# Improving Potential Energy Surfaces Using Experimental Feshbach Resonance
Tomography

Karl P. Horn  Dahlem Center for Complex Quantum Systems and Fachbereich
Physik, Freie Universität Berlin, Arnimallee 14, D-14195 Berlin, Germany  Luis
Itza Vazquez-Salazar  [  Christiane P. Koch  Dahlem Center for Complex Quantum
Systems and Fachbereich Physik, Freie Universität Berlin, Arnimallee 14,
D-14195 Berlin, Germany  Markus Meuwly  [<EMAIL_ADDRESS>
###### Abstract

The structure and dynamics of a molecular system is governed by its potential
energy surface (PES), representing the total energy as a function of the
nuclear coordinates. Obtaining accurate potential energy surfaces is limited
by the exponential scaling of Hilbert space, restricting quantitative
predictions of experimental observables from first principles to small
molecules with just a few electrons. Here, we present an explicitly physics-
informed approach for improving and assessing the quality of families of PESs
by modifying them through linear coordinate transformations based on
experimental data. We demonstrate this ”morphing” of the PES for the
He-H${}_{2}^{+}$ complex for reference surfaces at three different levels of
quantum chemistry and using recent comprehensive Feshbach resonance (FR)
measurements. In all cases, the positions and intensities of peaks in the
collision cross section are improved. We find these observables to be mainly
sensitive to the long-range part of the PES.

a an the of on at with for and in into from by without These authors
contributed equally University of Basel]Department of Chemistry, University of
Basel, Klingelbergstrasse 80 , CH-4056 Basel, Switzerland. These authors
contributed equally University of Basel]Department of Chemistry, University of
Basel, Klingelbergstrasse 80 , CH-4056 Basel, Switzerland.

## Teaser

We introduce a systematic approach based on experimental data to validate and
improve molecular potential energy surfaces.

##  1 Introduction

The potential energy surface (PES) representing the total energy of a molecule
is a fundamental concept for characterizing the dynamics both in the gas and
condensed phase1, 2. With high-quality PESs, the computation of experimental
observables becomes possible with predictive power at a quantitative level. On
the other hand, while all physical observables depend on it, the PES itself
cannot be observed. This raises the question of how to obtain the most
accurate PES for a given system. From an electronic structure perspective, it
is known that full configuration interaction (FCI) calculations with large
basis sets provide the highest quality for the total energies of a molecule.
However, the unfavourable scaling of FCI with the number of electrons and
basis functions prevents its routine use for constructing full-dimensional
PESs for any molecule consisting of more than a few light atoms.
Alternatively, one may approach the question from an experimentalist’s
perspective and argue that the “most accurate PES” is the one that best
describes physical observations. Such an approach has been developed for
diatomic molecules: the rotational Rydberg-Klein-Rees (RKR) method solves the
“inversion problem” of obtaining the potential energy curve given
spectroscopic information.3 Rotational RKR has also been applied to triatomic
van der Waals complexes4, 5 but cannot be extended to molecules of arbitrary
size. Indeed, solving the “inverse problem”, i.e., determining the PES given
experimental observables and an evolution equation from which these
observables are calculated has in general turned out to be very difficult in
chemical physics.6 This concerns both the choice of observables as well as the
actual inversion procedure.

An alternative that is not particularly sensitive to the dimensionality of the
problem is ”morphing” PESs.7, 8 This method exploits the topological
relationship between a reference and a target PES. Provided that calculations
with the reference PES yield qualitatively correct observables
$\mathcal{O_{\rm calc}}$ when compared with experimental observations
$\mathcal{O_{\rm exp}}$, the squared difference $\mathcal{L}=|\mathcal{O_{\rm
calc}}-\mathcal{O_{\rm exp}}|^{2}$ can be used to reshape the PES through
linear or non-linear coordinate transformations (”morphing”).7 It capitalizes
on the correct overall topology of the reference PES and transmutes it into a
new PES by stretching or compressing internal coordinates and the energy
scale, akin to machining a piece of rubber. Alternatives for reshaping PESs
are machine learning-based methods such as $\Delta-$ML9, transfer learning10,
11, or differential trajectory reweighting.12 Morphing has been applied
successfully to problems in spectroscopy,13 state-to-state reaction cross
sections,14 and reaction dynamics15 for systems with up to 6 atoms16. A near-
quantitative reference PES has, however, so far not been available for direct
comparison. For scattering experiments with He–H${}_{2}^{+}$ such a PES is now
available.

The He–H${}_{2}^{+}$ molecular complex is an ideal proxy for the present work
owing to the fact that the PES can be calculated rigorously at the highest
level of quantum chemistry (FCI). The complex is also interesting in itself,
and the current status of experimental and computational spectroscopy and
reaction dynamics has recently been reviewed.17 He–H${}_{2}^{+}$, which is
isoelectronic to H3, is stable in its electronic ground state and features a
rich reaction dynamics and spectroscopy. Experimentally, near-dissociative
states18, 19 and the low-resolution spectroscopy were reported for both,
He–H${}_{2}^{+}$ and He–D${}_{2}^{+}$.20 Assignments of the vibrational bands
were possible by comparing with bound state calculations utilizing a FCI
PES.21 Only recently, it was possible to estimate the dissociation energy of
$\sim 1800$ cm-1 from spectroscopic measurements.20 This compares with earlier
bound state calculations using the FCI PES predicting a value of $D_{0}=1784$
cm-1.21 This value was confirmed from a subsequent focal point analysis
resulting in $D_{0}=1789(4)$ cm-1 for para-H${}_{2}^{+}$.22 Furthermore, a
range of reactive collision experiments was carried out which yielded total
and differential cross sections, depending on the vibrational state of the
diatomic,17 but with marked differences between observed and computed results.
In particular, computationally predicted sharp reactive scattering resonances
have not been found experimentally as of now.17 Finally, the role of
nonadiabatic couplings is of considerable current interest as a means to
clarify the role of geometric phase in reaction outcomes and as a source of
friction in the formation of the He–H${}_{2}^{+}$ complex in the early
universe. This provides additional impetus for a comprehensive
characterization of this seemingly “simple” system.

The present work uses experimentally measured Feshbach resonances for
He–H${}_{2}^{+}$23 to morph potential energy surfaces. Feshbach(-Fano)
resonances arise if a bound molecular state on a potential energy surface of a
closed channel couples to scattering states in an open channel. 24, 25 The
recoil translational energy is determined from measurements which are expected
to probe large spatial areas of a PES and the underlying intermolecular
interactions.25 The redistribution of energy due to the Feshbach resonances
has recently been mapped out comprehensively for He–H${}_{2}^{+}$ and
Ne–H${}_{2}^{+}$ with coincidence velocity map imaging of electrons and
cations, yielding very favourable agreement between theory and experiment.23
In these experiments, the ionic molecular complexes are generated at
separations of up to 10 a0 between the rare gas atom and the molecular ion,
confirming that the experiment indeed probes a large spatial extent of the
PES, including its long-range part.

