Anharmonic States Research Articles

First-Principles Anharmonic Infrared and Raman Vibrational Spectra of Materials: Fermi Resonance in Dry Ice.

We introduce a computational tool for the quantum-mechanical simulation of anharmonic infrared and Raman vibrational spectra of materials. The approach, implemented in the CRYSTAL software, stems from Taylor's expansion of the potential energy surface (PES) on the basis of normal modes up to cubic and quartic terms. The PES can be sampled with four different numerical schemes at the level of density functional theory (DFT), with local, generalized-gradient, and hybrid density functional approximations. Anharmonic states are obtained by solving Shrödinger's nuclear equation with either the vibrational self-consistent field (VSCF) or vibrational configuration interaction (VCI) methods. Nuclear quantum effects (NQEs) are thus fully accounted for. Infrared intensities are computed numerically through a Berry phase approach or analytically through a coupled-perturbed (CP) approach. Raman intensities are computed analytically via the CP approach. A variety of anharmonic features of vibrational spectra of materials can be simulated, including band shifts, combination bands, overtones, resonances (first-order Fermi, second-order Darling-Dennison), and hot bands. We showcase the effectiveness of the approach on the description of a first-order Fermi resonance (FR) in CO2 dry ice: a challenging test-case given that the FR occurs in the Raman spectrum, requires NQEs, and involves two- and three-mode couplings. Fundamental mechanistic differences with respect to the well-known FR in molecular CO2 are addressed. This application represents the first quantum-mechanical, periodic description of FR in dry ice.

Accelerating and stabilizing the convergence of vibrational self-consistent field calculations via the direct inversion of the iterative subspace (vDIIS) algorithm.

The vibrational self-consistent field (VSCF) method yields anharmonic states and spectra for molecular vibrations, and it serves as the starting point for more sophisticated correlated-vibration methods. Convergence of the iterative, non-linear optimization in VSCF calculations can be erratic or altogether unsuccessful, particularly for chemical systems involving low-frequency motions. In this work, a vibrational formulation of the Direct Inversion of the Iterative Subspace method of Pulay is presented and investigated. This formulation accounts for distinct attributes of the vibrational and electronic cases, including the expansion of each single-mode vibrational wavefunction in its own basis set. The resulting Direct Inversion of the Iterative Subspace method is shown to substantially accelerate VSCF convergence in all convergent cases as well as rectify many cases where Roothaan-based methods fail. Performance across systems ranging from small, rigid molecules to weakly bound molecular clusters is investigated in this analysis.

Anharmonic state counts and partition functions for molecules via classical phase space integrals in curvilinear coordinates

An algorithm is presented for calculating fully anharmonic vibrational state counts, state densities, and partition functions for molecules using Monte Carlo integration of classical phase space. The algorithm includes numerical evaluations of the elements of the Jacobian and is general enough to allow for sampling in arbitrary curvilinear or rectilinear coordinate systems. Invariance to the choice of coordinate system is demonstrated for vibrational state densities of methane, where we find comparable sampling efficiency when using curvilinear z-matrix and rectilinear Cartesian normal mode coordinates. In agreement with past work, we find that anharmonicity increases the vibrational state density of methane by a factor of ∼2 at its dissociation threshold. For the vinyl radical, we find a significant (∼10×) improvement in sampling efficiency when using curvilinear z-matrix coordinates relative to Cartesian normal mode coordinates. We attribute this improved efficiency, in part, to a more natural curvilinear coordinate description of the double well associated with the H2C-C-H wagging motion. The anharmonicity correction for the vinyl radical state density is ∼1.4 at its dissociation threshold. Finally, we demonstrate that with trivial parallelizations of the Monte Carlo step, tractable calculations can be made for the vinyl radical using direct ab initio potential energy surface evaluations and a composite QCISD(T)/MP2 method.

