This document presents a method for solving the coupled-channels time-independent Schrödinger equation for bound states of the A1Σ+ − b3Π0 electronic states in NaCs, which are coupled by spin-orbit interaction. The method expands the coupled-channel eigenstates over a basis of rovibrational eigenstates of the uncoupled potentials. This leads to a system of equations for the expansion coefficients that can be solved by diagonalizing a 260x260 matrix. Plots of the bound-state matrix elements of the spin-orbit coupling operator show they decrease for more highly-excited vibrational states. Based on this, the method approximates the problem by neglecting couplings to continuum states

1. Solving the coupled-channels time independent Schr¨odinger equation for
bound states of the A1
Σ+
− b3
Π0 electronic states coupled by spin-orbit
coupling in NaCs
v. 1.5
Goal Present the method used to solve the coupled-channels time dependent Schr¨odinger equation for bound
states of the A1
Σ+ −b3
Π0 electronic states coupled by spin-orbit coupling in NaCs, show the resulting rovibrational
coupled-channels wave functions obtained, and present the assessment of the validity of the results.
Preliminary All notations and concepts mentioned here, unless otherwise specified, are defined in [1]. It is worth
your time to reread Ref. [1] to refresh your memory before jumping into the current notes.
1 Introduction
At the end of my notes on the inclusion of the spin-orbit coupling effect in NaCs [1], I noticed that I had to analyze
the strength of the mixing-angle couplings between rovibrational states of the hybrid potentials. Through this
analysis, I expected to be able to neglect the effect of the mixing angle for some states, and proceed to solve my
problem without recoursing to a coupled-channel calculations.
However, after obtaining the mixing-angle matrix elements (MAMEs) defined at the end of said notes, I con-
cluded that the coupling could not be neglected, and that running a coupled-channel calculation was mandatory
The last research meeting I had with Profs. Parker and Morrison comforted this conclusion.
As I was reading about the coupled-channel propagator methods reported by Hutson [2], and as I was going
through James Dizikes coupled-channel scattering solvers implemented in Mathematica package, I remarked that
there was another way to proceed given the data I had already calculated. These notes present the method I tried,
and show that the results I obtained are indeed solutions to the spin-orbit coupled-channel time-independent
Schr¨odinger equation. Section 2 outlines the problem at hand, the necessary background for the derivation required
to solve the problem, the approximations used, and the derivation itself. Section 3 shows my assessment of the
validity of the results obtained by implementing the method, and my physical interpretation of the results.
2 Method
2.1 Identifying the problem
I want to find the eigenenergy Ecc
vcc
and the reduced radial eigenfunction |Φcc
vcc
of the Time-Independent Schr¨odinger
Equation
H |Φcc
vcc
= Ecc
vcc
|Φcc
vcc
, (1)
with
H = Tn + Vnn + Vne + Vee + Te
He
+HSO, (2)
where
Tn – kinetic energy operator for the nuclei,
Vnn – nucleus-nucleus Coulomb interaction,
Vne – nucleus-electron Coulomb interaction,
Vee – electron-electron Coulomb interaction,
Te – kinetic energy operator for the electrons,
1

2. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
HSO ≡ i ˆai
#
i · #si – spin-orbit interaction (see [3, Eq. (3.4.3) p. 182]. Also the sum runs only over open shells
electrons, [3, last sentence p. 183]).
In Ref. [1] I demonstrated the existence of a hybrid electronic basis H = {|V1/2 ,|V3/2 } that diagonalizes He + HSO +
Vnn.
In the hybrid basis H, the full hamiltonian H of Eq. (2) can be recast as
H = T +
R2
2µR2
+ V tot
+ γ, (3)
where in basis H using projectors
T = −
2
2µ
d2
dR2
ˆ1 (4a)
R2
2µR2
= 2
2
2µR2
ˆ1 (4b)
V tot
= V1/2(R)|V1/2 V1/2| + V3/2(R)|V3/2 V3/2| + 2
2
2µR2
ˆ1 (4c)
γ =
2
2µ


dγ
dR
2
ˆ1 +
d2
γ
dR2
+ 2
dγ
dR
d
dR
(|V1/2 V3/2| − |V3/2 V1/2|)