Here, morphing is applied to initial PESs ranging from essentially exact FCI
(apart from non-Born-Oppenheimer and remaining basis set effects) to medium-
and lower-level methods, that is, Multi-Reference Configuration Interaction
including the Davidson correction (MRCI+Q) and second-order Møller-Plesset
perturbation theory (MP2). This allows us to determine the sensitivity of the
PES and information content in the experimental observables about local and
global features of the PES and to assess the performance of lower-level
methods (e.g. MP2) compared with FCI. We found that starting from a PES of
sufficiently high quality, the changes introduced by morphing can be related
to parts of the PES that are probed by the experiments. At the same time,
additional experimental observables, probing primarily the bound region for He
interacting with H${}_{2}^{+}$, will be required for morphing at the lower
levels of quantum chemical theory.

##  2 Results

The three PESs considered in the present work, in decreasing order of rigour,
were determined at the FCI, MRCI+Q, and MP2 levels of theory, using Jacobi
coordinates $R$ (distance between the centre of mass of the H${}_{2}^{+}$ and
He), $r$ (distance between the two hydrogen atoms), and $\theta$ (the angle
between the two vectors $\vec{R}$ and $\vec{r}$), see Figure 1A. To set the
stage, scattering calculations with the FCI PES are considered which give
quite satisfactory results when compared with the experimental data (Figure 2
A and Table 1). The measured kinetic energy distributions feature a series of
peaks which reflect the rovibrational quenching associated with the decay of
the Feshbach resonances.23 On average, the positions of the peak maxima are
reproduced to within 10.8 cm-1 whereas the maximum intensities, $I_{\rm max}$,
of $P(E)$ differ by 20.9 arb. u. (blue squares in Figure 2A).

Figure 1: Morphing of ab initio potentials based on experimental data. General
flowchart of the morphing procedure (A): Module (1) implements the calculation
of ab-initio points for the system under study, the triatomic HeH${}_{2}^{+}$
with the definition of the respective coordinates indicated. Module (2)
represents the fitting of the points obtained from the previous step using the
Reproducing Kernel Hilbert Space Method, with the functional form used to
approximate the given PES. Module (3) corresponds to the scattering
calculations performed with the potential obtained in module (2), calculating
the eigenstates of the Hamiltonian. Module (4) post-processes the results of
the scattering calculations, resulting in the cross sections, with examples
for three values of $j^{\prime}$ displayed. Module (5) evaluates the loss
function for morphing, comparing the experimental values of the cross sections
with the results of the scattering calculations. Module (6) carries out the
actual morphing procedure, as explained in panel B. Morphing results in a new
potential, and the procedure continues until the value of the loss function in
module (5) does not improve further. The termination conditions are
$\mathcal{L}/\mathcal{L}_{0}\leq\lambda_{\rm M1}=0.3$ or
$\mathcal{L}/\mathcal{L}_{0}\leq\lambda_{\rm M2}=0.4$ for M1 and M2,
respectively where $\mathcal{L}_{0}$ is the loss function of the unmorphed
cross section, see Figure S6. Panel B: Morphing module (6) for procedures M1
(3-body) and M2 (global).

Next, morphing is applied to all three PESs, including the FCI PES. The FCI
PES has been validated with respect to experiment18, 19, 20, 23 and therefore
can serve as a suitable proxy for changes required for PESs at the MRCI+Q and
MP2 levels of theory. Two morphing strategies were considered (Figure 1B): For
Morphing M1, the total energy was decomposed into one-body
($\mathcal{V}_{i}^{(1)}$), two-body ($\mathcal{V}_{i}^{(2)}$) and three-
body($\mathcal{V}^{(3)}$) terms ,

$V(R,r,\theta)=\mathcal{V}^{(1)}_{\rm He}+\mathcal{V}^{(1)}_{\rm
H}+\mathcal{V}^{(1)}_{\rm H^{+}}+\mathcal{V}_{\rm HeH}^{(2)}(r_{\rm
HeH})+\mathcal{V}_{\rm HeH^{+}}^{(2)}(r_{\rm HeH^{+}})+\mathcal{V}_{\rm
H_{2}^{+}}^{(2)}(r_{\rm H_{2}^{+}})+\mathcal{V}^{(3)}(R,r,\theta)\,,$ (1)

and the morphing transformation was applied only to the three-body
contribution $\mathcal{V}^{(3)}(R,r,\theta)$. Approach M1 is motivated by the
assumption that all diatomic potentials $\mathcal{V}_{i}^{(2)}$ are of high
quality so that changes are only required in the higher-order correction
three-body term. In the second approach, called “M2”, the morphing
transformation maps $(R,r)\rightarrow(\alpha R,\beta r)$ and the total energy
is multiplied by $\varepsilon$ to obtain the morphed energy. In this case, the
PES is globally modified, including the two-body contributions.

Morphing M1 applied to the FCI PES leaves most of the peak positions
unchanged, see filled vs. open blue symbols in Figure 2D, but improves the
peak heights considerably (by 30 %) as demonstrated in Figure 2E and Table 1
(rightmost column). These improvements are accommodated by reshaping the
underlying PES as shown in Figure 3A: In the long-range ($R>3.0$ a0), the
anisotropy of the morphed PES is somewhat decreased due to reshaping the PES
around $\theta=90^{\circ}$ (T-shaped geometry). On the other hand, the
curvature along the angular coordinate is increased. One-dimensional cuts
along the $r_{\mathrm{HH}}$ and $R$ coordinates for a given angle $\theta$
show that changes in the PES become more significant at larger values of
$r_{\mathrm{HH}}$ with small changes in the depth of the potential wells but
maintaining the overall shape of the curves (Figures S7 and S8). The changes
with respect to $R$ are noticeable for $R<3.0$ a0 with small distortions of
the energy contours at different angles $\theta$, but maintaining the overall
shape of the curves. For larger values of $R$, the changes are negligible
compared with the original PES, reflecting the accurate treatment of the long-
range interaction (Figure S7). 2D projections of the combined changes of
$r_{\mathrm{HH}}$ and $R$ at different angles show that the most pronounced
modifications in the shape of the PES concern regions of $r_{\rm HH}$ larger
than the equilibrium geometry of H${}_{2}^{+}$, i.e., $r_{\rm HH}>2.1$ a0, and
$R=2-3$ a0 (Figures S9A S10A and S11A).

