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15 Dec 1971

Volume 55, Issue 12, pp. 5425-5847

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Collision Induced Infrared Absorption of Gaseous Nitrogen at Low Temperatures

De T. Sheng and George E. Ewing

J. Chem. Phys. 55, 5425 (1971); http://dx.doi.org/10.1063/1.1675703 (6 pages) | Cited 13 times

Online Publication Date: 9 December 2003

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The collision induced infrared absorption spectrum of gaseous nitrogen was studied over the temperature range 77–297°K. The integrated absorption coefficient depends quadratically on the gas density and indicates as have previous studies on this and other nonpolar diatomic molecules that the transition inducing mechanism involves a two molecule interaction. A profile analysis of the absorption band at various temperatures assigns the observed features to colliding pairs of freely rotating nitrogen molecules. No evidence for bound state dimers was found although our data do not preclude their existence. Since the integrated absorption coefficient is related to the intermolecular potential, a comparison between the theoretically calculated and experimentally measured values at different temperatures provides a means for studying the potential. It was found that the Lennard‐Jones potential was adequate from 297°K to about 130°K with the traditionally accepted σ and ϵ∕k parameters, but for lower temperatures the agreement with spectroscopic measurements was poor. The discrepancy can be accounted for by realizing that the Lennard—Jones potential is isotropic, but at low temperatures angularly dependent quadrupole interactions become relatively more important and need to be included in the description of the total intermolecular interaction.

Valence‐Bond Studies of AH2 Molecules. I. BeH2

Robert G. A. R. Maclagan and Gary W. Schnuelle

J. Chem. Phys. 55, 5431 (1971); http://dx.doi.org/10.1063/1.1675704 (5 pages) | Cited 21 times

Online Publication Date: 9 December 2003

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The valence‐bond method is discussed and is compared with the molecular‐orbital SCF method from a computational standpoint. A procedure is outlined for performing valence‐bond calculations making use of the Prosser and Hagstrom biorthogonalization technique to evaluate the cofactors required when evaluating the energy using Löwdin's formula. A minimal basis set (STO) valence‐bond calculation on BeH2 including all possible configurations which do not involve excitation of the Be 1s electrons is reported. This is compared with a molecular‐orbital calculation using the same basis set. A valence‐bond wavefunction neglecting ionic configurations is shown to give an energy better than the molecular‐orbital wavefunction and almost as good as a ``full'' valence‐bond calculation. The perfect pairing and Craig and Thirunamachandran's resonance approximation are examined. A scheme for performing semiempirical valence‐bond calculations analogous to the Pople—Santry—Segal CNDO∕2 MO method is described. Ionic configurations cannot be neglected in this ZDO approximation, but this is not a serious limitation. The method is very simple computationally. Hybrid and localized orbitals can be used.

Applicability of SCF Theory to Some Open‐Shell States of CO, N2, and O2

John B. Rose and Vincent McKoy

J. Chem. Phys. 55, 5435 (1971); http://dx.doi.org/10.1063/1.1675705 (7 pages) | Cited 26 times

Online Publication Date: 9 December 2003

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By working with the real functions πx and πy instead of π+ and π−, we show how to express the SCF Hamiltonians for the Σ states of the configurations (πu)3(πg),(πu)3(πg)3, and (1π)3 (2π) of diatomic molecules in terms of the Coulomb and exchange operators only. With these results, we have used conventional SCF programs to solve for the wavefunctions of many interesting states of N2, O2, and CO, e.g., the B 3Σu− state of O2. For many states, the SCF results are in good agreement with experiment. However, SCF theory runs into serious trouble if electron correlation is important in determining the relative locations of excited states.

Geminate Recombination of X‐Ray Excited Electron—Hole Pairs in Anthracene

R. C. Hughes

J. Chem. Phys. 55, 5442 (1971); http://dx.doi.org/10.1063/1.1675706 (6 pages) | Cited 40 times

Online Publication Date: 9 December 2003

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A recent controversy is discussed concerning the relative importance of geminate and track recombination of charge carriers excited by ionizing radiation in single crystals of anthracene. Data is presented which strongly supports the geminate recombination hypothesis for bulk excitation by x rays in the 50–500−keV range. In agreement with the earlier work of Kepler and Coppage, it is found that it requires about 2700 eV to create a pair of charge carriers in anthracene at an applied field of 10 kV∕cm.

