# Positron annihilation and binding in aromatic and other ring molecules

E. Arthur-Baidoo<EMAIL_ADDRESS>J. R. Danielson
<EMAIL_ADDRESS>C. M. Surko<EMAIL_ADDRESS>Department of
Physics, University of California San Diego, La Jolla, CA, 92093 USA.  J. P.
Cassidy1 1Centre for Light-Matter Interactions, School of Mathematics and
Physics, Queen’s University Belfast,
University Road, Belfast BT7 1NN, Northern Ireland, United Kingdom.
2School of Physics, Trinity College Dublin, Dublin 2, Ireland.  S. K. Gregg1
1Centre for Light-Matter Interactions, School of Mathematics and Physics,
Queen’s University Belfast,
University Road, Belfast BT7 1NN, Northern Ireland, United Kingdom.
2School of Physics, Trinity College Dublin, Dublin 2, Ireland.  J. Hofierka1
1Centre for Light-Matter Interactions, School of Mathematics and Physics,
Queen’s University Belfast,
University Road, Belfast BT7 1NN, Northern Ireland, United Kingdom.
2School of Physics, Trinity College Dublin, Dublin 2, Ireland.  B. Cunningham1
1Centre for Light-Matter Interactions, School of Mathematics and Physics,
Queen’s University Belfast,
University Road, Belfast BT7 1NN, Northern Ireland, United Kingdom.
2School of Physics, Trinity College Dublin, Dublin 2, Ireland.  C. H.
Patterson2 D. G. Green1<EMAIL_ADDRESS>1Centre for Light-Matter
Interactions, School of Mathematics and Physics, Queen’s University Belfast,
University Road, Belfast BT7 1NN, Northern Ireland, United Kingdom.
2School of Physics, Trinity College Dublin, Dublin 2, Ireland.

###### Abstract

Annihilation spectra are presented for aromatic and heterocyclic ring
molecules resolved as a function of incident positron energy using a trap-
based positron beam. Comparisons with the vibrational mode spectra yield
positron-molecule binding energies. Good to excellent agreement is found
between the measured binding energies and the predictions of an ab initio
many-body theory that takes proper account of electron-positron correlations
including virtual-positronium formation. The calculations elucidate the
competition between permanent dipole moments and $\pi$ bonds in determining
the spatial distribution of the bound-state positron density. The implications
of these results and the role of multimode features in annihilation in these
molecules, including Fermi resonances, are discussed.

##  I INTRODUCTION AND OVERVIEW

Positrons bind to many polyatomic molecules via the excitation of vibrational
Feshbach resonances (VFR) [1, 2, 3]. Binding energies for over 100 molecules
have been measured to date with magnitudes ranging from a few millielectron
volts to $>$ 0.3 eV. While positrons are also predicted to bind to atoms [4,
5, 6], they have yet to be observed experimentally due to the lack of low-
lying excitations to mediate attachment, though proposals exist [7, 8, 9, 10].
Thus the molecular studies play a special role in our understanding of
positron interactions with matter.

Experimentally, positron binding has been observed to depend upon molecular
parameters such as the polarizability $\alpha$, permanent dipole moment $\mu$,
the number of molecular $\pi$ bonds $N_{\pi}$, the ionization potential $I$,
and the geometry (e.g., upon isomerization) [11, 12, 13, 14, 15]. Theoretical
work successfully predicts many aspects of these chemical trends, but there
remain a number of important questions [16, 17, 18, 19, 13, 20, 21]. For
unsaturated ring molecules, enhancement of the positron-molecule binding
energy $\varepsilon_{\textit{\text{b}}}$ with increasing $N_{\pi}$ is observed
[15].

A recent many-body theory (MBT) [22] developed by some of us provides an ab
initio description of positron binding to molecules, importantly taking proper
account of electron-positron correlations including the nonperturbative
process of virtual-positronium formation (where a molecular electron
temporarily tunnels to the positron). To date it has, e.g., provided the first
calculated binding energies in agreement with measurements, quantified the
role of correlations and the contributions of individual molecular orbitals to
the positron-molecule correlation potential, and explained trends within
molecular families [22, 21]. It has recently been extended to positron
scattering and annihilation on small molecules [23]. Particularly relevant to
the present study, the MBT provides new insights through calculations of the
spatial distribution of the bound-state positron wave functions.

In this paper, measurement of the positron annihilation spectrum and binding
energy are presented for aromatic and substituted ring molecules and compared
to predictions of the many-body theory. This work extends earlier studies [15,
14, 11] using ring substitutions to study molecules with varying amplitudes of
permanent dipole moment and numbers of $\pi$ bonds. As a result, it provides a
sensitive test of the competition between these two mechanisms of positron
binding.

Results are presented for benzene, toluene, acetophenone, aniline,
phenylacetylene, benzonitrile, benzaldehyde, furan, pyrrole, pyridine,
pyridazine, cycloheptatriene, and cyclooctatetraene. The molecules studied
span a wide range of $N_{\pi}$ (2–5) and $\mu$ (0.3–4.5 D) and a factor of 1.7
in $\alpha$. Good to excellent agreement is found between the MBT predictions
and binding energy measurements for this new class of molecules, which include
the largest studied to date via the MBT.

As discussed below, the MBT shows that regions of the bound-state positron
density are localized near the $\pi$-bond electron density above and below
planes of ring molecules such as benzene and toluene. This exemplifies the
importance of the $\pi$-bonds in localizing the bound-state positron wave
function. Other distributions of positron density are localized close to
strong permanent dipole moments that typically lie in the planes of ring
molecules such as in benzaldehyde and pyridine. For molecules with weaker
dipole moments, including furan and pyrrole, the effects compete, and the
bound-state positron distribution is observed in regions of both the
$\pi$-bond electrons and adjacent the dipole moment. This contrasts to the
approximately uniform covering of positron density surrounding non-polar
molecules without $\pi$ bonds such as that seen in alkanes [3].

Positron annihilation rates for molecules are conventionally expressed in
terms of the quantity $Z_{\text{eff}}$ which is the measured rate normalized
by that expected for a gas of free electrons with the density equal to the
density of the molecular gas [2]. Positron attachment via VFR typically
results in annihilation rates $Z_{\text{eff}}\gg 1$ and greater than those
expected for a simple collision (e.g., $Z_{\text{eff}}\simeq$ the number of
valence electrons) [2]. The VFR theory of positron annihilation due to dipole-
and quadrupole-allowed fundamental vibrations can account for enhancements of
an order of magnitude or so in annihilation rate (e.g., $Z_{\text{eff}}\leq
2000$ for a single vibrational mode) [24, 25]. However, $Z_{\text{eff}}$ for
many molecules, including those studied here, is 2 – 3 orders of magnitude
larger, and this is not understood.

Of note in this study, and similar to that observed previously in benzene
[26], broad regions in the annihilation spectra are observed in aromatic
molecules such as pyridine and aniline at incident positron energies that do
not correspond to fundamental vibrations. The origins of these features are
likely multimode vibrations where the specific modes involved have yet to be
identified. Related to this, enhanced annihilation is observed in benzaldehyde
that is likely due to the multimode phenomenon of Fermi resonance (the
resonance of a fundamental vibration with the second harmonic of another
vibration). This provides an example of a multi-mode contribution to the
annihilation spectrum where the specific modes involved are known and is
worthy of further study.

