# Spectral line shape in the limit of frequent velocity-changing collisions

Nikodem Stolarczyk<EMAIL_ADDRESS>Piotr Wcisło  Roman Ciuryło
Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus
Copernicus University in Toruń, ul Grudziądzka 5, 87-100 Toruń, Poland

###### Abstract

The speed-dependent spectral line profiles collapse into a simple Lorentz
profile in the regime dominated by the velocity-changing collisions. We derive
general formulas for the effective width and shift of the Lorentzian for
arbitrary speed-dependent collisional broadening and shift and velocity-
changing collision operators. For a quadratic speed dependence of collisional
broadening and shift, and the billiard ball model of velocity-changing
collisions, we provide simple analytical expressions for the effective
Lorentzian width and shift. We show that the effective Lorentzian width and
shift split into components originating from the: well-known Dicke-narrowed
Doppler width, speed-averaged collisional broadening and shift, their speed
dependencies, and a product term that mixes the contributions of the
broadening and shift speed dependencies. We show how the components depend on
rates of speed-changing and velocity-changing collisions related to the
perturber/absorber mass ratio. We validate analytical formulas numerically on
example of H2 transition perturbed by He.
Keywords: molecular line shapes, inhomogeneous broadening and shift, velocity-
changing collisions, Dicke narrowing

##  I Introduction

The shape of molecular spectral line affected by Doppler broadening and
absorber-perturber collisions, in the general case, requires numerical
evaluation and cannot be represented by a simple analytical function, unless
some simplifications or assumptions are made [1]. The goal of this work is to
show that, in the case when the velocity-changing collisions dominate other
line-shape effects, the spectral line shape collapses to an ordinary Lorentz
profile and the expressions for its width and shift can be provided
analytically.

The Lorentz profile has been used for over a century to describe
collisionally-broadened atomic and molecular spectral lines. It is
particularly justified at high pressure of perturbers, much lighter than
absorbers, in microwave spectral range, where the speed dependence of
collisional broadening and shift [2, 3], as well as Doppler broadening, can be
neglected. Interestingly, also the Gaussian shape of the spectral line can
collapse to the Lorentz profile when the velocity-changing collisions reduce
the mean free path of the absorber well below the wavelength of absorbed
radiation as predicted by Dicke [4, 5] in the fifties. It was later
demonstrated numerically [6], that the weighted sum of Lorentz profiles (WSL)
[7, 8], under frequent velocity-changing collisions described by the billiard-
ball model [9, 10] approaches the Lorentz profile and the convergence is
faster for lower perturber/absorber mass ratio. It is related to relative
contribution of speed change during velocity-changing collisions which is
determined by perturber/absorber mass ratio [11, 12]. Recently, studies on
line shapes for which width is dominated by the speed-dependent collisional
shift led to formulating a simple analytical expression [13, 14] (see Eq. (15)
in Ref. [13]) for width of Lorentzian profile approximating such line shape
when speed-dependent collisional shift is described by a quadratic function
[15] and velocity-changing collisions are approximated by the hard-collision
model [16, 17]. The phenomenological finding from Ref. [13] got justification
in a derivation [18] coming from the speed-dependent hard-collision profile
[19, 20].

In this article, we generalize the results from Ref. [18]; we derive general
formulas for the effective width and shift of a Lorentzian to which a
sophisticated line shape model, based on any arbitrary speed-dependent
collisional broadening and shift and velocity-changing collision operator,
converges in the limit of frequent velocity-changing collisions. For quadratic
speed dependencies of collisional width and shift, and the billiard ball model
of velocity-changing collisions, we provide a simple analytical expression for
the effective Lorentzian width and shift. We show how their components depend
on rates of speed-changing and velocity-changing collisions related to the
perturber/absorber mass ratio. We validate the analytical formulas
numerically.

##  II Algebraic representation of a spectral line shape

In general, the shape of an isolated spectral line affected by Doppler
broadening and collisions with perturbers can be evaluated [21, 22] from a
function $h(\omega,\vec{v})$,

$I(\omega)=\frac{1}{\pi}\mbox{\rm
Re}\;\left\\{(1,h(\omega,\vec{v}))\right\\},$ (1)

where $(\cdot,\cdot)$ is defined as a product $(a(\vec{v}),b(\vec{v}))=\int
d^{3}\vec{v}f_{m_{A}}(\vec{v})a(\vec{v})b(\vec{v})$ of two functions
$a(\vec{v})$, $b(\vec{v})$ of absorber velocity $\vec{v}$,
$f_{m_{A}}(\vec{v})=(\pi v^{2}_{m_{A}})^{-3/2}\exp(-v^{2}/v^{2}_{m_{A}})$ is
Maxwellian distribution, $v_{m_{A}}=\sqrt{k_{B}T/(2m_{A})}$ is the most
probable speed of the absorber having mass $m_{A}$ at temperature $T$, and
$k_{B}$ is Boltzmann’s constant. Function $h(\omega,\vec{v})$ fulfils
transport-relaxation kinetic equation [22],

$1=-i(\omega-\omega_{0}-\vec{k}\cdot\vec{v})h(\omega,\vec{v})-\hat{S}^{f}h(\omega,\vec{v}),$
(2)

where $\omega_{0}$ is unperturbed transition frequency, $\vec{k}$ is a
wavevector of radiation, and operator $\hat{S}^{f}$ describes the effect of
collisions with perturbers.

Equations (1) and (2) can be converted into an algebraic form expanding
function $h(\omega,\vec{v})$,

$h(\omega,\vec{v})=\sum_{s=0}^{\infty}c_{s}(\omega)\varphi_{s}(\vec{v}),$ (3)

in a set of orthonormal functions fulfilling condition
$(\varphi_{s}(\vec{v}),\varphi_{s^{\prime}}(\vec{v}))=\delta_{s,s^{\prime}}$.
We set $\varphi_{0}(\vec{v})=1$. The expansion coefficients $c_{s}(\omega)$
depend only on frequency for given operator $\hat{S}^{f}$. In this basis, any
operator $\hat{A}$ can be represented by matrix A, having matrix elements
$[\mbox{\bf
A}]_{s,s^{\prime}}=(\varphi_{s}(\vec{v}),\hat{A}\varphi_{s^{\prime}}(\vec{v}))$.

The shape of an isolated spectral line can have algebraic representation by a
series of Lorentz profiles. Following the approaches from [21, 10, 23, 24] and
notation described in Refs. [22, 25, 6], the line shape can be written as [26,
25, 6]:

$I(\omega)=\frac{1}{\pi}\mbox{\rm Re}\;\left\\{c_{0}(\omega)\right\\},$ (4)

where the coefficient $c_{0}(\omega)$ can be evaluated by solving a set of
complex linear equations,

$\mbox{\bf\text@underline{b}}=\mbox{\bf
L}(\omega)\mbox{\bf\text@underline{c}}(\omega),$ (5)

for the coefficients $c_{s}(\omega)$, where $s=0,1,...,s_{\rm max}$. Here the
column b contains unity in the position 0 and zeros in other positions, i.e.
$[\mbox{\bf\text@underline{b}}]_{s}=\delta_{0,s}$, and the column
$\mbox{\bf\text@underline{c}}(\omega)$ consists of the coefficients
$c_{s}(\omega)$, i.e.
$[\mbox{\bf\text@underline{c}}(\omega)]_{s}=c_{s}(\omega)$. The matrix
$\mbox{\bf L}(\omega)$ depends on the frequency $\omega$ and has the following
form:

$\mbox{\bf L}(\omega)=-i(\omega-\omega_{0})\mbox{\bf 1}+i\mbox{\bf
K}-\mbox{\bf S}^{f},$ (6)

where $\omega_{0}$ corresponds to the unperturbed frequency of the transition,
1 is the unit matrix, $[\mbox{\bf 1}]_{s,s^{\prime}}=\delta_{s,s^{\prime}}$, K
is the matrix that represents the Doppler shift, $\mbox{\bf S}^{f}=\mbox{\bf
S}_{D}^{f}+\mbox{\bf S}_{VC}^{f}$ is the matrix which represents the collision
operator split into two components: $\mbox{\bf S}_{D}^{f}$ is the matrix which
represents the "dephasing and relaxation" [27, 28] collisional width and shift
and $\mbox{\bf S}_{VC}^{f}$ is the matrix which represents the "velocity-
changing" collision operator also affected by dephasing and relaxation. The
representation used here has a property $[\mbox{\bf
S}_{VC}^{f}]_{s,0}=[\mbox{\bf S}_{VC}^{f}]_{0,s}=0$ for any $s$ and
$[\mbox{\bf K}]_{0,0}=0$. Moreover, all matrices discussed in this work are
symmetric.

In practice, the coefficient $c_{0}(\omega)$ is calculated using the
diagonalization technique (c.f. [24, 29, 30]). To do it in this way, one needs
to find the full set of eigenvectors, $\mbox{\bf\text@underline{e}}_{j}$, and
corresponding eigenvalues $\varepsilon_{j}$ which fulfill the following
equation:

$(i\mbox{\bf K}-\mbox{\bf
S}^{f})\mbox{\bf\text@underline{e}}_{j}=\varepsilon_{j}\mbox{\bf\text@underline{e}}_{j},$
(7)

where $j=0,1,...,s_{\rm max}$. Once the eigenvectors and eigenvalues are
known, the coefficient $c_{0}(\omega)$ can be computed from the following
expression:

$c_{0}(\omega)=\sum_{j=0}^{s_{\rm
max}}\frac{\beta_{j}[\mbox{\bf\text@underline{e}}_{j}]_{0}}{\varepsilon_{j}-i(\omega-\omega_{0})},$
(8)

where the coefficients $\beta_{j}$ fulfill the relation
$\mbox{\bf\text@underline{b}}=\sum_{j=0}^{s_{\rm
max}}\beta_{j}\mbox{\bf\text@underline{e}}_{j}$. The main advantage of this
approach is that the time-consuming diagonalization can be carried out once
and this is sufficient to calculate the whole line shape.

##  III High frequency of the velocity-changing collisions limit

Our derivation is carried out in the limit where the velocity-changing
collisions dominate over the Doppler broadening and collisional broadening and
shift. We assume the absolute values of all matrix elements of K and
$\mbox{\bf S}_{D}^{f}$, as well as detuning, $\omega-\omega_{0}$, to be much
smaller than the absolute value of the effective optical frequency of velocity
changing collisions, $\nu_{\rm opt}$. In general, $\nu_{\rm opt}$ can be
complex due to the dephasing associated with optical velocity-changing
collisions. All nonzero matrix elements of $\mbox{\bf S}_{VC}^{f}$ are
proportional to $\nu_{\rm opt}$.

