# Dissociative recombination, and vibrational excitation of CO+: model
calculations and comparison with experiment

J. Zs Mezei1,2,3 R. D. Backodissa-Kiminou1 D. E. Tudorache1,4 V. Morel5 K.
Chakrabarti1,6 O. Motapon7 O. Dulieu2 J. Robert2 W.-Ü. L. Tchang-Brillet8 A.
Bultel5 X. Urbain9 J. Tennyson10 K. Hassouni11 I. F. Schneider1,2
<EMAIL_ADDRESS>1LOMC CNRS$-$Université du Havre$-$Normandie
Université, 76058 Le Havre, France  2LAC, CNRS$-$Université Paris-Sud$-$ENS
Cachan$-$Université Paris-Saclay, 91405 Orsay, France  3HUN-REN Institute for
Nuclear Research (ATOMKI), H-4001 Debrecen, Hungary  4Lab. EM2C, CNRS-École
Centrale Paris, 92295 Châtenay-Malabry, France  5CORIA CNRS$-$Université de
Rouen$-$Université Normandie, F-76801 Saint-Etienne du Rouvray, France  6Dept.
of Mathematics, Scottish Church College, Calcutta 700 006, India  7LPF, UFD
Math., Info. Appliq. Phys. Fondamentale, University of Douala, P. O. Box
24157, Douala, Cameroon  8LERMA, Observatoire de Paris, CNRS-Sorbonne
Universités-UPMC Univ Paris 06, F-92195 Meudon, France  9Institute of
Condensed Matter and Nanosciences, PAMO - Louvain-la-Neuve, 1348, Belgium
10Dept. of Physics and Astronomy, University College London, WC1E 6BT London,
UK  11LSPM, CNRS$-$Université Paris 13$-$USPC, 93430 Villetaneuse, France

###### Abstract

The latest molecular data - potential energy curves and Rydberg$/$valence
interactions - characterizing the super-excited electronic states of CO are
reviewed, in order to provide inputs for the study of their fragmentation
dynamics. Starting from this input, the main paths and mechanisms for CO+
dissociative recombination are analyzed; its cross sections are computed using
a method based on Multichannel Quantum Defect Theory. Convoluted cross
sections, giving both isotropic and anisotropic Maxwellian rate-coefficients,
are compared with merged-beam and storage-ring experimental results. The
calculated cross sections underestimate the measured ones by a factor of $2$,
but display a very similar resonant shape. These facts confirm the quality of
our approach for the dynamics, and call for more accurate and more extensive
molecular structure calculations.

###### pacs:

33.80. -b, 42.50. Hz

††preprint: APS/123-QED

##  I Introduction

Carbon monoxide (CO) is the most abundant molecule detected in the
interstellar medium after molecular hydrogen, and being heteronuclear is much
easier to be detected in emission. Its observation at long wavelengths, as in
radio astronomy, is the source of the much of the information obtained on
different interstellar environments, from molecular clouds in which protostars
form and irradiate the surrounding residual material, to the circumstellar
disks that surround stars at the end of their lives and to the cometary gases.
Its spectrum has been intensively studied in astrophysical context, since its
large binding energy implies that many electronically excited states possess
discrete line spectra, see rosen98 ; chakrabarti06 ; chakrabarti07 ;
eidelsberg2012  and references therein. In all cases, under the influence of
ultraviolet radiation produced during the stellar cycle, CO is the subject of
a rich photochemistry, implying the formation among other species, of its
cation, CO+.

CO+ was the first molecular ion discovered outside the Earth fowler1909 , has
been identified as a major constituent of dense interstellar molecular clouds
roueff2010 . CO+ also occurs in flames krupente65 ; seaton66 ; mitchell85 ,
and is expected to be very abundant in the plasmas formed by the hypersonic
entry of spacecrafts and comets in the Martian atmospheres bultel2006 ;
park1994 . Cold plasmas containing a large number of molecules and molecular
ions constitute a subject of rising scientific interest, involving more and
more technological applications. CO+ is of the most important of such
molecular ionic species, since it occurs in practically all air assisted
processes. Its dissociative recombination (DR) can be regarded as the main
photochemical carbon loss process from the Martian atmosphere fox1999  and it
is considered the major source of excited C$(^{1}D)$ atoms.

The kinetic description of all the above mentioned environments requires a
good knowledge of the rate coefficients of the dominant reactions, including
those between electrons and molecular ions. As for the CO+ ions, their
abundance is strongly affected by the DR,

$\mbox{CO}^{+}(v_{i}^{+})+e^{-}\rightarrow\mbox{C}+\mbox{O},$ (1)

but also by other related competitive processes, such as the inelastic
($v_{f}^{+}>v_{i}^{+}$) and the superelastic ($v_{f}^{+}<v_{i}^{+}$)
collisions with electrons:

$\mbox{CO}^{+}(v_{i}^{+})+e^{-}\rightarrow\mbox{CO}^{+}(v_{f}^{+})+e^{-}.$ (2)

Here $v_{i}^{+}$ and $v_{f}^{+}$ stand for the initial and final vibrational
quantum number of the target ion, and rotational structure was neglected.

The pioneering theoretical study of the DR rate coefficient made by guberman-
dr07  shows that the dominant routes for the DR process are curves of the
${}^{3}\Pi$, ${}^{1}\Pi$ and ${}^{1}\Sigma^{+}$ symmetries. Guberman
calculated the rate constants at $300$ K to be $4.2\times 10^{-7}$
cm${}^{3}/$s for 12C16O+ and $2.9\times 10^{-7}$ cm${}^{3}/$s for 13C16O+
respectively. The most recent storage ring experiments of rosen98  gave a rate
coefficient of $2.75\times 10^{-7}$ cm${}^{3}/$s for 13C16O+ which is in good
agreement with the calculations of Guberman. The earlier afterglow experiments
geoghegan91  for 12C16O+ gave a rate coefficient of $1.6\times 10^{-7}$
cm${}^{3}/$s, which does not agree with the theoretical calculations for
$v^{+}_{i}=0$ and may indicate that a proportion of the CO+ in that experiment
was vibrationally excited.

Recently, a series of ab initio calculations of the CO states relevant to DR
(and of the couplings between them) have been performed by Chakrabarti and
Tennyson chakrabarti06 ; chakrabarti07  using the $R$-matrix method. The data
obtained makes a new Multichannel Quantum Defect Theory (MQDT) investigation
of the process over the whole energy range explored by the experiments rosen98
; mitchell85  possible.

The main goal of the present work is to evaluate the DR cross sections and the
thermal rate using the previously mentioned ab initio data of chakrabarti06 ;
chakrabarti07 . The paper is structured in the following way: Section II
outlines the main ideas and steps of our MQDT approach. Section III presents
the molecular data used in the calculation. The main results are given in
section IV and the paper ends by conclusions.

##  II The MQDT-type approach to DR

The MQDT approach seaton83 ; gj85 ; jungen96 ; giusti80  has been shown to be
a powerful method for the evaluation of the cross sections of the DR process.
Although it was applied with a great success to several diatomic systems like
H${}_{2}^{+}$ and its isotopologues giusti83 ; ifs-a09 ; takagi93 ; tanabe95 ;
ifs-a18 ; amitay99 , O${}_{2}^{+}$ ggs91 ; guberman-dr99 , NO+ sn90 ; ifs-a22
; ifs-a36  and triatomics like H${}_{3}^{+}$ ifs-a26 ; kokoou01 ; kokoou03 ,
its application to vibrational transitions, mainly to superelastic collisions
for NO+ is relatively recent Ngassam1 ; motapon06b .

We aim to describe the sensitivity of the reactive electron-cation collisions
on the vibrational levels involved. At least to a first approximation,
rotational effects are known to be negligible forCO+ ifs-a22 . The reasons for
this rely on the weak Rydberg-valence interaction responsible for the indirect
process: since the rotational structures and interactions play their role
especially within the indirect mechanism - due to its resonant character - the
weakness of this latter process implies that the rotational effects - if of
any relevance - can be roughly restricted to the existence of the centrifugal
barrier, due to the rotational excitation. But even in this latter context,
rotation does not matter very much, since the target ion and the neutral are
equally excited and, consequently, the Franck-Condon overlaps (direct process)
do not change significantly with respect to those occurring for the case of
rotationally ground states. However, this does not mean that electron-impact
rotational excitation is negligible faure01 .

The theoretical summary given below is limited to an account of the
vibrational structure and coupling, illustrated mainly for DR. However, the
reader should keep in mind that the other competitive reactions – such as
superelastic collision (SEC) ($v^{+}_{i}>v^{+}_{f}$ in eq. (2)), elastic
collision (EC) ($v^{+}_{i}=v^{+}_{f}$) and inelastic collision (IC)
($v^{+}_{i}<v^{+}_{f}$) – also occur and can display quite similar features.

The DR can take place following two mechanisms:

  1. (i)

the direct process where the capture takes place into a dissociative state of
the neutral system (CO∗∗),

$\mbox{CO}^{+}(v_{i}^{+})+e^{-}\rightarrow\mbox{CO}^{**}\rightarrow\mbox{C}+\mbox{O},$
(3)

  2. (ii)

the indirect process where the capture occurs via a Rydberg state of the
molecule CO∗ which is predissociated by the CO∗∗ state,

$\mbox{CO}^{+}(v_{i}^{+})+e^{-}\rightarrow\mbox{CO}^{*}\rightarrow\mbox{CO}^{**}\rightarrow\mbox{C}+\mbox{O}.$
(4)

We note here, that we follow the standard nomenclature in this work, namely
CO∗∗ and CO∗ represent the doubly excited and singly excited states of CO
respectively. In both these processes autoionization is in competition with
the predissociation and leads, through the reaction (2), to SEC, EC or IC.

The MQDT treatment of DR involves ionization channels (describing the
electron-ion scattering) and dissociation channels (describing the atom-atom
scattering). Each ionization channel consists of a Rydberg series of excited
states, extrapolated above the continuum threshold \- a vibrational level
$v^{+}$ of the molecular ion. A channel is considered open if its
corresponding threshold is situated below the total energy of the system, and
closed in the opposite case. In the present work we have $N^{+}=N$, and as for
the dissociative channels, only open channels are used.

The present MQDT approach is based on a description of molecular states in
which only part of electronic Hamiltonian is diagonalized, within subspaces of
electronic states with similar nature. Moreover we use a quasidiabatic
representation of molecular states sidis  to cope with problems due to the
avoided crossings of the potential energy curves. The short-range electronic
interactions between states of different subspaces are then represented in
terms of an electronic coupling operator $\mathcal{V}$ given by

$\mathcal{V}_{d_{j},lv}^{\Lambda}(E^{\prime},E)=\langle\chi_{{d}_{j}}^{\Lambda}(R)|V_{{d}_{j},l}^{\Lambda}(R)|\chi_{v}^{\Lambda}(R)\rangle,$
(5)

which couples the ionization channels (labeled by $lv$) to the dissociative
channels (labeled by $d_{j}$). Starting from $\mathcal{V}$, one can build the
short-range reaction matrix $\mathcal{K}$, which is a solution of the
Lippmann-Schwinger integro-differential equation:

$\bm{{\cal K}}=\bm{{\cal V}}+\bm{{\cal V}}\frac{1}{E-\bm{H_{0}}}\bm{{\cal
K}}.$ (6)

$H_{0}$ being the zeroth order Hamiltonian associated to the molecular system,
i.e. the Hamiltonian operator excluding the interaction potential
$\mathcal{V}$. The short-range effects are valid in the region of small
electron-ion and nuclei-nuclei distances, namely in the ‘A-region’ ja77  where
the Born-Oppenheimer representation is appropriate for the description of the
colliding system. Here, the energy-dependence of the electronic couplings can
be neglected. In the case of weak coupling, a perturbative solution of
equation (6) can be obtained. This solution has been proved to be exact to
second order, in the case of energy-independent electronic couplings Ngassam2
. In the external zone, the ‘B-region’ ja77  represented by large electron-
core distances, the Born-Oppenheimer model is no longer valid for the
ionization channels and a close-coupling representation in terms of ‘molecular
ion $+$ electron’ is more appropriate. This corresponds to a frame
transformation defined by the projection coefficients:

${\cal
C}_{lv^{+},\Lambda\alpha}=\sum_{v}U_{lv,\alpha}^{\Lambda}\langle\chi_{v^{+}}|\cos(\pi\mu_{l}^{\Lambda}(R)+\eta_{\alpha}^{\Lambda})|\chi_{v}^{\Lambda}\rangle,$
(7) ${\cal
C}_{d_{j},\Lambda\alpha}=U_{d_{j}\alpha}^{\Lambda}\cos\eta_{\alpha}^{\Lambda},$
(8) ${\cal
S}_{lv^{+},\Lambda\alpha}=\sum_{v}U_{lv,\alpha}^{\Lambda}\langle\chi_{v^{+}}|\sin(\pi\mu_{l}^{\Lambda}(R)+\eta_{\alpha}^{\Lambda})|\chi_{v}^{\Lambda}\rangle,$
(9) ${\cal
S}_{d_{j},\Lambda\alpha}=U_{d_{j}\alpha}^{\Lambda}\sin\eta_{\alpha}^{\Lambda}.$
(10)

Here, $\chi_{v^{+}}$ is the vibrational wavefunction of the molecular ion, and
$\chi^{\Lambda}_{v}$ is a vibrational wavefunction adapted to the interaction
(A) region. The index $\alpha$ denotes the eigenchannels built through the
diagonalization of the reaction matrix $\mathcal{K}$ in equation (6) and
$U^{\Lambda}_{lv,\alpha}$ and $\eta_{\alpha}^{\Lambda}$ are related to the
corresponding eigenvectors and eigenvalues, while $\Lambda$ refers to the
electronic symmetry of the neutral species (${}^{1}\Sigma^{+}$, ${}^{1}\Pi$,
and ${}^{3}\Pi$ in the present study).

