMuX,00(0)0 of the light isotopomers XMuX(00(0)0) are below the threshold for dissociation, with wavefunctions centered at the transition state. This means that XMuX(00(0)0) are vibrationally bonded. This implies a fundamental change of the nature of chemical bonding, from vdW bonding of the heavy XHX, XDX to vibrational bonding of XMuX. For BrMuBr, the present results derived from experimental pds of BrHBr(-) and BrDBr(-) confirm the recent discovery of vibrational bonding based on quantum chemical ab initio calculations [D. G. Fleming, J. Manz, K. Sato, and T. Takayanagi, Angew. Chem., Int. Ed. 53, 13706 (2014)]. The extension from BrLBr to ILI means the discovery of a new example of vibrational bonding. These empirical results for the vibrational bonding of IMuI, derived from the photoelectron spectra of IHI(-) and IDI(-), are supported by ab initio simulations of the spectra and of the wavefunction representing vibrational bonding of IMuI.

Automatic determination of important mode–mode correlations in many-mode vibrational wave functions

We introduce new automatic procedures for parameterizing vibrational coupled cluster (VCC) and vibrational configuration interaction wave functions. Importance measures for individual mode combinations in the wave function are derived based on upper bounds to Hamiltonian matrix elements and/or the size of perturbative corrections derived in the framework of VCC. With a threshold, this enables an automatic, system-adapted way of choosing which mode-mode correlations are explicitly parameterized in the many-mode wave function. The effect of different importance measures and thresholds is investigated for zero-point energies and infrared spectra for formaldehyde and furan. Furthermore, the direct link between important mode-mode correlations and coordinates is illustrated employing water clusters as examples: Using optimized coordinates, a larger number of mode combinations can be neglected in the correlated many-mode vibrational wave function than with normal coordinates for the same accuracy. Moreover, the fraction of important mode-mode correlations compared to the total number of correlations decreases with system size. This underlines the potential gain in efficiency when using optimized coordinates in combination with a flexible scheme for choosing the mode-mode correlations included in the parameterization of the correlated many-mode vibrational wave function. All in all, it is found that the introduced schemes for parameterizing correlated many-mode vibrational wave functions lead to at least as systematic and accurate calculations as those using more standard and straightforward excitation level definitions. This new way of defining approximate calculations offers potential for future calculations on larger systems.

Fourier analysis of the nuclear flux density in diatomic molecules: A complementary tool to map potential-energy curves and to characterize vibrational wave functions

Pump-probe spectroscopy has allowed the construction of the nuclear probability density $\ensuremath{\rho}(R,t)$ as a function of the internuclear bond distance $(R)$ and the time $(t)$ in diatomic molecules and consequent deduction of the nuclear flux density $j(R,t)$. Thus, the two observables $[\ensuremath{\rho}(R,t),j(R,t)]$ comprise a very detailed description of the nuclear motion in ultrafast molecular dynamics. Here a Fourier analysis of $j(R,t)$ is proposed and compared with the already existing Fourier analysis of $\ensuremath{\rho}(R,t)$. It is shown that the two power spectra $|\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\rho}}{(R,\ensuremath{\omega};T)|}^{2}$ and $|\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{j}{(R,\ensuremath{\omega};T)|}^{2}$ provide the same information in the frequency domain $\ensuremath{\omega}$, but entirely different information in the spatial domain (i.e., along the $R$ coordinate).