# Construction and testing of sMDT tubes at the University of Michigan for the
ATLAS Muon Spectrometer upgrade

C. Wei  A. Chen  D. Amidei  N. Anderson  E. Carpenter  L. Cooperrider  T. Dai
E. Diehl  C. Ferretti  J. Li  P. Lugato  J. Minnella  S. Moskaitis  K. Nelson
V. Pillsbury  E. Salzer  L. Simpson  Z.C. Wang  Z.R. Wang  C. Weaverdyck  Z.
Yang  B. Zhou

###### Abstract

This paper reports on the design and construction of infrastructure and test
stations for small-diameter monitored drift tube (sMDT) assembly and testing
at the University of Michigan (UM) to prepare for the ATLAS Muon Spectrometer
upgrade for the high-luminosity program of the Large Hadron Collider.
Procedures of the tube assembly and quality assurance and control (QA/QC)
tests are described in detail. More than 99% of the tubes meet the tube QA/QC
specifications based on 2100 tubes built at UM. The UM test stations are also
used for QA/QC testing on the tubes constructed at Michigan State University.
These tubes are being used to construct the sMDT chambers which will replace
the current MDT chambers of the barrel inner station of the Muon Spectrometer.

##  1 Introduction

The ATLAS Muon Spectrometer will receive a major upgrade in the next long
shutdown (LS3) of the Large Hadron Collider (LHC) to cope with the the much
higher interaction rate of the High-Luminosity LHC (HL-LHC) [1]. The central
(barrel) region inner station of the Monitored Drift Tube (MDT) chambers will
be replaced by new chambers using small-diameter MDT (sMDT) tubes of 15 mm
diameter, half that of the previous MDT. The smaller tube radius reduces both
the maximum drift time (from 750 ns down to 180 ns) and the tube cross-section
which lowers the tube occupancy by a factor 8 for improved performance in the
HL-LHC. Furthermore the smaller volume of the sMDT detector makes space for
the installation of a new trigger tRPC (thin resistive plate chamber)
detectors to improve the Level-1 muon trigger efficiency in the HL-LHC
environment.

A total of 96 new sMDT chambers will be built to replace the Muon Spectrometer
BIS1 to BIS6 MDT chambers, where BIS stands for Barrel-Inner-Small (sectors).
Half of these will be constructed at the Max Plank Institute for Physics in
Munich (MPI), Germany, and the other half in the US at the University of
Michigan (UM) and at Michigan State University (MSU). The sMDT drift tube
design and parameters are described in detail in [2]. Table 1 lists the major
tube parameters.

Parameter | sMDT
---|---
Tube material | Aluminium AW6060-T6/AlMgSi
Tube surface | Surtec 650 chromatization
Tube outer diameter | 15.000 mm
Tube wall thickness | 0.4 mm
Tube length | 1615 mm
Tube straightness | 0.5 mm / tube
Wire material | W-Re (97:3)
Wire diameter | $\rm 50\,\mu m$
Wire resistance | 44 $\Omega$/m
Wire tension | 350 ${}\pm 15\;$ g
Gas mixture | Ar:CO2 (93:7)
Gas pressure | 3 bar (abs.)
Gas leak rate limit |  $<~{}1\times 10^{-8}~{}\frac{mbar~{}\times~{}cm^{3}}{sec}$ per tube
Gas gain | $\rm 2\times 10^{4}$
Wire potential | 2730 V
Table 1: sMDT tube materials and operating parameters

The endplugs (Figure 2) consist of a precision brass wire locator, called a
twister, inside a central brass insert (both manufactured by Poschl111Pöschl
Präzisionsteile GmbH, https://www.poeschl-gmbh.eu/home.html, Industriestr. 10,
82110 Germering Germany, Tel. 089/894454-0, Fax 089/89445444) which is molded
into a PBTP plastic body manufactured by the Institute for High-Energy Physics
(IHEP) at the Kurchatov Institute in Protvino, Russia. IHEP also makes the
crimp tubes and stoppers (which lock the locators in the endplugs). Two
grooves on the plastic outer surface hold o-rings which make a gas tight seal
between the endplug and the tube inner wall.

(a)

(b)

Figure 2: (a) Tube design, with an exploded view of the end-plug components
[2]. (b) The components and assembly of an endplug. All dimensions are in mm.

26000 precision tubes are needed for the Michigan chamber production. MSU is
responsible for tube assembly and initial Quality Assurance (QA) tests after
which tubes are delivered to UM. Tubes are tested again at UM to verify all
specifications are met before chamber assembly.

Early in the R&D period for sMDT construction tooling and the production cycle
UM also built and tested 2600 tubes, 500 for R&D in 2018 and 2105 for chamber
mass production during the summer of 2021. With three students working 8 hours
per day, a production rate of 50 tubes/day was reached consistently in summer
2021, including all the QA tests on the constructed tubes.

This document describes the infrastructure and tooling used in building tubes
at UM, and the methods used to assess the quality of all tubes used in sMDT
chamber production at Michigan.

##  2 Infrastructure

The ATLAS “tube room“ is a $12\times 8~{}m^{2}$ lab space dedicated to drift
tube assembly and QA testing. It is organized into multiple work stations for
the construction and testing of sMDT tubes.

(a)

(b)

Figure 3: (a) Tube construction stations: wiring and crimping (middle),
endplug swaging (right), and the QA station for tube length and wire tension
measurements (left). (b) Tube QA dark current test station to test up to 48
individual tubes in parallel using a CAEN HV system.

Figure 3(a) and (b) show the tube construction station and the dark current
test station in the tube room, respectively. Each station is set up on a
$3.66\times 1.22~{}m^{2}$ flat optical table. Once a tube is assembled, its
straightness, gas tightness, length, wire tension, and dark current are
measured. In addition, a large storage structure accommodating up to 3,000
tubes was built to handle the logistics for chamber mass production. Table 2
lists the UM tube room work stations. These stations are arranged so that all
of them can be operated at the same time, optimizing the tube construction and
test efficiencies.

Station name in the tube room | Purpose | Tubes per station
---|---|---
Wire stringing and tensioning | construction | single
Swaging endplug onto tube | construction | single
Length and tension measurement | QA | single
Straightness measurement | QA | single
Gas leak at 3 barA measurement | QA | single
Dark current measurement | QA | 48
Reverse HV treatment | QA | 48
Table 2: UM tube room stations.

Lab cleaning is performed twice a day to maintain a pristine environment.
Personal protection equipment (lab-coat, shoe covers, hair nets, gloves and
disposable masks) are easily accessible immediately upon entry and are
required at all times, with the exception of the gloves which are used only
when handling tubes.

##  3 Tube Construction

The tube construction infrastructure and test stations were designed and built
at UM during the R&D phase (2018-2020) of the ATLAS muon detector upgrade
project. Using raw tubes from the manufacturer Mifa Alumninum222Mifa
Aluminium, Rijnaakkade 6, 5928 PT Venlo, E, The Netherlands:
https://mifa.eu/en<EMAIL_ADDRESS>T: +31 77 389 88 88, F: +31 77 389 89 89 500
drift tubes were constructed and tested to commission and verify the
infrastructure and tooling. These tubes were used to build an sMDT prototype
chamber at UM to demonstrate the tube construction and test procedures.
Michigan sMDT chamber mass production started in May 2021. Knowledge and
technologies for tube production and testing were shared with MSU for tube
mass production. One batch of 2105 sMDT tubes was built at the UM site in the
summer of 2021 to supplement the tube construction effort at MSU to assure
that the early sMDT chamber production schedule at UM could be met.

The tube construction process involves three main steps: preparation of raw
aluminum tubes; tube assembly; and quality assurance tests of each constructed
tube. This section will focus on the construction process carried out at the
UM site.

###  3.1 Tube straightness measurement

Mifa supplied all raw sMDT tubes for both MPI and Michigan, delivered in
crates of 702 tubes each. Michigan’s tubes were sent by truck to CERN and then
by air freight to Michigan. All incoming tubes were inspected at CERN and at
UM before tube construction.

One requirement is the tube straightness: the specification calls for no more
than 0.5 mm deviation when a tube is resting on a flat surface. However,
nearly $20\%$ of early tube shipments exceeded this specification. To measure
tube straightness a system with an optical technique using a microscope was
developed. The system consists of a straight V-shape bar that has a flat
surface (flatness < 0.05 mm) with straight backstop to keep the center-line of
the tube at the focal point of the microscope. The microscope looks at the the
center bottom of the tube where it should touch the flat surface. The tube is
rotated in the V bar to allow measurement of the maximum distance between the
tube and the surface. The microscope has a built-in screen which is set to 20X
magnification (see picture on Figure 4 (a)).

(a)

(b)

(c)

Figure 4: Straightness test station (a) and results of straightness
measurements, the fraction of tubes accepted depending on a bending value, of
raw tubes delivered in 2020 (b) and 2021 (c).

This technique assumes that the greatest out-of- straightness deviation
(refereed to as sagitta) will be at the center along the tube length. Plots in
Figure 4 (b) and (c) show the fraction of tubes accepted in two lots as a
function of the cut used to reject bent tubes. About 87% of the tubes met the
specification of the tube straightness requirement. Based on these
measurements it was decided to relax the original cutoff specification and to
only reject tubes with more than 0.8 mm sagitta, about 6% of those delivered
in 2020 and 2021. The rejected tubes (slightly over 1,000) were sent back to
the manufacturer for straightening treatment and returned to Michigan at the
end of 2021. Three identical optical measurement devices were built at UM and
sent to the tube production company (MIFA in Netherlands), CERN, and MSU. This
test is now a routine check of the tube mass production. After using the UM
measurement technique the company adopted a stricter QA/QC protocol which
reduced the fraction of bent tubes delivered to about 2%.