The projection coefficients shown in (7)-(10) include the two types of
couplings controlling the process: the electronic coupling, expressed by the
elements of the matrices $\mathcal{U}$ and $\eta$, and the non-adiabatic
coupling between the ionization channels, expressed by the matrix elements
involving the quantum defect $\mu_{l}^{\Lambda}$. This latter interaction is
favored by the variation of the quantum defect with internuclear distance $R$.
The matrices $\cal C$ and $\cal S$ with the elements given by (7) to (10) are
the building blocks of the ‘generalized’ scattering matrix $X$:

$\mbox{\boldmath$X$}=\frac{\mbox{\boldmath$\cal C$}+i\mbox{\boldmath$\cal
S$}}{\mbox{\boldmath$\cal C$}-i\mbox{\boldmath$\cal S$}},$ (11)

whereas the ‘proper’ scattering matrix, restricted to the open channels, is
given by seaton83 :

$\mbox{
$S$}=\mbox{\boldmath$X$}_{oo}-\mbox{\boldmath$X$}_{oc}\frac{1}{\mbox{\boldmath$X$}_{cc}-\exp({\rm-i2\pi}\mbox{\boldmath$\nu$})}\mbox{\boldmath$X$}_{co}.$
(12)

More precisely the physical S-matrix is obtained from the $2\times 2$ sub-
matrices of $X$ involving the open (o) and closed (c) channels, and from the
diagonal matrix $\nu$ containing the effective quantum numbers
${\nu}_{v^{+}}=[2(E_{v^{+}}-E)]^{-1/2}$ (in atomic units) associated with each
vibrational threshold $E_{v^{+}}$ of the ion situated above the current energy
$E$. For a molecular ion initially in vibrational level $v_{i}^{+}$ and
recombining with an electron of energy $\varepsilon$, the cross section for
capture into all the dissociative states $d_{j}$ of the same symmetry
${\Lambda}$ is given by:

$\sigma^{{\Lambda}}_{{\rm diss}\leftarrow
v^{+}_{i}}=\frac{\pi{\rho}^{\Lambda}}{4\varepsilon}\sum_{j}\sum_{l}|S^{{\Lambda}}_{d_{j}\leftarrow
lv^{+}_{i}}|^{2},$ (13)

where $\rho^{\Lambda}$ is the ratio between the multiplicities of the neutral
system and of the ion. One has to perform the MQDT calculation for each group
of dissociative states of symmetry $\Lambda$, and the sum over the resulting
cross sections is the total DR cross section:

$\sigma_{{\rm diss}\leftarrow v^{+}_{i}}=\sum_{\Lambda}\sigma^{\Lambda}_{{\rm
diss}\leftarrow v^{+}_{i}}$ (14)

In a similar way, the cross section for a vibrational transition of a
molecular ion from the initial level $v_{i}^{+}$ to the final level
$v_{f}^{+}$ is (excitation and/or deexcitation):

$\sigma^{{\Lambda}}_{{v_{f}^{+}}\leftarrow
v^{+}_{i}}=\frac{\pi}{4}\frac{{\rho}^{{\Lambda}}}{\varepsilon}\sum_{l,l^{\prime}}|S^{{\Lambda}}_{l^{\prime}v_{f}^{+}\leftarrow
lv^{+}_{i}}|^{2},$ (15)

while the total cross section for this vibrational transition reads as:

$\sigma_{{v_{f}^{+}}\leftarrow
v^{+}_{i}}=\sum_{\Lambda}\sigma^{\Lambda}_{{v_{f}^{+}}\leftarrow v^{+}_{i}}.$
(16)

##  III Molecular data

###  III.1 Ab initio R-matrix calculations

The earliest attempts to compute excited states of CO relevant for the
dissociative recombination and their autoionization widths goes back to 1996
tennyson1996a ; tennyson1996b  but were restricted to a single geometry.

Ten years later, two of us chakrabarti06 ; chakrabarti07  performed a much
more detailed R-matrix calculation for a series of fixed geometries. These
calculations used up to the $14$ lowest CO+ states in a close coupling
expansion where each of these states were represented using a valence complete
active space configuration interaction expansion. The calculations were
repeated for $10$ different internuclear distances in the range of
$1.5~{}a_{0}$ to $3.5~{}a_{0}$. Moreover chakrabarti07  computed positions and
widths for a number of dissociative states, which appear as resonances in the
R-matrix scattering calculations, as a function of geometry. In particular,
resonance states of electronic symmetry ${}^{1}\Sigma^{+}$, ${}^{1}\Pi$ and
${}^{3}\Pi$ that have been used in the present work as the other symmetries
are only very weakly coupled.

Although the R-matrix resonance calculations chakrabarti07  for the CO
molecule have not been subject to comparisons and calibrations against all
available spectroscopic data, some Rydberg states previously were calculated
tennyson1996a ; chakrabarti06 . It was found that these Rydberg states are
placed higher than the observed values which is consistent with the
variational nature of the R-matrix calculations. As a consequence, for low
$\ell$ values, the estimated quantum defects were uniformly lower by about
$0.07$ compared to the available experimental data, see for example Tables 4
and 5 in chakrabarti06 . The geometry dependent study on CO also showed that
the quantum defects depend weakly on geometry except when the adiabatic curves
are perturbed by an avoided crossing. This approach was successfully used by
ifs-a36  to give greatly improved results for DR of NO+.

###  III.2 Modeling of the dissociative curves

Figure 1: Electronic states relevant for dissociative recombination with
${}^{1}\Sigma^{+}$ total symmetry. The ion ground state is shown in black;
$\square,\vartriangle$ are the R-matrix data chakrabarti07 : (Chakrabarti and
Tennyson 2007) for the lowest two CO∗∗ states. The two continuous colored
curves crossing the ion are the dissociative states obtained as outlined in
section III.2. The curves indicated in the legends refer to the curves from
guberman-dr07 : Guberman (2007), lefebvre07 : Lefebvre Brion and Lewis (2007)
and Vazquez : Vazquez et al (2009)

The DR cross section is extremely sensitive to the position of the potential
energy curves (PECs) of the neutral dissociative states with respect to that
of the target ion. More specifically, a slight change of the crossing point of
the PEC of a neutral dissociative state with that of the ion ground state can
lead to a significant change in the predicted DR cross section. The magnitude
of this change is roughly proportional to the square of the Franck-Condon type
integral appearing in equation (5). This is an important issue here as we use
the approximate PECs of the dissociative states above the ion ground state
produced by R-matrix calculations chakrabarti06 ; chakrabarti07 . Indeed, for
example, the crossing point of the lowest relevant ${}^{1}\Sigma^{+}$ state
was found to be at higher energy and at larger internuclear distance than
those obtained previously using a de-perturbation procedure tchang92  or ab-
initio quantum chemistry computations guberman-dr07 . Moreover, recent
calculations Vazquez ; lefebvre07 ; lefebvre10  give PECs for the _lowest_
neutral states relevant for DR, that only partially agree with those of the
other calculations mentioned above.

All the available lower dissociative neutral states for the ${}^{1}\Sigma^{+}$
symmetry is shown in Figure 1, where we have presented the PECs relevant for
the present study. The zero of energy was chosen to be the lowest vibrational
level of the ground electronic state of the molecular ion. At the same time,
apart from the dissociative curves, the MQDT calculation also requires the
couplings between the ion ground state and the neutral dissociative states.
The R-matrix calculations of chakrabarti07  systematically provide
autoionization widths for a large number of states which contain information
about the required couplings. Such comprehensive data on couplings are not
available from any other source. Thus use their data in our dynamical study on
DR but correct the PECs when it is needed. In this way we undertake a semi-
empirical calculation.

The dissociative (valence) states produce numerous avoided crossings when
crossing the series of Rydberg states. By turning to the diabatic
representation these avoided crossings will become ”true” crossings essential
for the indirect process. A quasi-diabatic PEC may be constructed, by smoothly
connecting the resonant states obtained by scattering calculations to the
asymptotic adiabatic states. A more detailed inspection of the available
material, both diabatic and adiabatic, confirms the quality of the diabatic
states obtained by guberman-dr07 , justifying their use in the construction of
quasi-diabatic PECs by matching them in the relevant range of internuclear
distances. As a general rule, we apply a slight shift to the R-matrix PECs
only if it is necessary in order to match them with the diabatic curves of
guberman-dr07 . This is clearly the case for the first state of
${}^{1}\Sigma^{+}$ symmetry, see Fig 1. No such a behavior is visible for the
second dissociative PEC, but we apply the same shift in quantum defect as we
applied to connect the first state to its quasi-diabatic counterpart.
Correlation to the proper asymptotic limits follows both the wigner28  rules
and the adiabatic curves of Vazquez , where they are available. In our model,
we construct the lowest dissociative states as a function of the internuclear
distance on the basis of a physically reasonable compromise between the
available valence curves coming from all the above cited authors. As an
example, consider the lowest state of ${}^{1}\Sigma^{+}$ symmetry: At the
smaller internuclear distances, our quasi-diabatic PEC follows the R-matrix
points of chakrabarti07 . At larger internuclear distances we shift
progressively the R-matrix dissociative curve in order to match them with
those of guberman-dr07 . This curve is used until its crosses the PEC of the
molecular ion, and beyond, followed by a smooth connection to the lowest
_diabatic_ PEC of lefebvre07 , which agrees very well with the asymptotic
behavior of the lowest _adiabatic_ potential energy curves of Vazquez at large
internuclear distances. The dissociative curves constructed in this way will
agrees well with the PEC used by Guberman. A similar procedure is followed for
all the other excited states with different symmetries.

Figure 2: The dissociative curves and couplings relevant for CO+ dissociative
recombination for the symmetries indicated in each figure. Top panel: the
adjusted potential energy curves (cf section III), the ion ground state and
first 10 vibrational levels are represented in black. Bottom panel: couplings
between the valence dissociative states and the ionization continua.

As for the highest R-matrix dissociative PECs, we believe that they are
robust, since the inherent errors in the R-matrix calculations decrease
significantly within the degree of excitation. Finally, Figure 2 shows the ion
ground state and vibrational levels, the adjusted dissociative curves used in
the present calculations and their corresponding autoionization widths coming
from the R-matrix computations.

##  IV Evaluation of the cross section using the MQDT-type approach

###  IV.1 Mechanisms and couplings

Using the set of molecular data (PEC and electric couplings) determined as
described in the previous section, we performed a series of MQDT calculations
of the DR cross section, assuming the molecular ion to be initially in its
electronic ground state ($X^{2}\Sigma^{+},v_{i}^{+}=0,1,2$) and by neglecting
rotational and spin-orbit effects. The calculations were performed for the
states with total symmetry of ${}^{1}\Sigma^{+}$, ${}^{1}\Pi$ and ${}^{3}\Pi$,
while the number of dissociative states considered for each symmetry are
indicated in Figure 2.

We consider incident electron energies from $0.01$ meV up to $3$ eV. Since the
dissociation energy of CO+($X^{2}\Sigma^{+}$) is about $8.5$ eV, the majority
of the $53$ vibrational levels of the ion lie above the total energy of the
CO${}^{+}+e$ system. These levels are associated with closed ionization
channels, as defined in section II, responsible for temporary resonant capture
into Rydberg states. As the energy increases, more and more ionization
channels become open, which can result in autoionization leading to
competitive processes like IEC or SEC, decreasing the flux of DR.

The direct electronic couplings between ionization and dissociation channels -
equation (5) - have been extracted from the autoionization widths of the
valence states chakrabarti07 .

Since a highly accurate solution of the Lippman-Schwinger system of integral
equation (6) is difficult to obtain takagi2000 ; pichl2000 , we take advantage
of the fact that the couplings (among ionization and dissociation channels)
involved are small and perturbative solution is acceptable. Following the main
ideas of the earlier studies ggs91 ; ifs-a09 , we adopt a second-order
perturbative expansion, which accounts for all the basic mechanisms involved
in DR, including indirect electronic interaction between the ionization
channels.

The non-adiabatic couplings between the ionization channels rely, see Eqs. (7)
and (8), on the $R$ dependence of the quantum defects, which have been
evaluated using the ab initio calculations described in Section III.1.

For each dissociative channel available, we have considered its interaction
with the most relevant series of Rydberg states, namely for ${}^{1}\Sigma^{+}$
symmetry the lowest $4$ partial waves (s, p, d and f) were considered, while
for the ${}^{1}\Pi$ and ${}^{3}\Pi$ symmetries only the lowest $3$ (s, p, d)
have been used. For the sake of simplicity, figure 2 shows the so-called
global couplings which were constructed using the autoionization widths of
chakrabarti06 ; chakrabarti07 , namely
$V^{\Lambda}_{d_{j}}=\left(\left(\sum_{l}\Gamma^{\Lambda}_{d_{j},l}\right)/2\pi\right)^{1/2}$.

According to Eq. (12), the interference between the direct process (involving
open channels exclusively and described by the first term: $\bm{X_{oo}}$) and
the indirect one (involving closed as well as open channels and accounted for
by the second term) results in what the total process.

