Energy Sudden Research Articles

The use of the energy sudden approximation in them-scaling of rotational cross-sections

We invoke energy sudden (ES) dynamics in the projection- or m-scaling of crossed-beam rotational energy transfer cross-sections. We identify the dynamical quantities which must be input to the scaling analysis and note their symmetry properties. Also we show that a partial m-summing of cross-sections is sufficient to effect significant simplifications of the scaling. Application of the scaling analysis is made to integral cross-sections in the scattering of He+CO. It is found that the ES scaling is very effective in describing the m-dependent transitions.

Collisional scaling within a multichannel square representation

A new approach is examined for the state-to-state scaling of collision problems. It derives from multichannel scattering at a square interaction and is referred to as mutlichannel square (MS) scaling. Since it retains the full internal energy spectrum while approximating the radial coupling elements, it may be viewed as complementing the conventional energy sudden (ES) scaling. Some general comments are made with regard to the structure of the MS relations and we show how several attractive features of ES scaling can be matched by the new scaling. Application is made to the analysis of rotationally inelastic integral cross sections for the scattering of He from p-H2. We compare MS scaled results with both ES and modified ES results. It is found that for the range of total energies examined, the MS scaling yields results which are in good agreement with exact ones and are dramatically improved over the basic ES ones. The modified ES procedure depends upon incorporating off-energy shell effects into the ES scaling and like the MS scheme is here set up to contain a single free parameter. We find that the MS results are roughly twice as accurate as the modified ES ones. A number of avenues for further development and application of the MS scaling are discussed.

Close coupling-wave packet formalism for gas phase nonreactive atom–diatom collisions

In this paper we discuss the adaptation of the close coupling-wave packet (CCWP) method for solving the time dependent Schrödinger equation for inelastic, nonreactive gas phase atom–diatom collisions. The approach is novel in that (a) it is an initial value rather than boundary value method, (b) it can be formulated to either avoid or include the partial wave expansion normally used for gas phase atom–diatom collisions, (c) it can be formulated to determine either a single column of the differential scattering amplitude matrix or S matrix rather than the full matrix, (d) the labor involved in a single calculation scales with the number of rotor states squared rather than cubed as in standard close coupling, (e) a single calculation yields numerically exact results over the full range of energies contained in the original wave packet, and (f) results for other initial states can be obtained by means of the energy sudden (ES) or energy corrected sudden (ECS) factorization relations. The analysis for extracting the differential scattering amplitude at fixed energies is given in detail because it differs markedly from that normally given in textbook treatments of the wave packet formulation of gas phase scattering. Finally, an example approximate version of the formalism (namely the energy sudden) is given.

Vibrational state distributions following the photodissociation of (collinear) triatomic molecules: The vibrational reflection principle in model calculations for CF3I

Vibrational state distributions following the direct photodissociation of a collinear, triatomic molecule is investigated with particular emphasis on the so-called final state interaction, i.e., the translational–vibrational coupling due to the excited state interaction potential. In order to separate the various effects which determine the state distribution we performed calculations on three levels of accuracy: The energy sudden (ES) approximation, the modified sudden (MS) approximation, and the exact close-coupling (CC) formulation. The pure ES distributions peak at high states and are very broad. They are explained within the semiclassical limit as a mapping of an amplitude onto the quantum number axis. We call this effect vibrational reflection principle in analogy to the equivalent effect in rotational excitation processes. It is a direct and sensitive probe of the parameters of the system, most importantly the potential energy surface. Energy conservation strongly modifies the ES distributions. The MS and CC distributions are much narrower and peak at considerably lower states. A detailed analysis is given within the MS approximation. Based on these general conclusions we suggest a particular excited state potential for the dissociation of CF3I which qualitatively reproduces the recently reported experimental CF3 distribution. The essential feature of this potential is a distance dependent local frequency which we find necessary to obtain distributions as broad as in the experiment. Because of the inherent difficulties with the time-of-flight technique if several energy transfer channels are involved we certainly do not know how realistic our final potential energy surface is. However, the general trends found in this study should be valid for a larger class of systems.

Scaling theory: Energy sudden and dynamically modified relations

An approach is described for dynamically modifying energy sudden (ES) collisional scaling relations. It is based upon a generalized form of perturbation theory (PT), which contains ES dynamics as the zeroth order approximate. The corresponding first order PT scattering matrix is further modified by exponential unitarization (EPT). Our scaling relations take on the following structure: an input column of S-matrix elements (back) projects through first order EPT (and hence in an approximate fashion), onto the corresponding column of ES elements; a set of ES scaling coefficients (forward) projects these elements onto a new column; the new column (forward) projects through again first order EPT, onto the corresponding scaled column. The effectiveness of this approach is illustrated by application to a simple classical path three-state problem. Two slightly different versions of the approach are compared. We also examine how ‘‘column based’’ scaling predictions compare with ‘‘single element based’’ predictions. Finally, a number of avenues for further development and application are discussed.