For the matrix $\mbox{\bf S}_{VC}^{f}$ we can calculate eigenvectors,
$\mbox{\bf\text@underline{e}}_{j}^{VC}$, and eigenvalues,
$\varepsilon_{j}^{VC}$, which fulfil the following equation:

$-\mbox{\bf
S}_{VC}^{f}\mbox{\bf\text<EMAIL_ADDRESS>(9)

It is easy to see that one of the eigenvectors of the matrix having property
$[\mbox{\bf S}_{VC}^{f}]_{s,0}=[\mbox{\bf S}_{VC}^{f}]_{0,s}=0$ is vector b
and its corresponding eigenvalue is zero. Therefore, we can set
$\mbox{\bf\text@underline{e}}_{0}^{VC}=\mbox{\bf\text@underline{b}}$ and
$\varepsilon_{0}^{VC}=0$. Consequently, for $j\neq 0$ we have
$[\mbox{\bf\text@underline{e}}_{j}^{VC}]_{0}=0$ and
$\varepsilon_{j}^{VC}\sim\nu_{\rm opt}$.

Now we can rewrite Eq. (7) in the form:

$(-\mbox{\bf S}^{f}_{VC}-\mbox{\bf S}^{f}_{D}+i\mbox{\bf
K})\mbox{\bf\text@underline{e}}_{j}=\varepsilon_{j}\mbox{\bf\text@underline{e}}_{j},$
(10)

where the matrix, which we want to diagonalize, is split into the dominating
part $\mbox{\bf S}^{f}_{VC}$ and the perturbation part containing $\mbox{\bf
S}^{f}_{D}$ and K. We take advantage of that and approximate eigenvectors
$\mbox{\bf\text@underline{e}}_{j}$ by $\mbox{\bf\text@underline{e}}_{j}^{VC}$
and eigenvalues $\varepsilon_{j}$ by $\varepsilon_{j}^{VC}$ and improve them
by perturbation corrections. Setting
$\mbox{\bf\text@underline{e}}_{j}\approx\mbox{\bf\text@underline{e}}_{j}^{VC}$
we get $\beta_{j}=\delta_{j,0}$. In this, way Eq. (8) can be approximated with
the Lorentz profile:

$c_{0}(\omega)\approx\frac{1}{\varepsilon_{0}-i(\omega-\omega_{0})},$ (11)

where the eigenvalue $\varepsilon_{0}$ is approximated by second-order
perturbation. The eigenvalues, $\varepsilon_{j}$, in the second-order
perturbation are given by the following expression:

$\varepsilon_{j}\approx\varepsilon_{j}^{VC}+\sum_{s,s^{\prime}=0}^{s_{\rm
max}}[\mbox{\bf\text@underline{e}}_{j}^{VC}]_{s}(-[\mbox{\bf
S}^{f}_{D}]_{s,s^{\prime}}+i[\mbox{\bf
K}]_{s,s^{\prime}})[\mbox{\bf\text@underline{e}}_{j}^{VC}]_{s^{\prime}}+\\\
+\sum_{\begin{subarray}{c}j^{\prime}=0\\\ j^{\prime}\neq
j\end{subarray}}^{s_{\rm max}}\frac{(\sum_{s,s^{\prime}=0}^{s_{\rm
max}}[\mbox{\bf\text@underline{e}}_{j}^{VC}]_{s}(-[\mbox{\bf
S}^{f}_{D}]_{s,s^{\prime}}+i[\mbox{\bf
K}]_{s,s^{\prime}})[\mbox{\bf\text@underline{e}}_{j^{\prime}}^{VC}]_{s^{\prime}})^{2}}{\varepsilon_{j}^{VC}-\varepsilon_{j^{\prime}}^{VC}}.$
(12)

This equation for $\varepsilon_{0}$ simplifies to the following form

$\varepsilon_{0}\approx-[\mbox{\bf S}^{f}_{D}]_{0,0}+\\\
+\sum_{j^{\prime}=1}^{s_{\rm max}}\frac{\\{\sum_{s^{\prime}=1}^{s_{\rm
max}}(-[\mbox{\bf S}^{f}_{D}]_{0,s^{\prime}}+i[\mbox{\bf
K}]_{0,s^{\prime}})[\mbox{\bf\text@underline{e}}_{j^{\prime}}^{VC}]_{s^{\prime}}\\}^{2}}{-\varepsilon_{j^{\prime}}^{VC}},$
(13)

remembering that $\varepsilon_{0}^{VC}=0$,
$[\mbox{\bf\text@underline{e}}_{0}^{VC}]_{s}=\delta_{0,s}$ and $[\mbox{\bf
K}]_{0,0}=0$. In this way, we get a single Lorentz profile,

$I(\omega)=\frac{1}{\pi}\frac{\Gamma_{\rm eff}}{\Gamma_{\rm
eff}^{2}+(\omega-\omega_{0}-\Delta_{\rm eff})^{2}},$ (14)

in the limit of frequent velocity-changing collisions, here

$\Gamma_{\rm eff}+i\Delta_{\rm eff}=\varepsilon_{0}.$ (15)

It should be noted that we can always use the basis in which matrix $\mbox{\bf
S}^{f}_{VC}$ is diagonal. In such case
$[\mbox{\bf\text@underline{e}}_{j}^{VC}]_{s}=\delta_{j,s}$,
$\varepsilon_{j}^{VC}=-[\mbox{\bf S}^{f}_{VC}]_{j,j}$ and Eq. (13) takes
simple form

$\varepsilon_{0}\approx-[\mbox{\bf S}^{f}_{D}]_{0,0}+\sum_{s=1}^{s_{\rm
max}}\frac{(-[\mbox{\bf S}^{f}_{D}]_{0,s}+i[\mbox{\bf
K}]_{0,s})^{2}}{[\mbox{\bf S}^{f}_{VC}]_{s,s}}.$ (16)

Furthermore, one can use a basis in which if a matrix element is nonzero,
$[\mbox{\bf S}^{f}_{D}]_{0,s}\neq 0$, then the corresponding matrix element is
zero, $[\mbox{\bf K}]_{0,s}=0$, and, if $[\mbox{\bf K}]_{0,s}\neq 0$ then
$[\mbox{\bf S}^{f}_{D}]_{0,s}=0$. It is a simple consequence of the symmetry
of the corresponding operators. The collisional broadening and shift depend on
the absolute value of the absorber velocity, $v=|\vec{v}|$, and are not
dependent on velocity direction. On the other hand, the Doppler shift is
proportional to the scalar product $\vec{k}\cdot\vec{v}$ which depends on
velocity direction. Taking this into account, we can rewrite Eq. (16) in the
following form:

$\varepsilon_{0}\approx-[\mbox{\bf S}^{f}_{D}]_{0,0}-\sum_{s=1}^{s_{\rm
max}}\frac{[\mbox{\bf K}]_{0,s}^{2}}{[\mbox{\bf
S}^{f}_{VC}]_{s,s}}+\sum_{s=1}^{s_{\rm max}}\frac{[\mbox{\bf
S}^{f}_{D}]_{0,s}^{2}}{[\mbox{\bf S}^{f}_{VC}]_{s,s}}.$ (17)

##  IV Application in case of quadratic speed-dependent collisional broadening
and shift

To get physical insight into the expressions derived above, we consider matrix
representation using Burnett functions [9] described in Appendix A. We assume
quadratic speed dependence of collisional broadening and shift [15]. It means
that the operator

$\hat{S}_{D}^{f}=-\Gamma(v)-i\Delta(v)$ (18)

is determined by $\Gamma(v)$ and $\Delta(v)$ given in the following form

$\Gamma(v)+i\Delta(v)=\Gamma_{0}+i\Delta_{0}+(\Gamma_{2}+i\Delta_{2})\left(\frac{v^{2}}{v_{m}^{2}}-\frac{3}{2}\right),$
(19)

where $\Gamma_{0}$ and $\Delta_{0}$ are the collisional broadening and shift
parameters, averaged over absorber velocity, respectively. Quadratic speed
dependencies of collisional broadening and shift are described by $\Gamma_{2}$
and $\Delta_{2}$ parameters, respectively.

Discussing the velocity-changing collisions, we will focus on the case where
matrix $\mbox{\bf S}^{f}_{VC}$ is diagonal or is approximated by a diagonal
matrix.

In the Burnett functions basis representation, instead of the index
$s=0,1,...,s_{\rm max}$, we prefer to use two other indices, $n=0,1,...,n_{\rm
max}$ and $l=0,1,...,l_{\rm max}$. The pair of indices $nl$ can be connected
with $s=n+(n_{\rm max}+1)l$ and $s_{\rm max}=n_{\rm max}+(n_{\rm max}+1)l_{\rm
max}$.

The properties of matrix elements in Burnett functions representation are
summarized in Appendix A. The assumptions made above constrain the number of
nonzero matrix elements which contribute to Eq. (17) to only a few,

$\varepsilon_{0}=-[\mbox{\bf S}^{f}_{D}]_{00,00}-\frac{[\mbox{\bf
K}]_{00,01}^{2}}{[\mbox{\bf S}^{f}_{VC}]_{01,01}}+\frac{[\mbox{\bf
S}^{f}_{D}]_{00,10}^{2}}{[\mbox{\bf S}^{f}_{VC}]_{10,10}}.$ (20)

Now we can explicitly express the matrix elements: $[\mbox{\bf
K}]_{00,01}=\omega_{D}\sqrt{1/2}$, where $\omega_{D}=kv_{m_{A}}$, $[\mbox{\bf
S}^{f}_{D}]_{00,00}=-(\Gamma_{0}+i\Delta_{0})$, and $[\mbox{\bf
S}^{f}_{D}]_{00,10}=-(\Gamma_{2}+i\Delta_{2})\sqrt{3/2}$.