Figure 3: Cross sections for the direct (dashed lines) and the total (direct
$+$ indirect) processes (continuous lines), for a vibrationally relaxed target
in electronic states with ${}^{1}\Sigma^{+}$ (black), ${}^{1}\Pi$ (red) and
${}^{3}\Pi$ (blue) symmetries.

The role of the direct and indirect mechanisms in the DR process can be seen
in figure 3. The full colored lines show the total cross sections for the
different symmetries used in the calculations, while dashed lines of the same
color represent the cross sections given by the direct process only. As one
could expect from the magnitude of the valence-Rydberg couplings, see figure
2, a strong dependence on the states and interactions can be observed, the
majority of the cross section is given by the ${}^{1}\Pi$ and
${}^{1}\Sigma^{+}$ symmetries, while the ${}^{3}\Pi^{+}$ symmetry contributes
only a small part. Moreover, for both states with $\Pi$ symmetry, the indirect
process plays a minor role for this ion; the magnitude of the total cross
section is given by the direct process and the indirect one is responsible for
the resonance structures. This is most striking for the ${}^{1}\Pi$ symmetry,
where the total and direct only cross sections lie on the top of each other.
The importance of the indirect mechanism is relevant for the ${}^{1}\Sigma$
symmetry, where due to the destructive interference between the direct and
indirect processes the total cross section is lowered by almost one order of
magnitude compared to the direct one, at certain collision energies. The
indirect processes give rise to broad resonance structures in the range of the
collision energy of the present study. Even though the indirect process
globally plays a minor role, evaluating its effect gives increased insight
into the recombination mechanism.

The importance of the different dissociation paths in our calculation is
summarized in table 1. Our calculated final state distribution are compared
with the experimental values of rosen98  for four relative collision
(detuning) energies. For the zero collision energy the statistical error was
evaluated to be about $5\%$, while for higher energy values, due to the poor
statistics, it is around $30\%$.

Figure 4: Cross sections for the total (direct $+$ indirect) DR process for a vibrationally relaxed target including the summed-up contributions of the states with ${}^{1}\Sigma^{+}$, ${}^{1}\Pi$ and ${}^{3}\Pi$ symmetry. The experimental results of mitchell85  are shown with a dotted line with full violet diamonds, while those of rosen98  with dotted line with full blue circles, respectively. Table 1: The calculated final state distribution and its comparison with the experimental values for four relative collision (detuning) energies. Dissociation | energy (eV)
---|---
path | 10-5 | 0.4 | 1 | 1.5
| calc | exp† | calc | exp | calc | exp | calc | exp
C(3P)+O(3P) | 86.56% | 76.1% | 80.85% | 53% | 43.25% | 39% | 50.93% | 38%
C(3P)+O(1D) | - | 9.4% | - | 8% | - | 15% | - | 11%
C(1D)+O(3P) | 13.41% | 14.5% | 19.15% | 34% | 9.61% | 35% | 4.50% | 35%
C(1S)+O(3P) |  | 0.0% |  |  | ¡$10^{-5}$% | 5% | ¡10-2% | 5%
C(1D)+O(1D) |  |  |  | 5% | 47.14% | 6% | 44.56% | 11%
C(3P)+O(1S) |  |  |  |  |  |  |  $\sim$10-11% |
C(1S)+O(1D) |  | - |  | - |  | - |  $\sim$10-7% | -
† rosen98  |  |  |  |  |  |  |  |

The dominant dissociation pathway for the DR of the CO+ is the one which
correlates with the C(3P)+O(3P) atomic limits. Table 1 shows a qualitative
agreement between the calculated and the measured final state distributions.
The agreement is good, particularly at low collision energies. At higher
energies, the two sets of data disagrees, mainly due to the increased
experimental statistical errors and the fact that our model lacks a
dissociating pathway correlating to the C(3P)+O(1D) atomic limits.

##  V Results and discussion

Figure 5: Anisotropic rate coefficients for a vibrationally relaxed target.
Experimental results rosen98  are in given in black with filled circles. Our
theoretical results, for all symmetries considered here, are shown in red.
Figure 6: Isotropic rate coefficients for a vibrationally relaxed target. Our
theoretical results, for all symmetries considered here, are represented by
magenta dotted line. The experimental results are shown as follows: mitchell85
, green dashed line; rosen98 , black solid line; geoghegan91 , blue diamonds,
mentzoni68 , red circles and laube98 , violet squares.

The DR cross section corresponding to the three symmetries contributing to the
process (${}^{1}\Sigma^{+}$, ${}^{1}\Pi$ and ${}^{3}\Pi$), summed according to
Eq. (14), are shown in figure 4. The calculations have been performed for the
12C16O+ isotopologue of the cation. The total cross section is characterized
by resonance structures, superimposed on a smooth background (direct process
only). The strong interaction between the closed channels, associated with
higher vibrational ion levels and the current dissociative states,
contaminates, via non-adiabatic or indirect electronic coupling, the direct
interaction between the entrance and the dissociative channels. This results
in a stronger capture probability, and consequently a higher DR cross section,
which leads to the rich resonance structures observed in the total cross
section. The direct DR cross section is proportional to the square of the
Franck-Condon overlap, of the type seen in Eq. (5), between the vibrational
wave functions of the ion and the dissociation state of the neutral. The
favorable crossings at lower collision energies of the dissociation states
correlating to the C(3P)+O(3P) and C(1D)+O(1D) atomic limits (blue and cyan
curves in Fig. 2) with the ionic ground state (black curve on the same figure)
and the significant couplings to the entrance channel are responsible for the
large contribution by the direct DR cross sections to all the three symmetries
used in the calculations. This is the smooth background cross section on which
the contribution from the indirect process is superimposed, resulting in the
total cross section shown in Fig. 4. The closed ionization channels are those
responsible for the temporary capture into the Rydberg states and consequently
to the indirect mechanism. There is an increasing number of resonance states
of all symmetries at higher energies, and some or all of these can participate
in the DR process via channel mixing mechanism.

The role of each dissociation channel depends on the energy of the incident
electron. At higher collision energies, when the other two dissociation
channels become open (magenta and red curves in figure 2) due to their less-
favorable crossings with the ionic ground state the indirect process gains
importance leading to the resonance structures in the total cross section. We
have included all the contributions of the dissociative channels in our
calculations.

In view of the comparison with the storage-ring measurements, we have
convoluted our MQDT cross sections with the anisotropic Maxwell velocity
distribution function for the electrons. The transversal and longitudinal
temperatures are $K_{B}T_{\perp}=1$ meV and $K_{B}T_{\parallel}=10$ meV,
respectively. The calculated anisotropic rate coefficient (magenta) together
with the measured one (green) rosen98  are shown in figure 5. The overall
agreement with experimental value is good, our calculated rate coefficient
reproduces perfectly the trends of the experimental curves, underestimating
them on average by a factor of $2$.

Figure 7: Total DR cross sections for different initial vibrational levels of
the ion, for all the considered symmetries. Black continuous line stand for
$v_{i}^{+}=0$, while the red dotted and green dashed lines for $v_{i}^{+}=1$
and $2$, respectively.  Figure 8:  DR and state-to-state VE and VdE isotropic
rate coefficients of CO+ in its ground electronic state, $v^{+}_{i}$ standing
for the vibrational quantum number of the target ion. Curves of same color
show the rate coefficients for the vibrational (de-)excitations corresponding
to the same $|\Delta v|=|v^{+}_{f}-v^{+}_{i}|$, $v_{f}>v_{i}$ for the VE and
$v_{f}<v_{i}$ for the VdE global rate coefficients, respectively. The final
vibrational quantum numbers of the ion are indicated for these processes.

In addition to the anisotropic rate coefficients, starting from the computed
cross section, we have evaluated the Maxwell isotropic rate coefficient as
well, for a broad range of electronic temperatures, relevant especially for
cold non-equilibrium plasmas. The rate for $v_{i}^{+}=0$ is displayed in
figure 6, and its thermal ($300$ K) value is in excellent agreement with the
measured values obtained in merged-beam mitchell85  and laube98  as well as
storage-ring based rosen98  collision experiments. The rate coefficients of
mentzoni68  presented as red circles in the figure are placed higher than the
other measurements, probably due to the presence of CO${}^{+}\cdot($CO$)_{n}$
clusters in the sample, as it was pointed out by whitaker81 . Moreover, our
calculations have shown much less isotopic effect than those found by
guberman-dr07 , having as mean deviation $0.43\%$ (due to the different
nuclear reduced masses) among the two isotopologues.

Finally, we have calculated dissociative recombination, vibrational excitation
(VE) and de-exciation (VdE) cross sections for different vibrational levels in
the entrance channels, namely for $v_{i}^{+}=1$ and $2$ and compared with the
cross sections found for $v_{i}^{+}=0$. The comparison of the DR cross
sections can be seen in figure 7. We find that the DR cross section for the
ground vibrational levels exceeds significantly, by more than a factor of $2$,
those of $v_{i}^{+}=1$ and $2$, whereas the later ones have similar rates.

Figure 8 shows the total DR and vibrational transition (VE, VdE) rate
coefficients for the three initial vibrational levels of the ion. We find the
highest rates for all the processes for $v^{+}_{i}=0$, their magnitude
decreases with excitation of the initial ionic vibrational level, except for
the $\Delta v=v^{+}_{f}-v^{+}_{i}=-1$ de-excitation transitions where the
$v_{i}^{+}=2$ gives higher rates.

The trends of the observables are the same for both the cross section and rate
coefficients. Thus one can conclude that whenever the initial sample can
contain ions with excited vibrational levels, fully accounting for them will
be important.

To aid the simple use of the rate coefficients shown in Fig. 8, we have fitted
them using simple formulas. The calculated DR rate coefficients for
CO${}^{+}(v=0,1,2)$ is represented by:

$k_{(CO^{+}),v}^{DR}(T_{e})=A_{v}\,T_{e}^{\alpha_{v}}\,\exp\left[-\sum_{i=1}^{7}\frac{B_{v}(i)}{i\,\,T_{e}^{i}}\right]$
(17)

over the electron temperature range $T_{min}<T_{e}<5000$ K with $T_{min}$ =
100 K. The values given by Eq. (17) depart from the reference values by only a
few percent. The parameters $A_{v}$, $\alpha_{v}$ and $B_{v}(i)$ are listed in
Table 2. The corresponding formula for the vibrational transitions (VE and
VdE) has the form:

$k_{(CO^{+}),v^{\prime}\to v^{\prime\prime}}^{VE,VdE}(T_{e})=A_{v^{\prime}\to
v^{\prime\prime}}\,T_{e}^{\alpha_{v^{\prime}\to
v^{\prime\prime}}}\,\exp\left[-\sum_{i=1}^{7}\frac{B_{v^{\prime}\to
v^{\prime\prime}}(i)}{i\,\,T_{e}^{i}}\right]$ (18)

over the electron temperature range $T_{min}<T_{e}<5000$ K. Again the values
calculated using Eq. (18) depart from the referenced values only by a few
percent. The parameters $T_{min}$, $A_{v^{\prime}\to v^{\prime\prime}}$,
$\alpha_{v^{\prime}\to v^{\prime\prime}}$ and the different $B_{v^{\prime}\to
v^{\prime\prime}}(i)$ for $i\in[1,7]$ are listed in Tables 3 to 5.

Table 2: Parameters used in Eq. (17) to represent the DR rate coefficients of
CO+.

$v$ | $A_{v}$ | $\alpha_{v}$ | $B_{v}(1)$ | $B_{v}(2)$ | $B_{v}(3)$ | $B_{v}(4)$ | $B_{v}(5)$ | $B_{v}(6)$ | $B_{v}(7)$
---|---|---|---|---|---|---|---|---|---
0 |  .281949503$\times 10^{-04}$ |  -.802579890$\times 10^{00}$ |  .710130957$\times 10^{03}$ |  -.698308895$\times 10^{06}$ |  .335177615$\times 10^{09}$ |  -.850620477$\times 10^{11}$ |  .117104453$\times 10^{14}$ |  -.826280436$\times 10^{15}$ |  .233919918$\times 10^{17}$
1 |  .715701893$\times 10^{-05}$ |  -.682706983$\times 10^{00}$ |  .843520677$\times 10^{03}$ |  -.713971932$\times 10^{06}$ |  .300490272$\times 10^{09}$ |  -.703053442$\times 10^{11}$ |  .920945293$\times 10^{13}$ |  -.629626035$\times 10^{15}$ |  .174554736$\times 10^{17}$
2 |  .158116294$\times 10^{-05}$ |  -.537454517$\times 10^{00}$ |  .740102970$\times 10^{03}$ |  -.910764930$\times 10^{06}$ |  .460910057$\times 10^{09}$ |  -.118698360$\times 10^{12}$ |  .163967837$\times 10^{14}$ |  -.115670469$\times 10^{16}$ |  .327041619$\times 10^{17}$

Table 3: Parameters used in Eq. (18) to represent the VE rate coefficients of
CO+ ($v^{\prime}$ = 0).

