# Fermi Resonance and the Quantum Mechanical Basis of Global Warming

R. Wordsworth  School of Engineering and Applied Sciences, Harvard, Cambridge,
MA 02138, USA  Department of Earth and Planetary Sciences, Harvard, Cambridge,
MA 02138, USA<EMAIL_ADDRESS>J. T. Seeley  School of
Engineering and Applied Sciences, Harvard, Cambridge, MA 02138, USA  K. P.
Shine  Department of Meteorology, University of Reading, Reading, RG6 6ET,
United Kingdom

###### Abstract

Although the scientific principles of anthropogenic climate change are well-
established, existing calculations of the warming effect of carbon dioxide
rely on spectral absorption databases, which obscures the physical foundations
of the climate problem. Here we show how CO2 radiative forcing can be
expressed via a first-principles description of the molecule’s key
vibrational-rotational transitions. Our analysis elucidates the dependence of
carbon dioxide’s effectiveness as a greenhouse gas on the Fermi resonance
between the symmetric stretch mode $\nu_{1}$ and bending mode $\nu_{2}$. It is
remarkable that an apparently accidental quantum resonance in an otherwise
ordinary three-atom molecule has had such a large impact on our planet’s
climate over geologic time, and will also help determine its future warming
due to human activity. In addition to providing a simple explanation of CO2
radiative forcing on Earth, our results may have implications for
understanding radiation and climate on other planets.

Earth atmosphere(437) — Planetary atmospheres(1244) — Planetary climates(2184)
— Greenhouse Effect(2314)

##  1 Introduction

Carbon dioxide is an essential greenhouse gas on all rocky planets in the
solar system with significant atmospheres (Venus, Earth and Mars). On Earth,
the carbonate-silicate cycle regulates atmospheric CO2 on geological
timescales, but the last 150 years has seen a rapid rise in concentrations
from approximately $280$ ppmv to $415$ ppmv (Tziperman, 2022), due to burning
of fossil fuels by humans and land use changes (Friedlingstein et al., 2022).
Earth’s global mean surface temperature has risen by approximately 1 K during
this same period, with most of the warming a direct result of this CO2
increase. CO2 affects surface temperature because it is a greenhouse gas: it
absorbs more effectively at thermal infrared frequencies than the near-
infrared and visible frequencies where solar radiation peaks. As a result,
increasing levels of atmospheric CO2 shifts the emission of thermal radiation
to space to higher altitude regions of the atmosphere, where air is less dense
and colder. This colder air releases less thermal radiation, so increasing CO2
decreases total emission to space for fixed surface and atmospheric
temperatures. The magnitude of this decrease is defined as the radiative
forcing of CO2.

Figure 1: Idealized plot of temperature vs. altitude in Earth’s atmosphere.
The solid black line shows the temperature structure used to derive (7) (fixed
tropospheric lapse rate of -6.5 K km1, isothermal stratospheric temperature of
217 K). The dashed black line shows the 1976 Standard Atmosphere temperature
profile (Minzner, 1977). Red and cyan dots show the values of $T_{s}$ and
$T_{t}$, respectively. Figure 2: High-accuracy absorption spectrum of CO2 in
the region of the $\nu_{2}$ band, in cross-section units of meters squared per
molecule of CO2, at pressure $p_{s}$ and temperature $T_{s}$ (see Table 1). a)
shows the band over a wide frequency range, while b) zooms in on the central
fundamental band, with P, Q and R branches labeled (see Section 3). Both plots
were produced using the HITRAN database, with line truncation at the standard
value of 25 cm-1(0.75 THz) (Gordon et al., 2022).

Atmospheric mixing is fast compared to the rate of CO2 emission and removal,
so to a first approximation the CO2 concentration is uniform in Earth’s lower
atmosphere. The stratosphere-adjusted radiative forcing111This differs
slightly from the _effective_ radiative forcing definition now used by the
IPCC (Masson-Delmotte et al., 2021), although the difference is not important
for our purposes. See also discussion in Section 5. due to an increase in the
atmospheric CO2 molar concentration from $x_{0}$ to $x$ is typically expressed
as

$\Delta F\approx+\alpha\ln[x/x_{0}]$ (1)

where $\alpha$ is defined as the CO2 radiative forcing parameter. Global-mean
calculations from detailed radiative transfer codes using tabulated
spectroscopic data yield $\alpha\approx 5.35$ W m-2 (Myhre et al., 1998), with
an estimated accuracy of around 10%. Once the atmosphere and ocean have
thermally equilibrated, this radiative forcing gives rise to a surface
temperature change of magnitude

$\Delta T_{s}\approx+\lambda^{-1}\Delta F$ (2)

where the parameter $\lambda$ is the climate feedback parameter, with units of
W m-2 K-1. Based on data from a range of observational and modeling sources,
the sixth assessment report of the Intergovernmental Panel on Climate Change
(IPCC AR6) stated that $\Delta T_{s}$ for a doubling of CO2 is“very likely”
(i.e., with greater than 90% probability) in the range 2 to 5 K, with a best
estimate of 3 K (Masson-Delmotte et al., 2021). Since the radiative forcing
for CO2 doubling $\Delta F_{2\times}=5.35\ln(2)=3.71$ W m-2, this indicates a
value of between 0.74 and 1.85 W m-2 K-1 for $\lambda$.

Given the clear correspondence between observations and the results of
sophisticated climate models, the scientific basis of climate change is
indisputable. In addition, many comprehensive descriptions of the physics of
climate and global warming, including the specifics of the radiative effects
of CO2 doubling, already exist (Pierrehumbert, 2011; Wilson & Gea-Banacloche,
2012; Zhong & Haigh, 2013; Mlynczak et al., 2016; Dufresne et al., 2020;
Jeevanjee et al., 2021b; Romps et al., 2022; Tziperman, 2022; Shine & Perry,
2023). Despite this, it is currently still not possible to derive (1) directly
starting from fundamental properties of the CO2 molecule. This is an important
objective, because analytic methods are a powerful tool to increase
understanding and elucidate the results of numerical simulations. Here, we
build on previous efforts and show how this can be achieved, via a synthesis
of molecular spectroscopy and climate physics. Our analysis here focuses on
Earth’s present-day climate, but potential applications to other planets in
the solar system and exoplanets are discussed in Section 6.

##  2 A simple empirical model of CO2 radiative forcing

In several recent papers (Wilson & Gea-Banacloche, 2012; Jeevanjee et al.,
2021b; Romps et al., 2022), it has been shown that key features of CO2
radiative forcing can be captured using a simplified representation of
atmospheric radiative transfer combined with an empirical approach to CO2
spectroscopy. Absorption by CO2 in the thermal infrared at Earth-like
concentrations is dominated by a broad collection of bands centered around 20
THz (667 cm-1 in wavenumber units)222Traditionally, one of the novelties of
atmospheric spectroscopy work is a continuous need to interconvert between
various non-SI units. Here, we stick to SI units as much as possible, but when
stating frequencies in Hz or THz we also report wavenumber values in cm-1, to
allow easy comparison with other work. that has a ‘triangular’ shape in
logarithmic units (Fig. 2). Because of this, this collection of bands can
approximately be represented using the formula

$\ln[\sigma/\sigma_{cen}]=-\frac{|\nu-\nu_{cen}|}{w}$ (3)

for absorption cross-section $\sigma$. Here $\sigma_{cen}$ is the value of
$\sigma$ at the band center, $\nu$ is frequency, $\nu_{cen}$ is the frequency
of the band center, and $w$ is a band structure coefficient, such that a
smaller value of $w$ leads to a narrower band. Numerical estimation of $w$
from HITRAN spectroscopic data or line-by-line spectral radiative forcing
calculations yields a value of around 0.37 THz (12.5 cm-1), with the value
somewhat dependent on the fitting approach chosen (Jeevanjee et al., 2021b;
Romps et al., 2022).

Taking the angle-averaged optical depth $\tau$ as increasing downwards from
the top of the atmosphere to the surface, we can write a small change in
$\tau$ as

$d\tau=\frac{x\sigma}{\overline{\mu}m_{a}}\frac{dp}{g}$ (4)

where $\overline{\mu}$ is the mean propagation angle of upwelling infrared
photons, $p$ is pressure, $g$ gravity and $m_{a}$ is the mean molecular mass
of air. If we assume a linear dependence of $\sigma$ on pressure due to line
broadening333This is a simplified approach, because pressure scaling at line
centers and line wings differs (Goody & Yung, 1995, see also Section 3.2).
However, the approach works well in practice (Romps et al., 2022), and the
specific choice of pressure scaling in this equation does not affect our
subsequent analysis. such that $\sigma=\sigma_{0}(p/p_{0})$, where $p_{0}$ is
a reference pressure (Table 1), then

$\tau=\frac{x\sigma_{0}p^{2}}{2g\overline{\mu}m_{a}p_{0}}.$ (5)

Next, approximating infrared emission to space at a given frequency as coming
from a narrow pressure range in the atmosphere, one can write an emission
pressure corresponding to $\tau(p,\nu)=\tau_{em}(\nu)$. Combining (3) and (5),
noting that $\tau$ and $\sigma$ are proportional, setting $\tau_{em}=1$,
rearranging in terms of $\nu$ and writing $\nu=\nu_{em}$ yields (Jeevanjee et
al., 2021b)

$\nu_{em}(p,x)=\nu_{cen}\pm
w\ln\left[\frac{x\sigma_{cen,0}p^{2}}{2g\overline{\mu}m_{a}p_{0}}\right].$ (6)

The logarithmic dependence of $\nu_{em}$ on $x$ is what gives rise to the
logarithmic dependence of $\Delta F$ on $x$. The relationship between
$\nu_{em}$ and $\Delta F$ can be determined by noting that increasing $x$ is
equivalent to swapping emission over a certain frequency range from the
surface to the stratosphere (Romps et al., 2022). The expression for the CO2
radiative forcing parameter that emerges from this analysis is

$\alpha=2\pi w[\mathcal{B}(\nu_{cen},T_{s})-\mathcal{B}(\nu_{cen},T_{t})].$
(7)

Here $\mathcal{B}(\nu,T)$ is the Planck spectral irradiance evaluated at
frequency $\nu$ and temperature $T$, $T_{s}$ is surface temperature and
$T_{t}$ is the tropopause temperature (see Fig. 1).

