Independent-mode Approximation Research Articles

Anharmonic Coupling of Stretching Vibrations in Ice: A Periodic VSCF and VCI Description.

The anharmonicity of O-H stretching vibrations of water ice is characterized by use of a periodic implementation of the vibrational self-consistent field (VSCF) and vibrational configuration interaction (VCI) methods, which take phonon-phonon couplings explicitly into account through numerical evaluation of high-order terms of the nuclear potential. The low-temperature, proton-ordered phase of water ice (namely, ice XI) is investigated. The net effect of a coupled anharmonic treatment of stretching modes is not just a rigid blue-shift of the respective harmonic spectral frequencies but rather a complex change of their relative spectral positions, which cannot be captured by simple scaling strategies based on harmonic calculations. The adopted techniques allow for a hierarchical treatment of anharmonic terms of the nuclear potential, which is key to an effective identification of leading factors. We show that the anharmonic independent-mode approximation─only describing the "intrinsic anharmonicity" of the O-H stretches─is unable to capture the correct physics, and that couplings among O-H stretches must be described. Inspection of harmonic normal coordinates allows identification of specific features of the O-H stretching motions which most likely enable strong mode-mode couplings. Finally, by coupling O-H stretches to all other possible modes of ice XI (THz collective vibrations, molecular librations, bendings), we identify specific types of motion which significantly affect O-H stretching states: in particular, molecular librations are found to affect the stretching states more than molecular bendings.

First-principles study of large-amplitude dynamic Jahn–Teller effects in vanadium tetrafluoride

Transition metal tetrahalides are a class of highly symmetric molecules for which very few spectroscopic data exist. Exploratory ab initio calculations of electronic potential energy functions indicate that the equilibrium molecular geometries of the vanadium, niobium, and tantalum tetrafluorides (i.e., VF4, NbF4, and TaF4) exhibit strong distortions from the tetrahedral configuration in their electronic ground state (2E) and first excited state (2T2) along the nuclear displacement coordinates of e symmetry. The distortions result from the E × e and T2 × e Jahn-Teller (JT) effects, respectively. In addition, there are weaker distortions in the 2T2 state along the coordinates of t2 symmetry due to the T2 × t2 JT effect. The description of the large-amplitude dynamics induced by these JT effects requires the construction of JT Hamiltonians beyond the standard model of JT theory, which is based on Taylor expansions up to second order in normal-mode displacements. These higher-order JT Hamiltonians were constructed in this work by expansions of the electronic potentials of the title molecule in terms of symmetry invariant polynomials in symmetry-adapted nuclear displacement coordinates for the bending modes of VF4. A multi-configuration electronic structure method was employed to determine the coefficients of these high-order polynomial expansions from first principles. Using these large-amplitude Jahn-Teller Hamiltonians, the vibronic spectra of VF4 were computed. The spectra illustrate the effects of large-amplitude fluxional nonadiabatic dynamics due to exceptionally strong E × e and T2 × e JT couplings. In addition, the vibronic spectrum of the T2 × (e + t2) JT effect, including the bending mode of t2 symmetry, was computed. The spectrum displays strong inter-mode coupling effects exhibiting a vibronic structure, which is substantially different from that predicted by independent-mode approximation. These results represent the first ab initio study of dynamical Jahn-Teller effects in VF4.

One-dimensional anharmonic oscillator: Quantum versus classical vibrational partition functions

We analyze vibrational partition functions of low vibrational modes within the independent mode approximation to gain insight pertinent to the development of anharmonic corrections for transition state rate constants. As an example, we use a transition state for hydrogen abstraction on a lignin model compound. We found that for low frequencies, anharmonic effects are far more important than quantum effects. Our results suggest a hybrid model, where low-frequency modes are treated fully coupled using the classical or the Wigner–Kirkwood approximation, while high frequencies are described by the harmonic approximation. Mid-range frequencies could be expressed as independent anharmonic quantum modes.