ESR of Free Radicals in Ferroelectric and Antiferroelectric Arsenate Single Crystals

J. Gaillard, O. Constantinescu, and B. Lamotte

J. Chem. Phys. 55, 5447 (1971); http://dx.doi.org/10.1063/1.1675707 (6 pages) | Cited 24 times

Online Publication Date: 9 December 2003

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The electron spin resonance of gamma irradiated single crystals of KH2AsO4, KD2AsO4, NH4H2AsO4, and ND4D2AsO4 has been investigated. New paramagnetic centers present in several varieties and which exhibit a strong coupling with one 75As atom have been described. At room temperature, their g tensor is nearly isotropic and their hyperfine couplings are axially symmetric. The annealing of these centers is examined and their behavior upon cooling down below Tc is described. All these centers which seem to belong to the same family have been identified as AsO32− species. Their different varieties must be ascribed to different environments coming either from various proton environments or from several associations with varied impurities.

Localized Orbitals for Polyatomic Systems. II. Open Shell Case

Herbert Schlosser

J. Chem. Phys. 55, 5453 (1971); http://dx.doi.org/10.1063/1.1675708 (6 pages) | Cited 10 times

Online Publication Date: 9 December 2003

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We present a generalization of the Adams—Gilbert approach for constructing localized Hartree—Fock (HF) orbitals from the closed shell single Slater determinant case to the case where one has several open shells of different symmetry in addition to closed shells of electrons. Our starting point is the open shell HF formalism discussed by Roothaan and Huzinaga. We then apply appropriate localization operators to construct the localized AG orbitals, and also develop the resulting equations in powers of the overlap. Finally, we present several perturbation‐iteration schemes for solving the open shell Adams—Gilbert equations, and comment on pseudopotential methods for construction of localized orbitals.

Localized Orbitals for Polyatomic Systems. III. Open Shell Extended Hückel Approximation

Herbert Schlosser

J. Chem. Phys. 55, 5459 (1971); http://dx.doi.org/10.1063/1.1675709 (6 pages) | Cited 6 times

Online Publication Date: 9 December 2003

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We present a rigorous derivation of the extended Hückel (EH) approximation for open shell polyatomic systems making use of the open shell generalization of the Adams—Gilbert (AG) equations for localized orbitals which we derived in a previous paper. The open shell EH theory is derived for the open shell filled band and open shell partly filled band cases from a systematic expansion of matrix elements of the one‐electron Fock operator in powers of the overlap using distorted atomic orbital and polarized atomic orbital solutions of the AG equations as bases, respectively.

Phosphorescence Enhancement in Phenyl‐Substituted Methanes

Fred H. Watson and M. Ashraf El‐Bayoumi

J. Chem. Phys. 55, 5464 (1971); http://dx.doi.org/10.1063/1.1675710 (7 pages) | Cited 10 times

Online Publication Date: 9 December 2003

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Experimental evidence is presented which shows that the natural phosphorescence lifetimes, τP°, decrease with increased phenylation in the molecular series: toluene, diphenylmethane, triphenylmethane. This indicates that a spin—orbit enhancement mechanism is operative. Natural fluorescence lifetime data support the exciton molecular model for diphenylmethane and triphenylmethane and show that the phenyl groups have little electronic overlap. The experimentally observed natural fluorescence lifetimes, τF°, at 77°K suggest that the exciton levels have similar transition moments with small forbidden character for the low‐energy component. The dramatic enhancement of phosphorescence in the series is due to several factors, namely, an increase in intersystem crossing rate constant kisc, an increase in phosphorescence rate constant kP, and a decrease in radiationless triplet decay constant kP′. Although the electronic overlap is small enough for the exciton model to be a valid approximation to explain singlet absorption intensities and fluorescence lifetimes, the overlap must be taken into consideration to account for the enhancement of spin—orbit coupling down the series. The molecular geometry of diphenylmethane consistent with the fluorescence lifetime data and the observed exciton splitting is described.

