Local normal modes and vibrational adiabatic potentials

Local normal-mode analysis for a collinear potential energy surface generates a system of curvilinear coordinates, which are orthogonal in the mass-skewed system. The motion is locally separable in these coordinates. We compare the utility of one of the normal modes as a transversal-vibration coordinate, with the conventional choice of the direction perpendicular to the reaction coordinate in the mass-skewed system. The comparison is done for two commonly used reaction coordinates: BEBO and the steepest-descent path. Results differ for different choices of directions and reaction coordinates. Future work should concentrate on a choice of a reaction coordinate which is itself one of the normal modes.     

Efficient computation of adiabatic populations in multi-mode Jahn-Teller systems through the use of effective vibrational modes

A highly efficient scheme for computing adiabatic electronic populations in multi-mode Jahn-Teller systems is presented. It relies on the transformation to an effective-mode vibrational basis in which the relevant quantities depend on the coordinates of a single mode only. In this way, the generally tedious numerical evaluation of high-dimensional integrals is avoided and replaced by one-dimensional integrations. The effective-mode scheme is applied to a variety of two-mode and three-mode Jahn-Teller systems and gives a typical speedup of about two to three orders of magnitude as compared to the direct evaluation of the adiabatic populations. The gain grows rapidly with the numbers of modes.     

Characterization of vibrational transition modes by use of normal forms

Summary In this paper we use the Birkhoff-Gustavson perturbation theory to analyze the vibrational modes of two linearly coupled Morse oscillators in the transition region from normal modes to local modes. Our study is based on: truncation of the Hamiltonian written in normal mode coordinates at the 4th order, transformation to normal form and analytical study; construction and use of the approximate integrals of motion of the exact Hamiltonian according to Birkhoff and Gustavson theory. By comparison with a previous analytical study, we demonstrate that perturbation theory, based either on local or normal modes can be used to accurately describe transition modes.     

Diabatic ordering of vibrational normal modes in reaction valley studies

Diabatic ordering of the normal model of a reaction complex along the reaction path has several advantages with regard to adiabatic ordering. The method is based on rotations of the vibrational normal modes at one point, s, of the reaction path to maximize overlap with the vibrational modes at a neighboring point. Global rotations precede the rotations of degenerate modes so that changes in the direction of the reaction path and changes in the force constant matrix, which represent the two major effects for changes in mode ordering, can be separated. Overlap criteria identify resolved and unresolved avoided crossings of normal modes of the same symmetry. Diabatic mode ordering (DMO) can be used to resolve the latter by reducing the step size, thus guaranteeing correct ordering of normal modes in dependence of s. DMO is generally applicable to properties of the reaction complex that depend on s such as normal mode frequencies, orbital energies, the energy of excited states, etc. Additional applications are possible using a generalized reaction path vector, which may describe the change in atom masses, geometrical parameters, and/or the force constant matrix. In this way, the vibrational spectra of isotopomers can be investigated or the vibrational frequencies of different molecules correlated. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1282–1294, 1997     

A new way of analyzing vibrational spectra. I. Derivation of adiabatic internal modes

A new way of analyzing measured or calculated vibrational spectra in terms of internal vibrational modes associated with the internal parameters used to describe geometry and conformation of a molecule is described. The internal modes are determined by solving the Euler–Lagrange equations for molecular fragments ϕ_n described by internal parameters ζ_n. An internal mode is localized in a molecular fragment by describing the rest of the molecule as a collection of massless points that just define molecular geometry. Alternatively, one can consider the new fragment motions as motions that are obtained after relaxing all parts of the vibrating molecule but the fragment under consideration. Because of this property, the internal modes are called adiabatic internal modes, and the associated force constants k_a, adiabatic force constants. Minimization of the kinetic energy of the vibrating fragment ϕ_n yields the adiabatic mass m_a (corresponding to 1/G_nn of Wilson's G matrix) and, by this, adiabatic frequencies ω_a. Adiabatic modes are perfectly suited to analyze and understand the vibrational spectra of a molecule in terms of internal parameter modes in the same way as one understands molecular geometry in terms of internal coordinates. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67 : 1–9, 1998     

First-principles-based calculations of vibrational normal modes in polyatomic materials with translational symmetry: application to PETN molecular crystal