This paper is organized as follows. Section II presents the description of the
experiment and data analysis procedures. Section III presents the measured
annihilation spectra as a function of incident positron energy and the
resulting binding energy analyses. Section IV describes the many-body theory
and the predicted binding energies for the molecules studied here and
comparison with the measurements. Further aspects of the results are discussed
in Sec. V, and Sec. VI presents a summary and concluding remarks.

##  II DESCRIPTION OF THE EXPERIMENT AND DATA ANALYSIS

The experimental techniques used in this positron annihilation study have been
described in detail previously [12]. Low-energy positrons are obtained from a
22Na radioisotope source and a solid neon moderator. Positrons leaving the
moderator are radially confined and guided using magnetic fields into a three-
stage buffer-gas trap (BGT) [27]. In the BGT, they are slowed by inelastic
collisions with N2 and CF4 resulting in positrons trapped and cooled to the
ambient temperature (300 K). The pulsed positron beam is formed by slowly
ramping the bottom of the potential well to a voltage higher than that of the
exit gate [27, 12]. The beam thus formed has a narrow energy spread with a
mean energy just slightly larger than the exit gate potential. The energy
distribution of the beam is measured using a retarding potential analyzer
(RPA).

For these experiments, the _parallel_ energy distribution of the beam is
approximately Gaussian with a mean energy $E_{\parallel}=$ 0.67–0.70 eV and
standard deviation $\sigma_{\parallel}$= 8–10 meV. The perpendicular energy of
the beam is obtained from measuring mean parallel energy at different RPA
magnetic fields. The slope of this measurement yields the mean perpendicular
energy E⟂= 20 $\pm$ 2 meV [27]. The distribution in total energies is the
convolution of the parallel and perpendicular energy distributions which
yields an exponentially modified Gaussian distribution (EMG) [27]. The guiding
magnetic field varies by a factor of five along the beam line, where the
annihilation gas cell is located at the end [28]. The magnetic field at the
gas cell is independently controlled. When it matches the value at the exit
gate of the BGT, both the parallel and perpendicular particle distributions
are known, and hence the total energy distribution is known for positrons
interacting with the test gas.

The positrons propagate downstream to a 26 cm long gas cell, the potential of
which is used to set the mean parallel energy of the beam interacting with the
test gas. During annihilation measurements, an isolated electrode after the
annihilation cell is biased to 6 V to reflect the beam back towards the BGT.
Positrons continue to bounce between the exit gate of the BGT and the
reflecting electrode while measurements are made. See Ref. [12] for details.

The measured annihilation rates are obtained by converting the annihilation
counts ($N_{c}$) to an annihilation cross section using the number of
positrons per pulse ($N_{p}$), the calibrated detector efficiency
($\eta_{D}$), the test-gas density ($n_{m}$) and the annihilation cell length
($L_{D}$). For a given molecule, the normalized annihilation rate is given by
[2]

$Z_{\text{eff}}(E_{\parallel})=\frac{N_{\text{c}}(E_{\parallel})}{2N_{\text{p}}\eta_{\text{D}}L_{\text{D}}n_{\text{m}}}\frac{\upsilon(E_{\parallel})}{\pi
r_{0}^{2}c}$ (1)

where $\upsilon(E_{\parallel})$ is the mean particle velocity, $r_{0}$ is the
classical radius of the electron, and c is the speed of light. The factor of
two in the denominator accounts for the two passes of the positrons through
the annihilation cell in a single bounce.

Shown in Fig. 1 are contemporary data for the $Z_{\text{eff}}$ of the alkane
octane plotted as a function of the mean parallel energy of the beam. Also
shown is the downshifted IR spectrum (arbitrarily scaled) [29], which
demonstrates the spread of the vibrational modes. The $Z_{\text{eff}}$
spectrum shows an asymmetric resonant peak near 200 meV. In order to obtain
the binding energy, a fit to the C-H stretch resonance is performed where each
IR-active mode is taken to be a delta function and convolved with the beam EMG
distribution. The resonances from all modes are summed to give a single
resonant peak with a width which is a combination of the measured beam
parameters and energies of the vibrational modes. The positron-molecule
binding energy $\varepsilon_{\textit{\text{b}}}$ is obtained using the total
energy $\varepsilon_{r}$ at which the resonance is observed and the
vibrational mode energy $\hbar\omega$ with the relationship,

$\varepsilon_{\textit{\text{b}}}=\hbar\omega-\varepsilon_{r}.$ (2)

For the octane data in Fig. 1, this procedure yields a binding energy of 150
$\pm$ 3 meV [12].

Figure 1:  Annihilation rate $Z_{\text{eff}}$ for octane as a function of mean
parallel energy of the positron beam. The green curve is the IR spectrum of
octane obtained from NIST [29], and vertical blue bars indicate the locations
of the IR active fundamental modes, with both downshifted by
$\varepsilon_{b}$. The solid black line shows the fit to the C-H modes used to
determine $\varepsilon_{b}$.

In contrast to octane and other alkane molecules [12], benzene and the other
aromatics studied here have appreciable annihilation amplitudes that are
believed to be due to unresolved non-fundamental modes [26]. This leads to
uncertainty in obtaining the resonances used in Eq. 2. As a result, for all
aromatic molecules, the choice was made to use only the dipole-active high-
energy phenyl-CH stretch modes to obtain $\varepsilon_{\textit{\text{b}}}$
values. These modes used are shown as solid vertical lines in the figures, and
the resulting fits are shown by the solid black lines. For most molecules,
these fits match the data reasonably well. However, in a number of cases,
including the other components of the spectra can modify the best-fit energy
of the resonance by several meV. Thus, the conservative error estimate of
$\pm$ 5 meV is associated with the $\varepsilon_{\textit{\text{b}}}$
measurements unless otherwise noted.

This analysis is based on the Gribakin and Lee (GL) theory for
$Z_{\text{eff}}$ from VFRs mediated by dipole-allowed fundamental vibrational
modes [24, 2]. If the IR activity for the fundamentals is known, and using
$\varepsilon_{b}$ obtained from above, the contribution from each mode can be
summed to predict the total expected $Z_{\text{eff}}$ as a function of the
mean parallel energy of the positron beam. For almost all molecular targets
this estimate is _much smaller_ than that observed. However, the spectral
shape is often quite accurate.

The amplitude of the fit can be related to the enhancement factor that the GL
prediction must be multiplied by to match the measurements. For the molecules
here, this factor varies from $\sim 220$ for cyclooctatetraene to $\sim 1.5$
for furan. The solid line in Fig. 1 is 75 times the GL prediction for octane.
This enhancement is believed to be due to the coupling of the fundamentals to
a large number of multi-mode resonances by intramolecular vibrational
redistribution (IVR) [30, 2]. However, this has yet to be confirmed. The
origin of this enhancement mechanism remains as a major topic of study, beyond
the scope of the work presented here.

