Effective Hamiltonian Formalism Research Articles

Fluorescence of supercooled molecules as a probe for intramolecular vibrational redistribution rates

A theory is developed for the fluorescence spectra of supercooled polyatomic molecules in a jet. A single molecular mode (the ’’system’’) is assumed to be optically active and the other modes constitute a ’’bath’’ which induces line broadening and relaxation. Since the bath is finite and is initially cold, its motion is strongly correlated with that of the system. The correlated reduced dynamics of the system and bath is treated using the Mori projection operator technique. The present ’’tetradic reduced description’’ becomes useful especially in large statistical molecules where many molecular states are involved in the fluorescence. This is precisely the limit where the effective Hamiltonian formalism commonly used for the intramolecular dynamics in intermediate size molecules becomes very tedious and impractical. The fluorescence spectra in large statistical molecules may be interpreted in terms of simple rate equations and may be used to directly monitor the intramolecular vibrational redistribution of energy (IVR). We further compare the spectra predicted for statistical and intermediate size molecules with the existing experimental results in supersonic beams.

Effects of uniaxial stress on excitons inCu2O

We have investigated the effects of uniaxial stress up to 2.5 kbar on the first four dipole-forbidden states of the yellow exciton series in cuprous oxide. The linear and nonlinear splittings and shifts as well as the symmetries of the stress-split components were determined using the quadrupole-dipole Raman-scattering technique involving odd-parity phonons. We have observed the triply degenerate $1S$ quadrupole state split into a singlet and a doublet with the singlet energy increasing and the doublet energy decreasing with stress as first seen by Gross and Kaplyanskii. For the higher quadrupole $S$ states, however, the sense of the splitting was reversed from that of the $1S$, while the magnitude of the $3S$ splitting exceeded that of the $1S$ as previously noted by Agekyan and Stepanov. The quadratic stress coefficients for all components of the $1S$, $3D$, and $4S$ states were found to be equal while those for the $3S$ state were markedly different. A theoretical analysis was carried out for [001] stress using the effective-Hamiltonian formalism including both deformation of the bands and electron-hole exchange. The theory accounts satisfactorily for the general features of the $1S$, $3D$, and $4S$ states only if both the magnitude and sign of the exchange constant are considered to be dependent on the exciton state. For the $3S$, however, the nonlinear stress dependence cannot be reconciled with the theory for yellow excitons. These results suggest that the "$3S$ yellow exciton" may actually belong to the green-exciton series, or that interactions between yellow and green excitons with different principal quantum numbers may be significant although not included in the theory.

Thermal and Magnetic Properties of Nd3+ in Doped Garnets

The Schottky specific heats and magnetic susceptibilities of the Nd3+ in five doped garnets, GGG, LuAG, LuGG, YAG and YGG, have been calculated using effective spin Hamiltonian formalism in connection with the crystal field theory in a temperature interval of 30 to 500 K. Both heat capacities and susceptibilities are determined by two different methods. The first method is dealing with the diagonalization of the effective spin Hamiltonian and then obtaining the quantities in a very general way of quantum statistical mechanics. The second method is based on the cumulant expansion of partition function and utilizing the invariance properties of the trace of the Hamiltonian matrices. It is thought that the specific heat and the susceptibility calculations may help to refine the crystal field parameters of such crystals.

Random coupling model for multiphoton photofragmentation of large molecules

In this paper we present a quantum mechanical theory of multiphoton photodissociation of large, collision-free, molecules, which rests on the notion that the radiative coupling terms between adjacent sets of congested bound molecular states in the quasicontinuum exhibit a wide variation both in terms of magnitude and of sign. Invoking the rotating-wave approximation, neglecting spontaneous infrared decay, and assuming that near-resonant radiative coupling prevails, the equations of motion were solved within the framework of the random radiative coupling model for the radiative interactions in the quasicontinuum. In the low energy range (range I) the equations of motion for the amplitudes are determined by the effective Hamiltonian formalism, while in the quasicontinuum (range II) the populations are governed by kinetic equations for sequential reversible decay. All the features of coherent excitation are preserved for range I, while in range II intramolecular erosion of phase coherence effects prevails. This model provides a set of reasonable predictions.

Magnetic and Thermal Properties of Nd3+ in Scheelite‐Type Crystals

AbstractUsing the effective spin Hamiltonian formalism, crystal field calculations of the temperature dependent susceptibilities and the Schottky specific heat are performed over the temperature range 10 to 300 K besides the dipolar and quadrupolar transition probabilities of Nd3+PbMoO4 and Nd3+CaWO4 using different sets of known parameters. Two different methods are employed for the susceptibility calculations and the Curie‐Weiss law is checked. Emphasis is given of the effects of different sets of crystal field parameters.

