Partial wave

Since we are dealing with spherical symmetric potentials, the solution to the Schrodinger equation inside the spheres can be written as  

The functions Y/(r) are the well known spherical harmonics [41] which are purely angular functions. The functions 4 rl (E, r) are the solutions to the radial Schrodinger 

What we are essentially looking for is the asymptotic behavior of the solution of Eq. (142) with F(r) = F(r). In this event tpj is a function only of 6 and r, as the scattering is symmetric about the polar axis z. But we delay consideration of the general spherically symmetric potential while we observe the effect of the simpler potential, F(r) = 0, since the solution of the equation 

Thus the A, are called phase shifts, as they are the difference in phase between the asymptotic behaviors of the solutions of the equations with and without U r). 

Explicitly then, the wave function y) (r,d), which represents an incident and scattered wave, is 

Bauer and Wu have applied a method involving the phase shifts A,. But they find that their results for the reaction of Eq. (162) are approximately the same whether the phase shifts are eliminated from or included in their computation. Setting the A, equal to zero is tantamount to a Bom approximation. Therefore, the negligible dependence of the results on A, supports in a back-handed way the prior work of Golden and Peiser. This support, however, is tenuous because of the differences in the models of the two approaches. 

Golden and Peiser use the Eyring-Polanyi potential, whereas Bauer and Wu use an even more simphfied potential. Furthermore, Bauer and Wu treat the linear case and do not consider rotation. A major difference is that whereas Golden and Peiser bypass the activated complex and compute transitions between initial and final states, Bauer and Wu have taken the activated complex made up of Hj and Br as an actual state and have computed probabilities from the initial state to this activated state . Then they have merely chosen a transmission coefficient of for the transition from the activated state to the final state. Undoubtedly, if the interaction potential is well known, the introduction of the activated state is an unnecessary complexity, but Bauer and Wu may well be justified in their claim that the interaction potential is sufficiently mysterious at present to warrant the crutch. 

---

Problems in chemical physics which involve the collision of a particle with a surface do not have rotational synnnetry that leads to partial wave expansions. Instead they have two dimensional translational symmetry for motions parallel to the surface. This leads to expansion of solutions in terms of diffraction eigenfiinctions. 

We call this a partial M/ave expansion. To detennine tire coefficients one matches asymptotic solutions to the radial Scln-ddinger equation with the corresponding partial wave expansion of equation (A3.11.106). It is customary to write the asymptotic radial Scln-ddinger equation solution as... 

The asymptotic partial wave expansion of equation (A3.11.106) can be developed using the identity... 

Equations A3.11.114(b) and A3.11.115(b) are in a fonn that is convenient to use for potential scattering problems. One needs only to detemiine the phase shift 5 for each i, then substitute into these equations to detemiine the cross sections. Note that in the limit of large i, finiist vanish so that the infinite sum over partial waves iwill converge. For most potentials of interest to chemical physics, the calculation of finiist be done numerically. 

The cross section a -is related to the partial wave reactive scattering matrix , tln-ough the partial wave sum... 

Theoretically, the asymptotic fonn of die solution for the electron wave fiinction is the same for low-energy projectiles as it is at high energy however, one must account for the protracted period of interaction between projectile and target at the intennediate stages of the process. The usual procedure is to separate the incident-electron wave fiinction into partial waves... 

The partial wave decomposition of the incident-electron wave provides the basis of an especially appealing picture of strong, low-energy resonant scattering wherein the projectile electron spends a sufficient period of time in the vicinity... 

B2.2.6.7 PARTIAL WAVE EXPANSION FOR TRANSPORT CROSS SECTIONS... 

Wlien the atom-atom or atom-molecule interaction is spherically symmetric in the chaimel vector R, i.e. V(r, R) = V(/-,R), then the orbital / and rotational j angular momenta are each conserved tln-oughout the collision so that an i-partial wave decomposition of the translational wavefiinctions for each value of j is possible. The translational wave is decomposed according to... 

A partial wave decomposition provides the frill close-coupling quantal method for treating A-B collisions, electron-atom, electron-ion or atom-molecule collisions. The method [15] is siumnarized here for the inelastic processes... 

MoCullough E A Jr 1975 The partial-wave self-oonsistent method for diatomio moleoules oomputational formalism and results for small moleoules J. Chem. Phys. 62 3991-9... 

We use s, p, and d partial waves, 16 energy points on a semi circular contour, 135 special k-points in the l/12th section of the 2D Brillouin zone and 13 plane waves for the inter-layer scattering. The atomic wave functions were determined from the scalar relativistic Schrodinger equation, as described by D. D. Koelling and B. N. Harmon in J. Phys. C 10, 3107 (1977). 

PECD in Camphor Convergence of the Partial Wave Expansion 1. Data Acquisition and Analysis Molecular Conformation and Substitution Effects 1. Data Acquisition and Treatment... 

In Eq. (12), l,m are the photoelectron partial wave angular momentum and its projection in the molecular frame and v is the projection of the photon angular momentum on the molecular frame. The presence of an alternative primed set l, m, v signifies interference terms between the primed and unprimed partial waves. The parameter ct is the Coulomb phase shift (see Appendix A). The fi are dipole transition amplitudes to the final-state partial wave I, m and contain dynamical information on the photoionization process. In contrast, the Clebsch-Gordan coefficients (CGC) provide geometric constraints that are consequent upon angular momentum considerations. 

The partial wave basis functions with which the radial dipole matrix elements fLv constructed (see Appendix A) are S-matrix normalized continuum functions obeying incoming wave boundary conditions. 

Substitution of Eq. (18) into the partial wave expansion formula [Eq. (A.3)] gives for the continuum state... 

The last two CGC in Eq. (12) evidently dictate that rather different partial wave interference contributions are made to each of the angular parameters. This will impact on the dynamical information conveyed by each one. Equally important, the phase subexpression... 

A common finding of computational PECD studies is that a relatively large partial wave expansion, typically running to niax > 15 is required. Chiral molecules necessarily are of very low, or no, symmetry, and hence are quite... 

Following normal practice, it is convenient to replace the continuum fimction in Eq. (A.l) or (A.2) with an incoming wave normalized partial wave expansion [39, 40, 118] ... 

---