# Interactions and cold collisions of AlF in the ground and excited electronic
states with He

Sangami Ganesan-Santhi  Faculty of Physics, University of Warsaw, Pasteura 5,
02-093 Warsaw, Poland  Matthew D. Frye  Faculty of Physics, University of
Warsaw, Pasteura 5, 02-093 Warsaw, Poland  Marcin Gronowski  Faculty of
Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland  Michał Tomza
<EMAIL_ADDRESS>Faculty of Physics, University of Warsaw, Pasteura 5,
02-093 Warsaw, Poland

###### Abstract

Aluminium monofluoride (AlF) is a promising candidate for laser cooling and
the production of dense ultracold molecular gases, thanks to its relatively
high chemical stability and diagonal Frank-Condon factors. In this study, we
examine the interactions and collisions of AlF in its $X^{1}\Sigma^{+}$,
$a^{3}\Pi$, and $A^{1}\Pi$ electronic states with ground-state He using state-
of-the-art ab initio quantum chemistry techniques. We construct accurate
potential energy surfaces (PESs) employing either the explicitly correlated
coupled-cluster CCSD(T)-F12 method augmented by the CCSDT correction or the
multireference configuration-interaction method for higher-excited electronic
states. Subsequently, we employ these PESs in coupled-channel calculations to
determine the scattering cross-sections for AlF+He collisions and bound states
of the complex. We estimate the uncertainty of the calculated PESs and apply
it to assess the uncertainty of the scattering results. We find a relatively
low sensitivity of the cross-sections to the variation of the PESs, but the
positions of shape resonances remain uncertain. The present results are
relevant for further improvements and optimizations of buffer-gas cooling of
AlF molecules.

##  I Introduction

Ultracold systems are an excellent platform for various experiments ranging
from controlled chemical reactions [1] and precision measurements [2] to
quantum simulation of many-body physics [3] and quantum information processing
[4]. Research into cold and ultracold molecules experienced a significant
surge following the first successful production of an ultracold gas of
molecules in its absolute ground state [5]. The complex architecture of energy
states within molecules, which comprise electronic, vibrational, rotational,
fine, and hyperfine degrees of freedom [6], provides opportunities for precise
control and manipulation of both internal and translational molecular motion.
This capability facilitates comprehensive explorations of chemical
interactions, spectroscopy, and fundamental physics [4].

Ultracold molecules can be produced by two broad strategies. The first
involves photo- or magneto-association of pre-cooled atoms at ultralow
temperatures [7, 8], usually followed by optical transfer to a deeply bound
state [9]. The alternative strategy involves direct cooling of molecules
themselves from higher temperatures. Buffer gas cooling [10] is a direct
cooling method in which the translational and internal energies are dissipated
through collisions with cold, inert gas atoms, usually helium. This typically
serves as one of the initial cooling steps since it is generally effective at
temperatures above a few hundred millikelvin [11] but can produce high
densities in preparation for a further cooling stage to ultracold
temperatures. In general, buffer-cell cooling allows the thermalization of
both translational and internal degrees of freedom; however, the balance
between them is determined by the ratio of elastic to inelastic collisions.
Laser cooling [12] is another direct cooling method and is frequently used as
a second-stage cooling method for molecules, paired with buffer gas cooling.
In this process, the directional absorption of photons results in slowing down
molecules. This technique requires nearly closed optical cycling and is
ubiquitous in atom cooling, but it has also been extended to a particular
class of molecules with highly diagonal Frank-Condon factors [13, 14].

The aluminium fluoride (AlF) molecule is an ideal candidate for cooling to
ultracold temperatures [15, 16]. This is primarily due to its highly diagonal
Frank-Condon factors, which are perfect for laser cooling [17, 18]. However,
another advantage of AlF is its chemical stability due to its relatively high
binding energy (about 7 eV). As the binding energy of simple monofluorides
grows, the efficiency of the reaction of the fluorinating reagent with hot
ablated metal also increases, as was demonstrated by theoretical computations
for AlF and CaF [19]. A comparison of buffer-gas-cooled beams of four
monofluorides showed that the one containing AlF was an order of magnitude
brighter than others due to this increased stability relative to MgF, CaF, and
YbF [20]. New accurate spectroscopic studies of AlF in molecular beams have
been recently reported [17, 21, 22], and the first magneto-optical traps with
AlF are under construction [15]. Once cooled, its relatively large dipole
moment of 1.5 D makes it a promising candidate for studying numerous types of
dipolar physics.

Before AlF became an object of research at ultralow temperatures, the
discovery of AlF in a protoplanetary circumstellar envelope sparked
astronomers’ interest in this molecule [23]. For this reason, the first
potential energy surface of the ground-state AlF+He complex was reported [24].
This potential exhibited one minimum only with the well depth of about 24
cm-1. Newer studies [16] suggested that this potential is shallower by about 2
cm-1.

In this work, we aim to investigate the prospects for buffer-gas cooling of
AlF molecules. As a theoretical foundation, we study in detail the
interactions between the AlF molecule in its three lowest electronic states
($X^{1}\Sigma^{+}$, $a^{3}\Pi$, and $A^{1}\Pi$) and ground-state He using
accurate ab initio quantum chemistry methods. We compute two-dimensional
potential energy surfaces for the five lowest electronic states of the AlF+He
complex. We evaluate the accuracy of our calculations by analyzing the
convergence with the wave function quality and basis set size and the impact
of the relativistic effects. Next, we employ the electronic stricture data in
coupled-channel scattering calculations of elastic and inelastic scattering
cross sections. We find the high ratio of elastic to inelastic collisions,
suggesting that buffer gas cooling of AlF molecules can be efficient.

The plan of the paper is as follows. The theory behind the construction of the
potential energy surface and a brief account of collision theory are discussed
in Section II. Potential energy surfaces and scattering results are presented
in Section III. We conclude our work in Section IV.

##  II Methods

The $X^{1}\Sigma^{+}$, $a^{3}\Pi$, and $A^{1}\Pi$ electronic states of AlF
upon interaction with ground-state He correspond to the following electronic
states of the interacting complex: $X^{1}A^{\prime}$, $a^{3}A^{\prime\prime}$,
$b^{3}A^{\prime}$, $A^{1}A^{\prime}$, and $B^{1}A^{\prime\prime}$, under the
$C_{\textrm{s}}$ point group, as collected in Table 1. We use advanced ab
initio quantum-chemical methods to describe all the mentioned electronic
states of AlF+He. Our calculations are based on the Born-Oppenheimer
approximation, in which a separate potential energy surface is defined for
each electronic state.

Table 1: The relation and symmetry of the electronic states of the AlF+He complex to the electronic states of interacting AlF and He. AlF | He | AlF+He
---|---|---
$X^{1}\Sigma^{+}$ | ${}^{1}S$ | $X^{1}A^{\prime}$
$a^{3}\Pi$ | ${}^{1}S$ |  $b^{3}A^{\prime}$, $a^{3}A^{\prime\prime}$
$A^{1}\Pi$ | ${}^{1}S$ |  $A^{1}A^{\prime}$, $B^{1}A^{\prime\prime}$

###  II.1 Ab initio electronic structure methods

The AlF molecule is considered as a rigid rotor with a fixed bond length
$r_{\textrm{AlF}}$. The Jacobi coordinates $R$ and $\theta$ are used to
describe the orientation of the molecule and the atom. $R$ is the distance
between the helium atom and the center of mass (c.m.) of the molecule and
$\theta$ is the angle between the molecular axis and vector from c.m. to He
($\theta=0^{\circ}$ and $\theta=180^{\circ}$ correspond to the linear HeAlF
and AlFHe arrangements, respectively). The coordinates are presented in Fig.
1. We use the vibrationally averaged value of the bond length
$r_{\textrm{AlF}}$ for the corresponding electronic states [25, 26] (3.136
bohr for $X^{1}\Sigma^{+}$, 3.124 bohr for $a^{3}\Pi$, and 3.126 bohr for
$A^{1}\Pi$). The interaction potential, $V_{\textrm{int}}$, depends on both
$R$ and $\theta$. We obtain $V_{\textrm{int}}(R,\theta)$ by using the
supermolecular method,

$V_{\textrm{int}}(R,\theta)=E_{\textrm{AlF+He}}(R,\theta)-E_{\textrm{AlF}}(R,\theta)-E_{\textrm{He}}(R,\theta),$
(1)

where $E_{\textrm{AlF+He}}$ is the total energy of the complex, while
$E_{\textrm{AlF}}$ and $E_{\textrm{He}}$ are the energies of the monomers. The
basis set superposition error is corrected using the counterpoise correction
[27], where the monomer energies are also calculated in the same basis set as
that of the whole complex.

