# Photoelectron and fragmentation dynamics of the H+ \+ H+ dissociative
channel in NH3 following direct single-photon double ionization

Kirk A. Larsen<EMAIL_ADDRESS>Graduate Group in Applied Science and
Technology, University of California, Berkeley, CA 94720, USA  Chemical
Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
USA  Thomas N. Rescigno<EMAIL_ADDRESS>Chemical Sciences Division,
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  Travis Severt
J.R. Macdonald Laboratory, Physics Department, Kansas State University,
Manhattan, Kansas 66506, USA  Zachary L. Streeter  Chemical Sciences Division,
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  Department of
Chemistry, University of California, Davis, CA 95616, USA  Wael Iskandar
Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley,
CA 94720, USA  Saijoscha Heck  Chemical Sciences Division, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, USA  Max-Planck-Institut für
Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  J.W. Goethe
Universität, Institut für Kernphysik, Max-von-Laue-Str. 1, 60438 Frankfurt,
Germany  Averell Gatton  Chemical Sciences Division, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, USA  Department of Physics, Auburn
University, Alabama 36849, USA  Elio G. Champenois  Graduate Group in Applied
Science and Technology, University of California, Berkeley, CA 94720, USA
Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley,
CA 94720, USA  Richard Strom  Chemical Sciences Division, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, USA  Department of Physics, Auburn
University, Alabama 36849, USA  Bethany Jochim  J.R. Macdonald Laboratory,
Physics Department, Kansas State University, Manhattan, Kansas 66506, USA
Dylan Reedy  Department of Physics, University of Nevada Reno, Reno, Nevada
89557, USA  Demitri Call  Department of Physics, University of Nevada Reno,
Reno, Nevada 89557, USA  Robert Moshammer  Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany  Reinhard Dörner  J.W. Goethe
Universität, Institut für Kernphysik, Max-von-Laue-Str. 1, 60438 Frankfurt,
Germany  Allen L. Landers  Department of Physics, Auburn University, Alabama
36849, USA  Joshua B. Williams  Department of Physics, University of Nevada
Reno, Reno, Nevada 89557, USA  C. William McCurdy  Chemical Sciences Division,
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  Department of
Chemistry, University of California, Davis, CA 95616, USA  Robert R. Lucchese
Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley,
CA 94720, USA  Itzik Ben-Itzhak  J.R. Macdonald Laboratory, Physics
Department, Kansas State University, Manhattan, Kansas 66506, USA  Daniel S.
Slaughter  Chemical Sciences Division, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA  Thorsten Weber<EMAIL_ADDRESS>Chemical Sciences
Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

###### Abstract

We report measurements on the H+ \+ H+ fragmentation channel following direct
single-photon double ionization of neutral NH3 at 61.5 eV, where the two
photoelectrons and two protons are measured in coincidence using 3-D momentum
imaging. We identify four dication electronic states that contribute to H+ \+
H+ dissociation, based on our multireference configuration-interaction
calculations of the dication potential energy surfaces. The extracted
branching ratios between these four dication electronic states are presented.
Of the four dication electronic states, three dissociate in a concerted
process, while the fourth undergoes a sequential fragmentation mechanism. We
find evidence that the neutral NH fragment or intermediate NH+ ion is markedly
ro-vibrationally excited. We also identify differences in the relative
emission angle between the two photoelectrons as a function of their energy
sharing for the four different dication states, which bare some similarities
to previous observations made on atomic targets.

###### pacs:

Valid PACS appear here

##  I Introduction

Photo-Double-Ionization (PDI) is a process in which two electrons are ejected
from an atom or molecule by absorption of a single photon. The resulting
dication can be produced through either an indirect or a direct process. In
the indirect process Lablanquie  _et al._ (1989); Sann  _et al._ (2011), the
target is first ionized to produce a photoelectron and a singly-charged,
excited cation. Subsequently, the cation decays by autoionization to produce a
second continuum electron. The secondary electrons in indirect PDI have a
unique signature, i.e. often a very narrow kinetic energy distribution and a
rather isotropic angular emission pattern, which allows the process to be
uniquely identified in a two-electron energy- or momentum-coincidence
spectrum. In contrast to the indirect process, direct PDI involves
simultaneous projection of two bound electrons to a correlated pair of
continuum states. The interaction of the two electrons makes PDI an ideal
process for studying electron-electron correlation Mergel  _et al._ (1998);
McCurdy  _et al._ (2004); Yip  _et al._ (2013); Weber  _et al._ (2004a);
Vanroose  _et al._ (2006).

Because of the repulsive Coulomb interaction between singly charged ions that
is active over very large internuclear distances, the vertical double
ionization thresholds of small molecules generally lie above the dissociation
limits corresponding to formation of singly charged fragments. Since the
dissociative electronic states of a polyatomic dication can possess various
fragmentation pathways involving different numbers of bodies, distinct
fragment species can be measured depending on various factors. Studying the
photoelectron pair and various ionic fragments in coincidence can provide
information on electron-electron correlation, the features of dication
potential energy surfaces, and the nuclear dynamics involved in the dication
breakup. The molecular fragmentation that typically follows direct PDI can be
broadly described as occurring in a single step (concerted), where all charged
and neutral fragments are born simultaneously, or occurring in multiple steps
(sequential), where first a portion of the charged and neutral fragments are
generated, leading to a metastable intermediate moiety, which then undergoes
further dissociation to produce the final set of fragments Gaire  _et al._
(2014); Rajput  _et al._ (2018).

In sequential fragmentation, the decay of the metastable intermediate(s) can
be facilitated by various mechanisms, such as internal conversion or
intersystem crossing to a dissociative state. Although spin-orbit coupling is
generally weak in low-Z systems, intersystem crossing can in certain instances
be the primary decay mechanism of metastable intermediates in a sequential
dissociation process. Due to the weak coupling, the rate of intersystem
crossing can be low, which leads to a significant period spent in the
intermediate, providing time for the metastable fragments to rotate between
the two fragmentation steps.

Distinguishing between concerted and sequential fragmentation channels is
crucial in certain types of measurements, as concerted fragmentation channels
can enable body-fixed frame photoelectron angular distributions to be
retrieved, which carry far more information content than laboratory frame
angular distributions. These body-fixed frame photoelectron angular
distributions can, in most cases, only be reconstructed if the dication
dissociates promptly along the relevant internuclear axes relative to rotation
of those axes, allowing the molecular orientation at the instant of the PDI to
be determined. This requirement is known as the axial recoil approximation
Zare (1972). Since measuring body-frame photoelectron angular distributions
following PDI poses a great experimental challenge, there exists only a small
body of literature covering this topic, primarily focused on H2 Weber  _et
al._ (2004a, b); Akoury  _et al._ (2007); Vanroose  _et al._ (2006); Reddish
_et al._ (2008). Various experimental methods such as particle coincidence 3-D
momentum imaging, including COLd Target Recoil Ion Momentum Spectroscopy
(COLTRIMS), allow measurements to be made in the molecular frame, but are
predicated on the axial recoil approximation, hence it is useful to first
determine which dication states exhibit concerted fragmentation mechanisms.
The body-fixed frame electron emission pattern, or Molecular Frame
Photoelectron Angular Distributions (MFPADs), can be established if the
complete structure of the molecule at the time of dissociation can be
reconstructed from the detected heavy ionic fragments. However, if a
dissociative channel produces more than two (undetected) neutral fragments, or
results in a polyatomic fragment with unknown orientation, only the Recoil
Frame Photoelectron Angular Distribution (RFPAD) can be reconstructed. The
latter represents the electron emission pattern with respect to a
distinguished axis or plane spanned by the (detected) charged fragments.
R/MFPADS are particularly sensitive to electron-electron correlation in both
the initial and final states.

Various experimental and theoretical studies spanning a few decades have
investigated the different dication electronic states and dissociation
channels present in NH3 following PDI, electron impact double ionization, and
double ionization via double-charge-transfer spectroscopy Winkoun and Dujardin
(1986); Stankiewicz  _et al._ (1989); Locht  _et al._ (1989); Locht and
Momigny (1987a); Eland (2006); Samson  _et al._ (1987); Appell and Horsley
(1974); Märk  _et al._ (1977); Langford  _et al._ (1992); Griffiths and Harris
(1990); Locht  _et al._ (1988); White  _et al._ (1977); Camilloni  _et al._
(1977); Økland  _et al._ (1976); Jennison (1981); Boyd  _et al._ (1985). Most
of these studies have focused on determining the appearance energies of the
different fragments and the energetic locations of the dication electronic
states. Among these investigations, no study, to our knowledge, has examined
the H+ \+ H+ fragmentation channels of ammonia.

