In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time−dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many−particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton’s equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State−specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H2 system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.

Time‐dependent approach to semiclassical dynamics

A fourier method solution for the time dependent Schrödinger equation as a tool in molecular dynamics

A simple model is presented for calculating the forces between closed‐shell atoms and molecules in the regions both of the attractive well and of the repulsive wall at shorter distances. Account is taken of both the overlap of the separate atomic densities and of electron correlation. Applications to pairs of rare gas atoms and to alkali halide molecules demonstrate quantitative agreement with empirically determined intermolecular potentials for these systems over the whole range of separations inside and including the potential minimum.

Theory for the Forces between Closed‐Shell Atoms and Molecules

The Arthurs and Dalgarno space‐fixed (SF) axes formulation of the quantum theory of atom‐diatom scattering is compared with the body‐fixed (BF) axes formulation of Curtiss using consistent notation to facilitate the comparison. While equivalent, the two theories are not always equally convenient. When rotation is treated in a sudden approximation, the BF formulation has a tremendous conceptual and computational advantage: It allows an infinite‐order sudden approximation, independent of the form of the potential energy, which should be very helpful in vibrationally inelastic and reactive scattering problems. Also, a rapid procedure for calculating WKB phase shifts is presented.

Space‐fixed vs body‐fixed axes in atom‐diatomic molecule scattering. Sudden approximations

New coupled equations describing collisions of an atom and a diatomic molecule are derived in this paper. By utilizing a description of the collision in terms of rotating coordinates, all coupling in the z component of angular momentum is isolated into purely kinematic effects. By neglecting these couplings, one is led to approximate equations for which the jz component of angular momentum for the molecule is conserved. In addition, the scattering cross sections are formulated by neglecting the effect on the wavefunction of the rotation of the coordinate axes so that in place of Wigner rotation matrices dmmJ (Θ) appearing, one deals with simple Legendre polynomials and the orbital angular momentum l2 is approximated by l(l + 1) ℏ2. It is noted that the procedure involves no approximations so far as the potential matrix elements are concerned. Furthermore, the number of equations remaining coupled is drastically reduced and a completely quantum mechanical description of the dynamics of both internal states and relative motion is retained. The physical implications of the approximations are examined, and it is seen that the neglect of intermultiplet coupling gives rise to consideration of only transitions where both the orientation and magnitude of the rotor angular momentum change. Further, the neglect of transformation effects on the wavefunction is expected to be least accurate for the inelastic forward scattering and best for backward scattering and the j =0→0 elastic scattering. Finally, the present simplest version of the approximation obviously is not intended for treating processes dependent on mj transitions, e.g., NMR relaxation in He–H2. Next the formalism is applied in test calculations to He–H2 collisions using the Krauss‐Mies potential energy surface. Numerical results for elastic and inelastic integral and differential cross sections are compared with exact quantum mechanical close coupling solutions of the standard coupled channel equations. Over the energy range studied (from 0.1 eV up to 0.9 eV), agreement to within a few percent is obtained. Additional coupled states calculations are reported at 1.2 eV and computation times are compared against those required for a full close coupling solution. Calculations for the Roberts He–H2 surface are also reported to illustrate the independence of the approximations on the strength of the coupling (so long as the inelastic scattering is predominantly in the backward direction). The dramatic savings afforded by the present approach are such as to make possible fully converged calculations at collision energies typically studied in molecular beam experiments. Thus, for elastic and inelastic nonreactive collisions, involving a repulsive‐type interaction, the approach makes the a priori quantum mechanical description of the scattering of a diatom by an atom practical.

Quantum mechanical close coupling approach to molecular collisions. jz ‐conserving coupled states approximation

Exact quantum‐mechanical calculations of the transition probabilities for the collinear collision of an atom with a diatomic molecule are performed. The diatomic molecule is treated as a harmonic oscillator. A range of interaction potentials from very hard to very soft are considered. It is found that for ``realistic'' interaction potentials the approximate calculations of Jackson and Mott are consistently high, even when the transition probabilities are low and good approximate results are expected. In some cases double and even triple quantum jumps are more important than single quantum jumps. Comparisons are made with exact classical calculations. A semiempirical formula is given for computing quantum‐mechanical transition probabilities from classical calculations.