The explicitly correlated coupled cluster method [28] restricted to a single,
double, and non-iterative triple excitations (CCSD(T)-F12b) [29], is used to
calculate the potential energy surfaces for the $X^{1}A^{\prime}$,
$a^{3}A^{\prime\prime}$, and $b^{3}A^{\prime}$ electronic states of the AlF +
He complex. In CCSD(T)-F12b computations, we use aug-cc-pV6Z [30] as the
orbital basis set, aug-cc-pV6Z-RIFIT as the density fitting and resolution of
the identity basis set, and aug-cc-pV5Z-JKFIT as the density fitting basis set
for the exchange and Fock operators. We obtain aug-cc-pV5Z-JKFIT by augmenting
cc-pV5Z-JKFIT [31]. cc-pV5Z-JKFIT for He is based on the unpublished work
[32]. aug-cc-pV6Z-RIFIT is based on unpublished work available in the Basis
Set Exchange repository [33, 34, 35]. We improve the description of the
electronic correlation in the $X^{1}A^{\prime}$, $a^{3}A^{\prime\prime}$, and
$b^{3}A^{\prime}$ states by including the full triple correction:

$\delta
V_{\textrm{int}}^{\textrm{CCSDT}}=V_{\textrm{int}}^{\textrm{CCSDT}}-V_{\textrm{int}}^{\textrm{CCSD(T)}}\,,$
(2)

where $V_{\textrm{int}}^{\textrm{CCSDT}}$ and
$V_{\textrm{int}}^{\textrm{CCSD(T)}}$ are the interaction energies calculated
using the the CCSDT and CCSD(T) methods, respectively, with the aug-cc-pVTZ
basis set [36, 37, 38].

Figure 1: The Jacobi coordinates for the AlF+He system.

We describe $A^{1}A^{\prime}$ and $B^{1}A^{\prime\prime}$ states of the
complex with the internally contracted multiconfiguration reference
configuration interaction (MRCI) method with the Davidson correction [39, 40,
41] and the aug-cc-pV5Z basis set [38, 37, 36]. We obtain appropriate orbitals
by multi-state multiconfiguration self-consistent field (MCSCF) calculations
[42, 43, 44, 45]. The active space is composed of 8 electrons distributed over
5 orbitals in the $A^{\prime}$ symmetry and 1 orbital in the
$A^{\prime\prime}$ symmetry.

The potential energy surfaces $V_{\textrm{int}}(R,\theta)$ are anisotropic and
can be expanded in the basis of Legendre polynomials,
$P_{\lambda}(\cos\theta)$, as

$V_{\textrm{int}}(R,\theta)=\sum_{\lambda=0}^{\lambda_{\textrm{max}}}V_{\lambda}(R)P_{\lambda}(\cos\theta),$
(3)

where $V_{\lambda}(R)$ are the expansion coefficients dependent on $R$.
Potential energy surfaces $V_{\textrm{int}}(R,\theta)$ are calculated on a
two-dimensional grid of 50 points in $R$ between 5 and 35 Bohr and 15 points
in angle $\theta$ between $0^{\circ}$ and $180^{\circ}$ chosen to be the roots
of the Legendre polynomial of the order of 15.

At long range, the atom+molecule potential is dominated by attractive van der
Waals interactions of the form

$V(R,\theta)\approx-\frac{C_{6,0}}{R^{6}}-\frac{C_{6,2}}{R^{6}}P_{2}(\cos\theta)\,,$
(4)

where other Legendre components have leading $C_{n,m}$ with $n>6$. The
isotropic $C_{6,0}$ coefficient is the sum of the dispersion and induction
contributions. The dispersion is an intermonomer correlation effect and
corresponds to the interaction of fluctuating instantaneous dipole moments.
The isotropic dispersion coefficient calculated from the Casimir-Polder
formula is [46]

$C_{6,0}^{\textrm{dis}}=\frac{3}{\pi}\int_{0}^{\infty}\bar{\alpha}^{\textrm{AlF}}(i\omega)\bar{\alpha}^{\textrm{He}}(i\omega)d\omega\,,$
(5)

where $\bar{\alpha}^{X}$ is the mean dynamic dipole polarizability of $X$. The
anisotropic $C_{6,2}^{\textrm{dis}}$ coefficient is given by

$C_{6,2}^{\textrm{dis}}=\frac{1}{\pi}\int_{0}^{\infty}\Delta\alpha^{\textrm{AlF}}(i\omega)\bar{\alpha}^{\textrm{He}}(i\omega)d\omega\,,$
(6)

where $\Delta\alpha^{\text{AlF}}$ is the anisotropy of the polarizability of
the AlF molecule defined as the difference between its parallel
$\alpha^{\text{AlF}}_{\parallel}$ and the perpendicular
$\alpha^{\text{AlF}}_{\perp}$ dynamic dipole polarizabilities:
$\Delta\alpha^{X}=\alpha^{\text{AlF}}_{\parallel}-\alpha^{\text{AlF}}_{\perp}$.

For AlF in the ground state, we calculate polarizabilities using the coupled
cluster polarization propagator method [47, 48, 46, 49, 50] and obtain
$C_{6,0}^{\textrm{dis}}=18.97\,E_{h}a_{0}^{6}$ and
$C_{6,2}^{\textrm{dis}}=-0.13\,E_{h}a_{0}^{6}$. For AlF in the excited states,
we scale the ground-state isotropic and anisotropic dispersion coefficients by
the ratio of the corresponding molecular static isotropic and anisotropic
polarizabilities.

The induction effect is due to the polarization of one monomer due to the
static field of the other monomer, in this case, AlF. The induction energy is
determined by the permanent multipole moments and static polarizabilities of
the monomers. The induction contribution to the $C_{6,m}$ coefficients can be
written as

$C_{6,0}^{\textrm{ind}}=C_{6,2}^{\textrm{ind}}=\mu_{\textrm{AlF}}^{2}\bar{\alpha}^{\textrm{He}}(0)$
(7)

where $\mu_{\textrm{AlF}}$ is the permanent electric dipole moment of AlF. The
static polarizability of He, $\bar{\alpha}^{\textrm{He}}(0)$, is taken from
Ref. [51]. The value of $C_{6,0(2)}^{\textrm{ind}}$ is calculated to be 0.48
$E_{h}a_{0}^{6}$.

All electronic structure calculations are performed with the Molpro [52, 53,
54] and mrcc [55, 56, 57, 58, 59] packages of ab initio programmes.

###  II.2 Collision theory

The essence of buffer gas cooling is the thermalization of AlF molecules by
colliding with He atoms. Thus, scattering calculations are necessary to
understand the possible outcomes of cooling AlF by He.