Photoionization and Absorption Spectrum of Formaldehyde in the Vacuum Ultraviolet

J. E. Mentall, E. P. Gentieu, M. Krauss, and D. Neumann

J. Chem. Phys. 55, 5471 (1971); http://dx.doi.org/10.1063/1.1675711 (9 pages) | Cited 70 times

Online Publication Date: 9 December 2003

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Absorption and photoionization coefficients have been measured for H2CO in the 600–2000‐Å region. Integrated oscillator strengths were determined for a number of strong Rydberg transitions above 1200 Å. From the photoionization curve the first adiabatic ionization potential was found to be 10.87±0.01 eV. As an aid in interpreting the absorption spectrum, theoretical calculations were made using a single‐configuration self‐consistent field procedure for the Rydberg states and a model which included mixing between the Rydberg and valence states. On this basis, weak absorption features between 1340 and 1430 Å have been assigned to the 1B1(σ→π*) valence state. The 1A1(π→π*) valence state is deduced to be strongly autoionized just above the 2B2 ionization limit.

Determination of Proton Affinity from the Kinetics of Proton Transfer Reactions. I. Relative Proton Affinities

A. E. Roche, M. M. Sutton, D. K. Bohme, and H. I. Schiff

J. Chem. Phys. 55, 5480 (1971); http://dx.doi.org/10.1063/1.1675712 (5 pages) | Cited 29 times

Online Publication Date: 9 December 2003

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A study of the kinetics of selected proton transfer reactions proceeding in a flowing afterglow at 300°K has led to the proton affinity order PA(CO) > PA(N2O) > PA(CH4) > PA(CO2) > PA(NO) > PA(N2) > PA(H2) > PA(O2) > PA(Ar) and the order PA(CH4) > PA(CF4) > PA(N2). A comparison of these orders with established values for proton affinity reported in the literature has led to the following proton affinity limits: 3.4 eV<PA(Ar) <4.2 eV; 4.9 eV<PA(N2) <5.5 eV; 5.5 eV<PA(N2O) <6.1 eV; and 4.9 eV<PA(CF4) <5.5 eV. In most cases these limits represent a considerable improvement over previously available information on proton affinities.

Electron Spin Resonance Line Shapes in Partially Oriented Systems

Pier Luigi Nordio and Paola Busolin

J. Chem. Phys. 55, 5485 (1971); http://dx.doi.org/10.1063/1.1675713 (6 pages) | Cited 115 times

Online Publication Date: 9 December 2003

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The relaxation theory for paramagnetic molecules dissolved in liquid crystals, having the same elongated shape of the solvent molecules so as to act as probes of their dynamical behavior, has been revised. The diffusion equation is solved as a series expansion of Wigner rotation matrix elements, employing a simplified orientating potential, and the solution is used to evaluate the effects of the molecular fluctuations on the electron spin resonance line shapes. The irreducible components of the dipole and g interaction tensors relax with a spectrum of characteristic times, which are functions of an alignment parameter and cause any contribution to the linewidth to vanish in the limit of complete ordering.

Diatomic Vibrational Potential Functions from Integration of a Poisson Equation

Alfred B. Anderson and Robert G. Parr

J. Chem. Phys. 55, 5490 (1971); http://dx.doi.org/10.1063/1.1675714 (4 pages) | Cited 14 times

Online Publication Date: 9 December 2003

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It is shown how to generate a diatomic vibrational potential W(R) from a knowledge of the effective electron density function F(R) which enters the Poisson equation ∇2W = F(R). Namely,
math
Using the fact that F(R) is at least roughly interpretable as proportional to the electron density at one nucleus due to the electron density following the other nucleus during vibration [A. B. Anderson and R. G. Parr, J. Chem. Phys. 53, 3375 (1970)], various reasonable forms for F(R) are proposed and tested. F(R) proportional to R−4 leads to a Fues potential. F(R) proportional to R−1exp(−ηR) leads to a Hellmann potential. The simple assumption that F(R) is proportional to exp (−ζR) leads to a new potential which is a linear combination of terms R−1, exp(−ζR), and R−1 exp(−ζR). Predictions of these three potentials are compared.

Nanosecond Time‐Resolved Emission Spectroscopy of 1‐Anilino‐8‐Naphthalene Sulfonate

Saroj K. Chakrabarti and William R. Ware

J. Chem. Phys. 55, 5494 (1971); http://dx.doi.org/10.1063/1.1675715 (5 pages) | Cited 46 times

Online Publication Date: 9 December 2003

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Nanosecond time‐resolved emission spectroscopy and fluorescence lifetime measurements are reported for 1‐anilino‐8‐naphthalene sulfonate. Time‐dependent spectral shifts are observed that are interpreted in terms of dipole reorientation of the solvent around the electronically excited molecule. Both the radiative and nonradiative transitions probabilities are observed to be strongly solvent dependent.