The analytic model represented by (7) is highly simplified, but it allows
greater insight into the mechanism of CO2 radiative forcing than is possible
from numerical approaches. Like all current approaches to understanding
climate change, however, it still requires us to assume that the infrared
spectrum of CO2 is available as a pre-determined input. Our aim here is to
relax this requirement, and derive $w$ and $\nu_{cen}$, and hence equations
(1) and (7), from the basic properties of the CO2 molecule.

##  3 CO2 infrared spectroscopy

CO2 absorbs in the infrared due to combinations of vibrational and rotational
quantum transitions (Pierrehumbert, 2011). As a three-atom linear molecule,
CO2 has $3N-5=4$ vibrational degrees of freedom (Fig. 3), with four quantum
numbers $V_{1}$, $V_{2a}$, $V_{2b}$, and $V_{3}$ corresponding to excitation
of an symmetric stretch mode, two degenerate bending modes, and an asymmetric
stretch mode, respectively444The use of $\nu$ as the symbol for frequency and
$v$ for vibrational quantum number is common in the modern spectroscopic
literature, but to avoid confusion we have chosen to use the upper-case $V$
notation of Adel & Dennison (1933) for quantum number here.. Superposition of
the in-plane and out-of-plane bending motions corresponding to $V_{2a}$ and
$V_{2b}$ results in an excitation where the three atoms perform circular
motions about the molecule’s major axis (Fig. 3). Such motion has angular
momentum, which can be represented via introduction of a new quantum number
$l$. Because $V_{2a}$ and $V_{2b}$ are degenerate, vibrational CO2 states can
therefore be characterized555As always, the Dirac or bra-ket notation
represents quantum states, such that $|\psi\rangle=\int
d^{3}\mathbf{x}\psi(\mathbf{x})|\mathbf{x}\rangle$, and
$\langle\psi|\phi\rangle=\int\psi(\mathbf{x})^{*}\phi(\mathbf{x})d^{3}\mathbf{x}$
when written in terms of position $\mathbf{x}$ and wavefunctions $\psi$ and
$\phi$. as $|V_{1}V_{2}^{l}V_{3}\rangle$, where $V_{2}=V_{2a}+V_{2b}$ and
$l=V_{2},V_{2}-2,...,1$ or 0. Hence for $V_{2}=3$, for example, $l$ can be 3
or 1, while for $V_{2}=4$, $l=4$, 2 or 0 are all allowed.

Figure 3: a) Schematic of the three vibrational modes of carbon dioxide. The
two degenerate bending modes superimpose to produce a motion where each atom
rotates around the major axis of the molecule, which is represented via the
quantum number $l$. b) Mass-weighted coordinate system used to express the
three fundamental modes of CO2 as simple harmonic oscillations (see Section
4).

The frequency of each vibrational mode of CO2 can be calculated
approximately666More accurate determination of vibrational frequencies can be
achieved via _ab initio_ methods (e.g., Rodriguez-Garcia et al., 2007). by
treating the molecule as three point masses connected by spring-like bonds. In
particular, using a valence force approach, which assumes restoring forces act
to oppose changes in the distance and angles of local atom-atom bonds only
(e.g., Herzberg, 1945, p. 168), the molecular vibrational potential energy $U$
can be written as

$U=U_{e}+\frac{1}{2}k(Q_{1}^{2}+Q_{2}^{2})+\frac{1}{2}k_{\delta}\delta^{2}.$
(8)

Here $U_{e}$ is the equilibrium potential energy corresponding to zero
vibrational motion, while $Q_{1}$ and $Q_{2}$ are the change in separation
between the C and the first and second O atoms during vibrational motion,
respectively. $\delta$ is the change in bond angle that occurs during bending
motion, and $k$ and $k_{\delta}$ are linear and bending bond force constants,
respectively.

The form of the terms in (8) allow the normal mode frequencies of the molecule
to be simply expressed. We focus on the first two modes here. While the
antisymmetric stretch mode $\nu_{3}$ gives rise to a strong absorption band,
it is located outside the thermal infrared spectral region and hence is not
relevant to our discussion. Additional combination bands involving $\nu_{3}$
are present around 30 THz (1000 cm-1) but only contribute around 5% to
radiative forcing at present-day CO2 concentrations (Mlynczak et al., 2016).

For the symmetric stretch mode, we simply have oscillatory motion of each O
atom, leading to

$\nu_{1}=\frac{1}{2\pi}\sqrt{\frac{k}{m_{\ce{O}}}}$ (9)

where $m_{\ce{O}}$ is the mass of an oxygen atom. The value $k=1680$ N m-1
yields $\nu_{1}=40.1$ THz (1337 cm-1). This is fairly close to the C-O bond
force constant of 1860 N m-1 in carbon monoxide (CO). The bending mode
frequency is

$\nu_{2}=\frac{1}{2\pi}\sqrt{\frac{2}{m_{\ce{O}}}\left(1+2\frac{m_{\ce{O}}}{m_{\ce{C}}}\right)\frac{k_{\delta}}{a_{e}^{2}}}$
(10)

where $m_{\ce{C}}$ is the mass of a carbon atom (Herzberg, 1945). For
$k_{\delta}=7.7\times 10^{-19}$ N m and $a_{e}=116$ pm (Table 1), the value of
$\nu_{2}$ is 20.0 THz (667 cm-1). $\nu_{2}$ is of course the well-known
bending mode associated with the center of the thermal infrared band,
$\nu_{cen}$, as seen in Fig. 2. The fact that $\nu_{2}$ is (apparently
coincidentally) about half the value of $\nu_{1}$ will be very important in
the discussion that follows.

Parameter | Symbol | Value | Units
---|---|---|---
Mass of oxygen atom | $m_{\ce O}$ | 16 | $m_{u}$
Mass of carbon atom | $m_{\ce C}$ | 12 | $m_{u}$
Equilibrium C–O separation in CO2 | $a_{e}$ | $1.16\times 10^{-10}$ | m
CO2 symmetric stretch force constant | k | $1680$ | N m-1
CO2 bending mode force constant | $k_{\delta}/a_{e}^{2}$ | $57$ | N m-1
CO2 transition dipole moment magnitude | $|\langle m|\mathbf{d}|n\rangle|$ | $3.35\times 10^{-31}$ | C m
CO2 Fermi coupling term | $|b|$ | $2.14\times 10^{12}$ | Hz
Collision cross-section (CO2 in N2) | $\sigma_{c}$ | $0.44\times 10^{-18}$ | m2
Line broadening coefficient | $n_{b}$ | 0.5 | []
Earth mean surface pressure | $p_{s}$ | $10^{5}$ | Pa
Earth mean surface temperature | $T_{s}$ | 288 | K
Earth stratospheric temperature | $T_{t}$ | 217 | K

Table 1: Key parameters used in the analysis. Throughout this paper, we assume
that reference pressure and temperature are equal to their surface values,
$p_{0}\equiv p_{s}$ and $T_{0}\equiv T_{s}$. $m_{u}$ is the atomic mass unit
such that 1 $m_{u}=1.66\times 10^{-27}$ kg. Values for $a_{e}$, $k$,
$k_{\delta}/a_{e}^{2}$ and $|b|$ are taken from Herzberg (1945), pp. 21, 173 &
218; $\sigma_{c}$ is from Chapman & Cowling (1990), p. 263. Values for other
quantities are justified in the main text.

###  3.1 Line positions

Now we have an expression for the spectral location of the $\nu_{2}$ band in
Fig. 2, we can start to build up a description of the band structure. For
this, we need to know what determines the location, shape and intensity of all
the most important spectral lines. We tackle this problem step by step,
starting with line location. We focus on explaining the most abundant
isotopologue of CO2, ^12C^16O2, on the basis that the contribution of all
other isotopologues to CO2 radiative forcing is minor (Shine & Perry, 2023).

In the infrared, spectral lines appear because absorption or emission of
photons causes quantized changes in the vibrational and/or rotational state of
molecules. The center of the $\nu_{2}$ band777We use the term ‘$\nu_{2}$ band’
here and elsewhere for simplicity, but as will become clear soon, transitions
involving additional vibrational modes are also important to the larger band
structure. at 20 THz (667 cm-1) corresponds to a vibrational transition from
the ground state $|00^{0}0\rangle$ to the first excited bending mode with
angular momentum number $l=1$, $|01^{1}0\rangle$. The band has a strong
central peak called the Q-branch (Fig. 2) due to purely vibrational
transitions, and P and R branches at lower and higher frequencies,
respectively, due to combinations of vibrational and rotational transitions.
Solution of the time-independent Schrödinger equation in spherical coordinates
(Levine, 1975) shows that the location of P- and R-branch lines relative to
$\nu_{2}$ is given by

$\nu_{J,J+1}=(E_{J+1}-E_{J})/h=2B(J+1).$ (11)

Here $h$ is Planck’s constant, $J=0,1,2,...$ is the rotational quantum number,
$E_{J}$ is the energy of rotational state $J$, and the rotational constant
$B=h/8\pi^{2}I$, with moment of inertia $I=2m_{\ce{O}}a_{e}^{2}$. In addition,
$a_{e}$ is the equilibrium C–O separation in CO2 (Table 1). For lines in the Q
branch, there is no change in $J$, while in the P and R branches, $\Delta
J=\pm 1$. For ^12C^16O2, transitions involving odd values of $J$ are missing,
because of selection rules arising from the zero spin of the oxygen atoms
(Levine, 1975). Given that $a_{e}=116$ pm,
$B=h/16\pi^{2}m_{\ce{O}}a_{e}^{2}=11.7$ GHz (0.39 cm-1), so the spacing
between the P- and R-branch lines is about $2\times 2B=47$ GHz (1.6 cm-1). In
the Q-branch, transitions with different rotational energies are not exactly
colocated because of subtle effects such as Coriolis interactions, but for our
purposes the lines can be treated as unseparated in frequency. Rotational line
spacing is further affected by centrifugal and anharmonic effects, but these
complications are not important to climate forcing and so will be ignored
here.

