3.1 Quantum chemical data
The adiabatic potentials and the radial nonadiabatic couplings for the CH
4+ and the CD
4+ col- lisional systems are calculated
ab initio by means of the MOLPRO code [
33] for the internuclear distance from
R = 0.2 a.u. till
R = 10000 a.u. The electronic coordinates are measured from the centre of nuclear mass. A Huzinaga Gaussan basis set (9
s4
p) [
35] has been used for the hydrogen atom. For carbon, a 3
d orbital optimized on the 1
s2 3
d 2D state of C
3+ and a 4
f orbital from the correlation-consistent polarized cc-pVTZ basis of Dunning [
36] have been added to the Huzinaga basis set (11
s, 7
p) [
35]. The molecular orbitals have been optimized in a state-average CASSCF calculation [
37,
38] on the first six
2Σ
+ states. The seven lowest
σ, the four lowest
π and the two lowest
δ were chosen as active orbitals. This level of calculation corresponds to the calculation labelled CASSCF-2 in Ref. [
30]. The radial coupling matrix elements have been calculated by the standard numerical three points differentiation method implemented in MOLPRO [
33]. A detailed description of the quantum chemical calculations is given in Ref. [
30]. The calculations of Ref. [
30] were restricted to internuclear distances
R ≤ 32 a.u., where the nonadiabatic couplings do not reach yet their asymptotic values. For this reason, the present calculations are performed in the extended internuclear distance range. The potentials and the nonadiabatic couplings are in good agreement with other quantum chemical calculations (see, e.g., [
19]).
The quantum chemical data relevant to the process (1) are shown in
Figure 1. The potentials for the CH
4+(
2Σ
+) states are plotted in
Figure 1a
1. It is seen that the potentials of the adiabatic states
and
asymptotically (
R → ∞) correlated to the H(1
s) + C
4+ and H
+ + C
3+(3
d) states, respectively, are closely approaching each other at the internuclear distances
R ≈ 4.46 a.u. and
R ≈ 7.93 a.u., apparently showing two avoided crossings in these regions. Indeed the radial coupling matrix element
depicted in
Figure 1b shows sharp peaks at the same dis- tances confirming the avoided crossing interpretation. The area under each of the peaks is close to
π/2, the situation is expected to correspond closely to the Landau-Zener model. Other radial couplings are smaller by at least one order of magnitude, as shown in
Figure 3 of Ref. [
30].
In order to avoid such rapid variations in the coupling matrix elements, we transformed the adiabatic electronic basis
to a partly diabatic basis
. The new diabatic electronic states
and
are constructed as linear combinations of
and
[see Eq. (6)] by the requirement
. The other states
and
are left unchanged. The construction of the new basis is straightforward taking into account the fact that
goes to zero at
R → ∞. The avoided crossings between the adiabatic potentials become real crossings between the diabatic curves
V1(
R) and
V2(
R), the other potentials are not changed.
Figure 1a shows the diabatic potentials
Vj (
R) obtained in this manner. The coupling between the states
and
is now provided by an off-diagonal matrix element
(
Figure 1c, the thick solid line) of the electronic Hamiltonian, the couplings between the states
and
to
and
occurs by new derivative couplings (
Figure 1c), the coupling between the
and
states is unchanged.
Nonvanishing values of the asymptotic nonadiabatic couplings are clearly recognized in the insert of
Figure 1c. They are in agreement with the analytical expression [
4,
5,
19]
where
zat is the projection of the active electron coordinate onto the molecular axis, and
m is the reduced mass of an electron and the nuclei. If one of the nuclei is replaced by its isotope, the scalar factor
γk is changed [see Eq. (11)], and the nonadiabatic couplings have another asymptotic limits, which is shown in the insert of
Figure 1c, where the thick and thin dot-dashed lines are the
and
nonadiabatic matrix elements for the CD
4+ quasimolecule, while the thin dashed and solid lines denote the same matrix elements for the CH
4+ quasimolecule.
3.2 Dynamics
The program used in the present work for numerical integration of the coupled channel equations (8) is described in Ref. [
34].
Figure 2 shows typical numerical results for the transition probabilities from the initial to the final states calculated as the square of the absolute values of corresponding S-matrix elements in H + C
4+ (
Figure 2a) and D + C
4+ (
Figure 2b) collisions for collisional energy
E = 10 eV/amu and the impact parameter
= 1.06 a.u. (
Figure 2a) and b = 1:87 a.u. (
Figure 2b). The computed results are shown as functions of the upper limit
Rend of the numerical integration. As long as Eq. (13) is used, the computed transition probabilities (the solid lines in
Figure 2) are essentially independent from
Rend, when
Rend is sufficiently large. Some minor oscillations are still exist, but they are typically by more than three orders of magnitude smaller than a corresponding transition probability. We ascribe this to the neglection of second order terms in the derivation of Eq. (15), which appears to be well justified in this way. There is no other detectable variation of the computed result, supporting the use of the WKB approximation to derive Eq. (13).