##  III ANNIHILATION SPECTRA AND BINDING-ENERGY ANALYSES

The current study focuses on the manner in which additions to, and
substitutions in the benzene ring affect binding energies. The molecules
studied can be grouped in three categories: (i) benzene (C6H6) and its
derivatives, toluene (-CH3), acetophenone (-COCH3), phenylacetylene (-CCH),
benzonitrile (-CN), and benzaldehyde (-CHO); (ii) the heterocyclic aromatic
molecules furan, pyrrole, pyridine, and pyridazine; and (iii) non-aromatic
cyclic molecules, cycloheptatriene (CHT) and cyclooctatetraene (COT). The
results for each category and molecule are discussed separately. For easy
referral during this discussion, the molecular parameters for all molecules
and the measured positron-molecule binding energies are summarized in Table I.

Table 1: Experimental (exp.) binding energies for the molecules studied, together with their molecular properties, polarizability ($\alpha$) [31, 32], dipole moment ($\mu$) [31, 32], number of pi-bonds ($N_{\pi}$), and ionization potential ($I$) [31, 32]. The corresponding many-body theory calculated binding energies from the present work are shown in Table 2 Molecule | Structure | Nπ |  $\mu$ [D] |  $\alpha$ [$\text{\AA}^{3}$] |  $I$ [eV] |  $\varepsilon_{b}$ (exp.) [meV]
---|---|---|---|---|---|---
Benzene and its substituted derivatives
Benzene |  C6H6 | 3 | 0.00 | 10 | 9.24 | 132$\pm$3 111Value taken from Ref. [26].
Toluene |  C6H5CH3 | 3 | 0.38 | 11.86 | 8.83 | 173$\pm$5
Acetophenone |  C6H5COCH3 | 4 | 3.02 | 14.4 | 9.28 | 288$\pm$5
Aniline |  C6H5NH2 | 3 | 1.53 | 12.1 | 7.72 | 233$\pm$5
Phenylacetylene |  C6H5CCH | 5 | 0.66 | 13.8 | 8.82 | 230$\pm$5
Benzonitrile |  C6H5CN | 5 | 4.52 | 12.5 | 9.73 | 298$\pm$5
Benzaldehyde |  C6H5CHO | 4 | 3.14 | 12.80 | 9.50 | 220$\pm$10
Heterocyclic aromatic molecules
Furan |  C4H4O | 2 | 0.66 | 7.23 | 8.88 | 52$\pm$5
Pyrrole |  C4H4NH | 2 | 1.77 | 7.9 | 8.2 | 165$\pm$10
Pyridine |  C5H5N | 3 | 2.19 | 9.5 | 9.26 | 186$\pm$5
Pyridazine |  C4H4N2 | 3 | 4.22 | 9.27 | 8.74 | 330$\pm$5
Non-aromatic cyclic molecules
Cycloheptane |  C7H14 | 0 | 0.00 | 12.8 | 9.97 | 104$\pm$4 222From Ref. [12]
Cycloheptatriene |  C7H8 | 3 | 0.25 | 12.6 | 8.29 | 190$\pm$8
Cyclooctane |  C8H16 | 0 | 0.00 | 14.5 | 9.75 | 128$\pm$4 b
Cyclooctatetraene |  C8H8 | 4 | 0.00 | 13.76 | 8.43 | 225$\pm$5

###  III.1 Benzene and its substituted derivatives

Figure 2:  Measured $Z_{\text{eff}}$ values for benzene and its substituted
derivatives (a) benzene, (b) toluene, (c) acetophenone, (d) aniline, (e)
phenylacetylene, (f) benzonitrile. The green curves are the IR spectra
obtained from NIST [29], and vertical blue bars indicate the locations of the
IR active fundamental modes, with both downshifted by $\varepsilon_{b}$. The
solid black curves are fits to the C-H peaks. Vertical dashed green lines
indicate dipole active fundamental modes not used in the analysis.

A high-resolution annihilation spectrum for benzene was measured previously
using the cryogenic trap-based beam, yielding $\varepsilon_{b}$ = 132 $\pm$ 3
meV [26]. However, to better compare to the other molecules, data from the
room-temperature trap-based beam is shown in Fig. 2 (a), along with the fit to
the single dipole-allowed C-H vibrational mode. As discussed in Ref. [26], an
unusual feature of the spectrum is a broad range (i.e., $\sim$ 0.3 - 1.5 eV)
of enhanced annihilation in a region of energies absent fundamental
vibrational modes. As discussed below, these features appear to be quite
common in aromatic molecules.

Benzene derivative molecules with side-group substitutions typically have
dipole moments due to the addition or substitution. The new molecule with the
smallest dipole moment studied here is toluene ($\mu$ = 0.33 D), which
consists of a benzene ring with an attached methyl (CH3) group. It has three
$\pi$ bonds in the ring. While the net dipole moment of benzene is zero due to
symmetry, the methyl group both creates a dipole moment and increases the
polarizability of the molecule.

Shown in Fig. 2 (b) is the annihilation spectrum of toluene. It has a sharp
peak due to the five IR-active C-H stretch modes from benzene and three others
from the methyl group that are visible in the IR spectrum [33]. The binding
energy is found to be 173 $\pm$ 5 meV. Toluene has a factor of $\sim$ 30
increase in annihilation rate as compared to benzene, and is comparable to
enhancements seen in alkane molecules [12].

Shown in Fig. 2 (c) is the annihilation spectrum of acetophenone. It is a
benzene ring with an acetyl group (CH3CO) attached, making it the simplest
aromatic ketone. It has a net dipole moment of about $\sim 3$ D and four $\pi$
bonds. Using the five phenyl and three methyl IR active vibration modes [34],
the binding energy is 288 $\pm$ 5 meV.

Aniline is an aromatic compound comprised of a benzene ring attached to an
amide group (-$\ce{NH2}$). It has a net dipole moment of about 1.5 D and three
$\pi$ bonds. Figure 2 (d) shows the annihilation spectrum of aniline as a
function of mean parallel energy. There is an enhanced resonant peak around
0.2 eV which is due to the phenyl C-H modes. Using these modes, the resulting
binding energy of aniline is 233 $\pm$ 5 meV. In comparison with toluene, both
molecules show similar levels of enhancement of $Z_{\text{eff}}$ (i.e., to
within an order of magnitude). The difference in $\varepsilon_{b}$ of about 57
meV can likely be associated with the larger dipole moment in aniline. Similar
to other molecules, aniline spectrum displays broad spectral weight below 0.1
eV. While the shoulder near 0.19 eV on the high-energy side of the main peak
is likely associated with the N-H mode, the current energy resolution is not
good enough to make a firm identification.

Phenylacetylene (PA) and benzonitrile were also studied. In the PA molecule,
one hydrogen on benzene is replaced with an alkyne, C$\equiv$C-H that includes
a CC triple bond. In benzonitrile, one hydrogen is replaced with a cyanide
(C$\equiv$N) group, which also includes a triple bond. Thus, both molecules
have an additional two $\pi$ bonds from the triple bonds, in addition to the
three from the phenyl group, for a total of five $\pi$ bonds. One difference,
however, is that the net dipole of PA ($\mu$ = 0.66 D) is about six times
smaller than that of benzonitrile ($\mu$ = 4.52 D).