###  3.2 Line shape and width

All spectral lines have a shape that is determined by a combination of natural
broadening, Doppler and collisional effects (Goody & Yung, 1995). Line shape
is dependent on both temperature and pressure. A full analysis of this problem
could become complicated quickly, but in keeping with our aim of getting a
rough estimate of CO2 radiative forcing only, we take a simple approach here.
In the troposphere, thermal infrared absorption lines are Lorentzian to a
close approximation, with lineshape

$f(\nu-\nu_{mn})=\frac{1}{\pi}\frac{\gamma}{(\nu-\nu_{mn})^{2}+\gamma^{2}}$
(12)

where $\nu_{mn}$ is the frequency of the transition from state $m$ to $n$ and
$\gamma$ is the linewidth (halfwidth at half maximum) in Hz. $\gamma$ scales
approximately linearly with pressure. This can be shown by noting that the
root mean square speed of a molecule in a gas of temperature $T$ is

$\overline{v}=\sqrt{3k_{B}T/\overline{m}}$ (13)

where $k_{B}$ is Boltzmann’s constant, and $\overline{m}$ is the mean
molecular mass of air. The mean relative collision speed is larger than this
by a factor of $\sqrt{2}$, $\overline{v}_{rel}=\sqrt{2}\overline{v}$. The mean
free path, or average distance travelled by a molecule between collisions, is
(Chapman & Cowling, 1990, p. 88)

$l_{mfp}=\frac{1}{\sqrt{2}n\sigma_{c}}=\frac{k_{B}T}{\sqrt{2}\sigma_{c}p}$
(14)

where $\sigma_{c}$ is intermolecular collision cross-section, $n$ is number
density, $T$ is temperature, $p$ is pressure, and $p=nk_{B}T$ from the ideal
gas law. The linewidth in frequency units can be written as
$\gamma=1/(2\pi\tau_{c})=\overline{v}_{rel}/(2\pi l_{mfp})$, where $\tau_{c}$
is the mean collision time between molecules888The factor of $2\pi$ in the
relationship between $\gamma$ and $\tau_{c}$ can be obtained from a Fourier
analysis of an ensemble of radiating oscillators (e.g., Stamnes et al., 2017,
p. 88).. Finally we have

$\gamma(p,T)=\gamma_{0}(p/p_{0})(T/T_{0})^{-n_{b}}.$ (15)

Here $n_{b}=0.5$, the reference values $p_{0}$, $T_{0}$ are taken to be
surface values (Table 1), and

$\gamma_{0}=\frac{\sigma_{c}p_{0}}{\pi}\sqrt{\frac{3}{\overline{m}k_{B}T_{0}}}.$
(16)

For CO2 in N2, $\sigma_{c}=\pi\times(3.75\times 10^{-10}\mbox{ m})^{2}=0.44$
(nm)2 (Chapman & Cowling, 1990), so $\gamma_{0}=1.76$ GHz ($0.06$ cm-1). The
HITRAN database (Gordon et al., 2017) gives values for $\gamma_{0}$ in the CO2
$\nu_{2}$ band that range between 1.5 and 3 GHz (0.05 cm-1and 0.1 cm-1), and
$n_{b}$ between 0.5 and 0.8, so our simple method slightly underestimates
$\gamma_{0}$ and $n_{b}$. These differences are not important for the analysis
that follows.

###  3.3 Line intensity

Determining line intensity is one of the most challenging aspects of quantum
spectroscopy. However, in the range of CO2 concentrations over which radiative
forcing scales approximately logarithmically according to (1), $\Delta F$ is
not sensitive to the absolute values of line intensity in the $\nu_{2}$ band
(e.g., Jeevanjee et al., 2021b). Hence we can also take an approximate
approach here. The most important quantities in line intensity calculations
are the Einstein coefficients, which express the rate of absorption or
emission of a photon by a CO2 molecule. Analysis of the Schrödinger equation
in the presence of a time-dependent perturbation due to an oscillating
electric field leads to the following expression for the Einstein coefficient
for spontaneous emission999In many textbooks, the Einstein-A coefficient is
written in cgs units as $A_{mn}=64\pi^{4}\nu_{mn}^{3}|\langle
m|\mathbf{d}|n\rangle|^{2}/3hc^{3}$. To convert, note that in cgs units
$\epsilon_{0}=1/4\pi$, which when substituted into the previous expression
yields (17). :

$A_{mn}=\frac{16\pi^{3}\nu_{mn}^{3}|\langle
m|\mathbf{d}|n\rangle|^{2}}{3hc^{3}\epsilon_{0}}.$ (17)

Here $m$ and $n$ are any two quantum states of energy $E_{m}$ and $E_{n}$ such
that $E_{m}>E_{n}$ and $\nu_{mn}=(E_{m}-E_{n})/h$. In addition, $c$ is the
speed of light and $\epsilon_{0}$ is the vacuum permittivity. The term
$\langle m|\mathbf{d}|n\rangle$ is the _transition dipole moment_ , which is
defined as

$\langle m|\mathbf{d}|n\rangle\equiv\langle
n|\mathbf{d}|m\rangle=\int\psi_{m}(\mathbf{x})^{*}\mathbf{d}\psi_{n}(\mathbf{x})d^{3}\mathbf{x}.$
(18)

Here $\mathbf{d}$ is the dipole moment operator of the molecule and $\psi_{k}$
is the eigenfunction of quantum state $k$. The transition dipole moment has
typical magnitude $3.34\times 10^{-31}$ C m ($0.1$ D) for CO2 transitions in
the $\nu_{2}$ vibration-rotation band. This is similar to the permanent dipole
of the ground state of CO, $4.07\times 10^{-31}$ C.m (0.122 D) (Muenter,
1975). Taking $\nu_{mn}=20$ THz yields $A_{mn}\approx 1$ s-1. Comparison with
HITRAN data shows that this is a reasonable approximation at the center of the
$\nu_{2}$ band, although the values of $A_{mn}$ decrease away from the band
center.

Figure 4: a) Analytic CO2 $\nu_{2}$ band, including P, Q and R branches for
the fundamental vibrational transition from $|00^{0}0\rangle$ to
$|01^{1}0\rangle$, but neglecting all additional vibrational transitions. b)
shows a close-up of the same band. Pressure and temperature are $p_{s}$ and
$T_{s}$ in Table 1, as in Fig. 2.

Line intensity expresses the frequency-integrated absorption cross-section of
a line, independent of lineshape. In SI units of Hz m2 molecule-1, it is
defined as101010In HITRAN notation (Šimečková et al., 2006), we would write
$\tilde{S}_{mn}=({A_{mn}}/{8\pi c\tilde{\nu}_{mn}^{2}})\mathcal{O}(T)$, where
$\tilde{S}_{mn}$ has units of cm-1 (cm2 molecule-2). To convert, we use
$\tilde{\nu}=\nu/c$, and in addition $S_{mn}=c\tilde{S}_{mn}$.

$S_{mn}=\frac{c^{2}A_{mn}}{8\pi\nu_{mn}^{2}}\mathcal{O}_{mn}(T).$ (19)

The dimensionless term $\mathcal{O}_{mn}(T)$ incorporates multiple additional
effects, of which the most important here is energy level occupancy. The rate
of absorption of photons by molecules in a given energy state must depend on
the number density of molecules in that state, relative to the total number
density. At temperatures in the 250-290 K range, most CO2 molecules are in the
$\nu_{2}$ vibrational ground state111111This can be seen by calculating the
Boltzmann factor $\mathrm{e}^{-h\nu_{2}/k_{B}T}$, which equals 0.036 given
$T=288$ K. This also indicates that neglecting stimulated emission is a good
approximation in this case., so the fundamental band vibrational occupancy
factor can be approximated as 1 for our purposes. For the rotational state,
however, occupancy up to $J$ values of around 50 is significant at Earth-like
temperatures. We take this into account by writing

$\mathcal{O}_{mn}(T)=(2J+1)\mathrm{e}^{-J(J+1)hB/k_{B}T}/q_{r}$ (20)

with the rotational partition function approximated as $q_{r}\approx
k_{B}T/2hB$ (Bernath, 2020, p. 172). The $(2J+1)$ term in this expression
accounts for degeneracy: i.e., for a given $J$ value, there are $2J+1$ states
with the same energy.

The final step is to use line intensities to calculate the absorption spectrum
itself. This is done by noting that the absorption cross-section for a single
line, in units of m2 molecule-1, is simply

$\sigma_{mn}=S_{mn}f(\nu-\nu_{mn}).$ (21)

and the total absorption cross-section for the entire band is the sum of
$\sigma_{mn}$ over all transitions,

$\sigma(\nu)=\sum_{mn}\sigma_{mn}.$ (22)

Now we have all the pieces required to begin building absorption spectra.
Using the line locations from Section 3.1, lineshape and width definitions
from Section 3.2, and line intensity formulae from this section, we can plot
the P, Q and R branches of the $\nu_{2}$ fundamental band (Fig. 4). Here, we
have included lines up to $J=100$, on the basis that lines from higher
rotational number transitions are so weak that they contribute little further
to absorption. Comparison with Fig. 2 shows that line peaks are a little
higher than in the HITRAN spectrum, mostly because we are somewhat
underestimating the line broadening coefficient $\gamma$ via (15) and (16).
Otherwise, the overall form of the band center is reproduced fairly
accurately. At the sides of the band, however, it is clear that our approach
fails completely. Correctly incorporating these sidebands will be the task of
the next section.

##  4 Fermi Resonance

The multiple sidebands present in the spectrum in Fig. 2 arise due to
vibrational transitions between higher energy states than the
$|00^{0}0\rangle$ fundamental (Fig. 5). In a perfect simple harmonic
oscillator, the spacing of vibrational energy levels is uniform, and all
transitions occur at the same frequency. This is clearly not the case for the
CO2 $\nu_{2}$ band. The reason for the difference is _Fermi resonance_ (Fermi,
1931; Adel & Dennison, 1933). Fermi resonance is far better known in quantum
spectroscopy than in climate physics, but it is key to understanding CO2
radiative forcing. Recently, Shine & Perry (2023) extracted individual bands
from the HITRAN database for the purpose of numerical calculations of the
radiative forcing and found that Fermi resonance contributes approximately
half of the total forcing magnitude. Here we begin by discussing the physical
nature of Fermi resonance, and then show how incorporating it in our
derivation allows us to write down a quantum analytic formula for CO2
radiative forcing.