The total Hamiltonian of the atom+rigid-rotor system in the center-of-mass
frame under the Born-Oppenheimer approximation is

$\hat{H}=-\dfrac{\hbar^{2}}{2\mu}\nabla^{2}_{R}+\dfrac{\hbar^{2}\hat{L}^{2}}{2\mu
R^{2}}+V_{\textrm{int}}(R,\theta)+B_{0}\hat{j}^{2}.$ (8)

The first term is the component of the kinetic energy operator in the
scattering coordinate $R$. The second term is the centrifugal component, where
$\hat{L}$ is the rotational angular momentum of He and AlF around each other,
with the quantum number $L$. $V_{\textrm{int}}(R,\theta)$ is the interaction
potential of the colliding systems, as calculated in the previous section, and
$\mu$ is the collisional reduced mass. The states of the free molecule are
eigenfunctions of the rotational Hamiltonian, $H_{\textrm{rot}}$ =
$B_{0}\hat{j}^{2}$, with the rotational quantum number $j$ and energy
$j(j+1)B_{0}$, where $B_{0}=0.55$ cm-1 [60] is the rotational constant of AlF.
We neglect the small differences in rotational constant for different
electronic states.

The expansion coefficients $V_{\lambda}(R)$ of the interaction potentials are
interpolated and extrapolated using a reciprocal power reproducing kernel
Hilbert space method (RP-RKHS) [61]. This produces potentials with asymptotic
forms as a sum of terms with different inverse powers depending on the
parameters used. Terms with different values of $\lambda$ in the expansion
have different leading terms, and we have chosen parameters in the method to
give the correct leading powers up to $R^{-10}$. For the long-range
coefficients calculated in Sec. II.1, we use the method of Ref. [62] to fix
the extrapolation to these values.

We perform coupled-channel scattering calculations using the molscat package
[63, 64]. The angular component of the wave function is expanded in the total
angular momentum basis [65], limited by $j_{\textrm{max}}=12$. Coupled
equations are propagated using the Manolopoulos diabatic modified log-
derivative [66] and the Alexander-Manolopoulos Airy propagator [67]. The
solutions are matched to asymptotic boundary conditions. $S$ matrices are
extracted, and cross-sections are calculated using the usual methods.

Bound states are calculated using the bound package [64, 68]. This is closely
related to molscat and uses similar coupled-channels methods, but matches to
bound-state boundary conditions and solves for eigenenergies.

##  III Results and Discussion

###  III.1 Potential energy surfaces

Table 2: Equilibrium geometries $R_{e}$, $\theta_{e}$ and equilibrium well depths $D_{e}$ of the AlF+He complex in different electronic states. Values for global (gm) and local (lm) minima are reported. state | minima |  $R_{e}\,$(bohr) |  $\theta_{e}$ (degree) |  $D_{e}\,$(cm-1)
---|---|---|---|---
$X^{1}A^{\prime}$ | gm | 7.71 | 180 | 24.9
|  | 7.1 [16] | 180 [16] | 22 [16]
|  | 7.75 [24] | 180 [24] | 24.056 [24]
| lm | 10.06 | 0 | 8.0
$b^{3}A^{\prime}$ | gm | 7.17 | 141.4 | 27.5
| lm | 8.68 | 0 | 21.6
| lm | 7.59 | 180 | 27.2
$a^{3}A^{\prime\prime}$ | gm | 6.00 | 85.4 | 54.1
| lm | 7.59 | 180 | 27.2
$A^{1}A^{\prime}$ | gm | 7.62 | 180 | 24.0
| lm | 9.05 | 0 | 22.1
$B^{1}A^{\prime\prime}$ | gm | 7.62 | 180 | 24.0
| lm | 9.05 | 0 | 22.1

The equilibrium geometries and corresponding well depths for the studied
electronic states of AlF+He are collected in Table 2. The calculated well
depths for global minima range from 21 to 28 cm-1, except for the
$a^{3}A^{\prime\prime}$ state. They are only slightly deeper than that of a
comparable YbF+He complex (21.88 cm-1 [69]) and lie on the lower end of the
typical interaction strength scale for interactions between neutral molecules
and He [70]. The $a^{3}A^{\prime\prime}$ state is a notable exception, with a
potential well depth approximately twice as deep, placing it near the higher
end of the He+neutral molecule interaction strength scale [70]. This
significant difference indicates that the interaction strength between AlF and
He can be controlled by electronic excitation.

Figure 2: One-dimensional cuts through the potential energy surfaces for the
(a) $X^{1}A^{\prime}$, (b) $a^{3}A^{\prime\prime}$ and $b^{3}A^{\prime}$, and
(c) $A^{1}A^{\prime}$ and $B^{1}A^{\prime\prime}$ electronic states of AlF+He
in linear and perpendicular orientations.

Figure 2 presents one-dimensional cuts through the potential energy surfaces
for the $X^{1}A^{\prime}$, $a^{3}A^{\prime\prime}$, $b^{3}A^{\prime}$,
$A^{1}A^{\prime}$, and $B^{1}A^{\prime\prime}$ electronic states of AlF
interacting with He in two linear configurations ($\theta=0^{\circ}$ and
$180^{\circ}$) and one perpendicular configuration ($90^{\circ}$). For the
ground state of AlF, the linear configuration with helium near fluorine
displays a potential well that is twice as deep as that of both the
perpendicular and alternate linear configurations, occurring at a
significantly larger intermonomer separation. Electronic excitation of AlF
allows helium to come closer to aluminum and reduces the differences in the
well depths for two linear configurations. This illustrates how the shape of
the interaction potential can be controlled by varying the electronic state of
AlF.

Figures 3 and 4 show two-dimensional contour plots and corresponding Legendre
components of the interaction potentials for the $X^{1}A^{\prime}$,
$a^{3}A^{\prime\prime}$, $b^{3}A^{\prime}$, $A^{1}A^{\prime}$, and
$B^{1}A^{\prime\prime}$ electronic states of AlF+He. The anisotropic nature of
the studied interatomic interactions is clearly visible, with all surfaces
showing a strong orientation dependence. The global minima of the
$a^{3}A^{\prime\prime}$ and $b^{3}A^{\prime}$ states occur at non-linear
geometries in contrast to the singlet states that have linear equilibrium
geometries. At the global minima of the $X^{1}A^{\prime}$, $b^{3}A^{\prime}$,
$A^{1}A^{\prime}$, and $B^{1}A^{\prime\prime}$ states, the helium atom is near
the fluorine, which is the electronegative side of the molecule. A similar
situation was reported for YbF+He [69].

For the $X^{1}A^{\prime}$, $b^{3}A^{\prime}$, $A^{1}A^{\prime}$, and
$B^{1}A^{\prime\prime}$ states, there is a local minimum in the linear
configuration where He reaches the molecule from the side of Al. The depths of
these minima are approximately 8 cm-1, 21 cm-1, 22.1 cm-1 and 22.1 cm-1 for
$X^{1}A^{\prime}$, $b^{3}A^{\prime}$, $A^{1}A^{\prime}$ and
$B^{1}A^{\prime\prime}$ states, respectively. Only the $a^{3}A^{\prime\prime}$
state does not show a minimum in this orientation. The local minimum for the
$X^{1}A^{\prime}$ state was not reported in the previous studies, [24, 16] as
it only emerges after including the full triple correction. This minimum is
quite shallow, with the barrier between global and local minima having energy
only a fraction of cm-1 higher than the local minimum. For the
$a^{3}A^{\prime\prime}$ and $b^{3}A^{\prime}$ states, we see a local minimum
with a depth of 27.2 cm-1 in the linear configuration, where He reaches the
molecule from the side of F, similar to the global minima of the ground
states. The examination of the potential energy surfaces provides insights
into the nature of interactions within the AlF+He complex.