###  3.2 Aluminum tube and endplug preparation

Although tubes are cleaned at the factory, it was found that additional
cleaning reduced the dark current drawn by tubes. In the initial stage of R&D
tube production $\sim$12% of tubes exceeded the dark current limit. To reduce
this high dark current failure rate the 2105 aluminum tubes used for summer
2021 tube production at UM were each re-cleaned by swabbing the interior with
isopropyl alcohol (IPA) soaked wipes prior to the tube construction. The
interior of each tube was swept three times with fresh lint-free wipes wrapped
around a small cylindrical piece of foam and soaked in IPA for each sweep.
This cleaning plug fit snugly inside the tube and was pushed through the tube
with a clean acrylic rod to ensure the entire length of the interior was wiped
clean. The exterior was also cleaned with IPA-soaked lint-free wipes. The
cleaned tubes were stacked on foam saddles from the original packaging and the
ends were lightly covered with lint-free wipes to allow the IPA to evaporate
completely. After all tubes were cleaned, the two ends of each tube were
temporarily closed with clean rubber caps to keep the interior free of dust.
The thoroughly cleaned tubes had a failure rate of only $\sim$3% in the dark
current test, significantly lower than that observed in early tube tests. In
addition, it was found that negative HV treatment, described in Section 4.4.3,
could recover any tubes failing the initial dark current test.

The endplug is assembled with two o-rings, a twister, and a stopper (see
Figure 2 (b)). Individual parts were given a 15 minute ultrasonic bath in IPA
before assembly and a further 5 minute ultrasonic bath after assembly. After
air drying the endplug assemblies were stored in clean plastic packaging until
used.

###  3.3 Tube assembly

This section describes the tube construction stations for tube wiring and
swaging, and the tube assembly procedures.

####  3.3.1 Wiring station

The tooling for wire stringing consists of two vacuum chucks that hold the
tube rigidly in a straight line and aligned with the wire feed. Each chuck is
fitted with a pneumatically controlled end-plate on which is mounted a pair of
cam-operated crimp jaws (Figure 6a) which squeeze the 1.0 mm O.D. crimp pin
down to 0.7 mm. The spool of wire333Luma Metall Fine Wire, Kalmar, Sweden is
mounted on a magnetically damped spindle 444Warner Electric Precision Tork
MC2, 1-20 oz-in torque to allow smooth pulling. At the other end is a three
wheel tension meter 555Electromatic Equipment Co., Tension Sensor model
TE-500-22 and a linear actuator 666Parker Linear Motion Drive Model No.
ETS32-802QA21-GK200-A for stretching the wire to the desired tension (Figure
6b).

(a)

(b)

Figure 6: (a) Wire crimp jaw. (b) The wire pulling actuator and tension meter
used to stretch the wire before crimping.

####  3.3.2 Swaging station

The second station used in tube construction is the swaging station (Figure
7), where the thin walls of the tube are deformed (swaged) to compress and
seal against the o-rings on the endplugs. The swaging head is a custom built
rotary device that gradually forces 3 roller heads radially inward as they are
rotated around the tube (Figure 7(a)). The head, designed at MPI and built at
MSU, is driven by an Isel rotary motor 777Isel Model Drehachse ZD 30, Belt
drive with stepper motor, Model No. ETS32-802QA21-GK200-A. Figure 7(b) shows
the side view of tube swaging set up, including a structure to hold the tube
when inserting it into a rotational head.

(a) Swaging rotational head

(b) Side view of swaging set up

Figure 7: The rotary swaging head for creating indents around the tube at the
o-ring locations

####  3.3.3 Assembly procedure

The tube construction process starts by placing an aluminum tube on the vacuum
chuck to hold the tube securely. The tube interior is vacuumed to remove any
residual dust. The $\rm 50\,\mu m$ tungsten-rhenium wire is secured to a
shuttle which is carefully pulled through the tube using a gentle vacuum. The
wire is drawn over pulleys which prevent the wire from contacting the tube
wall. A magnetic brake controls the un-spooling wire to prevent tangles while
wiring the tube. The wire is then cut, leaving 0.5 m excess wire at each end.
The wire ends are threaded through endplugs at each end of the tube and then
the endplugs are inserted into the tube ends until seated. Next, crimp pins
are threaded onto each wire end and inserted into the endplugs. One end of the
wire is then secured to a stationary clamp. The other end is fed through the
tension monitor, stretched by hand to around 200 g and then secured to another
clamp attached to the actuator (Figure 6b). The fixed-end crimp pin (soft
copper, 1 mm OD) is crimped (pinched by jaws to 0.7 mm, see Figure 6a).

An online control program based on LabView [3] moves the actuator and monitors
the wire tension while the wire is gradually pulled until the tension reaches
400 g. This tension is held for 30 seconds, and then the tension is slowly
reduced to 325 g. Note that later in the process the tube swaging will
increase the wire tension to 350 g. The second crimp pin is pushed with
tweezers into the endplug until it touches the brass core to insure the
tension is not altered by the movement of the crimp pin. The crimped thickness
of each crimp pin is measured to ensure that it is no more than 0.71 mm which
is the value needed to hold the wire securely. Excess wire protruding from the
crimp-pins is cut off leaving a stub of about 2 mm.

After the tube is wired, the wire tension is measured in the tension test
station (details in Section 4.3) to verify the tension value as set. At this
point, a tube with an insufficient wire tension can easily be disassembled and
all components, except the wire and crimp-pins, can be salvaged for another
construction attempt.

Each end of the wired tube is then mechanically deformed in the swaging
station to lock the endplugs in place and produce a tight o-ring seal between
the plastic endplug and the interior wall of the tube. This is done with the
rotating set of three rounded rollers that create smooth indents around the
circumference of the tube at each of the two o-ring locations (Figure 7(a)).
The innermost indent is made such that the width of the tube in the groove
measures 13.7 mm or less. This swaging process introduces an additional 25 g
of tension in the wire due to the slight stretching of the tube. After swaging
a final tension test is done to confirm that the wire inside the constructed
tube has a tension of $350\pm 20$ g. Once the two endplugs are swaged the tube
construction is done and a bar code sticker is put on the tube with a unique
serial number for identification.

##  4 Quality Control

Both individual parts and assembled tubes must pass a number of quality
checks. Random checks of the endcap brass precision brass surface are done on
each batch received to ensure the diameter is within the specification of
5.000${}^{+0.00}_{-0.01}$ mm. Tube roundness and diameter are verified with
the precision endplugs to be well within tolerances. The straightness was
discussed in section 3.1.

The QA tests on all constructed tubes, whether built at MSU or UM, include
components visual inspection, tube straightness, leak tightness, assembled
tube length, wire tension, and dark current. The detailed procedures and test
results are described in the following sections.

###  4.1 Tube visual inspection

Constructed tubes are first given an visual inspection to check for defects
such as dents or cracks in the tube wall, badly swaged endplugs, or severely
bent crimp pins. Bent crimp pins are carefully straightened. Tubes with
unrecoverable problems are rejected and flagged in the database.

###  4.2 Tube leak tightness test

The constructed tube is required to be leak tight with an upper limit of $\rm
1\times 10^{-8}cm^{3}\times mbar/s$ with the tube at 3 bar (absolute)
pressure. As shown in Figure 8(a),

(a)

(b)

(c)

Figure 8: (a) Set up of the tube leak test station. (b) Interior end of the
tube is sealed with this brass piece. (c) Feed-through end of the vacuum
vessel with a tube connected. Most of the tube is inside the vacuum vessel.
The tube is pressurized with Helium to 3 barA.

the leak rate test is performed with a helium leak detector connected to a
vacuum vessel 25 mm ID x 1.8 m long. One end of the tube being tested is
sealed with a brass cap and an o-ring (as shown in Figure 8(b), the other is
screwed into the cap of the vacuum vessel, which allows the tube to be
pressurized with 3 bar Helium (see Figure 8(c)). O-ring seals are lubricated
with IPA to prevent twisted o-rings. IPA is used as a lubricant since it
completely evaporates and leaves no residue that might contaminate the drift
gas. The tube is inserted into the vessel, the end flange is sealed, and the
leak detector set to pump out the vessel. When the background He level reads
about $\rm 1\times 10^{-6}cm^{3}\times mbar/s$ the tube is filled with Helium
gas to a pressure of 3 barA. Leaks can be found within 30 seconds of pump-out.
When this test is completed the Helium is vented through a snorkel to the
outdoors.

The goal of this test step is to establish the tightness of the seal. No data
was stored for this test, only pass/fail. All the tubes built at UM passed the
ATLAS gas tightness requirement.

###  4.3 Tube length and wire tension measurements

The tube length and the wire tension are measured using the station shown in
Figure 9. A tube is placed in the V-shaped aluminum bar with its midpoint
inside a U-shape magnet, and a linear digital scale 888Mitutoyo ABSOLUTE
Digimatic Scale Units Series 572, https://www.mitutoyo.com/webfoo/wp-
content/uploads/E316-572R_ABS_Digi_ScaleUnits.pdf is used for the length
measurement. Before any session of measurements, the digital linear gauge
(precise to ${}\pm 10\;\mu$m) is calibrated with an aluminum rod of length
$1624.5\pm 0.1$ mm. Then a tube is placed into a V-shaped bar with the two
ends resting in 3D printed saddles (in blue in Figure 9) so that one end can
be pressed against a stopper. The operator makes sure that a good electrical
contact is made at both extremities (tube grounded and wire connected the
electrical circuit).