Fermi resonance occurs in CO2 because $\nu_{1}\approx 2\nu_{2}$: the symmetric
stretch frequency happens to be very close in value to double the bending
frequency. As a result, nonlinear interactions between the two modes shift the
energy levels of states $|10^{0}0\rangle$ and $|02^{0}0\rangle$ and cause
their wavefunctions to mix (Fig. 5). The best way to get an intuitive
understanding of Fermi resonance is by analogy with the classical coupled
pendulum experiment. This analogy is noted in passing in Herzberg (1945), but
we explore it in more detail here.

In the coupled pendulum experiment, two pendulums of almost equal natural
frequencies exchange energy with each other via some nonlinear coupling (most
commonly, torsion of the string to which they are attached). Without this
nonlinear interaction, a Fourier transform of their motion would yield a
single peak. However, when the pendulums are coupled, the same transform
yields two peaks that are shifted from the original central value by an amount
that depends on the strength of the interaction.

Figure 5: The key vibrational energy levels and transitions of the CO2
molecule that are most important to radiative forcing and climate change. Many
levels and transitions have been neglected for simplicity. Comparing with the
notation of Shine & Perry (2023), our A $\equiv$ Fu; B, C $\equiv$ F1; D, E
$\equiv$ FF, and H1, H2 and F2 are neglected. Numerical values for the energy
levels were obtained from Oberly et al. (1968).

Mathematically, we can understand this process by first writing the classical
Hamiltonian for a simple harmonic oscillator in isolation:

$H(x,p)=\frac{1}{2}\frac{p^{2}}{m}+\frac{1}{2}kx^{2}.$ (23)

Here $p$ is momentum, $m$ is mass, $k$ is a force constant and $x$ is
position. For two oscillators with a nonlinear coupling constant $\alpha$, the
Hamiltonian is

$H(x_{1},p_{1},x_{2},p_{2})=\frac{1}{2}\frac{p_{1}^{2}}{m}+\frac{1}{2}\frac{p_{2}^{2}}{m}+\frac{1}{2}kx_{1}^{2}+\frac{1}{2}kx_{2}^{2}+\alpha
x_{1}x_{2}$ (24)

Using $d\mathbf{p}/dt=-\partial H/\partial\mathbf{x}$ and
$d\mathbf{x}/dt=+\partial H/\partial\mathbf{p}$, where
$\mathbf{x}=(x_{1},x_{2})$ and $\mathbf{p}=(p_{1},p_{2})$, we can derive
equations of motion for each oscillator in the coupled case:

$\displaystyle\frac{d^{2}x_{1}}{dt^{2}}$ $\displaystyle=$
$\displaystyle-\omega_{0}^{2}x_{1}-\tilde{\omega}^{2}x_{2},$ (25)
$\displaystyle\frac{d^{2}x_{2}}{dt^{2}}$ $\displaystyle=$
$\displaystyle-\omega_{0}^{2}x_{2}-\tilde{\omega}^{2}x_{1}.$ (26)

Here $\omega_{0}=\sqrt{k/m}$ and $\tilde{\omega}=\sqrt{\alpha/m}$. The
coupling term $\tilde{\omega}$ causes energy to be repeatedly exchanged
between the first and second oscillators (Fig. 6). Taking the sum and
difference of (25) and (26) and writing $y_{1}=x_{1}+x_{2}$,
$y_{2}=x_{1}-x_{2}$, we have

$\frac{d^{2}y_{1}}{dt^{2}}=-\omega_{+}^{2}y_{1}$ (27)
$\frac{d^{2}y_{2}}{dt^{2}}=-\omega_{-}^{2}y_{2}.$ (28)

where $\omega_{-}^{2}=\omega_{0}^{2}-\tilde{\omega}^{2}$ and
$\omega_{+}^{2}=\omega_{0}^{2}+\tilde{\omega}^{2}$. With this change of
variables, it can be seen that the nonlinear coupling gives rise to two new
eigenfunctions whose frequencies are shifted away from the unperturbed
frequency $\omega_{0}$ by an amount dependent on the coupling constant
$\alpha$. This is directly analogous to the way in which nonlinear coupling
between the symmetric stretch and bending modes of CO2 give rise to the Fermi
bands. Figure 6 shows non-dimensional numerical solutions of (25) and (26) for
$k=2$, $m=1$ and $\alpha=0.1$, alongside the resulting power spectrum for
$x_{1}$.

Figure 6: Classical analogy to Fermi resonance. a) Numerical solution of (25)
and (26) and b) the resulting power spectrum of $x_{1}$, with the analytical
values of $\omega_{0}$, $\omega_{-}$ and $\omega_{+}$ also shown. The
simulation was run from $t=0$ to $t=10^{4}$ and solved using the 5(4) order
Runge-Kutta method. In the analogy, $\omega_{0}$ is equivalent to $\nu_{2}$
(times a factor of 2$\pi$), and $\omega_{-}$ and $\omega_{+}$ are equivalent
to the centers of the first Fermi sidebands.

To represent Fermi resonance in a quantum framework, a degenerate first-order
perturbation analysis can be used (Dennison, 1940). For this, the molecular
vibrational potential energy $U$ discussed earlier is expressed in terms of
dimensionless mass-weighted normal coordinates $\zeta$, $\rho$ and $\sigma$
(see Fig. 3) and expanded as $U=U^{0}+\epsilon U^{1}+...$, where $\epsilon$ is
a perturbation parameter. This results in the expressions

$\displaystyle U^{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}h(\nu_{1}\sigma^{2}+\nu_{2}\rho^{2}+\nu_{3}\zeta^{2}),$
(29) $\displaystyle\epsilon U^{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}h(a\sigma^{3}+b\sigma\rho^{2}+c^{\prime}\sigma\zeta^{2}).$
(30)

Here $a$, $b$ and $c^{\prime}$ have units of frequency. (29) allows for simple
harmonic oscillation of the CO2 molecule at its three normal frequencies
$\nu_{1}$, $\nu_{2}$ and $\nu_{3}$, and nothing else. (30) contains anharmonic
terms, of which the second, $b\sigma\rho^{2}$, describes interaction between
the bending and symmetric stretch modes. Because the potential energy must
remain the same if the coordinates are reflected across a plane of symmetry of
the molecule, all other cubic terms ($\rho^{3}$, $\zeta^{3}$ etc.) in $U^{1}$
are forbidden for a linear molecule like CO2.

With $U$ defined, the Hamiltonian of the system is written in the form
$H=H^{0}+\epsilon H^{\prime}$, and first-order degenerate perturbation theory
(e.g., Robinett, 2006) can be employed to derive a formula for the perturbed
energy levels and wavefunctions. For clarity, we focus on the nearly
degenerate states $|10^{0}0\rangle$ and $|02^{0}0\rangle$, which we assign
wavefunctions $\psi_{m}$ and $\psi_{n}$. Writing
$\psi=a_{m}\psi_{m}+a_{n}\psi_{n}$, we can express the time-independent
Schrödinger equation in the matrix form $\mathbf{H}\mathbf{a}=E\mathbf{a}$,
where the individual elements of $\mathbf{H}$ are
$\langle\psi_{i}|H|\psi_{j}\rangle=\int{\psi_{i}}^{*}H\psi_{j}d^{3}\mathbf{x}$.
With the expansion $H=H^{0}+\epsilon H^{\prime}$, we then have

$\begin{pmatrix}E_{m}^{0}+\epsilon W_{mm}&\epsilon W_{mn}\\\ \epsilon
W_{nm}&E_{n}^{0}+\epsilon W_{nn}\end{pmatrix}\begin{pmatrix}a_{m}\\\
a_{n}\end{pmatrix}=E\begin{pmatrix}a_{m}\\\ a_{n}\end{pmatrix}$ (31)

where $W_{nm}=\langle\psi_{n}^{0}|H^{\prime}|\psi_{m}^{0}\rangle$ and
$E^{0}_{m}$ and $E^{0}_{n}$ are the unperturbed energies. Because of the
symmetry of the system, $W_{mm}=W_{nn}=0$ and $W_{nm}=W_{mn}^{*}$. Hence

$\begin{pmatrix}E_{m}^{0}-E&\epsilon W_{mn}\\\ \epsilon
W_{mn}^{*}&E_{n}^{0}-E\end{pmatrix}\begin{pmatrix}a_{m}\\\
a_{n}\end{pmatrix}=0.$ (32)

Taking the determinant of the matrix to solve the eigenvalue equation, we have

$(E_{m}^{0}-E)(E_{n}^{0}-E)-\epsilon^{2}W_{mn}^{*}W_{mn}=0.$ (33)

Writing $(E_{m}^{0}+E_{n}^{0})/2=\overline{E}^{0}$, setting $\epsilon=1$ and
solving for $E$ yields

$E_{\pm}=\overline{E}^{0}\pm\frac{1}{2}\sqrt{(E_{m}^{0}-E_{n}^{0})^{2}+4|W_{mn}|^{2}}.$
(34)

Explicit evaluation of the perturbation matrix element for states
$|m\rangle=|10^{0}0\rangle$ and $|n\rangle=|02^{0}0\rangle$ (Adel & Dennison,
1933; Herzberg, 1945) yields the result that
$|W_{10^{0}0;02^{0}0}|=hb/\sqrt{2}$. Writing the difference between state
frequencies in the absence of resonance interaction as
$\Delta_{0}=(E_{m}^{0}-E_{n}^{0})/h$, we have the result that resonance makes
two new states emerge with energy levels separated by
$\Delta_{F}=\pm\sqrt{\Delta_{0}^{2}+2b^{2}}$. At the level of approximation we
are using, the frequency difference $|\Delta_{0}|=0.5$ THz (16.7 cm-1), and
the strength of coupling between the two modes $b=2.14$ THz (71.3 cm-1), so
$\Delta_{F}=3.07$ THz (102.3 cm-1). Interestingly, these values of
$|\Delta_{0}|$ and $b$ were originally first determined from the spectra of
CO2 itself (both absorption and Raman; Fermi, 1931; Dennison, 1940).

