Wave functions

The initial and final wave functions $\Psi _i , \Psi _f$ in the molecular formation matrix element caculation are usually approximated as:

$$\Psi _i = \psi ^{\mu t} _{1s} ({\bf r} ) \; \Psi ^{D_2} _{\... ...i K _i} ( {\bf r_1, r_2, \rho _2}) \; e^{i {\bf p \cdot r}},$$ (43)

where $\psi ^{\mu t} _{1s} ({\bf r} )$ is the $\mu t$ ground state wave function with ${\bf r}$ being the interparticle vector from t to $\mu$, and $\Psi ^{D_2} _{\nu _i K _i} ({\bf r_1, r_2, \rho _2})$ is the initial D₂ wave function with internuclear vector $\rho _2$ and electron vectors ${\bf r_1, r_2}$. The factor $e^{i {\bf p \cdot r}}$describes the relative motion of the centre of mass of the $\mu t$ and D₂.

The final wave function is written:

$$\Psi _f = \psi ^{d\mu t} _{11} ({\bf r, R} ) \; \Psi ^{Ndee} _{\nu _f K _f} ( {\bf r_1, r_2, \rho }),$$ (44)

where $\psi ^{d\mu t} _{11} ({\bf r, R} )$ is the $(d\mu t)_{11}$ wave function, and $\Psi ^{Ndee} _{\nu _f K _f} ( {\bf r_1, r_2, \rho })$ is the final state molecular complex wave function where $d\mu t$ is replaced by a fictitious particle of mass and charge equal to the sum of $d\mu t$ and located at the centre of mass of $d\mu t$.

Following Menshikov [149], most authors use an analytical approximation of $\psi ^{d\mu t} _{11} ({\bf r, R} )$ with its asymptotic form valid in the limit of infinite $\mu t+ d$ separation, since the main contribution to the matrix element integral comes from the region where the $\mu t+ d$ separation is large. The asymptotic wave function is characterized by a normalization constant C [149,150]. Recently Kino et al. [114] obtained a new value of C using Kamimura's variational wave function (Section 2.1.4, page ), calculated with the Gaussian basis functions in the Jacobi co-ordinate. As we have discussed, Kamimura's wave function treats the break up channel into $\mu t+ d$ directly, hence it gives an accurate description of asymptotic behaviour, while overcoming the difficulty of the Gaussian basis (``fast dumping'') by using a sufficiently large number of basis functions. The formation rates are proportional to the square of the constant C, and its new value decreased the predicted formation rates by 14% compared to an earlier calculation in Ref. [149] using the Adiabatic Representation, but increased them by 33% compared to Ref. [150] using a variational wave function with the Slater-type basis functions [110]. It is interesting to note that the Slater-type basis method, which was used so successfully in the energy level calculations (see Table 2.1), gives a less accurate description (assuming Kino et al. are correct) of the asymptotic wave function compared even to the Adiabatic Representation method, perhaps illustrating the difficulty in achieving a unified description of the non-adiabatic three body problem. In our analysis, the new value of C by Kino et al. is used.