Figure 2: Cross sections obtained from experiment (black, data taken from Ref. 23) and full coupled channels calculations using the unmorphed and M1-morphed PESs for FCI (A), MRCI+Q (B), and MP2 (C). Computed results for the initial (blue, green, red dashed) and best (blue, green, red solid) morphed PESs are reported, with the residuals for the peak positions ($E_{\rm exp}-E_{\rm calc}$) and fraction of error in the peak heights ($\frac{{P(E)_{\rm exp}-P(E)_{\rm calc}}}{P(E)_{\rm calc}}$) for each PES shown in Panels D and E. The statistical measures for all models are summarized in Table 1. Surface |  $D_{e}$ (cm-1) |  $R_{e}/$a0 |  $r_{e}/$a0 | RMSE$(E)$ | RMSE$(I)$
---|---|---|---|---|---
|  |  |  | (cm-1) | (arb. u.)
FCI Initial | 2818.9 | 2.97 | 2.07 | 10.8 | 20.9
FCI Morphed (M1) | 2772.0 | 2.95 | 2.07 | 11.9 | 13.7
FCI Morphed (M2) | 2819.1 | 2.99 | 2.07 | 10.8 | 13.8
MRCI+Q Initial | 2557.3 | 2.98 | 2.07 | 10.3 | 23.9
MRCI+Q Morphed (M1) | 3414.7 | 2.98 | 2.08 | 12.2 | 21.9
MRCI+Q Morphed (M2) | 2557.0 | 3.00 | 2.03 | 8.9 | 17.6
MP2 Initial | 2494.0 | 2.99 | 2.07 | 13.1 | 22.4
MP2 Morphed (M1) | 1685.6 | 2.93 | 2.12 | 12.8 | 10.9
MP2 Morphed (M2) | 2492.8 | 2.97 | 1.74 | 10.0 | 11.8
MP2 Morphed (PES-to-PES) | 2502.3 | 2.98 | 2.06 | 13.0(7) | 22.9
Table 1: Dissociation energies ($D_{e}$ in cm-1) for He+H${}_{2}^{+}$,
coordinates for the minimum energy structures, $R_{e}$ and $r_{e}$, and root
mean squared errors (RMSE) for the peak positions and heights of the kinetic
energy spectra for all initial and morphed PESs using both M1 and M2 methods.
In all cases, the equilibrium geometry is linear He–H${}_{2}^{+}$, i.e.
$\theta=0$ or $\theta=180^{\circ}$.

FCI calculations of entire PESs are only feasible for the smallest chemical
systems, i.e. for diatomic or triatomic molecules. For larger systems, quantum
chemical methods such as multi-reference configuration interaction or Møller-
Plesset perturbation theory need to be used instead. As reported in the two
rightmost columns of Table 1, the initial MRCI+Q and MP2 PESs perform almost
on par compared with the initial FCI PES for the positions and intensities of
the kinetic energy distributions. On the other hand, the dissociation energy
is smaller by more than 10% compared with the FCI PES due to partial neglect
of correlation energy in the MRCI+Q and MP2 methods. This confirms that
Feshbach resonances are not particularly informative with regards to features
of the PES around the minimum energy structure ($R\sim 3.0$ a0), although the
wavefunctions sample this region extensively, see Figure S20. In other words,
although an important characteristic of a PES such as the stabilization energy
of the complex differs by 10 % or more, the energies and intensities measured
in collision experiments are matched within similar bounds.

Figure 3: Projections of the PESs for $r_{\mathrm{HH}}=2.0$ a0 for the three
methods studied here. Isocontours for unmorphed PESs (FCI (blue), MRCI+Q
(green) and MP2 (red) from left to right) are shown as dashed lines, whereas
the M1-morphed PESs are solid lines. The zero of energy is set by the value at
$r=2.0$ a0 and $R=\infty$. Energies are in cm-1.

Morphing M1 applied to the MRCI+Q and MP2 PESs supports this observation. The
loss function evaluated in module (5) of the optimization, see Figure 1,
decreased by 74% and 88% for the two cases, with improvements in the
intensities by up to 50% for the MP2 PES, see Table 1 (rightmost column).
However, the resulting PESs are clearly unphysical, with pronounced
distortions in particular for the MP2 PES, see Figure 3C and dissociation
energies either increased by 40 % for MRCI+Q or decreased by 30 % for MP2,
respectively. Low-resolution experiments20 provide an estimate for the
dissociation energy $D_{0}\sim 1800$ cm-1, compared with $D_{0}=1794$ cm-1
from bound state calculations on the initial FCI PES21 which features a well
depth of $D_{e}\sim 2820$ cm-1. This value of $D_{e}$ serves as a reference
for the remainder of the present work.

The percentage changes of the parameters $[\alpha,\beta,\varepsilon]$ scaling
$(R,r,V)$ provide further information about the transformation induced by
morphing the initial PESs. For the FCI PES they are $(-0.6,-3.6,0.0)\%$
compared with $(-0.6,11.6,1.0)$% for the MRCI+Q and $(0.3,-9.7,0.1)$% for the
MP2 PES. The most significant changes concern the H${}_{2}^{+}$ vibrational
coordinate $r_{\mathrm{HH}}$ for MRCI+Q ($+12.0$%) and MP2 ($-10.0$%). Such
large changes are problematic since the many-body expansion used for morphing
M1, cf. Eq. (1), relies on the quality of the two-body contributions, i.e.,
the H${}_{2}^{+}$ and HeH+ potential energy curves. However, MP2
underestimates the experimentally determined dissociation energy of the HeH+
two-body interaction by 285 cm-1 (Figure S12) and accounts for an overall
error of $\sim 500$ cm-1 in $D_{e}$ for He–H${}_{2}^{+}$. On the other hand,
the two-body term for H${}_{2}^{+}$ agrees to within 3 cm-1 between the three
methods with remaining differences compared with the experiment primarily due
to neglect of non-Born-Oppenheimer contributions (Figure S13). To summarize:
while M1-morphing improves the match between experimentally measured and
calculated observables, it modifies the PES for the lower-level methods in an
unphysical way. This is attributed to the fact that M1-morphing operates on
the three-body term only and can thus not compensate for inaccuracies in the
two-body contributions to the overall PES. In contrast, for FCI the changes
for all characteristics of the morphed PES are moderate, underscoring the
accuracy of both, the initial and morphed PESs from FCI calculations.

Figure 4: Cross sections (A,C) obtained from experiment (black, data taken
from Ref. 23) and full coupled channels calculations using the unmorphed
(dashed lines) and M2-morphed (solid lines) PESs (B,D) for MRCI+Q (A,B), and
MP2 (C,D). The RMSE for the peak positions and heights are reported in Table
1. The projections of the PES (B,D) are shown for $r=r_{e}$ a0 (See Table 1)
with the zero of energy set for the $r-$value considered and $R=\infty$.
Energies are in cm-1. The changes in the PES suggest that the observables are
primarily sensitive to the long-range part and the repulsive wall of the PES.

To reduce the dependence of the morphed PESs on the quality of the two-body
contributions, morphing M2 was carried out. M2-morphing acts globally and
independently on each of the internal degrees of freedom, see Figure 1. This
makes M2 less prone to overcompensatory effects as observed for M1-morphing.
For the MRCI+Q PES the improvement in the observables amounts to $\approx 14$
% for the peak positions and $\approx 26$ % for the peak heights. At the same
time the changes in the PES are moderate, see Figure 4B, and the dissociation
energy does not change (Table 1) although the energy scaling parameter,
$\varepsilon$ was allowed to vary. Similarly, for MP2, the RMSE for the
positions and heights of the peaks improve by about 22 % and 47 %,
respectively. Contrary to M1, morphing M2 does not significantly modify the
well depth as reflected by the value of $D_{e}$, see Table 1.