Figure 2 (e) shows the measured annihilation spectrum for phenylacetylene.
Using the phenyl C-H modes to define the resonant peak, $\varepsilon_{b}=$ 230
meV $\pm$ 5 meV. It is surprising that there seems to be no obvious spectral
feature that can be associated with the C$\equiv$C-H group. This is similar to
the lack of an enhanced spectral feature that can be associated with the N-H
modes in aniline.

The annihilation spectrum of benzonitrile is shown in Figure 2 (f).
Substitution of the highly polar C$\equiv$N group creates a net dipole moment
of 4.5 D, which is the largest dipole moment of any molecule in the present
study. Analysis using the phenyl C-H stretch modes yields $\varepsilon_{b}=$
298 meV $\pm$ 5 meV. The enhanced $\varepsilon_{b}$ value relative to PA is
likely associated with increased dipole moment. The peak $Z_{\text{eff}}$ in
BA is a factor of 3 _lower_ than in PA.

Also considered in this group is benzaldehyde (BA), which is benzene with an
attached aldehyde (CHO) group, making it the simplest aromatic aldehyde.
Substitution of the aldehyde group results in a relatively large dipole
moment, $\mu$ = 3.14 D [32, 35]. The BA molecule has a total of four $\pi$
bonds, three from benzene and the fourth due to the C=O double (carbonyl)
bond. The annihilation spectrum for BA is shown in Fig. 3. The spectrum shows
two large peaks that are not completely resolved, one near 0.1 eV and one near
0.15 eV. The peak at 0.15 eV is due to the phenyl C-H stretch modes.

The lower-energy peak is in the region of the aldehyde C-H stretch mode.
However, for BA, this mode is in a strong Fermi Resonance [36] with the
aldehyde C-H bend overtone. The result is two peaks with comparable amplitude,
both of which are visible in the IR spectrum (cf. Fig. 3). The energy
resolution of the 300 K beam is not sufficient to determine whether the
annihilation resonance is from one or both modes, but including this spectral
feature is necessary to determine $\varepsilon_{b}$.

The binding energy analysis used the vibrational frequencies of the two FR
modes and the C-H vibrational modes from the phenyl group [37, 36]. The result
is $\varepsilon_{b}$ = 220 $\pm$ 10 meV. The larger error bar arises from the
uncertainty in identifying the correct mode frequencies for use with Eq. 2.
This value of $\varepsilon_{b}$ for BA is a factor of $\sim 1.7$ times larger
than that for benzene ($\varepsilon_{b}$ = 132 meV; $\mu$ = 0 D) and $\sim
1.2$ times larger than that for toluene ($\varepsilon_{b}$ = 173 meV; $\mu$ =
0.3 D). The difference likely reflects the larger dipole moment of BA. In
contrast, BA is a factor of $\sim 0.75$ times smaller than that of
acetophenone ($\varepsilon_{b}$ = 288 meV; $\mu$ = 3.0 D), which has a
comparable dipole moment but a larger $\alpha$ due to the added methyl group.

Figure 3:  $Z_{\text{eff}}$ measurements for benzaldehyde. The green curves
are the IR spectra obtained from NIST [29], and vertical blue bars indicate
the locations of the IR active fundamental modes, with both downshifted by
$\varepsilon_{b}$. The solid black curve is the fit (see text for details).

###  III.2 Heterocyclic aromatics

Four heterocyclic aromatic compounds were also studied: furan, pyrrole,
pyridine and pyridazine. Heterocyclic molecules are compounds with at least
two different atoms in the ring. This can change the molecular polarizability
and electronic structure, and it often adds a significant dipole moment.

Furan, which is the smallest heterocyclic aromatic, is a five-member ring
comprised of four carbon atoms and an oxygen. It has a dipole moment of 0.66
D. Figure 4 (a) shows the $Z_{\text{eff}}$ spectrum. It exhibits a broad set
of resonances below $0.15$ eV and a well resolved peak near $0.33$ eV. This
later peak is identified as due to the phenyl C-H stretch modes [38]. Using
Eq. 2, $\varepsilon_{b}=$ 52 $\pm$ 5 meV. From comparison to the downshifted
IR spectrum, the unresolved lower-energy resonances correspond to a large
number of low-energy fundamental modes, but were not analyzed in detail here.
Such a spectrum is common for molecules with small $\varepsilon_{b}$ values
where the low-energy vibrational mode resonances are prominent [39].

Figure 4:  $Z_{\text{eff}}$ measurements for (a) furan, (b) pyrrole, and (c)
pyridine. The green curves are the IR spectra obtained from NIST [29], and
vertical blue bars indicate the locations of the IR active fundamental modes,
with both downshifted by $\varepsilon_{b}$. The solid black curves are fits to
the C-H peaks. Vertical green dashed lines indicate dipole active modes not
used in the analysis.

Figure 4 (b) shows the annihilation spectrum of pyrrole. Pyrrole is a five-
member ring similar to furan but with the oxygen atom replaced by an N-H
group. Similar to furan, pyrrole has two $\pi$ bonds, but it has a larger
dipole moment of $1.77$ D. The most distinct peak is near $0.2$ eV which is
identified with the four phenyl C-H modes. The result is $\varepsilon_{b}$ =
165 $\pm$ 10 meV. Similar to benzene, pyrrole has significant annihilation
over a broad energy range where there are no fundamental modes. This leads to
a larger uncertainty in the fit, and hence a larger error bar for
$\varepsilon_{b}$. There is a small structure in the annihilation spectrum
near $0.26$ eV that is likely associated with the single N-H vibration. This
is consistent with the measured $\varepsilon_{b}$ but was not included in the
analysis. As shown previously for benzene [26], the higher-resolution cryobeam
will be necessary to resolve these features and obtain a more precise
measurement of $\varepsilon_{b}$.

Shown in Figure 4 (c) is the measured annihilation spectrum for pyridine.
Pyridine has a benzene-like structure with a C-H group replaced by a nitrogen.
Similar to benzene, it has three $\pi$ bonds. Due to the substitution, it has
a dipole moment of 2.19 D. The spectrum has a single prominent peak near
$0.19$ eV which is associated with the phenyl C-H modes. The result is
$\varepsilon_{b}=$ 186 $\pm$ 5 meV. Although the peak is well resolved,
similar to that in benzene, there is a broad, relatively flat, region of
enhanced annihilation at energies below the C-H peak extending to the low-
energy limit of the measurements. This is likely due to multimode VFRs.
Surprisingly, the IR spectrum does not show significant IR activity over much
of this region. Understanding these features is a topic of current research.

Figure 5:  $Z_{\text{eff}}$ measurements for pyridazine. The vertical blue
bars indicate the locations of the IR active fundamental modes, with both
downshifted by $\varepsilon_{b}$. The solid black curves are fits to the C-H
peaks. Hollow circles are data where systematic errors cannot be ruled out
(see text for details).