Numerical studies of vibrational energy transport and associated (non)linear infrared and Raman response in polyatomic materials require knowledge of the multidimensional vibrational potential-energy surface and the ability to perform normal-mode analysis on that potential. The presence of translational symmetry, as in crystals, leads to the observed dispersion of the unit cell normal modes and has to be accounted for in calculations of energy transfer rates and other spectroscopic quantities. Here we report on the implementation of a computational approach that combines the generalized supercell method and density functional theory electronic structure calculations to investigate the vibrational structure in translationally symmetric materials containing relatively large numbers of atoms in the unit cell (58 atoms in the present study). The method is applied to calculate the phonon and vibron dispersion relations and the vibrational density of states in pentaerythritol tetranitrate (PETN) molecular crystal which is an important energetic material. The results set the stage for future investigations of vibrational energy transport and associated nonlinear spectroscopic signatures in this class of materials.     

Analysis of the transition from normal modes to local modes in a system of two harmonically coupled Morse oscillators

Summary The system consisting of two Morse oscillators coupled via either a potential or a kinetic quadratic term is considered. The corresponding classical equations of motion have been numerically integrated and the initial conditions have been systematically analyzed in the regime of low total excitation energy of the system. Particular attention was paid to the full characterization of an intermediate type of motion, herein called transition mode, which appears at total energy values in between those typical of normal modes and those where local and normal modes coexist. A previously proposed perturbative approach (Jaffé C, Brumer P (1980) J Chem Phys 73:5646) is reanalyzed and compared with the results of numerical experiments, with the purpose of lending further support to the existence of transition modes.     

Fragment mode analysis and its application to the vibrational normal modes of boron trichloride-ammonia and boron trichloride-pyridine complexes

A method for expressing quantitatively the vibrational normal modes of a molecule in a basis set consisting of the normal vibrations (plus translations and rotations) of its constituent fragments is presented. The method is illustrated by describing the vibrational modes of BCl3-NH3 and BCl3-pyridine electron donor-acceptor complexes in terms of motions of BCl3 and either NH3 or pyridine. These complexes show examples of mixing between modes located on different fragments, mixing between modes of one fragment due to symmetry lowering, and the transformation of six fragment translations/rotations into vibrations of the complex. Although perturbation theory has been proposed to explain such examples of mode mixing, calculations imply that interactions between fragments of both complexes are too strong for perturbation theory to be generally applicable. In addition, the transformation of fragment rotations and/or translations into vibrations of the composite molecule will always occur and cannot be understood in detail by using perturbation theory. For the BCl3-pyridine complex, a band observed at 1107 cm(-1) is re-assigned as a combination of C-H in-plane bending and a ring-breathing mode of the pyridine fragment.     

A study of the adiabatic connection for two-electron systems

Some aspects of the adiabatic connection method are studied for two-particle spherically symmetric systems. Ground-state wave functions that are constrained by means of a set of moments to have the same density as a corresponding fully interacting system are obtained for noninteracting or partially interacting systems. Local one-body potentials that support these constrained wave functions are generated using a simple method. We examine an interacting two-particle system with a parameter-dependent one-body potential, which for a particular value of that parameter exhibits an intersection between the (3)S and the (3)P states, whereas the 2s and 2p eigenvalues of the corresponding Kohn-Sham potentials do not intersect along with the total energies. These results show that there do exist cases where occupying the orbitals from below in energy may not lead to the ground state, and that the inherent assumptions behind the adiabatic connection can sometimes be violated.     

Contributions of the 8-methyl group to the vibrational normal modes of flavin mononucleotide and its 5-methyl semiquinone radical