For the optimal morphing parameters, M2 applied to the MRCI+Q PES yields an
enlargement of $R$ by $\sim 1$ % whereas $r_{\mathrm{HH}}$ is reduced by
$1.9\%$ and $\varepsilon$ remains unaffected. The reduction in
$r_{\mathrm{HH}}$ leads to a small increase in the height of the barrier
between the two wells of the potential (Figure 4B) and a corresponding
increase in the energy of the transition state, as observed in the minimum
energy path (MEP), see Figure S14, for the angular coordinate. This effect is
compensated by a positive displacement of the values of $R$ (Figure S15) for
the MEP. On the other hand, for the MP2 surface, the morphing parameters are
($+0.6$, $+19.0$, $-0.04$) %. The large positive value for $\beta$ results in
a displacement of the H${}_{2}^{+}$ bond length to a shorter equilibrium value
(Figure S16 and S17). For the $R$ coordinate, the values are also reduced
while the barrier height remains unchanged (Figure S15). As for M1, in the MP2
and MRCI+Q PESs the largest changes are observed in the $r_{\mathrm{HH}}$
coordinate. However, in the M2 method, scaling of the global PES results in a
better performance for the calculation of the observable and a better physical
description.

Finally, morphing one PES into another one can probe the flexibility of the
morphing transformation as a whole. To this end, the MP2 PES was morphed to
best compare with the FCI PES in a least squares sense according to method M2,
i.e., by finding parameters $(\alpha,\beta,\varepsilon)$ that minimize
$(V_{\rm FCI}(R,r,\theta)-\varepsilon V_{\rm MP2}(\alpha R,\beta
r,\theta))^{2}$. This optimization procedure reduces the RMSE between the FCI
and unmorphed vs. morphed PES by about 30% (from 138 cm-1 to 97 cm-1, see
Figure S18). The changes in the topology of the surface in Figure 5C indicate
that the morphed MP2 PES is ”pulled towards” the FCI PES: Consider, for
example, the isocontours for $-400$ cm-1 for which the original MP2 isocontour
(blue) is far away from the FCI target contour (red), whereas the morphed PES
(grey) is deformed towards the grey target isocontour. Closer inspection
reveals this to occur for all the other isocontours in Figure 5C as well. The
barrier separating the [He–HH]+ and [HH–He]+ minima is reduced, which is also
seen in the minimum energy path (see Figure S19).

Figure 5: Morphing PES-to-PES. Panel A: Cross-sections obtained from
experiments (black, data taken from Ref. 23) and scattering calculations on
the unmorphed MP2 (dashed light red) and the morphed (grey) PESs for M2 PES-
to-PES morphing procedure with the FCI PES as target. Panel B: Same as Panel A
but comparing the best morphed PES (grey) to the unmorphed FCI surface (solid
blue). Panel C: 2D projections of the PES for $r=2.0$ a0 for unmorphed FCI
(solid blue), unmorphed MP2 (dashed light red) and best-morphed PES (grey).
The zero of energy is set to the value of the PES at $r_{\mathrm{HH}}=2.0$ a0
and $R=\infty$. Energies are in cm-1.

The results of the scattering calculations performed with the surface from the
PES-to-PES morphing procedure (Figure 5A) are overall slightly inferior to
those obtained from the initial FCI and MP2 PESs, when compared with the
experimental data: a negligible increase of the RMSE for the peak positions
($<1\%$) and intensities (2.2 %) is found. Moreover, the fact that the
morphing transformation increases the well depth by merely 10 cm-1 indicates
that the simple morphing transformation scaling only distances and the energy
is not sufficiently flexible to accommodate global changes between topologies
as different as FCI vs. MP2.

The results indicate that at all levels of theory improvements in describing
the experimental observables are possible. At the same time morphing applied
in the fashion done here provides a stringent test to probe the quality of an
initial PES at a quantitative level - with higher initial levels of theory,
the changes that need to be accommodated decrease and specific deficiencies of
a particular quantum chemical approach can be unveiled.

##  3 Discussion and Outlook

Given that essentially exact quantum calculations are possible for the
He–H${}_{2}^{+}$ complex,21, 20, 23 the present results highlight what can and
cannot be learned about molecular PESs — the central concept in classical and
quantum molecular dynamics — from accurate and rather comprehensive
experimental data based on Feshbach resonances. One hallmark of such quantum
scattering resonances is the large spatial extent of the PES which the
resonance wavefunction probes (Figure S20 and discussion in SI). In this
regard, the kinetic energy spectrum obtained from the decay of the Feshbach
resonances differs from spectroscopic observables, typically involving bound
states sensitive to smaller spatial regions of the PES.7

In addition to the actual changes of the PES, a comparison of the two morphing
procedures employed provides insight into the relationship between the PES,
the information provided by specific observables, and how this information can
be used to improve an initial PES. First, the much better performance of
morphing the global interaction energy instead of restricting to the three-
body contributions reveals the importance of corrections already at the level
of two-body interactions. Moreover, the physically meaningful changes to the
PES identified by the global morphing concern essentially the anisotropy in
the long range. To this end, comparatively small changes of the PESs result in
significant improvements in the agreement between calculated and measured
observables. This is in line with the expectation that Feshbach resonance
wavefunctions mainly probe the anisotropy of the PES in the long-range. Both
observations taken together suggest extending the morphing transformation to
include higher-order (and nonlinear) terms as well as the angular degree of
freedom $\theta$ for further improvements.

At a fundamental level, the present findings raise the question how much and
what experimental data is required to completely characterize a molecular PES.
Indeed, the present work proposes several PESs with comparable average
performance on the scattering observables, even though the shapes and local
characteristics of the PESs differ greatly, illustrating that the information
contained in the Feshbach resonances is not sufficient to uniquely define the
PES. In particular, information on the bound-state region is missing. One
possible way to answer the question which combination of observables is suited
to completely characterize the dynamics of a molecular system has been
developed in quantum information science and is referred to as quantum process
tomography.26 It has been adapted to molecular systems for example in the
context of ultrafast spectroscopy. In future work, quantum process tomography
could be applied to the quest of uniquely defining a PES by making use of the
mapping between the real-space representation of the molecular Hamiltonian and
qubits.27 This should allow for a systematic approach to identify the most
important measurements which would then provide additional data for morphing
PES.

##  4 Methods

###  4.1 Potential Energy Surfaces

For the present work, three PESs were employed. Full-dimensional PESs for
He–H${}_{2}^{+}$ were previously determined at the FCI/aug-cc-pV5Z and
MRCI+Q/aug-cc-pV6Z levels of theory, respectively.21 The reference data was
represented as a reproducing kernel Hilbert space (RKHS)28, 29 which provides
a highly accurate interpolation and allows to encode the leading order long-
range behaviour for large separations. In addition, a third PES using the same
underlying grid for determining reference energies at the MP2/aug-cc-pV5Z
level and also represented as a RKHS, was constructed for the present work.
These calculations were carried out using the MOLPRO suite of codes.30 All
PESs are represented as a sum of diatomic potential energy curves together
with an explicit three-body interaction. The complete many-body expansion for
the He–H${}_{2}^{+}$ system is given in Eq. (1), where distances $r_{i}\in$
{$r_{\rm HeH},r_{\rm HeH^{+}},r_{\rm H_{2}^{+}}\\}$ in the two-body terms
$\mathcal{V}^{(2)}_{i}$ are the distances between the respective atoms,
whereas for the three-body term $\mathcal{V}^{(3)}(R,r,\theta)$ the coordinate
$r$ is the H${}_{2}^{+}$ separation $r_{\rm H_{2}^{+}}$, $R$ the distance
between He and the centre of mass of the diatomic, and $\theta$ the angle
between the two distance vectors $\vec{r}$ and $\vec{R}$. Finally,
$\mathcal{V}^{(1)}$ corresponds to the respective atomic energies. The
energies $\mathcal{V}_{i}^{(1)}$ and $\mathcal{V}^{(2)}_{i}$ were also
determined at the respective level of theory from electronic structure
calculations and the contributions $\mathcal{V}^{(2)}_{i}$ were fitted to
analytical expressions described in Ref. 21. The fitting parameters for the
FCI and MRCI levels of theory were published before and those for the MP2
level of theory are provided in the supporting information. Combining all this
information, the three-body contribution $\mathcal{V}^{(3)}(R,r,\theta)$ was
obtained on the grid used in the electronic structure calculations for
$V(R,r,\theta)$ and represented as a RKHS.