Classical Study of Rotational Excitation of a Rigid Rotor: Li+ + H2

Robert A. La Budde and Richard B. Bernstein

J. Chem. Phys. 55, 5499 (1971); http://dx.doi.org/10.1063/1.1675716 (18 pages) | Cited 91 times

Online Publication Date: 9 December 2003

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Classical calculations are presented for a model problem simulating the rotational excitation of H2 (and D2) in collision with Li+, based upon the ab initio potential surface of Lester. The hydrogen molecule was taken to be a rigid rotor, even though this assumption becomes progressively worse as the collision energy is increased and the vibrational excitation threshold is crossed. However, the results should serve as a set of reference calculations on a well‐defined system, suitable for comparison with semiclassical and quantal treatments. The influence of initial rotational state, molecular mass, and collision energy upon the angular distribution of the average rotational energy transfer and total inelasticity is evaluated and discussed. Most of the trends are due to simple dynamical effects and they also occur in computations using a rigid ellipsoid model for the diatomic molecule.

Molecular Collisions. XVI. Comparison of GPS with Classical Trajectory Calculations of Rotational Inelasticity for the Ar☒N2 System

M. D. Pattengill, R. A. La Budde, R. B. Bernstein, and C. F. Curtiss

J. Chem. Phys. 55, 5517 (1971); http://dx.doi.org/10.1063/1.1675717 (6 pages) | Cited 85 times

Online Publication Date: 9 December 2003

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The classical limit of the ``infinite order'' generalized phase shift (GPS) treatment of rotationally inelastic atom‐molecule collisions was put into computationally feasible form in Paper XV of this series. It is now applied to a model problem intended to approximate thermal scattering of the Ar☒N2 system (collisional energy of mathkT at 300°K), at the same time comparing its predictions with exact classical trajectory (CT) results. This comparison indicates that the present version of the GPS method over‐estimates the rotational excitation and underestimates the de‐excitation, while maintaining the total inelasticity at approximately the correct (CT) value. An approximate ``quantization'' of the classical results leads to an estimate of the quantal cross sections corresponding to changes by ±2, ±4, and ±6 from an initial rotor quantum state math = 10. It is found that most of the total inelastic crosssection (of some 32 Å2) arises from the first‐order‐allowed transitions (Δl = ±2).

Continuous Wave Carbon Monoxide Chemical Laser

Curt Wittig, J. C. Hassler, and P. D. Coleman

J. Chem. Phys. 55, 5523 (1971); http://dx.doi.org/10.1063/1.1675718 (10 pages) | Cited 16 times

Online Publication Date: 9 December 2003

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A cw carbon monoxide chemical laser is discussed. Experimental results are given for a longitudinal flow system in which partially dissociated oxygen, in a helium carrier, is mixed with carbon disulfide in an optical cavity to produce vibrationally excited CO. Laser action occurs in highly excited vibrational levels of CO with the transitions originating from v=10 and v=9 being strongest. The system is analyzed using a static model with the result that the vibrational level pairs that exhibit the strongest laser action have average population densities that are roughly equal under lasing conditions. A dynamic model is proposed that invokes chemical excitation and V‐V relaxation as the dominant excitation and relaxation processes, respectively. Calculations using this model explain limitations imposed on the system by the fluid mixing geometry and agree qualitatively with experiment.

Spin‐Free Computation of Matrix Elements. II. Simplifications Due to Invariance Groups

B. R. Junker and D. J. Klein

J. Chem. Phys. 55, 5532 (1971); http://dx.doi.org/10.1063/1.1675719 (11 pages) | Cited 9 times

Online Publication Date: 9 December 2003

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The spin‐free equivalents of the antisymmetrized Löwdin projector, the valence‐bond structure projector, and a number of other symmetry projectors are shown to be invariant to within a phase under the operation of elements of certain subgroups of the spin‐free symmetric group, SN, associated with a spin‐free N‐electron system. These so‐called invariance groups are used to demonstrate equalities among the Pauling numbers thereby reducing the number of Pauling numbers which must be computed. In addition the subgroup of SN whose elements leave a basis ket invariant to within a phase factor are described and used to eliminate redundancy of molecular integrals in the expression for expectation values. By the use of invariance groups of both the projectors and kets, matrix elements over spin‐free operators are simplified by eliminating redundancy with respect to both the Pauling numbers and the molecular integrals.