Accuracy of the energy-corrected sudden (ECS) scaling procedure for rotational excitation of CO by collisions with Ar

We examine the question, given a set of state-to-state rate constants k( j=0 → j′‖T) for collision-induced rotational transitions in a diatomic molecule, where j and j′ are initial and final rotational quantum numbers and T is the translational temperature, can one use scaling analyses to predict a full set of j → j′ rate constants? To answer this we consider a rigid rotator model of CO in a bath of Ar at 500 K, and we calculate accurate quasiclassical k( j → j′‖T) for j=0, 10, and 20 and j′=0–29. These are used to test the energy sudden (ES) and energy-corrected sudden (ECS) scaling procedures. Both procedures are used to predict the j=10 and j=20 rate constants from the j=0 values. The ES procedure, which has no adjustable parameters, overestimates the rates out of excited states by a factor of about 1.5 with a rms error of about 60%. The ECS procedure, in contrast, when the one parameter bc (a critical impact parameter) is about 1.75–2.0 Å, yields excellent excited-state rates on the average and has a rms error of less than 20%. The value of bc can be estimated by a weighted average impact parameter leading to inelastic collisions from a j=0 initial state.

Classical trajectory calculations of diffusion and viscosity for He-N2mixtures

The authors present classical trajectory (CT) calculations for the diffusion and viscosity coefficients in the temperature range 77.3-1100K using the HFD1 surface of Fuchs et al. (1984). They have also performed energy-sudden (ES) calculations for this surface and classical infinite-order sudden (MM) calculations for both this surface and their HFD2 surface. The authors CT results agree with the measurements of Kestin et al. (1979) for the viscosity coefficient over the entire range of temperatures reported (298-973K), and for the diffusion coefficient in the temperature range 298-763K. Their error limits are +or-1% for viscosity and +or-2% for diffusion. However the authors' values consistently exceed by about 2% the measurements of Trengove and Dunlop (1982) for diffusion in the temperature range 277-323K, where their error estimate is +or-0.2%. Comparison of the ES results with the CT and the MM results indicates that the ES and MM (IOS) approximations are reasonable for viscosity but less reliable for diffusion. The MM results for both diffusion and viscosity are generally consistent with the quantal infinite-order sudden results for both surfaces, which suggests that quantal effects for these coefficients are relatively small. From the MM results it may be inferred that the HFD1 surface is better than the HFD2 surface for these properties.

Classical trajectory calculations of transport properties for a model Ar-N2potential surface

Classical trajectory calculations were performed using the Ar-N2 potential surface of Patterngill et al. (1979) for temperatures of 77.3 and 300K. Diffusion and viscosity cross sections, the production and relaxation cross sections governing the viscomagnetic effect and cross sections for both depolarised Rayleigh scattering and the rotational relaxation time were calculated. Comparison was made with classical energy-sudden (ES), classical infinite-order sudden (MM) and quantal infinite-order sudden (IOS) calculations. For both the viscosity and diffusion cross sections the CT and the ES, MM and IOS calculations agree within 3%, except for the IOS results at 77.3K. For the relaxation and DPR cross sections the IOS results are found to differ by about 40-80% from the CT results. The ES result for the DPR cross section is about 20% larger than the CT result, while for the production and relaxation cross sections it is very poor. The authors suggest that the main reason for these discrepancies is the breakdown of the sudden approximation for the non-sudden collisions always arising in thermal averages.

Energy sudden dissociative collisions: Structure and applications of factorization relations

In the energy sudden (ES) approximation for nonreactive molecular collisions, there exist factorization relations by which an arbitrary T-matrix element can be predicted as a spectroscopic linear combination of those out of some other, input state. These were first discovered for ground state input but this restriction was later removed. This general form of the spectroscopic factorization relations is straightforwardly extended here to ES dissociative collisions. One finds that in predicting dissociation amplitudes out of some state, it is necessary to use input data out of a higher (energy) bound state. Thus ground state factorization relations cannot be used. The structure of two natural forms of the factorization relation coefficients (equivalent by virtue of ES consistency conditions among T-matrix elements out of a single state) are analyzed in detail for a collinear atom-truncated square-well diatomic oscillator system. Relevance of these results to the prediction of (dissociative) state specific vibrational enhancement/inhibition is discussed.

A simple procedure for incorporating off-energy-shell effects into energy sudden scaling relations

In this paper, a new approach for incorporating off-energy-shell effects into energy sudden (ES) scaling relations is described. The framework for the new approach is provided by the first order Born approximation which allows for the relating in an approximate fashion of off-energy-shell and on-energy-shell T-matrix elements. The relationship between this approach and a former procedure is discussed. Application of the off-shell corrected ES relation is made to the vibrational-rotational scattering of He–H2.