###  IV.1 Hard collision model

To discuss the velocity-changing collisions we first consider the hard
collision (HC) model [16, 17], discussed recently in Ref. [18] in a similar
context. The HC model is frequently used due to its simplicity, despite its
shortcomings. In many situations, it helps to get some analytical results. The
operator $\hat{S}_{HC}^{f}$ describing hard velocity changing collisions has
the following form:

$\hat{S}_{HC}^{f}h(\omega,\vec{v})=-\nu_{\rm opt}h(\omega,\vec{v})+\nu_{\rm
opt}\int
d^{3}\vec{v}^{\prime}f_{m_{A}}(\vec{v}^{\prime})h(\omega,\vec{v}^{\prime}),$
(21)

where $\nu_{\rm opt}$ is the effective optical frequency of velocity-changing
collisions, which, in the general case, can be a complex number [17, 31, 32,
33]. It can be shown that in any orthonormal base assuming
$\varphi_{0}(\vec{v})=1$, the matrix representation of the $\hat{S}_{HC}^{f}$
operator is diagonal, $[\mbox{\bf S}^{f}_{HC}]_{s,s^{\prime}}=-\nu_{\rm
opt}\delta_{s,s^{\prime}}(1-\delta_{0,s})$. All its diagonal elements are
equal to $-\nu_{\rm opt}$, except $[\mbox{\bf S}^{f}_{HC}]_{0,0}=0$.
Therefore, also in the Burnett functions representation, we can write that
$[\mbox{\bf S}^{f}_{HC}]_{01,01}=[\mbox{\bf S}^{f}_{HC}]_{10,10}=-\nu_{\rm
opt}$. Inserting these matrix elements into Eqs. (20) and (15) we got

$\Gamma_{\rm eff}^{HC}+i\Delta_{\rm
eff}^{HC}=\Gamma_{0}+i\Delta_{0}+\frac{\omega_{D}^{2}}{2\nu_{\rm
opt}}-\frac{3\Gamma_{2}^{2}}{2\nu_{\rm opt}}+\frac{3\Delta_{2}^{2}}{2\nu_{\rm
opt}}-i\frac{3\Gamma_{2}\Delta_{2}}{\nu_{\rm opt}}.$ (22)

The above result has also been obtained in our recent paper with a different
method, see Eqs. (19a)–(19d) in Ref. [18], where we have shown that speed-
dependent hard-collision profile [20] collapses into a simple Lorentz profile
in the limit of frequent velocity-changing collisions.

###  IV.2 Soft collision model

In the case when perturbers are much lighter then absorbers, the velocity
changing collisions are described by soft collision (SC) model [34]. This
model was introduced by Galatry [35] into the theory of Dicke narrowed
spectral line shapes. By that time, the Galatry profile (GP) became one of the
most frequently used expression to describe collisionally-narrowed spectra.
The exact speed-dependent Galatry profile (SDGP) with quadratic speed
dependence of collisional width and shift was given in Ref. [36] and should be
not confused with its approximated expression provided by Prime et al [37]
using appproach from Ref. [38]. It was shown in Refs. [9, 10] that the
velocity changing soft collisions operator $\hat{S}_{HC}^{f}$ is represented
with a diagonal matrix operator in the Burnett functions basis. The relevant
matrix elements are given by the following expressions: $[\mbox{\bf
S}^{f}_{SC}]_{00,00}=0$, $[\mbox{\bf S}^{f}_{SC}]_{01,01}=-\nu_{\rm opt}$,
$[\mbox{\bf S}^{f}_{SC}]_{10,10}=-2\nu_{\rm opt}$, see Refs. [9, 10]. In
contrast to the HC model, the matrix elements $[\mbox{\bf
S}^{f}_{SC}]_{01,01}$ and $[\mbox{\bf S}^{f}_{SC}]_{10,10}$ are not identical.
Inserting these matrix elements into Eqs. (20) and (15) yields:

$\Gamma_{\rm eff}^{SC}+i\Delta_{\rm eff}^{SC}=\\\
\Gamma_{0}+i\Delta_{0}+\frac{\omega_{D}^{2}}{2\nu_{\rm
opt}}-\frac{3\Gamma_{2}^{2}}{4\nu_{\rm opt}}+\frac{3\Delta_{2}^{2}}{4\nu_{\rm
opt}}-i\frac{3\Gamma_{2}\Delta_{2}}{2\nu_{\rm opt}}.$ (23)

The three last terms of this equation are two times smaller then the ones
derived in case of the HC model, Eq. (22) [18].

Table 1: Coefficients $f_{D}$, $f_{v^{2}}$ and their ratio $f_{D}/f_{v^{2}}$ evaluated for different $\alpha$ within billiard ball model. $\alpha$ | $f_{D}$ | $f_{v^{2}}$ | $f_{D}/f_{v^{2}}$
---|---|---|---
0 | 1 | 1 | 1
1/100 | 1.00000331561954 | 1.00000991995975 | 0.99999339572530
1/50 | 1.00001319212471 | 1.00003935941918 | 0.99997383373542
1/20 | 1.00008114950708 | 1.00023998438455 | 0.99984120323127
1/10 | 1.00031613831745 | 1.00091992503335 | 0.99939676821212
1/5 | 1.00119948726834 | 1.00336387252056 | 0.99784287105456
1/4 | 1.00182496341562 | 1.00501525070746 | 0.99682563295473
1/3 | 1.00310178272716 | 1.00822551613664 | 0.99491806810334
1/2 | 1.00636139570140 | 1.01565719613513 | 0.99084750202223
2/3 | 1.01027159637697 | 1.02349086931942 | 0.98708413202431
1 | 1.01895378488101 | 1.03806540223391 | 0.98158919725889
3/2 | 1.03178515946401 | 1.05522030753985 | 0.97779122718887
2 | 1.04294734723661 | 1.06757150938264 | 0.97693441429486
3 | 1.05994565861 | 1.0835515889 | 0.9782142996
4 | 1.0717514674 | 1.093265697 | 0.980321134
5 | 1.08028126 | 1.09975074 | 0.982296461
10 | 1.101660 | 1.114423 | 0.9885474
20 | 1.11523 | 1.122707 | 0.99334
50 | 1.12464 | 1.1280 | 0.997
100 | 1.1280 | 1.1280 | 1
$\infty$ | $32/9\pi\approx 1.131$ | 1.131 | 1

###  IV.3 Billiard ball model

The billiard ball (BB) model [10] provides a more realistic description of the
velocity-changing collisions. This model properly accounts for the
perturber/absorber mass ratio, $\alpha=m_{p}/m_{a}$. The matrix describing
such velocity-changing collisions $[\mbox{\bf
S}^{f}_{BB}]_{nl,n^{\prime}l^{\prime}}=\nu_{\rm
opt}f_{D}M_{nl,n^{\prime}l^{\prime}}^{E*}$ is determined by coefficients
$M_{nl,n^{\prime}l^{\prime}}^{E*}$ [9, 10] defined in Appendix A. This matrix
is diagonal and the factor $f_{D}=1$ in case of $\alpha=0$, which corresponds
to the soft collision (SC) model. For nonzero $\alpha$, the matrix $[\mbox{\bf
S}^{f}_{BB}]_{nl,n^{\prime}l^{\prime}}$ is not diagonal and $f_{D}$ becomes
greater than one, reaching $32/(9\pi)\approx 1.132$ for $\alpha=\infty$ [10].
Nevertheless, we can approximate the original matrix by its diagonal
simplification with the same diagonal elements $[\mbox{\bf
S}^{f}_{BB}]_{nl,nl}$. The coefficients $M_{01,01}^{E*}=-1$ and
$M_{10,10}^{E*}=-2M_{1}$, where $M_{1}=m_{A}/(m_{A}+m_{B})$ are of our
particular interest since these allow to provide the explicit form of the
matrix elements $[\mbox{\bf S}^{f}_{BB}]_{01,01}=-\nu_{\rm opt}f_{D}$ and
$[\mbox{\bf S}^{f}_{BB}]_{10,10}=-\nu_{\rm opt}f_{D}2/(1+\alpha)$. In contrast
to the HC model, these matrix elements are not identical. The matrix element
$[\mbox{\bf S}^{f}_{BB}]_{01,01}\approx\nu_{\vec{v}}$ describes relaxation
rate, $\nu_{\vec{v}}$, of the velocity vector, $\vec{v}$. On the other hand,
$[\mbox{\bf S}^{f}_{BB}]_{10,10}\approx\nu_{v^{2}}$ describes relaxation rate,
$\nu_{v^{2}}$, of $v^{2}$ which is directly related to speed, $v$. As it was
shown in Refs. [12, 39], the ratio between the speed, $v$\- or its square,
$v^{2}$-changing collisions and velocity, $\vec{v}$-changing collisions rate
$\nu_{v^{2}}/\nu_{\vec{v}}=[\mbox{\bf S}^{f}_{BB}]_{10,10}/[\mbox{\bf
S}^{f}_{BB}]_{01,01}=2/(1+\alpha)$ varies with $\alpha$. This ratio agrees
well with results obtained from the classical molecular dynamics simulations
based on realistic molecular interaction potentials [11, 39]. Inserting the
matrix elements $[\mbox{\bf S}^{f}_{BB}]_{01,01}$ and $[\mbox{\bf
S}^{f}_{BB}]_{10,10}$ into Eqs. (20) and (15) we got

$\Gamma_{\rm eff}^{BB}+i\Delta_{\rm
eff}^{BB}=\Gamma_{0}+i\Delta_{0}+\frac{\omega_{D}^{2}}{2\nu_{\vec{v}}}-\frac{3\Gamma_{2}^{2}}{2\nu_{v^{2}}}+\frac{3\Delta_{2}^{2}}{2\nu_{v^{2}}}-i\frac{3\Gamma_{2}\Delta_{2}}{\nu_{v^{2}}}.$
(24)

We do not need to limit our discussion to the diagonal approximation in this
place. We can take into account the full matrix, $[\mbox{\bf
S}^{f}_{BB}]_{nl,nl}$, and use Eq. (13). However, to derive the exact
analytical expressions, we used another method described in Appendix B, where
we found that $\nu_{\vec{v}}=\nu_{\rm opt}$, $\nu_{v^{2}}=\nu_{\rm
opt}(2/(1+\alpha))(f_{D}/f_{v^{2}})$, and got the following

$\Gamma_{\rm eff}^{BB}+i\Delta_{\rm
eff}^{BB}=\Gamma_{0}+i\Delta_{0}+\frac{\omega_{D}^{2}}{2\nu_{\rm opt}}+\\\
\left(-\frac{3\Gamma_{2}^{2}}{2\nu_{\rm opt}}+\frac{3\Delta_{2}^{2}}{2\nu_{\rm
opt}}-i\frac{3\Gamma_{2}\Delta_{2}}{\nu_{\rm
opt}}\right)\frac{1+\alpha}{2}\frac{f_{v^{2}}}{f_{D}}.$ (25)

$\Gamma_{\rm eff}^{BB}$ and $i\Delta_{\rm eff}^{BB}$ are the effective width
and shift of the Lorentzian, which the quadratic speed-dependent billiard ball
profile (SDBBP) [6] collapses to, under frequent velocity-changing collisions.
This expression is a generalization of Eq. (22) [18] to the arbitrary mass
ratio $\alpha$ case. It should be noted that the factor $f_{v^{2}}/f_{D}$
(except $\alpha=0$ and $\alpha=\infty$ for which is equal to one), is slightly
greater than unity but not more than 2.36% in the worst case of $\alpha=2$,
see Table 1.