Figure 7: Analytic CO2 $\nu_{2}$ band, including the fundamental vibrational
transition and four additional Fermi sidebands. The red line in panel a) shows
the approximate band shape predicted using (3) and (35), given $T=T_{s}$.
Labels in a) correspond to the transitions in Fig. 5. b) shows a close-up of
the fundamental band. Pressure and temperature are the same as in Fig. 2.

##  5 Radiative Forcing

Using the results of the previous two sections, we can now plot a revised
synthetic spectrum for the CO2 $\nu_{2}$ band. The new Fermi resonance bands B
and C (Fig. 5) are modeled just like the fundamental band, with the difference
that they are shifted to lower/higher frequencies by an amount
$|\Delta_{F}|/2$, and to lower line intensities by addition of a vibrational
occupancy factor to $\mathcal{O}_{mn}(T)$. We take this to be the ratio of
occupancy of the ground states for each transition, or
$\mathrm{e}^{-h\nu_{2}/k_{B}T}$. We add the second set of Fermi bands D and E
in a similar way, shifting them to lower/higher frequencies by a factor
$|\Delta_{F}|$ and to lower lines intensities by
$\mathrm{e}^{-2h\nu_{2}/k_{B}T}$. Recent analysis of radiative transfer model
results shows that the fundamental band A contributes about 50% of CO2
radiative forcing, while the additional Fermi bands B to E contribute most of
the remaining 50% (Shine & Perry, 2023). The contribution of other bands, such
as the hot bands of the $\nu_{2}$ fundamental and the ‘second Fermi’ bands of
Shine & Perry (2023), are neglected here on the basis that they do not cause
sufficiently large changes in absorption in the spectral regions where they
are present. Examples of such bands include the H1, H2 and F2 transitions
mentioned in the caption to Fig. 5.

As can be seen from Fig. 7, inclusion of all the labeled transitions in Fig. 5
results in a far more realistic mid-infrared absorption band for CO2.
Reference to Fig. 2 shows that bands B and C are well captured by our
approach. For the bands D and E, the limitations of our simple methodology
becomes more apparent, as they are both about 0.6 THz (20 cm-1) too close to
the fundamental band and too strong compared to HITRAN data. These differences
arise due to our neglect of higher order anharmonic terms, and our assumption
of a constant value for the Einstein-A coefficient.

Overall, however, inclusion of these four new Fermi bands results in a far
more realistic mid-infrared absorption band for CO2 than was shown in Fig. 4.
It is worth emphasizing that despite the fact it reproduces the main P, Q and
R branches visible in the HITRAN CO2 spectrum (Fig. 2), the spectrum in Fig. 4
is vastly simpler. The total number of lines in this spectrum is 750, whereas
in the HITRAN CO2 dataset over the 15–25 THz (500 to 833 cm-1) spectral range,
around 100,000 lines are present if all isotopologues are included. Many of
these additional lines are not visible in Fig. 2 because they are overlaid by
the five strong bands incorporated in Fig. 7.

We can also use the understanding developed in the last section to write down
an analytic expression for the band structure coefficient $w$, and hence for
CO2 radiative forcing itself. Clearly, $\nu_{cen}=\nu_{2}$, and $w$ must
approximately equal half of $|\Delta_{F}|$ divided by the negative log of the
occupancy factor:

$w=\frac{|\Delta_{F}|}{2}\frac{k_{B}T}{h\nu_{2}}.$ (35)

Given $T=250$ K and $\nu_{2}=20$ THz, $w=0.40$ THz (13.3 cm-1). The equivalent
value obtained by fitting a full HITRAN spectrum121212Romps et al. (2022)’s
$b$, which has a value of $0.04$ cm, is equal to 1/2 times the inverse of our
$w$ expressed in cm-1. is remarkably close at 0.375 THz (12.5 cm-1) (Romps et
al., 2022).

Our value for $w$ yields $\alpha=7.39$ W m-2 when substituted into (7). This
is somewhat larger than the 5.35 W m-2 value derived from comprehensive
radiative transfer codes, or the slightly higher effective radiative forcing
values derived from Earth system models131313Technical note: There are
different definitions of radiative forcing (e.g., Masson-Delmotte et al.,
2021), which represents the perturbation to the planetary radiation budget due
to a change in (in our case) of CO2, in the absence of any surface temperature
change. The simplest definition is the “instantaneous radiative forcing” in
which only CO2 is changed. The “stratosphere-adjusted radiative forcing”
allows stratospheric temperatures to adjust to the CO2 perturbation; this is a
better predictor of surface temperature change, particularly when comparing
the effect of changes in different constituents. IPCC’s now preferred
definition (“effective radiative forcing”) allows additional atmospheric
adjustments (e.g., to tropospheric temperatures, humidity and clouds); it is
an even better predictor, although detailed Earth-System Model calculations
are needed to compute these adjustments. For the explanatory purposes of this
paper, the differences between these definitions are of secondary importance.
(Pinnock et al., 1995; Myhre et al., 1998; Masson-Delmotte et al., 2021). The
difference is primarily a result of the assumptions made in deriving (7), and
not our approach to calculating the CO2 spectrum, as demonstrated by the close
correspondence of our $w$ value with that in Romps et al. (2022). Refinement
of the derivation leading up to (7) could be interesting to investigate in
future.

Finally, we can use the result just obtained to write an approximate
expression for the radiative forcing from CO2 doubling as

$\Delta
F_{2\times}\approx\frac{\pi\ln(2)k_{B}T|\Delta_{F}|}{h\nu_{2}}[\mathcal{B}(\nu_{2},T_{s})-\mathcal{B}(\nu_{2},T_{t})]\approx
5.1~{}\mbox{W~{}m${}^{-2}$}.$ (36)

Interestingly, almost all the background theory required to derive (36) has
existed since the first numerical CO2 radiative forcing calculations were
performed in the 1960s. While this equation does not provide us with any truly
new information, it does show us that the key driver of anthropogenic climate
change can be expressed purely in terms of measured properties of Earth’s
atmosphere, and fundamental properties of the CO2 molecule, without the need
for any numerical calculations.

##  6 Climate Sensitivity

The preceding analysis shows that the value of CO2 radiative forcing can be
understood in terms of the quantum properties of the CO2 molecule, with the
influence of Fermi resonance on the band structure coefficient $w$
particularly critical. To tie this analysis back to (2) in the introduction
and link global warming to radiative forcing, the last thing we need is an
expression for the climate feedback parameter $\lambda$. Analytic approaches
to this problem have been discussed in detail in other recent works (Ingram,
2010; Seeley & Jeevanjee, 2021; Jeevanjee et al., 2021a; Koll et al., 2022;
Stevens & Kluft, 2022; Jeevanjee, 2023), so we only provide a very brief
summary here, in the interests of completeness.

The simplest possible definition of $\lambda$ occurs on an airless planet,
where all thermal emission comes from the surface. If the planet is also a
blackbody, we have

$\lambda=\frac{dF}{dT_{s}}=4\sigma T_{s}^{3}.$ (37)

This yields $\lambda\approx 5.4$ W m-2 K-1 for $T_{s}=288$ K. This gives us an
upper limit on $\lambda$ on Earth, because of course in many spectral regions
the atmosphere is optically thick, and radiation to space occurs much higher,
where air is colder and thermal emission is lower.

In-depth analysis in the papers cited above shows that under present-day
conditions, emission in the mid-infrared H2O window region is the dominant
contributor to $\lambda$, so (37) can be approximated by

$\lambda=\frac{d}{dT_{s}}\int_{\nu_{A}}^{\nu_{B}}\pi\mathcal{B}(\nu,T_{s})d\nu.$
(38)

where $\nu_{A}$ and $\nu_{B}$ are the frequency limits of the window region.
Using the Wien approximation, this integral can be written as

$\lambda\approx\frac{2\pi
h^{2}\nu_{w}^{4}\Delta\nu_{w}}{k_{B}c^{2}T^{2}}\mathrm{e}^{-h\nu_{w}/k_{B}T},$
(39)

where $\nu_{w}=(\nu_{A}+\nu_{B})/2$ and $\Delta\nu_{w}=\nu_{B}-\nu_{A}$.

Writing141414We leave the derivation of $\nu_{A}$ and $\nu_{B}$ from the
quantum mechanical properties of the H2O molecule to future work. $\nu_{A}=21$
THz ($700$ cm-1) and $\nu_{B}=36$ THz (1200 cm-1), we obtain $\lambda=2.3$ W
m-2 K-1, which is within the range for the clear-sky longwave feedback from
observations and complex climate models (Andrews et al., 2012; McKim et al.,
2021; Roemer et al., 2023). Combining this estimate with the value for $\Delta
F_{2\times}$ calculated in the last section yields $\Delta T_{s}=2.2$ K. This
is also reasonably close to values calculated using much more complicated
models, despite the approximate nature of the calculation. As mentioned in the
Introduction, the range of modern estimates for $\Delta T_{s}$ extends to
higher values (2 to 5 K), in part because cloud feedbacks are predicted to
increase warming in the future.

##  7 Discussion

We have shown using mostly first-principles reasoning how the radiative
forcing of CO2 emerges from the quantum mechanical properties of the CO2
molecule. This result has implications for our understanding of both
contemporary global warming and the long-term evolution of Earth’s climate.
There are of course many things that our analysis misses out. Many
spectroscopic details, including anharmonic interactions, line mixing, and
additional weak bands have been neglected, as have overlap with other gaseous
absorbers and the radiative effects of clouds. In common with many other 1D
calculations, atmospheric vertical temperature structure has been treated
crudely, and 3D dynamics is neglected entirely. Given all this, it is
remarkable that our analysis and others like it still allows a reasonably
accurate estimate of clear-sky radiative forcing and climate sensitivity. This
outcome provides further evidence, if such evidence were needed, of the rock-
solid foundation of the physics of global warming and climate change.

The work presented here should not be seen as a substitute for accurate
numerical calculations, but instead a way to understand the fundamental
physics that underpins them. As our approach to spectroscopy in this paper is
rather specific to the problem at hand, it is not immediately clear if there
are many future applications for doing calculations from first principles in
this way. Nonetheless, it would be interesting to see if our approach could be
extended to CO2 radiative forcing on other solar system planets. Mars is
likely to be an easier case than Venus in the solar system, because of the
importance of CO2 collision-induced absorption (CIA) in Venus’s thick
atmosphere, although in principle a simplified approach to CIA along the lines
we have pursued here should also be possible.