Theory of Separability of Many‐Electron Systems

S. Huzinaga and A. A. Cantu

J. Chem. Phys. 55, 5543 (1971); http://dx.doi.org/10.1063/1.1675720 (7 pages) | Cited 117 times

Online Publication Date: 9 December 2003

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Atomic and molecular systems are often intuitively separated into almost independent subsystems as, for example, the core and valence parts of an atom. Consequently, if this separation provides a good approximation, one can obtain the states of the system from the states of the subsystems which best represent the entire system. In the light of the work of McWeeny, in which one assumes strong orthogonality among subsystem wavefunctions, we determine an effective Hamiltonian for a given subsystem which should properly describe the states of that subsystem. Previous work is shown to have dealt with an improper effective Hamiltonian.

Electron Transfer and Ion—Atom Interchange in C+�CO Collisions

K. E. Maher, G. J. McClure, and J. J. Leventhal

J. Chem. Phys. 55, 5549 (1971); http://dx.doi.org/10.1063/1.1675721 (2 pages) | Cited 2 times

Online Publication Date: 9 December 2003

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Kinetic‐energy distributions of CO+ and C2+ ions resulting from C+�CO collisions were measured over the incident ion kinetic‐energy range 13–30 eV. These distributions were used for calculation of the reaction energetics. The CO+ kinetic‐energy distributions provide strong evidence that this product is formed by both electron transfer and ion—atom interchange. In addition to the major (forward scattered) peak in the C2+ distribution small yields of C2+ were detected at 180° in the center‐of‐mass system.

Vibrational Relaxation of Hydrogen Bromide in Gaseous Hydrogen Halide Mixtures

Hao‐Lin Chen

J. Chem. Phys. 55, 5551 (1971); http://dx.doi.org/10.1063/1.1675722 (6 pages) | Cited 48 times

Online Publication Date: 9 December 2003

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A pulsed HBr chemical laser has been used to vibrationally excite HBr molecules in gaseous mixtures to their first vibrational excited state. The decay of infrared fluorescence following the laser pulse was studied. Cross sections for vibrational deactivation of HBr by HBr, HCl, n‐H2, p‐H2, and rare gases at room temperature have been found:
math
The cross sections for vibration‐to‐vibration energy transfer from HBr to HCl and HI are σHBr�HCl = (4.56±0.43)×10−2 Å2 and σHBr�HI = (1.86±0.24)×10−1 Å2. The enhancement of probability for hydrogen halides at room temperature in Landau—Teller plots evident from extrapolating shock tube data available between 1800 and 800°K shows the importance of intermolecular forces. Direct comparison of vibrational relaxation rates among hydrogen halides (HCl, HBr, and HI) shows that the HCl�HCl interaction possesses a deeper attractive and steeper repulsive potential than HBr�HBr and HI�HI interactions. The small difference in cross section among rare gases indicates the unimportance of translational motion. Since equal cross sections are observed for n‐H2 and p‐H2, it is unlikely that energy is transferred into rotation of H2.

Vibration‐to‐Vibration Energy Transfer in Hydrogen Bromide Mixtures

Hao‐Lin Chen

J. Chem. Phys. 55, 5557 (1971); http://dx.doi.org/10.1063/1.1675723 (4 pages) | Cited 16 times

Online Publication Date: 9 December 2003

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The laser‐excited vibrational fluorescence technique has been used to determine the rate constants for vibration‐to‐vibration energy transfer from HBr to other molecules in gaseous mixtures. Cross sections at room temperature have been found:
math
The cross section for vibration‐to‐rotation or translation energy transfer between HBr and D2 has also been determined as σHBr�D2 = (0.96±0.1)×10−4 Å2. The dependence of cross section on V→V energy level differences is about the same magnitude as in the analogous HCl systems but considerably larger than it is in other diatomic—diatomic systems. Strong intermolecular forces and larger amplitudes of hydrogen atom stretching vibration are likely to be the dominant reason for the observed phenomena.

