# Magnesium doped helium nanodroplets

Alberto Hernando  Departament ECM, Facultat de Física, and IN2UB, Universitat
de Barcelona. Diagonal 647, 08028 Barcelona, Spain  Manuel Barranco
Departament ECM, Facultat de Física, and IN2UB, Universitat de Barcelona.
Diagonal 647, 08028 Barcelona, Spain  Ricardo Mayol  Departament ECM, Facultat
de Física, and IN2UB, Universitat de Barcelona. Diagonal 647, 08028 Barcelona,
Spain  Martí Pi  Departament ECM, Facultat de Física, and IN2UB, Universitat
de Barcelona. Diagonal 647, 08028 Barcelona, Spain  Francesco Ancilotto
Dipartimento di Fisica “G. Galilei”, Università di Padova, via Marzolo 8,
I-35131 Padova, and CNR-INFM-DEMOCRITOS National Simulation Center, Trieste,
Italy

###### Abstract

We have studied the structure of 4He droplets doped with magnesium atoms using
density functional theory. We have found that the solvation properties of this
system strongly depend on the size of the 4He droplet. For small drops, Mg
resides in a deep surface state, whereas for large size drops it is fully
solvated but radially delocalized in their interior. We have studied the
$3s3p$ 1P${}_{1}\leftarrow 3s^{2}$ 1S0 transition of the dopant, and have
compared our results with experimental data from laser induced fluorescence
(LIF). Line broadening effects due to the coupling of dynamical deformations
of the surrounding helium with the dipole excitation of the impurity are
explicitly taken into account. We show that the Mg radial delocalization
inside large droplets may help reconcile the apparently contradictory
solvation properties of magnesium as provided by LIF and electron-impact
ionization experiments. The structure of 4He drops doped with two magnesium
atoms is also studied and used to interpret the results of resonant two-
photon-ionization (R2PI) and LIF experiments. We have found that the two
solvated Mg atoms do not easily merge into a dimer, but rather form a weakly-
bound state due to the presence of an energy barrier caused by the helium
environment that keep them some $9.5$ Å apart, preventing the formation of the
Mg2 molecule. From this observation, we suggest that Mg atoms in 4He drops may
form, under suitable conditions, a soft “foam”-like aggregate rather than
coalesce into a compact metallic cluster. Our findings are in qualitative
agreement with recent R2PI experimental evidences. We predict that,
contrarily, Mg atoms adsorbed in 3He droplets do not form such metastable
aggregates.

###### pacs:

36.40.-c, 32.30.Jc, 78.40.-q, 47.55.D-, 71.15.Mb, 67.40.Yv

##  I Introduction

Optical investigations of atomic impurities in superfluid helium nanodroplets
have drawn considerable attention in recent years,Sti01 ; Sti06  as the shifts
of the electronic transition lines (atomic shifts) are a very useful
observable to determine the location of the foreign atom attached to a helium
drop.Bar06  In this context, the study of magnesium atoms attached to helium
drops has unraveled an interesting and somewhat unexpected solvation behaviour
as a function of the number ($N$) of helium atoms in the drop. Diffusion Monte
Carlo (DMC) calculationsMel05  carried out for small drops containing a number
of helium atoms up to $N=50$ indicate that a Mg atom is not fully solvated in
drops of sizes below $N\sim 30$. More recent quantum Monte Carlo calculations
Elh07 ; Elh08  suggest a surface Mg state for 4He clusters with up to $\sim
200$ atoms. Density Functional Theory (DFT) calculationsHer07  for 3HeN and
4HeN nanodroplets with $N\geq 300$ doped with alkaline earth atoms have shown
that Mg atoms are solvated in their interior, in agreement with the analysis
of Laser Induced Flourescence (LIF)Reh00  and Resonant Two-Photon-Ionization
(R2PI) experiments.Prz07  LIF experiments on the absorption and emission
spectra of Mg atoms in liquid 3He and 4He have been reported and successfully
analyzed within a vibrating atomic bubble model, where full solvation of the
impurity atom is assumed.Mor99 ; Mor06  A more recent experimentRen07  in
which electron-impact ionization data from Mg doped 4He drops with about 104
atoms seems to indicate that magnesium is instead at the surface of the
droplet, in disagreement with the above mentioned LIF and R2PI experiments.

There is some ambiguity associated with the notion of solvation in a helium
droplet. For not too small droplets, one may consider that Mg is fully
solvated when its position inside the droplet is such that its solvation
energy or atomic shift do not appreciably differ from their asymptotic values
in bulk liquid helium, as both quantities approach such limit fairly
alongside. However, for very small drops the energy or atomic shifts of an
impurity atom at the center of the drop may still differ appreciably from the
bulk liquid values because there is not enough helium to saturate these
quantities. This is the case, e.g., of Mg@4He50 studied in Ref. Mel05, .

Mella et al.Mel05  have discussed how the solvation properties of magnesium
are affected by the number of helium atoms in small 4He drops. Since DMC
calculations cannot be extended to the very large drops involved in LIF
experiments,Reh00  they could not carry out a detailed comparison between
their calculated atomic shifts and the experiments. They also pointed out the
sensitivity of the Mg solvation properties on apparently small differences in
the He-Mg pair potentials available in the literature.

