4.2 Methodology
4.3.1 Diabatization angle
The functionσ(ρ, θ,3φλ), needed to solve the Poisson equation, is displayed in Fig. 4.3,
with a multiplicative factor of sin2θ, as a function of θ and φ
λ for ρ = 2 bohr,
4 bohr, 6 bohr and 8 bohr. This factor is used to cancel out the pole behavior of σ associated with its 1/sin2θ dependence. This source term also displays the
permutation symmetry of the H3 system. For ρ = 2 bohr (panel (a)), it (sin2θ σ)
has sharp minima of about 0.75 bohr−2 at φ
λ = 0◦, 120◦ and 240◦ while displaying
small oscillations around a flat value in other regions. For ρ = 8 bohr (panel (d)), it displays sharp maxima of about 0.45 bohr−2 at φ
λ = 60◦, 180◦ and 300◦ while
staying flat with no oscillations in other regions. Both these extrema (sharp minima for ρ = 2 bohr and sharp maxima for ρ = 8 bohr) occur for all values of θ with the most pronounced behavior occuring at θ = 90◦ (collinear configurations). Its
characteristics at ρ = 4 bohr (panel (b)) are midway between those at ρ = 2 bohr andρ= 8 bohr, with no sharp features. Besides, at this hyperradius it has oscillations which show minima around φλ = 0◦, 120◦ and 240◦ and maxima around φλ = 60◦,
180◦ and 300◦. Atρ= 6 bohr (panel (c)), its behavior is similar to that atρ= 8 bohr.
In panels (e) through (h) we depict equatorial views of sin2θ σ(ρ, θ,3φ
λ) contours and
equatorial view because it corresponds to a non-perpendicular arc-length-preserving projection of the constant hyperradius hemisphere on a plane tangent to it at the point on its equator, defined byωλ =γλ =π/2. This permits the viewing of all three
possible atom-diatom arrangement channel regions (for the triatomic reaction) as well as the regions for which the three atoms are at comparable distances from each other, for a fixed hyperradius ρ. Maps of this kind have been used before [18, 52, 65, 66]. This view of sin2θ σ(ρ, θ,3φ
λ) contours confirms the sharp minima behavior for ρ=
2 bohr and sharp maxima behavior for ρ = 8 bohr in different regions of the φλ
space. They also confirm the flat regions of 0 bohr−2 for ρ = 6 bohr and ρ =
8 bohr around φλ = 0◦, 120◦ and 240◦ regions of configuration space. Analysis
of w(1)ad at these hyperradii (using their plots from Chapter 3) shows that in and
around these regions of configuration space the coupling vector is about two orders of magnitude smaller than in the regions of space around φλ = 60◦, 180◦ and 300◦. This
is also evident in the corresponding plots of w(1)lonad (Fig. 4.9) and w(1)traad (Fig. 4.10)
discussed in Sec. 4.3.4, since their sum gives the total coupling vector w(1)ad. In the
threeφλ regions mentioned above, the dominant wlon(1)ad is much smaller than in other
regions. This negligibly small magnitude of the coupling vector leads to its negligible divergence as indicated by the source termσ(ρ, θ,3φλ) plotted in panels (c), (d), (g)
and (h) of Fig. 4.3.
Solution of the Poisson Eq. (4.41), subject to the boundary condition of Eq. (4.43) and Eqs. (4.45) through (4.52) and the source term discussed above, furnishesγ(ρ, θ,3φλ)
in the domain of internal nuclear configuration space defined by Eq. (4.42), which is the entire ρand θ space but one-sixth of the fullφλ space due to the P3 symmetry of
H3. This can be extended to the full 0≤ φλ <2π space by symmetry considerations.
The diabatization angle β(ρ, θ, φλ) is then obtained over the full domain U by using γ(ρ, θ,3φλ) together with Eq. (4.36).
Fig. 4.4 (panels (a) through (d)) depicts the diabatization angle β(ρ, θ, φλ) as
a function of θ and φλ for four different hyperradii. Panels (a), (b), (c) and (d)
8 bohr (asymptotic geometries), respectively. Panels (e) through (h) display the equatorial views of β contours corresponding to panels (a) through (d) through their mapping onto the ¯xλy¯λ tangent plane of Fig. 4.1. Since near conical intersection
geometries and even at other geometries β(ρ, θ, φλ) is dominated by the φλ/2 term,
we plot in Fig. 4.5 (panels (a) through (d)) γ(ρ, θ,3φλ) as a function of θ and φλ for
the same hyperradii as before and the corresponding equatorial views in panels (e) through (h), respectively. In each of the panels in Figs. 4.4 and 4.5, the hyperradius ρis kept fixed and the hyperangle φλ is varied from 0 to 2π along the circle shown at
the bottom. In addition, the hyperangle θ is varied from 0 to π/2, from the center of that circle (corresponding to θ = 0 or conical intersection geometries) to its edge (corresponding to θ=π/2 or collinear geometries).
Panels (a) through (d) of Fig. 4.6 display cuts of the γ(ρ, θ,3φλ) plots shown in
panels (a) through (d), respectively of Fig. 4.5 at three values of θ (5◦, 45◦, and 90◦)
for the same four values of ρ. The corresponding DMBE angle, γDMBE(ρ, θ,3φ λ), is
displayed in panels (e) through (h) of Fig. 4.6 for the same θ cuts and hyperradii. SinceγDMBE(ρ, θ,3φ
λ) is accurate in the vicinity of the conical intersection, a quantity
∆γ is defined as the maximum difference between γ and γDMBE over all values of φλ,
keeping the values of ρ and θ fixed:
∆γ(ρ, θ) =max[γ(ρ, θ,3φλ)−γDMBE(ρ, θ,3φλ)]φλ (4.57)
In Fig. 4.7, ∆γ is depicted as a function ofθfor the same four values of the hyperradius ρ, since it provides an indication of theρ-dependent difference betweenγDMBE
andγas a function ofθ. As the latter increases from 0◦ to 90◦, the corresponding configuration
moves away from the conical intersection. Fig. 4.8 depicts γ6D obtained from the 6
Dirichlet boundary conditions calculationb2described in Sec. 4.2.4. Figs. 4.4 through 4.8 are further discussed in Sec. 4.3.4.