Equations (22) and (25) become equivalent when $\alpha=1$ and corresponding
$f_{v^{2}}/f_{D}=1.018756$ is approximated by unity. Some equivalence of the
HC model and BB model with $\alpha=1$ was already discussed in Ref. [6, 12,
39]. For $\alpha=0$ and the corresponding $f_{v^{2}}/f_{D}=1$, it means in
case of SC [35, 34], Eq. (25) provides the effective width and shift of
Lorentzian to which quadratic speed-dependent Galatry profile [36, 6]
collapses under frequent velocity-changing collisions. In this case Eq. (25)
is reduced to Eq. (23). It is worth mentioning that comparing the asymptotic
behavior of hard and soft collision models we can see that the components of
the effective width and shift related to suppressed Doppler effect [4] and
thermally-averaged collisional broadening and shift are the same for both
models. In contrast, the components related to the speed dependence of
collisional broadening and shift are two times smaller in the case of the soft
collision model. It is a natural consequence of the fact that
$\nu_{v^{2}}^{SC}=2\nu_{v^{2}}^{HC}$, when we keep the same
$\nu_{\vec{v}}^{SC}=\nu_{\vec{v}}^{HC}$ for both models.

##  V Components of effective width and shift

In line with our previous paper [18], it is important to note that the Dicke
parameter can take on complex values, $\nu_{\rm opt}=\nu_{\rm
opt}^{r}+i\nu_{\rm opt}^{i}$ [17, 31, 33, 32]. Therefore, we can decompose the
effective Lorentzian width and shift of the asymptotic quadratic correlated
speed-dependent billiard ball profile [6, 25, 40, 27] into several
contributions:

$\Gamma_{\rm
eff}=\Gamma_{0}+\Gamma_{\gamma}+\Gamma_{\delta}+\Gamma_{\gamma\delta}+\Gamma_{\omega_{D}},$
(26a) $\Delta_{\rm
eff}=\Delta_{0}+\Delta_{\gamma}+\Delta_{\delta}+\Delta_{\gamma\delta}+\Delta_{\omega_{D}}.$
(26b)

Assuming quadratic speed-dependent collisional broadening and shift as well as
velocity-changing collisions described by the billiard ball model, these
contributions are:

$\Gamma_{\gamma}=-\frac{f_{v^{2}}}{f_{D}}\frac{1+\alpha}{2}\frac{\nu_{\rm
opt}^{r}}{|\nu_{\rm opt}|^{2}}\frac{3}{2}\Gamma_{2}^{2},$ (27a)
$\Gamma_{\delta}=\frac{f_{v^{2}}}{f_{D}}\frac{1+\alpha}{2}\frac{\nu_{\rm
opt}^{r}}{|\nu_{\rm opt}|^{2}}\frac{3}{2}\Delta_{2}^{2},$ (27b)
$\Gamma_{\gamma\delta}=-2\frac{f_{v^{2}}}{f_{D}}\frac{1+\alpha}{2}\frac{\nu_{\rm
opt}^{i}}{|\nu_{\rm opt}|^{2}}\frac{3}{2}\Gamma_{2}\Delta_{2},$ (27c)
$\Gamma_{\omega_{D}}=\frac{\nu_{\rm opt}^{r}}{|\nu_{\rm
opt}|^{2}}\frac{1}{2}\omega_{D}^{2},$ (27d)
$\Delta_{\gamma}=\frac{f_{v^{2}}}{f_{D}}\frac{1+\alpha}{2}\frac{\nu_{\rm
opt}^{i}}{|\nu_{\rm opt}|^{2}}\frac{3}{2}\Gamma_{2}^{2},$ (27e)
$\Delta_{\delta}=-\frac{f_{v^{2}}}{f_{D}}\frac{1+\alpha}{2}\frac{\nu_{\rm
opt}^{i}}{|\nu_{\rm opt}|^{2}}\frac{3}{2}\Delta_{2}^{2},$ (27f)
$\Delta_{\gamma\delta}=-2\frac{f_{v^{2}}}{f_{D}}\frac{1+\alpha}{2}\frac{\nu_{\rm
opt}^{r}}{|\nu_{\rm opt}|^{2}}\frac{3}{2}\Gamma_{2}\Delta_{2},$ (27g)
$\Delta_{\omega_{D}}=-\frac{\nu_{\rm opt}^{i}}{|\nu_{\rm
opt}|^{2}}\frac{1}{2}\omega_{D}^{2}.$ (27h)

For real $\nu_{\rm opt}=\nu_{\rm opt}^{r}$ ($\nu_{\rm opt}^{i}=0$) the
components of effective width are: $\Gamma_{\omega_{D}}$ $\Gamma_{0}$,
$\Gamma_{\gamma}$, $\Gamma_{\delta}$. The first two terms represent
collisionally-suppressed Doppler broadening and velocity-averaged collisional
width, respectively. $\Gamma_{\gamma}$ can be seen as a reduction of line
width caused by the speed-dependence of collisional broadening. It is
qualitatively coherent with findings in the other context of the speed-
dependent Voigt profile (SDVP) [2, 3], where also the narrowing of the
spectral line was triggered by speed-dependence of collisional broadening. On
the other hand, $\Gamma_{\delta}$ represents an additional broadening of the
line caused by the speed-dependent spread of collisional shift [7, 8, 41]. The
character of these contributions does not depend on the sign of $\Gamma_{2}$
and $\Delta_{2}$ parameters. The effective shift, in such circumstances, has
only two components, $\Delta_{0}$ and $\Delta_{\gamma\delta}$. The first one
represents an ordinary velocity-averaged collisional shift, however, the
second term is less obvious and is related to the product of speed
dependencies of collisional broadening and shift or their correlation [18].

The other contributions, namely $\Gamma_{\gamma\delta}$, $\Delta_{\gamma}$,
$\Delta_{\delta}$, and $\Delta_{\omega_{D}}$, only come into play if $\nu_{\rm
opt}$ has a nonzero imaginary component, i.e., $\nu_{\rm opt}^{i}\neq 0$.
While we will not delve into a detailed discussion of these contributions, we
will touch on the last one, $\Delta_{\omega_{D}}$. This shift arises due to
Dicke-suppressed Doppler broadening, and it should decrease inversely with
pressure, similar to the well-known $\Gamma_{\omega_{D}}$ [4]. This is because
$\nu_{\rm opt}$ is proportional to pressure, while $\omega_{D}$ remains
constant at a given temperature. Therefore, in a moderate range of pressures,
this term can potentially impact the precise determination of the line
position. In fact, as we demonstrate in the following section, under certain
conditions, $\Delta_{\omega_{D}}$ can contribute up to 2 MHz to the effective
line shift, $\Delta_{\rm eff}$.

To describe the relative contribution of each effect
($x=\gamma,\;\delta,\;\gamma\delta\;{\rm or}\;\omega_{D}$) under specific
physical conditions, we introduce dimensionless parameters
$\Gamma_{x}/\Gamma_{0}$ and $\Delta_{x}/\Delta_{0}$. It is worth noting that
all collisional parameters, including $\nu_{\rm opt}$, $\Gamma_{0}$,
$\Gamma_{2}$, $\Delta_{0}$, and $\Delta_{2}$, are proportional to gas
pressure. Therefore, the dimensionless parameters $\Gamma_{x}/\Gamma_{0}$ and
$\Delta_{x}/\Delta_{0}$ for $x=\gamma,\;\delta,\;{\rm and}\;\gamma\delta$ are
independent of pressure. In fact, as we show in the following section, under
certain conditions, $\Gamma_{\delta}$ can account for approximately 35% of
$\Gamma_{0}$, and the absolute value of $\Delta_{\gamma\delta}$ can reach 5%
of $\Delta_{0}$ or 20% of $\Gamma_{0}$. For further details, please refer to
the next section.

##  VI Numerical validation

Figure 1: Numerical verification of asymptotic formulas for $\Gamma_{\gamma}$,
$\Gamma_{\delta}$, $\Gamma_{\omega_{D}}$, and $\Delta_{\gamma\delta}$.Panels
(a)-(c) show ratios of $\Gamma_{\rm exact}/\Gamma_{\gamma}$, $\Gamma_{\rm
exact}/\Gamma_{\delta}$, $\Gamma_{\rm exact}/\Gamma_{\omega_{D}}$ as functions
of $\nu_{\rm opt}/\Gamma_{2}$, $\nu_{\rm opt}/\Delta_{2}$, $\nu_{\rm
opt}/\omega_{D}$, respectively, where $\Gamma_{\rm exact}$ is numerically
evaluated half width at half maximum (HWHM) of the quadratic speed-dependent
billiard ball profile. Panel (d) shows the ratio of $\Delta_{\rm
exact}/\Delta_{\gamma\delta}$ as a function of $\nu_{\rm
opt}/\sqrt{\Gamma_{2}\Delta_{2}}$ where $\Delta_{\rm exact}$ is numerically
evaluated frequency corresponding to a maximum of the quadratic speed-
dependent billiard ball profile. Presented ratios were evaluated for
$\alpha=1/3,\;1,\;3,\;10$ as well as for $\alpha=0$ it means soft collisions
(SC) and hard collisions (HC).

We validated the analytical expressions obtained in the previous section. To
achieve this, we compared our results with the numerical half width at half
maximum (HWHM) $\Gamma_{\rm exact}$ and the frequency $\Delta_{\rm exact}$
corresponding to the position of the maximum of the quadratic speed-dependent
billiard ball profile [6]. Specifically, we focused on validating the formulas
for $\Gamma_{\gamma}$, $\Gamma_{\delta}$, $\Gamma_{\omega_{D}}$, and
$\Delta_{\gamma\delta}$ given by Eqs. (27a), (27b), (27d), and (27g),
respectively. To carry out our calculations, we considered four values of
$\alpha=1/3,1,3,10$, assuming real $\nu_{\rm opt}=\nu_{\rm opt}^{r}$,
$\nu_{\rm opt}^{i}=0$.