Exact Quantum-Mechanical Calculation of a Collinear Collision of a Particle with a Harmonic Oscillator

A systematic method is discussed for decoupling the internal angular momentum of molecules involved in a collision from their relative angular momentum. This leads to a large class of rotational approximations of varying degrees of complexity and accuracy. These approximations may be used directly for computing rotational transitions or they may be used for reducing the rotational complexity involved in accurate vibrational calculations. It is shown how this approach may be used to study the infinite−order sudden approximation and how that approximation may be extended to more complex potentials. It is shown also how one may use results of the jz−conserving approximation to obtain more complete information on the scattering matrix. The present approach may be used to deduce new angular momentum decoupling approximations and analyze such approximations arrived at through other considerations.

Theory of angular momentum decoupling approximations for rotational transitions in scattering

Wavefunctions and interaction energies are calculated for the helium‐atom–hydrogen‐molecule system at a wide range of internuclear separations. Letting X represent the distance measured along a line drawn from the helium nucleus to the midpoint of the hydrogen molecule, R the H2 bond distance, and θ the angle between X and R, results are reported for 2.8 ≤ X ≤ 7.0 a.u., 1.0 ≤ R ≤ 1.8 a.u. (Re = 1.4 a.u.), and 0° ≤ θ ≤ 90°. The orbital basis set consisted of Slater s and p orbitals, and all integrals are accurately evaluated. The wavefunctions are constructed by each of the two major approaches to molecular bonding; the self‐consistent‐field molecular‐orbital method of Roothaan and the configuration‐interaction method. The value of the two methods in atom–diatomic‐molecule interactions is compared extensively. For each individual method, the effect of the size of the basis set on the resultant interaction energy is examined. The values for the He–H2 interaction energies obtained by the self‐consistent‐field and configuration‐interaction method are very similar. The self‐consistent‐field method appears adequate for the calculation of interaction energies for closed‐shell noninteracting systems except at large center‐of‐mass separations. The interaction energies are accurately fit to an analytic expression for the intermolecular potential. Configuration‐interaction calculations performed at large center‐of‐mass separations fail to locate the He–H2 van der Waals minimum.

Helium-atom--hydrogen-molecule potential surface employing the lcao--mo-- scf and ci methods.

A practical method is presented for computing the T‐matrix elements for a rotating, vibrating oscillator. A simple model is used which approximates the features of the He–H2 system. It is found that the rigidrotor approximation is in error even at energies well below the threshold for vibrational excitation. For computation of rotational transitions from the ground to the first accessible excited rotational state, many of the excited rotational transitions may be neglected but some of the excited vibrational transitions must be included. At high energies, it is shown that for any particular transition many of the states not strongly coupled to the states involved in the transition may be neglected. It is found that the computation of T‐matrix elements for vibrational transitions in the presence of rotational transitions is not prohibitively time consuming. In computing the total cross section, it is shown that a calculation including only the ground state gives remarkably good results.

Calculation of Rotational and Vibrational Transitions for the Collision of an Atom with a Rotating Vibrating Diatomic Oscillator

A generalization of the log‐derivative method is presented which is useful for both reactive and nonreactive scattering problems. In the coupled system of radial equations for this problem a first derivative term is included for complete generality. Thus, this method may be used when, as is often the case in reactive or curve crossing problems, the equations contain a first derivative term. When no first derivative term is present and no reactive channels are present, the method reduces to the standard log‐derivative method. A reactive scattering problem is solved as an example.

Don Secrest

Papers

Exact Quantum-Mechanical Calculation of a Collinear Collision of a Particle with a Harmonic Oscillator

Theory of angular momentum decoupling approximations for rotational transitions in scattering

Helium-atom--hydrogen-molecule potential surface employing the lcao--mo-- scf and ci methods.

Calculation of Rotational and Vibrational Transitions for the Collision of an Atom with a Rotating Vibrating Diatomic Oscillator

The generalized log‐derivative method for inelastic and reactive collisionsa)