$v^{\prime}\to v^{\prime\prime}$ |  $T_{min}$(K) | $A_{v^{\prime}\to v^{\prime\prime}}$ | $\alpha_{v^{\prime}\to v^{\prime\prime}}$ | $B_{v^{\prime}\to v^{\prime\prime}}(1)$ | $B_{v^{\prime}\to v^{\prime\prime}}(2)$ | $B_{v^{\prime}\to v^{\prime\prime}}(3)$ | $B_{v^{\prime}\to v^{\prime\prime}}(4)$ | $B_{v^{\prime}\to v^{\prime\prime}}(5)$ | $B_{v^{\prime}\to v^{\prime\prime}}(6)$ | $B_{v^{\prime}\to v^{\prime\prime}}(7)$
---|---|---|---|---|---|---|---|---|---|---
0 $\to$ 1 | 100 |  .641552103$\times 10^{-05}$ |  -.771371374$\times 10^{00}$ |  .414475403$\times 10^{04}$ |  -.555697606$\times 10^{06}$ |  .177113555$\times 10^{09}$ |  -.379319033$\times 10^{11}$ |  .499340108$\times 10^{13}$ |  -.353310994$\times 10^{15}$ |  .101763661$\times 10^{17}$
0 $\to$ 2 | 200 |  .748711072$\times 10^{-05}$ |  -.848613994$\times 10^{00}$ |  .808028559$\times 10^{04}$ |  -.314102151$\times 10^{07}$ |  .267788469$\times 10^{10}$ |  -.125332604$\times 10^{13}$ |  .323013435$\times 10^{15}$ |  -.429933877$\times 10^{17}$ |  .230779467$\times 10^{19}$
0 $\to$ 3 | 250 |  .653782258$\times 10^{-05}$ |  -.889871881$\times 10^{00}$ |  .100400799$\times 10^{05}$ |  -.153427057$\times 10^{07}$ |  .166204703$\times 10^{10}$ |  -.968343280$\times 10^{12}$ |  .307963790$\times 10^{15}$ |  -.505267521$\times 10^{17}$ |  .334692067$\times 10^{19}$
0 $\to$ 4 | 300 |  .410991022$\times 10^{-04}$ |  -.111309015$\times 10^{01}$ |  .144578277$\times 10^{05}$ |  -.425122442$\times 10^{07}$ |  .457572230$\times 10^{10}$ |  -.279947663$\times 10^{13}$ |  .974452298$\times 10^{15}$ |  -.180068315$\times 10^{18}$ |  .137000155$\times 10^{20}$
0 $\to$ 5 | 350 |  .673816604$\times 10^{-04}$ |  -.121225593$\times 10^{01}$ |  .186129714$\times 10^{05}$ |  -.751669966$\times 10^{07}$ |  .865504846$\times 10^{10}$ |  -.581314625$\times 10^{13}$ |  .227948002$\times 10^{16}$ |  -.482391689$\times 10^{18}$ |  .424701037$\times 10^{20}$
0 $\to$ 6 | 400 |  .678304142$\times 10^{-04}$ |  -.126655170$\times 10^{01}$ |  .216968564$\times 10^{05}$ |  -.932715458$\times 10^{07}$ |  .125788319$\times 10^{11}$ |  -.980812594$\times 10^{13}$ |  .440562841$\times 10^{16}$ |  -.105632400$\times 10^{19}$ |  .104556888$\times 10^{21}$
0 $\to$ 7 | 450 |  .935214058$\times 10^{-04}$ |  -.135011215$\times 10^{01}$ |  .242759946$\times 10^{05}$ |  -.866103794$\times 10^{07}$ |  .131034124$\times 10^{11}$ |  -.117137348$\times 10^{14}$ |  .603374022$\times 10^{16}$ |  -.165375862$\times 10^{19}$ |  .186623905$\times 10^{21}$
0 $\to$ 8 | 500 |  .127223673$\times 10^{-03}$ |  -.143309661$\times 10^{01}$ |  .268650737$\times 10^{05}$ |  -.655138124$\times 10^{07}$ |  .888503051$\times 10^{10}$ |  -.729741009$\times 10^{13}$ |  .338502966$\times 10^{16}$ |  -.803004985$\times 10^{18}$ |  .732964756$\times 10^{20}$
0 $\to$ 9 | 550 |  .106418750$\times 10^{-03}$ |  -.146917756$\times 10^{01}$ |  .289593294$\times 10^{05}$ |  -.336743387$\times 10^{07}$ |  .171303281$\times 10^{10}$ |  .173009739$\times 10^{13}$ |  -.296792983$\times 10^{16}$ |  .156774183$\times 10^{19}$ |  -.292893172$\times 10^{21}$
0 $\to$ 10 | 600 |  .144300838$\times 10^{-04}$ |  -.135396575$\times 10^{01}$ |  .296834933$\times 10^{05}$ |  .806531715$\times 10^{07}$ |  -.280264726$\times 10^{11}$ |  .444214735$\times 10^{14}$ |  -.379357647$\times 10^{17}$ |  .167825913$\times 10^{20}$ |  -.301076067$\times 10^{22}$
0 $\to$ 11 | 650 |  .867766831$\times 10^{-07}$ |  -.958697496$\times 10^{00}$ |  .257440709$\times 10^{05}$ |  .409137189$\times 10^{08}$ |  -.119011764$\times 10^{12}$ |  .189804083$\times 10^{15}$ |  -.170031924$\times 10^{18}$ |  .799092375$\times 10^{20}$ |  -.152981923$\times 10^{23}$
0 $\to$ 12 | 700 |  .325017591$\times 10^{-32}$ |  .446113412$\times 10^{01}$ |  -.512283310$\times 10^{05}$ |  .493759852$\times 10^{09}$ |  -.147156377$\times 10^{13}$ |  .246380214$\times 10^{16}$ |  -.232988126$\times 10^{19}$ |  .115912196$\times 10^{22}$ |  -.235418187$\times 10^{24}$

Table 4: Parameters used in Eq. (18) to represent the VE and VdE rate
coefficients of CO+ ($v^{\prime}$ = 1).

$v^{\prime}\to v^{\prime\prime}$ | $T_{min}$ | $A_{v^{\prime}\to v^{\prime\prime}}$ | $\alpha_{v^{\prime}\to v^{\prime\prime}}$ | $B_{v^{\prime}\to v^{\prime\prime}}(1)$ | $B_{v^{\prime}\to v^{\prime\prime}}(2)$ | $B_{v^{\prime}\to v^{\prime\prime}}(3)$ | $B_{v^{\prime}\to v^{\prime\prime}}(4)$ | $B_{v^{\prime}\to v^{\prime\prime}}(5)$ | $B_{v^{\prime}\to v^{\prime\prime}}(6)$ | $B_{v^{\prime}\to v^{\prime\prime}}(7)$
---|---|---|---|---|---|---|---|---|---|---
1 $\to$ 0 | 100 |  .633920326$\times 10^{-05}$ |  -.770044378$\times 10^{00}$ |  .104178437$\times 10^{04}$ |  -.551232678$\times 10^{06}$ |  .175018067$\times 10^{09}$ |  -.374062491$\times 10^{11}$ |  .492117834$\times 10^{13}$ |  -.348198826$\times 10^{15}$ |  .100308337$\times 10^{17}$
1 $\to$ 2 | 100 |  .112757539$\times 10^{-05}$ |  -.667253557$\times 10^{00}$ |  .301658328$\times 10^{04}$ |  .265128201$\times 10^{06}$ |  -.167485020$\times 10^{09}$ |  .460282036$\times 10^{11}$ |  -.652022601$\times 10^{13}$ |  .465696778$\times 10^{15}$ |  -.132656980$\times 10^{17}$
1 $\to$ 3 | 150 |  .562989138$\times 10^{-06}$ |  -.651039420$\times 10^{00}$ |  .526482012$\times 10^{04}$ |  .134397812$\times 10^{07}$ |  -.848709209$\times 10^{09}$ |  .288214188$\times 10^{12}$ |  -.547449693$\times 10^{14}$ |  .544047009$\times 10^{16}$ |  -.219710957$\times 10^{18}$
1 $\to$ 4 | 200 |  .130991873$\times 10^{-06}$ |  -.537069718$\times 10^{00}$ |  .875368149$\times 10^{04}$ |  .339794737$\times 10^{06}$ |  -.146345601$\times 10^{09}$ |  .270246030$\times 10^{11}$ |  .183890443$\times 10^{12}$ |  -.736617748$\times 10^{15}$ |  .699602764$\times 10^{17}$
1 $\to$ 5 | 250 |  .209439988$\times 10^{-05}$ |  -.858043793$\times 10^{00}$ |  .147483446$\times 10^{05}$ |  -.617683461$\times 10^{07}$ |  .628994423$\times 10^{10}$ |  -.343234925$\times 10^{13}$ |  .103740710$\times 10^{16}$ |  -.163563141$\times 10^{18}$ |  .104899664$\times 10^{20}$
1 $\to$ 6 | 300 |  .439087022$\times 10^{-05}$ |  -.950117793$\times 10^{00}$ |  .180805100$\times 10^{05}$ |  -.780708040$\times 10^{07}$ |  .975729438$\times 10^{10}$ |  -.654274154$\times 10^{13}$ |  .239495154$\times 10^{16}$ |  -.451595265$\times 10^{18}$ |  .342947398$\times 10^{20}$
1 $\to$ 7 | 350 |  .990587644$\times 10^{-05}$ |  -.107079534$\times 10^{01}$ |  .209321916$\times 10^{05}$ |  -.904877766$\times 10^{07}$ |  .124117280$\times 10^{11}$ |  -.929195541$\times 10^{13}$ |  .387627434$\times 10^{16}$ |  -.844294279$\times 10^{18}$ |  .746823381$\times 10^{20}$
1 $\to$ 8 | 400 |  .290364496$\times 10^{-05}$ |  -.976365340$\times 10^{00}$ |  .209545755$\times 10^{05}$ |  .435177075$\times 10^{07}$ |  -.136200658$\times 10^{11}$ |  .175763114$\times 10^{14}$ |  -.114936867$\times 10^{17}$ |  .372696383$\times 10^{19}$ |  -.475130034$\times 10^{21}$
1 $\to$ 9 | 450 |  .553151534$\times 10^{-05}$ |  -.107588250$\times 10^{01}$ |  .228931552$\times 10^{05}$ |  .102630980$\times 10^{08}$ |  -.292944159$\times 10^{11}$ |  .393966744$\times 10^{14}$ |  -.277151718$\times 10^{17}$ |  .982366990$\times 10^{19}$ |  -.138325077$\times 10^{22}$
1 $\to$ 10 | 500 |  .295872921$\times 10^{-18}$ |  .190367144$\times 10^{01}$ |  -.111794851$\times 10^{05}$ |  .184960763$\times 10^{09}$ |  -.439761036$\times 10^{12}$ |  .564818219$\times 10^{15}$ |  -.399526307$\times 10^{18}$ |  .146086706$\times 10^{21}$ |  -.215320891$\times 10^{23}$
1 $\to$ 11 | 550 |  .860719793$\times 10^{-14}$ |  .818099637$\times 10^{00}$ |  .202216367$\times 10^{04}$ |  .144200890$\times 10^{09}$ |  -.371106005$\times 10^{12}$ |  .518902243$\times 10^{15}$ |  -.402326107$\times 10^{18}$ |  .162115803$\times 10^{21}$ |  -.264323393$\times 10^{23}$
1 $\to$ 12 | 600 |  .238429782$\times 10^{-24}$ |  .309243990$\times 10^{01}$ |  -.287784203$\times 10^{05}$ |  .327916264$\times 10^{09}$ |  -.895346410$\times 10^{12}$ |  .135575406$\times 10^{16}$ |  -.114959696$\times 10^{19}$ |  .509491570$\times 10^{21}$ |  -.917135223$\times 10^{23}$
1 $\to$ 13 | 650 |  .171475246$\times 10^{-53}$ |  .946512823$\times 10^{01}$ |  -.118535342$\times 10^{06}$ |  .845804438$\times 10^{09}$ |  -.241202045$\times 10^{13}$ |  .384508952$\times 10^{16}$ |  -.344818061$\times 10^{19}$ |  .162171803$\times 10^{22}$ |  -.310615978$\times 10^{24}$

Table 5: Parameters used in Eq. (18) to represent the VE and VdE rate
coefficients of CO+ ($v^{\prime}$ = 2).