Figure 3: Two-dimensional potential energy surfaces and corresponding Legendre
components for the (a,d) $X^{1}A^{\prime}$, (b,e) $b^{3}A^{\prime}$, and (c,f)
$a^{3}A^{\prime\prime}$ electronic states of AlF+He. Figure 4: Two-dimensional
potential energy surfaces and corresponding Legendre components for the (a,c)
$A^{1}A^{\prime}$ and (b,d) $B^{1}A^{\prime\prime}$ electronic states of
AlF+He. Figure 5: The convergence of the interaction energy of AlF+He in the
$X^{1}A^{\prime}$ state at $\theta=180^{\circ}$ (a) for different ab initio
methods using the aug-cc-pV5Z basis set and (b) for the aug-cc-pV$X$Z basis
set with increasing cardinal number $X$ using the CCSD(T) method. Figure 6:
The effect of including noniterative quadruple excitations within the CCSDT(Q)
method and the relativistic correction within the Douglas-Kroll-Hess
Hamiltonian on the interaction energy of AlF+He in the $X^{1}A^{\prime}$ state
at $\theta=180^{\circ}$.

The accuracy of our electronic structure calculations may be affected by (a)
an incomplete orbital basis set, (b) an inadequate description of the
correlation energy, and (c) the neglect of relativistic effects [71, 72]. The
correct description of the electron correlation is crucial for AlF+He, similar
to other systems dominated by the van der Wals interaction. The convergence of
interaction energy for the $X^{1}A^{\prime}$ state with the quality of the
wavefunction is analyzed in Fig. 5(a). The Hartree-Fock method, which is a
mean-field method, produces a repulsive potential. The second-order
perturbation theory, MP2, reproduces about 75% of the correlation energy. The
depth of the potential predicted at the CCSD level is 19.4 cm-1. Further
perturbative inclusion of triple excitation deepens the potential by 3.7 cm-1
while the full triple correction improves it further by 0.62 cm-1. The
perturbative treatment of the triple excitation is responsible for
approximately 85% of the total triple contribution. When calculating the
entire potential energy surface, we neglect the excitations higher than triple
in the coupled cluster expansion. We estimate that contribution as a
difference between CCSDT(Q) and CCSDT potential, and it turns out to be
relatively small, as presented in Fig. 6. It alters the interaction energy by
0.5 cm-1 near the classical turning point and fastly decreases with the
intermonomer distance to more than ten times lower value (0.045 cm-1) at the
global minimum. The reported potential energy surface also neglects the core
and core-valence correlation. Including the core and core-valence correlation
changes the interaction energy by 0.02 cm-1 at the equilibrium geometry, as we
estimate at the CCSD(T) level of theory.

Basis sets with increasing cardinal numbers are used to study convergence
toward the complete basis limit. As shown in Fig. 5(b), the interaction
strength for the $X^{1}A^{\prime}$ state increases monotonically as the basis
set size increases. The difference in the well depth of the potentials
predicted by CCSD(T)-F12 using aug-cc-pV5Z and aug-cc-pV6Z basis sets is 0.28
cm-1. For sextuple zeta basis sets, the difference between CCSD(T) and
CCSD(T)-F12 is about 0.5 cm-1. The complete basis set limit estimated using
CCSD(T) is deeper by 0.3 cm-1 than the depth of the well depth estimated by
CCSD(T)-F12.

The system under study is relatively light, so the relativistic effects are
relatively small. This can be seen in the CCSD(T)/aug-cc-pV5Z calculations
with the Douglas-Kroll-Hess Hamiltonian included up to the third order
presented in Fig. 6. The relativistic correction ranges from $-0.1$ to $-0.05$
cm-1 around the global minimum and decreases with increasing intermonomer
distance.

Finally, we test the accuracy of the rigid rotor approximation by optimizing
the structures of AlF and AlF+He using CCSD(T)/aug-cc-pV5Z. The interaction
with He decreases the equilibrium bond length of AlF by 0.008 bohr. This small
relaxation of the bond length changes the interaction energies by a negligible
0.02 cm-1.

The accuracy of our calculations is predominantly limited by the basis set
incompleteness in the description of the valence electron correlation. The
overall uncertainty of our potential at the global minimum is estimated to be
0.3 cm-1, which is 1.3 % of the well depth of the ground state potential. Our
ground state interaction potential exhibits well depth similar to that in Ref.
[24], but it is deeper by 2.9 cm-1 (12%) than that reported in Ref. [16]. This
discrepancy is not surprising, as the calculations reported in Ref. [16] do
not reach the complete basis set limit. The older results reported in Ref.
[24] used the mid-bond functions, which improve the convergence of the
interaction energy calculations with the basis set size. The basis set
incompleteness hampers the accuracy of our potential in the $b^{3}A^{\prime}$
state by 1 cm-1, which is 3.9% of the well depth of the $b^{3}A^{\prime}$
potential and 2% of the well depth of the $a^{3}A^{\prime\prime}$ potential,
the same for the $A^{1}A^{\prime\prime}$ and $B^{1}A^{\prime}$ state
(MRCI/aug-cc-pV5Z) is 0.4 cm-1 which is 1.6 % of the depth of the potential.
We expect that the MRCI computations are affected mainly by the lack of higher
excitations in the configuration interaction expansion and the size
inconsistency of the method. We do not have a good estimation of those
contributions. Still, we may expect that it can be even of the order of
10-20%, similar to the difference between the CCSD and CCSDT interaction
energies in the ground state.

###  III.2 Collision dynamics

When AlF collides with He, there are two possible outcomes: elastic collision,
where there is no change in AlF’s internal state, and inelastic collision,
where there is a change in AlF’s rotational state. Elastic collisions
determine the translational thermalization of AlF, whereas inelastic
collisions determine rotational thermalization. Figure 7 shows the elastic and
inelastic cross sections for the AlF+He collision for the $X^{1}A^{\prime}$,
$a^{3}A^{\prime\prime}$ and $b^{3}A^{\prime}$ states. To better understand
these outcomes, we can examine the cross sections of these events.

Figure 7: Elastic and inelastic cross sections as a function of the collision energy for AlF+He scattering in the (a,d) $X^{1}A^{\prime}$, (b,e) $b^{3}A^{\prime}$, and (c,f) $a^{3}A^{\prime\prime}$ states. The cross sections of the scattering channels for the rotational states $j=0,1,2$ of AlF are shown. The corresponding curves in the same color are the thermally averaged cross sections. Table 3: Ratio of elastic to inelastic collision cross sections at 1 K for the AlF+He collisions. State | $\sigma_{j=1}/\sigma_{j=1\rightarrow 0}$ | $\sigma_{j=2}/\sigma_{j=2\rightarrow 0}$ | $\sigma_{j=2}/\sigma_{j=2\rightarrow 1}$
---|---|---|---
$X^{1}A^{\prime}$ | 7.6 | 10.8 | 3.1
$a^{3}A^{\prime\prime}$ | 18.6 | 40.3 | 17.6
$b^{3}A^{\prime}$ | 36.6 | 8.9 | 25.5

Let us focus on the temperatures between 100 mK to 10 K, which are the most
interesting from the buffer-cell cooling perspective. A significant value of
elastic cross-section is necessary to provide effective translational
thermalization. In systems where inelastic cross-sections are about one to two
orders of magnitude lower than elastic ones [73, 11] both translational and
rotational degrees of freedom can be thermalized. The thermally-averaged
elastic cross sections for the AlF+He collision (Fig. 7) lie in the range
between $0.2\times 10^{3}$ and $2\times 10^{3}$ Å2, regardless of the quantum
state of AlF. These calculated values are larger than the $0.1\times 10^{3}$
Å2 experimentally estimated for CaH [74], but comparable to the values
calculated for that system [75]. On average, buffer gas cooling can achieve a
temperature range of a few hundred mK. For the $X^{1}A^{\prime}$ state of
AlF+He, the ratio of elastic to inelastic collision is around 7 for
$\sigma_{j=1}/\sigma_{j=1\rightarrow 0}$ at 1 K gradually decreasing to 0.9 at
0.2 K, around 10 for $\sigma_{j=2}/\sigma_{j=2\rightarrow 0}$ at 1 K
increasing to 15 near 0.5 K to 0.9 at 10 mK, around 3 for
$\sigma_{j=2}/\sigma_{j=2\rightarrow 1}$ at 1 K to 0.9 at 90 mK. With higher
elastic collision cross sections, translational thermalization looks more
prospective near 1 K. The trend is similar for other states of the AlF. The
values of the ratio of elastic to inelastic collisions at 1 K are reported in
Table 3.