Alternatively, there are different empirical approaches to solve the coupled system. The simplest approach consists in cutting off the asymptotic couplings at an arbitrary distance
; formally, this is the same as integrating numerically up to
Rend =
and then using Eq. (13), however with
τ±- matrices equal to the unit matrix. Results obtained in this way (the dashed lines in
Figure 2) show large oscillations as a function of
Rend; this represents the expected inelastic transitions between atomic states at large distance. The amplitudes of the oscillations vary from a few to nearly 100 percents with respect to corresponding rigorous probabilities depending on the collision conditions and the final states. Obviously the errors from the neglection of electron translation are larger for higher collisional energies [see Eq. (15)] and in cases of small transition probabilities.
Another approach consists in constructing a smooth transformation from the actual value of the coupling matrix element to zero. The numerical tests carried out in the present work show that the oscillations of the transition probabilities with the upper integration limit calculated by using this approach still exist. However, they are smaller than the ones obtained by simple cutting off the nonzero asymptotic couplings. The transition probabilities in this case are in better agreement with the rigorous results. This seems to justify numerous calculations, which have been carried out in this way in the past. However, this is a purely empirical justification, and there is no guarantee that the procedure works similarly well in other cases.
In the calculation of cross sections the influence of the electron translation neglection is expected to be reduced by integration over an impact parameter.
Figure 3 shows the result of using the present quantum approach (the solid lines) and neglecting electron translation (the dashed curves) for calculation of cross sections as a function of the upper integration limit. It is seen that the neglection of electron translation leads to variation of the cross sections, while the cross sections obtained within the
t/τ-matrix approach practically do not depend on the upper integration limit, when it is large enough.
Figure 4 shows our results for the partial and total cross sections as functions of the collisional energy for the charge transfer processes (1) and (2), the thick and the thin lines, respectively. They are compared to the data from Ref. [
30], the plusses, the stars, the crosses, and the diamonds; these results were obtained by using coupling matrix elements extrapolated to zero. The good agreement is to be expected in view of the good agreement of the dashed lines in
Figure 2 and
Figure 3 with the rigorous result. Except for the large distance extrapolation, the same potentials and couplings are used. However, the rest of the work was quite different: The result of Ref. [
30] is obtained with a completely diabatic basis and a wave packet approach, in contrast to our time independent coupled channels approach with a partly diabatic basis. We interpret the good agreement as an indication of reliability of the treatments.
The present calculations are also in agreement with the previous calculations of Ref. [
19,
20,
26,
27,
29] and the available experimental data for the total cross sections of the charge exchange processes (1) and (2) [
27,
39,
40]. As the example, the experimental data [
39] for H + C
4+ collisions are shown in
Figure 4 by circles. The recent experimental data [
41] for D + C
4+ collisions at low energies are shown by squares; it is seen that these data are in good agreement with the partial cross section for the excitation of the C
3+(3
p) state. As discussed in the literature, the cross section [
41] are smaller than both experimental [
27,
39,
40] and theoretical [
19,
20,
26,
27,
29,
30] total cross sections. For the detailed discussion of the theoretical results and the comparison with experiment for H, D + C
4+ collisions see Ref. [
30].
It is usually expected that cross sections for different isotopes should coincide for equal colli- sional velocities or collisional energies expressed in the eV/amu units.
Figure 4 shows that for the processes (1) and (2) this is indeed the case at relatively high collisional energies, but for energies below 20 eV/amu the isotopic effect gets remarkable. This is understandable in terms of nonadia- batic models, e.g., the Landau-Zener model, where the transition probability depends on a radial velocity in a nonadiabatic region which is different from a collisional velocity at infinity due to different (diabatic) potentials in a nonadiabatic region and at infinity. In the present case nona- diabatic transitions occur in several nonadiabatic regions (see
Figure 1), which makes the dynamics more complicated than in case of only one nonadiabatic region. In particular, the most remarkable differences in the calculated cross sections for the H and D isotopes take place at the collisional energies around 10 eV (note different collisional energies measured in the eV/amu unit), which correspond to the nonadiabatic region around
R ≈ 4.46 a.u. for H + C
4+ collisions, see
Figure 1. This results in the different nonadiabatic nuclear dynamics for the different isotopes and, hence, in different transition probabilities.
Figure 5 clearly shows different types of behavior of the transition probabilities for the different isotopes for the collisional energy
E = 10 eV/amu especially for the excitation of the C
3+(3
d) state. For H + C
4+ collisions this energy (
E = 9.231 eV) is nearly equal to the crossing of the 1
s and 3
d diabatic potentials in the nonadiabatic region, and the turning points are inside the nonadiabatic region (for small impact parameters), while for D + C
4+ collisions at the energy
E = 10 eV/amu (the collisional energy
E = 17.143 eV) the turning points are outside of the nonadiabatic region. As a result, the transition probabilities and the partial cross sections for excitation of the C
3+(3
d) state are significantly different in H + C
4+ and D + C
4+ collisions, reflecting, therefore, the isotopic effect for the charge exchange processes (1) and (2) at low collisional energies. The experimental evidence of violation of the scaling law is given in Ref. [
42].