$v^{\prime}\to v^{\prime\prime}$ | $T_{min}$ | $A_{v^{\prime}\to v^{\prime\prime}}$ | $\alpha_{v^{\prime}\to v^{\prime\prime}}$ | $B_{v^{\prime}\to v^{\prime\prime}}(1)$ | $B_{v^{\prime}\to v^{\prime\prime}}(2)$ | $B_{v^{\prime}\to v^{\prime\prime}}(3)$ | $B_{v^{\prime}\to v^{\prime\prime}}(4)$ | $B_{v^{\prime}\to v^{\prime\prime}}(5)$ | $B_{v^{\prime}\to v^{\prime\prime}}(6)$ | $B_{v^{\prime}\to v^{\prime\prime}}(7)$
---|---|---|---|---|---|---|---|---|---|---
2 $\to$ 0 | 100 |  .143264325$\times 10^{-05}$ |  -.676398119$\times 10^{00}$ |  .770399508$\times 10^{03}$ |  -.649235829$\times 10^{06}$ |  .280763127$\times 10^{09}$ |  -.687619887$\times 10^{11}$ |  .943978108$\times 10^{13}$ |  -.672803202$\times 10^{15}$ |  .193189057$\times 10^{17}$
2 $\to$ 1 | 100 |  .110150559$\times 10^{-05}$ |  -.664642895$\times 10^{00}$ |  -.475690563$\times 10^{02}$ |  .274737488$\times 10^{06}$ |  -.172309109$\times 10^{09}$ |  .472936617$\times 10^{11}$ |  -.669923671$\times 10^{13}$ |  .478615271$\times 10^{15}$ |  -.136383154$\times 10^{17}$
2 $\to$ 3 | 100 |  .296299450$\times 10^{-06}$ |  -.540421763$\times 10^{00}$ |  .322902717$\times 10^{04}$ |  -.185353201$\times 10^{06}$ |  .102324472$\times 10^{09}$ |  -.295771623$\times 10^{11}$ |  .446770807$\times 10^{13}$ |  -.336457775$\times 10^{15}$ |  .998160611$\times 10^{16}$
2 $\to$ 4 | 150 |  .278155705$\times 10^{-05}$ |  -.783269363$\times 10^{00}$ |  .680722101$\times 10^{04}$ |  -.103878379$\times 10^{07}$ |  .741843999$\times 10^{09}$ |  -.276056904$\times 10^{12}$ |  .549666157$\times 10^{14}$ |  -.558745281$\times 10^{16}$ |  .228035409$\times 10^{18}$
2 $\to$ 5 | 200 |  .266683567$\times 10^{-04}$ |  -.107862034$\times 10^{01}$ |  .110175934$\times 10^{05}$ |  -.322418905$\times 10^{07}$ |  .245112696$\times 10^{10}$ |  -.101947642$\times 10^{13}$ |  .237960237$\times 10^{15}$ |  -.292641709$\times 10^{17}$ |  .147503229$\times 10^{19}$
2 $\to$ 6 | 250 |  .542147000$\times 10^{-05}$ |  -.983012338$\times 10^{00}$ |  .122608040$\times 10^{05}$ |  .321533385$\times 10^{06}$ |  -.729487979$\times 10^{09}$ |  .465646557$\times 10^{12}$ |  -.147373795$\times 10^{15}$ |  .236308026$\times 10^{17}$ |  -.153015638$\times 10^{19}$
2 $\to$ 7 | 300 |  .317786833$\times 10^{-06}$ |  -.721222477$\times 10^{00}$ |  .142487517$\times 10^{05}$ |  .102722134$\times 10^{07}$ |  -.582000169$\times 10^{09}$ |  .484888182$\times 10^{11}$ |  .644040556$\times 10^{14}$ |  -.223384907$\times 10^{17}$ |  .220068495$\times 10^{19}$
2 $\to$ 8 | 350 |  .273643357$\times 10^{-06}$ |  -.744726391$\times 10^{00}$ |  .184645633$\times 10^{05}$ |  -.445860723$\times 10^{07}$ |  .783952518$\times 10^{10}$ |  -.677237281$\times 10^{13}$ |  .310300559$\times 10^{16}$ |  -.723338935$\times 10^{18}$ |  .674944000$\times 10^{20}$
2 $\to$ 9 | 400 |  .161814112$\times 10^{-05}$ |  -.947134778$\times 10^{00}$ |  .222431469$\times 10^{05}$ |  -.163473342$\times 10^{07}$ |  -.438401719$\times 10^{10}$ |  .980397656$\times 10^{13}$ |  -.775362755$\times 10^{16}$ |  .276484946$\times 10^{19}$ |  -.372814588$\times 10^{21}$
2 $\to$ 10 | 450 |  .857949106$\times 10^{-05}$ |  -.113992880$\times 10^{01}$ |  .247605739$\times 10^{05}$ |  .513853850$\times 10^{07}$ |  -.233874625$\times 10^{11}$ |  .348626787$\times 10^{14}$ |  -.254552925$\times 10^{17}$ |  .917457255$\times 10^{19}$ |  -.130303783$\times 10^{22}$
2 $\to$ 11 | 500 |  .222319239$\times 10^{-11}$ |  .337546441$\times 10^{00}$ |  .802866148$\times 10^{04}$ |  .966824466$\times 10^{08}$ |  -.237713634$\times 10^{12}$ |  .308999968$\times 10^{15}$ |  -.219586955$\times 10^{18}$ |  .804421605$\times 10^{20}$ |  -.118654044$\times 10^{23}$
2 $\to$ 12 | 550 |  .967268411$\times 10^{-11}$ |  .141405228$\times 10^{00}$ |  .108888796$\times 10^{05}$ |  .100995726$\times 10^{09}$ |  -.264030143$\times 10^{12}$ |  .370381461$\times 10^{15}$ |  -.287172365$\times 10^{18}$ |  .115625983$\times 10^{21}$ |  -.188366916$\times 10^{23}$
2 $\to$ 13 | 600 |  .859235026$\times 10^{-11}$ |  .675737504$\times 10^{-01}$ |  .117098302$\times 10^{05}$ |  .110881879$\times 10^{09}$ |  -.305681884$\times 10^{12}$ |  .463253664$\times 10^{15}$ |  -.392591768$\times 10^{18}$ |  .173889056$\times 10^{21}$ |  -.312870721$\times 10^{23}$
2 $\to$ 14 | 650 |  .164458016$\times 10^{-23}$ |  .280896167$\times 10^{01}$ |  -.257956296$\times 10^{05}$ |  .332078837$\times 10^{09}$ |  -.946887885$\times 10^{12}$ |  .150871571$\times 10^{16}$ |  -.135222037$\times 10^{19}$ |  .635565472$\times 10^{21}$ |  -.121648562$\times 10^{24}$

##  VI Conclusion

This paper presents a theoretical study of DR of CO+ over a broad range of
electron energies including several dissociation states with different
symmetries. Although our basic approach is ab initio, using such methods it is
not yet possible to compute the position of the dissociative curves involved
in the DR process accurately enough to give reliable results. We have
therefore used potential curves derived from spectroscopic data on CO to
calibrate the ab initio data. Our DR calculations using these calibrated
curves give good agreement with the low-energy experimental data, up to about
$3$ eV, significantly improving the results of existing previous studies. We
note however the very recent study of DR N${}_{2}^{+}$ performed by jtx  based
on fully ab initio potential energy curves and couplings computed with the
R-matrix method jt560 ; jt574 . N${}_{2}^{+}$ is isoelectronic with CO+,
although its higher symmetry makes it somewhat easier to treat. These studies
suggest that a similar, accurate, fully ab initio treatment of the CO+ problem
should be possible in the fairly near future.

A major motivation of this paper was to show and test the applicability of our
method, whenever there are significant lacks of reliable quantum chemistry
calculations. We have performed calculations making use of $4$ dissociation
states in three different symmetry accounting Rydberg states up to four
partial waves. The calculations were done in the highest order of complexity
possible.

On overall, one can say that the agreement achieved between theory and
experiment over a significant range of energies is satisfactory, the trends
being qualitatively reproduced for this diatomic system with many-electrons.
This suggests that our calibrated ab initio approach provides a suitable
procedure for studying other many-electron systems for which pure ab initio
calculations are not reliable.

The found results are of fundamental importance and very promising, suggesting
for a need of further studies especially for improved and more accurate
potential energy curves and couplings not just for the molecular ion but for
neutral molecule as well.

## Acknowledgments

The authors thank Ch. Jungen and S. L. Guberman for numerous and fruitful
discussions. They acknowledge support from the International Atomic Energy
Agency via the Coordinated Research Project ”Light Element Atom, Molecule and
Radical Behaviour in the Divertor and Edge Plasma Regions,” from Agence
Nationale de la Recherche via the projects ”SUMOSTAI” (No. ANR-09-BLAN-020901)
and ”HYDRIDES” (No. ANR-12-BS05-0011-01), from the IFRAF-Triangle de la
Physique via the project ”SpecoRyd,” and from the Centre National de la
Recherche Scientifique via the programs ”Physique et Chimie du Milieu
Interstellaire,” and the PEPS projects ”Physique théorique et ses interfaces”
TheMS and TPCECAM. They also acknowledge generous financial support from La
Région Haute-Normandie via the CPER ”THETE” project, and the GRR Electronique,
Energie et Matriaux, from the ”Fédération de Recherche Energie, Propulsion,
Environnement,” and from the LabEx EMC3, via the project PicoLIBS (No.
ANR-10-LABX-09-01). K.C. thanks the department Institut des Sciences de
l’Ingénierie et des Systémes (INSIS) of CNRS for a research grant in 2013, and
Laboratoire Ondes et Milieux Complexes (LOMC) of Le Havre University for
hospitality.

## Data availability

Upon a reasonable request, the data supporting this article will be provided
by the corresponding author.

## References

  * (1) Rosen S, Peverall R, Larson M, Le Padellec A, Semaniak J, Larson Å, Strömholm C, Zande W J Vn der, Danared H and Dunn G H 1998 Phys. Rev. A 57, 4462.
  * (2) Chakrabarti K and Tennyson J 2006 J. Phys. B: At. Mol. Opt. Phys. 39, 1485.
  * (3) Chakrabarti K and Tennyson J 2007 J. Phys. B: At. Mol. Opt. Phys. 40 ,2135.
  * (4) Eidelsberg M, Lemaire J L, Federman S R, Stark G, Heays A N, Sheffer Y, Gavilan L, Fillion J H, Rostas F, Lyons J R, Smith P L, de Oliveira N, Joyeux D, Roudjane M and Nahon L 2012 Astronomy & Astrophysics 543, A69.
  * (5) Fowler A 1909 Month. Not. R: Astr. Soc.  70, 176.
  * (6) Fuente A, Garcia-Burillo S, Gerin M, Rizzo J R, Usero A, Teyssier D, Roueff E and le Bourlot J 2010 Astrophys. J 641, L105.
  * (7) Krupente P H and Weissman S 1965 J. Chem. Phys. 43, 1529.
  * (8) Seaton M. J. 1966 Proc. Phys. Soc. (London) 88, 801.
  * (9) Mitchell J B A and Hus H 1985 J. Phys. B: At. Mol. Phys. 18, 547.
  * (10) Park C et al 1994 J. Thermophys. Heat Transfer 8, 9.
  * (11) Bultel A, Chéron B G, Bourdon A, Motapon O and Schneider I F 2006 Phys. Plasmas 13, 043502.
  * (12) Fox J L 1999 Bull. Am. Astron. Soc. 31, 1584.
  * (13) Guberman S L 2007, Workshop on Planetary Atmospheres, communication.
  * (14) Geoghegan M, Adams N G and Smith D 1991 J. Phys. B 24, 2589.
  * (15) Giusti-Suzor A 1980 J. Phys. B: At. Mol. Phys. 13, 3867.
  * (16) Seaton M J 1983 Rept. Prog. Phys. 46,167.
  * (17) Greene C H and Jungen Ch 1985 Adv. At. Mol. Phys. 21, 51.
  * (18) Jungen C 1996, ed., Molecular Applications of Quantum Defect Theory, Institute of Physics Publishing Bristol.
  * (19) Giusti-Suzor A, Bardsley J. N and Derkits C 1983 Phys. Rev. A 28, 682.
  * (20) Schneider I F, Dulieu O and Giusti-Suzor A 1991 J. Phys. B: At. Mol. Phys. 24, L289.
  * (21) Takagi H 1993 J. Phys. B: At. Mol. Opt. Phys. 26, 4815.
  * (22) Tanabe T, Katayama I, Kamegaya H, Chida K, Arakaki Y, Watanabe T, Yoshizawa M, Saito M, Haruyama Y, Hosono K, Hatanaka K, Honma T, Noda K, Ohtani S and Takagi H 1995 Phys. Rev. Lett. 75, 1066.
  * (23) Schneider I F, Strömholm C, Carata L, Urbain X, Larsson M and Suzor-Weiner A 1997 J. Phys. B: At. Mol. Opt. Phys. 30, 2687.
  * (24) Amitay Z, Baer A, Dahan M, Levin J, Vager Z, Zajfman D, Knoll L, Lange M, Schwalm D, Wester R, Wolf A, Schneider I F, and Suzor-Weiner A 1999 Phys. Rev. A 60, 3769.
  * (25) Guberman S L and Giusti-Suzor A 1991 J. Chem. Phys. 95, 2602.
  * (26) Guberman S L 2000 in M Larsson, J B A Mitchell and I F Schneider, eds., Dissociative Recombination: Theory, Experiment and Applications IV, World Scientific Singapore p. 111.
  * (27) Sun H and Nakamura H 1990 J. Chem. Phys. 93, 6491.
  * (28) Vâlcu B, Schneider I F, Raoult M, Strömholm C, Larsson M and Suzor-Weiner A 1998 European Physical Journal D 1, 71.
  * (29) Schneider I F, Rabadán, Carata L, Andersen L H, Suzor-Weiner A and Tennyson J 2000a J. Phys. B: At. Mol. Opt.Phys.33, 4849.
  * (30) Schneider I F, Orel A E and Suzor-Weiner A 2000b Phys. Rev. Lett. 85, 3785.
  * (31) Kokoouline V, Greene C H and Esry B D 2001 Nature 412, 891.
  * (32) Kokoouline V and Greene C H 2003 Phys. Rev. A 68, 012703.
  * (33) Ngassam V, Motapon O, Florescu A, Pichl L, Schneider I F and Suzor-Weiner A 2003a Phys. Rev. A 68, 032704.
  * (34) Motapon O, Fifirig M, Florescu A, Waffeu Tamo F O, Crumeyrolle O, Varin-Bréant G, Bultel A, Vervisch P, Tennyson J and Schneider I F, 2006 Plasma Sources Sci. Technol. 15, 23.
  * (35) Faure A and Tennyson J 2001 Month. Not. R: Astr. Soc.  325, 443.
  * (36) Sidis V and Lefebvre-Brion H 1971J. Phys. B 4, 1040.
  * (37) Jungen C and Atabek O 1977 J. Chem. Phys. 66, 5584.
  * (38) Ngassam V, Florescu A, Pichl L, Schneider I F, Motapon O, and Suzor-Weiner A 2003b Eur. Phys. J. D 26, 165.
  * (39) Tennyson J 1996a J. Phys. B: At. Mol. Opt. Phys. 29, 1817.
  * (40) Tennyson J 1996b J. Phys. B: At. Mol. Opt. Phys. 29, 6185.
  * (41) Tchang-Brillet W-Ü L., Julienne P S, Robbe J-M , Letzeltel C and Rostas F 1992 J. Chem. Phys. 96, 6735.
  * (42) Lefebvre-Brion H and Lewis B R 2007 Mol. Phys. 105, 1625.
  * (43) Vázquez G J, Amero J M, Liebermann H P and Lefebvre-Brion H 2009 J. Chem. Phys. 113, 13395.
  * (44) Lefebvre-Brion H, Liebermann H P and Vázquez G J 2010 J. Chem. Phys. 132, 024311.
  * (45) Wigner E and Witmer E E 1928 Z. Physik 51, 859.
  * (46) Pichl L, Nakamura H and Horacek J 2000 CPC 124, 1.
  * (47) Takagi H 2000 in M Larsson, J B A. Mitchell and I F Schneider, eds., Dissociative Recombination: Theory, Experiment and Applications IV World Scientific Singapore p. 180.
  * (48) Mentzoni M H and Donohoe J 1968 Phys. Lett. 26A, 330.
  * (49) Laubé S, Lehfaoui L, Rowe B R and Mitchell J B A 1998 J. Phys. B: At. Mol. Opt. Phys. 31, 4181.
  * (50) Whitaker M, Biondi M A, and Johnsen R 1981 Phys. Rev. A 23, 1481.
  * (51) Little D A, Chakrabarti K, Mezei J Zs, Schneider I F, and Tennyson J 2014b Phys. Rev. A 90, 052705.
  * (52) Little D A and Tennyson J 2013 J. Phys. B: At. Mol. Opt. Phys. 46, 145102.
  * (53) Little D A and Tennyson J 2014a J. Phys. B: At. Mol. Opt. Phys. 47, 105204.