###  4.2 Scattering calculations

Integral scattering cross sections and scattering wave functions for
He-H${}_{2}^{+}$, resulting from a spatially distributed input wave packet,
were evaluated using a home-written coupled-channels collision simulation
based on the renormalized Numerov method.31, 32 Details on these calculations
have been given in earlier work 23 and only the salient features are presented
here. The wavepacket simulations use Jacobi coordinates with $\vec{r}$ the
vector between the hydrogen atoms, $\vec{R}$ the vector from the dihydrogen
centre of mass to the helium atom and $\theta$ the angle between the two
vectors. With $R=|\vec{R}|$ and $r=|\vec{r}|$, the total Hamiltonian is then

$H_{\mathrm{tot}}=-\frac{\hbar^{2}}{2\mu_{\mathrm{cmplx}}}\nabla^{2}_{\vec{R}}-\frac{\hbar^{2}}{2\mu_{\mathrm{diat}}}\nabla^{2}_{\vec{r}}+V(R,r,\theta)\,\,,$
(2)

where $\mu_{\mathrm{cmplx}}$ is the reduced mass of the three-body complex,
$\mu_{\mathrm{diat}}$ the reduced mass of the dihydrogen molecule, and
$V(R,r,\theta)$ the three-dimensional PES. The total wavefunction of the
system $\Psi(\vec{R},\vec{r})$ is written as a product of $R-$, $r-$, and
angularly dependent terms,

$\Psi^{JMvj\ell}(\vec{R},\vec{r})\propto\sum_{v^{\prime}j^{\prime}\ell^{\prime}}G_{v^{\prime}j^{\prime}\ell^{\prime}}^{Jvj\ell}(R)\chi_{\mathrm{diat},v^{\prime}j^{\prime}}(r)\sum_{m_{j}=-j}^{j}\sum_{m_{\ell}=-\ell}^{\ell}C_{m_{j}m_{\ell}}^{JM}Y_{\ell,m_{\ell}}(\theta_{R},\varphi_{R})Y_{j,m_{j}}(\theta_{r},\varphi_{r})\,\,,$
(3)

see Ref.23 for more detail. Channels consist of tuples of quantum numbers
$v,j$, and $\ell$, corresponding to diatomic vibration, rotation and orbital
angular momentum, respectively. In Eq. (3), $\chi_{\mathrm{diat},v,j}(r)$
designates the rovibrational eigenstates of the molecule. Starting from a
given entrance channel, the Schrödinger equation is solved numerically to
obtain the radial wave functions $G(R)$ for the exit channel with quantum
numbers $(v^{\prime},j^{\prime},\ell^{\prime})$ connected with the entrance
channel $(v,j,\ell)$. The total angular momentum,
$\vec{J}_{\mathrm{tot}}=\vec{j}+\vec{L}$ obtained from coupling diatomic and
orbital rotation, and parity are conserved under the Hamiltonian (2).

In the experiments, the He–H${}_{2}^{+}$ complex (plus a leaving electron) is
formed by Penning ionization (He∗+H2), and the scattering calculations
considered in the present work describe the half-collision on the
He–H${}_{2}^{+}$ PES. The initial wavepacket $\phi(R)$ along the
$R-$coordinate is approximated by Gaussian distributions centered around
$R\approx 8~{}a_{0}$.23 The experiment prepares the input wavepacket with
$j_{\mathrm{wp}}=0,1$ for para- and ortho-H${}_{2}^{+}$, respectively.
However, as the system is prepared in a superposition of $J-$states,
individual simulations need to be carried out for each possible value of $J$
and partial wave $\ell$. Then, the integral cross section is calculated as a
weighted sum over the individual contributions for a given collision energy
$E_{\mathrm{col}}/\mathrm{k_{B}}\approx 2.5$ K. The $J-$weights, which were
calculated separately,33 are shown in Figure S21.

Evaluation of the collision cross section due to the spatially distributed
input wavepacket can be accomplished by expanding $\phi(R)$ in a basis of
eigenfunctions of $H_{\mathrm{tot}}$. To this end, the time-independent
Schrödinger equation was solved on a discretized interval of 1002 energies
ranging from $100~{}\mathrm{cm}^{-1}$ below to $100~{}\mathrm{cm}^{-1}$ above
the dissociation threshold of the given entrance channel. Because full
coupled-channel calculations are computationally demanding, the considered set
of initial wavepacket quantum numbers $J$ and $\ell$ was limited to
$(\ell/J)\in\\{(0/0),(1/1),(2/2),(3/3),(4/4)\\}$ for para- and
$(\ell/J)\in\\{(0/1),(1/1,2),(2/1,2,3)$ $,(3/2,3,4),(4/3,4,5)\\}$ for ortho-
dihydrogen, respectively. For each coupled channel calculation a converged
basis set of diatomic rotational states up to $j_{\rm max}=19$ and diatomic
vibrational states up to $v_{\max}=5$ was used.

Solving the Schrödinger equation in this fashion allows for calculating the
channel-resolved integral cross section for each energy in the discretized
interval. For a given output channel, the eigenenergy
$E_{v^{\prime}j^{\prime}\ell^{\prime}}=E_{\mathrm{int},v^{\prime}j^{\prime}\ell^{\prime}}+E_{\mathrm{kin},v^{\prime},j^{\prime},\ell^{\prime}}$
can be decomposed into its internal and kinetic parts, respectively. By
generating a histogram for all output channels
($v^{\prime}$,$j^{\prime}$,$\ell^{\prime}$), the cross-section can be
expressed as a function of kinetic energy, which can be compared with the
experimental results. Next, the kinetic energy histogram is convoluted using a
Gaussian envelope to account for the finite resolution in the experiments.23
Before convolution, and as shown in Figure S22, the computed peaks are sharp
in $E_{\mathrm{kin}}$ which is a signature of Feshbach resonances. It should
be noted that experimental peaks are clearly distinguishable and energetically
match the theoretical predictions. However, the peak shapes and heights can
vary, dependent on the histogram resolution and convolution width. In this
work, only single initial vibrational excitations ($v=1$) were considered, in
order to exploit the experimental resolution of separate $j^{\prime}$ peaks in
the cross-section as a function of kinetic energy 34.