To provide further insight, we also compared our results with those obtained
from the soft collision model, which corresponds to the billiard ball model
with $\alpha=0$ and the speed-dependent Galatry profile [36] and the hard
collision model [16, 17] which we discuss in detail in Ref. [18]. To simplify
our analysis, we focused on one effect at a time and set the appropriate
values for the line shape parameters: $\Gamma_{0}$, $\Delta_{0}$,
$\Gamma_{2}$, $\Delta_{2}$, $\omega_{D}$, $\nu_{\rm opt}$ in the quadratic
SDBBP.

Our results, shown in Fig. 1, demonstrate that $\Gamma_{\rm exact}$ converges
to $\Gamma_{\gamma}$, $\Gamma_{\delta}$, $\Gamma_{\omega_{D}}$ as well as
$\Delta_{\rm exact}$ approaches $\Delta_{\gamma\delta}$ in the limit of high
$\nu_{\rm opt}$. However, we observed that the convergence is slower for
higher values of $\alpha$ in the case of $\Gamma_{\gamma}$, $\Gamma_{\delta}$,
and $\Delta_{\gamma\delta}$. This observation is directly related to the fact
that $\nu_{v^{2}}$ decreases with increasing $\alpha$, while $\nu_{\vec{v}}$
remains constant. On the other hand, the situation is different for
$\Gamma_{\omega_{D}}$, which is determined by $\nu_{\vec{v}}$ and is not
dependent on $\alpha$. As shown in Fig.1(c), the variations for $\alpha$ are
weak in this case. However, for extremely large values of $\alpha$,
significant discrepancies for different $\alpha$ are observed. This topic has
been discussed in detail in Ref. [6].

Figure 2: Ratio of $\Gamma_{\rm exact}(\alpha)/\Gamma_{\rm exact}(\alpha=0)$
evaluated for several values $\nu_{opt}/\Delta_{2}$ within quadratic speed-
dependent billiard ball profile in which parameters $\Gamma_{0}$,
$\Delta_{0}$, $\Gamma_{2}$, $\nu_{\rm opt}^{i}$, and $\omega_{D}$ were set to
zero and its comparison with ratio
$\Gamma_{\delta}(\alpha)/\Gamma_{\delta}(\alpha=0)\approx 1+\alpha$ obtained
for $\nu_{\rm opt}/\Delta_{2}\rightarrow\infty$.

The discussion of the effective broadening component $\Gamma_{\delta}$, which
arises from the speed dependence of the collisional shift $\Delta(v)$, is
important in this context. As mentioned earlier, an increase in $\alpha$
results in a decrease in $\nu_{v^{2}}$. Therefore, we anticipate that, with
other parameters held constant, an increase in $\alpha$ will result in an
increase in $\Gamma_{\delta}$. In the frequent velocity-changing collisions
regime, where $\nu_{\rm opt}$ dominates, an asymptotic analytical relation for
quadratic SDBBP [6] can be written as:

$\frac{\Gamma_{\delta}(\alpha)}{\Gamma_{\delta}(\alpha=0)}=\frac{f_{v^{2}}}{f_{D}}(1+\alpha)\approx
1+\alpha.$ (28)

To simplify the calculation, the factor $f_{v^{2}}/f_{D}$ in the asymptotic
relation can be approximated as unity, introducing an error of less than 2%.
We verified the accuracy of this approximation for finite values of $\nu_{\rm
opt}/\Delta_{2}$ by simulating quadratic SDBBP for various values of $\nu_{\rm
opt}/\Delta_{2}$ and $\alpha$, while setting all other parameters
($\Gamma_{0}$, $\Delta_{0}$, $\Gamma_{2}$, $\nu_{\rm opt}^{i}$, and
$\omega_{D}$) to zero. We then calculated $\Gamma_{\rm exact}$ for the
simulated profiles, and plotted the ratios $\Gamma_{\rm
exact}(\alpha)/\Gamma_{\rm exact}(\alpha=1)$ as a function of $\alpha$ for
several values of $\nu_{\rm opt}/\Delta_{2}$ (3, 10, 100, 1000) in Fig.2. As
shown in the Figure, for $\nu_{\rm opt}/\Delta_{2}=10$, the simulation results
are reproduced by Eq.(28) for small to moderate values of $\alpha$ up to
$\alpha=3$.

Table 2: Comparison of the Lorentz profile effective parameters for Q(1) 1-0 transition in H2 perturbed by He ($\alpha=2$) at 10 atm and 296 $K$ obtained with Eqs. (27a)-(27h) derived from the billiard ball model (BB) and effective parameters from Ref. [18, 42] derived from hard collision model (HC). The line-shape parameters are $\Gamma_{0}=22.346$, $\Delta_{0}=90.971$, $\Gamma_{2}=10.856$, $\Delta_{2}=38.192$, $\nu_{\rm opt}^{r}=404.148$, $\nu_{\rm opt}^{i}=-54.697$, $\omega_{D}=21.659$ [42] are given in the units of $10^{-3}\;{\rm cm}^{-1}$. $\Gamma_{x}$ | $\Gamma^{BB}_{x}/(10^{-3}\;{\rm cm}^{-1})$ | $\Gamma^{BB}_{x}/\Gamma_{0}$ | $\Gamma^{HC}_{x}/(10^{-3}\;{\rm cm}^{-1})$ | $\Gamma^{HC}_{x}/\Gamma_{0}$
---|---|---|---|---
$\Gamma_{0}$ | 22.346 | 1.000 | 22.346 | 1.000
$\Gamma_{\gamma}$ | -0.645 | -0.029 | -0.430 | -0.019
$\Gamma_{\delta}$ | 7.974 | 0.357 | 5.316 | 0.238
$\Gamma_{\gamma\delta}$ | 0.613 | 0.027 | 0.409 | 0.018
$\Gamma_{\omega_{D}}$ | 0.572 | 0.026 | 0.572 | 0.026
$\Gamma_{\rm eff}$ | 30.860 | 1.381 | 28.215 | 1.263
$\Delta_{x}$ | $\Delta^{BB}_{x}/(10^{-3}\;{\rm cm}^{-1})$ | $\Delta^{BB}_{x}/\Delta_{0}$ | $\Delta^{HC}_{x}/(10^{-3}\;{\rm cm}^{-1})$ | $\Delta^{HC}_{x}/\Delta_{0}$
$\Delta_{0}$ | 90.971 | 1.000 | 90.971 | 1.000
$\Delta_{\gamma}$ | -0.087 | 0.001 | -0.058 | -0.0006
$\Delta_{\delta}$ | 1.079 | 0.012 | 0.719 | 0.008
$\Delta_{\gamma\delta}$ | -4.533 | -0.050 | -3.022 | -0.033
$\Delta_{\omega_{D}}$ | 0.077 | 0.0008 | 0.077 | 0.0008
$\Delta_{\rm eff}$ | 87.507 | 0.962 | 88.688 | 0.975
Figure 3: A comparison of the reference speed-dependent billiard ball profiles
(SDBBP) calculated with $\alpha=2,\;1,\;0$ and parameters listed in the
caption of Tab. 2 for Q(1) 1-0 transition in H2 perturbed by He at 10 atm and
296 $K$ [18, 42] and Lorentz profiles (LPBB) derived as the collapsed SDBBP
calculated with parameters given in this work by Eqs. (26a)-(27h) as well as
Lorentz profile (LPHC) derived as the collapsed hard collision profile from
Ref. [18]. The area under each line is normalized to 1. The lower graph
presents the residuals against reference SDBBP ($\alpha=2$) having $\alpha$
corresponding to H2-He system with the same color notation like in upper
panel.

We compared the collapsed Lorentz profile to the SDBBP [6]. To do this, we
examined the Q(1) 1-0 transition in H2 perturbed by He at a pressure of 10 atm
and a temperature of 296 K. We used the same line-shape parameters as in Ref.
[18], which were obtained from Ref. [42]. The values of the line shape
parameters can be found in the caption of Tab. 2. This table lists all
contributions $\Gamma_{x}$ and $\Delta_{x}$ to the effective Lorentzian width
$\Gamma_{\rm eff}$ and shift $\Delta_{\rm eff}$, respectively. These values
were calculated using Eqs. (27a)-(27h) under the assumption of a billiard ball
model ($\alpha=2$) for velocity-changing collisions and the hard collision
model, as described in Ref. [18].

The ratios $\nu^{r}_{\rm opt}/\omega_{D}=18.7$, $\nu^{r}_{\rm
opt}/\Gamma_{2}=37.2$, and $\nu^{r}_{\rm opt}/\Delta_{2}=10.6$ demonstrate
that our assumption from Sec. III, that the velocity-changing collisions
dominate over Doppler broadening and collisional broadening and shift, is
valid for the chosen conditions (pressure, temperature, molecular system). It
is important to note that the values of $\Gamma_{0}$ and $\Delta_{0}$ are not
relevant in this context, as they only result in a simple convolution with the
Lorentz profile determined by these parameters, which are additive in the case
of Lorentz profiles.

The H2 spectral line perturbed by He, for which $\alpha=2$, allows us to
observe the importance of the perturber/absorber mass ratio for the collapse
of the spectral line shape to a Lorentz profile due to frequent velocity-
changing collisions. As shown in Tab. 2, the use of the BB model results in a
Lorentz profile that is 9.4% wider than the profile obtained from the HC model
[18]. This is mainly due to the fact that $\Gamma_{\delta}$, the second most
significant contribution to $\Gamma_{\rm eff}$ after $\Gamma_{0}$, is 50%
larger in the case of the BB model compared to the HC model.

Figure 3 compares the SDBBP [6] with $\alpha=2,\;1,\;0$ with Lorentz profiles
derived from the BB model (LPBB) and the HC model (LPHC). It shows that
LPBB($\alpha=2$) provides a good approximation of SDBBP($\alpha=2$), with a
difference of about 2% at the maximum of the profile. In contrast, LPHC
exhibits a much larger deviation. The LPHC follows closely LPBB($\alpha=1$)
and SDBBP($\alpha=1$). The discrepancy between SDBBP($\alpha=2$) and LPHC is
caused by the fact that, in the case of the HC model, the frequency of speed-
changing collisions $\nu_{v^{2}}$ does not capture its dependence on $\alpha$,
which is about 2/3 times smaller for the BB model with the same parameter
values and $\alpha=2$. On the other hand the frequency of speed-changing
collisions $\nu_{v^{2}}$ for HC model agrees within 2% with those from BB
model with $\alpha=1$. A comparison of SDBBP($\alpha=2$) with soft collision
profiles LPSC and SDGP [36] which are equivalent to LPBB($\alpha=0$) and
SDBBP($\alpha=0$), respectively, looks even worse. It is caused by the fact
that the BB model with the same parameter values and $\alpha=2$ has the
frequency of speed-changing collisions $\nu_{v^{2}}$ only about 1/3 of this in
SC model (BB model with $\alpha=0$).