In this work, we investigate H+ \+ H+ dissociation following direct valence
PDI of neutral NH3 at 61.5 eV, approximately 27 eV above the PDI threshold
Locht and Momigny (1987b), where both the photoelectron and proton pairs are
measured in coincidence using COLTRIMS. Based on Multi-Reference
Configuration-Interaction (MRCI) calculations of dication Potential Energy
Surfaces (PESs), we identify four dication electronic states that contribute
to the H+ \+ H+ fragmentation. Our measurement provides the branching ratios
between the four involved dication electronic states. As will be detailed
below, of these four states, one appears to dissociate via a sequential
mechanism and three dissociate in a concerted mechanism. Two of the three
concerted dissociative states fragment at geometries near that of the ground
state of neutral NH3, where the axial recoil approximation appears valid,
while the third state undergoes a significant change in nuclear geometry prior
to fragmentation. By measuring the correlated electron and ion fragment
momenta, we determine that the neutral NH fragment or charged intermediate NH+
cation is ro-vibrationally excited with considerable internal energy, in some
cases more than 2 eV.

##  II Experiment

The H+ \+ H+ fragmentation channel following valence PDI at 61.5 eV was
investigated using COLTRIMS Dörner  _et al._ (2000); Ullrich  _et al._ (2003),
where the two photoelectrons and two protons were collected with full $4\pi$
solid angle, and their 3-D momenta were measured in coincidence, on an event-
by-event basis. These four charged particles were guided using weak static
parallel electric and magnetic fields, 11.4 V/cm and 10.0 G, respectively, to
multi-hit position- and time-sensitive detectors at opposite ends of the
spectrometer. Each detector comprised a Multi-Channel Plate (MCP) stack in
chevron configuration for time readout, together with a delay-line anode,
which decoded the hit position of each particle Jagutzki  _et al._ (2002). The
electron and ion delay-line detectors were a hex-anode with an 80 mm MCP stack
and a quad-anode with a 120 mm MCP stack, respectively. This system encodes a
charge particle’s 3-D momentum into its hit position on the detector and Time-
of-Flight (TOF) relative to each ionizing extreme ultraviolet (XUV) pulse
emitted by the synchrotron. These detectors have a small but significant dead-
time following each detected particle, therefore they are subject to limited
multi-hit capability Jagutzki  _et al._ (2002). This problem is most prominent
in the electron pair detection, due to the small differences in the electron
arrival times and hit positions at the detector. This dead-time effect can
influence measured relative electron-electron angular distributions and is
thus important to quantify, in order to distinguish real features from those
that may emerge due to the detection scheme. We point out that the photoions
do not suffer from this dead-time problem to the same degree as the electrons,
as they are much more spread out in TOF and hit position on the ion detector.
The electron-pair resolution is estimated by simulating the charged particle
motion in the spectrometer fields with various sum kinetic energies and in
various energy sharing conditions of the electron pair. For each pair of
trajectories, the relative hit position and time-of-flight is computed, which
is used to determine the fraction of simulated electron-pair events lost due
to an estimated detector response, and thus approximate the fraction of actual
losses.

The PDI experiment was performed using a tunable monochromatic linearly
polarized beam of XUV photons produced at beamline 10.0.1.3. at the Advanced
Light Source (ALS) synchrotron located at Lawrence Berkeley National
Laboratory. The beamline monochromator was configured to provide 61.5 eV
photons to the experiment, with an energy resolution narrower than $\pm$50
meV. The photon energy of 61.5 eV was chosen to be near the maximum of the PDI
cross section of NH3, while at the same time providing electron kinetic
energies that can be detected with full solid angle and adequate energy
resolution (around 1:10). Moreover, it is beneficial to keep the electron sum
energy greater than $\sim 5$ eV in order to utilize a large region of the 3D
electron pair detection phase space, minimizing losses due to the electron
detector dead-time (this will be apparent in Fig. 4 later in the discussion).

A beam of rotationally and vibrationally cold neutral NH3 ($\sim$80 K) was
produced by an adiabatic expansion of the pressurized target gas ($\sim$35
psi) through a 50 $\mu$m nozzle, and collimated by a pair of downstream
skimmers. The first skimmer has a diameter of 0.3 mm and the second skimmer
has a diameter of 0.5 mm. The first skimmer is placed 8 mm downstream of the
nozzle and in the zone of silence of the supersonic expansion. The second
skimmer is 10 mm downstream of the first skimming stage. The resulting
supersonic jet of target molecules propagated perpendicular to the photon
beam, where the two beams crossed at the interaction region ($\sim 0.15\times
0.15\times 1.0$ mm3) inside the 3-D momentum imaging spectrometer, where PDI
of the neutral ammonia in its ground state occurs at an average rate of less
than 0.01 events per XUV pulse, assuring unambiguous coincidence conditions.

The TOF and hit position of the charge particles produced by PDI were recorded
in list mode on an event-by-event basis, enabling relevant events to be
selected and examined in a detailed off-line analysis. For each PDI event, the
kinetic energies and emission angles of the photoelectrons were determined
from the 3-D photoelectron momenta, while the orientation of the recoil frame
and the kinetic energy release (KER) of the fragmentation were determined
using the measured 3-D momenta of the two protons. We infer the momentum of
the center of mass of the remaining neutral NH radical by assuming momentum
conservation between it and the two measured protons, treating the
fragmentation as three-body breakup (even if the NH diatom fragments to N +
H).

##  III Theory

Most previous work on the ammonia dication have been experimental in nature.
Of the earlier theoretical studies, most have focused on computing the
vertical double ionization energy of neutral ammonia Pope  _et al._ (1983);
Locht and Momigny (1987b). Tarantelli et al. Tarantelli  _et al._ (1985)
computed excited state excitation energies of NH${}_{3}^{2+}$ at the
equilibrium geometry of NH3 (see also Table 1), but to our knowledge no
earlier calculations of NH${}_{3}^{2+}$ potential surfaces have been reported.
The electron configuration of NH3 in its ground-state is
$(1a_{1})^{2}(2a_{1})^{2}(1e)^{4}(3a_{1})^{2}$. At a photon energy of $61.5$
eV, there are nine dication states which are energetically accessible
following a vertical transition. In order to determine which of these states
correlate with the three-body NH + H+ \+ H+ fragmentation channel, we carried
out a series of electronic structure calculations. At each molecular geometry
considered, we generated a set of molecular orbitals from a two-state,
Complete Active Space (CAS) Multi-Configuration Self-Consistent Field (MCSCF)
calculation on the lowest triplet (3E) states of the dication. We kept one
orbital (N 1s) frozen and included seven orbitals in the active space. We then
performed MRCI calculations including all single and double excitations from
the CAS reference space to generate 1-D cuts through the PESs. All bond angles
were frozen at the equilibrium geometry of neutral ammonia (107o), as was one
hydrogen (H${}_{\text{III}}$) bond length (1.9138 Bohr), while two hydrogen
bonds (H${}_{\text{I}}$) and (H${}_{\text{II}}$) were symmetrically stretched.
The results of the calculations are shown in Fig. 1 with the electron
configuration and state labels of each dication PES cut identified in the
legend. The PES cuts were calculated out to a symmetric stretch
N-H${}_{\text{I}}$/N-H${}_{\text{II}}$ distance of 50.0 bohr and extrapolated
to infinity under the assumption of a purely repulsive Coulomb interaction
between the positively charged fragments. The vertical energies at the neutral
NH3 geometry and the energies at the asymptotic limits are given in Table 1.
Note that here we do not explicitly consider cuts through the dication PESs
where only one NH bond is stretched, as that is the subject of a future paper.