###  4.3 Morphing

The morphing transformation considered here is

$V_{\mathrm{morphed}}(R,r,\theta)=\varepsilon V_{\mathrm{ab-initio}}(\alpha
R,\beta r,\theta)\,\,.$ (4)

In Eq. (4), the three parameters ($\alpha,\beta,\varepsilon$) are used for
energy- ($\varepsilon$) and geometry-($\alpha,\beta$) related scalings. For
the purpose of this work, the angle $\theta$ was not modified. The morphing
procedure described further below optimizes the values of
$(\alpha,\beta,\varepsilon)$ such that the difference between observed and
computed features of the resonances is minimized. Application of such a
procedure modifies local features (e.g. slope, curvature) of the PES but
maintains its global shape.

For morphing M1 and M2 the refinement with respect to experimental values is
formulated as an optimization problem with a loss function,

$\mathcal{L}=\min_{\alpha,\beta,\varepsilon}\left[\sum_{j^{\prime}}|E_{\mathrm{exp}}^{(j^{\prime})}-E_{\mathrm{calc}}^{(j^{\prime})}(\alpha,\beta,\varepsilon)|+\sum_{j^{\prime}}\delta_{h(j^{\prime})}^{\kappa}\right]\,\,,$
(5)

to be minimized. Here, $E^{(j^{\prime})}$ is the kinetic energy of each cross-
section corresponding to an exit-channel $j^{\prime}$, and
$\delta_{h(j^{\prime})}^{\kappa}$ accounts for the difference in the peak
heights between experimental and calculated values:

$\delta_{h(j^{\prime})}^{\kappa}=\left\\{\begin{array}[]{cr}(\Delta
h(j^{\prime})-h_{\mathrm{noise}})^{\kappa},&(\Delta
h(j^{\prime})-h_{\mathrm{noise}})^{\kappa}>0\,\,,\\\ 0,&(\Delta
h(j^{\prime})-h_{\mathrm{noise}})^{\kappa}\leq 0\,\,,\end{array}\right.$ (6)

where, $\delta_{h}(j^{\prime})$ is regularized by subtracting
$h_{\mathrm{noise}}=10.0$ to avoid fitting experimental noise. By design, only
values $\delta_{h}(j^{\prime})>0$ contribute to the error. Here $\Delta
h(j^{\prime})=|h^{(j^{\prime})}_{\mathrm{exp}}-\gamma
h^{(j^{\prime})}_{\mathrm{calc}}(\alpha,\beta,\varepsilon)|$, where
$h^{(j^{\prime})}$ is the peak height of the cross section corresponding to an
exit-channel $j^{\prime}$. The parameter $\gamma$ is recalculated after each
iteration to best match the experiment by performing an additional 1d
minimization over the squared difference in peaks heights.

The workflow to perform the optimization of Eq. (5) is shown schematically in
Figure 1. In the first step, ab initio points of the PES are used to generate
a RKHS kernel. Depending on the morphing procedure chosen, a new RKHS needs to
be generated (for M1) or the existing kernel will be reused (for M2). All
kernels are constructed and evaluated using the “fast” method.28 The obtained
PES is passed to the scattering code to perform the wavepacket propagation.
Next, the resulting cross-sections are processed and then compared with the
available experimental data. If the difference between experimental and
calculated values matches a given tolerance the cycle finishes; otherwise, the
PES is modified by three parameters as described in Eq. (4) following the
chosen morphing approach. The values of the parameters $\alpha$, $\beta$ and
$\varepsilon$ were obtained by a non-linear optimization using the NLopt
package35.

## Data Availability Statement

The data needed for the PESs is available at
https://github.com/MeuwlyGroup/morphing.

## Acknowledgments

This work was supported by the Swiss National Science Foundation grants
200021-117810, 200020-188724, the NCCR MUST, and the University of Basel which
is gratefully acknowledged. The authors thank B. Margulis and Prof. E.
Narevicius for providing the experimental data used in this work.

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##  Appendix S1 Two-body potential fitting

In this work, the two-body interaction energy for a molecule AB was expressed
as21, 36, 37

$\mathcal{V}^{(2)}_{AB}(R_{AB})=\dfrac{c_{0}e^{-\alpha_{AB}R_{AB}}}{R_{AB}}+\sum_{i=1}^{M}C_{i}\rho_{AB}^{i}+\mathcal{V}_{long}(\tilde{r})$
(S7)

In equation S7, $C_{i}$ are linear coefficients with $C_{0}>0$ to assure that
the diatomic potential remains repulsive ($V_{\rm AB}(r_{\rm
AB})\rightarrow\infty$) for $r_{\rm AB}\rightarrow 0$ and $\rho_{\rm AB}^{i}$
is defined as:

$\rho_{AB}^{i}=R_{AB}e^{-\beta^{(2)}_{AB}R_{AB}}$ (S8)

The long range part of equation S7 is written as38:

$\mathcal{V}_{long}(\tilde{r})=-\dfrac{\alpha_{d}q^{2}}{2\tilde{r}^{4}}-\dfrac{\alpha_{q}q^{2}}{2\tilde{r}^{6}}-\dfrac{\alpha_{o}q^{2}}{2\tilde{r}^{8}}-\dfrac{\beta_{ddq}q^{3}}{6\tilde{r}^{7}}-\dfrac{\gamma_{d}q^{4}}{24\tilde{r}^{8}}$
(S9)

In equation S9, $q$ is the charge, $\alpha_{i},\ i\in\\{d,q,o\\}$ are the
dipole, quadrupole and octupole polarizabilities for H and He atoms,
respectively, and $\beta_{ddq}$ and $\gamma_{d}$ are the first and second
hyperpolarizabilities, respectively. Values for these parameters were taken
from Refs. 38, 39. Finally, the coordinate $\tilde{r}$, whose objective is to
remove the divergence of the long-range terms at short separations of H-H and
H-He, is defined as40

$\tilde{r}=r+r_{l}\exp{(-(r-r_{e}))}$ (S10)

Here, $r_{l}$ is a distance parameter, and $r_{e}$ is the equilibrium bond
distance of the diatomic molecule. The linear coefficients $C_{i}$ and the
parameters $\alpha_{AB}$ and $\beta_{AB}^{(2)}$ in equations S9 and S8 were
taken from Ref. 21 for FCI and MRCI potentials. For the MP2 potential, the
values were determined using the same method as described in Ref. 21 and are
given in Table S2.

Coefficients | H${}_{2}^{+}$ | HeH+
---|---|---
C0 | 1.01921689 | 10.7720718
$\alpha$ | 1.64361518 | 2.37373920
C1 | -0.73449753 | -4.46348296
C2 | 5.10324938 | 59.1487168
C3 | -81.5643228 | -3857.67751
C4 | 847.329344 | 104277.881
C5 | -5377.78872 | -1890643.12
C6 | 21730.0685 | 21015000.8
C7 | -56454.3034 | -1.3583E+08
C8 | 91470.4779 | 4.1364E+08
C9 | -84131.3637 | -29244093.5
C10 | 33516.3571 | -2.0736E+09
$\beta$ | 0.99789130 | 2.23414441
Table S2: Coefficients for the MP2/aug-cc-pV5Z diatomic potentials.