Resonance Raman spectroscopy is a powerful tool to investigate flavins and flavoproteins, and a good understanding of the flavin vibrational normal modes is essential for the interpretation of the Raman spectra. Isotopic labeling is the most effective tool for the assignment of vibrational normal modes, but such studies have been limited to labeling of rings II and III of the flavin isoalloxazine ring. In this paper, we report the resonance and pre-resonance Raman spectra of flavin mononucleotide (FMN) and its N5-methyl neutral radical semiquinone (5-CH 3FMN(*)), of which the 8-methyl group of ring I has been deuterated. The experiments indicate that the Raman bands in the low-frequency region are the most sensitive to 8-methyl deuteration. Density functional theory (DFT) calculations have been performed on lumiflavin to predict the isotope shifts, which are used to assign the calculated normal modes to the Raman bands of FMN. A first assignment of the low-frequency Raman bands on the basis of isotope shifts is proposed. Partial deuteration of the 8-methyl group reveals that the changes in the Raman spectra do not always occur gradually. These observations are reproduced by the DFT calculations, which provide detailed insight into the underlying modifications of the normal modes that are responsible for the changes in the Raman spectra. Two types of isotopic shift patterns are observed: either the frequency of the normal mode but not its composition changes or the composition of the normal mode changes, which then appears at a new frequency. The DFT calculations also reveal that the effect of H/D-exchange in the 8-methyl group on the composition of the vibrational normal modes is affected by the position of the exchanged hydrogen, i.e., whether it is in or out of the isoalloxazine plane.     

Contribution to the vibrational normal modes analysis of d8-THF by neutron inelastic scattering

The Neutron Inelastic Scattering (NIS) spectrum of deuterated tetrahydrofuran d₈-THF has been recorded between 20 and 2000 cm⁻¹. The lowest frequency internal vibration is located at 120cm⁻¹, in good agreement with the solid state Raman spectrum. The NIS spectrum has been simulated using the results of a Quantum Mechanical calculation (E. Gallinella, B. Cadioli, J.P. Flament and G. Berthier, J. Mol. Struct (Theochem), in press) which also agrees with the experimental frequencies using a scale factor of 0.95.     

A normal coordinate analysis of the vibrational modes of the three redox forms of methylviologen: Comparison with experimental results

A normal coordinate analysis of the three redox forms of methylviologen (MV) was accomplished using a simplified valence force field. The calculated band frequencies are compared with those observed experimentally. The plausibility of the assignments for the Raman A_g modes, as inferred from the normal coordinate analysis, is discussed. The assignments of several Raman A_g modes were checked by a comparison of the observed Raman spectra of viologens with different substituents, such as methyl group deuterated methyl viologen (DV), N-octyl-N′-methyl viologen (C₈MV), N-hexadecyl-N′-methyl viologen (C₁₆MV) and dibenzyl viologen (BV), with MV. The values of the root mean square (r.m.s.) amplitudes of vibrations between the bonded and the nonbonded atom pairs, and r.m.s. Cartesian displacements from the equilibrium geometry for MV²⁺ were computed. These data provide important insight with respect to the molecular geometry.     

In-plane vibrational modes of cytosine from an ab initio MO calculation

Harmonic force constants, in-plane vibrational frequencies, and in-plane vibrational modes of cytosine were calculated by an ab initio Hartree—Fock SCF MO method. The force contants were calculated by the use of an energy gardient method with the STO-3G basis set, and then they were corrected into “4-31G force constants” by the scaling factors given by us previously for the case of uracil. The corrected set of force constants can produce a calculated vibrational spectra of cytosine and cytosine-1,amino-d₃, that can be well corrected with the observed Raman and infrared spectra of these compounds, with little ambiguity. Thus, the assignments of all the in-plane vibrations are now practically established. The calculated vibrational modes, in addition, can account for the recently published resonance Raman effects of cytosine residue.     

Correcting model energies by numerically integrating along an adiabatic connection and a link to density functional approximations

Model Hamiltonians are considered for which electrons interact via long-range forces. It is assumed that their eigenvalues can be obtained with satisfying accuracy. Extrapolation techniques using asymptotic behavior considerations provide estimates for the energy of the physical system. Results for the uniform electron gas and some two-electron systems show that very few quadrature points can already produce good quality results. Connections to the density functional theory are discussed.     

Vibrational projection analysis: New tool for quantitatively comparing vibrational normal modes of similar molecules