Spin-label and deuterium order parameter discrepancies in bilayers: one possible explanation

We have simulated electron spin resonance spectra of anisotropically immobilized spin labels of the type seen in lipid and soap-like bilayers using a rigorous formalism which explicitly includes the effects of spin-label motion. In most bilayer systems, spin-label experiments have shown lower order parameters then deuterium-label experiments. In the past this apparent decrease in the order parameters was thought to reflect the distortion of the bilayer by the doxyl ring of the spin probes. We wish to report that this type of discrepancy may be due to the neglect of important motional effects in the time-independent effective Hamiltonian formalisms used in previous interpretations of anisotropically immobilized spin label spectra. That the true order parameters may be the same can be shown by including slow motional corrections in the effective Hamiltonian formalism. The larger volume of the doxyl ring may change the apparent order parameter by increasing the importance of the slow motional effects, as opposed to causing a real decrease in the order parameter, as previously proposed.

Quantum number and energy scaling of rotationally inelastic scattering cross sections

A general formula for rotationally inelastic cross sections in atom-diatomic collisions is derived. This result is achieved by assuming the transition probability is a function of rotational quantum number differences, the kinetic energy in the upper states, and is inversely proportional to the number of accessible states within an effective hamiltonian formalism. The scaling law is able to predict all rows, or columns, of the inelastic cross section matrix, σ jj, given any one row, or column, as a function of energy. Finally, we applied this scaling theory to a variety of collision systems; in general, good agreememt between predicted and exact results is exhibited. We have developed and tested a scaling law for rotationally inelastic transitions which predicts all rows or columns of the inelastic cross section matrix, σ jj, given any one row or column. The agreement between exact and predicted cross sections was good for the HeHF [20], HeHCl [21], H 2CS [17], HeCO [16] and Li +H 2 [18] systems which vary in reduced mass from approximately 2–4 an and rotor constant from 0.82–60.0 cm −1. However, for ArN 2 [13,14, 19,24] collisions (μ = 16.5 au, B e = 2.01 cm −1) the error increased with increasing j and/or Δ. In every case the disagreement was in the opposite direction from the prediction of eqs. (15a)–(15c) which are based upon fully degenerate statistics instead of the effective hamiltonian non-degenerate statistics of eqs. (2a) and (2b). Finally, we state two general conclusions: 1. (1) Whenever the initizil kinetic energy ( E—ϵ j) is greater than approximately five times the energy gap (ϵ j+Δ—ϵ j), the scaling theoretic cross sections are accurate within 20%; and 2. (2) at total energies much larger than that of the internal states of interest, all cross sections with the same change in rotational quantum number become approxinutely equal.

Multiphoton molecular dissociation in intense laser fields

In this paper we advance a model for multiphoton photofragmentation of an ’’isolated,’’ collision-free, polyatomic molecule on the ground state potential surface. The molecular energy levels are separated into three regions. In the low energy range the level structure is sparse and only dynamic Stark shifts will be exhibited. In the intermediate energy range the density of bound vibrational states is high and the level structure can be described in terms of mixed (zero-order) states. We argue that in this energy range intrastate anharmonic scrambling may be of central importance in the excitation process, but that intrastate vibrational relaxation and energy redistribution is not encountered for medium-sized molecules. In the high energy region dissociative channels open up and reactive intramolecular decay is handled in terms of resonance theory. The time evolution of a multilevel system, whose highest energy levels are metastable, and which is driven by an intense laser field, is handled by the effective Hamiltonian formalism. Explicit expressions are derived for the photofragmentation yields and their dependence on the molecular parameters and on the field parameters. Specific applications to two distinct model systems are presented. First, we treat the quasidiatomic model, which disregards level scrambling in the intermediate energy range, whereupon near-resonant radiative coupling prevails between the states of a truncated anharmonic oscillator. Second, we have considered the two-ladder model where in the low energy range near-resonant radiative coupling occurs within an anharmonic ladder, while in the intermediate energy range resonant radiative coupling between mixed states prevails. We present numerical simulations of the photofragmentation yields and their dependence on the molecular parameters, such as the diagonal anharmonicity, the molecular dissociation energy, the predissociative widths, and the isotopic shift. We have also explored the dependence of the photofragmentation yields on the pulse parameters, such as the off-resonance energy, the field intensity, and the pulse duration. The quasidiatomic model seems to overestimate the power onset for photodissociation and the power required for the onset of saturation effects, while the two-ladder model is quite adequate to account for the gross features of coherent multiphoton molecular photofragmentation.

Unified theory of symmetry-breaking effects on excitons in cubic and wurtzite structures

A general group-theoretical scheme is developed to discuss any symmetry-breaking effects on the 12-fold $1s$-exciton states originating from an $s$-like and $p$-like electron-hole pair. All the possible effects on the conduction and valence bands are taken into account by an effective Hamiltonian formalism. For the electronhole exchange interaction, the simplest term (including dipole-dipole interaction) is considered. The symmetry-breaking effects treated here can be anything; e.g., a magnetic field, an electric field, stress, the finiteness of exciton wave vector, etc., and even the products of them. As a group-theoretical treatment, this theory includes all the relevant previous theories as special cases.