Study of the Lifetimes of Individual Vibronic States of the Isolated Benzene Molecule

Kenneth G. Spears and Stuart A. Rice

J. Chem. Phys. 55, 5561 (1971); http://dx.doi.org/10.1063/1.1675724 (21 pages) | Cited 180 times

Online Publication Date: 9 December 2003

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In this paper we report measurements of the lifetimes of 22 vibronic states of benzene and the relative quantum yields of fluorescence for transitions originating from these states. The measurements, which are all made in the free molecule limit, are used to test the theory of radiationless processes. It is found that the available theory provides a moderately good description of the observations. Our principal observation is that the nonradiative lifetime of the prepared state decreases with increasing excitation over the origin in such a way that the decrement in nonradiative lifetime per quantum of vibrational energy is approximately constant. The excitation of progressions in low frequency out‐of‐plane modes decreases the nonradiative lifetime per unit energy more than do excitations of progressions in, say, the totally symmetric mode. We also discuss the pressure dependence of the lifetime of a prepared state and its relationship to molecular properties.

Dissociative Excitation of Vacuum‐Ultraviolet Emission Features by Electron Impact on Molecular Gases. II. N2

M. J. Mumma and E. C. Zipf

J. Chem. Phys. 55, 5582 (1971); http://dx.doi.org/10.1063/1.1675725 (7 pages) | Cited 38 times

Online Publication Date: 9 December 2003

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Atomic‐nitrogen multiplets which radiate in the vacuum ultraviolet have been excited by electron impact on N2. Their excitation cross sections have been measured over the energy range from threshold to 350 eV. The transition arrays, mean wavelengths, and cross sections for dissociative excitation of these multiplets at 100 eV are: N(2p34S0−2p23s4 4P) 1200 Å−6.70×10−18 cm2±18%; N[2p3 2D0−2p2(1D)3s′2D] 1243 Å−1.45×10−18 cm2±22%; and N(2p3 2D0−2p23s 2P) 1493 Å−2.53×10−18 cm2±21%. In addition, the cross sections of the following multiplets have been measured at 100 eV only: N(2p3 2D0−2p24s 2P) 1177 Å−5.16×1019 cm2±22%; N(2p3 2D0−2p23d 2D) 1164 Å−1.63×10−19 cm2±23%.

Statistical Mechanics of Systems with Internal Constraints: Rubber Elasticity

Karl F. Freed

J. Chem. Phys. 55, 5588 (1971); http://dx.doi.org/10.1063/1.1675726 (12 pages) | Cited 5 times

Online Publication Date: 9 December 2003

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The general formulation of the statistical mechanical description of systems with internal constraints is recast into a form in which the ensemble average is written as a summation over the initial and final microscopic states of the system. The functional integral formulation previously developed to treat polymer networks is generalized to include these permanent constraints. The statistical mechanics of rubber elasticity is considered using a simple model in which the cross linkages are taken to be permanent topological constraints on the polymer network; these linkages are taken to be at the same places along the chains in both the initial and the deformed states. Entanglements are ignored, and the elasticity of this ``phantom'' network is obtained by using a variational principle for the free energy. The analog of the average affine deformation of the network is obtained as a variational result of the calculation. The elastic modulus obtained is in agreement with Flory's theory; however, the present statistical mechanical formulation is capable of also considering the effects of interactions between different polymer chains, chain stiffness, etc., and closed circuits in the polymer network.

Theory of Inelastic Collisions: Uniform Asymptotic (WKB) Solutions and Semiclassical S‐Matrix Elements for Two‐Channel Problems

Byung Chan Eu

J. Chem. Phys. 55, 5600 (1971); http://dx.doi.org/10.1063/1.1675727 (10 pages) | Cited 26 times

Online Publication Date: 9 December 2003

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Semiclassical scattering matrix elements for two‐channel problems are obtained by means of uniform asymptotic (WKB) solutions and solving a set of coupled first‐order differential equations generated from the radial Schrödinger equations. By solving such first‐order coupled equations, Stueckelberg's mathematically unsatisfactory procedure of tracing solutions in a complex plane to obtain the S‐matrix elements is completely removed and the expression thus obtained for the transition probability is devoid of physically unreasonable shortcomings of Stueckelberg's result. The S‐matrix elements obtained are given in terms of well‐defined quadratures easy to calculate and valid for any types of the interaction matrix elements, Vii (x) and w (x). The transition probability obtained from the S‐matrix elements is shown to reduce essentially to the well‐known Landau—Zener expression as a limiting case when w (x) is small.
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