##  Appendix S2 Discussion of morphing M1 for MRCI and MP2 PESs

Multi-Reference CI: Figure 2B compares the cross sections from experiments
with the results from computations with PESs before and after morphing M1 for
the MRCI+Q PES. Overall, the RMSE for the energies changes from 10.3 to 12.2
cm-1, whereas the intensities improve from an RMSE of 23.9 to 21.9 arb. u. The
results indicate that M1 has the most pronounced impact on intermediate values
of $j^{\prime}$ (i.e. $j^{\prime}=4,5$); see Figures 2D and E. Changes in the
peak energies do not show a clear trend. The largest improvements are observed
for $j^{\prime}=5$ and for $j^{\prime}=[0,1]$. Errors for peaks with
$j^{\prime}=8$ and $j^{\prime}=6$ do not reduce using M1. The remaining peaks
showed an increase in the error after applying M1. For the peak intensity,
again, the largest improvement is observed for the $j^{\prime}=[0,1]$ peak.
For most other peaks, with the exception of $j^{\prime}=5$ and $j^{\prime}=8$,
there is clearly an improvement in the intensities.

The initial and morphed MRCI PESs are compared in Figure 3B. In this case,
morphing increases the anisotropy at long-range compared to the initial PES.
However, changes are more pronounced than for the FCI PES. One-dimensional
cuts along the $r_{\rm HH}$ and $R$ coordinates for given angle $\theta$ are
provided in Figures S23 and S24. As for the FCI PES, the difference between
the initial surface and the morphed surface are more pronounced as $r_{\rm
HH}$ increases. The 1D cuts of the surface at different values of $r_{\rm HH}$
(Figure S24) show further evidence of the change in the depth of the potential
well. The modifications of the energy curves with respect to the $r_{\rm HH}$
coordinate follow the same trend as the FCI surface.

MP2: The results for the lowest-quality surface (MP2) are shown in Figures 2C
and 3C. The RMSE for the energies improves from 13.1 to 12.8 cm-1 whereas for
the intensities, it changes from 22.4 to 10.9 arb. u. Particularly notable is
the improvement in the intensities by more than a factor of two. Overall, the
changes in the position of the energies and the intensities of the peaks for
the calculated cross sections are more pronounced than for the FCI and MRCI+Q
PESs. The energy position for peaks with large $j^{\prime}$ ($j^{\prime}=7$
and $j^{\prime}=8$) improve by $\approx 5$ cm-1. Another difference is that
the shoulder of the peak at $j^{\prime}=8$ that appears for the two previously
described surfaces is not visible for the MP2 surface. For the peaks with
$j^{\prime}=4$ and $j^{\prime}=5$, the error with respect to the experimental
spectra upon morphing increases slightly.

The original MP2 PES and its morphed variant for a H${}_{2}^{+}$ separation of
$r_{\rm HH}=2.0$ a0 are reported in Figure 3C. Because Møller-Plesset second-
order theory is a single-reference method and makes further approximations,
the changes in the topology of the PES are considerably larger than for the
FCI and MRCI+Q PESs. Most of the isocontours are compressed compared with the
initial MP2 surface, and the well depth is reduced from 2493 cm-1 to 1684 cm-1
(Table 1), see Figure S25. The one-dimensional cuts along the $r_{\rm HH}$ and
$R$ coordinates for given $\theta$, see Figures S26 and S27, show that as
$r_{\rm HH}$ increases the single-reference assumption of the method, leading
to convergence problems for small $R$. As a consequence of the contraction of
the potential wells, the barrier of the transition state at $\theta\approx
90^{\circ}$ is increased, which is further confirmed by the Minimum Energy
Path (MEP) shown in Figure S28C. A more detailed analysis of the MEP (Figure
S29C) reveals a small increase in the energy of the transition state along the
angular coordinate $\theta$. On the other hand, for the $R-$coordinate a non-
physical barrier emerges at around 3.5 $a_{0}$.

##  Appendix S3 Resonances under Morphing

The cross sections depending on the binding energy between He and
H${}_{2}^{+}$ as opposed to the relative kinetic energy of the two reactants
shows distinct peaks that are no longer separated by final states
($j^{\prime}$) of the H${}_{2}^{+}$ fragment but rather appear as one or
several Feshbach Resonances per input $J$ and $\ell$ at certain values of the
binding energy. Both the energy at which a Feshbach Resonance appears, and the
distribution of intensities in all exit channels, depend sensitively on the
topography of the PES. In consequence, the effect of morphing on the PES can
influence the number, energy and intensities of the Feshbach resonances. To
illustrate this, it is instructive to consider projections of wave functions
for particular resonances to characterize how changes in the PES, which lead
to changes in the collision cross-section, are reflected in the radial and
angular behaviour of the wave function.

Figure S20 shows the square of the $(v^{\prime}=v)$ and $(j^{\prime}=j)$
components of the resonance wave functions (first and third rows of panels)
and corresponding resonances in the cross-section (second and fourth rows of
panels) for the dominant $\ell$ and $J$ contributions for para- and
ortho-H${}_{2}^{+}$ for all three unmorphed and morphed PESs, respectively.
The number, position(s) and intensities of the spectroscopic features respond
to morphing in a largely unpredictable way. As an example, the unmorphed and
morphed PESs at the FCI level are considered for para-H${}_{2}^{+}$ with
$(\ell=4,J=4)$ (left column, rows 1 and 2 in Figure S20). Although M1 changes
the topology of the morphed PES only in a minor fashion, the effect on the
wavefunctions and resulting spectroscopic feature is clearly visible. For the
unmorphed FCI PES there is one resonance at –8.1 cm-1 which splits into two
resonances at –2.1 cm-1 and –16.3 cm-1 of approximately equal height upon
morphing the PES. Accordingly, the wavefunctions also differ, in particular in
the long-range part, i.e. for large $R$. Similar observations were made for
the wavefunctions on the MP2 PES, whereas for the MRCI PESs the changes in the
wavefunctions are comparatively smaller.

Conversely, for ortho-H${}_{2}^{+}$ the resonances of both FCI and MRCI PESs
are affected in a comparable fashion and more noticeable changes to the
resonance wave function are observed than for para-H${}_{2}^{+}$. Whilst the
resonance wave functions are shifted to larger $R$ in the cases of FCI and
MP2, the MRCI resonance wave function only experiences a small shift.
Significantly, even though the anisotropy of the PESs only changes in a minor
fashion under morphing, all three resonance wave functions respond owing to a
change in the superposition of outgoing partial wave (quantum number
$\ell^{\prime}$). For the FCI and MP2 PESs angular/radial coupling is enhanced
by morphing, which leads to elongation of certain lobes in the wavefunctions
along the $(R,\theta)-$direction for ortho-H${}_{2}^{+}$–He. This contrasts
with para-H${}_{2}^{+}$–He for which unique assignments of the ro-vibrational
quantum numbers is possible from conventional node-counting.