Substituent Effects on the Reaction Rates of Hydrogen Abstraction in the Pyrolysis of Phenethyl Phenyl Ethers

We report reaction profiles and forward rate constants for hydrogen abstraction reactions occurring in the pyrolysis of methoxy-substituted derivatives of phenethyl phenyl ether (PhCH2CH2OPh, PPE), where the substituents are located on the aryl ether ring (PhCH2CH2OPh-X). We use density functional theory in combination with transition-state theory, and anharmonic corrections are included within the independent mode approximation. PPE is the simplest model of the abundant β-O-4 linkage in lignin. The mechanism of PPE pyrolysis and overall product selectivities have been studied experimentally by one of us, which was followed by computational analysis of key individual hydrogen-transfer reaction steps. In the previous work, we have been able to use a simplified kinetic model based on quasi-steady-state conditions to reproduce experimental α/β selectivities for PPE and PPEs with substituents on the phenethyl ring (X-PhCH2CH2OPh). This model is not applicable to PPE derivatives where methoxy substituents are located on the phenyl ring adjacent to the ether oxygen because of the strongly endothermic character of the hydrogen abstraction by substituted phenoxy radicals as well as the decreased kinetic chain lengths resulting from enhanced rates of the initial C−O homolysis step. Substituents decelerate the hydrogen abstraction by the phenoxy radical, while the influence on the benzyl abstraction is less homogeneous. The calculations provide insight into the contributions of steric and polar effects in these important hydrogen-transfer steps. We emphasize the importance of an exhaustive conformational space search to calculate rate constants and product selectivities. The computed rate constants will be used in future work to numerically simulate the pyrolysis mechanism, pending the calculation of the rate constants of all participating reactions.

The Functional Callan–Symanzik Equation for the Coulomb Gas

A non-perturbative scheme, based on the functional generalization of the Callan–Symanzik equation is developed to treat the Coulomb interaction in an electron gas. The one-particle irreducible vertex functions are shown to satisfy an evolution equation whose initial condition is given by means of the classical action and the final point corresponds to the physical system. This equation is truncated by expanding it in momenta and excitation energies, leaving the electric charge as an arbitrary, not necessarily small parameter. Exact coupled partial differential equations up to first order in the frequencies and excitation energies are derived. The numerical integration of these equations is left to a later stage. Nevertheless, in order to demonstrate the relation with the perturbation expansion the one-loop Lindhard function and screening are reproduced in the independent mode approximation of the evolution equation.

Crossover between 2D and 3D systems for the discrete φ4 model

Ferroelectric phase transitions in layered systems are studied in the frame of discrete φ4 model numerically and by simple analytical methods. The single parameter a of the discrete φ4 model determines the type of transition: a → +0 and a → +∞ limit cases correspond to the displacive and order-disorder transition, respectively. Monte-Carlo simulations are performed for three values of a: a = 0.98 (almost displacive transition), a = 4000 (almost order-disorder case), and a = 44.2 (an intermediate case). Transition temperature Tc is obtained as a function of the number of layers N. Data show 2D value at N = 1 and tend to 3D with an increase of N. Monte-Carlo results are compared with predictions of the mean-field and independent-mode approximation. Mean-field result conserves the qualitative behaviour of Tc(N), but overestimates the value of Tc by 30–100%. Independent-mode approximation coincides the Monte-Carlo result for the displacive system with 5% accuracy, but is invalid for larger values of a.

