2 Theoretical considerations

Because the population of the excited states of atoms is negligible in typical interstellar clouds, only molecular states correlating to the ground state of the separated atoms need be considered. The largest cross sections for direct radiative association result when X⁺ and H approach along an excited state potential energy curve and then radiate via a strong dipole transition to the ground state (Babb & Kirby [1998]). Transitions take place from the vibrational continuum of the excited state to the bound vibrational levels of the lower state. In this way, the molecules SiH⁺ and PH⁺ are formed through the $A~^1\Pi - X~^1\Sigma^+$ and the $1~^2\Sigma^- - X~^2\Pi$ transitions, respectively. In SiH⁺, although the $1~^3\Sigma^+$ and the $a~^3\Pi$ states correlate also to the the ground state separated atom limit, the $1~^3\Sigma^+$ is repulsive enough that there is little overlap of its continuum wave functions with the bound vibrational wave functions in the $a~^3\Pi$ state (Sannigrahi et al. [1995]). This same argument holds true for the $^4\Pi$ and $^4\Sigma^-$states of PH⁺ (Gu et al. [1999]). Therefore, for both SiH⁺ and PH⁺, radiative association through these states of higher multiplicity will be negligible.

For SH⁺, only the $X~^3\Sigma^-$ and $1~^5\Sigma^-$ states correlate to ground state atoms. Dipole transitions are not allowed between these states, and the $1~^5\Sigma^-$ is highly repulsive (Kimura et al. [1997]), so radiative association can only occur within the $X~^3\Sigma^-$ state itself, driven by the first and higher order derivatives of the $X~^3\Sigma^-$ dipole moment function. As will be seen in Sect. 4, the cross sections for this latter process are much smaller than for the two-state process.

The radiative association cross section is given quantum mechanically by (cf. Zygelman & Dalgarno [1990]; Babb & Kirby [1998])

$$\sigma (E) = \sum\limits_{N^{\prime}} \sum\limits_{v^{\prime \prime}} \sigma_{N^{\prime}} (v^{\prime\prime},E),$$ (2)

where

 

$$\sigma_{N^{\prime}}(v^{\prime\prime},E)= {64\over 3}{\pi^{5}\over c^{3}} {\nu^{3}\over k^{2}} p [S_{N^\prime-1,N^{\prime}} M_{v^{\prime\prime}, N^{\prime}-1;k,N^{\prime}}^2$$

$$+ S_{N^\prime,N^{\prime}} M_{v^{\prime\prime},N^{\prime};k,N^{\prime}}^2$$

$$+ S_{N^{\prime}+1,N^\prime} M_{v^{\prime\prime}, N^{\prime}+1;k,N^{\prime} }^2].$$ (3)

In Eq. (3), E is the relative collision energy, $\nu$ is the photon frequency, k is the wave number of relative motion, p is the probability of approach in the initial electronic state, $N^{\prime}$ is the initial rotational quantum number, $v^{\prime\prime}$ is the final vibrational quantum number, S is the Hönl-London factor, and M is the electric dipole matrix element connecting the initial and final nuclear and electronic states. The probability pfor SiH⁺, PH⁺, and SH⁺ is respectively ${1\over 6}$, ${1\over 9}$, and ${3\over 8}$ according to the degeneracy of the molecular state correlating to the relevant separated atom limits.

 

 

Table 1: Asymptotic separated-atom and united-atom limits

Molecular	Separated-atom	United-atom
State	Atomic states
SiH+:
$X~^1\Sigma^+$	Si+(3p 2P$^\circ$) + H	3s23p2 1D_ 2
$A~^1\Pi$	Si+(3p 2P$^\circ$) + H	3s23p2 1D_ 2
PH+:
$X~^2\Pi$	P+(3p2 3P) + H	3s23p3 2D $^{\rm o}_{\rm J}$
$1~^2\Sigma^-$	P+(3p2 3P) + H	3s23p3 2D $^{\rm o}_{\rm J}$
SH+:
$X~^3\Sigma^-$	S+(3p3 4S$^\circ$) + H	3s23p4 3 ${\rm P}_{\rm J}$