Pyridazine adds a second nitrogen to the pyridine structure and thus is a
diazine. In this case, the dipole moment almost doubles to $4.2$ D, and so it
is expected to have a significantly higher $\varepsilon_{b}$ value than
pyridine. The measured annihilation spectrum is shown in Fig. 5. The spectrum
shows an apparent peak near $0.04$ eV, which is at the low-energy limit of our
measurements. In order to resolve the peak position, the data must be extended
down to near $E_{\parallel}\approx 0$. Due to non-ideal effects, including
approaching the beam cutoff and scattering effects at low energies, both the
magnitude of the peak and position are not assured to be reliable. Neglecting
this consideration, and ascribing the peak to the 4 phenyl CH mode yields
$\varepsilon_{b}$ = 330 meV. Assuming the mode identification is correct, we
can use the fit to the high-energy side of the peak to state that
$\varepsilon_{b}\geq$ 330 meV (i.e., a lower bound on $\varepsilon_{b}$).
Above the main peak, there is a broad range of enhanced annihilation extending
to $\sim 0.25$ eV. This feature is similar to that seen in benzene and several
deuterated benzenes [26]. It is also prominent in many other aromatics [e.g.,
phenylacetylene (Fig. 2 (e))], and is an important topic for future study.

###  III.3 Non-aromatic cyclic molecules

All of the molecules discussed above were aromatics. This raises the question
as to whether, or how, aromaticity effects $\varepsilon_{b}$. In a previous
study, we measured $\varepsilon_{b}$ for several 5- and 6-carbon alkane rings
at different levels of saturation [12], all of which had zero or very small
dipole moments ($\mu<0.5$). Those molecules exhibited modest increases of
$\varepsilon_{b}$ with increasing $N_{\pi}$, but they each had only one or two
$\pi$ bonds. In order to extend the comparison to non-aromatic molecules with
larger $N_{\pi}$, we measured the spectra for cycloheptatriene (C7H8, 3 $\pi$
bonds), and cyclooctatetraene (C8H8, 4 $\pi$ bonds). The results and
comparison to the analogous saturated 7 and 8 carbon alkanes are included in
Table 1.

Figure 6:  $Z_{\text{eff}}$ measurements for the non-aromatic and ring
molecules (a) cycloheptatriene and (b) cyclooctatetraene. The green curves are
the IR spectra obtained from NIST [29], and vertical blue bars indicate the
locations of the IR active fundamental modes, with both downshifted by
$\varepsilon_{b}$. The solid black curves are fits to the C-H peaks.

Spectra for cycloheptatriene (CHT) and cyclooctatetraene (COT) are shown in
Figs. 6(a) and (b). In the case of CHT, a VFR peak is observed at $\simeq$
0.17 eV, dominated by the C-H stretch vibrational modes. Here, the peak is
broadened because there are two types of C-H modes with different
characteristic energies. Using all 8 modes, we obtain $\varepsilon_{b}=$ 190
$\pm 8$ meV. The curve fits reasonably well, but there appears to be extra
spectral weight that is not accounted for on both sides of the peak. Further,
the model assumed equal amplitude for all 8 modes. If that is incorrect, it
could shift the measured $\varepsilon_{b}$ value by several milli electron
volts leading to a larger uncertainty for this molecule.

For COT, there are 6 distinct C-H modes (2 degenerate) with 3 non-dipole
active. Using only the dipole-active modes, $\varepsilon_{b}$ = 225 $\pm$ 5
meV. There also appears to be some broadening on the low energy side of the
peak, but in contrast to CHT, the peak is well fit. Thus, this broadening does
not appear to significantly affect $\varepsilon_{b}$. For both molecules (more
so COT), there is a broad region of enhanced annihilation above the CH
resonance. This is very similar to what was seen in the aromatics discussed
above. While not presently understood, it appears that this extra structure is
most prominent in molecules with appreciable symmetry.

With regard to the $\varepsilon_{b}$ values, although COT has no dipole
moment, the presence of the extra $\pi$ bond and more symmetric structure of
the molecule leads to a larger binding energy (i.e., a difference of $>$ 35
meV) compared with CHT. We can also compare the binding energies of these
molecules with the saturated rings cycloheptane and cyclooctane Ref. [12]. For
CHT as compared to cycloheptane, adding 3 $\pi$ bonds increases
$\varepsilon_{b}$ by $\sim 80$ meV. For COT compared to cyclooctane, a larger
increase of $\sim$ 100 meV is observed by adding the 4 $\pi$ bonds. This trend
extends the results observed for the smaller rings [12] to larger numbers of
$\pi$ bonds and shows the strong impact that $N_{\pi}$ has on
$\varepsilon_{b}$.

##  IV THEORY OF POSITRON BINDING AND COMPARISONS WITH EXPERIMENTAL
MEASUREMENTS

We use the _ab initio_ many-body theory [22] to calculate positron binding
energies for the molecules considered, assuming fixed nuclei 333Note that
vibrational and geometry relaxation effects are known to typically provide a
few percent correction to fixed-nuclei calculations of binding energies and
wave function densities [56, 17, 57, 58, 59, 2].. The approach is implemented
in our EXCITON+ code [22], which is heavily adapted from the original EXCITON
all-electron molecular many-body code [41] to include positrons. We solve the
Dyson equation [42, 43]

$\left(H^{(0)}+\hat{\Sigma}_{\varepsilon}\right)\psi_{\varepsilon}(\bf{r})=\varepsilon\psi_{\varepsilon}(\bf{r}),$
(3)

where $\psi_{\varepsilon}(\bf{r})$ is the quasiparticle positron wave function
with energy $\varepsilon$, $H^{(0)}$ is the Hamiltonian of the positron in the
static (Hartree-Fock) field of the ground-state molecule and
$\hat{\Sigma}_{\varepsilon}$ is the energy-dependent correlation potential
(self energy of the positron in the field of the molecule) that accounts for
electron-positron correlations. The energy dependence of $\Sigma$ demands a
self-consistent solution of Eqn. (3) for the bound state with
$\varepsilon_{b}=|\varepsilon|$ for negative eigenvalues $\varepsilon$. Our
approach includes three main diagrammatic contributions to the positron-
molecule self energy (see Fig. 1 of [22]). The first is the $GW$ contribution,
$\Sigma^{GW}$, which accounts for polarization of the molecular electron cloud
by the positron, screening of the electron-positron Coulomb interaction (the
random phase approximation ring series), and electron-hole attraction (time-
dependent-Hartree-Fock or Bethe-Salpeter-Equation approximation, depending on
whether bare or screened electron-hole Coulomb interactions are used within
the ring series). The second contribution describes the nonperturbative
process of virtual-positronium formation (where a molecular electron
temporarily tunnels to the positron, leading to an attractive interaction),
via calculation of the infinite ladder series of electron-positron
interactions $\Sigma^{\Gamma}$. The third contribution is an analogous diagram
containing the ladder series for positron-hole repulsion, $\Sigma^{\Lambda}$.
We calculate the total self energy via their sum as:
$\Sigma=\Sigma^{GW}+\Sigma^{\Gamma}+\Sigma^{\Lambda}$.