|
---|---
Figure 1: PES cuts of the NH3 dication generated from MRCI calculations as described in the text. Here, two protons are symmetrically stretched while the third remains fixed , with all internal angles frozen at the geometry of neutral ammonia. The zero of energy is set at the ground-state (${}^{1}A_{1}$) of the ammonia dication at the geometry of neutral ammonia, which lies 34.8 eV below the dication Locht and Momigny (1987b). On this energy scale, the 61.5 eV photon energy lies at 26.7 eV. The dashed vertical line indicates the equilibrium geometry of neutral ammonia. (a) Cuts of the experimentally identified relevant states; (b) detail of PES cuts for selected NH3 dication triplet states. The inset indicates a region of large symmetric stretch distances where charge exchange may occur, as discussed in the text. State | Vertical Energy (eV) | Asymptote | Adiabatic Limit Energy (eV)
---|---|---|---
(1e-2)3A2 (cyan) | 8.64 (8.23)* | NH(${}^{3}\Sigma^{-}$)+H++H+ | 0.96
(1e-2)1E (magenta) | 9.94 (9.91)* | NH+(${}^{2}\Pi$)+H+H+ | 0.52
(1e-2)1A1 (green) | 11.94 (11.77)* | NH(${}^{1}\Delta$[${}^{1}\Sigma^{+}$])+H++H+ | 2.69 [3.74]
(2a${}_{1}^{-1}$,3a${}_{1}^{-1}$)1A1 (red) | 18.94 (19.33)* | NH(${}^{1}\Sigma^{+}$)+H++H+ | 3.74
Table 1: Ammonia dication vertical energies at neutral NH3 geometry and
asymptotic three-body limits extrapolated from ab initio calculations at
N-H${}_{\text{I}}$/N-H${}_{\text{II}}$ distances of 50.0 bohr. Note that for
the 1A1 state (green), two possible asymptotic limits are given (see text).
*Values in parentheses are configuration interaction results from Ref.
Tarantelli  _et al._ (1985).

Our calculations reveal that there are only three three-body proton-proton
dissociative limits. Of the three-body proton-proton channels, two are singlet
states and one is a triplet state. The two singlet states leave the remaining
neutral NH molecule in a ${}^{1}\Delta$ or a ${}^{1}\Sigma^{+}$ state, while
the triplet leaves the neutral NH fragment in a ${}^{3}\Sigma^{-}$ state. In
order to produce the two experimentally observed protons in the fragmentation,
the implication is that an excitation must access one of these three
dissociative limits, or undergo a four-body fragmentation mechanism that
yields two protons, i.e. results in the fragments N + H + H+ \+ H+.

We identify three relevant singlet states, ($1e^{-2}$) 1E, ($1e^{-2}$) 1A1,
and ($2a_{1}^{-1},3a_{1}^{-1}$) 1A1, shown in Fig. 1 (a) as solid curves
(magenta, green and red), and a fourth relevant triplet state, ($1e^{-2}$)
3A2, shown as a dashed curve (cyan). The curves in Fig. 1 (a) are color-coded
to be consistent with the experimental features to be discussed in the
following section. Since spin-orbit coupling, required for an intersystem
crossing, is expected to be weak, the triplet state must dissociate to a
triplet fragment state. However, Fig. 1 shows that the 3A2 state (cyan dashed)
actually correlates with the NH+(${}^{2}\Pi$)+H++H(2S) dissociation channel
(cyan dashed in the Fig. 1 (b) inset). To reach the NH(${}^{3}\Sigma^{-}$)+2H+
limit (black curve in the inset) requires a charge exchange, which is possible
at N-H separations greater than 18 Bohr where the 3E (3A′′) and 3A2 (3A′′)
states become nearly degenerate in energy across a range of geometries (see
cyan dashed and black curves in Fig. 1 (b)). This can result in charge
exchange over a large range of distances along the asymmetric stretch
coordinate that the dissociating wave packet traverses. We have observed an
analogous asymptotic charge-exchange mechanism at such large N-H distances in
an earlier study of dissociative electron attachment to ammonia Rescigno  _et
al._ (2016).

For singlet states accessible in the Franck-Condon (FC) region as depicted in
Fig. 1 (a), there are two different proton-proton limits (red and green
curves). The ($1e^{-2}$) 1A1 state (green) is seen to cross two other
dissociative 1A1 states (green and red), which correlate with the products NH
(${}^{1}\Delta$) or NH (${}^{1}\Sigma^{+}$) plus two protons, respectively.
Conical intersections (CIs) between the dissociative states and the initially
excited 1A1 state can result in dissociation to either of the singlet limits.
Since the location of the CIs cannot be determined from 1-D energy cuts
(although numerous avoided crossing are observed), we must rely on the
experimental findings to see which of the singlet limits are populated.

Previous experimental observations have indicated that PDI to the ($1e^{-2}$)
1E state is associated with the NH+ \+ H+ \+ H fragmentation channel
Stankiewicz  _et al._ (1989). Since the dissociative limit of the ($1e^{-2}$)
1E state does not directly yield two protons, excitation to this state must
undergo a nonadiabatic transition to either of the two 1A1 excited dication
states, or the NH+ fragment it produces must dissociate to N + H+, in order to
result in the measured two-proton coincidence.

The ($1e^{-2}$) 1E state, doubly degenerate in C3v geometry, splits into A′
and A′′ states when two N-H bonds are symmetrically stretched. Of these two
states, the upper state has A′′ symmetry. Accordingly, internal conversion to
either of the 1A′ states that have limits producing NH + H+ \+ H+ is
unfavorable. Dissociation on the lower curve yields an NH+ fragment in its
X${}^{2}\Pi$ ground state. If the NH+ fragment is produced with sufficient
internal energy, it can dissociate to N+(3P$)+$ H(2S) or through intersystem
crossing to another NH+ state, to N(4S) + H+. In the latter case this results
in the production of two protons via a sequential four-body breakup
NH${}_{3}^{++}\rightarrow$ NH+ \+ H+ \+ H $\rightarrow$ N + 2H+ \+ H. This
sequential breakup process will be examined in detail below.

##  IV Results and Discussion

Using the insights gained from the calculations on dication electronic states
described in the previous section, we provide a detailed discussion of the
experimental results below, which has been divided into three sub-sections. In
the first sub-section, we present and discuss the energetics of the
photoelectrons and photoions, identifying features which correspond with the
states outlined in the previous section. In the second sub-section, we address
the details of the dissociation dynamics by analyzing the relative emission
angle between the two protons in each of these states. Lastly, we present
results on the photoelectron dynamics via an analysis of the relative emission
angle between the two photoelectrons for the four dication states in different
energy sharing conditions of the electron pair.

###  IV.1 Photoelectron and photoion energetics

The H+ \+ H+ fragmentation following PDI of NH3 at 61.5 eV, $\sim$27 eV above
the PDI threshold, is identified and isolated by selecting the two charged
fragments in the time-of-flight spectrum and then in momentum space, and by
enforcing that two electrons are measured in coincidence with the two ionic
fragments. First, we plot the PDI yield as a function of the energy difference
between the two particles of the proton pair and the energy sum of the
photoelectron pair. This plot is shown in Fig. 2.

Figure 2: The yield of H+ \+ H+ after valence PDI of NH3 as a function of the
energy difference of the proton pair and the energy sum of the photoelectron
pair. The four color-coded ellipses guide the eye to the relevant features and
dication states discussed in the text. The data has been mirrored about zero
proton energy difference, as there is no physical meaning to the order in
which the two proton are detected.

Here we are able to identify four features, which we attribute to the four
different dication electronic states calculated and tabulated in the previous
section, resulting in photoelectron pairs with energy sums centered around 7.3
eV, 14.1 eV, 16.7 eV, and 17.6 eV. These features are indicated by ellipses to
guide the eye and color-coded to be consistent with the calculated values of
7.8, 14.8, 16.8 and 18.1 eV listed in Table 1. The measured and calculated
values are in excellent agreement and are consistent with the state
assignments. Note that the ellipses do not reflect the actual software gates
used in the data analysis. In the offline analysis, we choose each of these
states by selecting carefully around the center of each feature in Fig. 2,
while additionally placing constraints on the proton energy sum (which aids in
separating the low and high KER features). Enforcing conditions in a multitude
of dimensions in this fashion enables us to separate these four features for
subsequent analysis.