Another interesting extension could be to use the analysis in Section 3 to
provide quick estimates of the warming potential of different greenhouse gases
in a planetary context. For exoplanet and paleoclimate applications, this
might be a particularly useful way of increasing intuition and providing a
reality check on the results of complex climate models. It might also be
interesting to investigate in more detail why resonances appear in certain
molecules and bands but not in others.

Finally, carbon dioxide has likely been a key greenhouse gas throughout
Earth’s history. Given this, the dependence of CO2 radiative forcing on the
accidental resonance between $\nu_{1}$ and $\nu_{2}$ is particularly
interesting. One can imagine that with minor differences in the quantum
structure of CO2, this resonance might be changed or inhibited, and the past
and future evolution of our planet’s climate would be very different. In this
sense, the CO2 Fermi resonance may be seen as somewhat analogous to the
nuclear resonances in astrophysics that give rise to the production of heavy
elements in stellar interiors (Hoyle, 1954; Livio et al., 1989).

We thank the organizers of the PCTS 2022 ‘From Spectroscopy to Climate:
Radiative Constraints on the General Circulation’ workshop for thought-
provoking discussions that helped to motivate this article. We also thank
Nadir Jeevanjee and an anonymous reviewer for insightful comments that
significantly improved the quality of this manuscript. RW acknowledges
additional discussions with Iouli Gordon on HITRAN unit conversion and related
topics. RW and JS acknowledge funding from NSF awards AST-1847120 (CAREER) and
AGS-2210757. Code to reproduce the plots in the paper is available open-source
at _https://github.com/wordsworthgroup/quantum_global_warming_.

## References

  * Adel & Dennison (1933) Adel, A., & Dennison, D. M. 1933, Physical Review, 43, 716
  * Andrews et al. (2012) Andrews, T., Gregory, J. M., Webb, M. J., & Taylor, K. E. 2012, Geophysical Research Letters, 39
  * Bernath (2020) Bernath, P. F. 2020, Spectra of atoms and molecules (Oxford University Press)
  * Chapman & Cowling (1990) Chapman, S., & Cowling, T. G. 1990, The mathematical theory of non-uniform gases: An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases (Cambridge University Press)
  * Dennison (1940) Dennison, D. M. 1940, Reviews of Modern Physics, 12, 175
  * Dufresne et al. (2020) Dufresne, J.-L., Eymet, V., Crevoisier, C., & Grandpeix, J.-Y. 2020, Journal of Climate, 33, 3827
  * Fermi (1931) Fermi, E. 1931, Zeitschrift für Physik, 71, 250
  * Friedlingstein et al. (2022) Friedlingstein, P., Jones, M. W., O’Sullivan, M., et al. 2022, Earth System Science Data, 14, 1917
  * Goody & Yung (1995) Goody, R. M., & Yung, Y. L. 1995, Atmospheric radiation: Theoretical basis (Oxford University Press)
  * Gordon et al. (2017) Gordon, I. E., Rothman, L. S., Hill, C., et al. 2017, Journal of Quantitative Spectroscopy and Radiative Transfer, 203, 3
  * Gordon et al. (2022) Gordon, I. E., Rothman, L. S., Hargreaves, R. J., et al. 2022, Journal of Quantitative Spectroscopy and Radiative Transfer, 277, 107949
  * Herzberg (1945) Herzberg, G. 1945, Molecular spectra and molecular structure. Vol. 2: Infrared and Raman spectra of polyatomic molecules (Van Nostrand Reinhold Company Inc.)
  * Hoyle (1954) Hoyle, F. 1954, Astrophysical Journal Supplement, 1, 121
  * Ingram (2010) Ingram, W. 2010, Quarterly Journal of the Royal Meteorological Society, 136, 30
  * Jeevanjee (2023) Jeevanjee, N. 2023, Authorea Preprints
  * Jeevanjee et al. (2021a) Jeevanjee, N., Koll, D. D. B., & Lutsko, N. J. 2021a, Geophysical Research Letters, 48, e2021GL093699
  * Jeevanjee et al. (2021b) Jeevanjee, N., Seeley, J. T., Paynter, D., & Fueglistaler, S. 2021b, Journal of Climate, 34, 9463
  * Koll et al. (2022) Koll, D. D. B., Jeevanjee, N., & Lutsko, N. J. 2022, Authorea
  * Levine (1975) Levine, I. N. 1975, Molecular spectroscopy (John Wiley & Sons)
  * Livio et al. (1989) Livio, M., Hollowell, D., Weiss, A., & Truran, J. W. 1989, Nature, 340, 281
  * Masson-Delmotte et al. (2021) Masson-Delmotte, V., Zhai, P., Pirani, A., et al. 2021, Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change, 2
  * McKim et al. (2021) McKim, B. A., Jeevanjee, N., & Vallis, G. K. 2021, Geophysical Research Letters, 48, e2021GL094074
  * Minzner (1977) Minzner, R. A. 1977, Reviews of Geophysics, 15, 375
  * Mlynczak et al. (2016) Mlynczak, M. G., Daniels, T. S., Kratz, D. P., et al. 2016, Geophysical research letters, 43, 5318
  * Muenter (1975) Muenter, J. S. 1975, Journal of Molecular Spectroscopy, 55, 490
  * Myhre et al. (1998) Myhre, G., Highwood, E. J., Shine, K. P., & Stordal, F. 1998, Geophysical Research Letters, 25, 2715
  * Oberly et al. (1968) Oberly, R., Rao, K. N., Hahn, Y. H., & McCubbin Jr, T. K. 1968, Journal of Molecular Spectroscopy, 25, 138
  * Pierrehumbert (2011) Pierrehumbert, R. T. 2011, Principles of Planetary Climate (Cambridge University Press)
  * Pinnock et al. (1995) Pinnock, S., Hurley, M. D., Shine, K. P., Wallington, T. J., & Smyth, T. J. 1995, Journal of Geophysical Research: Atmospheres, 100, 23227
  * Robinett (2006) Robinett, R. 2006, Quantum mechanics: Classical results, modern systems, and visualized examples (Oxford University Press)
  * Rodriguez-Garcia et al. (2007) Rodriguez-Garcia, V., Hirata, S., Yagi, K., et al. 2007, The Journal of Chemical Physics, 126, 124303
  * Roemer et al. (2023) Roemer, F. E., Buehler, S. A., Brath, M., Kluft, L., & John, V. O. 2023, Nature Geoscience, 1
  * Romps et al. (2022) Romps, D. M., Seeley, J. T., & Edman, J. P. 2022, Journal of Climate, 35, 4027
  * Seeley & Jeevanjee (2021) Seeley, J. T., & Jeevanjee, N. 2021, Geophysical Research Letters, 48, e2020GL089609
  * Shine & Perry (2023) Shine, K., & Perry, G. 2023, Quarterly Journal of the Royal Meteorological Society, 1856
  * Šimečková et al. (2006) Šimečková, M., Jacquemart, D., Rothman, L. S., Gamache, R. R., & Goldman, A. 2006, Journal of Quantitative Spectroscopy and Radiative Transfer, 98, 130
  * Stamnes et al. (2017) Stamnes, K., Thomas, G. E., & Stamnes, J. J. 2017, Radiative transfer in the atmosphere and ocean (2nd ed.) (Cambridge University Press)
  * Stevens & Kluft (2022) Stevens, B., & Kluft, L. 2022, EGUsphere, 1
  * Tziperman (2022) Tziperman, E. 2022, Global Warming Science: A Quantitative Introduction to Climate Change and Its Consequences (Princeton University Press)
  * Wilson & Gea-Banacloche (2012) Wilson, D. J., & Gea-Banacloche, J. 2012, American Journal of Physics, 80, 306
  * Zhong & Haigh (2013) Zhong, W., & Haigh, J. D. 2013, Weather, 68, 100


[4]Zhu Rong

1]Wuhan Maritime Communication Research Institute, Wuhan, China

2]Chongqing Jiaotong University, Chongqing, China

3]Hong Kong-Zhuhai-Macao Bridge Authority, Zhuhai, China

4]Wuhan University, Wuhan, China

# Skipped Feature Pyramid Network with Grid Anchor for Object Detection

Li Pengfei<EMAIL_ADDRESS>Wei Wei<EMAIL_ADDRESS>Yan Yu<EMAIL_ADDRESS><EMAIL_ADDRESS>Zhou Liguo<EMAIL_ADDRESS>[  [  [  [

###### Abstract

CNN-based object detection methods have achieved significant progress in
recent years. The classic structures of CNNs produce pyramid-like feature maps
due to the pooling or other re-scale operations. The feature maps in different
levels of the feature pyramid are used to detect objects with different
scales. For more accurate object detection, the highest-level feature, which
has the lowest resolution and contains the strongest semantics, is up-scaled
and connected with the lower-level features to enhance the semantics in the
lower-level features. However, the classic mode of feature connection combines
the feature of lower-level with all the features above it, which may result in
semantics degradation. In this paper, we propose a skipped connection to
obtain stronger semantics at each level of the feature pyramid. In our method,
the lower-level feature only connects with the feature at the highest level,
making it more reasonable that each level is responsible for detecting objects
with fixed scales. In addition, we simplify the generation of anchor for
bounding box regression, which can further improve the accuracy of object
detection. The experiments on the MS COCO and Wider Face demonstrate that our
method outperforms the state-of-the-art methods.

###### keywords:

CNN, Feature Pyramid Network, Object Detection, Face Detection

##  1 Introduction

Object detection is an important task in computer vision and has been widely
studied in the past decades. Nowadays, many emerging applications, such as
autonomous driving [1, 2] and face recognition [3, 4], hinge on object
detection. In recent years, Convolutional Neural Networks (CNN) [5] have
achieved significant progress in object detection. An object detection network
can usually be divided into two parts. One part is used to extract features,
and the other part is used to predict the objects’ bounding boxes. The
performance of object detection relies on better feature extraction. For
improving the performance of CNN in feature extracting, more and more
complicated components are proposed. However, the most popular networks always
have a similar structure to the very original LeNet [6], which contains
several pooling operations and produces pyramid-like feature maps. In the
feature pyramid, the higher-level features, which have lower-resolution, have
stronger semantics, while the lower-level features, which have higher-
resolution, have weaker semantics. R-CNN series [7, 8, 9] use the highest-
level feature, which has the strongest semantics, to detect objects. For
detecting multi-scale objects accurately, the image is resized to different
sizes and then input the network repeatedly, which increases the time cost.
The SSD-based methods [10] use features of multi-levels to detect objects, and
this method is more effective for detecting multi-scale objects.