Let us focus on temperatures below 100 mK, which are difficult to obtain by
pure buffer-cell cooling; however, we see higher sensitivity of the cross-
sections on the rotational state. The elastic cross-sections are relatively
constant at these temperatures, as the shape resonances gradually disappear
with a decreasing temperature. The inelastic cross section is most significant
for the state $X^{1}A^{\prime}$, resulting in fast rotational relaxation from
$j=1$ to $j=0$ of AlF. For the $X^{1}A^{\prime}$ state of AlF+He,
$\sigma_{j=1\rightarrow 0}$ is more than an order of magnitude higher than
$\sigma_{j=1}$. This means that the molecules should be cooled into the
rotational ground state efficiently. Lower values of inelastic cross-sections
for triplet states can be related to lower anisotropy of the potentials,
visible as a smaller difference in the shape of the potential energy curves
for $\theta=0\degree$ and $\theta=180\degree$ in Fig. 3(b) and (c). Still, the
inelastic and elastic cross-sections are usually in comparable range. Panel
(c) of Fig. 7 shows that for the $a^{3}A^{\prime\prime}$ state, the elastic
cross section for $j=1$ is negligibly low in the low-energy limit; this is due
to the scattering length being accidentally close to zero. This will inhibit
translational thermalisation in this state, but the inelastic cross section is
much larger, so the molecules will rapidly reach a different state where
cooling could continue.

Table 4: Bound rovibrational levels of AlF+He in the $X^{1}A^{\prime}$ state. $J$ is the total angular momentum, $p$ is the parity of the state. Energies are in cm-1. $J^{p}$ |  |  |  |  |
---|---|---|---|---|---
0+ | -7.59 | -2.44 | -0.63 |  |
1+ | -4.77 | -0.03 |  |  |
1- | -7.22 | -4.84 | -2.01 | -0.81 |
2+ | -6.50 | -4.13 | -1.52 | -1.16 | -0.43
2- | -3.96 | -1.45 |  |  |
3+ | -2.75 | -0.41 |  |  |
3- | -5.43 | -3.07 | -0.71 |  |
4+ | -4.01 | -1.68 |  |  |
4- | -1.16 |  |  |  |
5- | -2.27 |  |  |  |
6+ | -0.21 |  |  |  |

The bound rovibrational levels of the AlF+He complex in the
X${}^{1}A^{\prime}$ state are reported in Table 4. The lowest bound state
supported by this potential is at -7.59 cm-1. There are 26 bound states for
the ground state in total. For the $a^{3}A^{\prime\prime}$ state with a well
depth of 54 cm-1, there are 124 bound states, which are tabulated in the
Supplemental Material.

Figure 8: (a) The sensitivity of the elastic scattering cross section to the
scaling of the potential for the $X^{1}A^{\prime}$ state of AlF+He for the
collision energies 5 K, 1 K, 0.01 K, and 0.0001 K. (b) Elastic scattering
cross-sections as a function of the collision energy for the $X^{1}A^{\prime}$
potential scaled by $\pm 1.5$%.

We study the sensitivity of our scattering results to interaction energy by
scaling potential by $\pm 10\%$ and observe changes in the elastic cross-
sections. Figure 8(a) shows how the cross section varies with a scaling of the
potential for four energies. Here, $\lambda$ scales the potential energy as
$V\rightarrow(1+\lambda)\times V$. We observe a peak in the 1 K line (Fig.
8(a)), which can be attributed to the sensitivity of the shape resonances
visible around 1 K in Fig. 7,(a). The uncertainty of the potential energy
surface is about 2 %, so the scattering results are not very far from the
truth for all four energies. In Fig. 8(b) we show the shift in elastic cross
sections for a $\pm 1.5\%$ scaling for two rotational states of AlF. For the
$j=0$ rotational state of AlF, the first few shape resonances are more
sensitive to the scaling of potential, but their relative sensitivity
decreases for higher energies. For the $j=1$ rotational state of AlF, the
curves are shifted while preserving the general shape. The scattering cross
sections generally lie in the same range as predicted recently in Ref. [16],
with a shift in the position of shape resonances. Although our ab initio
calculations are more accurate, the scattering results from Ref. [16] are
similar, which suggests that the details of the potential energy surface have
a moderate effect on the scattering outcomes for this system.

##  IV Conclusion

We have constructed accurate potential energy surfaces using the explicitly
correlated coupled-cluster and multireference configuration interaction
methods for the AlF+He interactions with AlF in the ground and excited
electronic states. We have found that interactions between AlF and He are weak
and anisotropic. For all electronic states, we have identified global and
local minima. For most of the states, the global minimum is linear and appears
when He approaches AlF from the F side. The only exception from this rule is
the $a^{3}A^{\prime\prime}$ state, however, it exhibits deep local minima for
the linear AlFHe geometry, too. AlF+He is a relatively light system
uninfluenced by relativistic effects. We have demonstrated that the shape and
depth of the AlF+He interaction potential can be controlled by changing the
electronic state of AlF. The calculated potential energy surfaces are provided
in numerical form in the Supplemental Material.

Next, we have used coupled-channel scattering theory to calculate the elastic
and inelastic scattering cross sections of the AlF+He collision outcomes. The
uncertainties of the potential energy surfaces have been estimated, and the
sensitivity of the scattering results to the scaling of the potential has been
studied to give us a better idea of the predictability of the results. The
presented data can help to determine the optimal conditions for buffer gas
cooling. The high ratio of elastic to inelastic collisions suggests that
buffer gas cooling of AlF molecules can be efficient.

###### Acknowledgements.

We gratefully acknowledge the Foundation for Polish Science within the First
Team programme co-financed by the European Union under the European Regional
Development Fund for the financial support and the Poland’s high-performance
computing infrastructure PLGrid (HPC Centers: ACK Cyfronet AGH) for providing
computer facilities and support within computational grant no.
PLG/2023/016115. Dr. Piotr Gniewek helped us with calculating long-range
coefficients and we are grateful for his support.

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# Estimating the lifetime risk of a false positive screening test result

Tim White∗ and Sara Algeri School of Statistics, University of Minnesota,
Minneapolis, MN

Correspondence<EMAIL_ADDRESS>
###### Abstract

False positive results in screening tests have potentially severe
psychological, medical, and financial consequences for the recipient. However,
there have been few efforts to quantify how the risk of a false positive
accumulates over time. We seek to fill this gap by estimating the probability
that an individual who adheres to the U.S. Preventive Services Task Force
(USPSTF) screening guidelines will receive at least one false positive in a
lifetime. To do so, we assembled a data set of 116 studies cited by the USPSTF
that report the number of true positives, false negatives, true negatives, and
false positives for the primary screening procedure for one of five cancers or
six sexually transmitted diseases. We use these data to estimate the
probability that an individual in one of 14 demographic subpopulations will
receive at least one false positive for one of these eleven diseases in a
lifetime. We specify a suitable statistical model to account for the
hierarchical structure of the data, and we use the parametric bootstrap to
quantify the uncertainty surrounding our estimates. The estimated probability
of receiving at least one false positive in a lifetime is 85.5% ($\pm$0.9%)
and 38.9% ($\pm$3.6%) for baseline groups of women and men, respectively. It
is higher for subpopulations recommended to screen more frequently than the
baseline, including more vulnerable groups such as pregnant women and men who
have sex with men. Since screening technology is imperfect, false positives
remain inevitable. The high lifetime risk of a false positive reveals the
importance of educating patients about this phenomenon.