Each of these four features possesses a Full Width at Half Maximum (FWHM) in
electron energy sum of roughly 6.2 eV, 2.1 eV, 4.2 eV, and 2.4 eV,
respectively. The FWHM of the electron energy sum of each dication state
roughly indicates the magnitude of the gradient of the PES in the FC region,
provided the electron detector energy resolution is smaller than the width of
the feature in question. To estimate the expected spread of observed
photoelectron energies for the various dication states, we use a variant of
the so-called reflection approximation Rescigno and Schneider (1988). The
range of detectable KERs is determined by the FC envelope of the initial
(neutral) vibrational state reflected onto the final dication PESs. We
approximate the initial vibrational wavefunction with a harmonic oscillator
function $\chi_{0}$, obtained from a fit of the ground state energy of ammonia
as a function of the symmetric stretch coordinate. If we assume that the PDI
cross section varies little over the FC region and that the final continuum
vibrational wavefunctions can be approximated by delta functions about the
classical turning points on the dication PESs Vanroose  _et al._ (2006), then
the envelope of the expected photoelectron energies is given by the values of
the vertical PDI energies as a function of the symmetric-stretch coordinate,
weighted by the square of the symmetric-stretch vibrational wavefunction. We
find that $|\chi_{0}|^{2}$ reaches half its maximum value at a symmetric-
stretch displacement of approximately $\pm$0.11 Bohr from equilibrium, and we
have used these values to calculate the FWHM of the photoelectron
distributions. According to this procedure we find widths of 5.1 eV, 1.9 eV,
3.1 eV and 2.2 eV, respectively, which are in good agreement with the
measurement (see also Table 2). From this we find that, given our
photoelectron spectral resolution of roughly $\Delta E/E\sim 0.1$, the
measured FWHM of each state does indeed roughly correspond with the gradient
of its PES in the FC region.

We present the 1-D photoelectron energy sum spectrum in Fig. 3, where each
feature we identified in Fig. 2 has been indicated by the color-coded
distribution. The peak value of each distribution has been indicated in Table
2, where it is also compared with the theoretically calculated value. We find
good agreement between the measurement and calculations. We can clearly
identify the feature with a photoelectron energy sum centered near 7.3 eV,
while the three higher photoelectron energy features appear clustered
together. The branching ratios between the four measured features that
correspond with the four dication states are estimated from the relative yield
of these four features, and they are presented in Table 3. The method for
extracting these branching ratios is discussed later.

Figure 3: The NH3 PDI yield of the H+ \+ H+ channel as a function of the
photoelectron energy sum integrated over all features (black) as well as for
the four color-coded features corresponding with the identified dication
states. The electron energy sum distributions for the four features have been
scaled by a factor of four, for better visibility.

The yields of the H+ \+ H+ channels as a function of the kinetic energy of the
first and second detected electron are plotted in the electron-electron energy
correlation map shown in Fig. 4. Since the two electrons are indistinguishable
particles, the labeling (as 1 and 2) is arbitrary and the figure has been
symmetrized across the diagonal (the line E2 = E1) to account for this.

Figure 4: Electron-electron energy correlation map for the H+ \+ H+ channels
of the PDI of NH3. The four identified dication states are color-coded and
indicated by diagonal lines.

The four different features that correspond to the dication electronic states
identified in Figs. 2 and 3 are indicated as color-coded diagonal lines (which
take the form E2 = -E1 \+ Esum, where Esum is the photoelectron energy sum
corresponding to that feature) in Fig. 4. We point out that the red diagonal
line appears to be off the center of the diagonal feature, even though this
location represents the peak. This is because there are fewer bins along a
given constant electron energy sum (i.e. a diagonal of the form E2 = -E1 \+
Esum) as the photoelectron energy sum decreases. Since the length of a
constant energy diagonal line scales as $\sqrt{2}E_{e^{-}}$, the number of
available bins that events can populate decreases with decreasing
E${}_{e^{-}}$. This leads to the counts at low constant electron energy sums
to be concentrated in just a small number of bins, which can render the true
location of the peak obscured in this 2-D spectrum, while it is well
represented in Fig. 3.

All four dication states are accessed via direct PDI, as indicated by the
uniform diagonal features (taking the form E2 = -E1 \+ Esum) and the absence
of any Auger or autoionization lines, which would appear with vertical or
horizontal characteristics at very unequal energy sharing due to the
autoionization electron possessing a narrow constant (low) energy. The
uniformity of the diagonal features in Fig. 4 indicates that the two
photoelectrons do not exhibit a strong preference towards either equal or
unequal energy sharing, but rather exhibit roughly constant H+ \+ H+ yield as
a function of the electron energy sharing (see also Fig. 12). The
photoelectron energy sharing distributions for each of the four states will be
presented and discussed in more detail in the final sub-section C.

The same four features, corresponding with those seen in Fig. 2, are present
in the proton-proton energy correlation map given in Fig. 5. As in the
electron-electron energy correlation map of Fig. 4, the two protons are
indistinguishable particles, hence the labeling is arbitrary and the figure
has been symmetrized across the diagonal (the line E2 = E1). We have removed
events that lie in the low energy corner of Fig. 5, as the events that lie
within this region originate from false coincidences. For each proton-pair we
compute the KER by treating the process as a three-body fragmentation and by
inferring the momentum of the N-H center of mass via momentum conservation.
Each feature seen in Fig. 5 possesses a different KER distribution centered
around 12.7 eV, 5.9 eV, 7.7 eV, and 5.5 eV, each with a FWHM of roughly 6.1
eV, 2.2 eV, 3.0 eV, and 2.0 eV, respectively. These KER distributions are
discussed in more detail later. The three KER features we have associated with
the $(2a_{1}^{-1},3a_{1}^{-1})$ 1A1, $(1e^{-2})$ 1A1 and 3A2 states exhibit a
tendency towards equal energy sharing between the two protons, consistent with
a concerted breakup mechanism. The fourth KER feature, associated with the 1E
state, exhibits highly unequal energy sharing between the two protons,
indicative of a sequential breakup mechanism.

State | Photoelectron Energy Sum (eV) | KER (eV)
---|---|---
| Experiment (FWHM) | Theory (FWHM) | Experiment (FWHM) | Theory* (FWHM)
(1e-2)3A2 (cyan) | 17.6 (2.4) | 18.1 (2.2) | 5.5 (2.0) | 7.7 (2.2)
(1e-2)1E (magenta) | 16.7 (4.2) | 16.8 (3.1) | 7.7 (3.0) | 9.4 (3.1)
(1e-2)1A1 (green) | 14.1 (2.1) | 14.8 (1.9) | 5.9 (2.2) | 8.2 (1.9)
(2a${}_{1}^{-1}$,3a${}_{1}^{-1}$)1A1 (red) | 7.3 (6.2) | 7.8 (5.1) | 12.7 (6.1) | 15.2 (5.1)
Table 2: The measured and calculated photoelectron energy sum and KER
centroids for each of the four identified features from H+ \+ H+ fragmentation
following PDI of NH3 at 61.5 eV. The asterisk marking the theoretical KER
values indicates that these are calculated assuming ro-vibrational ground
state fragments, i.e. assuming maximum KER with no energy channeled into
internal excitations. The theoretical KER values are all roughly 2 eV higher
than the measured values, which is consistent with the dissociation producing
fragments possessing approximately 2 eV of ro-vibrational energy (as explained
in the text).

Theoretical KER values are obtained by subtracting the asymptotic energies
from the associated vertical PDI energies in Table 1, while theoretical
photoelectron energy sum values are computed by subtracting these vertical PDI
energies and the double ionization threshold from the photon energy. These
results are displayed in Table 2. For the concerted breakup channels theory
gives 15.2, 8.2, and 7.7 eV for the $(2a_{1}^{-1},3a_{1}^{-1})$ 1A1,
$(1e^{-2})$ 1A1, and $(1e^{-2})$ 3A2 dication states, respectively. (Note that
the NH(${}^{1}\Sigma^{+}$) asymptote has been used for both 1A1 states.) These
values are uniformly higher, by 2.5, 2.3 and 2.2 eV, respectively, than the
measured values. This discrepancy is either due to calculated dissociation
energies that are all uniformly too small by approximately 2 eV, or can arise
if the NH fragment in all three concerted breakup channels is produced with
approximately 2 eV of ro-vibrational energy. The energy balance of the
sequential breakup is consistent with high internal energy of the NH fragment.
For the sequential 1E breakup channel, theory gives a KER value of 9.4 eV,
which is 1.7 eV higher than the measured value. This corresponds to a four-
body breakup mechanism, discussed in detail in Section IV.2.

