Theory of Dissociative Ionization Processes

A quantum mechanical theory for the ℏ ω + A B → A + B + + e - three body breakup is 
presented. The theory is based on the Continuum Coupled Channels expansion (CCC) 
in which the three body wave-function is expanded in terms of square integrable and 
(flux carrying) continuum states. The theory consistently treats the accompanying 
processes of three body inelastic collisions, dissociative attachment and Penning ionization. 
Applications to the dissociative photoionization of H2 and HD are presented. 
The relevant equations are solved exactly using the Artificial Channel method. Excellent 
agreement is obtained with experiment, concerning the proton kinetic energy 
distribution.


INTRODUCTION
When molecular hydrogen absorbs a photon with energy above the dissociative ionization threshold, four distinct outcomes are energeti- (Here E, denotes the kinetic energy of either a hydrogen atom or proton, and e the kinetic energy of an electron.) Indeed there is experimental evidence for the participation of the first two of these channels above the H2 three body threshold Eth 18.1 eV. [1][2][3][4] In view of these possibilities, a theoretical analysis of the fate of molecular hydrogen excited at these energies must consider interactions between varieties of two body and three body processes such as H(ls)+H(ns) --, H++H(ls)+e-Penning ionization H + 2 + e ----H(ls) + H(ns) Dissociative attachment H(ls)+H + +e-(e) -, H(ls)+H + +e-(e') Three-body inelastic scattering in the course of assigning a probability to an individual dissociation channel. These subtleties were confronted in a preliminary study of the dissociative ionization of Hz out of which emerged the theory of continuum coupled channels. 5 The theory was found to account successfully for the distribution of proton kinetic energies measured for the three body breakup of Hz at 26.9 eV. Here we consider some of the salient features of the continuum coupled channels method, and explore further its application to the dissociative photoionization of Hz and HD.

THEORY
In the center of mass coordinate system (whose origin is fixed to good approximation along the HE or HD internuclear distance) we write the total radiation-matter Hamiltonian as H(qq2lR)= T(R)+Hl(qlq2lR)+Hr"a(to)+o, (

1]
Eth + Vt(R) + V +----+ 4t (qlR) 0 (5) qIA qB R where Vt(R) is the R-dependent total electronic energy, Eth is the threshold energy for H + H / production. Xe,(q2) is an/-wave (p-wave in the present case) coulomb wavefunction 2 Ee + v2 + 1/Iq2l-lx,,,(q2) 0 is now expanded as follows, (qlq2, R) In),#g (qq2lg),g (R) + In 1)b, (qlq21e)@r(n) +In 1) de,' 4,,(qlqzlR) (R) where In) are n photon eigenstates satisfying (Hraa(o) n ho)ln) 0 (9) and emax=E-E,h. We now substitute expansion (8) into Eq. (7) premultiply and integrate by In),#, and In--1), in turn. Using the orthonormality of 4g and 4r we obtain for the discrete components of the wavefunction the following equations, with denoting integration over ql and q2 only. In Eq. (10a) we have neglected the back-coupling of the initial state to the host of In-1) photon states. This neglect is not essential but is completely justified for most radiation fields employed in the photoionization experiments. [1][2][3] In extracting the continuum components we note that the continuum-continuum matrix elements assume the form, The short range potential, V,(R) (tX,t 1 1 1 1 q2A q2B q12 Iq21 CtX', + X'.tCr) (1 3) can be made diagonal if a more exact representation of X,t is given. in the present application it is neglected. When we do this and premultiply Eq. (7) by In-1) with , expanded as in Eq.
Our goal remains the calculation of absorption cross sections from the ground state to individual exit channels. In the case of transition to I-r) a state which evolves to the asymptotic limit of the resonance state, tr(r +-g) 87raF I<I,t(-r)lF l[,lIg>12 (14) 3c For absorption to a state IW -*)) which evolves asymptotically to a three body state with definite electron kinetic energy e we seek to determine 83v (-In the representation of expansion (8) -'(R, q)= ,-' + de -(16b) 0 (-(R, )= ,-+ e' (1c) and the superscript (-) refers to outgoing spherical wave boundary conditions in the internuclear coordinate R. The total dissociative photoionization cross section a({e} g) is given by the sum Bound-continuum matrix elements of the form appearing in Eq. (14), (1) are obtained directly using the "artificial channel method, '' and to this end we discretize the indenumerable set of equations (I0c). This is accomplished by replacing the integral in (10b) by a quadrature: where Ae are products of quadrature weights (repeated midpoint9) and quadrature intervals. Our criteria of convergence are the overall shape of the distribution of partial absorptions, and total photoioni-  Figure 2), (---A---) tz,g neglected. uniform or varying density across the range of available energy without affecting the shape of the distribution or the integrated cross section.
Again for 65 < N < 67, integration of the high proton kinetic peak alone affords O'dion(Ep > 2 eV) =4.2 x 10-4a02(+0.003%). This is in reaonable agreement with the only experimental estimate of this cross section in the literature" 2.5 x 10-4a20<o-(E p > 2 eV)<5.4 x 10-4ao 2 and serves as the basis for the scaling of the experimental and theoretical distributions in Figures 2 and 3.
The role of direct tX,g and autoionization Vr, processes in the production of protons with low and high kinetic energies was deduced by alternatively suppressing/Z,g and Vr in the calculation. When Vr is suppressed, the high proton kinetic energy peak disappears and the low energy peak remains, although reduced in intensity by 7%. Similarly, when/x,g is suppressed the dashed line spectrum of Figure  3 is obtained' here the low energy peak is absent, and the high energy peak is reduced by 13% from Figure 2. All compound processes which originate in direct absorption to the continuum, undergo dissociative attachment and secondary autoionization are represented in this factor of 13%. Most significantly, these compound processes preserve the kinetic energy distribution characteristic of the final ionization.
The distinct low and high energy features of Figures 2 and 3 appear when we demand that V,, 0 for R > R , where R , is the internuclear distance where the resonant state and continuum channel potentials cross. In the Born-Oppenheimer approximation this is rigorously true, for at distances R, + dtR where dlR > 0, the nuclei moving under the potential V,(R) have acquired more kinetic energy than will permit production of an electron with energy ei and conserve total energy. pairs and, consequently, include both ground and excited d* basis functions in expansion (8). This, in turn, will produce ground and excited V' potential functions in Eqs. (10c). Some experimental H-/HD / and D-/HD / ratios of order unity have been reported for lower energy photons, 12 and we shall assume here that the H2 coupling scheme presented above is appropriate for the dissociative photo-  presently under investigation. A rigorous proof for the existence of convergent cross sections based on continuum Coulomb wavefunctions is also being sought. The full scope of the continuum coupled channels method will emerge as these questions are clarified.