Figure 9: Test Station for tube length and wire tension measurements.

The digital scale head is brought into contact with the endplug of the tube
and the length of the (fully assembled) tube is recorded. The measured length
distribution of the tubes built at UM is shown in Figure 10. A maximum
difference in tube length slightly larger than 1 mm is observed. None of the
tubes built at UM were rejected due the length being out of specification.

Figure 10: Distribution of the measured length of the tubes built at UM.

Wire tension is then measured using the same set up with the following
procedure. The aluminum tube is tied to the ground on the V-shaped bar and the
wire is connected to the custom multiplexing circuit shown in Figure 11. The
circuit consists of an analog multiplexer, signal generator, instrumentation
amplifier, and the sMDT itself. The role of the multiplexing circuit is to
trigger wire vibrations in the sMDT by directing pulses from the signal
generator to the sMDT and then to send the signal from the discharging sMDT to
the instrumentation amplifier. The instrumentation amplifier is connected to a
data acquisition (DAQ) system which consists of an multipurpose I/O device (NI
USB-6008999National Instruments USB-6008 Multifunction I/O Device,
https://www.ni.com/en-us/support/model.usb-6008.html) interfaced to a
computer. A software program with a graphic user interface was developed to
allow setting of wire tension measurement parameters, such as the measured
tube length, and to perform real time analysis and measurements.

Figure 11: Circuit for wire tension test.

After the external pulse is stopped the wire oscillates at its natural
frequencies (see eq. 4.1 from [4]):

$f_{n}=\frac{n}{2L}\sqrt{\frac{T}{\rho}},$ (4.1)

that converted to the tension expressed in grams is $T=\pi
L^{2}d^{2}f_{1}^{2}\rho/g$. With the nominal sMDT wire parameters $\rm
L=1597\;mm$, $\rm d=50\;\mu m$, $\rm\rho=19.7\;g/cm^{3}$ and $\rm
g=9.81\;m/s^{2}$, the first resonant frequency of 93.3 Hz corresponds to a
tension of 350 g. The frequency is measured with a custom LabView program
which uses a fast Fourier transform (FFT) to find the first resonant frequency
and thus the wire tension.

In order to have a standard definition of tension and compare the results
measured at UM with those measured by MSU during the tube construction, the
value quoted uses the nominal (fixed) length of 1597 mm, so that only the
frequency ($f_{1}$) is a measured parameter.

A second tension measurement is made at least two weeks after the first
measurement to check for any significant drop in tension. If a drop larger
than 18 g in 2 weeks is found, the tube is rejected.

The results of the wire tension measurements for tubes made at UM are shown in
Figure 12. A total of 2 tubes out of the 2105 built (0.09%) did not pass the
tension tests.

(a)

(b)

Figure 12: (a) Tube wire tension, and (b) tension drop after at least 2 weeks.

###  4.4 Dark current measurement

The tube, at 3 atmospheres absolute pressure of the working gas mixture,
Ar:CO2 (93:7), operated at 2800 volts and must draw less than 2 nA of dark
current. The high voltage (HV) test set up (shown in Fig. 13) and procedure
are described in this section.

Figure 13: Test Stand with 48 tubes with the CAEN SY5527 mainframe, with the
CAEN GECO monitoring program running in the connected computer.

####  4.4.1 HV test station

The HV test station at UM holds 4 groups of 12 tubes. Tubes are inserted into
4 sets of gas-manifold blocks to pressurize them. The gas-manifold blocks,
machined from acytal plastic (polyoxymethylene POM), allow 12 tubes to be
plugged in with a simple push of a handle to make the tube mounting process
simple and quick. The 5 mm cylindrical precision brass part of the endplugs of
the 12 tubes are inserted into thick o-rings which are built inside the gas-
manifold to make a gas-tight seal. The design drawing of the set up is shown
in Figure 14.

Figure 14: Tube mounting diagram for dark current test station

The input gas pressure is regulated to 3 barA and fanned out to the 4 gas
distribution blocks, each with its own input and output shutoff valves. The
output of the 4 blocks is fanned into a single flow meter and a bubbler.
Initially the flow is set high (3 SCFH) for an hour to flush at least 3
volumes through the tubes. It is then turned down to a few bubbles per second
(flowing around one gas volume per day) for testing.

####  4.4.2 Power supply system

The power supply system consists of a CAEN SY5527 101010Universal Multi-
channel Power Supply System with 4 Channel 3.5 kV, 1.5/0.15 mA (4W) Common
Ground Dual Range Boards, https://www.caen.it/products/sy5527/ mainframe which
hosts two 24-channel AG7326 HV modules, able to deliver up to 3.5 kV per
channel with 500 pA resolution for current measurements, with GECO (GEneral
COntrol) monitoring and data acquisition software. Two places along the tubes
are held rigidly to make good contact with a grounding strip. One of the two
end manifolds is fitted with spring-loaded contacts for applying high voltage
to the wire. The spring contacts are hard-wired to HV cables plugged into 48
individual channels of two modules of the CAEN HV power supply. For a correct
dark current measurement, the pedestal current level for each channel must be
subtracted from the current measurement. The pedestals are found by turning on
the high voltage with no tubes connected and measuring the average current on
each channel for a period of one hour. The pedestals are stable except if the
humidity becomes too high and must be re-calibrated if the relative humidity
exceeds 50%.

####  4.4.3 Dark current measurement

To measure the dark current the tubes are placed in the gas manifolds and the
voltage is ramped up to +2,900 V and dark current readings are are recorded
for at least 4 hours. If the average current over the last hour of the test is
below 2 nA, the tube passes the test. Some tubes show dark current above the
limit in the first hours after the HV was turned on, very often with a
decreasing trend as shown in Figure 15 (a), but sometimes the dark current
increases in time leading to a tube failing the test (see Figure 15 (b)). Dark
current are retested for all tubes received from MSU to insure they are fully
burned-in before being used in chamber construction.

(a)

(b)

Figure 15: (a) Example of a tube recovered from initial high dark current, and
(b) one that instead developed higher and higher dark current in time. Note
the different scales.

####  4.4.4 Tube burn-in with negative HV

Tubes that failed the dark current test are treated with negative HV (-3 kV
for 30–60 minutes) on a separate test station, and then reassessed via the
standard HV test. If the tube fails again, then treatment with negative HV is
repeated. All tubes with high dark current have responded to this treatment so
that not a single tube had to be rejected in the dark current test and
treatment procedure.

In the 2105 tubes built at UM in the summer of 2021, a total of 67 tubes
(3.2%) were initially measured to have dark current above 2 nA. All of them
were recovered with either longer HV burn-in time or via the negative HV
treatment. Figure 16 (a) and (b) show the measured dark current distributions
of the UM built tubes, and the tube burn-in time distributions, respectively.

(a)

(b)

Figure 16: (a) Distribution of the measured final tube dark current, and (b)
the burn-in time (right), where the three spikes corresponds to the most
common test lengths: day time (just a few hours), overnight (typically about
15-16 hours) and over the weekend (normally around 64 hours).

##  5 Tube Production Database

All the information gathered in tube production and testing is recorded into a
local database (DB) for production QA/QC monitoring, and a subset of the data
is uploaded to a master chamber production DB encompassing both US and German
chambers. UM receives the QA measurements for the tubes delivered by MSU, used
in part as a first data-point in the history of the tubes tests at UM, and the
UM results are shared back with MSU for cross-checks and validation. All the
test results from the UM test stations are saved in text files, with a format
that can be read by different databases (MS Access, MySQL, Oracle, Jason,
etc). The most natural choice is a relational database, since all information
to be saved is relative to many single tubes, each one with a history of
production and test results. In the tube DBs the barcode serial number is used
as the primary key for retrieving information. The barcode is also the link
between the entries of the tubes layout in a sMDT chamber and the associated
test results.

ROOT [5], an analysis framework used commonly in high energy physics, is
chosen locally to store, retrieve, and quickly plot the distributions of the
test results. A few C++ classes have been developed to read the station
measurement results and convert them into a set of text files for each tube.
At the end of the process all the test conditions and data relative to a tube
are saved into a "TTree" of the ROOT file with an entry for each tube. To ease
access to tube test data, separate "TTrees" have been created for each
station. In addition, there is a "master" tree with all the tube test
information.

While filling the "TTree", a set of standard plots for each test is
automatically saved and a local web page is automatically created to show the
results history of all tests.

Eventually, all the production and test information are uploaded to the CERN
sMDT production database which records information for all of the detectors to
be installed in the ATLAS experiment.

##  6 Conclusions

Fully functional stations for sMDT tube production and testing were designed,
built, and successfully used at the University of Michigan for the ATLAS HL-
LHC muon detector upgrade project. A total of 2105 tubes were built at UM and
tested for sMDT chamber production. The rejection rate of the UM constructed
tubes is below 0.1% from the QA tests. In addition, over 14,000 tubes built at
MSU have been fully tested at UM.

Tubes passing all tests have been used to construct 30 sMDT chambers so far
out of the 50 of the whole production task. All the chambers meet the
stringent precision and performance requirements, which will be reported in a
separate paper.

## Acknowledgements

We would like to thank Hubert Kroha (MPI) and Reinhard Schwienhorst (MSU) and
their team members for their close collaborations and valuable technical
discussions; Toni Baroncelli (USTC) and Massimo Corradi (INFN Roma) for
coordinating the contract with MIFA for aluminum tube production and delivery;
and Rinat Fakhrutdinov (IHEP, Protvino) for producing the endplugs and
associated components. We also thank the UM faculty, Shawn McKee, Tom Schwarz,
Junjie Zhu, and Jianming Qian, for their strong support; the UM electrical
engineers, Yuxiang Guo, Xiangting Meng, and Jinhong Wang, and former graduate
student Christopher Grud, for their help in the initial R&D phase to design
and build the required electronics; and the UM undergraduate students, Tyler
Coates, Greg Kondas, Kathryn Ream, and Noah West, for their great technical
assistance.