The FWHM of the KER distribution associated with each dication state carries
similar information as the electron sum energy FWHM (see also Table 2),
indicating the steepness of the potential energy surfaces in the FC region,
convoluted with the energy resolution of the ion spectrometer (estimated to be
on the order of 100 meV). These values are indicated in Table 2.

We show the 1-D KER spectrum in Fig. 6, where each feature we identified in
Fig. 5 has been indicated by the color-coded distribution. The peak value of
each distribution is listed in Table 2, where it is also compared with our
theoretical results. The differences between the measured and calculated
values in Table 2 are consistent with the molecular fragments containing
roughly 2 eV of internal energy (or the aforementioned four-body breakup
mechanism, which is discussed below), not explicitly accounted for in our
theory which only considers fragments in their rotational and vibrational
ground states.

Figure 5: Proton-proton energy correlation map for the H+ \+ H+ fragmentation
channels of the valence PDI of NH3. The four identified dication states are
color-coded and indicated by ellipses to guide the eye. Figure 6: The yield of
H+ \+ H+ fragmentation channel of the valence PDI of NH3 as a function of KER,
shown for the total yield (black), as well as for the four color-coded
features corresponding to the identified relevant dication states. The KER
distributions for the four features have been scaled by a factor of five, for
better visibility. Figure 7: The PDI yield of NH3 at 61.5 eV as a function of
the energy above the double ionization threshold at the adiabatic limit
following dissociation, Einf, and the energy above the double ionization
threshold, Ei, for each of the four identified relevant dication states from
the H+ \+ H+ fragmentation channel. The four identified dication states are
color-coded and indicated by ellipses to guide the eye.

The estimated branching ratios between these four dication states are
displayed in Table 3. These branching ratios are approximated by
simultaneously fitting each feature in Fig. 2 with a 2-D Gaussian distribution
(although the distributions may not be explicitly Gaussians, this is
nonetheless a good approximation). The fitting procedure varied the widths
along each dimension independently, while also including a varying constant
background offset. Following this fitting procedure, we integrate the fit for
each feature individually to estimate its contribution to the total H+ \+ H+
yield. The main contribution to the uncertainty of the branching ratio is
rooted in the aforementioned electron pair deadtime, which influences the
detection yield of the electron-ion coincidences for each dication state as a
function of the electron sum energy. Applying the simulation mentioned above,
we estimate the total possible loss in PDI yield for electron sum energies of
7.3 eV (($2a_{1}^{-1},3a_{1}^{-1}$) ${}^{1}A_{1}$), 14.1 eV (($1e^{-2}$)
${}^{1}A_{1}$), 16.7 eV (($1e^{-2}$) ${}^{1}E$), and 17.6 eV (($1e^{-2}$)
${}^{3}A_{2}$) to be 27.2%, 10%, 8.1%, and 7.5%, respectively. This translates
to an error of up to 5% in the branching ratio. Errors due to deviations from
the assumed Gaussian shape of each feature in the fitting process and the
quality of the fit are estimated to be small ($<$1% and $<$0.3%,
respectively).

State | Branching Ratio
---|---
$(2a_{1}^{-1},3a_{1}^{-1})$ 1A1 | 14.6%
$(1e^{-2})$ 1A1 | 4.5%
$(1e^{-2})$ 1E | 18.1%
$(1e^{-2})$ 3A2 | 62.8%
Table 3: The branching ratios for the four dication states contributing to the
H+ \+ H+ dissociation channel following PDI of NH3 at 61.5 eV. The errors on
these fractions are estimated to be up to 5% (see text).

Last, we plot the H+ \+ H+ yield as a function of the energy at the adiabatic
limit Einf and the energy above the double ionization threshold Ei. This plot
is shown in Fig. 7, with $E_{i}=\hbar\omega-DIP-(E_{e_{1}}+E_{e_{2}})$ and
$E_{inf}=\hbar\omega-DIP-(E_{e_{1}}+E_{e_{2}}+KER)$, where DIP is the Double
Ionization Potential. As a guide to the eye, each of the four identified
features have been indicated by ellipses. This plot indicates for each state
and its dissociative limit where the NH${}_{3}^{++}$ is excited to upon PDI,
relative to the dication ground state. The circled features can be directly
compared with the calculated vertical energy and adiabatic energy values shown
in Table 1, which show good agreement with our theoretical results. As
mentioned above, the measured energies Einf are each approximately 2 eV higher
than what is theoretically predicted for rotationally and vibrationally cold
fragments, whereas the molecular fragments in the experiment can carry away
this amount of energy internally, as we think is plausible from our analysis
presented in section IV.2.

###  IV.2 Photodissociation dynamics: distinguishing concerted and sequential
fragmentation

To examine the connection between the measured KER and the molecular geometry
in each dication electronic state, we plot the yield as a function of cosine
of the measured angle between the momenta of the two protons,
$\cos\theta_{p_{1},p_{2}}=\bm{p_{1}}\cdot\bm{p_{2}}/|\bm{p_{1}}||\bm{p_{2}}|$,
and the KER, as shown in Fig. 8. It should be mentioned that due to the
Coulomb repulsion between the two photoions, the measured proton-proton angle
is an asymptotic dissociation angle, hence its value will be slightly larger
than the true angle at which the fragmentation transpires. Although we do not
have an exact estimate of how significantly the asymptotic dissociation angles
differ from the true bond angles, our analysis carries useful information that
differentiates the dissociation dynamics for each of the four features. In
Fig. 8, the neutral ground state geometry of NH3 (specifically the H-N-H bond
angle) is indicated by the vertical black dashed line. First, we point out
that of the four dication states three - the $(2a_{1}^{-1}3a_{1}^{-1})$ 1A1,
$(1e^{-2})$ 1A1, and $(1e^{-2})$ 3A2 states - exhibit decreasing KER with
increasing measured dissociation angle between the protons, as seen in Fig. 8.
Qualitatively, if the angle between the two protons increases due to nuclear
motion in the dication, e.g. the NH3 umbrella opening, their separation
increases and their Coulomb repulsion correspondingly decreases, resulting in
the negative bivariate correlation between the KER and the proton-proton
angle, $\theta_{p_{1},p_{2}}$. Although this type of nuclear motion was not
addressed in our calculations (which kept bond angles frozen), we still bring
forward this qualitative picture as a possible explanation for the observed
correlation. This also gives further support to the notion that these three
dication states dissociate via a concerted mechanism, where the two protons
are simultaneously eliminated from the dication.

Figure 8: The PDI yield of NH3 as a function of cosine of the measured proton-
proton angle, $\cos\theta_{p_{1},p_{2}}$, and KER for each of the four
dication states from the H+ \+ H+ fragmentation channel at 61.5 eV. The dashed
black vertical line indicates the neutral ground state H-N-H angle.

We also point out that the features associated with the
$(2a_{1}^{-1},3a_{1}^{-1})$ 1A1 state and the $(1e^{-2})$ 1A1 state dissociate
at angles closer to the neutral ground state geometry of the NH3 molecule
(H-N-H bond angle $\sim 107^{\circ}$) than the feature associated with the
$(1e^{-2})$ 3A2 state, which tends to fragment at angles approaching 180∘.
Although the distributions for the $(1e^{-2})$ 1A1 and $(1e^{-2})$ 3A2 states
appear similar in shape, each state’s fragmentation dynamics can be
distinguished as different by the location of their respective peaks in the
measured proton-proton angle. This suggests that the
$(2a_{1}^{-1},3a_{1}^{-1})$ 1A1 and the $(1e^{-2})$ 1A1 states exhibit prompt
fragmentation, while the molecular structure in the $(1e^{-2})$ 3A2 state
evolves further away from the neutral configuration, driven towards larger
bond angles between the two protons, prior to dissociation. This is indeed
consistent with the asymptotic charge exchange mechanism, described in Section
III, that couples the 3A2 (3A′′) and 3E (3A′′) states (PES cuts inset in Fig.
1(b)). The dissociation on the $(2a_{1}^{-1},3a_{1}^{-1})$ 1A1 and $(1e^{-2})$
1A1 states result in the direct elimination of two protons, which are light
and depart fast, providing little time for the molecular structure to evolve
away from the neutral equilibrium geometry during the fragmentation. In
contrast, the fragmentation on the $(1e^{-2})$ 3A2 state initially involves a
heavier NH+ ion preceding the charge exchange mechanism that produces a light
proton. Thus the initial dissociation on the $(1e^{-2})$ 3A2 state (prior to
the charge exchange) is slower due to the increased mass of one of the charged
fragments.