Keywords: false positive, screening test, lifetime risk, U.S. Preventive
Services Task Force

##  1 Introduction

Healthcare professionals encourage individuals to get screened regularly for
certain cancers, sexually transmitted diseases (STDs), and other medical
conditions. Programs of repeated screening are beneficial because they allow
for the early detection of these diseases, which, in turn, increases the
likelihood of successful treatment. However, since even the most sophisticated
modern screening technology falls short of 100% accuracy, some test results
will inevitably be incorrect. Specifically, the result of any screening test
might wind up being a false negative, indicating that the patient does not
have the disease when they do, or a false positive, indicating that the
patient has the disease when they do not.

Incorrect screening test results occur for a variety of reasons, ranging from
improper specimen collection to transcription and reporting inaccuracies.1
Regardless of their cause, screening test errors have serious implications.
False negatives may delay the detection of potentially life-threatening
diseases, reduce public confidence in screening, and, in some cases, induce
legal action by the affected party.2 False positives, on the other hand, take
a toll on the recipient’s mental health, as they can generate stress and
strain personal relationships.3-7 There is also evidence that false positives
reduce compliance with subsequent screenings.8-9 Further, false positives
often require follow-up tests, and they can even prompt individuals to undergo
unnecessary and costly invasive procedures.10

Despite these adverse psychological, medical, and financial effects, efforts
to quantify and communicate the risk of a false positive have remained scarce.
For most recommended procedures, reliable data exist regarding the false
positive rate of a single screening occasion. However, these rates are far
from common knowledge among the general public. There have also been
relatively few attempts to estimate the probability of receiving at least one
false positive when a particular procedure is repeated over time.11-14 Even
less pursued is the estimation of this probability across multiple screening
procedures for different diseases.15-16 Still absent in the literature is an
effort to compute the probability of receiving at least one false positive in
a lifetime when an individual adheres to a program of repeated screening for
several diseases.

In this manuscript, we estimate the lifetime probability of a false positive
for individuals adhering to the screening guidelines of the United States
Preventive Services Task Force (USPSTF),17 which are widely considered the
gold standard among medical practitioners. We consolidate the USPSTF
recommendations and evidence into a data set that reports the results of 116
studies, each of which summarizes the accuracy of a screening procedure for
one of five cancers or six STDs. We use these data to estimate the probability
of receiving at least one false positive in a lifetime from any of the
screening procedures considered, and we replicate our analysis for 14
demographic and behavioral subpopulations. Our findings provide patients and
healthcare providers with an individualized layer of information that sheds
new light on the long-term significance of screening test imprecision.

##  2 Methods

###  2.1 Diseases and screening procedures

Many different procedures are used to screen individuals for cancers, STDs,
and other diseases like diabetes and osteoporosis. However, most procedures
are recommended only for individuals at high risk for the disease, often only
when risk factors emerge or intensify. Our analysis focuses on the set of
diseases for which routine screening is advocated for large segments of the
population. In addition, we consider only those diseases for which the
negative consequences of false positives are well-documented or can be
reasonably inferred based on the existing literature.

We formalize these principles by using the following criteria to identify the
diseases relevant to our analysis: (1) The disease must be a cancer18 or an
STD19 and (2) the USPSTF must have assigned a grade of C or higher to the
screening service for the disease in its most recent recommendation — i.e., at
minimum, it must recommend the service to be offered selectively to patients
based on their unique circumstances. As detailed in Table S1 of the
supplementary material, there are five cancers (breast, cervical, colorectal,
lung, and prostate)20-24 and six STDs (chlamydia, gonorrhea, hepatitis B,
hepatitis C, HIV, and syphilis)25-31 that satisfy these criteria.

Some of these eleven diseases have only one recommended screening modality,
while others have several viable procedures. However, for nearly all diseases
where multiple screening procedures are available, one is more common than the
others in practice. With this in mind, we focus on only the primary screening
procedure for each disease. The only two diseases for which the most common
procedure is unclear are HIV and syphilis; for these, we select the primary
screening modality based on the quantity and recency of available data.

###  2.2 Subpopulations and screening intervals

Not every individual is recommended to get screened for all eleven diseases
listed above. For each cancer and STD, the USPSTF guidelines target only those
segments of the population for which there are proven benefits to screening.
These groups are defined according to demographic characteristics like age and
sex or behavioral considerations like sexual activity and smoking status.

We account for this heterogeneity in screening protocols by replicating our
analysis for 14 demographic and behavioral subpopulations. The six female
subpopulations are defined according to the individual’s smoking status and
the number of pregnancies they expect to experience in their lifetime. The
eight male subpopulations are determined by the individual’s smoking status,
whether or not they are a man who has sex with men (MSM), and whether or not
they intend to get screened for prostate cancer, which is optional according
to the USPSTF.24 Baseline females are non-smokers who anticipate zero
pregnancies in their lifetime, while baseline males are non-smoking, non-MSM
individuals who do not intend to receive routine prostate exams. For both
females and males, we define smoking status according to the USPSTF
eligibility criteria for lung cancer screening — i.e., smokers are individuals
with a 20 pack-year history who currently smoke or have quit within the past
15 years.23

Since the necessity and frequency of screening vary by subpopulation for each
disease, so too does the number of times that individuals are recommended to
get screened in a lifetime. For some diseases, the lifetime number of
screening occasions follows clearly from the USPSTF guidelines. For others,
the USPSTF guidelines lack either an age range, an interval at which screening
should be repeated, or both, which makes it more complicated to derive the
lifetime number of screening occasions for each subpopulation. Section S1
explains our approach to these more ambiguous cases, and Table S2 reports the
assumed lifetime number of screening occasions for each disease by
subpopulation. Figure 1 lists the primary screening procedure and summarizes
the USPSTF guidelines for each disease considered in our analysis.

###  2.3 Data collection

Given the number of times an individual is recommended to get screened for a
particular disease in a lifetime and provided that we have an estimate of the
false positive rate for each screening occasion, we can estimate the
probability that at least one of these occasions will result in a false
positive. Let $p_{d}$ denote the probability that an individual tests positive
for disease $d$ when the disease is not present. Our first objective is to
obtain an estimate, $\widehat{p}_{d}$, of $p_{d}$.

Figure 1: Timeline of screening guidelines

To compile the data necessary to obtain $\widehat{p}_{d}$, we first identified
the most recent USPSTF recommendation statement for each disease as of August
31, 2021 (see Table S1). Next, we extracted all sources among those cited by
the USPSTF that reported the number of true positives, false negatives, true
negatives, and false positives from a study involving the primary screening
procedure for that disease. If no such studies were cited in the USPSTF
recommendation statement for a particular disease, we referred to the
corresponding USPSTF evidence review and proceeded similarly. In cases where
neither of the two current documents contained relevant studies, we applied
the same strategy to the next most recent set of USPSTF guidelines. See Table
S3 for more details about the data collection procedure for each disease.

This process yielded 116 studies. Let $S_{d}$ denote the number of studies
collected for disease $d$ using the above methodology. The estimated false
positive rate of the primary screening procedure for disease $d$ is given by

$\widehat{p}_{d}=\frac{\sum_{s=1}^{S_{d}}\text{FP}_{s}}{\sum_{s=1}^{S_{d}}n_{s}},$
(1)

where $\text{FP}_{s}$ is the number of false positives in study
$s\in\\{1,...,S_{d}\\}$ and $n_{s}$ is the number of false positives plus the
number of true negatives. It follows from (1) that $\widehat{p}_{d}$ can be
used to estimate the conditional probability of testing positive for disease
$d$ given that the disease is not present in the individual — i.e., the false
positive probability of a single screening occasion for disease $d$. Table 1
reports the estimates obtained from (1) for each disease. Table 2 summarizes
the notation used in Sections 2.3 and 2.4.

###  2.4 Statistical methods

We seek to model the probability $p_{i}$ that a healthy individual in
subpopulation $i$ will receive at least one false positive in a lifetime for
at least one of the five cancers or six STDs considered, assuming that they
adhere to the USPSTF screening guidelines. We define healthy individuals as
those who remain negative for all five cancers and six STDs throughout their
lifetime.