* [BO21] Florian Beyer and Todd A. Oliynyk.  Localized big bang stability for the Einstein-scalar field equations, 2021.  arXiv:2112.07730v2.
  * [Cal94] Patrick James Callahan.  Spectral geometry of hyperbolic 3-manifolds.  ProQuest LLC, Ann Arbor, MI, 1994.  Thesis (Ph.D.)–University of Illinois at Urbana-Champaign.
  * [CBC02] Yvonne Choquet-Bruhat and Spiros Cotsakis.  Global hyperbolicity and completeness.  J. Geom. Phys., 43(4):345–350, 2002.
  * [CBCLN06] B. Chow, P.L.L.N. Bennett Chow, P. Lu, and L. Ni.  Hamilton’s Ricci Flow.  Graduate studies in mathematics. American Mathematical Society/Science Press, 2006.
  * [CBG69] Yvonne Choquet-Bruhat and Robert Paul Geroch.  Global aspects of the Cauchy problem in general relativity.  Communications in Mathematical Physics, 14:329–335, 1969.
  * [CBIM04] Y. Choquet-Bruhat, J. Isenberg, and V. Moncrief.  Topologically general U(1) symmetric vacuum space-times with AVTD behavior.  Nuovo Cimento Soc. Ital. Fis. B, 119(7-9):625–638, 2004.
  * [CBM01] Y. Choquet-Bruhat and V. Moncrief.  Future global in time Einsteinian spacetimes with $\rm U(1)$ isometry group.  Ann. Henri Poincaré, 2(6):1007–1064, 2001.
  * [CCM+23] Michele Cicoli, Joseph P. Conlon, Anshuman Maharana, Susha Parameswaran, F. Quevedo, and Ivonne Zavala.  String cosmology: from the Early Universe to Today.  arXiV, 2023.  2303.04819v2.
  * [CI93] Piotr T. Chruściel and James Isenberg.  Nonisometric vacuum extensions of vacuum maximal globally hyperbolic spacetimes.  Phys. Rev. D (3), 48(4):1616–1628, 1993.
  * [CK90] Demetrios Christodoulou and Sergiu Klainerman.  Asymptotic properties of linear field equations in minkowski space.  Comm. Pure Appl. Math., 43:137–199, 1990.
  * [CK93] Demetrios Christodoulou and Sergiu Klainerman.  The global nonlinear stability of the Minkowski space, volume 41 of Princeton Mathematical Series.  Princeton University Press, Princeton, NJ, 1993.
  * [CR95] P.T. Chrusciel and A.D. Rendall.  Strong Cosmic Censorship in vacuum space-times with compact, locally homogeneous cauchy surfaces.  Annals of Physics, 242(2):349–385, Sep 1995.
  * [CS99] Neil J. Cornish and David N. Spergel.  On the eigenmodes of compact hyperbolic 3-manifolds, 1999.  arXiv:9906017v1.
  * [CZ95] Xu Cheng and Detang Zhou.  First eigenvalue estimate on Riemannian manifolds.  Hokk. Math. J., 24(3):453 – 472, 1995.
  * [DHRW02] Thibault Damour, Marc Henneaux, Alan D. Rendall, and Marsha Weaver.  Kasner-Like Behaviour for Subcritical Einstein-Matter Systems.  Ann. Henri Poincaré, 3:1049–1111, 2002.
  * [Dra03] Sever Silvestru Dragomir.  Some Gronwall type inequalities and applications.  Nova Science Publishers, Inc., Hauppauge, NY, 2003.
  * [Faj17] David Fajman.  The nonvacuum Einstein flow on surfaces of negative curvature and nonlinear stability.  Comm. Math. Phys., 353(2):905–961, 2017.
  * [Faj20] David Fajman.  Future attractors in $2+1$ dimensional $\Lambda$ gravity.  Phys. Rev. Lett., 125(12):121102, 6, 2020.
  * [FB52] Yvonne Fourès-Bruhat.  Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires.  Acta Mathematica, 88:141–225, 1952.
  * [FK20] David Fajman and Klaus Kröncke.  Stable fixed points of the Einstein flow with positive cosmological constant.  Comm. Anal. Geom., 28(7):1533–1576, 2020.
  * [FL20] Grigorios Fournodavlos and Jonathan Luk.  Asymptotically Kasner-like singularities, 2020.  arXiv:2003.13591v2.
  * [FOW21] David Fajman, Maximilian Ofner, and Zoe Wyatt.  Slowly expanding stable dust spacetimes, 2021.  arXiv:2107.00457v1.
  * [FRS23] Grigorios Fournodavlos, Igor Rodnianski, and Jared Speck.  Stable big bang formation for Einstein’s equations: the complete sub-critical regime.  J. Amer. Math. Soc., 36(3):827–916, 2023.
  * [FU22] David Fajman and Liam Urban.  Blow-up of waves on singular spacetimes with generic spatial metrics.  Lett. Math. Phys., 112(2):Paper No. 42, 31, 2022.
  * [FW21] David Fajman and Zoe Wyatt.  Attractors of the Einstein-Klein-Gordon system.  Comm. Part. Diff. Eq., 46(1):1–30, 2021.
  * [GaNS19] Pedro M. Girão, José Natário, and Jorge Drumond Silva.  Solutions of the wave equation bounded at the big bang.  Classical Quantum Gravity, 36(7):075016, 10, 2019.
  * [Haw67] S. W. Hawking.  The occurrence of singularities in cosmology. iii. Causality and Singularities.  Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 300(1461):187–201, 1967.
  * [IM02] James Isenberg and Vincent Moncrief.  Asymptotic behaviour in polarized and half-polarized U$(1)$ symmetric vacuum spacetimes.  Classical Quantum Gravity, 19(21):5361–5386, 2002.
  * [Ino01] Kaiki Taro Inoue.  Numerical study of length spectra and low-lying eigenvalue spectra of compact hyperbolic 3-manifolds.  Classical Quantum Gravity, 18(4):629–652, 2001.
  * [Krö15] Klaus Kröncke.  On the stability of Einstein manifolds.  Ann. Global Anal. Geom., 47(1):81–98, 2015.
  * [Mon08] Vincent Moncrief.  Relativistic Teichmüller theory—a Hamilton-Jacobi approach to $2+1$-dimensional Einstein gravity.  In Surveys in differential geometry. Vol. XII. Geometric flows, volume 12 of Surv. Differ. Geom., pages 203–249. Int. Press, Somerville, MA, 2008.
  * [Mon20] Puskar Mondal.  Thurston boundary of the Teichmüller space is the space of big bang singularities of 2+1 gravity, 2020.  arXiv:2002.03551v6.
  * [O’N83] Barrett O’Neill.  Semi-Riemannian geometry with applications to relativity.  Academic Press, 1983.
  * [Pen65] Roger Penrose.  Gravitational collapse and space-time singularities.  Phys. Rev. Lett., 14:57–59, Jan 1965.
  * [Ren08] Alan D. Rendall.  Partial differential equations in general relativity, volume 16 of Oxford Graduate Texts in Mathematics.  Oxford University Press, Oxford, 2008.
  * [Rin08] Hans Ringström.  Future stability of the Einstein-non-linear scalar field system.  Invent. Math., 173(1):123–208, 2008.
  * [Rin09] Hans Ringström.  The Cauchy problem in general relativity.  ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zürich, 2009.
  * [Rin19] Hans Ringström.  A unified approach to the Klein-Gordon equation on Bianchi backgrounds.  Comm. Math. Phys., 372(2):599–656, 2019.
  * [Rin20] Hans Ringström.  Linear systems of wave equations on cosmological backgrounds with convergent asymptotics.  Astérisque, (420):xi + 510, 2020.
  * [Rin21] Hans Ringström.  Wave equations on silent big bang backgrounds, 2021.  arXiv:2101.04939v2.
  * [RS18a] Igor Rodnianski and Jared Speck.  A regime of linear stability for the Einstein-scalar field system with applications to nonlinear big bang formation.  Ann. of Math. (2), 187(1):65–156, 2018.
  * [RS18b] Igor Rodnianski and Jared Speck.  Stable big bang formation in near-FLRW solutions to the Einstein-scalar field and Einstein-stiff fluid systems.  Selecta Math. (N.S.), 24(5):4293–4459, 2018.
  * [RS22] Igor Rodnianski and Jared Speck.  On the nature of Hawking’s incompleteness for the Einstein-vacuum equations: the regime of moderately spatially anisotropic initial data.  J. Eur. Math. Soc. (JEMS), 24(1):167–263, 2022.
  * [Spe18] Jared Speck.  The maximal development of near-FLRW data for the Einstein-Scalar Field system with spatial topology $\mathbb{S}^{3}$.  Comm. Math. Phys., 364(3):879–979, Oct 2018.
  * [Wan19] Jinhua Wang.  Future stability of the $1+3$ Milne model for the Einstein-Klein-Gordon system.  Classical Quantum Gravity, 36(22):225010, 65, 2019.
  * [Wan21] Jinhua Wang.  Nonlinear wave equation in a cosmological Kaluza Klein spacetime.  J. Math. Phys., 62(6):Paper No. 062504, 15, 2021.


$\mathcal{S}_{1},\dots,\mathcal{S}_{m_{r}}$ that contains more than
$\nicefrac{{\ell}}{{\chi}}$ elements, and consider the standard (unweighted)
maximum $k$-set coverage problem, that ignores constraint (_i_) of the above
observation, on these remaining collection of sets $\mathcal{T}$ over the
universe $\mathcal{U}_{r}$. Since we have guessed the correct value of $\ell$,
there is at least one valid solution and thus the following assertions hold:
(I) there exists a set of $k_{r}$ sets that covers
$\frac{{\mathsf{OPT}}_{\\#}}{\chi}$ elements, and  (II) $|\mathcal{T}|\geq
k_{r}$.  Let $\nu_{k}$ denote the _maximum_ number of elements that can be
covered by selecting $k$ sets from $\mathcal{T}$. There are the following two
well-known algorithm algorithms for the maximum $k$-set coverage problem both
of which select exactly $k$ sets: the greedy algorithm covers at least
$\varrho(k)\nu_{k}$ elements (F98, , Proposition 5.1), where the pipage-
rounding algorithm (based on the ${\mathsf{LP}}$-relaxation in Fig. 2) covers
at least $\varrho(f)\nu_{k}$ elements AS04 . Note that we do _not_ know the
exact value of $k_{r}$ and we _cannot_ guess by enumerating every possible
$k_{r}$ values for every $r\in\\{1,\dots,\chi\\}$ in polynomial time. To
overcome this obstacle, we use the following steps.

  1. $\triangleright$

We run both the algorithms for maximum $k$-set coverage for $k=1,2,\dots$
until we find the first (_smallest_) index $\widehat{k_{r}}\leq k_{r}$ such
that the better of the two algorithms cover _at least_
$\max\\{\varrho(\widehat{k_{r}}),\,\varrho(f)\\}\frac{{\mathsf{OPT}}_{\\#}}{\chi}\geq\rho\frac{{\mathsf{OPT}}_{\\#}}{\chi}$
elements.