This work was supported in part by the U.S. Department of Energy grant DE-
SC0012704 and in part from the U.S. National Science Foundation PHY -1948993.

## References

  * [1] ATLAS collaboration, T. A. Collaboration, _Technical Design Report for the Phase-II Upgrade of the ATLAS Muon Spectrometer_ , CERN-LHCC-2017-017, https://cds.cern.ch/record/2285580, CERN, 2017.
  * [2] H. Kroha, R. Fakhrutdinov and A. Kozhin, _New High-Precision Drift-Tube Detectors for the ATLAS Muon Spectrometer_ , _JINST_ 12 (2017) 12.
  * [3] R. Bitter, T. Mohiuddin and M. Nawrocki, _LabVIEW: Advanced programming techniques_.  Crc Press, 2006.
  * [4] J. K.Lang and V. Vassilakopoulos, _A technique of direct tension measurement of a strung fine wire_ , _Nucl. Instr. and Meth. A 420_ (1999) 392.
  * [5] R. Brun and F. Rademakers, _Root - an object oriented data analysis framework_ , _Nucl. Inst. & Meth. in Phys. Res. A 389 (1997) 81-86_ (1996) .


The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg,
Germany The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761
Hamburg, Germany The Hamburg Center for Ultrafast Imaging, Luruper Chaussee
149, 22761 Hamburg, Germany Center for Computational Quantum Physics, Flatiron
Institute, 162 5th Avenue, New York, NY 10010, USA Nano-Bio Spectroscopy
Group, University of the Basque Country (UPV/EHU), 20018 San Sebastián, Spain

# Cavity-Born-Oppenheimer Hartree-Fock Ansatz: Light-matter Properties of
Strongly Coupled Molecular Ensembles

Thomas Schnappinger  Department of Physics, Stockholm University, AlbaNova
University Center, SE-106 91 Stockholm, Sweden  Dominik Sidler  Max Planck
Institute for the Structure and Dynamics of Matter and Center for Free-
Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany  Michael
Ruggenthaler  Max Planck Institute for the Structure and Dynamics of Matter
and Center for Free-Electron Laser Science, Luruper Chaussee 149, 22761
Hamburg, Germany  Angel Rubio  Max Planck Institute for the Structure and
Dynamics of Matter and Center for Free-Electron Laser Science, Luruper
Chaussee 149, 22761 Hamburg, Germany  Markus Kowalewski
<EMAIL_ADDRESS>Department of Physics, Stockholm University,
AlbaNova University Center, SE-106 91 Stockholm, Sweden

###### Abstract

Experimental studies indicate that optical cavities can affect chemical
reactions, through either vibrational or electronic strong coupling and the
quantized cavity modes. However, the current understanding of the interplay
between molecules and confined light modes is incomplete. Accurate theoretical
models, that take into account inter-molecular interactions to describe
ensembles, are therefore essential to understand the mechanisms governing
polaritonic chemistry. We present an ab-initio Hartree-Fock ansatz in the
framework of the cavity Born-Oppenheimer approximation and study molecules
strongly interacting with an optical cavity. This ansatz provides a non-
perturbative, self-consistent description of strongly coupled molecular
ensembles taking into account the cavity-mediated dipole self-energy
contributions. To demonstrate the capability of the cavity Born-Oppenheimer
Hartree-Fock ansatz, we study the collective effects in ensembles of strongly
coupled diatomic hydrogen fluoride molecules. Our results highlight the
importance of the cavity-mediated inter-molecular dipole-dipole interactions,
which lead to energetic changes of individual molecules in the coupled
ensemble.

The strong coupling of light and matter within optical cavities provides an
innovative way to alter and design matter properties, making it a rapidly
evolving field 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 at the intersection of chemistry,
quantum optics, and materials science. In polaritonic chemistry, depending on
whether the quantized cavity modes are coupled via their characteristic
frequencies to electronic or vibrational degrees of freedom of molecules, the
situation is described as electronic-strong coupling (ESC) or vibrational-
strong coupling (VSC), respectively. Under ESC, it becomes possible to modify
the photochemistry/photophysics of molecules including charge transfer
processes and electronic spectroscopy, and photoinduced reactions can be
influenced 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28. Similarly, for VSC, the vibrational spectra of molecules are altered by
the formation of light-matter hybrid states, and even the chemical reactivity
of the ground state can be modified 29, 30, 31, 32, 33, 34, 35, 36, 34, 27.
The observed effects of molecular ESC and VSC are often discussed in a
phenomenological way, and understanding of the underlying microscopic and
macroscopic physical mechanisms, especially with respect to the effects of
VSC, is still incomplete 37, 38, 39, 6, 40, 41. In our recent work 42 we have
shown numerically that the interaction between an optical cavity and ensembles
of molecules not only leads to cavity detuning and a change of the optical
length, but also allows for a local molecular polarization mechanism under
strong collective vibrational coupling in the thermodynamic limit. The
interplay of microscopic and macroscopic polarization is due to a cavity-
mediated dipole-dipole interaction. A deep understanding of this effect may
bridge the gap between existing simplified models for a macroscopic ensemble
of molecules and experiments. We have been able to study this cavity-mediated
dipole-dipole interaction for very large ensembles by using simple Shin-Metiu
molecules. As a next step to go beyond this simple molecular model, we present
here a formulation of the well-known Hartree-Fock ansatz in the context of the
cavity Born-Oppenheimer approximation (CBOA) 43, 44, 45 derived from the
complete non-relativistic Pauli-Fierz Hamiltonian 46, 2, 47, 48. We refer to
the resulting method as cavity Born-Oppenheimer Hartree-Fock (CBO-HF) approach
and, to the best of our knowledge, this is the first wave function-based
method to describe strong coupling of real molecules in a cavity in the CBOA
framework. The CBO-HF method allows us to study cavity-mediated dipole-dipole
interactions for realistic molecular systems.

The first part of this work describes the CBO-HF formalism. Next, the CBO-HF
approach is used to explore the effects of collective cavity-mediated coupling
in small ensembles of diatomic hydrogen fluoride (HF) molecules. By explicitly
simulating the molecular systems in a self-consistent calculation, we are able
to study the interactions within the ensemble beyond what can be captured by
scaled single-molecule model Hamiltonians. In the following we study static
ensembles of molecules but do not address the effects of vibrational
resonances 49, 50, 48. Consistent with recently reported results 51, 42, we
observe non-negligible cavity-induced energy changes at the local molecular
level. By a detailed analysis of these energy changes at the microscopic
(single-molecule) and macroscopic (molecular ensemble) level, we can show how
the size of the ensemble, the individual molecular orientation, and the change
in nuclear configuration can affect these collective interactions.

The physics of a cavity-embedded molecular system is described using the non-
relativistic Pauli–Fierz Hamiltonian 46, 2, 47, 48, which is represented in
the length gauge, assuming the dipole approximation and the CBOA 43, 44, 45.
Within the CBOA, the cavity modes and nuclei are considered to be ”slow”
degrees of freedom compared to the ”fast” electrons and, consequently, only
the electrons are treated quantum mechanically. In the following bold symbols
denote vectors and atomic units ($\hbar=4\pi\varepsilon_{0}=m_{e}=1$) are used
throughout the paper, unless otherwise indicated. For a single-mode cavity,
the CBOA Hamiltonian takes the form

$\begin{split}\hat{H}_{CBO}&=\hat{H}_{el}+\frac{1}{2}\omega_{c}^{2}q_{c}^{2}-\omega_{c}q_{c}\left(\bm{\lambda}_{c}\cdot\bm{\hat{\mu}}\right)+\frac{1}{2}\left(\bm{\lambda}_{c}\cdot\bm{\hat{\mu}}\right)^{2}\,,\end{split}$
(1)

where

$\begin{split}\quad\bm{\hat{\mu}}=\bm{\hat{\mu}}_{el}+\bm{\mu}_{Nuc}=-\sum_{i=1}^{N_{el}}\bm{\hat{r}}_{i}+\sum_{A=1}^{N_{Nuc}}Z_{A}\bm{R}_{A}\,,\end{split}$
(2)

represents the molecular dipole operator, which is defined by the operators of
the $N_{el}$ electron coordinates $\bm{\hat{r}}$, the classic coordinates
$\bm{R}$ of the $N_{Nuc}$ nuclei and the nuclear charge $Z$. $\hat{H}_{el}$ is
the Hamiltonian for the field-free many-electron system, and the second term
defines the harmonic potential introduced by the cavity-mediated displacement
field, with the photon displacement coordinate $q_{c}$ and $\omega_{c}$ being
the frequency of the cavity mode. The third term of Eq. 1 describes the dipole
coupling between the molecular system and the photon displacement field, which
is characterized by the coupling strength $\bm{\lambda}_{c}$. The last term is
the dipole self-energy (DSE) operator 52, 6, which is an energy contribution
that describes the self-polarization of the molecule-cavity system. Note that
the inclusion of the DSE contribution is strictly necessary to obtain a finite
polarization and a bounded solution 53. Since in practice a finite basis, such
as Gaussian basis sets used in most of quantum chemistry methods, also limits
the polarization, the lack of a stable ground state is not always observed in
numerical calculations. In the following, we will show that the DSE term is
not only formally needed, but yields inter-molecular interactions in an
ensemble of molecules. The coupling parameter $\bm{\lambda}_{c}$ for a cavity
with an effective mode volume $V_{c}$ is defined as follows:

$\bm{\lambda}_{c}=\bm{e}\lambda_{c}=\bm{e}\sqrt{\frac{4\pi}{V_{c}}}\,.$ (3)

The unit vector $\bm{e}$ denotes the polarization axis of the cavity mode. In
the context of a Fabry-Pérot type cavity, $\bm{\lambda}_{c}$ can be directly
related to the electric vacuum field strength $\epsilon_{c}$ via
$\bm{\lambda}_{c}=\bm{e}\sqrt{\frac{2}{\omega_{c}}}\epsilon_{c}$. Without loss
of generality, we will use this relation to quantify $\lambda_{c}$ by
$\epsilon_{c}$.