(a) FPN

(b) SFPN

Figure 1: Comparison between FPN and Skipped FPN (SFPN).

In order to further improve the accuracy of object detection, Feature Pyramid
Network (FPN) [11] is introduced to enhance the semantics of features at all
levels. As shown in Figure 1(a), FPN up-scales the feature of highest-level
and fuses it with the feature of the lower-level. Then the fusion result is
up-scaled to fuse with the feature of more lower-level. All the needed low-
level features are fused with the feature of the highest-level and obtain
stronger semantics. The Path Aggregation Network (PAN) [12] extents FPN by
fusing the features again from bottom to top. In PAN, the lowest-level feature
of FPN is down-scaled to fuse with the features of higher-levels. Although PAN
is just a reverse operation of FPN, it can also enhance the semantics in the
features which are used to detect multi-scale objects.

In this paper, we modify the FPN by reconnecting features. Figure 1 shows the
difference between FPN and our method. We connect the feature of highest-level
with the features of lower-levels directly. The feature of each lower-level is
fused with the feature of highest-level, but has no relationship with other
levels. The semantics of the highest-level can be preserved to the greatest
extent. In addition, the features with different resolutions are supposed to
detecting objects of different sizes. The points in the features of lower-
levels have smaller Receptive Field [13]. It is suitable to use lower features
to detect smaller objects. The points in the features of higher-levels have
larger Receptive Field, so the higher features are dedicated to detecting
larger objects. Our method can reduce mutual interference by cutting the
relationship between the features of different levels. In addition, we modify
the anchor generation and bounding-box regression to get a better accuracy of
object detection. We test our method on MS COCO [14] and Wider Face [15]. The
experiments demonstrate that our method outperforms the state-of-the-art
methods.

##  2 Related Works

###  2.1 Object detection

Since the R-CNN [7] succeeded in object detection, various excellent object
detection algorithms have emerged one after another. Object detection
algorithms can be broadly divided into two categories: one-stage and two-
stage. R-CNN and its improved versions, Fast R-CNN [8] and Faster R-CNN [9],
all belong to the two-stage. They firstly search the region proposals and then
classify the contents in the proposals. SSD [10] and YOLO [16] are two
representative one-stage methods. They are based on global regression or
classification in which the image pixels are mapped directly to bounding box
coordinates and class probabilities. The two-stage methods are more time-
consuming than the one-stage methods because of the existence of the region
proposal part. Both one-stage and two-stage methods contain bounding-box
regression for accurate detection.

###  2.2 Face detection

As a special object detection task, the progress of face detection benefits
from the development of object detection. Many CNN-based object detection
methods are used in face detection. For improving the accuracy on the popular
face benchmarks [17, 15], loads of complicated components are added to the
networks, which result in the degradation of speed. PyramidBox [18] proposes a
context-assisted single-shot face detector. DSFD [19] introduces a feature
enhance module to extend the single-shot detector [10] to the dual shot
detector. EXTD [20] generates the feature maps by iteratively reusing a shared
lightweight and shallow backbone network instead of a single backbone network.
These SSD-based [10] face detectors all get high accuracy on the Wider Face
benchmark. Based on the YOLOv5 [21] object detector, YOLO5Face [22] achieves a
better performance on the Wider Face benchmark by adding a five-point landmark
regression head and using the Wing loss function [23].

##  3 Method

Figure 2: The object detection network with Skipped FPN (SFPN). The  indicates
the connections are cut off and the dotted arrow lines are the new
connections.

###  3.1 Network

We use CSPNet [24] in YOLOv5 [21] as the base framework of our method. Our
network is shown in Fig. 2. The network outputs three feature maps for object
detection. The feature extracting part includes a backbone, FPN [11], and PAN
[12]. The Focus [25], CSP [24], and SPP [26] are introduced for better feature
extraction. The Conv with S=2 and Upscale operation layers downscale the
feature map to 1/4 and enlarge the feature map to 4 times, respectively. The
operation layers with the same color output feature maps with the same size.
At the end of the network, there are three branches for detecting objects. The
widths and heights of Feature1-3 are 1/8, 1/16, and 1/32 of the width and
height of the input image.

According to the description of SFPN in Figure 1(b), we cut some connections
and rebuild some connections as the  and dotted arrow lines indicate in Figure
2.

(a) Image and Its Feature Map Pyramid Produced by CNN.

(b) The White Squares are the Anchors Selected According to the Feature Maps

Figure 3: (a) An image and its feature map pyramid produced by CNN. The green
box is the ground-truth bounding-box. (b) The image is divided into grid
images according to the sizes of the feature maps. In each grid image, one
grid that includes the center point of the ground-truth and two grids closest
to the center point of the ground-truth are selected as the anchors.

###  3.2 Grid Anchor

We first divide the input image into grids according to the sizes of the
feature maps. As shown in Fig. 3(b), if the size of input image is $w\times
h$, Feature1-3 divide the image into four grid images with $w/8\times h/8$,
$w/16\times h/16$, and $w/32\times h/32$ grids. In each grid image, the three
white grids are selected as the anchors. One includes the center point of the
ground-truth, and two are closest to the center point of the ground-truth. For
Feature1-3, the sizes of the anchors are $8\times 8$, $16\times 16$, and
$32\times 32$. If the anchor size is $d\times d$ in one branch, the ground-
truth whose longer side is longer than $d^{2}$ should be ignored when
selecting anchors.

###  3.3 Bounding-box Regression

Like YOLOv5 [21], we code the ground-truth bounding-box first. In pixel domain
of an image, the ground-truth bounding-box can be denoted as
$(t_{x},t_{y},t_{w},t_{h})$ and the anchor is
$(g_{x},g_{y},d,d),d\in\\{8,16,32\\}$. $(t_{x},t_{y})$ is the center point of
the ground-truth and $(g_{x},g_{y})$ is the upper-left corner of the anchor.
$(t_{w},t_{h})$ and $(d,d)$ are the widths and heights. We code the ground-
truth to $T_{c}(t_{x\\_c},t_{y\\_c},t_{w\\_c},t_{h\\_c})$ by (1).

$\begin{split}&t_{x\\_c}=\frac{t_{x}-g_{x}}{d}\\\
&t_{y\\_c}=\frac{t_{y}-g_{y}}{d}\\\ &t_{w\\_c}=\frac{t_{w}}{d}\\\
&t_{h\\_c}=\frac{t_{h}}{d}\end{split}$ (1)

The $(t_{x\\_c},t_{y\\_c})$ is the relative position to the upper-left corner
of the anchor in the feature domain and the values are between -0.5 and 1.5.
$(t_{w\\_c},t_{h\\_c})$ is the width and height of the ground-truth in the
feature domain.

At the end of the network, the channel of the feature map is transformed to 5.
If a feature point, which is a 5-dimension vector
$(z_{0},z_{1},z_{2},z_{3},z_{4})$, corresponds to a selected anchor, it is
used for predicting the bounding-box. The vector should be divided into two
parts. The first four are transformed to denote the predicted coded bounding-
box. The last one represents the score of the bounding-box. We first normalize
the values to between 0 and 1 by Sigmoid (2), and the result is
$(z_{0}’,z_{1}’,z_{2}’,z_{3}’,z_{4}’)$.

$z^{\prime}=\frac{1}{1+e^{-z}}$ (2)

The predicted coded bounding-box
$P_{c}(p_{x\\_c},p_{y\\_c},p_{w\\_c},p_{h\\_c})$ can be get by (3).

$\begin{split}&p_{x\\_c}=2z_{0}^{\prime}-0.5\\\
&p_{y\\_c}=2z_{1}^{\prime}-0.5\\\ &p_{w\\_c}=\frac{d^{2z_{2}^{\prime}}}{d}\\\
&p_{h\\_c}=\frac{d^{2z_{3}^{\prime}}}{d}\end{split}$ (3)

We use CIoU [27] to measure the predicted bounding-box. The regression loss of
each branch in the network is defined in (4),

$L_{reg}^{(d)}=\frac{1}{N^{(d)}}\sum_{n=1}^{N^{(d)}}1-CIoU[(P{{}_{c}}^{(n)},T{{}_{c}}^{(n)})]$
(4)

where $N^{(d)}$ is the number of anchors that are selected for bounding-box
regression in one batch. The regression loss of the whole network is (5).

$L_{reg}=L_{reg}^{(8)}+L_{reg}^{(16)}+L_{reg}^{(32)}$ (5)

The score of the bounding-box can be represented by CIoU. The score loss of
each branch in the network is defined in (6).

$\displaystyle
L_{score}^{(d)}=-\frac{1}{N^{(d)}}\sum_{n=1}^{N^{(d)}}[(1-CIoU^{(n)})\log(1-z_{4}^{{}^{\prime}(n)})$
(6) $\displaystyle+CIoU^{(n)}\log(z_{4}^{{}^{\prime}(n)})]$

The score loss of the whole network is

$L_{score}=L_{score}^{(8)}+L_{score}^{(16)}+L_{score}^{(32)}$ (7)

The loss function of the whole network is (8),

$L=L_{score}+L_{reg}+\lambda||W||^{2}$ (8)

where $W$ are the weights in the network and $\lambda||W||^{2}$ is added to
avoid over-fitting. In testing, the predicted bounding-box
$(p_{x},p_{y},p_{w},p_{h})$ can be obtained by (9).