Study of the vibronic effects from the Zeeman-field-induced Mössbauer spectra ofFe2+ions in octahedral symmetry. I. The effective Hamiltonian formalism

The importance of the Zeeman-field-induced quadrupole and hyperfine interactions in Jahn-Teller active ions is demonstrated with particular reference to ${\mathrm{Fe}}^{2+}$ in octahedral crystals such as CaO and KMg${\mathrm{F}}_{3}$. An effective electron nuclear Hamiltonian is derived from general symmetry principles, whose parameters are calculated by perturbation procedure, incorporating the vibronic effects in terms of Ham reduction factors between spin-orbit states. Relevant theoretical expressions for the M\ossbauer transitions are given so that these parameters can be determined experimentally from the analysis of the M\ossbauer spectra.

Multiphoton transitions: Approximations for the effective Hamiltonian

The problem of calculating transition probabilities for an atom interacting with an intense radiation field is set up within the context of an effective Hamiltonian formalism. This has the virtue that resonance effects are properly accounted for at the outset; more generally, it provides a convenient framework for approximations. Through the restriction of the class of intermediate states to be included in the construction of the effective potential, a unitary (nonperturbative) model is obtained. A successive approximation procedure for determining the effective potential, which is based on continued-fraction representations, is described. While very likely intractable for the true problem, this successive approximation procedure can be implemented, for the model effective potential, with the aid of a generalized Rayleigh-Ritz procedure described here. The level-shift matrix is constructed from the effective potential; it determines the transition probabilities, linewidths, and energy-level displacements induced by the field. We derive variational expressions of the Kohn type for each of the elements of the model level-shift matrix.

A model for isotope separation via molecular multiphoton photodissociation

Multiphoton photofragmentation of an “isolated” molecule on the lowest potential surface is considered in terms of a truncated anharmonic oscillator driven by a pulsed, intense laser field. Intramolecular vibrational relaxation and indirect photodissociation are accounted for by assignment of decay widths to the appropriate levels. The effective-hamiltonian formalism is applied to derive explicit expressions for the photofragmentation yield and for the isotopic separation factor.

Investigation of spin quintet states by electron paramagnetic resonance

EPR studies of spin quintet states are reviewed. A general effective Hamiltonian formalism forS=2 is given and applied to the phenomenological energy structure description of spin quintet states. Microscopic models for the phenomenological constants of this Hamiltonian are given for paramagnetic ions (3d4 and 3d6), free radicals and aggregates consisting of exchange-coupled paramagnetic ions. The fourcopper-ion aggregate with effective spinS=2 in TGFB: Cu2+ is given particular attention.

Phenomenological Description of Aggregates of Exchange‐Bonded Paramagnetic Ions in Crystalline Environment. I. Effective Hamiltonian Formalism

AbstractA general effective Hamiltonian formalism is proposed and applied to the phenomenological description of energetic structure of aggregates consisting of several exchange‐bonded paramagnetic ions in a crystalline environment. The low‐lying energetic multiplets of such a system, resulting from isotropic exchange (superexchange) interaction, split under the further interactions and, for a variety of ions, can be described in terms of effective spin Hamiltonians. Such a description is highly useful from the electron paramagnetic resonance point of view, in which the transitions inside multiplets are observed. The microscopic model for the phenomenologic constants of these Hamiltonians is given in the second order of perturbation treatment, and the case of ions with orbitally nondegenerate ground crystal term is discussed in some detail.

Role of Generalized Wannier Functions in the Effective-Hamiltonian Formalism for Magnetic Fields

Generalized Wannier functions appropriate to magnetic fields are constructed by group-the-oretic methods for nearly free electrons. A one-to-one correspondence is indicated with the zero-field band states. The diagonal elements of the effective Hamiltonian are discussed and are shown to resemble closely the energy-band function $E(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})$ when expressed in the basis set of these Wannier functions.

Effective Schrödinger equations for the elastic interaction of heavy ions

Starting from the many-body Schrödinger equation and using completely antisymmetric state vectors, an effective Hamiltonian formalism is developed which is appropriate for a description of elastic scattering of heavy ions. The additional potential terms arising from the elimination of all non-elastic channels are non-local and non-hermitean. They can be written in terms of transition amplitudes and state densities, so that an approximate evaluation seems possible within the two-center shell model. Special attention is paid to the imaginary part of the potential. Some qualitative features of the imaginary potential are discussed, taking integrals of motion into account. Finally, the possible extension of our treatment to reaction channels is sketched.