As in the standard Hartree-Fock approach, the CBO-HF electronic wave function
of an $N_{el}$-electron molecular system is a Slater determinant of mutually
orthonormal spin orbitals $\varphi_{i}$ 54:

$\Psi(\bm{\tau}_{1},\bm{\tau}_{2},\ldots,\bm{\tau}_{N})=\frac{1}{\sqrt{N_{el}!}}|\varphi_{1},\varphi_{2},\dots\varphi_{i}\rangle\,.$
(4)

Here $\bm{\tau}_{N}$ is used to denote the complete set of coordinates
associated with the $N$-th electron, comprised of the spatial coordinate
$\bm{r}_{N}$ and a spin coordinate. Note that the $N_{el}$-electron system
described by $\Psi$ can be a single molecule or an ensemble of many molecules.
Thus, it is possible to treat cavity-induced interactions and standard Coulomb
interactions between molecules in the ensemble in the same way. For the
special case of the dilute gas limit, that is, the situation in which the
electronic structures of different molecules do not overlap and interact, the
ensemble Slater determinant may be replaced by a product 42 of individual
molecular Slater determinants. Note that the displacement coordinate $q_{c}$
of the electric field mode is treated as a parameter in CBO-HF ansatz, analog
to the nuclear coordinates, and is thus not part of the wave function.

Using $\hat{H}_{CBO}$ and $\Psi$, the CBO-HF energy expectation value
$E_{CBO}$ can be determined using the standard self-consistent field (SCF)
procedure 54:

$\bigl{\langle}E_{CBO}\bigr{\rangle}=\bigl{\langle}\Psi\big{|}\hat{H}_{el}-\omega_{c}q_{c}\left(\bm{\lambda}_{c}\cdot\bm{\hat{\mu}}\right)+\frac{1}{2}\left(\bm{\lambda}_{c}\cdot\bm{\hat{\mu}}\right)^{2}\big{|}\Psi\bigr{\rangle}+\frac{1}{2}\omega_{c}^{2}q_{c}^{2}.$
(5)

The resulting energy expectation $E_{CBO}$ consists of four energy
contributions:

$E_{CBO}=E_{el}+E_{lin}+E_{dse}+E_{dis}\quad\text{ with}\quad
E_{dis}=\frac{1}{2}\omega_{c}^{2}q_{c}^{2}\,.$ (6)

The detailed derivation of all new energy contributions in the CBO-HF ansatz
is given in section S1 of the supporting information. The following discussion
of these energies is based on Hartree-Fock matrix elements formulated in the
basis of orthonormal spin orbitals. The first term $E_{el}$ contains all
Hartree-Fock energy components of the many-electron system 54 and is only
indirectly affected by the cavity via the SCF procedure. The second term
$E_{lin}$ describes the linear part of the light-matter coupling and is
obtained from the dipole coupling between the photon displacement field, the
electrons, and the nuclei. It can be written as a sum of one-electron
integrals formulated in terms of spin orbitals $\varphi_{i}$ and a parametric
nuclear contribution:

$E_{lin}=-\omega_{c}q_{c}\bigl{\langle}\Psi\big{|}\bm{\lambda}_{c}\cdot\bm{\hat{\mu}}\big{|}\Psi\bigr{\rangle}=-\omega_{c}q_{c}\sum_{i=1}^{N_{oc}}\bigl{\langle}\varphi_{i}\big{|}\hat{x}\big{|}\varphi_{i}\bigr{\rangle}-\omega_{c}q_{c}\left(\bm{\lambda}_{c}\cdot\bm{\mu}_{Nuc}\right)\
\text{ with}\ \hat{x}=-\bm{\lambda}_{c}\cdot\bm{\hat{r}}\,.$ (7)

The remaining component $E_{dse}$ can be decomposed into a purely electronic
term, a mixed electron-nuclear term, and a pure nuclear contribution:

$\begin{split}E_{dse}&=\frac{1}{2}\bigl{\langle}\Psi\big{|}\left(\bm{\lambda}_{c}\cdot\bm{\hat{\mu}}\right)^{2}\big{|}\Psi\bigr{\rangle}=E^{(el)}_{dse}+E^{(e\text{-}n)}_{dse}+E^{(nuc)}_{dse}\\\
&=\frac{1}{2}\bigl{\langle}\Psi\big{|}\left(\bm{\lambda}_{c}\cdot\bm{\hat{\mu}}_{el}\right)^{2}\big{|}\Psi\bigr{\rangle}+\left(\bm{\lambda}_{c}\cdot\bm{\mu}_{Nuc}\right)\bigl{\langle}\Psi\big{|}\left(\bm{\lambda}_{c}\cdot\bm{\hat{\mu}}_{el}\right)\big{|}\Psi\bigr{\rangle}+\frac{1}{2}\left(\bm{\lambda}_{c}\cdot\bm{\mu}_{Nuc}\right)^{2}\,.\end{split}$
(8)

To simplify the electronic structure calculations in this work, the classical
nuclei are arranged so that their contributions to the total dipole moment are
zero ($\bm{\mu}_{Nuc}=0$, center of charge). Thus, the nuclear contribution in
Eq. 7 as well as $E^{(e\text{-}n)}_{dse}$ and $E^{(nuc)}_{dse}$ are zero by
definition. More details on the latter two can be found in section S1 of the
Supporting Information. The pure electronic contribution $E^{(el)}_{dse}$,
contains a squared electronic position operator
$\hat{x}^{2}=\hat{x}_{i}\hat{x}_{j}$, which is a two-electron operator but can
be decomposed into one-electron contributions and a two-electron contributions
(for details see Eq. S7 in the supporting information) 55, 56, 57, 58:

$E^{(el)}_{dse}=E^{(1e)}_{dse}+E^{(2e)}_{dse}=\frac{1}{2}\sum_{i=1}^{N_{oc}}\bigl{\langle}\varphi_{i}\big{|}\hat{x}^{2}\big{|}\varphi_{i}\bigr{\rangle}+\frac{1}{2}\left[\sum_{i,j}^{N_{oc}}\bigl{\langle}\varphi_{i}\big{|}\hat{x}\big{|}\varphi_{i}\bigr{\rangle}\bigl{\langle}\varphi_{j}\big{|}\hat{x}\big{|}\varphi_{j}\bigr{\rangle}-\left|\bigl{\langle}\varphi_{i}\big{|}\hat{x}\big{|}\varphi_{j}\bigr{\rangle}\right|^{2}\right]\,.$
(9)

The one-electron $E^{(1e)}_{dse}$ term retains the quadratic nature of the
$\hat{x}$ operator and behaves like scaled quadrupole tensor elements and
describes a localized energy contribution. The expansion of the two-electron
part, $E^{(2e)}_{dse}$ in terms of spin orbitals, follows a logic similar to
that of the derivation of the Coulomb and exchange integrals in the regular
Hartree-Fock ansatz. The terms here can be factorized and further decomposed
into a dipole-dipole interaction component $E^{(2J)}_{dse}$ and an exchange-
like component $E^{(2K)}_{dse}$. This exchange-like quantity $E^{(2K)}_{dse}$
vanishes if $\varphi_{i}$ and $\varphi_{j}$ have no spatial overlap and
therefore describes a localized interaction:

$E^{(2K)}_{dse}=-\sum_{i,j}^{N_{oc}}\left|\bigl{\langle}\varphi_{i}\big{|}\hat{x}\big{|}\varphi_{j}\bigr{\rangle}\right|^{2}$
(10)

The $E^{(2J)}_{dse}$ part can be rewritten as a product of the scaled
electronic dipole moment:

$E^{(2J)}_{dse}=\sum_{i,j}^{N_{oc}}\bigl{\langle}\varphi_{i}\big{|}\hat{x}\big{|}\varphi_{i}\bigr{\rangle}\bigl{\langle}\varphi_{j}\big{|}\hat{x}\big{|}\varphi_{j}\bigr{\rangle}=\left(\bm{\lambda}_{c}\cdot\bigl{\langle}\bm{\hat{\mu}}_{el}\bigr{\rangle}\right)\left(\bm{\lambda}_{c}\cdot\bigl{\langle}\bm{\hat{\mu}}_{el}\bigr{\rangle}\right).$
(11)

Unlike $E^{(2K)}_{dse}$, the dipole-dipole interaction $E^{(2J)}_{dse}$ does
not require a spatial overlap of the spin orbitals $\varphi_{i}$ and
$\varphi_{j}$ and thus results in a delocalized interaction. These two
properties are of special interest when ensembles of molecules are described.
The last energy contribution $E_{dis}$ in Eq. 6 is the energy resulting from
the photon displacement field 52.