##  VII Conclusions

In summary, we demonstrated that under the limit of frequent velocity-changing
collisions, the speed-dependent Dicke-narrowed profile of the spectral line
collapses to the Lorentz profile. Our work provides formulas for effective
Lorentzian width and shift, which take into account the arbitrary speed
dependence of collisional broadening and shift, as well as the velocity-
changing collision operator. Specifically, for the asymptotic behavior of the
quadratic speed-dependent billiard ball profile [6], we obtained simple
analytical expressions for Lorentzian width and shift. Our results generalize
those recently reported in Ref. [18] for the speed-dependent hard collision
profile [20] to the case of arbitrary perturber/absorber mass ratio, $\alpha$.
We verified the applicability of our formulas by comparing them with numerical
calculations of quadratic speed-dependent billiard ball profiles over a wide
range of line-shape parameters.

Finally, our comparison with the numerical calculation of the H2 spectral line
perturbed by He showed good agreement with the profile derived in this work.
This is thanks to properly accounting for the perturber/absorber mass ratio,
which was not possible with the hard collision model. Our results offer a more
accurate and general description of spectral line shapes under the limit of
frequent velocity-changing collisions, with potential applications in various
fields including atmospheric and astrophysical spectroscopy.

## Acknowledgements

N.S. was supported by Polish National Science Centre Project No.
2019/35/N/ST2/04145. P.W. was supported by Polish National Science Centre
Project No. 2019/35/B/ST2/01118. R.C. was supported by Polish National Science
Centre Project No. 2021/41/B/ST2/00681. The research is a part of the program
of the National Laboratory FAMO in Toruń, Poland. Calculations have been
partially carried out using resources provided by the Wroclaw Centre for
Networking and Super-computing (http://wcss.pl), Grant No. 546. We gratefully
acknowledge Poland’s high-performance computing infrastructure PLGrid (HPC
Centers: ACK Cyfronet AGH, PCSS, CI TASK) for providing computer facilities
and support within computational Grant No. PLG/2022/015576.

##  Appendix A The Burnett functions representation

For evaluation of the speed-dependent Dicke-narrowed spectral line shape from
the transport/relaxation equation we use a subset of the Burnett functions
having axial symmetry about the wave vector $\vec{k}$ [9]. A detailed
discussion of the application of this set of basis functions, as well as
matrix representation of the operators present in the transport/relaxation
equation, can be found in Ref. [6]. For convenience we recall these results
here, using the original formulation.

We assume the basis functions in the following form [10]:

$\varphi_{nl}(\vec{v})=N_{nl}\;(v/v_{m_{A}})^{l}\;L_{n}^{l+1/2}(v^{2}/v_{m_{A}}^{2})\;P_{l}(\vec{e}_{k}\cdot\vec{e}_{v}),$
(29)

where a normalization factor,

$N_{nl}=\sqrt{\frac{\pi^{1/2}\;n!\;(2l+1)}{2\;\Gamma(n+l+3/2)}}\;,$ (30)

$\Gamma(...)$ is the gamma-Euler function. The associated Laguerre polynomials
are defined as

$L_{n}^{l+1/2}(x^{2})=\sum_{m=0}^{n}\frac{(-1)^{m}\;\Gamma(n+l+3/2)}{m!\;(n-m)!\;\Gamma(m+l+3/2)}\;x^{2m}\;,$
(31)

where $x=v/v_{m_{A}}$ is the reduced speed of the active molecule. The
Legendre polynomials are defined as

$P_{l}(y)=\frac{1}{2^{l}}\sum_{k=0}^{[l/2]}\frac{(-1)^{k}\;(2l-2k)!}{k!\;(l-k)!\;(l-2k)!}\;y^{l-2k}\;,$
(32)

where $y=\vec{e}_{k}\cdot\vec{e}_{v}$ is the cosine of the angle between the
velocity vector $\vec{v}=v\vec{e}_{v}$ and the wave vector
$\vec{k}=k\vec{e}_{k}$, $\vec{e}_{v}$ and $\vec{e}_{k}$ are unit vectors.

The basis functions, $\varphi_{nl}(\vec{v})$, are eigenfunctions of the
velocity-changing collision operator in the case of the soft collisions, where
perturber/absorber mas ratio $\alpha=0$. Like in [6], when calculating matrix
elements of an operator, $\hat{A}$, we use the following notation $[\mbox{\bf
A}]_{nl,n^{\prime}l^{\prime}}=(\varphi_{nl}(\vec{v}),\hat{A}\varphi_{n^{\prime}l^{\prime}}(\vec{v}))=\langle
nl|\hat{A}|n^{\prime}l^{\prime}\rangle$.

The Doppler shift operator,
$\vec{k}\cdot\vec{v}=\omega_{D}\vec{e}_{k}\cdot\vec{v}/v_{m_{A}}$, is
represented by a matrix, K, the elements of which are: [6]

$[\mbox{\bf K}]_{nl,n^{\prime}l^{\prime}}=\omega_{D}\langle
nl|\vec{e}_{k}\cdot\vec{v}/v_{m}|n^{\prime}l^{\prime}\rangle\;,$ (33)

where $\omega_{D}=kv_{m_{A}}$ and

$\langle nl|\vec{e}_{k}\cdot\vec{v}/v_{m}|n^{\prime}l^{\prime}\rangle=\\\
\left[\sqrt{n+l+3/2}\;\delta_{n,n^{\prime}}-\sqrt{n}\;\delta_{n,n^{\prime}+1}\right]\\\
\times\sqrt{\frac{(l+1)^{2}}{4(l+1)^{2}-1}}\;\delta_{l,l^{\prime}-1}\\\
+\left[\sqrt{n+l+1/2}\;\delta_{n,n^{\prime}}-\sqrt{n+1}\;\delta_{n,n^{\prime}-1}\right]\\\
\times\sqrt{\frac{l^{2}}{4l^{2}-1}}\;\delta_{l,l^{\prime}+1}.$ (34)

The speed-dependent collisional width and shift operator,
$\hat{S}_{D}^{f}=-\Gamma(v)-i\Delta(v)$, is represented by a matrix,
$\mbox{\bf S}_{D}^{f}$, the elements of which are: [6, 24]

$[\mbox{\bf S}_{D}^{f}]_{nl,n^{\prime}l^{\prime}}=\\\
\frac{4}{\sqrt{\pi}(2l+1)}\;N_{nl}\;N_{n^{\prime}l}\;\delta_{l,l^{\prime}}\int_{0}^{\infty}dx\;e^{-x^{2}}x^{2l+2}\;\times\\\
\times
L_{n}^{l+1/2}(x^{2})\;L_{n^{\prime}}^{l+1/2}(x^{2})\;\left[\Gamma(xv_{m_{A}})+i\Delta(xv_{m_{A}})\right].$
(35)

In the case of quadratic speed-dependent collisional broadening and shift
[15],

$\Gamma(v)+i\Delta(v)=\Gamma_{0}+i\Delta_{0}+(\Gamma_{2}+i\Delta_{2})\left(\frac{v^{2}}{v_{m}^{2}}-\frac{3}{2}\right),$
(36)

the matrix elements can be written as

$[\mbox{\bf
S}_{D}^{f}]_{nl,n^{\prime}l^{\prime}}=-\left[\Gamma_{0}+i\Delta_{0}-(\Gamma_{2}+i\Delta_{2})3/2\right]\delta_{n,n^{\prime}}\delta_{l,l^{\prime}}\\\
-\left[\Gamma_{2}+i\Delta_{2}\right]\langle
nl|v^{2}/v_{m}^{2}|n^{\prime}l^{\prime}\rangle,$ (37)

where

$\langle nl|v^{2}/v_{m}^{2}|n^{\prime}l^{\prime}\rangle=\\\
\left[(2n+l+3/2)\;\delta_{n,n^{\prime}}-\sqrt{(n+l+1/2)n}\;\delta_{n,n^{\prime}+1}\right.\\\
\left.-\sqrt{(n+l+3/2)(n+1)}\;\delta_{n,n^{\prime}-1}\right]\delta_{l,l^{\prime}}.$
(38)

The velocity-changing collisions operator $\hat{S}_{BB}^{f}$ in the billiard
ball model is represented by a matrix, $\mbox{\bf S}_{BB}^{f}$, the elements
of which are: [9, 10]

$[\mbox{\bf
S}_{BB}^{f}]_{nl,n^{\prime}l^{\prime}}=\nu^{(0)}\;M_{nl,n^{\prime}l^{\prime}}^{E*},$
(39)

where $\nu^{(0)}=v_{m}^{2}/(2D^{(0)})$ and

$D^{(0)}=\frac{3}{8}\left(\frac{k_{B}T}{2\pi\mu}\right)^{1/2}\frac{1}{N\sigma^{2}}$
(40)

is the first-order self-diffusion coefficient for rigid spheres, $\sigma$ is
the average of the rigid sphere diameter of the absorber and perturber, $N$ is
the number density of perturbers, $\mu=m_{A}m_{P}/(m_{A}+m_{P})$ is the
reduced mass. The analytical expressions for coefficients
$M_{nl,n^{\prime}l^{\prime}}^{E*}$ for billiard ball model were derived by
Lindenfeld and Shizgal [9, 10]

$M_{nl,n^{\prime}l^{\prime}}^{E*}=\\\
-\delta_{l,l^{\prime}}\;\frac{3\;l!}{8M_{2}}\sqrt{\frac{n!\;n^{\prime}!}{\Gamma(n+l+3/2)\Gamma(n^{\prime}+l+3/2)}}\times\\\
\Bigg{\\{}\sum_{p=0}^{\tilde{n}}\;\sum_{s=0}^{\tilde{n}-p}\;\sum_{m=0}^{\tilde{n}-p-s}\;\sum_{q=0}^{l}\;\sum_{r=0}^{l-q}\;\left[\frac{4^{p}\;(r+s+p+q+1)!}{(p+q+1)!\;r!\;s!}\right]\times\\\
\left[\frac{\Gamma(n+n^{\prime}-2s-2p-m+l-r-q-1/2)\;B_{p,q}^{(1)}(\infty)}{(n-m-
s-p)!\;(n^{\prime}-m-s-p)!\;(l-r-q)!\;m!}\right]\times\\\
\left[M_{1}^{l+p-r-q}M_{2}^{n+n^{\prime}+q-2m-2s-p}(M_{1}-M_{2})^{m+r+2s}\right]\Bigg{\\}},$
(41)

where $M_{1}=m_{A}/(m_{A}+m_{P})=1-M_{2}$, $\tilde{n}={\rm min}(n,n^{\prime})$
and

$B_{p,q}^{(1)}(\infty)=\frac{(2p+q+1)!}{2q!\;(2p+1)!}-\frac{2^{q-1}(p+q+1)!}{p!\;q!}\;.$
(42)

The exact diffusion coefficient $D$ differs from $D^{(0)}$ and they are
related by coefficient $f_{D}=D/D^{(0)}$ which evaluation is explained in
Appendix B. Using these quantities the matrix elements can be written as

$[\mbox{\bf S}_{BB}^{f}]_{nl,n^{\prime}l^{\prime}}=\nu_{\rm
diff}\;f_{D}\;M_{nl,n^{\prime}l^{\prime}}^{E*}$ (43)

where

$\nu_{\rm diff}=\frac{v_{m}^{2}}{2D}$ (44)

is the effective frequency of velocity-changing collisions.

