Local and hyperspherical mode approximations to resonances in collinear atom–diatom reactions

Abstract

Approximate hindered asymmetric top wavefunctions for atom-diatomic molecule reactionsA collinear, symmetric reactive scattering system which exhibits a highly oscillatory reaction probability as a function of collision energy has been studied. The A + BA model consists of two coupled Morse oscillators. The peaks in the reaction probability are due to resonances which have been analyzed using local and hyperspherical mode bound state expansions. In a small range of collision energy, resonances are found some of which are local and some hyperspherical in character. Resonance energies calculated using bound state techniques are compared to those from exact quantum mechanical scattering calculations.3088

Isotope Effects of Reactions in Quantum Solids Initiated by IR + UV Lasers: Quantum Model Simulations for Cl(2P3/2) + X2(ν) → XCl + X in X2 matrices (X = H, D)

“…Here, one makes the ansatz 60 with quantum numbers ν φ ) 0, 1, 2, ... and ν r ) 0, 1, 2, ... for motions along φ and r, respectively. Solving the hyperspherical angular equation for fixed value of r, this ansatz yields the wave functions Ψ ν φ (φ;r) as well as the adiabatic potential curves V ν φ (r) for motions along the hyperspherical radius r. 27,29,31,[59][60][61][62][63][64][65][66][67]69,70,72 The V ν φ (r) are ordered with increasing energies…”

“…31,73 Otherwise it can decay only by transitions to lower V ν′ φ (r) with ν′ φ < ν φ ; then it is called Feshbach resonance. Bound states or resonances with levels labeled E ν φ ,ν r )0 are called hyperspherical modes; [60][61][62]64,66,67,70,72,74 their BOA2 wave functions (43) extend along periodic orbits [75][76][77][78] (see also refs 79 and 80) that resemble arcs with approximately constant values of r, and ν φ nodes along φ but without any nodes along r. Of course, the exact solutions include couplings between the BOA2 states labeled ν φ and ν r , and these introduce level shifts as well as distortions of the BOA2 wave functions (43). Nevertheless, often one of them is dominant, and then the quantum numbers ν φ and ν r may be assigned to the exact wave functions as well.…”

“…In contrast, for large values of r , the angle ϕ describes vibrations of X 2 as well as XCl. Moreover, since motions along r involve all atoms including the heavy Cl, wheras motion along ϕ involves exclusively the light atom X, it is convenient to employ a separation of r and ϕ that may be called the second Born−Oppenheimer separation, analogous to the here called first BO separation of the coordinates Q and q describing the motions of the nuclei and electrons, respectively; see also refs , , , and – . The well-known first BO separation of Q and q suggests the first Born−Oppenheimer approximation (BOA1): here the wave functions for electrons and nuclei are approximately written as products of electronic times nuclear wave functions Ψ el,nu ( q , Q ) ≈ Ψ el ( q ; Q ) × Ψ el,nu ( Q ), with quantum numbers “el” and “nu”, respectively.…”

“…Here, one makes the ansatz Ψ ν ϕ , ν r ( ϕ , r ) ≈ Ψ ν ϕ ( ϕ ; r ) × Ψ ν ϕ , ν r ( r ) with quantum numbers ν ϕ = 0, 1, 2, ... and ν r = 0, 1, 2, ... for motions along ϕ and r , respectively. Solving the hyperspherical angular equation for fixed value of r , this ansatz yields the wave functions Ψ ν ϕ (ϕ; r ) as well as the adiabatic potential curves V ν ϕ ( r ) for motions along the hyperspherical radius r . ,,,– ,,, The V ν ϕ ( r ) are ordered with increasing energies V 0 ( r ) < V 1 ( r ) < V 2 ( r ) < ... for ν ϕ = 0, 1, 2, .... Asymptotically, for r → ∞, the potential curves V ν ϕ ( r ), approach the vibrational levels of the reactants E X 2 (ν) and products E XCl (ν′); see eqs and . Accordingly, these levels may be renamed E ν ϕ , in energetic order.…”

Application of box quantization to the determination of the dissociation rates of a system of two coupled morse oscillators

“…(4) is designated by $I ( p , a ) , it obeys for a bound state the normalization condition It is convenient [26] to define an auxiliary function +( p , a ) by the relation $4 P , a ) = P-"2+( P9 a> (7) which is fulfilling the normalization. (9) It is convenient to scale the coordinate p according to with h2 = h/8m2pc i, = hp c being the velocity of light.…”