  2. $\triangleright$

Suppose that this algorithm selects the $\widehat{k_{r}}$ sets (after possible
re-numbering of set indices)
$\mathcal{S}_{1},\dots,\mathcal{S}_{\widehat{k_{r}}}$, where we have ordered
the sets such that for every $j\in\\{2,3,\dots,\widehat{k_{r}}\\}$ the number
of elements covered by $\mathcal{S}_{j}$ and _not_ covered by any of the sets
$\mathcal{S}_{1},\dots,\mathcal{S}_{j-1}$ is _at least_ as many as the number
of elements covered by $\mathcal{S}_{\ell}$ and not covered by any of the sets
$\mathcal{S}_{1},\dots,\mathcal{S}_{j-1}$ for _any_ $\ell>j$. Remember that
$\max_{j\in\\{1,\dots,\widehat{k_{r}}\\}}\\{|\mathcal{S}_{j}|\\}\leq\nicefrac{{{\mathsf{OPT}}_{\\#}}}{{\chi}}$.
Let $j$ be the _smallest_ index such
$|\cup_{\ell=1}^{j-1}\mathcal{S}_{j}|<\varrho\frac{{\mathsf{OPT}}_{\\#}}{\chi}$
but
$|\cup_{\ell=1}^{j}\mathcal{S}_{j}|\geq\varrho\frac{{\mathsf{OPT}}_{\\#}}{\chi}$.
We have the following cases.

    1. $\triangleright$

If $|\mathcal{S}_{j}|\geq\varrho\frac{{\mathsf{OPT}}_{\\#}}{\chi}$ then we
select $\mathcal{S}_{j}$ as our solution since
$\varrho\frac{{\mathsf{OPT}}_{\\#}}{\chi}\leq|\mathcal{S}_{j}|\leq\frac{{\mathsf{OPT}}_{\\#}}{\chi}<2\,\varrho\frac{{\mathsf{OPT}}_{\\#}}{\chi}$.

    2. $\triangleright$

Otherwise $|\mathcal{S}_{j}|<\varrho\frac{{\mathsf{OPT}}_{\\#}}{\chi}$ and in
this case we select the $j\leq\widehat{k_{r}}\leq k_{r}$ sets
$\mathcal{S}_{1},\dots,\mathcal{S}_{j}$ in our solution since
$\varrho\frac{{\mathsf{OPT}}_{\\#}}{\chi}\leq|\cup_{\ell=1}^{j}\mathcal{S}_{j}|\leq
2\,\varrho\frac{{\mathsf{OPT}}_{\\#}}{\chi}$.

###  11.2 $\Delta$-bal-fmc: improved deterministic approximation

###### Proposition 2

There exists a polynomial-time deterministic algorithm Alg-greedy that, given
an instance of unweighted $\Delta$-bal-fmc $\\!\\!(\chi,k)$ outputs a solution
with the following properties:

(_a_)

The number of selected sets is $($exactly$)$ $k$.

(_b_)

The approximation ratio is at least
$\varrho>1-\nicefrac{{1}}{{{\mathbf{e}}}}$.

(_c_)

The coloring constraints are $O(\Delta f)$-approximately satisfied $($cf.  _(
5)_$)$, _i.e._ ,
$\forall\,i,j\in\\{1,\dots,\chi\\}:\,\nicefrac{{p_{i}}}{{p_{j}}}\leq(2+2\Delta)f$.

###### Proof

As already mentioned in the proof of Theorem 11.1 and elsewhere, there is a
deterministic polynomial-time algorithm for the maximum $k$-set coverage
problem with an approximation ratio of $\varrho$. For the given instance of
$\Delta$-bal-fmc $\\!\\!(\chi,k)$, we run this algorithms (ignoring element
colors) selecting $k$ sets, say $\mathcal{S}_{1},\dots,\mathcal{S}_{k}$.
Obviously, the total weight of all the elements covered in the selected
solution is at least $\varrho\,{\mathsf{OPT}}$. Let
$\alpha^{+}=\sum_{i=1}^{k}\lceil\nicefrac{{|\mathcal{S}_{i}|}}{{\chi}}\rceil+\Delta$
and
$\alpha^{-}=\sum_{i=1}^{k}\max\big{\\{}1,\,\lfloor\nicefrac{{|\mathcal{S}_{i}|}}{{\chi}}\rfloor-\Delta\big{\\}}$.
Note that $k\leq\alpha^{-}\leq\alpha^{+}\leq\alpha^{-}+(2\Delta+1)k$. Since
each of the sets in the solution is balanced, an upper bound for the number
$p_{i}$ of elements of color $i$ in the solution is given by
$p_{i}\leq\alpha^{+}$. Also note that by definition of $f$ we have
$p_{i}\geq\frac{\alpha^{-}}{f}$. It thus follows that for any $i$ and $j$ we
have $\nicefrac{{p_{i}}}{{p_{j}}}\leq
f\times\frac{\alpha^{+}}{\alpha^{-}}\leq(2+2\Delta)f$.

##  12 Approximating Geom-fmc via randomized shifting

We refer the reader to textbooks such as V01  for a general overview of the
randomized shifting technique (textbook V01  illustrates the technique in the
context of Euclidean travelling salesperson problem).

###### Theorem 12.1

For any constant $0<\varepsilon<1$, we can design a randomized algorithm Alg-
geom for Geom-fmc with the following properties:

  1. _(_ a _)_

Alg-geom runs in $O((\Delta/d)^{d}2^{(Cd/\varepsilon)^{O(d)}}k^{d})$ time.

  2. _(_ b _)_

Alg-geom satisfies the following properties with probability $1-o(1)$ $($cf.
Inequality (5)′–(5)${}^{\prime\prime})$:

    1. $\triangleright$

The algorithm covers at least
$(1-O(\varepsilon))({\mathsf{OPT}}-\varepsilon\chi)$ points.

    2. $\triangleright$

The algorithm satisfies the $(1+\varepsilon)$-approximate coloring constraints
$($cf. Inequality (5)${}^{\prime\prime})$, _i.e._ , for all
$i,j\in\\{1,\dots,\chi\\}$, $p_{i}\leq(1+\varepsilon)p_{j}$.

###### Remark 7

In many geometric applications, the dimension parameter $d$ is a _fixed_
constant. For this case, Alg-geom runs in polynomial time, and moreover, under
the _mild_ assumption of $\frac{{\mathsf{OPT}}}{\chi}\geq\eta$ for _some_
constant $\eta>1$, Alg-geom covers at least $(1-O(\varepsilon)){\mathsf{OPT}}$
points, i.e., under these conditions Alg-geom behaves like a randomized
polynomial-time approximation scheme.

###### Proof

Fix an optimal solution having $k$ unit balls
$\mathcal{B}_{1}^{\ast},\ldots,\mathcal{B}_{k}^{\ast}\subset{\mathbb{R}}^{d}$,
such that for all $i,j\in\\{1,\dots,\chi\\}$,
$\mu_{i}(\mathcal{B}^{\ast})=\mu_{j}(\mathcal{B}^{\ast})$, where
$\mathcal{B}^{\ast}=\bigcup_{i=1}^{k}\mathcal{B}_{i}^{\ast}$. Thus, we need to
show that our algorithm Alg-geom computes in
$(\Delta/d)^{d}2^{(Cd/\varepsilon)^{O(d)}}k^{d}$ time a set of unit balls
$B_{1},\ldots,B_{k}\subset\mathbb{R}^{d}$ such that the following assertions
hold with probability $1-o(1)$ (where $B=\bigcup_{i=1}^{k}B_{i}$):

$\displaystyle\sum_{i=1}^{\chi}\mu_{i}(B)>(1-O(\varepsilon))\sum_{i=1}^{\chi}(\mu_{i}(\mathcal{B}^{\ast})-\varepsilon)$
$\displaystyle\forall\,i,j\in\\{1,\ldots,\chi\\}:\,\mu_{i}(B)\leq(1+\varepsilon)\mu_{j}(B)$

Set $L=8d/\varepsilon$. Let $G\subset\mathbb{R}^{d}$ be an axis-parallel grid
such that every connected component of ${\mathbb{R}}^{d}\setminus G$ is an
open $d$-dimensional hypercube isometric to $(0,L)^{d}$. In other words, $G$
is the union of $d$ infinite families of axis-parallel $(d-1)$-dimensional
hyperplanes, spaced apart by $L$ in each orthonormal direction. Let
$\alpha\in[0,L)$ be chosen uniformly at random, and let $G^{\prime}=G+\alpha$
be the _random_ translation of $G$ by $\alpha$, i.e.,

$(p_{1}+\alpha,\dots,p_{d}+\alpha)\in
G^{\prime}\,\,\equiv\,\,(p_{1},\dots,p_{d})\in G$

Let $F$ is the set of indices of all balls $\mathcal{B}_{i}^{\ast}$ that has a
non-empty intersection with the randomly shifted grid $G^{\prime}$, i.e.,

$F=\\{i\in\\{1,\dots,k\\}:\mathcal{B}^{\ast}_{i}\cap
G^{\prime}\neq\emptyset\\}$

Let $\mathcal{B}^{\ast,F}=\bigcup_{i\in F}\mathcal{B}_{i}^{\ast}$. Any point
$p\in{\mathbb{R}}^{d}$ is contained in $\mathcal{B}^{\ast,F}$ only if it is
contained in some unit ball intersecting $G^{\prime}$. Therefore,
$p\in\mathcal{B}^{\ast,F}$ only if it is at distance at most $2$ from
$G^{\prime}$ (in other words, $\mathcal{B}^{\ast,F}$ is contained in the
$2$-neighborhood of $G^{\prime}$). The probability that any particular point
$p$ is at distance at most $2$ from any family of parallel randomly shifted
hyperplanes in $G^{\prime}$ is exactly $4/L$. By the union bound over all
dimensions, $\Pr[p\in\mathcal{B}^{\ast,F}]\leq 4d/L$. Therefore, by the
linearity of expectation,

${\mathbb{E}}[\mu_{i}(\mathcal{B}^{\ast,F})]=\sum_{p\in\cup_{i=1}^{k}\mathcal{B}_{i}^{\ast}}\\!\\!\\!\\!\Pr\left[p\in\mathcal{B}^{\ast,F}\,\right]\,\mu_{i}(p)\leq\frac{4d}{L}\mu_{i}(\mathcal{B}^{\ast})$

Consequently, by Markov’s inequality, we get

$\Pr\left[\textstyle\mu_{i}(\mathcal{B}^{\ast,F})\geq\frac{8d}{R}\mu_{i}(\mathcal{B}^{\ast})\right]\leq\nicefrac{{1}}{{2}}$

Set $\delta=2^{-\Theta(d)}\varepsilon/C$. Let
$B_{1}^{\ast\ast},\ldots,B_{k}^{\ast\ast}$ be the collection of unit balls in
${\mathbb{R}}^{d}$ obtained as follows. For each $i\in\\{1,\dots,k\\}\setminus
F$, obtain a unit ball by translating $\mathcal{B}_{i}^{\ast}$ such that its
center has coordinates that are _integer multiples_ of $\delta$, i.e., it is
an element of the dilated integer lattice $\delta\cdot\mathbb{Z}^{d}$. For
every $i\in F$, we obtain a unit ball by picking an arbitrary ball obtained
for some $j\in\\{1,\dots,k\\}\setminus F$ as described above. Essentially, the
new solution $B^{\ast\ast}_{1},\dots,B^{\ast\ast}_{k}$ is missing all the
balls that intersect $G^{\prime}$, and rounds every other ball so that its
center is contained in some integer lattice. In this construction, each ball
$\mathcal{B}_{i}^{\ast}$, with $i\notin F$, gets translated by at most some
distance $\sqrt{d}\delta$. Since each for all $j\in\\{1,\ldots,\chi\\}$,
$\mu_{j}$ is $C$-Lipschitz, it follows that

$|\mu_{j}(\mathcal{B}^{\ast}_{i})-\mu_{j}(B^{\ast\ast}_{i})|\leq\mathrm{vol}(\mathcal{B}^{\ast}_{i})\,\sqrt{d}\,\delta\,C\leq
2^{\Theta(d)}\delta C$

Letting $B^{\ast\ast}=\bigcup_{i=1}^{k}B^{\ast\ast}_{i}$, we get that for all
$i\in\\{1,\dots,\chi\\}$,

$|\mu_{j}(\mathcal{B}^{\ast})-\mu_{j}(B^{\ast\ast})|\leq k2^{\Theta(d)}\delta
C$

Let ${\cal I}$ be the set of connected components of $[0,\Delta]^{d}\setminus
G^{\prime}$. We refer to the elements of ${\cal I}$ as _cells_. For each
$A\in{\cal I}$, we enumerate the set, ${\cal S}_{A}$, of all possible subsets
of at most $k$ unit balls with centers in $A\cap(\delta\cdot\mathbb{Z}^{d})$.
There are at most $(L/\delta)^{d}$ lattice points in $A$, and thus there are
at most $2^{(L/\delta)^{d}}$ such subsets of unit balls. Since $|{\cal
I}|\leq(\lceil\Delta/L\rceil)^{d}$, it follows that this enumeration takes
$O((\Delta/L)^{d}\,2^{(L/\delta)^{d}})$ time.