##  Appendix B Non-diagonal corrections to effective Lorentzian width and
shift

Aiming at calculations of spectral line shape with quadratic speed-dependent
collisional broadening and shift in the limit dominated by velocity-changing
collisions, Eq. (5) can be rewritten in the Burnett functions representation,

$1=\left(-i(\omega-\omega_{0})-[\mbox{\bf
S}_{D}^{f}]_{00,00}\right)c_{00}(\omega)\\\ +i[\mbox{\bf
K}]_{00,01}c_{01}(\omega)-[\mbox{\bf S}_{D}^{f}]_{00,10}c_{10}(\omega),$ (45a)
$0=i[\mbox{\bf
K}]_{01,00}\delta_{n,0}c_{00}(\omega)-\sum_{n^{\prime}=0}^{\infty}[\mbox{\bf
S}_{VC}^{f}]_{n1,n^{\prime}1}c_{n^{\prime}1}(\omega),$ (45b) $0=-[\mbox{\bf
S}_{D}^{f}]_{10,00}\delta_{n,1}c_{00}(\omega)-\sum_{n^{\prime}=1}^{\infty}[\mbox{\bf
S}_{VC}^{f}]_{n0,n^{\prime}0}c_{n^{\prime}0}(\omega).$ (45c)

We also take advantage from the fact that $c_{00}(\omega)$ is not coupled to
other coefficients by matrix $\mbox{\bf S}_{VC}^{f}$ and it dominates two
other matrices $\mbox{\bf S}_{D}^{f}$ and K. Therefore, in the limit of
frequent velocity-changing collisions, we are allowed to use matrix $\mbox{\bf
S}_{VC}^{f}$ and only those matrix elements of $\mbox{\bf S}_{D}^{f}$ and K
which provide coupling of $c_{00}(\omega)$ to other coefficients in the
transport/relaxation equation. With the velocity-changing collision operator
in the form from Eq. (39), we can find $c_{00}(\omega)$ by solving two other
sets of linear equations,

$\delta_{n,0}=-\sum_{n^{\prime}=0}^{\infty}M^{E*}_{n1,n^{\prime}1}a_{n^{\prime}1},$
(46a)
$\delta_{n,1}=-\sum_{n^{\prime}=1}^{\infty}M^{E*}_{n0,n^{\prime}0}a_{n^{\prime}0}.$
(46b)

Equations (46a) and (46b) are equivalent to Eqs. (45b) and (45c) when we set

$c_{n1}(\omega)=-\frac{i[\mbox{\bf
K}]_{01,00}}{\nu^{(0)}}c_{00}(\omega)a_{n1},$ (47a)
$c_{n0}(\omega)=\frac{[\mbox{\bf
S}_{D}^{f}]_{10,00}}{\nu^{(0)}}c_{00}(\omega)a_{n0}.$ (47b)

Now we can rewrite Eq. (45a) in the following form,

$1=\bigg{(}-i(\omega-\omega_{0})-[\mbox{\bf S}_{D}^{f}]_{00,00}\bigg{.}\\\
\bigg{.}+\frac{[\mbox{\bf K}]_{01,00}^{2}}{\nu^{(0)}}a_{01}-\frac{[\mbox{\bf
S}_{D}^{f}]_{10,00}^{2}}{\nu^{(0)}}a_{10}.\bigg{)}c_{00}(\omega).$ (48)

It is convenient to introduce the coefficient
$f_{D}=a_{01}(-M^{E*}_{01,01})=a_{01}$, were $a_{01}$ can be found by solving
Eq. (46a) [10]. Analogically, we can introduce
$f_{v^{2}}=a_{10}(-M^{E*}_{10,10})=a_{10}2/(1+\alpha)$, were $a_{01}$ can be
found by solving Eq. (46b).

Table 1 collects the coefficients $f_{D}$ and $f_{v^{2}}$ as well as their
ratios for the same set of $\alpha$. We repeat the calculations of $f_{D}$
performed by Lindenfeld in 1980 [10] for the billiard ball model but with
higher numerical precision (our results are in good agreement for all but the
last digit of the Table 1 of Ref.[10]). With these coefficients we define
effective rates $\nu_{\vec{v}}=\nu^{(0)}/f_{D}$ of change of velocity vector
$\vec{v}$ and $\nu_{v^{2}}=\nu^{(0)}(2/(1+\alpha))/f_{v^{2}}$ of change of
$v^{2}$. From that we can obtain the relation
$\nu_{v^{2}}=\nu_{\vec{v}}(2/(1+\alpha))(f_{D}/f_{v^{2}})$ between rates of
change of $\vec{v}$ and $v^{2}$.

Solving Eq.(48) yields

$c_{00}(\omega)=\frac{1}{\Gamma_{\rm eff}+i\Delta_{\rm
eff}-i(\omega-\omega_{0})},$ (49)

where the effective Lorentzian width and shift are

$\Gamma_{\rm eff}+i\Delta_{\rm eff}=-[\mbox{\bf
S}_{D}^{f}]_{00,00}+\frac{[\mbox{\bf
K}]_{01,00}^{2}}{\nu_{\vec{v}}}-\frac{[\mbox{\bf
S}_{D}^{f}]_{10,00}^{2}}{\nu_{v^{2}}}\;.$ (50)

This equation can be also written in the following form,

$\Gamma_{\rm eff}+i\Delta_{\rm
eff}=\Gamma_{0}+i\Delta_{0}+\frac{1}{2}\frac{\omega_{D}}{\nu_{\vec{v}}}-\frac{3}{2}\frac{(\Gamma_{2}+i\Delta_{2})^{2}}{\nu_{\vec{v}}}\frac{1+\alpha}{2}\frac{f_{v^{2}}}{f_{D}},$
(51)

which gives Eq. (25).

To take into account any arbitrary speed-dependence of collisional broadening,
$\Gamma(v)$, and shift, $\Delta(v)$, we need to include other matrix elements,
$[\mbox{\bf S}_{D}^{f}]_{n0,00}$, besides $[\mbox{\bf S}_{D}^{f}]_{10,00}$. In
such case, Eqs. (45a)-(45b) should be generalized to the following form,

$1=\left(-i(\omega-\omega_{0})-[\mbox{\bf
S}_{D}^{f}]_{00,00}\right)c_{00}(\omega)\\\ +i[\mbox{\bf
K}]_{00,01}c_{01}(\omega)-\sum_{n^{\prime}=1}^{\infty}[\mbox{\bf
S}_{D}^{f}]_{00,n^{\prime}0}c_{n^{\prime}0}(\omega),$ (52a) $0=i[\mbox{\bf
K}]_{01,00}\delta_{n,0}c_{00}(\omega)-\sum_{n^{\prime}=0}^{\infty}[\mbox{\bf
S}_{VC}^{f}]_{n1,n^{\prime}1}c_{n^{\prime}1}(\omega),$ (52b) $0=-[\mbox{\bf
S}_{D}^{f}]_{n0,00}c_{00}(\omega)-\sum_{n^{\prime}=1}^{\infty}[\mbox{\bf
S}_{VC}^{f}]_{n0,n^{\prime}0}c_{n^{\prime}0}(\omega).$ (52c)

The coefficient $c_{00}(\omega)$can be found in analogical way like in Eqs.
(45a)-(45b). The only difference is that the set of linear equations, Eq.
(46b), is replaced by

$\frac{[\mbox{\bf S}_{D}^{f}]_{n0,00}}{\sum_{n^{\prime}=1}^{\infty}[\mbox{\bf
S}_{D}^{f}]_{n^{\prime}0,00}}=-\sum_{n^{\prime}=1}^{\infty}M^{E*}_{n0,n^{\prime}0}a_{n^{\prime}0},$
(53)

and scaling Eq. (47b) is replaced by

$c_{n0}(\omega)=\frac{\sum_{n^{\prime}=1}^{\infty}[\mbox{\bf
S}_{D}^{f}]_{n^{\prime}0,00}}{\nu^{(0)}}c_{00}(\omega)a_{n0}.$ (54)

Now from Eq. (52a) we get

$1=\left(-i(\omega-\omega_{0})-[\mbox{\bf S}_{D}^{f}]_{00,00}+\frac{[\mbox{\bf
K}]_{01,00}^{2}}{\nu^{(0)}}a_{01}\right.\\\
\left.-\frac{1}{\nu^{(0)}}\sum_{n=1}^{\infty}[\mbox{\bf
S}_{D}^{f}]_{n0,00}\sum_{n^{\prime}=1}^{\infty}[\mbox{\bf
S}_{D}^{f}]_{n^{\prime}0,00}a_{n^{\prime}0}\right)c_{00}(\omega),$ (55)

and, as a consequence, the effective Lorentzian width and shift:

$\Gamma_{\rm eff}+i\Delta_{\rm eff}=-[\mbox{\bf
S}_{D}^{f}]_{00,00}+\frac{[\mbox{\bf K}]_{01,00}^{2}}{\nu_{\vec{v}}}\\\
-\frac{1}{\nu^{(0)}}\sum_{n=1}^{\infty}[\mbox{\bf
S}_{D}^{f}]_{n0,00}\sum_{n^{\prime}=1}^{\infty}[\mbox{\bf
S}_{D}^{f}]_{n^{\prime}0,00}a_{n^{\prime}0}.$ (56)