A new method for quantitatively comparing calculated vibrational modes is described that relies on projecting the vectors describing the normal modes of one molecule onto those of a basis molecule. The procedure virtually automates the assignment of vibrational modes from one molecule to a second, structurally similar one. We illustrate the concept by using the classical Wilson modes of benzene as a basis for describing normal modes of the monosubstituted benzene derivatives phenol, phenol-d5, and phenol radical cation. These examples demonstrate the method's power for accurately assigning and comparing the normal modes of molecules perturbed by chemical substitution, isotopic substitution, or oxidation state. The vibrational projection analysis method—a special case of vector projection analysis—is compared and contrasted with total energy distribution analysis, perhaps the most commonly used technique for quantitatively comparing vibrational modes, and is found superior in each case when comparing normal modes. Vibrational projection analysis need not be limited to single molecules and quantum calculations, because normal modes may be obtained for much larger systems using molecular mechanics or molecular dynamics techniques. The method should therefore prove useful for interpreting the normal modes of ordered periodic solids and structures perturbed by noncovalent contacts, including proteins and polymers. The method may also prove useful in evaluating new computational methods by allowing direct, quantitative comparison of the vibrational modes obtained from different techniques. The power of this technique will make vibrational projection analysis a valuable tool for computational chemists as well as those using calculations to complement vibrational spectroscopic measurements. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1663–1674, 1998     

Application of adiabatic theory of vibrational predissociation to T-shaped triatomic van der Waals' molecules

An expression for the rate constant of the vibrational predissociation (VPD) of T-shaped triatomic van der Waals' (vdW) molecules is derived on the basis of the adiabatic separation between the high-frequency intramolecular and low-frequency vdW modes. The intramolecular and vdW modes are assumed to be characterized by the Morse-type interaction potentials. The dependence of the VPD line width on the intramolecular vibrational quantum number of a T-Shaped I₂-He vdW molecule is calculated by using the expression derived. The magnitudes of the calculated VPD line width are of the same order as those of the experimental. It is shown that the Condon approximation is insufficient and the non-Condon treatment is necessary to evaluate quantitatively the VPD rate constant within the adiabatic theory.     

A joint local mode and normal mode interpretation of vibrational anharmonicity in methyl bromide

The gas-phase i.r. spectrum of CH₃Br has been studied up to 14 000 cm⁻¹. Some 32 new vibration levels are located accurately, involving up to 5 quanta (V= 5) of excitation in CH stretching. Reproduction of a total of 72 vibration levels is achieved through a joint treatment of CH stretching vibrations in a local mode basis, other vibrations in a normal mode basis, and with the inclusion of two Fermi resonances. The local mode approach satisfactorily accounts for the effects of the large Darling—Dennison vibrational interactions which occur between CH stretching modes. Fermi resonances between CH stretching and overtones of both methyl group deformation vibrations (ν₂ and ν₅) are treated explicitly. Interacting levels become quasi-degenerate at V = 3 in the case of 2ν₅, and at V = 5 in the case of 2ν₂. The analysis permits the determination of a complete set of 27 physically representative anharmonicity constants for CH₃Br, four of which are interrelated through the local mode model. Data reproduction between 600 and 14 000 cm⁻¹ is achieved with an r.m.s. error of 2.55 cm⁻¹.     

Study of adiabatic connection in density functional theory with an accurate wavefunction for two‐electron spherical systems

An accurate semianalytic wavefunction is proposed for the Hookium and two‐electron atoms for varying strength of where is the strength parameter and is coulomb interaction between two electrons. The wavefunction leads to energies that are as accurate as those from the Coupled cluster singles and doubles (CCSD) calculations. Using this wavefunction, we construct the external potential such that the density of the system remains unchanged as is varied. The work thus gives a unified picture of adiabatic connection for these systems based on an easy to use wavefunction and complements the past investigations done in this direction. Using the potential obtained, we explicitly calculate the energy of the corresponding positive ions and show that the chemical potential—calculated as the difference between the energies of the two‐electron system and its positive ion—is equal to the experimental ionization energy and remains unchanged as is varied. Furthermore, using total energies of these systems as a function of , we provide a new perspective into a variety of hybrid functionals.     

Assessment and formal properties of exchange-correlation functionals constructed from the adiabatic connection

We examine the development and investigate the performance of exchange-correlation functionals constructed from the adiabatic connection. Our method is based on a direct modeling of the adiabatic connection curve in the coupling-constant space and is very flexible in the models. Several different models are investigated in the construction of new families of exchange-correlation functionals. Also the performance of two of these models (MCY1 and MCY2) is investigated over a wider range of systems and properties, with comparison made to the performance of established functionals. Overall, the adiabatic functionals improve upon widely used hybrid and generalized gradient approximation functionals, particularly in correctly describing one-electron systems and reaction energy barriers.