The positron and electron wave functions are expanded in distinct Gaussian
basis sets, centred at each atom in the molecule and at five to ten additional
‘ghost’ centres approximately 1$\mathring{\rm{A}}$ from the molecule. This
combination enables a good description of the positron wave function at the
nuclei and in the regions away from but close to the molecule where virtual-
positronium formation occurs (see [22] for more details). For both the
electron and positron bases, we use diffuse-function-augmented correlation-
consistent polarized aug-cc-pVXZ (X=T, Q) Dunning basis sets [44]. An
additional large even-tempered basis set is included for the positron at the
region of highest positron density (for polar molecules), or at the center of
the molecule (for non polars), to account for long-range correlations. The
even-tempered bases have exponents $\zeta_{k}=\zeta_{0}\beta^{k-1}$,
$k=1,2,\ldots$ and take the form $10s\,9p\,8d\,7\\!f\,3g$ with $\beta=2.2$,
$\zeta_{0}=10^{-3}$ for all molecules except COT, benzene and PA, which have
$10s\,9p\,8d\,7\\!f\,6g$ with $\beta=2.0$ and $\zeta_{0}=10^{-3}$ (benzene and
PA) or $\zeta_{0}=10^{-4}$ (COT). Solution of the Dyson equation for the
molecules considered required diagonalization of Casida matrices of size
0.5M$\times$0.5M, requiring $\sim$15 TB RAM. They were performed on the UK
Tier-2 HPC cluster Kelvin2, and the UK National Supercomputer ARCHER2.

The calculated binding energies are presented in Table 2 at three successively
more sophisticated levels of many-body theory which differ in the
approximation used to calculate the ladder series for the virtual positronium
formation and positron-hole repulsion self-energy contributions: the first
(second) uses bare (screened) Coulomb interactions in the ladder diagrams and
Hartree-Fock molecular orbital energies, and the third, our most sophisticated
approach, uses screened Coulomb interactions and molecular orbital energies
calculated at $GW$ level [41]. Additionally, results are shown for a model
approach [22], which approximates the computationally-expensive virtual-
positronium formation self energy, $\Sigma^{\Gamma}$, by scaling the bare
polarization self energy, $\Sigma^{(2)}$, by a factor $g$ so that
$\Sigma\approx g\Sigma^{(2)}+\Sigma^{\Lambda}$. In the molecules studied to
date [22, 21] and here, setting $g=1.4$ and $g=1.5$ provides good lower and
upper bounds for the positron binding energy, respectively. Table 2 also
contains calculated values of the dipole moment $\mu$, dipole polarizability
$\alpha$ and ionization energy $I$ which are in good agreement with the
reference values in Table 1. The ionization energies are calculated at the HF
and $GW$ levels. We typically find that it is only the energy of the highest
occupied molecular orbital (HOMO) that is larger, and that the MOs below the
HOMO have smaller energies at the $GW$ level compared to HF; as such the
latter are more easily perturbed by the positron leading to an increase in the
strength of the positron-molecule correlation potential, and ultimately
increased binding energies compared to the calculations that use the HF
energies in the diagrams.

Table 2:  Calculated MBT positron binding energies $\varepsilon_{b}$ (meV) for ringed molecules compared with experiment. Also shown are dipole moments $\mu$ (calculated at HF), dipole polarizabilities $\alpha$ (calculated at the $GW$@BSE level of MBT), ionization energies $I$ (calculated at HF and the $GW$@RPA level of MBT) and annihilation contact densities (calculated at the $GW+\Gamma+\Lambda$ level of MBT with enhancement factors [45]).  |  | Molecular properties | Positron binding energy [meV] |
---|---|---|---|---
Molecule | Formula |  $\mu$ [D] |  $\alpha$ [Å3] |  $I$ [eV]444The first (second) number is that calculated at HF ($GW$@RPA). | MBT555Many-body calculations at three levels of $\Sigma^{GW+\Gamma+\Lambda}$: (i) using bare Coulomb interactions within the ladders and HF energies; (ii) using dressed Coulomb interactions within the ladders and HF energies; and (iii) using dressed Coulomb interactions and $GW$@RPA energies. The latter (highlighted in bold) is the most sophisticated calculation. | Exp. | MBT model666 Using a scaled self energy to account for virtual positronium formation [22], $\Sigma\approx g\Sigma^{(2)}+\Sigma^{\Lambda}$, with $g=1.4$ and $g=1.5$. |  $\delta$ [a.u.]777Figures in square brackets indicate powers of 10.
Benzonitrile |  C6H5CN | 4.9 | 12.1 | 9.86, 10.07 | 326, 303, –888Our screened ladder with $GW$ energies calculation for benzonitrile is forthcoming; as such, in the text we compare theory and experiment only for the other 10 molecules in the table. | 298$\pm$5 | 284, 343 | 1.60[$-$2]
Pyridazine |  C4H4N2 | 4.3 | 8.2 | 10.57, 10.87 | 334, 308, 333 | 330$\pm$5 | 327, 398 | 1.89[$-$2]
Benzaldehyde |  C6H5CHO | 3.5 | 11.9 | 9.64, 9.90 | 210, 192, 213 | 220$\pm$10 | 198, 256 | 1.46[$-2$]
Pyridine |  C5H5N | 2.3 | 8.9 | 9.52, 9.91 | 184, 155, 181 | 186$\pm$5 | 149, 211 | 1.58[$-2$]
Pyrrole |  C4H4NH | 1.9 | 7.6 | 8.22, 8.64 | 157, 127, 148 | 165$\pm$10 | 94, 148 | 1.55[$-$2]
Aniline |  C6H5NH2 | 1.4 | 11.0 | 8.07, 8.26 | 220, 180, 209 | 233$\pm$5 | 172, 251 | 1.92[$-$2]
Phenylacetylene |  C6H5CCH | 0.8 | 13.3 | 8.86, 9.15 | 231, 187, 220 | 230$\pm$5 | 183, 266 | 1.87[$-2$]
Furan |  C4H4O | 0.7 | 6.7 | 8.85, 9.35 | 45, 32, 42 | 52$\pm$5 | 20, 49 | 7.32[$-$3]
Toluene |  C6H5CH3 | 0.4 | 11.4 | 8.87, 9.14 | 174, 135, 161 | 173$\pm$5 | 128, 203 | 1.87[$-2$]
Benzene |  C6H6 | 0.0 | 9.8 | 9.21, 9.57 | 158, 122 148 | 132$\pm$3 |  | 1.61[$-2$]
Cyclooctatetraene |  C8H8999In the ‘tub’ ($D_{2\rm d}$) geometry. | 0.0 | 13.0 | 8.37, 8.66 | 202, 158, 190 | 225$\pm$5 | 151, 238 | 2.03[$-2$]

Comparison of the predicted and measured binding energies is shown in Fig. 7.
Agreement is good to excellent. For ten of the molecules for which
$\varepsilon_{b}$ is given in Table 2, the rms difference in binding energy
between theory (using values in bold) and experiment is 13 meV. This
corresponds to an rms fractional deviation in the experimental and calculated
binding energies, $[\varepsilon_{b}$(calc.)
$-\varepsilon_{b}$(exp.)]/$\varepsilon_{b}$(exp.) of 9%. One noticeable trend
is that, of the comparisons presented here, the measured binding energies of
six of the ten molecules are larger than the theoretical prediction, three are
within the experimental error bars, and only one measured value (benzene) is
smaller than that predicted 11footnotemark: 1.