By replacing the electronic dipole moment $\bm{\hat{\mu}}_{el}$ of the total
ensemble with the sum over the individual molecular dipole moments in Eq. 11
we obtain:

$E^{(2J)}_{dse}=\sum_{m=1}^{N_{mol}}\left[\left(\bm{\lambda}_{c}\cdot\bigl{\langle}\bm{\hat{\mu}}_{el}^{(m)}\bigr{\rangle}\right)\left(\bm{\lambda}_{c}\cdot\bigl{\langle}\bm{\hat{\mu}}_{el}^{(m)}\bigr{\rangle}\right)+\sum_{n\neq
m}^{N_{mol}}\left(\bm{\lambda}_{c}\cdot\bigl{\langle}\bm{\hat{\mu}}_{el}^{(m)}\bigr{\rangle}\right)\left(\bm{\lambda}_{c}\cdot\bigl{\langle}\bm{\hat{\mu}}_{el}^{(n)}\bigr{\rangle}\right)\right]\,,$
(12)

where the summations run over all molecules $N_{mol}$ in the ensemble. The
first term is the local or intra-molecular contribution to $E^{(2J)}_{dse}$
for each individual molecule $m$, while the second product describes the
interaction of the molecule $m$ with all other molecules in the ensemble. This
inter-molecular interaction depends only on the orientation and size of the
individual dipole moments, but not on their distance. This molecular dipole-
dipole interaction term $E^{(2J)}_{dse}$ in combination with $E^{(nuc)}_{dse}$
is commonly used as a first-order approximation 59, 60, 61 of the DSE energy.
Note that the full $E_{dse}$ can also be approximated with the help of
permanent dipole moments and transition dipole moments in a resolution of
identity approach by summing over excited-electronic states 62, 22, 27.

By solving Eq. 5 with a SCF approach, $E_{CBO}$ is minimized for a given
configuration of classic nuclei and a fixed photon displacement coordinate
(parametric photon field). The ground state for the combined electronic-
photonic subsystem is obtained by minimizing $E_{CBO}$ with respect to the
photon displacement coordinate, which leads to the following expression:

$\frac{\partial}{\partial
q_{c}}E_{CBO}=\omega_{c}^{2}q_{c}-\omega_{c}\bigl{\langle}\Psi\big{|}\bm{\lambda}_{c}\cdot\bm{\hat{\mu}}\big{|}\Psi\bigr{\rangle}=\omega_{c}^{2}q_{c}-\omega_{c}\left(\bm{\lambda}_{c}\cdot\bigl{\langle}\bm{\hat{\mu}}\bigr{\rangle}\right)\stackrel{{\scriptstyle\text{!}}}{{=}}0\,.$
(13)

The resulting minimum of $E_{CBO}$ is:

$q_{c}=q_{min}=\frac{\left(\bm{\lambda}_{c}\cdot\bigl{\langle}\bm{\hat{\mu}}\bigr{\rangle}\right)}{\omega_{c}}\,.$
(14)

Because we work in the length gauge, the electric field $\bm{\mathcal{E}}$ 52
is:

$\displaystyle\tfrac{1}{4\pi}\bm{\mathcal{E}}=\bm{\mathcal{D}}-\bm{\mathcal{P}}=\tfrac{1}{4\pi}\bm{\lambda}_{c}\omega_{c}q_{c}-\tfrac{1}{4\pi}\bm{\lambda}_{c}\left(\bm{\lambda}_{c}\cdot\bigl{\langle}\bm{\hat{\mu}}\bigr{\rangle}\right)=0\,,$
(15)

where $\bm{\mathcal{D}}$ is the cavity-mediated displacement field and
$\bm{\mathcal{P}}$ is the polarization. Requiring that the transverse electric
field vanishes in the ground state, $\bm{\mathcal{E}}=0$ also leads for Eq.
14. This demonstrates that minimizing $E_{CBO}$ with respect to $q_{c}$ is
equivalent to fulfilling the zero transverse electric field condition 52, 6 in
the semi-classical limit, which guarantees a non-radiating ground state. From
Eq. (15) we also find that
$E_{dis}+E_{lin}+E_{dse}=\tfrac{1}{8\pi}\int_{V_{c}}\hat{\bm{\mathcal{E}}}^{2}\mathrm{d}\,{\bm{r}}$,
which is in agreement with Maxwell’s equations 52. Thus, by making the CBOA,
we discard the magnetic contribution to the photonic energy 63

The main aim of this work is to study an ensemble of well-separated molecules
interacting with the cavity field. We thus assume that the wave functions of
different molecules in the ensemble do not overlap and that the Coulomb
interaction between them is negligible. By satisfying the zero transversal
electric field condition (Eq. 14) for such a molecular ensemble, the photon
displacement coordinate $q_{min}$ becomes a function of the total ensemble
dipole moment. Thus, $E_{lin}$ and $E_{dis}$ and $E_{dse}$ are not exclusively
dependent on local molecular properties, but rather on the total ensemble.

The CBO-HF method has been implemented in the Psi4NumPy environment 64, which
is an extension of the PSI4 65 electronic structure package. All calculations
were performed using the aug-cc-pVDZ basis set 66 and the geometry of the
isolated single HF molecule has been optimized at the Hartree-Fock level of
theory. Note that we have not re-optimized the geometries of the molecular
systems in the cavity; as such, our calculations do not account for any
geometry relaxation effects originating from the presence of the cavity. In
all CBO-HF calculations performed in this work, we consider a single mode,
lossless cavity. We keep the collective coupling strength $\bm{\lambda}_{c}$
constant by applying a scaling factor of scaling factor of
${1}/{\sqrt{N_{mol}}}$ to obtain a fixed Rabi splitting for different ensemble
sizes and treat $\lambda_{0}$ as a tunable coupling parameter:

$\bm{\lambda}_{c}=\frac{\lambda_{0}}{\sqrt{N_{mol}}}\bm{e}$ (16)

Here $\lambda_{0}$ is equivalent to $\lambda_{c}$ in Eq. 3 in the single
molecule case. As a result, we increase the mode volume $V_{c}$ of the cavity,
but by including more molecules, we keep the average density of molecules
$N_{mol}/V_{c}$ fixed. For $N_{mol}\gg 1$, we would be approaching the
thermodynamic limit. Since the main goal of this work is to demonstrate the
CBO-HF ansatz and to understand how the different energy contributions to the
cavity-mediated interaction under VSC behave when increasing the size of the
ensemble but keeping $N_{mol}/V_{c}$ fixed, it is enough to simulate small
ensembles. In this work, we restrict ourselves to up to eight HF molecules. We
use an artificially increased coupling strength $\lambda_{0}$ in the range of
0.004 to 0.04, which corresponds to effective mode volumes (Eq. 3) in the
single-molecule case as large as $125.27\text{\,}{\mathrm{nm}}^{3}$ (for
$\lambda_{0}$ 0.004) or as small as $1.25\text{\,}{\mathrm{nm}}^{3}$ (for
$\lambda_{0}$ 0.04). To refer to a more intuitive physical quantity, the
unscaled coupling strength $\lambda_{0}$ is quantified in this work by the
vacuum electric field strength $\epsilon_{c}$. The fundamental cavity
frequency $\omega_{c}$ is set to be identical with the first vibrational mode
of the uncoupled HF molecule, which is $4467\text{\,}{\mathrm{cm}}^{-1}$ for
the chosen level of theory. This value of $\omega_{c}$ is chosen arbitrarily
in our static setup, since resonance effects are not likely to play a role in
our analysis. All calculations were performed in a reproducible environment
using the Nix package manager together with NixOS-QChem 67 (commit f5dad404)
and Nixpkgs (nixpkgs, 22.11, commit 594ef126).

In this work, we study fixed ensembles of perfectly aligned, well-separated HF
molecules in an optical cavity. We chose HF because of its large permanent
dipole moment of $1.8\text{\,}\mathrm{\text{D}}$, which is advantageous for
the interaction between the molecule and the cavity mode, but its
polarizability is small with $0.8\text{\,}{\mathrm{\text{Å}}}^{3}$. 68
Furthermore, the HF simulation results directly extend our previous results in
Ref. 42, whereas the molecular setup connects to earlier ab-initio studies on
(collective) electronic strong coupling 51. To define these ensembles, the
optimized structure of a single HF molecule is replicated $N_{mol}$ times. All
these replicas are separated by $800\text{\,}\mathrm{\text{Å}}$ and placed
inside the cavity to avoid interactions via longitudinal electric fields. In
general, three different orientations of the molecular HF ensembles are
studied in this work and are visualized in Fig. 1. In the first orientation,
called all-parallel, the HF molecules are aligned parallel to the cavity mode
polarization axis. In the antiparallel case, the $N_{mol}$ HF molecules are
pairwise antiparallel, resulting in the ensemble dipole moment being zero
(even number of molecules) or equal to $\bm{{\mu}}$ of a single HF (odd
$N_{mol}$). The third configuration of the ensemble, labeled defective,
represents the situation where the dipole moments of $N_{mol}-1$ HF molecules
point in the opposite direction to the remaining molecule. Note that for all
three ensemble configurations, the individual dipole moment vectors are
aligned with the cavity polarization axis, and the zero transverse electric
field condition (Eq. 14) is satisfied for the entire ensemble. This aligned
orientation is not the energetically most favorable configuration for the
individual molecule (see the supporting information section S2 for a detailed
discussion of the single-molecule situation), but it allows us to set an upper
bound on all effects, as it guarantees the maximum molecular cavity
interaction.

Figure 1: Sketch of the all-parallel (p) orientation, the antiparallel (a)
configuration, and the defective (d) orientation shown for an ensemble of five
HF molecules as an example. All molecules are separated by a distance of
$800\text{\,}\mathrm{\text{Å}}$. The scan along the bond length is performed
for the HF molecule highlighted in red and the cavity polarization axis is
shown in blue. Note that except for the defective case, the choice of the
highlighted molecule is arbitrary.