Time-independent energy-sudden transformationa)

The time-independent energy sudden (ES) representation is defined through application of the energy shift operator S=exp[−(h−ωn)∂/∂ε], where h is the internal (molecular) Hamiltonian. Our introduction of S follows from an earlier study by Chang, Eno, and Rabitz where exp[−iht], which ‘‘factors out’’ internal motion, was used to define the time-dependent ES representation. Exact integral equations for the scattering wave function within the ES representation are derived, the leading terms being the approximate ES wave function. Corrections to the ES wave function are nonsingular and involve the generalized potential increment V=S−1VS−V, where V is the interaction potential. Boundary conditions and transition amplitudes are discussed, as is the connection between wave functions in the ES and the original representations.

Exact scattering solutions in an energy sudden (ES) representation

In this paper, we lay down the theoretical foundations for computing exact scattering wave functions in a reference frame which moves in unison with the system internal coordinates. In this frame the (internal) coordinates appear to be fixed and its adoption leads very naturally (in zeroth order) to the energy sudden (ES) approximation [and the related infinite order sudden (IOS) method]. For this reason we call the new representation for describing the exact dynamics of a many channel scattering problem, the ES representation. Exact scattering solutions are derived in both time dependent and time independent frameworks for the representation and many interesting results in these frames are established. It is shown, e.g., that in a time dependent frame the usual Schrödinger propagator factorizes into internal Hamiltonian, ES, and energy correcting propagators. We also show that in a time independent frame the full Green’s functions can be similarly factorize. Another important feature of the new representation is that it forms a firm foundation for seeking corrections to the ES approximation. Thus, for example, the singularity which arises in a conventional perturbative expansions of the full Green’s functions (with the ES Green’s function as the zeroth order solution) is avoided in the ES representation. Finally, a number of both time independent and time dependent ES correction schemes are suggested.

On the validity of the energy sudden approximation

This paper contains an examination of the conditions under which the energy sudden (ES) approximation may be expected to be valid. Our approach involves using dimensional analysis to identify (dimensionless) quantities which control energy suddenness and in this fashion three sets of ES criteria emerge. One involves the relative kinetic energy between collision partners and the energy spacing of the internal states of interest; another the strength of the coupling interaction and the same spacing; and a third involves the masses of the colliding molecules and component atoms. We discuss the relationship between these conditions and the justifications given by earlier workers for adopting the ES approximation and then the mass conditions in particular are used as the basis for certain broad statements concerning the applicability of the ES method within nonreactive diatom–diatom and reactive atom–diatom collisions. Finally, a number of avenues for further development of this work are discussed.

Factorization relations and consistency conditions in the sudden approximation

Linear factorization relations are derived for the matrix elements of quantum mechanical operators defined on some space ℋ = ℋ1⊕⋅2 which are diagonalizable on ℋ1. The coefficients in these relationships do not depend on the operators per se but do depend on the representations in which the operators are diagonal. The formulation is very general with regard to the nature of the ’’input’’ information in the factorization. With each choice of input information there are associated consistency conditions. The consistency conditions, in turn, give rise to a flexibility in the form of the factorization relations. These relations are examined in detail for the operators of scattering theory which are local in the internal molecular coordinates. In particular, this includes S and T matrices in the energy sudden (ES) approximation. A similar development is given for the square of the magnitude of operator matrix elements appropriately averaged over ’’symmetry classes.’’ In the ES these relations apply to transition cross sections between symmetry classes. In particular, they apply to degeneracy averaged cross sections in situations where the symmetry classes correspond to energy levels.

Time reversal symmetry for magnetic transitions in rotationally inelastic scattering. II. Angular momentum decoupling approximations

The consequences of the generalized microscopic reversibility derived in the preceding paper are considered for the jz-conserving coupled states or centrifugal sudden (CS) and energy sudden (ES) approximations. It is shown that the two most popular choices of the CS parameter ? lead to violation of the generalized microscopic reversibility. However, it is also shown that these two choices of ? play the role of time reversal partners to one another. The simplest choice of the ES parameter k leads to an approximation that preserves the generalized microscopic reversibility.

Interpretation of ES, CS, and IOS approximations within a translational–internal coupling scheme. I. Atom–diatom collisions

Rotational invariance is applied to the description of atom–diatom collisions in a translational–internal coupling scheme, to obtain energy sudden (ES), centrifugal sudden (CS), and infinite order sudden (IOS) approximations to the reduced scattering S matrix S (??;L;jλ). The method of presentation emphasizes that the translational–internal coupling scheme is actually the more natural description of collision processes in which one or more directions are assumed to be conserved.