Finally, having the positron bound-state wave function enables calculation of
the electron-positron annihilation contact density $\delta$ as

$\delta=\sum_{n=1}^{N_{\rm
e}}\gamma_{n}\int\left|\phi_{n}(\mathbf{r})\right|^{2}\left|\psi_{\varepsilon}(\mathbf{r})\right|^{2}d\mathbf{r},$
(4)

where $\phi_{n}(\mathbf{r})$ are the occupied electronic molecular orbitals,
the sum is over the $N_{\rm e}$ occupied electron orbitals and $\gamma_{n}$
are enhancement factors which account for short-range electron-positron
attraction, and depend on the $GW$ ionization energy $\varepsilon_{n}$ of each
molecular orbital as follows [45]:
$\gamma_{n}=1+\sqrt{1.31/\left|\varepsilon_{n}\right|}+(0.834/\left|\varepsilon_{n}\right|)^{2.15}$.

Figure 7:  Comparison of the calculated binding energies versus the measured
binding energies. Green diamonds: benzene and benzene substitutions; blue
diamonds: heterocyclic aromatic molecules; black square: COT. Error bars on
the calculated values are the maximum difference between results from the
three many-body calculations in Table 2.

##  V FURTHER DISCUSSION

####  V.0.1 Localization of the bound-state positron wave function

Early ab initio and model-potential calculations predicted $\varepsilon_{b}$
values and enabled the investigation of the shape of the positron wave
function [46, 47, 19, 48]. When compared to the experimental measurements for
$\varepsilon_{b}$, these calculations yielded considerable insight into roles
of $\alpha$, $\mu$, and the molecular geometry for specific molecular series
including saturated chain and ring alkanes [19, 12] and simple halogen
substitutions [13, 49]. However, these theories were unable to adequately
treat electron-positron correlations, including the effects of virtual
positronium formation, and thus they were unable to achieve quantitative
agreement with the experimental measurements.

While the model-potential calculations have been effective in achieving
agreement with the experimental measurements, they required fit parameters;
and thus they have not been very successful in making predictions beyond the
molecular series for which the parameters were tuned [19, 13, 17, 18, 16].
Further, most of the previous molecules studied that included $\pi$ bonds also
had large dipole moments. This strongly localized the bound positron and
dominated the binding energy (e.g., acetonitrile), and hence it tended to mask
the influence of the $\pi$ bonds. Generally, the previous calculations either
showed a more-or-less uniform positron cloud around the molecular periphery
(e.g. alkanes), or a strongly localized cloud near the negative end of the
molecule (e.g. acetonitrile), although other shape dependences have also been
noted [13, 50].

In contrast, the MBT calculations are in quantitative agreement with
experimental measurements, and they provide further insight into the effects
that contribute to positron binding [22, 21]. Beyond the influence of $\alpha$
and $\mu$, the calculations investigated the possible dependence on other
parameters such as $I$ and $N_{\pi}$. The effect of $I$ on binding is
predicted to be weak, since binding depends on positron interactions with many
orbitals, not just the highest occupied orbital that sets $I$. Successively
deeper lying molecular orbitals generally contribute less to the positron-
molecule correlation potential, but the decrease is not monotonic due to the
enhanced strength of $\pi$ bonds. The $\pi$-bond orbital electron density is
localized away from the ionic cores, allowing these electrons interact
strongly with the positron [11, 22]. Thus $N_{\pi}$ can be an important
parameter in determining $\varepsilon_{b}$, especially if the $\pi$ bonds are
HOMOs (as in COT).

Figure 8:  Correlated positron Dyson wave functions (positive and real as
displayed here) for several heterocyclic molecules at the $GW+\Gamma+\Lambda$
level of many-body theory, shown at 80$\%$ of the wave function maximum unless
otherwise stated. (a) benzonitrile; (b) pyridazine; (c) benzaldehyde; (d)
pyridine; (e) pyrrole; (f) aniline (left, 80$\%$; right, 60$\%$); (g)
phenylacetylene (two orientations, both 80$\%$); (h) furan; (i) toluene; (j)
benzene; (k) cyclooctatetraene ($D_{\text{2d}}$ geometry).

The bound-state positron wave functions for the molecules studied here are
shown in Fig. 8. For molecules where the aromatic $\pi$ bonds dominate, the
positron wave function is localized in planes above and below the plane of the
molecule where the $\pi$-bond electron density is largest. Examples of this
effect are seen in Figs. 8 (j) and (i), benzene and toluene. In contrast,
similar to what was seen with non-aromatic molecules [22], the calculations
show that the introduction of a large permanent dipole moment results in
localization of the wave function very near the dipole, as seen in Figs. 8
(c), (b) and (d) for benzaldehyde, pyridazine and pyridine.

However, the effects of attraction of the positron to the $\pi$-bond electrons
and to a permanent dipole compete when both are present on the same molecule.
As examples, aromatic $\pi$-bond localization is seen to be (weakly) perturbed
by the dipoles in Figs. 8 (f) and (e), aniline and pyrrole. A stronger effect
is seen in (h), furan, where the 80% positron contour extends from the region
of the $\pi$-bond electrons all the way to the dipole. The C$\equiv$C triple-
bond $\pi$ orbitals in (g) phenylacetylene result in concentration of the
positron around this bond, which extends continuously into the region of the
aromatic $\pi$-bond electrons. Lower-density contours would show larger
regions of both the $\pi$-bond and triple bond localization.

Changes in molecular symmetry can also affect $\pi$-bond localization. An
example of this is the distorted positron wave function contour resulting from
the broken planar symmetry of cyclooctatetraene. Figure 8 (f) for aniline
shows that the strong, slightly-out-of-plane dipole results in restricting the
80% positron wave function contour to the region above the plane, whereas the
65% contour shows some amplitude below the plane as well. For all molecules,
the long-range $\alpha$ potential results in a lower-amplitude, more uniform
cloud of positron density further from, and entirely surrounding the molecule.

####  V.0.2 Relationship of $\varepsilon_{b}$ to molecular parameters

It has been natural to try to relate binding energies to global molecular
parameters such as $\alpha$, $\mu$, $I$ and $N_{\pi}$, and much has been
written on this topic. It has also been shown by comparison of isomers that
molecular geometry can play an important role in positron binding,
irrespective of other global parameters [51, 48]. Shown in Fig. 9 are
$\varepsilon_{b}$ values for the molecules studied here and selected previous
data as a function of $\alpha$, $\mu$, $I$ and $N_{\pi}$, extending the ranges
of parameters studied beyond those done previously.