Although our calculations keep the bond angles frozen, it is known that for
molecules of the form AH3, ionization from the $1e$ orbital (as in the case of
the $(1e^{-2})$ 3A2 state) drives the molecule towards a planar configuration,
i.e. larger H-N-H bond angles (this can be seen in a Walsh diagram, see Ref.
Higuchi (1956)). The increased fragmentation time leads to an increased
likelihood for processes such as the aforementioned charge exchange to take
place, as well as more time for the molecular geometry to evolve away from the
neutral equilibrium geometry towards larger H-N-H angles, preceding the
dissociation. The timescale for a wave packet in the $(1e^{-2})$ 3A2 state to
reach the geometry where charge exchange can occur, as well as other details
of the dissociation dynamics, precisely explaining the propensity towards
fragmentation at H-N-H angles approaching 180∘ (beyond our qualitative
description), would need to be addressed in a future study requiring time-
dependent calculations that include non-adiabatic coupling.

In contrast to the three states in Fig. 8 discussed above, the ($1e^{-2}$) 1E
state in Fig. 8 (c) displays a band of KER over a wide distribution of
$\theta_{p_{1},p_{2}}$ extending all the from zero to 180∘ and smoothly peaked
towards 180∘. This distribution is consistent with the sequential dissociation
mechanism discussed below in detail, namely
$\textrm{NH}_{3}^{++}\rightarrow\textrm{NH}^{+}+\textrm{H}^{+}+\textrm{H}\rightarrow\textrm{N}+2\textrm{H}^{+}+\textrm{H}$.
If prior to the second step of this process the NH+ fragment rotates freely
before dissociating via a crossing with another electronic state, the H+ is
ejected in a random direction in the body frame of the NH+ molecule. However
that is not a random direction in the laboratory frame because the NH+
fragment is translating with a center of mass momentum opposite to the sum of
the momenta of the H and H+ atoms produced in the first step, presumably
ejected near the directions of the original NH bonds. The diatom’s center of
mass is therefore moving away from the H+ ion produced in the first step, and
consequently the random angular distribution of the proton ejected from the
moving NH+ shifted in the direction opposite the direction of the first H+
ion. A similar effect has been seen in dissociation of the water dication
following one-photon double ionization, in which a sequential dissociation
channel involving dissociation of OH+ is seen Streeter  _et al._ (2018); Ben-
Itzhak and Severt (2020).

Other evidence also suggests that the different fragmentation dynamics of the
$(1e^{-2})$ 1E state can be specifically attributed to a sequential
dissociation mechanism involving four bodies in the final set of fragments.
Here, we do not consider the possibility of a sequential dissociation process
first resulting in NH${}_{2}^{+}$ \+ H+ fragmentation, with the NH${}_{2}^{+}$
subsequently dissociating to NH + H+ or N + H + H+. Our interpretation does
not include these channels, as we have analyzed the NH${}_{2}^{+}$ \+ H+
dissociation channel (which is the subject of a future paper) and did not
observe any electron-ion momentum correlation consistent with shared dication
electronic states producing both NH + H+ \+ H+ or NH${}_{2}^{+}$ \+ H+
fragments. However, we cannot totally rule out these possibilities, as the
lifetime of the intermediate NH${}_{2}^{+}$ fragment may be too short for
these fragments to survive the flight time to the ion detector. However, if
intermediate NH${}_{2}^{+}$ fragments dissociate during their flight to the
detector, the secondary ion momenta should exhibit a broad spread in momentum.
Since this is not observed, we argue in favor of a different sequential
dissociation mechanism.

Previous measurements have found that PDI to the $(1e^{-2})$ 1E state produces
the fragments NH+ \+ H+ \+ H, where the bound NH+ ion is in its ground state,
i.e. the X ${}^{2}\Pi$ state Stankiewicz  _et al._ (1989). Although the
dissociative limit of the NH+ ${}^{2}\Pi$ state results in N+(${}^{3}P$) +
H(${}^{2}S$) fragmentation, it has been shown that the X ${}^{2}\Pi$ state
crosses the a ${}^{4}\Sigma^{-}$ state in the FC region and that population
transfer between the X and a states can occur via spin-orbit coupling Liu and
Verhaegen (1970); Ame ; Kawaguchi and Amano (1988); Colin and Douglas (1968);
Tarroni  _et al._ (1997). As seen in Fig. 9, the NH+ a ${}^{4}\Sigma^{-}$
state dissociates to H+ \+ N(${}^{4}S$) with a dissociation energy that is
roughly 1 eV smaller than the X ${}^{2}\Pi$ state dissociation energy. Thus,
high-lying vibrational states of the NH+ fragment that are initially bound in
the X ${}^{2}\Pi$ state can undergo intersystem crossing to the a
${}^{4}\Sigma^{-}$ state, yielding the final fragments of the reaction
NH${{}_{3}^{++}}\rightarrow$ N(${}^{4}S$) + H(${}^{2}S$) + H+ \+ H+. In the
present context, population transfer can occur along the inner wall of the
quasi-degenerate NH+ states when the initial breakup of the ($1e^{-2}$) 1E
state produces NH+(${}^{2}\Pi$) ions with internal energy that lies within the
appearance window shown in Fig. 9. We can estimate the location of the four-
body limit by first extrapolating the MRCI energy for the 3E state (Fig. 1
blue curve) to infinite separation of the N-H bonds. This places the
NH+(${}^{4}\Sigma^{-}$) + H + H+ limit at 0.63 eV. Adding to this the 3.66 eV
dissociation energy of NH+(${}^{4}\Sigma^{-}$) places the four-body limit at
4.29 eV, directly in the center of the appearance window. This four-body
breakup mechanism also explains why the theoretical KER value of 9.42 eV
gleaned from Table 1 is higher than the measured value of 7.7 eV. From Fig. 9
we see that the NH+ fragment must have a minimum internal energy of 3.7 eV to
dissociate to N + H+ at the lower end of the appearance window to produce a
fast proton with (9.42–3.7 eV) = 5.72 eV and a zero energy proton. At the
upper end of the appearance window we get a fast proton with (9.42–4.5 eV) =
4.92 eV and a slow proton with 1 eV. This interpretation appears to be
consistent with the measured particle energy balance and prompts us to believe
that each NH fragment in the three concerted dissociation channels was
produced with a distribution of ro-vibrational energy around 2 eV, while the
NH+ fragment in the sequential dissociation channel was produced with a
distribution of ro-vibrational energy that extends well beyond 3.7 eV,
enabling the second fragmentation step. These results are also consistent with
a previous theoretical treatment of the dissociation of H2O++ Gervais  _et
al._ (2009), where the internal energy distribution of the OH+ fragment in the
H+ \+ OH+ two-body dissociation channel was observed to span approximately 3–5
eV.

Although the initial set of photoions produced via excitation to the
$(1e^{-2})$ ${}^{1}E$ state would not produce the four-particle (two-electron,
two-proton) coincidence we measure, highly vibrationally excited ground state
NH+ fragments (lying within the appearance window) can spin-orbit couple to a
state where a fragmentation, producing a second proton, is possible, yielding
the necessary two-proton coincidence. Since the spin-orbit coupling is weak,
and the ensuing dissociation is not instantaneous, the intermediate NH+
fragment can rotate prior to coupling to the dissociative state, which results
in a proton-proton angular distribution that differs from the other three
dication states that involve fewer fragmentation steps. The lifetime of the
excited intermediates in the appearance window in the X(${}^{2}\Pi$) state is
determined by the strength of the spin-orbit coupling but not deduced in our
experiment. It could potentially be measured using a different detection
scheme or calculated using a different theoretical approach than the one taken
in this study.

Figure 9: The potential energy curves for the X${}^{2}\Pi$ ground state and
the a${}^{4}\Sigma^{-}$ state of NH+, extracted from Ref. Ame . Population
transfer may occur between these states via spin-orbit coupling, where
initially bound excitations on the X${}^{2}\Pi$ state can dissociate on the
a${}^{4}\Sigma^{-}$ state. Only diatomic NH+ fragments with internal energy
within (or above) the appearance window will dissociate.

