Bimolecular recombination reactions: K-adiabatic and K-active forms of the bimolecular master equations and analytic solutions

Both Fredholm and Hilbert start from the corresponding linear system $fˆ=(I−λKˆ)gˆ$⁠, where $Kˆ$ is a square matrix, and $fˆ$⁠, $gˆ$ are vectors. Fredholm investigated when λ = − 1 is an eigenvalue of arbitrary multiplicity of the kernel using operation on functions. Unlike Fredholm, Hilbert first developed a theory for linear systems and eigensystems and then by a limiting process generalized the theory to the Fredholm equation. Schmidt worked directly with the integral equation and introduced what is now called the singular value decomposition for unsymmetric kernels.

Information on the intermolecular collisional energy transfer ΔE in bimolecular reactions may be obtained from classical trajectories to obtain the relevant parameters for the transfer probability function, P(E′, E) = N(E′, E)/N_eff/δE, where N(E′, E) is the number of trajectories with energy transfer ΔE = E′ − E, δE is the energy interval whereupon N(E′, E) is evaluated, and N_eff are the number of trajectories within a desired cross section, as noted and investigated in Refs. 40 and 41.

A recurrence relation is an equation that expresses a function f_n as some combination of f_i with i < n. Once the initial condition(s) are specified, then each further term in the sequence or array is specified as a function of the preceding terms in a recursive manner. The term “difference equation” is also used. For further discussion, please see Refs. 44 and 45.

Upon noting the definition of λ from Eq. (19), then $λ∫Eo∞Z(E,E′)dE<1$ when k_d≠0 in λ, and so $1−λ∫Eo∞Z(E,E′)dE′≠0$⁠. k_d > 0 is a possibility for energies above the critical energy limit, E > E^*(JK), whereby energy in excess of that locked in the K-motion is available for a bond dissociation, and where the energy condition, E = E_rot + E_orb + E_vib + E_trans + E_o is satisfied, e.g., for an A + BC → ABC process and E_rot, E_orb, E_vib, E_trans, and E_o are the rotational, orbital, vibrational, translational, and zero point energies, respectively, at the TS and elsewhere. For the center-to-center distance R, a reaction coordinate, as R → ∞, then E_orb → 0.

In Eq. (27), f(EJK) > 0, $λAˆ>0$⁠, and $λ1Cˆ>0$⁠, where necessarily E > E^*(JK) so to have k_d(EJK) > 0 to avoid a singularity in $Aˆ$⁠, and for k_r(EJK) > 0 so to have both $Aˆ>0$ and f(EJK) > 0, where otherwise if k_d = 0 then, undesirably, $f(EJK)=0, Aˆ=0$⁠, and hence $f(EJK)+λAˆ=0$⁠, so to yield a singularity in the left hand side of Eq. (27).

In this case, k_d = 0 and k_r > 0. For a given (EJK), it may be assumed that both the associative and dissociative channels are open for energies greater than the critical energy E^*(JK), unless subtle geometric and dynamical effects would render either one open and the other closed to association or recombination for the same (EJK).

For the evaluation of the integral in the second term of Eq. (32), for the energy region E′ ≤ E, the lower limit of integration is the least value of E such that E > E^*(JK), so to validate the condition k_d(EJK) > 0, and its upper limit is E. For region E′ > E, the lower limit of the integral is a selected E, where the lowest permissible selected value would be again that given earlier for region E′ ≤ E, and its upper limit is ∞, or can be set, for practical purposes, to such a value that would ensure the convergence of the integral.

For a physical situation obeying the condition A(t)∘BC(t) ≫ g(EJK, t), whereby relatively the greater concentration of reactants A(t)∘BC(t) appears as a constant compared to a concentration of product molecules, then dA(t)∘BC(t)/dt ≅ 0 and thence for simplicity of a numerical calculation assumes A(t)∘BC(t) be independent of time in Eq. (35), and likewise via a similar argument for the K-active case, Eq. (36).

The analysis is as follows in obtaining the test cases: we compare the ratio of g(EJ)₊, Eq. (33) to g(EJK)₊, Eq. (29). In the analysis, for simplicity, the primes were dropped in k_r(E′J′K′), upon making an assumption offered following Eq. (32); however these simpler test cases (a) and (b) in the text give the same result as the ratio of g(EJ)₊, Eq. (33), to g(EJK)₊ when retaining all terms in the calculation. As Z_LJ → 0, then $limZLJ→0g(EJ)+=kr(EJ)/kd(EJ)$⁠, where the second term in g(EJ)₊ is zero. Similarly for the K-adiabatic case $limZLJ→0g(EJK)+=kr(EJK)/kd(EJK)$⁠. And so, $limZLJ→0[g(EJ)+/g(EJK)+]=[kr(EJ)/kd(EJ)]/[kr(EJK)/kd(EJK)]$⁠. Upon inquiring if the latter ratio is ≥1 then yields the test cases (a) and (b). The case for <1 was discussed in the text. As Z_LJ → ∞, the same ratio is also obtained, where now the first term is zero for either g(EJ)₊ or g(EJK)₊, and upon using L’Hopital’s rule (Ref. 80) on the second terms of g(EJ)₊ and g(EJK)₊. For the intermediate values of Z_LJ, again the same ratio [k_r(EJ)/k_d(EJ)]/[k_r(EJK)/k_d(EJK)] emerges as a viable test case, for physically realizable values of k_r, k_d, and Z_LJ, and the ratio may be used as a test case to obtain results on the ratio of populations for cases (a) and (b) in the text.

In the low pressure limit, the first and the second terms in the master equations, Eqs. (13) and (14), are each zero, since Z_LJ → 0 appears there, in which it leaves the third term in the master equations which is either k_d(EJK) g(EJK)₊ or k_d(EJ) g(EJ)₊ in Eqs. (13) and (14), respectively. In turn, the k_rec(EJK) A∘BC = k_d(EJK) g(EJK)₊ reduces to k_r(EJK) A∘BC, where the latter arises from the first term of g(EJK)₊, when the k_d(EJK) in the numerator canceled the k_d(EJK) in the denominator with Z_LJ = 0, while the second term in g(EJK)₊ is zero, since a Z_LJ appears in the Z(E, E′) there. Averaging over K in the above analysis then yields, similar, results for the K-active case.

For example, in Ref. 27, the analytic solution (population) from Wiener-Hopf technique was inserted in ∫∫g(E) Z(E′, E) dE′dE to obtain the low pressure recombination rate constant. The J and K degree of freedoms may be introduced in the latter formula.

In the case for comparing k_rec to that from classical trajectories, the N^* and ρ would be determined classically, e.g., discussed for the K-active and K-adiabatic cases in Ref. 73.

The average downward energy is defined as $ΔEEd=∫0E(E′−E)Z(E,E′)dE′/∫0EZ(E,E′)dE′$ for E′ < E. May see Ref. 36.

The L’Hopital’s rule (infinity-over-infinity case)⁸⁰ states that if f and g are defined and differentiable for all x larger than some fixed number, then, if $limx→∞f(x)=∞, limx→∞g(x)=∞$⁠, and $limx→∞f′(x)g′(x)=L$⁠, then it follows that $limx→∞f(x)g(x)=L$⁠. For further discussion, please see Ref. 80. The application of the rule to Eq. (G1) gives $limZLJ→∞g(EJK)+=dd ZLJ[∑J′K′∫Eo∞Z(E′,E)g(E′J′K′)dE′]/dd ZLJ[∫Eo∞Z(E,E′)dE′+kd(EJK)]$⁠, where the numerator in the latter equation yields Eq. (G2), and the denominator is equal to 1.