There is naturally a large scatter in the data in the one-parameter-at-a-time
representations in Fig. 9, since wide variations of the other parameters (not
shown in a particular plot) produce changes in $\varepsilon_{b}$ as well.
Despite the large scatter in the data, the largest changes of
$\varepsilon_{b}$ are associated with changes in $\mu$ and $N_{\pi}$ [cf.
panels (b) and (d)]. This highlights the role of $\mu$ and $N_{\pi}$ in
determining $\varepsilon_{b}$ for the molecules studied here, similar to that
seen in the spatial distribution of the positron wave function.

From the data in Fig. 9 (c), $\varepsilon_{b}$ appears to be independent of
$I$, to within the scatter in the values. This is made clearer when one
considers separately $I$ for aromatic and non-aromatic rings. They span
narrower bands of $I$ at differing mean $I$ (i.e., separated by $\sim$ 0.8 eV)
but exhibit similar broad ranges of $\varepsilon_{b}$. This tends to confirm
that $\varepsilon_{b}$ depends only weakly, if at all, on $I$ for the
molecules studied here.

Regarding the dependence of $\varepsilon_{b}$ on $N_{\pi}$, the molecules
shown in Fig. 9 include different types of $\pi$ bonds: aromatic and non-
aromatic $\pi$ bonds in rings, $\pi$ orbitals in C=O double bonds, and $\pi$
bonds in the C$\equiv$C triple bond in phenylacetylene. When all are included
[cf. Fig. 9 (d)], they indicate that $\varepsilon_{b}$ increases with
increasing $N_{\pi}$. While these limited data constrain detailed analysis, it
also appears that the change in $\varepsilon_{b}$ with $N_{\pi}$ is similar
for the aromatic and non-aromatic $\pi$ bonds. Thus, the present data are
consistent with all $\pi$ bonds contributing approximately equally to
$\varepsilon_{b}$, but more precise conclusions will require further study.

Figure 9:  Measurements of $\varepsilon_{\text{{b}}}$ for aromatic and ring
molecules vs. (a) molecular polarizability $\alpha$, (b) permanent dipole
moment $\mu$, (c) ionization potential $I$, and (d) number of $\pi$ bonds Nπ.
Diamonds: six-member carbon aromatic rings: green, benzene and benzene
substitutions; blue, heterocyclic molecules. Squares: five-member non-
saturated and non-aromatic carbon rings. Red triangles, alkanes. Black
symbols, cyclic molecules. Solid symbols: one or more $\pi$ bonds and open
symbols, no $\pi$ bonds. Data for alkanes from Ref. [12] and other non-
saturated rings from Ref. [15].

####  V.0.3 Multimode phenomena in ring molecules

A theory of coupling to dipole-allowed transitions is successful in explaining
VFR due to fundamental modes in small molecules [24], and there were two
observations of similar quadrupole coupling [25]. Going further, it is natural
to try to associate annihilation VFR not associated with dipole- and
quadrupole-allowed fundamental vibrations with other infrared activity in the
molecule. However, a detailed explanation is lacking of many of the observed
regions of enhanced annihilation that do not appear to correspond to
fundamental vibrational modes. The working assumption is that these resonances
are due to as yet unidentified multimode resonances (e.g., combinations and
overtones) but this has yet to be confirmed. The special case of the Fermi
resonance (FR) in benzaldehyde offers an opportunity to study such multimode
resonances, for a case in which the individual modes have been identified.
Better positron beam resolution will be needed to study this FR in further
detail.

There are also regions in the annihilation spectra where broad VFR activity is
observed that do not appear to correspond to appreciable IR activity. Examples
presented here are the spectra of benzene, pyridine and pyrrole. This raises
the question as to whether there is some other, presently unrecognized,
annihilation VFR mechanism (i.e., not visible in IR spectral measurements),
but this is beyond the scope of the present work.

##  VI SUMMARY AND CONCLUDING REMARKS

There has been much progress in understanding positron binding to molecules,
see e.g., [24, 3, 2, 16, 17, 18, 19, 13, 20, 22, 21], but many questions
remain. This paper presents a study of resonant positron annihilation and
positron binding in selected aromatic and ring compounds. It enabled an
investigation of the effects of permanent dipole moment and $\pi$ orbitals,
and the competition of the two, in determining positron annihilation and
binding. Experimental measurements for $\varepsilon_{\textit{b}}$ were
compared with the predictions of a recent _ab initio_ many body theory. Good-
to-excellent agreement was found (i.e., 9% rms fractional deviation between
measurements and predictions for 10 molecules). This is was made more
significant by the fact that the compounds span a wide range of $N_{\pi}$ (2 -
5) and $\mu$ (0.3 - 4.5 D) and a factor of 1.7 in $\alpha$.

An important result of the theory is predictions for the spatial distribution
of the bound positron wave function. This can be used to distinguish the
different character of the bound states due to specific molecular parameters.
As established previously, the polarizability results in a more-or-less
uniform bound state density surrounding the molecule, while a permanent dipole
moment localizes the bound state density near the dipole. Aromatic $\pi$ bonds
produce maxima in positron density above and below the planes of the ring
molecules. A related $\pi$-bond localization effect is seen in the non-valence
anion C6F${}_{6}^{-}$ in which the excess (weakly bound) electron has a very
similar charge distribution to that of a positron attached to benzene, C6H6
[52, 53].

This qualitative picture of positron wave function localization provides
insight into positron-molecule bound states complementary to the quantitative
predictions of the theory. An interesting question for future study is whether
one could distinguish experimentally the $\pi$ and $\mu$ components of the
bound state wave function using gamma-ray spectroscopy [i.e., Doppler-
broadening or angular correlation of annihilation radiation (ACAR)] [54].

Understanding the observed enhancements in $Z_{\text{eff}}$ beyond the
predictions of simple theory [24] is also a major gap in our understanding.
One example of the chemical specificity discussed here include the increase in
$Z_{\text{eff}}$ by a factor of 30 when a methyl group is added to benzene.
Another example is the large amount of likely multimode spectral weight at low
energies in pyrrole, relative to that due to the C-H stretch mode. That the
observed annihilation VFR can be smaller, or orders of magnitude larger, than
the predictions of simple theory has been attributed to vibrational coupling
to other modes (i.e., IVR) [2], however attempts to pursue this have had only
modest success [55]. Similarly, there is now appreciable evidence of multimode
annihilation VFR (e.g., in benzene, benzaldehyde and furan), but little
understanding of when and how this occurs. The observation of the annihilation
Fermi resonance in benzaldehyde may provide further insight into this
phenomenon.

###### Acknowledgements.

We wish to acknowledge helpful conversations with G. F. Gribakin and K. D.
Jordan. The work at UCSD was supported by the U. S. NSF, grant PHY-2306404 and
the U. C. San Diego Foundation. The QUB theoretical work was funded by the
European Research Council, grant 804383 ‘ANTI-ATOM’ and used the Northern-
Ireland HPC Service and the ARCHER2 UK National Supercomputing Service.

## Author Contributions

J. P. Cassidy, S. K. Gregg and J. Hofierka contributed equally to the
theoretical work.

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