. (4d)
For the electronic states of interest, the selection rules of my problem constrain J = 1, yielding Eq. (4b) (see [1,
§2.4.2]). Note that the 2
2
2µR2 term in Eq. (4c) spawns from the van Vleck pure precession hypothesis, as explained
in §2.4.2 in [1], and not from the rotational operator. The operator γ originates from changing from the Hund’s case
(a) basis to the hybrid basis, again see [1].
It is possible to adapt the coupled-channels solver developed by James Dizikes to handle bound states boundary
conditions, using methods from Hutson [2] to home in on the eigenenergies.
However, in order to analyse the matrix elements of γ in Eq. (4d) between rovibrational functions, I determined
with LEVEL [4] the rovibrational eigenenergies and eigenfunctions of T +V tot, and calculated with Mathematica
the matrix elements
χv
dγ
dR
2
χv , Ξq
dγ
dR
2
Ξq , (5a)
Ξq
d2
γ
dR2
+ 2
dγ
dR
d
dR
χv , (5b)
where |χv is a rovibrational eigenstate of the J = 1,V1/2(R) electronic state, and |Ξq is a rovibrational eigenstate of
the J = 1,V3/2(R) electronic state. With such data at hand, I can recourse to a standard basis set expansion to solve
for |Φcc
vcc
and Ecc
vcc
.
2.2 Derivation
Let’s define the orthonormal basis B = {|χv |V1/2 }v,{|e |V1/2 }e,{|Ξq |V3/2 }q,{|E |V1/2 }E , where v = 0,...,145 and
q = 0,...,113, e represents a scattering energy above the asymptote of V1/2(R), and E is a scattering energy above the
SV-coupledChannelSONaCs-v1.5.tex 2

3. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
asymptote of V3/2(R). The various kets in B are eigenstates of the corresponding time-independent hamiltonians:
T + V tot
|χv |V1/2 = −
2
2µ
d2
dR2
+ 4
2
2µR2
+ V1/2(R) |χv |V1/2 = ev |χv |V1/2 (6a)
T + V tot
|e |V1/2 = e|e |V1/2 (6b)
T + V tot
|Ξq |V3/2 = −
2
2µ
d2
dR2
+ 4
2
2µR2
+ V3/2(R) |Ξq |V3/2 = Eq |Ξq |V3/2 (6c)
T + V tot
|E |V3/2 = E |E |V3/2 (6d)
The coupled-channel eigenstate |Φcc
vcc
satisfies Eq. (3) and may be expanded over the basis B
|Φcc
vcc
=
v
av,vcc
|χv |V1/2 +
e
ae,vcc
|e |V1/2 de +
q
bq,vcc
|Ξq |V3/2 +
E
bE,vcc
|E |V3/2 dE. (7)
Note that all spatial dependencies are carried within the kets, and thus all expansion coefficients are constants.
Plugging Eq. (7) into Eq. (3), and using the definitions of Eqs. (6) yields the following system of equations for the
expansion coefficients1:
∀v ∈ 0,145 , Ecc
vcc
av,vcc
= evav,vcc
+
v
av ,vcc
χv| V1/2 γ V1/2 |χv +
e
ae ,vcc
χv| V1/2 γ V1/2 |e de (8a)
+
q
bq ,vcc
χv| V1/2 γ V3/2 |Ξq +
e
be ,vcc
χv| V1/2 γ V3/2 |e de ,
∀q ∈ 0,113 , Ecc
vcc
bq,vcc
=
v
av ,vcc
Ξq| V3/2 γ V1/2 |χv +
e
ae ,vcc
Ξq| V3/2 γ V1/2 |e de (8b)
+ Eqbq,vcc
+
q
bq ,vcc
Ξq| V3/2 γ V3/2 |Ξq +
e
be ,vcc
Ξq| V3/2 γ V3/2 |e de ,
∀e > e∞, Ecc
vcc
ae,vcc
= eae,vcc
+
v
av ,vcc
e| V1/2 γ V1/2 |χv +
e
ae ,vcc
e| V1/2 γ V1/2 |e de (8c)
+
q
bq ,vcc
e| V1/2 γ V3/2 |Ξq +
e
be ,vcc
e| V1/2 γ V3/2 |e de ,
∀e > e∞, Ecc
vcc
be,vcc
=
v
av ,vcc
e| V3/2 γ V1/2 |χv +
e
ae ,vcc
e| V3/2 γ V1/2 |e de (8d)
+ ebe,vcc
+
q
bq ,vcc
e| V3/2 γ V3/2 |Ξq +
e
be ,vcc
e| V3/2 γ V3/2 |e de ,
where e∞ = lim
R→∞
V1/2(R) and e∞ = lim
R→∞
V3/2(R). If all the bound-continuum and continuum-continuum matrix
elements of γ are ignored, obtaining the expansion coefficients amounts to diagonalize a 260 × 260 matrix.
2.3 Examining the bound-bound matrix elements of γ
Figures 1, 2, and 3 are plots of the bound-bound matrix elements of γ. One very important feature common to these
three figures is the trend of the matrix elements. The higher the vibrational quantum number, the less will γ cou-
ple this particular rovibrational state to other rovibrational states of either the same potential (Figs. 1 and 2), or the
other potential (Fig. 3). For example, in Fig. 1, the matrix elements
2
2µ V1/2| χ145
dγ
dR
2
χv1/2
|V1/2 is extremely
1As usual, ∀v ∈ a,b means that v can be any integer between a and b with (a,b) ∈ R2.
SV-coupledChannelSONaCs-v1.5.tex 3

4. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
105
110
115
120
125
130
135
140
145
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145
v12
v1 2
1.50
1.25
1.00
0.75
0.50
0.25
0
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
6420246
Figure 1: Diagonal bound-bound matrix elements
2
2µ χv1/2
dγ
dR
2
χv1/2
. The legend on the right is in atomic units
×10−4.
small for all values of v1/2 compared to other couplings: the values of the couplings always stay in the cyan range of
the legend, indicating closeness to zero. Figures 2 and 3 display the same behavior for
2
2µ V3/2| Ξv3/2
dγ
dR
2
Ξv3/2
|V3/2
and −
2
2µ V3/2| Ξv3/2
d2
γ
dR2 + 2
dγ
dR
d
dR χv1/2
|V1/2 for very high values of v3/2 and v1/2.
Continuum wave functions oscillate with very small amplitude—compared to bound states—until the internu-
clear separation exceeds the value of the right classical turning point of the highest bound state2. Given the shape
of the coupling functions (see Figs. 3–5 in [1]), the continuum-continuum and bound-continuum matrix elements
2What Londo˜no et al. [5] call RN is somewhat greater than the rightmost classical turning point. The rightmost classical turning point is thus
a good estimate for a lower bound on RN .
SV-coupledChannelSONaCs-v1.5.tex 4

5. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
105
110
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110
v32
v3 2
2.0
1.5
1.0
0.5
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6420246
Figure 2: (Color online) Diagonal bound-bound matrix elements
2
2µ Ξv3/2
dγ
dR
2
Ξv3/2
. The legend on the right is in
atomic units ×10−4.
of γ are therefore likely to be negligible.
Therefore, it seems reasonable to neglect all couplings to continuum states in Eqs. (8). This approximation
reduces the problem of finding the eigenstates of the coupled-channel Time-Independent Schr¨odinger Equation to
the diagonalization of a 260 × 260 matrix. Indeed the J = 1,V1/2 potential holds 145 + 1 = 146 rovibrational states,
and the J = 1,V3/2 potential holds 113 + 1 = 114, thus the total matrix to diagonalize has dimensions 260 × 260.
Results of that operation are examined in the next section.
SV-coupledChannelSONaCs-v1.5.tex 5

6. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
105
110
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145
v32
v1 2
6.5
6.0
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6420246
Figure 3: (Color online) Off-diagonal bound-bound matrix elements −
2
2µ Ξv3/2
d2
γ
dR2 + 2
dγ
dR
d
dR
χv1/2
. The legend on
the right is in atomic units ×10−4.
3 Results
3.1 Wave functions for each separated channels
The diagonalization of the real, symmetric 260 × 260 matrix takes about 0.03s to run on Mathematica 3. The
diagonalization gives the set of coefficients {{av,vcc
}v,{bq,vcc
}q} defined for each value of vcc. One may express the 2
components coupled-channel eigenket |Φcc
vcc
in a vector form:


|Φcc
vcc
V1/2| ψ
[1/2]
vcc
(R)
V3/2| ψ
[3/2]
vcc
(R)

 with ψ
[1/2]
vcc
(R) =
v
av,vcc
χv(R) and ψ
[3/2]
vcc
(R) =
q
bq,vcc
Ξq(R). (9)
To check that the kets |Φcc
vcc
I obtained by diagonalizing the system of equations (8) are indeed solutions of the
Time-Independent Schr¨odinger Equation, I calculate for each value of vccthe wave functions ψ
[1/2]
vcc
(R) and ψ
[3/2]
vcc
(R).
The ket |Φcc
vcc
is a solution of the Time-Independent Schr¨odinger Equation with energy Ecc
vcc
if and only if it satisfies
3Fun fact: it takes longer for the notebook to read-in the relevant information than to run the diagonalization.
SV-coupledChannelSONaCs-v1.5.tex 6

7. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
Eq. (3), or equivalently if ψ
[1/2]
vcc
(R) and ψ
[3/2]
vcc
(R) satisfy

−
2
2µ


d2
dR2
−
dγ
dR
2
 + V1/2(R) + 4
2
2µR2

ψ
[1/2]
vcc
(R) +
2
2µ
d2
γ
dR2
+ 2
dγ
dR
d
dR
ψ
[3/2]
vcc
(R) = Ecc
vcc
ψ
[1/2]
vcc
(R), (10a)
−
2
2µ
d2
γ
dR2
+ 2
dγ
dR
d
dR
ψ
[1/2]
vcc
(R) +

−
2
2µ


d2
dR2
−
dγ
dR
2
 + V3/2(R) + 4
2
2µR2

ψ
[3/2]
vcc
(R) = Ecc
vcc
ψ
[3/2]
vcc
(R), (10b)
⇔ ˆh11 ψ
[1/2]
vcc
+ ˆh12 ψ
[3/2]
vcc
= Ecc
vcc
ψ
[1/2]
vcc
, (10c)
ˆh21 ψ
[1/2]
vcc
+ ˆh22 ψ
[3/2]
vcc
= Ecc
vcc
ψ
[3/2]
vcc
. (10d)
By plotting on the same graph the left and right hand side of Eqs. (10c-10d), I can assess whether |Φcc
vcc
is actually
an eigenstate of the coupled-channel Time-Independent Schr¨odinger Equation with eigenenergy Ecc
vcc
. If the left
hand side of the equations superimposes on the right hand side, then |Φcc
vcc
is indeed a coupled-channel eigenstate
with energy Ecc
vcc
.
Figures 4–11 below show plots of Eqs. (10c) and (10d). On each figure, panel (a) is always a plot of Eq. (10c), and
panel (b) is always a plot of Eq. (10d). The red (resp. gray) solid line always represents the left hand side of Eq. (10c)
(resp. Eq. (10d)), while the dotted blue (resp. dashed green) line represents the right hand side of Eq. (10c) (resp.
Eq. (10d)). The legend on each figure is a reminder of this convention. The black horizontal line is the horizontal
axis, drawn to guide the eye.
Notice that on all figures, the continuous and the discontinuous lines always superimpose nicely, no matter
the rovibrational energy. Such graphical match strongly suggests that the basis expansion method used produces the
correct coupled-channel wave functions and rovibrational energies.
3.2 Probability density functions
Figures 12 to 19 plot the coupled-channel probability density functions (PDFs), on top of the potential energy
curves, for the same values of the coupled-channel vibrational index vcc as in Figs. 4 to 11. Note that the amplitude
of the PDFs is not to scale in any of the figures. The amplitude was adjusted in each figure to display as much of
the important features as possible, and these graphs should not be used to gain any quantitative information about
the PDFs. The base line of the PDF matches the value of the corresponding rovibrational energy.
For vcc = 0, figure 12 shows that the PDF has the expected behavior of a ground state rovibrational wave func-
tion: a single, sharp peak above the minimum of the potential. Likewise for vcc = 3, figure 13 displays the same
feature: the vcc = 3 rovibrational energy is barely above the minimum of the A1
Σ+ state, and not yet above the
potential energy crossing, thus the effect of spin-orbit coupling on this state is very small, and the bottom of the
well of the A1
Σ+ state dominates the behavior of |Φcc
vcc=3 .
The vcc = 6 rovibrational energy grazes the local maximum at the bottom of V1/2(R). Imagine that at this energy,
the b3
Π0 and the A1
Σ+ states each have a rovibrational state. The rightmost lobe of the PDF belonging to b3
Π0
would combine through the spin-orbit interaction with the leftmost lobe of the PDF belonging to A1
Σ+, thereby
producing the sharp peak in the middle of the coupled-channel state |Φcc
vcc=6 . In terms of the hybrid potentials,
Fig. 6 shows that the dominant single-channel components of |Φcc
vcc=6 are the χv=6(R) and Ξq=1(R) wave functions,
the two main peaks that occur at the avoided crossing produce the sharp peak in the resulting coupled-channel
PDF.
At vcc = 75 (see Figs. 7 and 15), the spin-orbit coupling disturbs the oscillation of the PDF near the inner wall of
the V3/2 potential, the rest of the function otherwise behaves like a standard single-channel probability density.
The vcc = 165 state in Fig. 16 displays prominent features characteristic of spin-orbit coupling, and similar to the
ones published by Londo˜no et al. [5]: 4 maxima above the classical turning points of that energy for both coupled
potentials, and a hole drilled in the PDF where the two potentials cross. The vcc = 194 state in Fig. 17 displays the
same features, but the external extrema on the V1/2 curve have a much smaller amplitude.
Figure 18 shows a probability density that differs from vcc = 194 by more than just one vibrational index. The
wave function now stretches from one end of the V1/2 potential to the other, almost ignoring the presence of the
SV-coupledChannelSONaCs-v1.5.tex 7

8. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
V3/2 potential, as the absence of local maxima at the corresponding turning points show. The effect of spin-orbit
still manifests through the pinching of the probability density above the potential crossing. This figure vividly
illustrates that simply changing from one vibrational index (194) to the next (195) causes important changes in the
probability density, a behavior we never saw for the single-channel case. I am convinced that when it comes to
using chirped lasers, such drastic changes will become very crucial.
The last probability density figure (19) for vcc = 235 looks very similar to the vcc = 195 case, in particular the
rightmost peak. However, notice that |Φcc
235 has no probability for small R values between the inner walls of the
potentials. Also, the pinching of the probability density above the potential crossing causes a bump up from the
base line on the PDF, rather than a dip down to the base line—the situation of vcc = 195. Furthermore, for R > 9a0,
the locus of the top of the arches of |Φcc
235 does not increase monotonically as in |Φcc
195 , the coupling between the
channels causes a short plateau of the top of the arches around R = 15a0. This behavior is more pronounced for
coupled-channel states with rovibrational energy above the asymptote of V1/2.
4 What’s next
The graphs above demonstrate that the basis expansion method provides the actual 260 eigenstates and eigenen-
ergies of the coupled-channel Time-Independent Schr¨odinger Equation. For all calculated coupled-channel eigen-
states, the percent relative error between the left and right hand side of Eqs. (10c) and (10d) in the classically al-
lowed region never exceeds 5%.
Given these coupled-channel wave functions, the immediate next steps I can see are:
1. Chore: polish documentation of the notebook that runs the calculation and examines the results, so that the
notebooks can be released to research group
2. Chore: divide above mentioned notebook in 2, one that runs the calculation, one that examines the results
3. Programming brainstorm: Find a way to export the coupled-channel wave functions so they are easy to
use with the ReadLevelWfn package of James Dizikes, and so that these wave functions can be imported to
calculate the Transition Dipole Moment Matrix Elements (TDMME)
4. IMPORTANT: Brainstorm the structure of the code that will read in the coupled-channel wave functions,
X1Σ+ ground electronic state wave functions, and the sine and cosine of the mixing angle γ, and then calcu-
late the necessary TDMMEs. The core of this code—calculating the TDMMEs—already exists and has been
carefully tested for my DAMOP 2012 presentation.
Things I should on the side to take a break from the above:
1. Search for values of typical chirp rates
2. Think about streamlining the calculation of continuum states above the asymptote of the X1Σ+ state to
prepare for the ultimate step: running the calculation for a wave packet.
As usual any feedback on these notes is more than welcome. I think that this calculation in itself constitutes a
least publishable unit, but when implemented with (even non chirped) laser pulses, PA+stabilization and STIRAP
from a single stationary continuum state, then that could constitute a good paper. Thoughts?
SV-coupledChannelSONaCs-v1.5.tex 8

9. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
energywavefunctionEha0
12
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
a
h11Ψ0
1 2
h12Ψ0
3 2
E0Ψ0
1 2
6 8 10 12 14 16
R a0
energywavefunctionEha0
12
0.0015
0.0010
0.0005
0.0000
0.0005
0.0010
0.0015
b
h21Ψ0
1 2
h22Ψ0
3 2
E0Ψ0
3 2
E0 0.0467792 Eh 10266.8 cm 1
Figure 4: (Color online) Graphical check of the validity of the coupled-channel wave function vcc = 0 calculated
with the basis expansion method. (a)—Left and right hand side of the coupled-channel Time-Independent
Schr¨odinger Equation for the V1/2 channel. (b)—Same as (a) for the V3/2 channel. Note how the continuous
and discontinuous lines superimpose, showing the Time-Independent Schr¨odinger Equation is verified.
SV-coupledChannelSONaCs-v1.5.tex 9

10. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
energywavefunctionEha0
12
0.00
0.01
0.02
0.03
0.04
0.05
0.06a
h11Ψ3
1 2
h12Ψ3
3 2
E3Ψ3
1 2
6 8 10 12 14 16
R a0
energywavefunctionEha0
12
0.002
0.001
0.000
0.001
0.002
0.003
b
h21Ψ3
1 2
h22Ψ3
3 2
E3Ψ3
3 2
E3 0.0479463 Eh 10523. cm 1
Figure 5: (Color online) Graphical check of the validity of the coupled-channel wave function vcc = 3 calculated
with the basis expansion method. (a)—Left and right hand side of the coupled-channel Time-Independent
Schr¨odinger Equation for the V1/2 channel. (b)—Same as (a) for the V3/2 channel. Note how the continuous
and discontinuous lines superimpose, showing the Time-Independent Schr¨odinger Equation is verified.
SV-coupledChannelSONaCs-v1.5.tex 10

11. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
energywavefunctionEha0
12
0.04
0.02
0.00
0.02
0.04
0.06a
h11Ψ6
1 2
h12Ψ6
3 2
E6Ψ6
1 2
6 8 10 12 14 16
R a0
energywavefunctionEha0
12
0.03
0.02
0.01
0.00
0.01
0.02
0.03b
h21Ψ6
1 2
h22Ψ6
3 2
E6Ψ6
3 2
E6 0.0482906 Eh 10598.6 cm 1
Figure 6: (Color online) Graphical check of the validity of the coupled-channel wave function vcc = 6 calculated
with the basis expansion method. (a)—Left and right hand side of the coupled-channel Time-Independent
Schr¨odinger Equation for the V1/2 channel. (b)—Same as (a) for the V3/2 channel. Note how the continuous
and discontinuous lines superimpose, showing the Time-Independent Schr¨odinger Equation is verified.
SV-coupledChannelSONaCs-v1.5.tex 11

12. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
energywavefunctionEha0
12
0.04
0.02
0.00
0.02
0.04
a
h11Ψ75
1 2
h12Ψ75
3 2
E75Ψ75
1 2
6 8 10 12 14
R a0
energywavefunctionEha0
12
0.03
0.02
0.01
0.00
0.01
0.02
0.03b
h21Ψ75
1 2
h22Ψ75
3 2
E75Ψ75
3 2
E75 0.0592234Eh 12998.cm 1
Figure 7: (Color online) Graphical check of the validity of the coupled-channel wave function vcc = 75 calculated
with the basis expansion method. (a)—Left and right hand side of the coupled-channel Time-Independent
Schr¨odinger Equation for the V1/2 channel. (b)—Same as (a) for the V3/2 channel. Note how the continuous
and discontinuous lines superimpose, showing the Time-Independent Schr¨odinger Equation is verified.
SV-coupledChannelSONaCs-v1.5.tex 12

13. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
energywavefunctionEha0
12
0.03
0.02
0.01
0.00
0.01
0.02
0.03
a
h11Ψ165
1 2
h12Ψ165
3 2
E165Ψ165
1 2
6 8 10 12 14 16 18 20
R a0
energywavefunctionEha0
12
0.06
0.04
0.02
0.00
0.02
0.04
b
h21Ψ165
1 2
h22Ψ165
3 2
E165Ψ165
3 2
E165 0.0706757Eh 15511.5cm 1
Figure 8: (Color online) Graphical check of the validity of the coupled-channel wave function vcc = 165 calcu-
lated with the basis expansion method. (a)—Left and right hand side of the coupled-channel Time-Independent
Schr¨odinger Equation for the V1/2 channel. (b)—Same as (a) for the V3/2 channel. Note how the continuous and
discontinuous lines superimpose, showing the Time-Independent Schr¨odinger Equation is verified.
SV-coupledChannelSONaCs-v1.5.tex 13

14. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
References
[1] S. Valladier, Including spin-orbit coupling of the first excited electronic states in the photoassociation and rovibrational relaxation
of NaCs (2013).
[2] J. M. Hutson, Coupled channel methods for solving the bound-state Schr¨odinger equation, Computer Physics Communications,
84(13), 1 (1994).
[3] H. Lefebvre-Brion and R. W. Field, The spectra and dynamics of diatomic molecules, Elsevier Academic Press, Amsterdam;
Boston (2004).
[4] R. J. LeRoy, LEVEL 8.0: A Computer Program for Solving the Radial Schr¨odinger Equation for Bound and Quasibound Levels,
Research Report CP-661, University of Waterloo Chemical Physics (2007).
[5] B. E. Londo˜no, J. E. Mahecha, E. Luc-Koenig, and A. Crubellier, Resonant coupling effects on the photoassociation of ultracold
Rb and Cs atoms, Phys. Rev. A, 80, 032511 (2009).
SV-coupledChannelSONaCs-v1.5.tex 14

15. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
energywavefunctionEha0
12
0.015
0.010
0.005
0.000
0.005
0.010
0.015
a
h11Ψ194
1 2
h12Ψ194
3 2
E194Ψ194
1 2
5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5
R a0
energywavefunctionEha0
12
0.06
0.04
0.02
0.00
0.02
0.04
b
h21Ψ194
1 2
h22Ψ194
3 2
E194Ψ194
3 2
E194 0.0730046 Eh 16022.7 cm 1
Figure 9: (Color online) Graphical check of the validity of the coupled-channel wave function vcc = 194 calcu-
lated with the basis expansion method. (a)—Left and right hand side of the coupled-channel Time-Independent
Schr¨odinger Equation for the V1/2 channel. (b)—Same as (a) for the V3/2 channel. Note how the continuous and
discontinuous lines superimpose, showing the Time-Independent Schr¨odinger Equation is verified.
SV-coupledChannelSONaCs-v1.5.tex 15

16. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
energywavefunctionEha0
12
0.06
0.04
0.02
0.00
0.02
0.04
0.06
a
h11Ψ195
1 2
h12Ψ195
3 2
E195Ψ195
1 2
5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5
R a0
energywavefunctionEha0
12
0.0100
0.0075
0.0050
0.0025
0.0000
0.0025
0.0050
0.0075
0.0100
b
h21Ψ195
1 2
h22Ψ195
3 2
E195Ψ195
3 2
E195 0.0730564Eh 16034.cm 1
Figure 10: (Color online) Graphical check of the validity of the coupled-channel wave function vcc = 195 calcu-
lated with the basis expansion method. (a)—Left and right hand side of the coupled-channel Time-Independent
Schr¨odinger Equation for the V1/2 channel. (b)—Same as (a) for the V3/2 channel. Note how the continuous and
discontinuous lines superimpose, showing the Time-Independent Schr¨odinger Equation is verified.
SV-coupledChannelSONaCs-v1.5.tex 16

17. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
energywavefunctionEha0
12
0.0100
0.0075
0.0050
0.0025
0.0000
0.0025
0.0050
0.0075
0.0100
a
h11Ψ235
1 2
h12Ψ235
3 2
E235Ψ235
1 2
5 10 15 20 25
R a0
energywavefunctionEha0
12
0.06
0.04
0.02
0.00
0.02
0.04b
h21Ψ235
1 2
h22Ψ235
3 2
E235Ψ235
3 2
E235 0.0753212Eh 16531.1cm 1
Figure 11: (Color online) Graphical check of the validity of the coupled-channel wave function vcc = 235 calcu-
lated with the basis expansion method. (a)—Left and right hand side of the coupled-channel Time-Independent
Schr¨odinger Equation for the V1/2 channel. (b)—Same as (a) for the V3/2 channel. Note how the continuous and
discontinuous lines superimpose, showing the Time-Independent Schr¨odinger Equation is verified.
SV-coupledChannelSONaCs-v1.5.tex 17

18. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
b3
0
A1
V3 2
V1 2
5 10 15 20 25
0.05
0.06
0.07
0.08
0.09
0.10
R a0
PotentialenergyEh
0
cc 2
Figure 12: (Color online) Coupled-channel probability density function for vcc = 0. Potential energy curves are
in the background. The base line for the probability density function matches the corresponding rovibrational
energy. As the ground coupled-channel state, with rovibrational energy barely above the lowest of all potentials
minima, the probability density function has the expected characteristic single peak centered in the middle of
the well.
SV-coupledChannelSONaCs-v1.5.tex 18

19. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
b3
0
A1
V3 2
V1 2
5 10 15 20 25
0.05
0.06
0.07
0.08
0.09
0.10
R a0
PotentialenergyEh
3
cc 2
Figure 13: (Color online) Coupled-channel probability density function for vcc = 3. Potential energy curves are
in the background. The base line for the probability density function matches the corresponding rovibrational
energy. This state has a rovibrational energy barely above the second minimum of the lowest potential, again the
probability density function has the expected characteristic single peak centered in the middle of this well.
SV-coupledChannelSONaCs-v1.5.tex 19

20. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
b3
0
A1
V3 2
V1 2
5 10 15 20 25
0.05
0.06
0.07
0.08
0.09
0.10
R a0
PotentialenergyEh
6
2
Figure 14: (Color online) Coupled-channel probability density function for vcc = 6. Potential energy curves are
in the background. The base line for the probability density function matches the corresponding rovibrational
energy. The rovibrational energy is barely grazing the local maximum of the potential, producing the pronounced
peak above the local maximum.
SV-coupledChannelSONaCs-v1.5.tex 20

21. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
b3
0
A1
V3 2
V1 2
5 10 15 20 25
0.05
0.06
0.07
0.08
0.09
0.10
R a0
PotentialenergyEh
75
cc 2
Figure 15: (Color online) Coupled-channel probability density function for vcc = 75. Potential energy curves are
in the background. The base line for the probability density function matches the corresponding rovibrational
energy. The probability density function resembles that of the single-channel V1/2 potential. However, the spin-
orbit interaction causes the disturbance in the oscillations around 7a0 near the inner wall of the V3/2 potential.
SV-coupledChannelSONaCs-v1.5.tex 21

22. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
b3
0
A1
V3 2
V1 2
5 10 15 20 25
0.05
0.06
0.07
0.08
0.09
0.10
R a0
PotentialenergyEh
165
cc 2
Figure 16: (Color online) Coupled-channel probability density function for vcc = 165. Potential energy curves
are in the background. The base line for the probability density function matches the corresponding rovibrational
energy. This probability density function better shows the features similar to those reported by Londo˜no et al.
[5]; 4 local maxima located above the corresponding classical turning points, and a pinch above the potential
crossing.
SV-coupledChannelSONaCs-v1.5.tex 22

23. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
b3
0
A1
V3 2
V1 2
5 10 15 20 25
0.05
0.06
0.07
0.08
0.09
0.10
R a0
PotentialenergyEh
194
cc 2
Figure 17: (Color online) Coupled-channel probability density function for vcc = 194. Potential energy curves
are in the background. The base line for the probability density function matches the corresponding rovibrational
energy. The same features as for vcc = 165 are present, except the amplitudes in the region between the V1/2 and
V3/2 potentials is much smaller. The dominant effect of spin-orbit coupling here is the characteristic pinching
above the potential crossing. The probability density function seems to ignore the presence of the V1/2, since the
probability is almost comparatively very small in the region between the classical turning points on the left and
on the right.
SV-coupledChannelSONaCs-v1.5.tex 23

24. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
b3
0
A1
V3 2
V1 2
5 10 15 20 25
0.05
0.06
0.07
0.08
0.09
0.10
R a0
PotentialenergyEh
195
cc 2
Figure 18: (Color online) Coupled-channel probability density function for vcc = 195. Potential energy curves
are in the background. The base line for the probability density function matches the corresponding rovibrational
energy. This probability density function exhibits spin-orbit coupling consequences only through the pinching
above the potential crossing. Compared to the v = 194 case, the function stretches from one end of V1/2 to the
other, almost ignoring the presence of the V3/2 potential.
SV-coupledChannelSONaCs-v1.5.tex 24

25. St´ephane
Valladier
SO coupled-channel rovibrational wave functions in NaCs
v. 1.5
4th November, 2013
b3
0
A1
V3 2
V1 2
5 10 15 20 25
0.05
0.06
0.07
0.08
0.09
0.10
R a0
PotentialenergyEh
235
cc 2
Figure 19: (Color online) Coupled-channel probability density function for vcc = 235. Potential energy curves
are in the background. The base line for the probability density function matches the corresponding rovibra-
tional energy, which lies above the asymptote of the V1/2 potential. The probability density function appears
to belong only to V3/2. The spin-orbit coupling still pinches the probability above the potential crossing. For
high-lying states, spin-orbit coupling replaces the local maximum in the probability density function above the
right classical turning point for V3/2 with a non monotonic increase of the locus of the top of the arches: the tops
form a plateau from 15a0 to ≈ 17a0.
SV-coupledChannelSONaCs-v1.5.tex 25