Anharmonic Electron−Phonon Coupling in Condensed Media: 1. Formalism

Three different schemes for calculating anharmonic line shape functions are reported and discussed for the first time in this article using eigenstate representation. First, the linear dipole-moment time correlation function (DMTCF), homogeneous (single-site) absorption line shape function, and the respective Franck-Condon factors (FCF) are derived and explored as a molecule makes a transition from a harmonic to an anharmonic (Morse potential) electronic state. Second, the linear DMTCF, homogeneous absorption line shape function, and FCFs are also derived as a molecule makes a transition from one anharmonic to another linearly displaced anharmonic state; FCFs in this case are reported in an exact closed-form expressed in terms of Appell's hypergeometric function. Third, same as the latter set of results are reported but with both linearly displaced and distorted shape of the upper Morse potential. FCFs of the zero-phonon line in all three cases are reported. The first case is rather mathematically complex as a result of taking the overlap integral of the Morse oscillator eigenfunctions, whose spatial decay is a simple exponential, with those of harmonic oscillator, whose decay is a Gaussian. This form of a functional disparity gives rise to some challenges. Model calculations are presented and discussed.

An effective Hamiltonian survey of the anharmonic vibrational state space of SCCl2 up to the dissociation energy

We survey about 3.5 million feature states in the anharmonic state space of SCCl(2) to answer three questions: how sharp is the threshold for intramolecular vibrational energy redistribution (IVR), how do specific resonances and regions of state space contribute to the threshold, and how many undiluted spectral features (or localized eigenstates) persist as energy increases? As quantitative measures we use the dilution factor sigma, Heller's F, and e, which describes how close to the edge of state space a feature lies. We find that an IVR threshold can be identified and lies between 250 and 300 THz. The threshold is softened because different resonances act at different energies in different parts of state space. According to our calculation, about 1 in 10(3) feature states remains undiluted near the dissociation limit of SCCl(2). This fraction matches the number of sharp spectral features observed recently by experiment when symmetry and Franck-Condon factors are taken into account, and it is in agreement with an analytical model that predicts an increasing fraction of undiluted features as molecular size increases.

The intramolecular tunnelling of hydrogen†

This paper reports an approach to the analysis of the local anharmonic states of motion and tunnelling of hydrogen illustrated with the molecule malonaldehyde. A potential energy function is used accurately to fit the local potential energy surface. The basis fitting functions in the potential resemble the Morse basis function, , but are intrinsically Gaussian: . After rearrangement to isolate the Gaussian operators, the new form of the potential allows the direct use of mathematically friendly (Gaussian) manipulations that are needed to carry out the quantum mechanical analysis of the anharmonic features, namely tunnelling. The quantal analysis uses Cartesian harmonic oscillator basis functions together with the Talmi–Moshinsky–Smirnov methods to evaluate matrix elements of the potential energy operator. The longer term objective of this kind of work is to develop practical methods of hybrid analysis with which to treat the local atom dynamics associated with a reactive species while the rest of the dynamics are modelled as simple harmonic oscillators. The goal is eventually to have a computational theory of proton/hydrogen or small molecular group transfer reactions in condensed-phase systems. This paper addresses only the detailed local dynamical analysis. †Dedicated to Pavel Rosmus on the occasion of the celebration of his scienti.c achievements, in Prague.

Molecular vibrational energy flow and dilution factors in an anharmonic state space

A fourth-order resonance Hamiltonian is derived from the experimental normal-mode Hamiltonian of SCCl2. The anharmonic vibrational state space constructed from the effective Hamiltonian provides a realistic model for vibrational energy flow from bright states accessible by pulsed laser excitation. We study the experimentally derived distribution PE(sigma) of dilution factors sigma as a function of energy. This distribution characterizes the dynamics in the long-time limit. State space models predict that PE(sigma) should be bimodal, with some states undergoing facile intramolecular vibrational energy redistribution (small sigma), while others at the same total energy remain "protected" (sigma approximately 1). The bimodal distribution is in qualitative agreement with analytical and numerical local density of states models. However, there are fewer states protected from energy flow, and the protected states begin to fragment at higher energy, shifting from sigma approximately 1 to sigma approximately 0.5. We also examine how dilution factors are distributed in the vibrational state space of SCCl2 and how the power law specifying the survival probability of harmonic initial states correlates with the dilution factor distribution of anharmonic initial states.

Semiclassical initial value treatment of correlation functions.