The aim of the present work is to re-examine the solvation of magnesium in 4He
nanodroplets from the DFT perspective, extending our previous
calculationsHer07  down to drops with $N\sim 50$ atoms and improving the DFT
approach (i) by treating the dopant as a quantal particle instead of as an
external field, and (ii) by fully taking into account the coupling of the
dipole excitation of the impurity with the dynamical deformations of the
helium around the Mg atom. Our results confirm that full solvation of a Mg
atom in 4He drops requires a minimum number of helium atoms, and disclose some
unusual results for small drops. We calculate the absorption spectrum of a Mg
atom attached to small and large drops, finding a good agreement with the
experiments for the latter. We discuss in a qualitative way the effect of the
impurity angular momentum on the electron-impact ionization yield and on the
absorption spectrum. We also address the structure of a two-magnesium doped
drop; the results are used to discuss the scenario proposed by Meiwes-Broer
and collaborators to interpret their experimental results on R2PI.Prz07

This work is organized as follows. In Sec. II we briefly present our density
functional approach, as well as results for the structure of Mg@4HeN drops.
The method we have employed to obtain the atomic shifts is discussed in Sec.
III, and applied to the case of Mg-doped 4He droplets in Sec. IV. In Sec. V we
study how the presence of a second magnesium atom in the droplet may alter
both the helium drop structure and the calculated atomic shift. A summary is
presented in Sec. VI, and some technical details of our calculations are
described in the appendices.

##  II DFT description of helium nanodroplets

###  II.1 Theoretical approach

In recent years, static and time-dependent density functional methodsDal95 ;
Gia03 ; Leh04  have become increasingly popular to study inhomogeneous liquid
helium systems because they provide an excellent compromise between accuracy
and computational effort, allowing to address problems inaccessible to more
fundamental approaches. Although DFT cannot take into account the atomic,
discrete nature of these systems, it can nevertheless address highly
inhomogenous helium systems at the nanoscale,Anc05  including the anisotropic
deformations induced by atomic dopants in helium drops (see Ref. Bar06,  for a
recent overview on the physics of helium nanodroplets).

Our starting point is the Orsay-Trento density functional,Dal95  together with
the Mg-He adiabatic ground-state potential $X^{1}\Sigma$ of Ref. Hin03, , here
denoted as $V_{Mg-He}$. To check the sensitivity of our results to the details
of different available pair potential describing the Mg-He interaction, we
also use the sligthly less attractive potential computed in Ref. Par01, . For
the sake of comparison, we plot both potentials in Fig. 1. Despite the
apparently minor differences between these two potential curves, they cause
very different solvation properties of Mg in small 4He drops, as we will show
in the following.

The energy of the Mg-helium system is written as a functional of the Mg wave
function $\Phi(\mathbf{r})$ and the 4He “order parameter”
$\Psi(\mathbf{r})=\sqrt{\rho(\mathbf{r})}$, where $\rho(\mathbf{r})$ is the
4He atomic density:

$\displaystyle E[\Psi,\Phi]$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2\,m_{He}}\int\mathrm{d}^{3}\mathbf{r}\,|\nabla\Psi(\mathbf{r}\,)|^{2}+\int\mathrm{d}^{3}\mathbf{r}\,{\cal
E}(\rho)$ (1) $\displaystyle+$
$\displaystyle\frac{\hbar^{2}}{2\,m_{Mg}}\int\mathrm{d}^{3}\mathbf{r}\,|\nabla\Phi(\mathbf{r}\,)|^{2}+\int\int\mathrm{d}^{3}\mathbf{r}\,\mathrm{d}^{3}\mathbf{r^{\prime}}\,|\Phi(\mathbf{r})|^{2}\,V_{Mg-
He}(|\mathbf{r}-\mathbf{r^{\prime}}|)\,\rho(\mathbf{r^{\prime}})\;.$

In this expression, ${\cal E}(\rho)$ is the 4He “potential energy
density”.Dal95  Minimizing $E$ under the constraints of a given $N$ and a
normalized Mg wave function, yields the ground state of the drop-impurity
complex. To address the solvation of the Mg atom, we have found it convenient
to minimize $E$ subjected to the additional constraint of a fixed distance
${\cal Z}_{0}$ between the centers of mass of the helium moiety and of the
impurity atom which, due to the symmetry of the problem, can both be taken on
the $z$ axis. This is done following a method -borrowed from Nuclear Physics-
similar to that used to describe the fission of rotating 3He drops.Gui93
Specifically, we minimize the expression

$E+\frac{\lambda_{C}}{2}[{\cal Z}-{\cal Z}_{0}]^{2}\;,$ (2)

where ${\cal Z}$ is the average distance between the impurity and the center
of mass of the helium droplet

${\cal Z}=\int d\mathbf{r}^{3}\,z\,|\Phi(\mathbf{r})|^{2}-\frac{1}{N}\int
d\mathbf{r}^{3}\,z\,\rho(\mathbf{r})\;.$ (3)

$\lambda_{C}$ is an arbitrary constant. The value $\lambda_{C}\sim$ 1000 K Å-2
has been used in our calculations, which ensures that the desired ${\cal
Z}_{0}$ value is obtained within a 0.1 % accuracy.