$\begin{split}&p_{x}=(2z_{0}^{\prime}-0.5)^{\prime}d+g_{x}\\\
&p_{y}=(2z_{1}^{\prime}-0.5)^{\prime}d+g_{y}\\\ &p_{w}=d^{2z_{2}^{\prime}}\\\
&p_{h}=d^{2z_{3}^{\prime}}\end{split}$ (9)

###  3.4 Post Processing

In testing, if an image is detected as containing objects, each branch will
output several predicted bounding-boxes for each object. We take three steps
to select the most appropriate bounding-boxes. Firstly, we filter out the
over-large bounding-boxes, whose longer side-lengths are longer than $d^{2}$,
in the outputs of the branches of Feature1-2. Secondly, we use the Non-Maximum
Suppression (NMS) to select the best bounding-box in each branch. Finally, we
use NMS again to select the best bounding-box among all branches.

##  4 Experiment

(a) MS COCO

(b) Wider Face

Figure 4: Examples in MS COCO and Wider Face.

We train and test our method on MS COCO and Wider Face benchmarks to compare
with the state-of-the-art methods. The MS COCO dataset contains more than
200,000 images and 80 object categories which are split into three subsets
train, validation, and test. All object instances are annotated with a
bounding box. Annotations on the training and validation sets (with over
500,000 objects) are publicly available. Figure 4(a) shows a sample of MS
COCO. The WIDER FACE contains 32,203 images and 393,703 labeled faces with a
high degree of variability in scale, pose, and occlusion, as shown in Figure
4(b). The whole dataset is divided into training, validation, and testing
sets. The validation and testing sets are split into three subsets (easy,
medium, and hard).

The network is trained on Nvidia V100 GPUs with PyTorch [28]. The hyper-
parameters of the backbone are the same as the YOLOv5x [21]. The hyper-
parameters of the other part are determined by the backbone. For data
preprocessing, we normalize the pixel values to [0,1] and use Mosaic data
augmentation [29]. The size of the image is randomly re-scaled from 0.5 to
1.5. The batch stochastic gradient descent [6] is used in training.

Model | Size | mAP val 50-95 | mAP val 50 | Params (M) | FLOPs @640 (B)
---|---|---|---|---|---
YOLOv5x | 640 | 50.7 | 68.9 | 86.7 | 205.7
YOLOv5xSfpn | 640 | 51.7 | 70.2 | 86.7 | 205.7
YOLOv5xSfpnGa | 640 | 52.0 | 70.8 | 86.7 | 205.7
Table 1: Comparison of Object Detection Results on MS COCO.

Table 1 shows the comparison between our method and YOLOv5x original method.
YOLOv5Sfpn is the modified YOLOv5 with SFPN, and YOLOv5SfpnGa is YOLOv5Sfpn
with grid anchor. The results show that SFPN can improve the Mean Average
Precision (mAP) of object detection and grid anchor can further improve mAP
slightly.

Wider Face releases the results of state-of-the-art methods. Figure 5 shows
the comparison between our method and state-of-the-art. YOLO5Face [22] uses
YOLOv5x’s method to do face detection. In Figure 5, YOLOv5Sfpn outperforms
YOLO5Face [22] and YOLOv5SfpnGa outperforms YOLOv5Sfpn. Both YOLOv5Sfpn and
YOLOv5SfpnGa reach the state-of-the-art and even outperform the state-of-the-
art in some subsets.

(a) Easy-Val

(b) Easy-Test

(c) Medium-Val

(d) Medium-Test

(e) Hard-Val

(f) Hard-Test

Figure 5: PR Curves on the Validation and Testing Sets of Wider Face.

##  5 Conclusion

CNN produces a feature pyramid, and features in different levels have
Receptive Field with different sizes. The features in different levels are
used to detect faces with different scales. To enhance the semantics of the
lower-level features, the highest-level feature is fused into the lower-level
features. In this paper, we change the FPN to SFPN by connecting lower-level
features with the highest-level feature directly. This method can enhance the
semantics of the lower-level features and reduce the mutual influence between
the lower-levels. Each level of the feature pyramid is more dedicated to
detecting objects with sizes corresponding to the resolution of the feature.
By combining the grid anchor, the Skipped FPN can further improve the
performance of object detection.

##  6 Acknowledgement

This research is funded by the National Key Research and Development Program
of China (2019YFB1600700).

## References

  *   * Cao et al. [2022] Cao, W., Zhou, L., Huang, Y., Knoll, A.: Autonomous driving simulator based on neurorobotics platform. arXiv preprint arXiv:2301.00089 (2022)
  * Zhou et al. [2023] Zhou, L., Song, Y., Gao, Y., Yu, Z., Liu, H., Liu, H., Liu, L., Ma, L., Liu, Y., Li, H., et al.: Garchingsim: An autonomous driving simulator with photorealistic scenes and minimalist workflow. In: 26th IEEE International Conference on Intelligent Transportation Systems (ITSC 2023) (2023)
  * Li and Wu [2022] Li, K., Wu, G.: Randomized approximate class-specific kernel spectral regression analysis for large-scale face verification. Machine Learning 111(6), 2037–2091 (2022)
  * Eliades et al. [2019] Eliades, C., Lenc, L., Král, P., Papadopoulos, H.: Automatic face recognition with well-calibrated confidence measures. Machine Learning 108, 511–534 (2019)
  * Krizhevsky et al. [2012] Krizhevsky, A., Sutskever, I., Hinton, G.E.: Imagenet classification with deep convolutional neural networks. In: Advances in Neural Information Processing Systems, pp. 1097–1105 (2012)
  * LeCun et al. [1998] LeCun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998)
  * Girshick et al. [2014] Girshick, R., Donahue, J., Darrell, T., Malik, J.: Rich feature hierarchies for accurate object detection and semantic segmentation. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 580–587 (2014)
  * Wang et al. [2017] Wang, X., Shrivastava, A., Gupta, A.: A-fast-rcnn: Hard positive generation via adversary for object detection. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2606–2615 (2017)
  * Ren et al. [2015] Ren, S., He, K., Girshick, R., Sun, J.: Faster r-cnn: Towards real-time object detection with region proposal networks. arXiv preprint arXiv:1506.01497 (2015)
  * Liu et al. [2016] Liu, W., Anguelov, D., Erhan, D., Szegedy, C., Reed, S., Fu, C.-Y., Berg, A.C.: Ssd: Single shot multibox detector. In: European Conference on Computer Vision, pp. 21–37 (2016). Springer
  * Lin et al. [2017] Lin, T.-Y., Dollár, P., Girshick, R., He, K., Hariharan, B., Belongie, S.: Feature pyramid networks for object detection. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2117–2125 (2017)
  * Liu et al. [2018] Liu, S., Qi, L., Qin, H., Shi, J., Jia, J.: Path aggregation network for instance segmentation. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 8759–8768 (2018)
  * Alonso and Chen [2009] Alonso, J.-M., Chen, Y.: Receptive field. Scholarpedia 4(1), 5393 (2009)
  * Lin et al. [2014] Lin, T.-Y., Maire, M., Belongie, S., Hays, J., Perona, P., Ramanan, D., Dollár, P., Zitnick, C.L.: Microsoft coco: Common objects in context. In: Computer Vision–ECCV 2014: 13th European Conference, Zurich, Switzerland, September 6-12, 2014, Proceedings, Part V 13, pp. 740–755 (2014). Springer
  * Yang et al. [2016] Yang, S., Luo, P., Loy, C.-C., Tang, X.: Wider face: A face detection benchmark. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5525–5533 (2016)
  * Bochkovskiy et al. [2020] Bochkovskiy, A., Wang, C.-Y., Liao, H.-Y.M.: Yolov4: Optimal speed and accuracy of object detection. arXiv preprint arXiv:2004.10934 (2020)
  * Jain and Learned-Miller [2010] Jain, V., Learned-Miller, E.: Fddb: A benchmark for face detection in unconstrained settings. Technical report, UMass Amherst technical report (2010)
  * Tang et al. [2018] Tang, X., Du, D.K., He, Z., Liu, J.: Pyramidbox: A context-assisted single shot face detector. In: Proceedings of the European Conference on Computer Vision (ECCV), pp. 797–813 (2018)
  * Li et al. [2019] Li, J., Wang, Y., Wang, C., Tai, Y., Qian, J., Yang, J., Wang, C., Li, J., Huang, F.: Dsfd: dual shot face detector. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 5060–5069 (2019)
  * Yoo et al. [2019] Yoo, Y., Han, D., Yun, S.: Extd: Extremely tiny face detector via iterative filter reuse. arXiv preprint arXiv:1906.06579 (2019)
  * Jocher [2020] Jocher, G.: ultralytics/yolov5: V3.1 - Bug Fixes and Performance Improvements. https://doi.org/10.5281/zenodo.4154370 . https://doi.org/10.5281/zenodo.4154370
  * Qi et al. [2021] Qi, D., Tan, W., Yao, Q., Liu, J.: Yolo5face: Why reinventing a face detector. arXiv preprint arXiv:2105.12931 (2021)
  * Feng et al. [2018] Feng, Z.-H., Kittler, J., Awais, M., Huber, P., Wu, X.-J.: Wing loss for robust facial landmark localisation with convolutional neural networks. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2235–2245 (2018)
  * Wang et al. [2020] Wang, C.-Y., Liao, H.-Y.M., Wu, Y.-H., Chen, P.-Y., Hsieh, J.-W., Yeh, I.-H.: Cspnet: A new backbone that can enhance learning capability of cnn. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pp. 390–391 (2020)
  * Ridnik et al. [2021] Ridnik, T., Lawen, H., Noy, A., Ben Baruch, E., Sharir, G., Friedman, I.: Tresnet: High performance gpu-dedicated architecture. In: Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pp. 1400–1409 (2021)
  * He et al. [2015] He, K., Zhang, X., Ren, S., Sun, J.: Spatial pyramid pooling in deep convolutional networks for visual recognition. IEEE transactions on pattern analysis and machine intelligence 37(9), 1904–1916 (2015)
  * Zheng et al. [2020] Zheng, Z., Wang, P., Liu, W., Li, J., Ye, R., Ren, D.: Distance-iou loss: Faster and better learning for bounding box regression. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol. 34, pp. 12993–13000 (2020)
  * Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. arXiv preprint arXiv:1912.01703 (2019)
  * Bochkovskiy et al. [2020] Bochkovskiy, A., Wang, C.-Y., Liao, H.-Y.M.: Yolov4: Optimal speed and accuracy of object detection. arXiv preprint arXiv:2004.10934 (2020)