## References

  * Hartmann  _et al._ [2008] J.-M. Hartmann, C. Boulet, and D. Robert,  _Collisional Effects on Molecular Spectra: Laboratory Experiments and Model, Consequences for Applications_ (Elsevier, Amsterdam, 2008).
  * Berman [1972] P. R. Berman, Speed-dependent collisional width and shift parameters in spectral profiles, J. Quant. Spectrosc. Radiat. Transfer 12, 1331 (1972).
  * Ward  _et al._ [1974] J. Ward, J. Cooper, and E. W. Smith, Correlation effects in the theory of combined Doppler and pressure broadening – I. Classical theory, J. Quant. Spectrosc. Radiat. Transfer 14, 555 (1974).
  * Dicke [1953] R. H. Dicke, The effect of collisions upon the Doppler width of spectral lines, Phys. Rev. 89, 472 (1953).
  * Wittke and Dicke [1956] J. P. Wittke and R. H. Dicke, Redetermination of the Hyperfine Splitting in the Ground State of Atomic Hydrogen, Phys. Rev. 103, 620 (1956).
  * Ciuryło  _et al._ [2002a] R. Ciuryło, D. A. Shapiro, J. R. Drummond, and A. D. May, Solving the line-shape problem with speed-dependent broadening and shifting and with Dicke narrowing. II. Application, Phys. Rev. A 65, 012502 (2002a).
  * Pickett [1980] H. M. Pickett, Effects of velocity averaging on the shapes of absorption lines, J. Chem. Phys. 73, 6090 (1980).
  * Farrow  _et al._ [1989] R. L. Farrow, L. A. Rahn, G. O. Sitz, and G. J. Rosasco, Observation of a speed-dependent collisional inhomogeneity in H2 vibrational line profiles, Phys. Rev. Lett. 63, 746 (1989).
  * Lindenfeld and Shizgal [1979] M. J. Lindenfeld and B. Shizgal, Matrix elements of the Boltzmann collision operator for gas mixtures, Chem. Phys. 41, 81 (1979).
  * Lindenfeld [1980] M. J. Lindenfeld, Self-structure factor of hard-sphere gases for arbitrary ratio of bath to test particle masses, J. Chem. Phys. 73, 5817 (1980).
  * Hoang  _et al._ [2002] P. N. M. Hoang, P. Joubert, and D. Robert, Speed-dependent line-shape models analysis from molecular dynamics simulations: The collision regime, Phys. Rev. A 65, 012507 (2002).
  * Ciuryło  _et al._ [2002b] R. Ciuryło, D. Lisak, and J. Szudy, Role of velocity- and speed-changing collisions on speed-dependent line shapes of H2, Phys. Rev. A 66, 032701 (2002b).
  * Wcisło  _et al._ [2016] P. Wcisło, I. E. Gordon, H. Tran, Y. Tan, S.-M. Hu, A. Campargue, S. Kassi, D. Romanini, C. Hill, R. V. Kochanov, and L. S. Rothman, The implementation of non-Voigt line profiles in the HITRAN database: H2 case study, J. Quant. Spectrosc. Radiat. Transfer 177, 75 (2016).
  * Martínez  _et al._ [2018] R. Z. Martínez, D. Bermejo, F. Thibault, and P. Wcisło, Testing the ab initio quantum-scattering calculations for the D2-He benchmark system with stimulated Raman spectroscopy, J. Raman Spectrosc. 49, 1339 (2018).
  * Rohart  _et al._ [1994] F. Rohart, H. Mäder, and H.-W. Nicolaisen, Speed dependence of rotational relaxation induced by foreign gas collisions: Studies on CH3F by millimeter wave coherent transients, J. Chem. Phys. 101, 6475 (1994).
  * Nelkin and Ghatak [1964] M. Nelkin and A. Ghatak, Simple Binary Collision Model for Van Hove’s ${G}_{s}(r,~{}t)$, Phys. Rev. 135, A4 (1964).
  * [17] S. G. Rautian and I. I. Sobelman, Usp. Fiz. Nauk 90, 209 (1966) [Sov. Phys. Usp. 9, 701 (1967)] .
  * Stolarczyk  _et al._ [2022] N. Stolarczyk, P. Wcisło, and R. Ciuryło, Inhomogeneous broadening, narrowing and shift of molecular lines under frequent velocity-changing collisions, J. Quant. Spectrosc. Radiat. Transfer 287, 108246 (2022).
  * Lance  _et al._ [1997] B. Lance, G. Blanquet, J. Walrand, and J.-P. Bouanich, On the Speed-Dependent Hard Collision Lineshape Models: Application to C2H2 Perturbed by Xe, J. Mol. Spectrosc. 185, 262 (1997).
  * Pine [1999] A. S. Pine, Asymmetries and correlations in speed-dependent Dicke-narrowed line shapes of argon-broadened HF, J. Quant. Spectrosc. Radiat. Transfer 62, 397 (1999).
  * Blackmore [1987] R. Blackmore, A modified Boltzmann kinetic equation for line shape functions, J. Chem. Phys. 87, 791 (1987).
  * Shapiro  _et al._ [2002] D. A. Shapiro, R. Ciuryło, J. R. Drummond, and A. D. May, Solving the line-shape problem with speed-dependent broadening and shifting and with Dicke narrowing. I. Formalism, Phys. Rev. A 65, 012501 (2002).
  * Podivilov  _et al._ [1994] E. V. Podivilov, A. I. Chernykh, and D. A. Shapiro, Ion line narrowing in a dense plasma, J. Exp. Theo. Phys. 78, 653 (1994).
  * Robert and Bonamy [1998] D. Robert and L. Bonamy, Memory effects in speed-changing collisions and their consequences for spectral lineshape: I. collision regime, Eur. Phys. J. D 2, 245 (1998).
  * Ciuryło  _et al._ [2002] R. Ciuryło, A. Bielski, J. R. Drummond, D. Lisak, A. D. May, A. S. Pine, D. A. Shapiro, S. J., and R. S. Trawński, High-resolution studies on the influence of velocity-changing collisions on atomic and molecular line shapes, in  _Spectral Line Shapes_ , Vol. 12, edited by C. A. Back (American Institute of Physics, New York, 2002) pp. 151–160.
  * Wehr  _et al._ [2002] R. Wehr, A. Vitcu, R. Ciuryło, F. Thibault, J. R. Drummond, and A. D. May, Spectral line shape of the P(2) transition in CO-Ar: Uncorrelated ab initio calculation, Phys. Rev. A 66, 062502 (2002).
  * Wcisło  _et al._ [2018] P. Wcisło, F. Thibault, M. Zaborowski, S. Wójtewicz, A. Cygan, G. Kowzan, P. Masłowski, J. Komasa, M. Puchalski, K. Pachucki, R. Ciuryło, and D. Lisak, Accurate deuterium spectroscopy for fundamental studies, J. Quant. Spectrosc. Radiat. Transfer 213, 41 (2018).
  * Słowiński  _et al._ [2020] M. Słowiński, F. Thibault, Y. Tan, J. Wang, A.-W. Liu, S.-M. Hu, S. Kassi, A. Campargue, M. Konefał, H. Jóźwiak, K. Patkowski, P. Żuchowski, R. Ciuryło, D. Lisak, and P. Wcisło, H2-He collisions: Ab initio theory meets cavity-enhanced spectra, Phys. Rev. A 65, 052705 (2020).
  * Dolbeau  _et al._ [1999] S. Dolbeau, R. Berman, J. R. Drummond, and A. D. May, Dicke narrowing as an example of line mixing, Phys. Rev. A 59, 3506 (1999).
  * Shapiro and May [2001] D. A. Shapiro and A. D. May, Dicke narrowing for rigid spheres of arbitrary mass ratio, Phys. Rev. A 63, 012701 (2001).
  * Hess [1972] S. Hess, Kinetic theory of spectral line shapes. The transition between Doppler broadening and collisional broadening, Physica 61, 80 (1972).
  * Pine [1994] A. Pine, Line shape asymmetries in Ar-broadened HF(v=1–0) in the Dicke-narrowing regime, Chem. Phys. 101, 3444 (1994).
  * Demeio  _et al._ [1995] L. Demeio, S. Green, and L. Monchick, Effects of velocity changing collisions on line shapes of HF in Ar, J. Chem. Phys. 102, 9160 (1995).
  * Chandrasekhar [1943] S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15, 1 (1943).
  * Galatry [1961] L. Galatry, Simultaneous effect of Doppler and foreign gas broadening on spectral lines, Phys. Rev. 122, 1218 (1961).
  * Ciuryło  _et al._ [2001] R. Ciuryło, R. Jaworski, J. Jurkowski, A. S. Pine, and J. Szudy, Spectral line shapes modeled by a quadratic speed-dependent Galatry profile, Phys. Rev. A 63, 032507 (2001).
  * Priem  _et al._ [2000] D. Priem, F. Rohart, J.-M. Colmont, G. Wlodarczak, and J.-P. Bouanich, Lineshape study of the J=3←2 rotational transition of CO perturbed by $N_{2}$ and $O_{2}$, J. Mol. Struct. 517-518, 435 (2000).
  * Ciuryło and Szudy [1997] R. Ciuryło and R. Szudy, Speed-dependent pressure broadening and shift in the soft collision approximation, J. Quant. Spectrosc. Radiat. Transfer 57, 411 (1997).
  * Wcisło  _et al._ [2014] P. Wcisło, H. Tran, S. Kassi, A. Campargue, F. Thibault, and R. Ciuryło, Velocity-changing collisions in pure H2 and H2-Ar mixture, J. Chem. Phys. 141, 074301 (2014).
  * Lisak  _et al._ [2006] D. Lisak, J. T. Hodges, and R. Ciuryło, Comparison of semiclassical line-shape models to rovibrational ${\mathrm{H}}_{2}\mathrm{O}$ spectra measured by frequency-stabilized cavity ring-down spectroscopy, Phys. Rev. A 73, 012507 (2006).
  * Wcisło  _et al._ [2015] P. Wcisło, F. Thibault, H. Cybulski, and R. Ciuryło, Strong competition between velocity-changing and phase- or state-changing collisions in h2 spectra perturbed by ar, Phys. Rev. A 91, 052505 (2015).
  * Wcisło  _et al._ [2021] P. Wcisło, F. Thibault, N. Stolarczyk, H. Jóźwiak, M. Słowiński, M. Gancewski, K. Stankiewicz, M. Konefał, S. Kassi, A. Campargue, Y. Tan, J. Wang, K. Patkowski, R. Ciuryło, D. Lisak, R. Kochanov, L. Rothman, and I. Gordon, The first comprehensive dataset of beyond-Voigt line-shape parameters from ab initio quantum scattering calculations for the HITRAN database: He-perturbed H2 case study, J. Quant. Spectrosc. Radiat. Transfer 260, 107477 (2021).