##  Appendix S4 Figures

Figure S6: Evolution of the values of the loss function ($\mathcal{L}_{i}$)
over the iterations with respect to the initial value ($\mathcal{L}_{0}$) for
the M1(left) and M2 (right) methods. The loss function is defined in the main
manuscript. Figure S7: 1D cuts of the FCI PES obtained from the M1 procedure
along $R$ for fixed $r_{\rm HH}$ and different angles ($\theta$). Figure S8:
1D cuts of the FCI PES obtained from the M1 procedure along $r_{\rm HH}$ for
fixed $R$ and different angles ($\theta$). Figure S9: Potential energy surface
2D projection $V(R,r)$ at $\theta=0^{\circ}$ obtained from the M1 procedure
for the three potentials studied in this work. The dotted lines represent the
unmorphed potential, complementary full lines show the morphed potential.
Isocontours are separated by 0.1 eV. The minimum of the potential is indicated
with a dot and a cross for the unmorphed and morphed potential. Figure S10:
Potential energy surface 2D projection $V(R,r)$ at $\theta=30^{\circ}$
obtained from the M1 procedure for the three potentials studied in this work.
The dotted lines represent the unmorphed potential; complementary full lines
show the morphed potential. Isocontours are separated by 0.1 eV.  Figure S11:
Potential energy surface 2D projection $V(R,r)$ at $\theta=60^{\circ}$
obtained from the M1 procedure for the three potentials studied in this work.
The dotted lines represent the unmorphed potential; complementary full lines
show the morphed potential. Isocontours are separated by 0.1 eV. Figure S12:
2-body potential for the HeH+ molecule. The inset shows the region close to
the equilibrium geometry. Zero of energy is at the He+H+ asymptote. For HeH+
the experimentally determined41 dissociation energy is $D_{e}=16456.23$ cm-1
compares with 16406.04 (FCI), 16403.96 (MRCI+Q), and 16171.65 (MP2) cm-1 from
the fitted 2-body potentials. The difference of 235 cm-1 between FCI and MP2
is substantial. Remaining differences between experiment and FCI are due to
basis set incompleteness and non-Born-Oppenheimer effects, not included in the
calculations. For other systems, such effects have been estimated at several
ten up to 100 cm-1 on total energies.42

Figure S13: 2-body potential for the H${}_{2}^{+}$ molecule. The zero of
energy is the H+H+ asymptote and the inset shows the region close to the
equilibrium geometry. The experimentally determined43 dissociation energy
$D_{0}=21379.36\pm 0.08$ cm-1 compares with 21428.5 (FCI), 21430.1 (MRCI+Q),
and 21427.5 (MP2) cm-1. The location of the ground states $(v=0,j=0)$ was
determined using the LEVEL code.44 Remaining differences between experiment
and FCI are due to basis set incompleteness and non-Born-Oppenheimer effects,
not included in the calculations. For other systems, such effects have been
estimated at several ten up to 100 cm-1 on total energies.42 Figure S14:
Minimum energy paths (MEPs) for the different surfaces obtained from the M2
method with respect to the variables $\theta$ and $R$. The zero of energy of
the path is the energy of the separated monomers. Figure S15: Minimum energy
paths (MEPs) for the different surfaces studied before and after applying the
morphing M2 method. In solid black, it is shown the MEP for the morph PES. In
dotted black, the MEP is shown for the initial PES. Figure S16: Potential
energy surface 2D projection $V(R,r)$ at $\theta=0^{\circ}$ obtained from the
M2 procedure for the three potentials studied in this work. The dotted lines
represent the unmorphed potential. Complementary full lines show the morphed
potential. Isocontours are separated by 0.1 eV. The minimum of the potential
is indicated with a dot and a cross for the unmorphed and morphed potential.
Figure S17: Schematic representation of the changes in the geometry by the
morphing transformation for an example Morse potential curve. If the value of
the scaling parameter $\beta>1$, the equilibrium minima will be displaced to
smaller values of coordinate $r$. The insight shows the change in the
curvature close to the value of the equilibrium geometry Figure S18: Scatter
plot of the energies of the FCI surface vs the initial and morphed MP2 surface
for the PES-to-PES procedure. In the inset of the figure, the distribution of
differences $E_{\rm FCI}-E_{\rm MP2}$ for unmorphed and morphed MP2 surfaces,
respectively. The Pearson correlation coefficients are 0.9984 and 0.9988 for
unmorphed and morphed MP2 PESs and the RMSE decreases from 138 cm-1 to 87 cm-1
upon morphing for energies spanning $\sim 7$ eV. Figure S19: Minimum energy
paths (MEPs) for the morph between the FCI and MP2 surface (PES to PES
morphing) with respect to the variables $\theta$ and $R$. In black, the MEP
for the FCI surface used as reference. The zero of energy of the path was
chosen as the energy of the transition state for the unmorphed potential.
Figure S20: Comparison of the unmorphed (red, dotted) and morphed (blue)
absolute value squared resonance wave functions in two dimensions $(R,\theta)$
in the case of para-H${}_{2}^{+}$ $\ell=4,J=4$ (upper two rows) and
ortho-H${}_{2}^{+}$ $\ell=4,J=5$ (lower two rows) for resonance energies as
marked and labelled in the corresponding cross sections are shown as a
function of binding energy (second and fourth rows for para and ortho,
respectively). The resonance wave functions have been scaled to have a maximal
value of one, and the contours occur at 0.01, 0.1, 0.25, 0.5, 0.75 and 0.99.
Figure S21: Relative contribution of each angular momentum state for collision
of He(3P) with para- and ortho-H2 (left and right panels, respectively) for
the experimental collision energy $E_{\mathrm{col}}/\mathrm{k_{B}}\approx
2.5\mathrm{K}$ and spread. Note that contributions for $\ell>5$ are not shown
since they are negligibly small. Figure S22: Theoretical kinetic energy
histograms for para (above) and ortho (below) H${}_{2}^{+}$ in their
convoluted (line) and unconvoluted (shaded area) forms. The unconvoluted
cross-sections have been scaled by a factor of $0.5$ relative to the
convoluted ones. The darkly shaded areas correspond to the dominant
contributions to the cross-section ($\ell=4,J=4$ and $\ell=4,J=5$ for para and
ortho respectively), whilst the lightly shaded areas include all initial
$\ell$ and $J$ contributions. The shown cross-sections were obtained using the
unmorphed FCI potential. Figure S23: 1D cuts of the MRCI PES obtained from the
M1 procedure along $R$ for fixed $r_{\rm HH}$ and different angles ($\theta$).
Figure S24: 1D cuts of the MRCI PES obtained from the M1 procedure along
$r_{\rm HH}$ for fixed $R$ and different angles ($\theta$). Figure S25: MP2
Potential energy surface from M1 procedure in 3D representation for fixed
values of $r_{\rm HH}$, on the top of the figure is indicated the value of
$r_{\rm HH}$. As zero of energy, it was considered the value at the given
$r_{\rm HH}$ and $R=\infty$. Energies are in cm-1 and separated by 100 cm-1
Figure S26: 1D cuts of the MP2 PES obtained from the M1 procedure along $R$
for fixed $r_{\rm HH}$ and different angles ($\theta$). Figure S27: 1D cuts of
the MP2 PES obtained from the M1 procedure along $r_{\rm HH}$ for fixed $R$
and different angles ($\theta$). Figure S28: Minimum energy paths (MEPs) for
the different surfaces obtained from the M1 method with respect to the
variables $\theta$ and $R$. The zero of energy of the path is the energy of
the separated monomers. Figure S29: Minimum energy paths (MEPs) for the
different surfaces studied before and after applying the morphing M1 method.
In solid black, it is shown the MEP for the morph PES. In dotted black, the
MEP is shown for the initial PES.