Table 1: Estimated false positive probability of one screening occasion Disease | Screening procedure | Estimate (SE)
---|---|---
Breast cancer | Mammogram | 4.9% (0.1%)
Cervical cancer | Pap test | 5.0% (0.1%)
Chlamydia | NAAT | 0.5% (<0.1%)
Colorectal cancer | Colonoscopy | 11.3% (1.3%)
Gonorrhea | NAAT | 0.2% (<0.1%)
Hepatitis B | HBsAg test | 2.0% (0.1%)
Hepatitis C | Anti-HCV antibody test | 1.0% (0.2%)
HIV | Antigen/antibody test | 0.2% (<0.1%)
Lung cancer | Low-dose CT scan | 20.7% (0.1%)
Prostate cancer | PSA test | 10.2% (0.3%)
Syphilis | RPR test | 0.3% (<0.1%)

In order to model $p_{i}$, we first model the probability $P_{id}$ that a
healthy individual in subpopulation $i$ will receive at least one false
positive for disease $d$ in a lifetime, assuming that they get screened the
recommended number of times $T_{id}$ with the primary screening procedure for
disease $d$. We also assume that, for each disease $d$, the results of each
screening occasion are independent from the results of the previous occasions.
Under these assumptions, the probability $P_{id}$ can be expressed as

$P_{id}=1-(1-p_{d})^{T_{id}}.$ (2)

Next, we extend this formula to incorporate more than one disease. Let
$\mathcal{D}_{i}$ denote the set of diseases for which an individual in
subpopulation $i$ is recommended to get screened at least once. We assume that
the event of receiving at least one false positive in a lifetime for each
disease $d\in\mathcal{D}_{i}$ is independent from the same event for each of
the other diseases in $\mathcal{D}_{i}$. It follows from this assumption that
the probability $p_{i}$, as defined above, is given by

$p_{i}=1-\prod_{d\in\mathcal{D}_{i}}(1-P_{id})=1-\prod_{d\in\mathcal{D}_{i}}(1-p_{d})^{T_{id}}.$
(3)

The derivations of (2) and (3) are provided in Sections S2 and S3,
respectively. These equations are informative in that they describe the
relationship between $p_{d}$, $P_{id}$, and $p_{i}$, but they are limited by
the fact that these quantities are unknown in the real world. However, we can
estimate (2) and (3) by plugging in the estimates $\widehat{p}_{d}$ from (1).
We obtain

$\widehat{P}_{id}=1-(1-\widehat{p}_{d})^{T_{id}}\qquad\text{and}\qquad\widehat{p}_{i}=1-\prod_{d\in\mathcal{D}_{i}}(1-\widehat{P}_{id})=1-\prod_{d\in\mathcal{D}_{i}}(1-\widehat{p}_{d})^{T_{id}},$
(4)

where the latter can be used to estimate, as desired, the probability that a
healthy individual in subpopulation $i$ will receive at least one false
positive in a lifetime.

Table 2: Notation and definitions $S_{d}$ |  Number of studies collected for disease $d$ |  $n_{s}$ |  $TN_{s}+FP_{s}$
---|---|---|---
$TP_{s}$ |  Number of true positives observed in study $s$ |  $N_{s}$ |  $TP_{s}+FN_{s}+TN_{s}+FP_{s}$
$FN_{s}$ |  Number of false negatives observed in study $s$ |  $\widehat{p}_{FP,s}$ |  Proportion of cases in study $s$ that are false positives
$TN_{s}$ |  Number of true negatives observed in study $s$ |  $\widehat{p}_{TN,s}$ |  Proportion of cases in study $s$ that are true negatives
$FP_{s}$ |  Number of false positives observed in study $s$ |  $\widehat{p}_{+,s}$ |  Proportion of cases in study $s$ that are true positives or false negatives
|
$Z_{s}$ |  Multinomial random variable representing the number of true positives, false negatives, true negatives, and false positives in study $s$
$B$ |  Number of bootstrap iterations
|
$p_{d}$ |  Probability of receiving a false positive on one screening occasion for disease $d$
$T_{id}$ |  Number of times an individual in subpopulation $i$ is recommended to get screened for disease $d$ in a lifetime
$P_{id}$ |  Probability that a healthy individual in subpopulation $i$ who adheres to the USPSTF screening guidelines will receive at least one false positive for disease $d$ in a lifetime
$\mathcal{D}_{i}$ |  Set of diseases for which an individual in subpopulation $i$ is recommended to get screened at least once in a lifetime
$p_{i}$ |  Probability that a healthy individual in subpopulation $i$ who adheres to the USPSTF screening guidelines will receive at least one false positive for any of the diseases in $\mathcal{D}_{i}$ in a lifetime

Finally, to quantify the uncertainty surrounding our estimators, we rely on
the assumption that the number of true positives (TP), false negatives (FN),
true negatives (TN), and false positives (FP) from each study $s$ can be
modeled by a multinomial random variable
$Z_{s}\sim\text{Multinomial}(N_{s}\text{ },\text{ }\widehat{p}_{FP,s}\text{
},\text{ }\widehat{p}_{TN,s}\text{ },\text{ }\widehat{p}_{+,s})$, where
$N_{s}$ is the total sample size of study $s$, $\widehat{p}_{FP,s}$ is the
proportion of cases in study $s$ that are false positives,
$\widehat{p}_{TN,s}$ is the proportion that are true negatives, and
$\widehat{p}_{+,s}$ is the proportion that are true positives or false
negatives. We incorporate the multinomial errors into our analysis by means of
the parametric bootstrap.32 Specifically, for each subpopulation $i$, we
simulate from the $\text{Multinomial}(N_{s}\text{ },\text{
}\widehat{p}_{FP,s}\text{ },\text{ }\widehat{p}_{TN,s}\text{ },\text{
}\widehat{p}_{+,s})$ distribution to account for the inherent randomness of
the results of each study $s\in\\{1,...,S_{d}\\}$ for each disease
$d\in\mathcal{D}_{i}$. We plug these simulated results into the equations for
$\widehat{p}_{d}$ and $\widehat{P}_{id}$ to obtain one realization of
$\widehat{p}_{d}$ and $\widehat{P}_{id}$ for each disease
$d\in\mathcal{D}_{i}$, as well as one realization of $\widehat{p}_{i}$. We
carry out this process $B=10,000$ times to obtain 10,000 realizations of each
estimator, and we use these realizations to compute their respective standard
errors. For $\widehat{p}_{i}$, the standard error is given by

$SE(\widehat{p}_{i})=\sqrt{\frac{1}{B-1}\sum_{b=1}^{B}(\widehat{p}_{i}^{\text{
}(b)}-\overline{\widehat{p}}_{i})^{2}},$ (5)

where
$\overline{\widehat{p}}_{i}=\frac{1}{B}\sum_{b=1}^{B}\widehat{p}_{i}^{\text{
}(b)}$. We use an analogous formula for $SE(\widehat{p}_{d})$ and
$SE(\widehat{P}_{id})$. These standard errors are reported along with the
corresponding point estimates.

##  3 Results

For each subpopulation $i$, Table 3 reports the estimated probability
$\widehat{p}_{i}$ that a healthy individual adhering to the USPSTF screening
guidelines will receive at least one false positive in a lifetime for one of
the five cancers or six STDs considered. Table S4 reports the lifetime
probability of a false positive separately for the cancers and the STDs, and
Table S5 breaks these probabilities down by disease for each subpopulation by
reporting the estimates $\widehat{P}_{id}$ from (4).