Two semiclassical, initial value representation (IVR) treatments are presented for the correlation function psi(f) e-iHt/h psi(i), where psi(i) and psi(f), are energy eigenfunctions of a "zero-order" Hamiltonian describing an arbitrary, integrable, vibrational system. These wave functions are treated semiclassically so that quantum calculations and numerical integrations over these states are unnecessary. While one of the new approximations describes the correlation function as an integral over all phase space variables of the system, in a manner similar to most existing IVR treatments, the second approximation describes the correlation function as an integral over only half of the phase space variables (i.e., the angle variables for the initial system). The relationship of these treatments to the conventional Herman-Kluk approximation for correlation functions is discussed. The accuracy and convergence of these treatments are tested by calculations of absorption spectra for model systems having up to 18 degrees of freedom, using Monte Carlo techniques to perform the multidimensional phase space integrations. Both treatments are found to be capable of producing spectra of excited, anharmonic states that agree well with quantum results. Although generally less accurate than full phase space or Herman-Kluk treatments, the half phase space method is found to require far fewer trajectories to achieve convergence. In addition, this number is observed to increase much more slowly with the system size than it does for the former methods, making the half-phase space technique a very promising method for the treatment of large systems.

Application of the anharmonic coherent states to the vibronic interaction

AbstractWe calculated the vibronic reduction factor (Ham factor) for the physical system (molecules, crystals) with octahedral symmetry. The vibrations of the nuclei of the systems are described by the anharmonic states of the Morse potential and also by the anharmonic coherent states of the same potential. The linear vibronic coupling of these vibration states with the electronic states of the system are considered.

Saddle-node states in the spectra of HCO and DCO: a periodic orbit classification of vibrational levels

The general structures of the vibrational states of HCO and its isotopomer DCO are analyzed in terms of periodic orbits (POs) and continuation/bifurcation diagrams. Both bound and resonance states are considered. It is shown that the members of the pure overtones are guided by POs, even in the continuum. In particular, it is demonstrated that the highly anharmonic states localized along the dissociation coordinate correspond to orbits originating from saddle-node bifurcations in the classical phase space.

Anharmonic states of the one-dimensional anisotropic Heisenberg model with free boundary conditions

States described by anharmonic functions are shown to exist in the one-dimensional anisotropic Heisenberg model with a finite number of spins 1/2 and free boundary conditions. Several such states are determined.

Conformational Dependence of Intramolecular Vibrational Energy Redistribution in the EthylenicCH2 Stretch Fundamental of Allyl Fluoride

The high-resolution (6 MHz) infrared spectra of the asymmetric ethylenic hydride stretches of both cis and gauche allyl fluoride have been measured. Rotational assignments of the eigenstates are made using the ground-state microwave-infrared double-resonance capabilities of an electric resonance optothermal spectrometer (EROS). The cis vibrational band near 3114 cm-1 is characterized by sparse, narrow IVR multiplets resulting from weak Coriolis or cross conformer interactions between the vibrational bath states. The IVR lifetime of the cis vibrational band is approximately 2 ns (2000 ps). The gauche vibrational band near 3100 cm-1 is qualitatively and quantitatively quite different from the cis band. The gauche band is characterized by significantly fragmented IVR multiplets with an average IVR lifetime of about 90 ps. The measured anharmonic state density of the gauche band is about 30 states/cm-1. The disagreement between the measured state densities of the gauche vibrational band and the calculated val...

Intramolecular vibrational energy redistribution in the acetylenic C–H and hydroxyl stretches of propynol