Two-dimensional and layered structures in the discrete φ4 model

The properties of phase transitions in two-dimensional and layered systems are investigated on the basis of a discrete φ4 model by numerical and analytical methods. The only parameter a of the discrete φ4 model determines the behavior of the system and makes it possible to investigate phase transitions ranging from transitions of the displacement type (a → +0) to order-disorder type (a → +∞). The behavior of a two-dimensional system is investigated in a wide range of values of the parameter a. The temperature dependences of the squared order parameter η2(T) and the phase transition temperature Tc as a function of the thickness N of the system are obtained for three characteristic values of the parameter a using the Monte Carlo method. The properties of phase transitions in the discrete φ4 model are investigated on the basis of the mean-field approximation and the independent-mode approximation. The results obtained in the numerical experiments are compared with the analytical approximations. It is shown that the mean-field approximation qualitatively describes the behavior of the phase-transition temperature Tc as a function of the thickness N of the system for a wide range of values of the parameter a, and the independent-mode approximation describes quantitatively, to within 5%, the results of the numerical simulation for small values of a.

Dynamical mean-field theory of realistic spin glasses beyond independent-mode approximation. II. Effect of reaction field

For pt.I see J. Phys. C., vol.17, p.5049, 1984. The author extends the previous calculation on a dynamical mean-field theory of realistic spin glasses (Chowdhury and Mookerjee) so as to incorporate the effect of the reaction field. The 'ordering temperature' of a mode is shown to depend not only on the corresponding eigenvalue of the random matrix J but also on the structure of the whole spectrum.

Dynamical mean-field theory of realistic spin glasses beyond the independent-mode approximation

The authors derive in detail a dynamical mean-field theory of spin freezing in spin glasses using the Fokker-Planck type of approach in the high-barrier/low-noise limit. Then they go beyond the independent-mode approximation (IMA) by incorporating, partly, the effects of overlap of modes. The dependence of relaxation time on size turns out to be weaker than that in the IMA.

SCF CI calculations for vibrational eigenvalues and wavefunctions of systems exhibiting fermi resonance

It is shown that configuration-mixing calculations with configurations built from SCF wavefunctions can be used to provide accurate and compact wavefunctions for systems in which the independent-mode approximation breaks down.

Experimental evidence for spin diffusion in the planar Heisenberg magnet (C 2H 5NH 3) 2MnCl 4

High temperature spin dynamics in the quasi-two-dimensional (2D) Heisenberg magnet ethylammonium-manganese-chloride have been investigated by EPR. Different experimental methods are presented which can be used to analyse secular and non-secular resonances of self-energy: (i) frequency dependence of the magic angle EPR linewidth, (ii) highly sensitive measurements of transverse absorption and dispersion, and (iii) lineshape analysis of half-field transitions. By each experimental method a logarithmic frequency or field dependence has been obtained which represents a characteristic feature of 2D spin diffusion. Within the bounds of the independent mode approximation the spin diffusion constant is evaluated to be D/a 2=(13±2) K.

Theory of exchange narrowing in one and two dimensions

We show that the theoretical approximations in the Kubo and Tomita theory of electron paramagnetic resonance are such as to make its application to one- and two-dimensional systems dubious. A self-consistent theory based upon the independent-mode approximation, that does not have the defects of the Kubo and Tomita theory, is described, and line shapes are calculated for the strongly narrowed limit. Both theories are in significant disagreement with the experimental results on tetramethyl manganese chloride.

Theory of motional narrowing of EPR spectral density of magnetic ions

A theory of the EPR line shape of a magnetic impurity coupled to a lattice is presented that is capable of treating both the "slow" and "fast" regimes. A self-consistent equation of motion for the propagator of the ion motion is obtained under the assumption that the fluctuations of the lattice are not affected by the presence of the impurity. The equation is shown to correspond to the choice of the lowest-order term in a renormalized perturbation theory for the "self-energy" operator of the system. A diagrammatic representation of the perturbation theory is given which one may readily use to include higher-order corrections. The lowest-order approximation is analogous to the mode-coupling, or independent-mode approximation in extended systems, and is equivalent to the Kubo-Tomita approximation in the "fast" regime. A solution of a model problem that exhibits the motional-narrowing phenomenon is given. The theory may be generalized to arbitrary finite-dimensional systems coupled to a bath.