Table 3: Estimated lifetime false positive probability by subpopulation Subpopulation | Estimate (SE)
---|---
Baseline females | 85.5% (0.9%)
Females, one pregnancy | 86.0% (0.8%)
Females, two pregnancies | 86.5% (0.8%)
Female smokers | 88.5% (0.7%)
Female smokers, one pregnancy | 88.9% (0.7%)
Female smokers, two pregnancies | 89.3% (0.6%)
Baseline males | 38.9% (3.6%)
Men who have sex with men (MSM) | 43.1% (3.4%)
Male smokers | 51.5% (2.9%)
MSM smokers | 54.9% (2.7%)
Males, routine prostate exams | 74.2% (1.7%)
MSM, routine prostate exams | 76.0% (1.6%)
Male smokers, routine prostate exams | 79.6% (1.3%)
MSM smokers, routine prostate exams | 81.0% (1.2%)

The estimated lifetime probability of a false positive is at least 38% for all
subpopulations and is greater than 50% for all but two. It exceeds 85% for all
female subpopulations and is only slightly higher for female smokers (88.5%
$\pm$ 0.7%) and pregnant women (86.0% $\pm$ 0.8%) than baseline females (85.5%
$\pm$ 0.9%). This similarity in lifetime false positive risk between baseline
and non-baseline females stems from the fact that women have a high
probability of receiving at least one false positive in a lifetime for breast
cancer and cervical cancer relative to the other cancers and STDs (see Table
S5), and the USPSTF guidelines for these two diseases are the same across all
female subpopulations. The USPSTF guidelines for breast cancer and cervical
cancer prescribe more screening occasions in a lifetime than the guidelines
for the other cancers and STDs (see Table S2), so it is not surprising that
these two diseases contribute so prominently to lifetime false positive risk
among females.

There is far more variability in lifetime false positive risk among males than
females, as the estimated lifetime probability of a false positive ranges from
around 39% for baseline males (38.9% $\pm$ 3.6%) to more than 80% for MSM
smokers who elect to get screened for prostate cancer (81.0% $\pm$ 1.2%). This
variation is attributable to the fact that the USPSTF screening guidelines
differ substantially across the male subpopulations. For instance, while
baseline males are only recommended to get screened for one cancer and two
STDs, MSM smokers who receive routine prostate exams are recommended to get
screened for three cancers and five STDs (see Table S2).

We can use odds ratios to compare the lifetime risk of a false positive
between subpopulations. The odds ratio for two subpopulations $i_{1}$ and
$i_{2}$ is given by
$(\widehat{p}_{i_{1}}/(1-\widehat{p}_{i_{1}}))/(\widehat{p}_{i_{2}}/(1-\widehat{p}_{i_{2}}))$,
where $\widehat{p}_{i_{1}}$ and $\widehat{p}_{i_{2}}$ are the respective
lifetime false positive probabilities for the two subpopulations. Applying
this formula to the estimates in Table 3, we find that the odds of receiving
at least one false positive in a lifetime are 9.30 times higher for baseline
females than baseline males and 7.27 times higher for female smokers than male
smokers. Much of the discrepancy in lifetime false positive risk between
females and males can be explained by the high lifetime false positive
probability for breast cancer and cervical cancer, as women are recommended to
get screened for these two diseases while men are not.

Among females, the odds of receiving at least one false positive in a lifetime
are 1.30 times higher for female smokers than baseline females. Relative to
baseline females, the odds are 1.04 times higher for women who experience one
pregnancy and 1.08 times higher for women who experience two pregnancies.
These odds ratios are consistent with the small amount of variation in
lifetime false positive risk among the female subpopulations.

Relative to baseline males, the odds of receiving at least one false positive
in a lifetime are 1.19 times higher for men who have sex with men, 1.67 times
higher for male smokers, and 4.53 times higher for males who undergo routine
prostate exams. This last result suggests that the decision to get screened
routinely for prostate cancer has a considerable effect on lifetime false
positive risk among males, which is not surprising considering the high
estimated probability of receiving a false positive from one prostate-specific
antigen (PSA) test (see Table 1) and the relatively frequent time interval for
prostate cancer screening (see Figure 1 and Table S2).

##  4 Discussion

To our knowledge, this manuscript is the first to quantify the lifetime risk
of a false positive for individuals adhering to a program of repeated
screening for multiple diseases. Our results indicate that healthy individuals
who follow the USPSTF screening guidelines for a particular set of five
cancers and six STDs have a high probability of receiving at least one false
positive in a lifetime. Notably, the estimated lifetime probability of a false
positive is high relative to the baseline for more vulnerable groups such as
pregnant women and men who have sex with men. Efforts by healthcare providers
to communicate the lifetime risk of a false positive are likely to be
particularly beneficial for these groups.

Our methodology relies on several assumptions about screening procedures,
participants, and intervals. While our conclusions are robust to minor
violations of these assumptions, we recognize that some of them may not hold
in practice for all individuals. First, the probabilities estimated in this
manuscript are conditional on the screening participant remaining negative for
all five cancers and six STDs throughout their lifetime. The estimates
reported in Tables 1, 3, S4, and S5 are not valid for individuals who contract
one or more of the eleven diseases throughout their lifetime.

Second, the accuracy of our estimates depends on the extent to which patients
and healthcare providers adhere to the USPSTF screening guidelines. We
maintain that our assumption of strict adherence to the USPSTF guidelines is
the most defensible way to determine the number of times an individual is
recommended to get screened for a particular disease in a lifetime.
Nevertheless, several studies suggest that the implementation of these
guidelines in clinical settings is imperfect.33-35 As such, the values
reported in Table S2 may overestimate or underestimate the number of times
individuals actually get screened for certain diseases in a lifetime.

Third, our assumption that individuals receive only the primary screening
procedure for each disease likely does not capture the nuances of actual
screening practices, which can vary across healthcare systems. Multiple
screening procedures are available for several of the diseases considered in
our analysis, and while one procedure tends to be more common than the others
for each disease, the proportion of individuals that get screened with the
other modalities may be nontrivial.

Finally, our determination of the lifetime number of screening occasions for
the STDs relies on a few assumptions that may oversimplify the individualized
nature of STD screening. The USPSTF guidelines for STD screening are highly
contingent on risk factors and personal circumstances, and they generally do
not provide both an age range and a time interval at which screening
procedures should be repeated.25-31 It is therefore possible, if not likely,
that two individuals in the same subpopulation who both adhere to the USPSTF
guidelines will get screened a different number of times for a particular STD
in a lifetime. While it would be reasonable to estimate the lifetime number of
STD screening occasions for each subpopulation empirically, the data required
for this task are not readily available.

Even with these limitations in mind, our results offer undeniable evidence
that the long-term burden of screening test imprecision is worth the attention
of patients and healthcare providers. Specifically, our findings reveal the
importance of communicating the pervasiveness of false positive results to
patients. This transparency is intended not to sow distrust in screening
procedures or the guidelines associated with them, but rather to improve the
ability of patients to respond sensibly to an unexpected positive result. An
individual who is informed about the estimated lifetime false positive
probability for someone with their demographic and behavioral characteristics
is likely to exhibit a more measured response in this scenario than an
uninformed individual.

To aid in this endeavor, we disseminate our findings via a web application
called The False Positives Calculator.36 This interactive tool, which is
openly accessible at https://falsepositives.shinyapps.io/calculator, allows
users to extract personalized information about screening test accuracy and
learn about the USPSTF guidelines for the eleven diseases considered in our
analysis. It also hosts our data set (free for download at
https://github.com/timwhite0/false-positives-calculator), which, to the best
of our knowledge, is the largest centralized, publicly available data source
on screening test accuracy.

False positive results are an inevitable feature of screening tests, as they
cannot be eliminated without simultaneously increasing the prevalence of false
negatives. For as long as screening technology remains imperfect, the best
course of action for patients and healthcare providers is to remain informed
about the risk of a false positive, paying particular attention to how this
risk accumulates over time. This manuscript provides a framework for
quantifying the lifetime risk of a false positive, one that could feasibly be
updated as screening technology evolves and more data become available.

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