The high-resolution infrared spectra of the acetylenic C–H and O–H stretches of propynol have been measured using an electric-resonance optothermal molecular beam spectrometer (EROS). Both spectra display extensive fragmentation of the hydride-stretch oscillator strength characteristic of the intramolecular vibrational energy redistribution (IVR) process. The IVR lifetime is strongly mode-specific. The IVR lifetime of the acetylenic C–H stretch is approximately 400 ps, with a slight increase in the lifetime with increasing values of the Ka quantum number. The lifetime of the O–H stretch is 60 ps and is independent of the rotational quantum numbers. The experimental upper limit for the anharmonic state densities are 30 and 40 states/cm−1 for the acetylenic C–H and O–H stretches, respectively. These values are in good agreement with the values obtained by a direct state count (19 and 32 states/cm−1, respectively). The measured density of states increases with an approximate (2J+1)-dependence. These results indicate that all energetically accessible states are involved in the IVR dynamics. However, neither the acetylenic C–H nor the O–H stretch shows a decrease in lifetime as the total angular momentum (J) increases. This result shows that Coriolis coupling of these two hydride stretches to the near-resonant bath states is much weaker than the anharmonic coupling. For the O–H stretch, we are able to obtain the root-mean-squared (rms) matrix element for the Coriolis coupling prefactor, 0.0015(5) cm−1. The rms anharmonic coupling matrix element is 0.03 cm−1. For the low J values measured in the O–H spectrum, the Coriolis-induced IVR rate is much slower than the initial redistribution rate resulting from the stronger anharmonic interactions leading to an IVR process with two distinct time scales.

An empirical rule relating fundamental to harmonic frequencies

An empirical approach to estimate harmonic frequencies from experimental fundamental vibrational frequencies is presented. It is based on a partitioning of the eigenvectors of each normal mode, s = 1, 2, 3,…, 3 N − 6, into internal motion components involving bond stretching, B s , angle bending, A s , torsional motion, T s , and out-of-plane motion, O s . This is called the BATO method. An empirical function, f s = (1250B s + 2000A s + 1300T s + 1600O s) A ̊ , is developed to assign relative anharmonic contributions to each kind of internal motion by relating harmonic frequencies, ω s , to the anharmonic or fundamental frequencies, ν s , with ν s = ω s + 2 χ s = ω s (1 − f s ω s ). The anharmonic states are assumed to satisfy a Morse expression, ν sn = ω s(n s + 1 2 ) + χ s(n s + 1 2 ) 2 , where n s = 0, 1, 2, ⋯ denote the vibrational states of mode s. The coefficients of f s are calibrated with the available experimental data for ν s exp and ω s exp with isotopes of 19 molecules which contain bonds and groups required for biochemical applications. The objective is to use the BATO method to estimate empirical harmonic frequencies, ω s emp, from experimental fundamental frequencies, ν s exp, in a self-consistent procedure in cases where ω s emp are not known. Of the 394 states in 19 molecules studied, the method yields results within 10 cm −1 for 232 states. 11–20 cm −1 for 86 states, 21–30 cm −1 for 45 states, 31–40 cm −1 for 19 states, 41–60 cm −1 for 9 states, and 61–81 cm −1 for 3 states tested. Detailed results are presented for the molecule containing the most common isotope. The maximum errors occur in C 2H 6 and C 3H 8, for which the method tends to overestimate the CH stretch in several cases where the comparison is against Dennison's rule. All molecules studied satisfy the product and summation rules well.

Harmonic and anharmonic states in glasses

(1987). Harmonic and anharmonic states in glasses. Philosophical Magazine B: Vol. 56, No. 2, pp. 213-214.

C–H vibrational states of benzene, naphthalene, and anthracene in the visible region by thermal lensing spectroscopy and the local mode model

Visible (red) absorption spectra of benzene (h6 and d6), methyl substituted benzenes, naphthalene, and anthracene are presented as seen by thermal lensing spectroscopy. It is shown how this highly sensitive laser-based technique can easily detect transitions having absorption coefficients as small as 10−6 cm−1. The observed transitions are principally due to C–H stretching vibrations at the sixth quantum of excitation. The local mode (LM) model for describing such molecular states is explored in some detail with reference to the C–H excitations in benzene. Based on a two parameter empirical fit, the LM model can give a true C–H stretch ’’mechanical’’ frequency; it successfully predicts isotope effects; and it fully determines the energy level scheme at any quantum of excitation. The LM energy level scheme for the 462 C–H stretching states at the sixth quantum of excitation is presented. A majority of these states are only mildly anharmonic; one state in fact is a perfect harmonic of the fundamental. However, only the most anharmonic state of the 462 is the spectroscopically allowed one. The LM model, already introduced in the literature, is seen to be an excellent zeroth order description of those very same highly excited vibrational states most readily accessible to a sensitive absorption technique such as that offered by the thermal lensing effect.