We discuss the cases of excitations below and above the appearance window
next. Excitations initially prepared in the X${}^{2}\Pi$ state that lie above
the appearance window directly dissociate to produce N+(${}^{3}P$) + H+ \+
H(${}^{2}S$) + H(${}^{2}S$). Indeed, this is supported by our measurements by
analyzing the N+ \+ H+ dissociation channel, which is briefly addressed here.
The same procedure used to select the H+ \+ H+ dissociation channel and
described at the beginning of this sub-section is used to select the N+ \+ H+
channel. We plot the PDI yield of the N+ \+ H+ fragmentation as a function of
the photoelectron energy sum and photoion energy sum, shown in Fig. 10. In
this fragmentation channel we observe a single feature (seen in Fig. 10),
which we attribute to a single contributing dication electronic state. We
argue that this feature corresponds with the magenta color-coded $(1e^{-2})$
1E state. This feature possesses an electron energy sum of 16.7 eV, which
exactly coincides with the electron energy sum measured for the feature in the
H+ \+ H+ dissociation channel corresponding with the $(1e^{-2})$ 1E state.
From this evidence we suggest that the single feature observed in the N+ \+ H+
channel corresponds with the same dication electronic state that contributes
to the sequential H+ \+ H+ dissociation mechanism. Comparing the H+ \+ H+ and
N+ \+ H+ yields following PDI to the $(1e^{-2})$ 1E state indicates that
roughly the same amount of population ends up above the appearance window as
compared to within it. As for excitations initially prepared in the X
${}^{2}\Pi$ state that lie below the appearance window, these will remain as
bound NH+ fragments. This is also supported by our measurements by analyzing
the NH+ \+ H+ dissociation channel (which is the topic of a future paper and
thus not presented here). In this dissociation channel we also identify a
feature corresponding with the $(1e^{-2})$ 1E state. These results are
entirely consistent with the explanation presented in the paragraph above,
where the PDI to the $(1e^{-2})$ 1E state produces the fragments NH+
${}^{2}\Pi$ \+ H+ \+ H for which the excitation in the NH+ ion can lie below,
within, or above the appearance window. All three of these cases are observed
in our measurement and illustrate the various levels of complexity in the
dissocation dynamics of simple polyatomic molecules that can occur following
valence PDI to just a single state.

Figure 10: The yield of N+ \+ H+ after valence PDI of NH3 as a function of
photoelectron pair energy sum and photoion pair energy sum for the $(1e^{-2})$
1E dication state.

To further support the claim that the H+ \+ H+ fragmentation on the
$(1e^{-2})$ 1E state occurs via the four-body mechanism discussed above, we
analyze the slow proton emerging from the dissociation on the $(1e^{-2})$ 1E
state, using its momentum to infer the KER of the dissociation of the NH+
fragment, shown in Fig. 11. Since two neutral particles are left undetected (N
and H), and simple conservation of momentum can thus not be applied, this is
realized by assuming that the momentum of the undetected neutral N atom is
approximately that of the N-H center of mass, inferred from the two proton
momenta. We find the inferred KER to peak at 0.61 eV (FWHM 0.71 eV), which
lies below the $\sim$1 eV maximum KER permitted by the locations of the two
adiabatic limits of the X${}^{2}\Pi$ and a${}^{4}\Sigma^{-}$ states, i.e. the
appearance window (see Fig. 9). This supports the assumption that the slow
proton emerges from a dissociation on the a${}^{4}\Sigma^{-}$ state. Since our
measurement also indicates that the $(1e^{-2})$ 1E state contributes to the
NH+ \+ H+ \+ H fragmentation channel (the topic of another manuscript,
currently in preparation), which is in agreement with previous measurements
Stankiewicz  _et al._ (1989), we believe that some small fraction of the NH+
fragments of this three-body fragmentation channel can decay through
intersystem crossing and feed into the four-body N + H + H+ \+ H+
fragmentation channel. This conclusion is also corroborated by our analysis of
the N+ \+ H+ dissociation channel, which shows that the $(1e^{-2})$ 1E state
also feeds into this four-body fragmentation channel and corresponds with the
initial excitations in the NH+ ${}^{2}\Pi$ ion that lie above the appearance
window.

Figure 11: The inferred KER from the dissociation of the NH+ fragment,
involving the measured slow proton and the nitrogen following PDI of NH3 at
61.5 eV to the $(1e^{-2})$ 1E state, resulting in the four-body fragmentation
N + H + H+ \+ H+. The KER peaks at 0.61 eV, with a FWHM of 0.71 eV.

###  IV.3 Photoelectron dynamics

Next, we display in Fig. 12 the photoelectron energy sharing distributions for
the four dication states. We define the electron energy sharing as:

$\rho=\frac{E_{e_{1}}}{E_{e_{1}}+E_{e_{2}}},$ (1)

where $E_{e_{1}}$ and $E_{e_{2}}$ are the energies of electron 1 and 2,
respectively. Values of $\rho$ near 0.5 indicate equal energy sharing between
the two photoelectrons, while values near 0 or 1 indicate unequal energy
sharing between the two photoelectrons. In all four dication states we do not
observe a strong enhancement in yield for any particular values of $\rho$. The
distributions are nearly flat. In the absence of autoionization, this is
similar to the PDI of atoms and molecules in this excess energy range (see
e.g. Andersson  _et al._ (2019); Reedy  _et al._ (2018)). The exception is the
($1e^{-2}$) 3A2 state (cyan) and perhaps the $(1e^{-2})$ 1E state (magenta),
which show some propensity towards increased yield at values of $\rho$ near
0.5. This is surprising since the ($1e^{-2}$) 3A2 and the $(1e^{-2})$ 1E state
dication states correspond to the highest electron sum energies (see Fig. 3).
A maximum PDI yield at equal energy sharing, if any, would be expected for the
lowest electron sum energies according the the Wannier threshold law Wannier
(1953), which favors the emission of two electrons with the same energy and
back-to-back close to the PDI threshold. However, the electron pair emission
patterns are subject to selection rules that are specific to each dication
state and the molecular orientation with respect to the polarization vector;
they inherently influence the electron energy sharing to a certain degree. The
detailed investigation of this complex problem requires M/RFPADs and is beyond
the scope of this work. These distributions have all been normalized to the
same value and have been placed in ascending order, based on the corresponding
photoelectron energy of the state (the state with the lowest photoelectron
energy sum is placed near the bottom and the state with the highest
photoelectron energy sum is placed at the top).

Lastly, we plot in Fig. 13 the yield of the H+ \+ H+ fragmentation as a
function of cosine of the relative emission angle between the two
photoelectrons and in different energy sharing conditions of the electron pair
for the four dication states, integrated over all molecular orientations
relative to the polarization vector of the incoming light and with no
restrictions on the emission direction of either electron. The relative
electron-electron angles are plotted for $0.425<\rho<0.575$ (shown in red) and
for $\rho<0.05$ or $\rho>0.95$ (shown in blue). We point out that our
measurement suffers from some multi-hit detector dead-time effects, which
influence the measured yields of the photoelectrons emitted in the same
direction with similar kinetic energies. For equal energy sharing between the
two emitted electrons and for $\theta_{e_{1},e_{2}}\leq 90^{\circ}$ (emission
into the same hemisphere), we can expect to fail to detect up to $\sim 52\%$
events for the ($1e^{-2}$) ${}^{3}A_{2}$ state, $\sim 27\%$ for the
($1e^{-2}$) ${}^{1}E$ state, $\sim 23\%$ for the ($1e^{-2}$) ${}^{1}A_{1}$
state, and $\sim 22\%$ for the ($2a_{1}^{-1},3a_{1}^{-1}$) ${}^{1}A_{1}$
state. Note that we estimate these losses for the ′′worst case′′ isotropic
relative electron-electron emission, which very well represents autoionization
processes that are sequential in nature and are subject to unequal energy
sharing between the electrons. The equal energy sharing case on the other hand
is dominated by knock-out processes with very few electron pairs emitted into
the same hemisphere. The actual losses are expected to be closer to the losses
for the case of unequal electron energy sharing reported below.

Figure 12: The yield of the H+ \+ H+ fragmentation after PDI of NH3 at 61.5 eV
as a function of photoelectron energy sharing for each of the four relevant
dication states. Here the y-axis indicates the PDI yield in arbitrary units on
a linear scale. The distributions are not internormalized. They have been
staggered in order based on their respective electron energy sum for better
visibility (i.e. the states are placed in ascending order with respect to
their respective photoelectron energy sum).