All calculations are carried out with rescaled values of $\bm{\lambda}_{c}$,
see Eq. 16. Analogous calculations without rescaling can be found in the
supporting information (see Figs. S5 and S6 in section S3). The energy change
of the all-parallel ensembles induced by the interaction with the cavity, as
well as the underlying energy components, is visualized in Fig. 2 as a
function of the size of the ensemble $N_{mol}$.

Figure 2: Influence of the cavity interaction on the collective energy of
different ensemble sizes and vacuum-field strengths $\epsilon_{c}$ of HF
molecules in the all-parallel configuration. a) The energy difference $\Delta
E$ between $E_{CBO}$ and the field-free energy $E_{HF}$, b) the potential
$E_{dis}$ introduced by the cavity-mediated displacement field, c) the
contribution of the linear dipole coupling $E_{lin}$, and d) the DSE
contribution $E_{dse}$ for optimized $q_{min}$ as a function of $N_{mol}$.
Individual dipole moments are aligned with the cavity polarization axis, and a
cavity frequency $\omega_{c}$ of $4467\text{\,}{\mathrm{cm}}^{-1}$ is used.
The strength of the cavity field $\epsilon_{c}$ increases from
$0.5\text{\,}\mathrm{V}\text{\,}{\mathrm{nm}}^{-1}$ to
$2.0\text{\,}\mathrm{V}\text{\,}{\mathrm{nm}}^{-1}$ (color-coded). The
coupling strength $\bm{\lambda}_{c}$ used is rescaled according to Eq. 16.

Let us first consider how the different total (ensemble) energies behave and
what we can learn from them. For simplicity, we focus on the all-parallel
configuration. In Fig. 2 a) we see that the proposed rescaling with an
increasing ensemble size of the coupling, i.e., the mode volume $V_{c}\propto
N_{mol}$, keeps the light and light-matter interaction energy constant. That
is, on the total energy level we see that the thermodynamic limiting procedure
is well-behaved, and we expect that approximately also for $N_{mol}\gg 1$ we
find such an energy difference. From a total energy contribution perspective,
one might be tempted to conclude that the photon and photon-matter interaction
contribution can be safely ignored, since $E_{el}$ increases linearly with
$N_{mol}$ and hence dominates. If we, however, in a next step consider the
different contributions of Eq. (6), we see that even for the total-ensemble
energy, a delicate balancing of macroscopically scaling energy contributions
appears. Indeed, in Fig. 2 b) we see that the energy of the displacement field
increases linearly even if we rescale the coupling strength. This
approximately linearly increasing term is countered by $E_{lin}$. That
$E_{lin}$ contributes negatively is simple to understand, since the
displacement field (interpreted as a constant external field) allows to lower
the total energy by separating particles of different charges. Without
$E_{dse}$ depicted in Fig. 2 d), we would find the well-known result that the
linear interaction would dissociate and ionize any bound system regardless of
the coupling strength 53, 52. We can thus conclude that in order to describe
an ensemble of molecules form first principles the dipole self-energy term
$E_{dse}$ is needed to find a stable and physical result.

So far, we have discussed the effect of the collective coupling on the total
molecular ensemble. However, the main question in polaritonic chemistry is to
understand how a collectively coupled ensemble can influence individual
molecules. In the next step, we will thus analyze the energy changes at the
level of a single molecule, which arise as a result of the collective
interactions of the entire ensemble. Such a local perspective is possible
since the ensemble CBO-HF density matrix is block diagonal, and individual
blocks can be used to create partial density matrices for each molecular
subsystem. These partial density matrices can be combined with the ensemble
Hamiltonian to calculate local energies (per molecule) and combine them
pairwise to calculate the interaction between the molecules (see Eq. 12). This
part of the DSE we denote inter $E_{dse}$, while all other contributions to
the DSE are summed and labeled local $E_{dse}$. The individual energy obtained
per molecule is equivalent to the eigenvalues of the cavity-Hartree equation
in our previous work 42. The change in individual molecular energy induced by
the interaction with the cavity, as well as the underlying energy components,
is visualized in Fig. 3 as a function of the size of the all-parallel ensemble
$N_{mol}$.

Figure 3: Influence of the cavity interaction on an individual HF molecule in
all-parallel ensembles of different sizes and vacuum field strengths
$\epsilon_{c}$. a) The energy difference $\Delta E$ between $E^{(1)}_{CBO}$
and the field-free energy $E^{(1)}_{HF}$, b) the local linear coupling
contribution $E_{lin}$, c) the local $E_{dse}$ and d) the inter $E_{dse}$ that
is part of $E^{(2J)}_{dse}$ (for definition, see Eq. 12) as a function of
$N_{mol}$. The individual dipole moments are aligned with the cavity
polarization axis, and a cavity frequency $\omega_{c}$ of
$4467\text{\,}{\mathrm{cm}}^{-1}$ is used. The strength of the cavity field
$\epsilon_{c}$ increases from
$0.5\text{\,}\mathrm{V}\text{\,}{\mathrm{nm}}^{-1}$ to
$2.0\text{\,}\mathrm{V}\text{\,}{\mathrm{nm}}^{-1}$ (color-coded). The
coupling strength $\bm{\lambda}_{c}$ used is rescaled according to Eq. 16.

The energy difference $\Delta E$ between the individual molecular energy
$E^{(1)}_{CBO}$ and $E^{(1)}_{HF}$ of the isolated HF molecule without cavity
interaction is shown in Fig. 3 a). Note that $E_{dis}$ is an ensemble
quantity, but since each molecule in the ensemble is affected by the same
potential, we include it in $E^{(1)}_{CBO}$. If a second molecule is included
in the cavity, the local $E^{(1)}_{CBO}$ decreases. As more and more HF
molecules are added, the initial trend reverses and $E^{(1)}_{CBO}$ increases
almost linearly, following the linear behavior of $E_{dis}$ shown in Fig. 2
b). Without $E_{dis}$, $E^{(1)}_{CBO}$ converges to a finite non-zero value
with $N_{mol}$ increasing, as can be seen in Fig. S4 a) in the supporting
information. The local dipole cavity interaction $E_{lin}$ converges to a
constant value with increasing $N_{mol}$, as shown in Fig. 3 b). This behavior
is a direct consequence of fulfilling the zero-field condition for the entire
all-parallel ensemble, and cannot be generalized to every nuclear
configuration. For this specific orientation, the ensemble dipole moment
$\bm{\mu}$ increases linearly with the number of molecules, and thus the
displacement induced by the cavity leads to higher values of $q_{min}$. This
effect, in combination with the rescaled coupling$\bm{\lambda}_{c}$ leads to
the constant value of $E_{lin}$ that depends only on the coupling strength. On
the contrary, the local $E_{dse}$, visualized in Fig. 3 c), decays with
$\nicefrac{{1}}{{N_{mol}}}$ and approaches zero in the large $N_{mol}$ limit.
The intermolecular dipole-dipole energy ( inter $E_{dse}$) shown in Fig. 3 d)
is part of the $E^{(2J)}_{dse}$ term and arises as a result of the cavity-
mediated interaction of the dipole moment of one molecule with all other
molecules in the ensemble (for the definition, see Eq. 12). This energy
contribution increases with an increasing number of molecules in the ensemble
and approaches a constant, non-zero value following the behavior of
$1-\nicefrac{{1}}{{N_{mol}}}$. All these results are clear indications that
the nontrivial interplay of the collective photon displacement effects
($E_{lin}$ in combination with $E_{dis}$) and the cavity-mediated dipole-
dipole interaction allow for local strong coupling to emerge.

In the following, we introduce a defect in perfectly aligned ensembles to
further study the effect of anisotropic ensembles on a single molecule. We
perform scans along the bond length of one HF molecule in fixed ensembles of
different sizes for all three configurations all-parallel, antiparallel and
defective. The scan along the bond length is performed for the HF molecule
encircled in red in Fig. 1. In the defective orientation, the fixed
$N_{mol}-1$ HF molecules point in the opposite direction to the perturbed
molecule. For the resulting two-dimensional cavity potential energy surfaces
(cPESs) spanned by the bond length coordinate and the photon displacement
coordinate, the minimum energy path along the bond length is determined. This
is equivalent to satisfying the zero transverse electric field condition (Eq.
14) for each nuclear configuration. The energy differences between these
minimum energy paths and the field-free one-dimensional potential energy
surface (PES) for all three orientations are visualized in Fig. 4. The
corresponding energy contributions $E_{lin}$, $E_{dis}$, the local $E_{dse}$
and the inter $E_{dse}$ are shown in Figs. S7, S8, and S9 in section S4 of the
supporting information.

Figure 4:  Cavity-induced energy change along the HF bond length for different
ensemble sizes and configurations. a) all-parallel configuration, b)
antiparallel configuration, and c) defective configuration, for definitions,
see Fig. 1. A cavity frequency $\omega_{c}$ of
$4467\text{\,}{\mathrm{cm}}^{-1}$ is used and the strength of the cavity field
$\epsilon_{c}$ is set to $1.5\text{\,}\mathrm{V}\text{\,}{\mathrm{nm}}^{-1}$.
The coupling strength $\bm{\lambda}_{c}$ used is rescaled according to Eq. 16
and the number of molecules in the ensemble is color-coded.