For each enumerated subset $J\in{\cal S}_{A}$ of unit balls, we record the
vector

$\left(|J|,\frac{\varepsilon}{k}\left\lfloor\mu_{1}(X)\frac{k}{\varepsilon}\right\rfloor,\dots,\frac{\varepsilon}{k}\left\lfloor\mu_{k}(X)\frac{k}{\varepsilon}\right\rfloor\right)$

where $X=\bigcup_{Y\in J}Y$. There are at most $(2^{O(d)}k/\varepsilon)^{d}$
such vectors for each cell in ${\cal I}$. Via standard dynamic programming, we
can inductively compute all possible sums of vectors such that we pick at most
one vector from each cell, and the total sum of the first coordinate, i.e.,
the number of unit balls, is at most $k$. This can be done in
$O((\Delta/L)^{d}2^{(L/\delta)^{d}}(2^{O(d)}k/\varepsilon)^{d})$ time. For the
correct choice of vectors that corresponds to the solution $B^{\ast\ast}$, the
sum of the vectors we compute is correct up to an additive factor of
$\varepsilon$ on each coordinate. This means that we compute a solution
$B_{1},\ldots,B_{k}$, with the following property:

$\sum_{i=1}^{k}\mu_{i}(B)\geq(1-\varepsilon)\sum_{i=1}^{\chi}\mu_{i}(B^{\ast\ast})\geq(1-\varepsilon)\Big{[}\sum_{i=1}^{\chi}\big{(}\,\mu_{i}(\mathcal{B}^{\ast})-2^{\Theta(d)}\delta
C-\mu_{i}(\mathcal{B}^{\ast,F})\,\big{)}\Big{]}\\\
\geq\big{(}\,1-\varepsilon-(8d/L)\,\big{)}\Big{[}\sum_{i=1}^{\chi}\big{(}\,\mu_{i}(\mathcal{B}^{\ast})-2^{\Theta(d)}\delta
C\,\big{)}\Big{]}=\big{(}\,1-2\varepsilon\,\big{)}\Big{[}\sum_{i=1}^{\chi}\big{(}\,\mu_{i}(\mathcal{B}^{\ast})-\varepsilon\,\big{)}\Big{]}$

with probability at least $\nicefrac{{1}}{{2}}$. Repeating the algorithm
$O(\log n)$ times and returning the best solution found, results in the high-
probability assertion, which concludes the proof.

##  13 Conclusion and open problems

In this paper we formulated a natural combinatorial optimization framework for
incorporating fairness issues in coverage problems and provided a set of
approximation algorithms for the general version of the problem as well as its
special cases. Of course, it is possible to design other optimization
frameworks depending on the particular application in hand, and we encourage
researchers to do that. Below we list some future research questions related
to our framework:

Eliminating the gap of factor $f$ in ${\mathsf{LP}}$-relaxation:

As noted in Section 9.7, all of our ${\mathsf{LP}}$-relaxations incur a gap of
factor $f$ in the coloring constraints while rounding. It seems non-trivial to
close the gap using additional linear inequalities while preserving the same
approximation ratio. However, it may be possible to improve the gap using
${\mathsf{SDP}}$-relaxations.

Primal-dual schema:

Another line of attack for the Fmc problems is via the primal-dual approach
V01 . For example, can the primal-dual approach for partial coverage problem
by Gandhi, Khuller and Srinivasan GKS04  be extended to Fmc? A key technical
obstacle seems to center around effective interpretation of the dual of the
coloring constraints. Our iterated rounding approach was able to go around
this obstacle but the case when $\chi=\omega(1)$ may be improvable.

Fixed parameter tractability:

As mentioned in Section 4 fixed-parameter tractability issues for $k$-node
coverage have been investigated by prior researchers such as Marx M08  and
Gupta, Lee and Li in GLL18a ; GLL18b . It would be interesting to extend these
results to Node-fmc.

## References

  * (1) A. Ageev and M. Sviridenko, _Pipage Rounding: A New Method of Constructing Algorithms with Proven Performance Guarantee_ , Journal of Combinatorial Optimization, 8(3), 307-328, 2004.
  * (2) J. Angwin and T. Parris, _Facebook Lets Advertisers Exclude Users by Race_ , Propublica, October 28, 2016 (www.propublica.org/article/facebook-lets-advertisers-exclude-users-by-race).
  * (3) N. Apollonio and B. Simeone, _The maximum vertex coverage problem on bipartite graphs_ , Discrete Applied Mathematics, 165, 37-48, 2014.
  * (4) S. Arora. _Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems_ , Journal of the ACM, 45(5), 753-782, 1998.
  * (5) A. Asudeh, Z. Jin and H. V. Jagadish, _Assessing and remedying coverage for a given dataset_ , $35^{\mathrm{th}}$ annual IEEE International Conference on Data Engineering, 554-565, 2019.
  * (6) P. Austrin, S. Khot and M. Safra, _Inapproximability of vertex cover and independent set in bounded degree graphs_ , Theory of Computing, 7(1), 27-43, 2011.
  * (7) P. Austrin and A. Stankovic, _Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder_ , in Achlioptas D. and Végh L. A. (eds),, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, Leibniz International Proceedings in Informatics, 145, 24:1-24:17, 2019.
  * (8) N. Bansal, _Constructive Algorithms for Discrepancy Minimization_ , $51^{\mathrm{st}}$ Annual IEEE Symposium on Foundations of Computer Science, 3-10, 2010.
  * (9) J. Beck and T. Fiala, _“Integer-making” theorems_ , Discrete Applied Mathematics, 3(1), 1-8, 1981.
  * (10) S. Bera, D. Chakrabarty, N. Flores and M. Negahbani, _Fair Algorithms for Clustering_ , in Advances in Neural Information Processing Systems, Wallach H., Larochelle H., Beygelzimer A., d'Alché-Buc F., Fox E. and Garnett R. (eds), 32, 4954-4965, 2019.
  * (11) T. Carnes and D. Shmoys, _Primal-Dual Schema for Capacitated Covering Problems_ , in Lodi A., Panconesi A. and Rinaldi G. (eds), Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, 5035, Springer, 288-302, 2008.
  * (12) R. D. Carr, L. K. Fleischer, V. J. Leung and C. A. Phillips, _Strengthening Integrality Gaps for Capacitated Network Design and Covering Problems_ , $11^{\mathrm{th}}$ Annual ACM-SIAM Symposium on Discrete Algorithms, 106-115, 2000.
  * (13) B. Chazelle, _The Discrepancy Method: Randomness and Complexity_ , Cambridge University Press, 2000.
  * (14) C. Chekuri, J. Vondrák and R. Zenklusen, _Dependent Randomized Rounding via Exchange Properties of Combinatorial Structures_ , $51^{\mathrm{st}}$ Annual IEEE Symposium on Foundations of Computer Science, 575-584, 2010.
  * (15) F. Chierichetti, R. Kumar, S. Lattanzi and S. Vassilvitskii, _Fair Clustering Through Fairlets_ , in Advances in Neural Information Processing Systems, Guyon I., Luxburg U. V., Bengio S., Wallach H., Fergus R., Vishwanathan S. and Garnett R. (eds), 30, 5029-5037, 2017.
  * (16) B. Doerr, _Discrepancy in different numbers of colors_ , Discrete Mathematics, 250, 63-70, 2002.
  * (17) B. Doerr and A. Srivastav, _Approximation of multi-color discrepancy_ , in Hochbaum D., Jansen K., Rolim J. D. P. and Sinclair A. (eds), Randomization, Approximation and Combinatorial Optimization, Lecture Notes in Computer Science, 1671, Springer, 39-50, 1999.
  * (18) U. Feige, _A Threshold of $\ln n$ for Approximating Set Cover_, Journal of the ACM, 45(4), 634-652, 1998.
  * (19) U. Feige and M. Langberg, _Approximation algorithms for maximization problems arising in graph partitioning_ , Journal of Algorithms, 41, 174-211, 2001.
  * (20) R. Gandhi, S. Khuller and A. Srinivasan, _Approximation algorithms for partial covering problems_ , Journal of Algorithms, 53(1), 55-84, 2004.
  * (21) M. R. Garey and D. S. Johnson, Computers and Intractability - A Guide to the Theory of NP-Completeness (W. H. Freeman & Co., San Francisco, CA, 1979).
  * (22) J. Guo, R. Niedermeier and S. Wernicke, _Parameterized complexity of vertex cover variants_ , Theory of Computing Systems, 41(3), 501-520, 2007.
  * (23) A. Gupta, E. Lee and J. Li, _An FPT algorithm beating 2 approximation for k-cut_ , $29^{\mathrm{th}}$ Annual ACM-SIAM Symposium on Discrete Algorithms, 2821-2837, 2018.
  * (24) A. Gupta, E. Lee and J. Li, _Faster exact and approximate algorithms for k-cut_ , $59^{\mathrm{th}}$ Annual IEEE Symposium on Foundations of Computer Science, 113-123, 2018.
  * (25) C. Guse, _Citi Bike neglects poor NYC neighborhoods and communities of color: report_ , New York Daily News, July 10, 2019 (https://bit.ly/3aEB4rz)
  * (26) Q. Han, Y. Ye, H. Zhang and J. Zhang, _On approximation of max-vertex-cover_ , European Journal of Operational Research, 143, 342-355, 2002.
  * (27) M. Hardt, E. Price and N. Srebro, _Equality of Opportunity in Supervised Learning_ , $30^{\mathrm{th}}$ International Conference on Neural Information Processing Systems, 3323-3331, 2016.
  * (28) D. S. Hochbaum, Approximating Covering and Packing Problems: Set Cover, Vertex Cover, Independent Set, and Related Problems, in Hochbaum, D. S. (ed.), Approximation Algorithms for NP-Hard Problems, 94-143, (PWS Publishing Company, 1997).
  * (29) A. Hunter, _Job ads show sexism still prevalent in most industries_ , HRD, February 22, 2019 (bit.ly/38eom11).
  * (30) D. Ingold and S. Soper, _Amazon Doesn’t Consider the Race of Its Customers. Should It?_ , Bloomberg, April 21, 2016.
  * (31) K. Jain, _A Factor 2 Approximation Algorithm for the Generalized Steiner Network Problem_ , Combinatorica, 21, 39-60, 2001
  * (32) T. Jan, _Redlining was banned 50 years ago. It’s still hurting minorities today_ , Washington Post, March 28, 2018.
  * (33) M. Langberg, _Approximation algorithms for maximization problems arising in graph partitioning_ , M. Sc. Thesis, Weizmann Institute of Science, 1998.
  * (34) L. C. Lau, R. Ravi and M. Singh, _Iterative Methods in Combinatorial Optimization_ , Cambridge University Press, 2011.
  * (35) J. Lee and C. Lubienski, _The impact of school closures on equity of access in Chicago_ , Education and Urban Society, 49(1), 53-80, 2017.
  * (36) A. Levy, H. Ramadas and T. Rothvoss, _Deterministic Discrepancy Minimization via the Multiplicative Weight Update Method_ , in Eisenbrand F., Koenemann J. (eds), Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, 10328, 380-391, Springer, 2017.
  * (37) P. Manurangsi, _A Note on Max k-Vertex Cover: Faster FPT-AS, Smaller Approximate Kernel and Improved Approximation_ $2^{\mathrm{nd}}$ Symposium on Simplicity in Algorithms, 15:1-15:21, 2019.
  * (38) D. Marx, _Parameterized complexity and approximation algorithms_ , The Computer Journal, 51(1), 60-78, 2008.
  * (39) J. Mitchell. _Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric tsp, k-mst, and related problems_ , SIAM Journal on Computing, 28(4), 1298-1309, 1999.
  * (40) R. Motwani and P. Raghavan, Randomized Algorithms (Cambridge University Press, 1995).
  * (41) M. Mulshine, _A major flaw in Google’s algorithm allegedly tagged two black people’s faces with the word ’gorillas’_ , Business Insider, July 1, 2015.
  * (42) N. Nisan, T. Roughgarden, E. Tardos and V. V. Vazirani, _Algorithmic Game Theory_ , Cambridge University Press, 2007.
  * (43) A. Olteanu, C. Castillo, F. Diaz and E. Kiciman, _Social data: Biases, methodological pitfalls, and ethical boundaries_ , Frontiers in Big Data, 2, pp. 13, 2019.
  * (44) A. Panconesi and A. Srinivasan, _Randomized Distributed Edge Coloring via an Extension of the Chernoff–Hoeffding Bounds_ , SIAM Journal of Computing, 26(2), 350-368, 1997.
  * (45) E. Petrank, _The hardness of approximation: Gap location_ , Computational Complexity, 4, 133-157, 1994.
  * (46) M. Simon, _HP looking into claim webcams can’t see black people_ , CNN, December 23, 2009.
  * (47) J. Spencer, _Six standard deviations suffice_ , Transactions of American Mathematical Society, 289, 679-706, 1985.
  * (48) S. Spencer, _Amazon to Bring Same-Day Delivery to Roxbury After Outcry_ , Bloomberg, April 26, 2016.
  * (49) A. Srinivasan, Distributions on level-sets with applications to approximation algorithms, $42^{\mathrm{nd}}$ Annual IEEE Symposium on Foundations of Computer Science, 588-597, 2001.
  * (50) A. Srivastav, _Multicolour Discrepancies_ , Combinatorics, Probability and Computing, 12, 365-399, 2003.
  * (51) V. Vazirani, Approximation Algorithms, Springer, 2001.
  * (52) A. Yan and B. Howe, _Fairness in Practice: A Survey on Equity in Urban Mobility_ , Data Engineering Bulletin, 42(4), 49, 2019.
  * (53) R. Zemel, Y. Wu, K. Swersky, T. Pitassi and C. Dwork, _Learning Fair Representations_ , $30^{\mathrm{th}}$ International Conference on Machine Learning, PMLR 28(3), 325-333, 2013.
  * (54) B. H. Zhang, B. Lemoine and M. Mitchell, _Mitigating Unwanted Biases with Adversarial Learning_ , AAAI/ACM Conference on AI, Ethics, and Society, 335-340, 2018.