Figure 13: The cosine of the relative emission angle between the two
photoelectrons for two different energy sharing conditions for each of the
four dication states of NH3 following PDI at 61.5 eV. Electron energy sharing
between 0.425 and 0.575 is shown in red, and energy sharing less than 0.05 or
greater than 0.95 is shown in blue.

The relative angles between the two photoelectrons under unequal energy
sharing conditions (blue circles in Fig. 13) are rather isotropic for all four
dication states, where there is a slight propensity towards back-to-back
emission (or in other words a lack of events with electrons emitted into the
same direction), which we partly attribute to the dead-time problem at
relative electron-electrons angles below 90∘ (emission into the same
hemisphere). The simulated losses of events with unequal energy sharing amount
to $\sim 26.1\%$ for the ($1e^{-2}$) ${}^{3}A_{2}$ state, $\sim 8.4\%$ for the
($1e^{-2}$) ${}^{1}E$ state, $\sim 5.2\%$ for the ($1e^{-2}$) ${}^{1}A_{1}$
state, and $\sim 4.2\%$ for the ($2a_{1}^{-1},3a_{1}^{-1}$) ${}^{1}A_{1}$
state. Evidently the small anisotropies in the relative angular distributions
for the unequal electron energy sharing case (blue), presented in Fig. 13 for
all four dication states, are accounted for by the detector dead-time
limitations, and are otherwise consistent with isotropic relative angular
distributions. As there is no hint for autoionzation visible in the electron-
electron energy correlation map depicted in Fig. 4, the unequal electron
energy sharing case is likely dominated by knock-out processes, as reasoned
below.

In contrast, the photoelectron dynamics for equal energy sharing conditions
(red in Fig. 13) reveals anisotropic angular distributions that are different
for all four dication states and exceed the anisotropy expected from dead-time
effects alone. For this case the relative angle between the two photoelectrons
producing the ($2a_{1}^{-1},3a_{1}^{-1}$) 1A1 state exhibits a preference
towards back-to-back emission. The emission angle between the two
photoelectons from the ($1e^{-2}$) 3A2 state increases starting at $0^{\circ}$
and peaks at an angle of roughly $125^{\circ}$ before decreasing as the angle
approaches $180^{\circ}$. The photoelectrons that produce the ($1e^{-2}$) 1E
state have relative emission angles that increase starting at $0^{\circ}$,
which then begin to level out at $100^{\circ}$, increasing at a slower rate as
the angle approaches $180^{\circ}$. Last, the relative electron-electron
emission angle of the ($1e^{-2}$) 1A1 state increases starting at $0^{\circ}$
and peaks at an angle of roughly $150^{\circ}$ before decreasing as the angle
approaches $180^{\circ}$. All four dication states show a non-vanishing PDI
yield for small electron-electron angles close to $0^{\circ}$. This
contribution is mainly due to the finite angular bin size accepting
differences in the relative emission angles of up to $27^{\circ}$ at these
values, as well as residual background from random coincidences underneath the
features visible in e.g. Fig. 2.

These trends in the relative electron-electron angular distributions as a
function of the electron energy sharing possess similarities to prior
observations made in the PDI of atomic and molecular targets Bräuning  _et
al._ (1998); Knapp  _et al._ (2005); Weber  _et al._ (2004a); Randazzo  _et
al._ (2020). In the valence PDI experiments for helium Bräuning  _et al._
(1998), which is dominated by knock-out processes, rich photoelectron angular
distributions emerge due to selection rules and symmetry considerations. It
has been seen that for equal energy sharing conditions and the first detected
electron fixed along the polarization vector of the ionizing field, the
relative emission angle between the photoelectrons can be quite anisotropic
and peaked at angles between $90^{\circ}$ and $180^{\circ}$ due to selection
rules for dipole allowed transitions, whereas in unequal energy sharing
conditions, the relative angle between the electrons can become more isotropic
with a smaller peak at $180^{\circ}$. In the atomic case for equal electron
energy sharing, there can be a node at a relative electron-electron angle of
$180^{\circ}$, regardless of the emission direction of either of the two
electrons. This is for instance true for the PDI of He and is due to a
selection rule based on parity conservation in one photon transitions. Such a
scenario is in general not well pronounced in the PDI of (polyatomic)
molecules, and rather resembles the distributions for all cases presented in
Fig. 13. In addition to the finite angular bin size, again accepting
differences in the relative emission angles of up to $27^{\circ}$ at
$180^{\circ}$, we attribute this to the fact that we have not enforced any
conditions on the molecular orientation or direction of the polarization
vector of the XUV field. Integrating over all molecular orientations and the
direction of the polarization vector is prone to wash out sharp features in
the electron relative angular distribution, since angular momentum can be
transferred to the nuclear systems and softens the aforementioned selection
rules (as seen and discussed in Refs. Weber  _et al._ (2004a); Vanroose  _et
al._ (2006)), in addition to other features. The limited number of events in
the present data set does not allow conditions to be enforced on the molecular
orientation or emission direction of one of the photoelectrons with high
statistical significance. Future COLTRIMS studies could be directed towards
the states that obey the axial recoil approximation to gather appreciable
statistics, in order to produce photoelectron angular distributions in the
molecular frame, which inter alia would help to study and understand the role
of selection rules in the PDI of a symmetric top molecule with respect to the
polarization vector of the incoming light.

##  V Conclusion

We have performed state-selective measurements on the H+ \+ H+ dissociation
channel of NH3 following direct valence PDI at 61.5 eV, where the two
photoelectrons and two protons were measured in coincidence on an event-by-
event basis using COLTRIMS. With the assistance of theoretical MRCI
calculations of dication PES cuts, we identified the four participating
dication electronic states that lead to H+ \+ H+ fragmentation, which
correspond with the four features we observed and have estimated their
branching ratios.

The PDI yield as a function of KER and the measured proton-proton angle
indicates that three of the four dication states dissociate in a concerted
mechanism, while the fourth state, the $(1e^{-2})$ 1E state, dissociates via a
sequential process, with the intermediate ro-vibrationally excited NH+
fragment ion decaying through an intersystem crossing that leads to a four-
body breakup. Two of the dication states, the $(2a_{1}^{-1},3a_{1}^{-1})$ 1A1
and $(1e^{-2})$ 1A1 states, exhibit concerted dissociation mechanisms that
fragment near the ground state geometry (axial recoil approximation applies).
The third state, the $(1e^{-2})$ 3A2 state, undergoes appreciable evolution in
its molecular geometry and an asymptotic electron transfer from H to NH+ at
distances greater than 18 Bohr in the dissociating dication, preceding the
three-body breakup. Differences between the MRCI calculations and the measured
KER suggest that the neutral NH fragment in each of the three-body
dissociation channels is highly ro-vibrationally excited.

The relative emission angle between the two photoelectrons as a function of
their energy sharing has some resemblance to prior measurements made on atomic
and molecular targets, in spite of integrating over all molecular orientations
and emission angles of the first photoelectron, relative to the XUV
polarization. While the present study has focused on PDI processes that result
in proton-proton breakup channels, we are presently analyzing the two- and
three-body PDI breakup channels that produce NH${}_{2}^{+}$ \+ H+ and NH+ \+ H
+ H+, which is the topic of a future manuscript.

##  VI Acknowledgments

Work at LBNL was performed under the auspices of the US Department of Energy
(DOE) under Contract DE-AC02-05CH11231, using the Advanced Light Source and
National Energy Research Computing Center, and was supported by the U.S. DOE
Office of Basic Energy Sciences, Division of Chemical Sciences. JRML personnel
were supported by the same US DOE funding source under Award No. DE-
FG02-86ER13491. A.G. was supported by the ALS through a Doctoral Fellowship in
Residence. Personnel from the University of Nevada, Reno was supported by the
National Science Foundation Grant No. NSF-PHY-1807017. The Frankfurt group
acknowledges the support of the Deutsche Akademische Austausch Dienst (DAAD)
and the Deutsche Forschungsgemeinschaft (DFG). We thank the staff at the
Advanced Light Source for operating the beamline and providing the photon
beam. Moreover, we thank the RoentDek GmBH for longtime support with detector
hardware and software.

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