The observed change on the energetics of the dissociation path due to the
cavity interaction is small (see Fig. 4), and interestingly, the same for all
three ensemble configurations. However, the effect is not negligible and the
cavity interaction shifts the ensemble cPESs to higher energies compared to
the cavity-free PES. With increasing bond length, the ensemble dipole moment
becomes larger and consequently the upshift mediated by the cavity interaction
increases. When comparing the different sizes of the ensemble, there is a
decreasing effect on the change in the energy of the ensemble with increasing
$N_{mol}$. As a second effect, the energy change is less and less dependent on
the bond length. It seems to converge to a non-zero finite value, which is
constant with respect to the bond length. A closer look at the individual
contributions $E_{lin}$, $E_{dis}$, the local $E_{dse}$ and the inter
$E_{dse}$ all shown in Figs. S7, S8, and S9 in section S4 of the supporting
information can explain these trends. The general shape of the cavity-induced
energy change shown in Fig. 4 is determined by the local $E_{dse}$. With an
increasing number of molecules in the cavity, this contribution becomes
dominated by the fixed ensemble of $N_{mol}-1$ molecules and therefore
constant. The other three contributions, $E_{lin}$, $E_{dis}$, and the inter
$E_{dse}$ are quite large and depend on both the size of the ensemble and the
bond length. However, when they are summed, they almost completely cancel each
other out, leaving only an almost negligible energy contribution. Since the
local $E_{dse}$ is the same for all three orientations and dominates the
energy change, all three ensemble configurations show the same behavior. It
should be noted that by imposing the zero transverse electric field condition
along the complete dissociation path, we assume that the whole ensemble
coupled to the cavity is in the electronic-photonic ground state. The behavior
discussed above may be different if this assumption no longer holds, for
example, if the system is coupled to a thermal bath.

In the last part of this work, we focus on the local perspective of the
dissociating HF molecule. Its field-free PES and the change in the local
energy induced by the cavity interaction in the presence of different
ensembles are shown in Fig. 5. Additional figures can be found in section S4
of the supporting information, showing the underlying energy components:
$E_{lin}$ in Fig. S10, $E_{dis}$ in Fig. S11, the local $E_{dse}$ in Fig. S12,
and the inter $E_{dse}$ in Fig. S13. The last two are calculated from the
perspective of the dissociating molecule.

Figure 5: a) Field-free PES of a single dissociating HF molecule. Cavity-
induced energy change on the individual dissociating HF molecule in ensembles
of different size in b) the all-parallel configuration, c) antiparallel
configuration, and d) the defective configuration, for definitions see Fig. 1.
The energy change is calculated as the difference of the locale energy
$E_{CBO}^{(1)}$ and the field-free energy $E_{HF}^{(1)}$. A cavity frequency
$\omega_{c}$ of $4467\text{\,}{\mathrm{cm}}^{-1}$ is used and the strength of
the cavity field $\epsilon_{c}$ is set to
$1.5\text{\,}\mathrm{V}\text{\,}{\mathrm{nm}}^{-1}$. The coupling strength
$\bm{\lambda}_{c}$ used is rescaled according to Eq. 16 and the number of
molecules in the ensemble is color-coded.

The most striking difference from the perspective of the ensemble is that at
the local level of the dissociating HF molecule, the three configurations all-
parallel (Fig. 5 b)), antiparallel (Fig. 5 c)) and defective (Fig. 5 d)) can
be distinguished in cavity-induced energy changes. In all three situations,
the cavity-induced changes depend on $N_{mol}$ as well as on the length of the
bond. In the all-parallel case, shown in Fig. 5 b), the energy change
increases with the bond length as well as with the number of molecules. Note
that the coupling is rescaled by $\nicefrac{{1}}{{\sqrt{N_{mol}}}}$ to keep
the collective Rabi-splitting constant. However, the effect on the individual
molecules grows with the number of molecules in the ensemble, even though the
single molecule coupling strength $\lambda_{0}$ becomes smaller. We thus
conclude, that collective coupling induces locally strong effects, as has been
reported for excited states in ESC previously 51. The local contribution
$E_{lin}$, which converges to a finite value, is not large enough to
compensate for $E_{dis}$, which grows linearly with the size of the ensemble
(see Figs. S10 a) and S11 a)). The resulting upshift in energy is further
amplified since inter $E_{dse}$ is also positive due to the all-parallel
configuration and grows with $1-\nicefrac{{1}}{{N_{mol}}}$ (see Fig. S13 a)).
Also in the antiparallel case (Fig. 5 c)) the local energy change due to the
cavity interaction increases with increasing bond length. However, the cavity-
induced energy change is generally smaller than for the all-parallel
configuration and becomes significantly smaller with increasing $N_{mol}$. The
curves, shown in Fig. 5 c), have a pairwise structure, where each ensemble
with an even value of $N_{mol}$ is very close in energy to the ensemble with
$N_{mol}+1$ molecules. The case of odd and even number of molecules in the
all-parallel configuration creates two different situations from a single-
molecule perspective. For even $N_{mol}$ the whole ensemble reduces to an
effective antiparallel bimolecular case, and the situation of odd $N_{mol}$ is
equivalent to the single-molecule case where both are scaled down by
$\nicefrac{{1}}{{N_{mol}}}$. Therefore, $E_{lin}$ and $E_{dis}$ become smaller
with increasing $N_{mol}$ for odd and even $N_{mol}$ (see Figs. S10 b) and S11
b)). Furthermore, for even $N_{mol}$ ensembles, there is a small negative
contribution from inter $E_{dse}$ (see Fig. S13 b)). In contrast to the all-
parallel and antiparallel configurations, the ensembles in the defective
orientation show clearly different trends. Similarly to the all-parallel case,
$\Delta E$ increases as $N_{mol}$ increases for a fixed nuclear configuration,
but for $N_{mol}>3$, it simultaneously decreases along the dissociation path.
This changing behavior is caused by the interplay of the local $E_{lin}$, the
ensemble quantity $E_{dis}$, and the cavity-mediated dipole-dipole
interaction. These three contributions are shown in Figs. S10 c), S11 c), and
S12 c). For $N_{mol}<4$ the ensemble, the dipole moment changes sign along the
dissociation path, whereas for a larger ensemble it is negative for the whole
pathway. The local dipole moment of the dissociating molecule is in the
studied configuration positive, which leads to a positive contribution of the
local $E_{lin}$ for $N_{mol}>3$ and a change in sign for $E_{lin}$ in smaller
ensembles. The inter $E_{dse}$ (see Fig. S13 c)) acting on the dissociating
molecule is always negative and is increasingly relevant for increasing bond
length. Consequently, the cavity-induced energy shift from the local
perspective decreases for the  defective configuration along the dissociation
pathway. In summary, for all three configurations studied, the interaction of
the cavity with the molecular ensembles modifies the local energy landscape of
the individual molecule. These changes are strongly dependent on the ensemble
properties (size and orientation of its components) and are due to the
interplay of the displacement field effects and the cavity-induced
polarization.

In conclusion, we have established an ab-initio Hartree-Fock ansatz in the
framework of the cavity Born-Oppenheimer approximation, capable of describing
the electronic ground state of molecular systems coupled to an optical cavity.
We have applied the CBO-HF ansatz to study the collective effects in small
ensembles of diatomic hydrogen fluoride (HF) molecules in an optical cavity.
The detailed analysis of the cavity-induced energy changes for the whole
ensemble and individual molecules shows that the self-consistent treatment and
the full dipole self-energy operator are crucial to capture relevant aspects
for the description of a strongly coupled molecular ensemble and its chemical
properties. The DSE terms are essential to describe cavity-induced
polarization at the level of individual molecules, as well as for the whole
ensemble. The observed interplay of displacement field effects and the cavity-
induced polarization enables energy changes for the individual molecule
because of collective coupling to an optical cavity. Consistent with our
previous work 42 we could identify a macroscopically induced microscopic
polarization mechanism based on intermolecular dipole-dipole interactions.
Although we have only studied the system in the electronic-photonic ground
state, we see indications that thermal fluctuations may play a decisive role
in polaritonic chemistry, in line with our previous work 42. Due to the nature
of the intermolecular dipole-dipole interactions, a local change/fluctuation
in the dipole moment and/or polarizability could affect the whole ensemble. A
pre-polarization of an ensemble with a static electric field, for example,
should lead to an observable effect in experiment. Another interesting topic
for further study is the interplay of this self-consistent polarization
mechanism with vibrational or electronic resonances.

The derivation of the CBO-HF equations demonstrates which molecular properties
are important for the DSE term and the couplings it introduces: molecular
dipole moments are important for inter-molecular interactions, while the
combination of dipole moments, quadrupole moments, and transition dipole
moments are important on an intra-molecular level. The CBO-HF ansatz and the
underlying CBOA formulation offers a suitable framework to derive post-
Hartree-Fock methods, such as configuration interaction or coupled cluster, or
potential self-consistent embedding schemes 69 for molecules under VSC or even
ESC 26. It may also provide potential energy surfaces that can be used for ab-
initio semiclassical dynamics or for nuclear-photonic quantum dynamics
simulations of molecular ensembles.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation program (grant
agreement no. 852286), the RouTe Project (13N14839), financed by the Federal
Ministry of Education and Research (Bundesministerium für Bildung und
Forschung (BMBF)) and supported by the European Research Council
(ERC-2015-AdG694097), the Cluster of Excellence “CUI: Advanced Imaging of
Matter” of the Deutsche Forschungsgemeinschaft (DFG), EXC 2056, project ID
390715994 and the Grupos Consolidados (IT1249-19). The Flatiron Institute is a
division of the Simons Foundation.

See the supplementary material for the details of the derivation of the CBO-HF
energy contribution, the discussion of the single-molecule case and additional
figures for the ensembles of HF molecules in an optical cavity. All data
underlying this study are available from the corresponding author upon
reasonable request.

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