We have solved the Euler-Lagrange equations which result from the variations
with respect to $\Psi^{*}$ and $\Phi^{*}$ of the constrained energy Eq. (2),
namely

$-\frac{\hbar^{2}}{2\,m_{He}}\Delta\Psi+\left\\{\frac{\delta{\cal
E}}{\delta\rho}+U_{He}-\lambda_{C}({\cal Z}-{\cal
Z}_{0})\frac{z}{N}\right\\}\Psi=\mu\Psi$ (4)
$-\frac{\hbar^{2}}{2\,m_{Mg}}\Delta\Phi+\left\\{U_{Mg}+\lambda_{C}({\cal
Z}-{\cal Z}_{0})\,z\right\\}\Phi=\varepsilon\Phi\;,$ (5)

where $\mu$ is the helium chemical potential and $\varepsilon$ is the lowest
eigenvalue of the Schrödinger equation obeyed by the Mg atom. The above
coupled equations have to be solved selfconsistently, starting from an
arbitrary but reasonable choice of the unknown functions $\Psi$ and $\Phi$.
The fields $U_{He}$ and $U_{Mg}$ are defined as

$\displaystyle U_{He}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{3}\mathbf{r^{\prime}}\,|\Phi(\mathbf{r^{\prime}})|^{2}\,V_{Mg-
He}(|\mathbf{r}-\mathbf{r^{\prime}}|)$ $\displaystyle U_{Mg}(\mathbf{r})$
$\displaystyle=$
$\displaystyle\int\mathrm{d}^{3}\mathbf{r^{\prime}}\,\rho(\mathbf{r^{\prime}})\,V_{Mg-
He}(|\mathbf{r}-\mathbf{r^{\prime}}|)\;.$ (6)

In spite of the axial symmetry of the problem, we have solved the above
equations in three-dimensional (3D) cartesian coordinates. The main reason is
that these coordinates allow to use fast Fourier transformation techniquesFFT
to efficiently compute the convolution integrals entering the definition of
${\cal E}(\rho)$, i.e. the mean field helium potential and the coarse-grained
density needed to compute the correlation term in the He density functional,
Dal95  as well as the fields defined in Eq. (6).

The differential operators in Eqs. (1,4,5) have been discretized using
13-point formulas for the derivatives. Eqs. (4-5) have been solved employing
an imaginary time method;Pre92  some technical details of our procedure are
given in Ref. Anc03, . Typical calculations have been performed using a
spatial mesh step of 0.5 Å. We have checked the stability of the solutions
against reasonable changes in the spatial mesh step.

###  II.2 Structure and energetics of Mg-doped helium nanodroplets

Equations (4-5) have been solved for $\lambda_{C}=0$ and several $N$ values,
namely $N=30$, 50, 100, 200, 300, 500, 1000, and 2000\. This yields the ground
state of the drop-impurity complex, and will allow us to study the atomic
shift for selected cluster sizes.

Figure 2 shows the energy of a magnesium atom in a drop, defined as the energy
difference

$S_{N}(Mg)=E(Mg@^{4}He_{N})-E(^{4}He_{N})\;.$ (7)

We compare in Fig. 2 the $S_{N}(Mg)$ values here obtained with those of Ref.
Her07, , where the Mg atom was treated as an infinitely massive particle
-i.e., as a fixed external field acting on the 4He drop. The neglect of the
quantum kinetic energy of the impurity overestimates the impurity solvation
energies by quite a large amount, about 19.7 K for $N=50$ and 18.8 K for
$N=2000$. The value we have found for $S_{50}(Mg)$, $-18.4$ K, compares well
with the DMC result of Ref. Elh08,  ($-21$ K), showing that DFT performs quite
well for small clusters,Bar06  far from the regime for which it was
parametrized. The DMC energy found for the same system is $\sim-168.2$ K,Mel05
whereas our DFT result is $\sim-157.0$ K, and the “asymptotic” DMC value for
$S_{200}(Mg)$, $-33.1$ K, compares well with the DFT value, still far from the
limit value corresponding to a very large helium drop (see Fig. 2).

The solvation properties of the Mg atom are determined by a delicate balance
between the different energy terms –surface, bulk and helium-impurity– in Eq.
(1), whose contribution is hard to disentangle and depends on the number of
atoms in the drop, as shown by DMC and DFT calculations. To gain more insight
into the solvation process of magnesium in small 4He droplets, we have
computed the energy of the doped droplet as a function of the impurity
position ${\cal Z}_{0}$.

The bottom panel in Fig. 3 shows $E({\cal Z}_{0})$ for $N=50$, as computed
using the two different Mg-He pair interactions shown in Fig. 1 (the
discussion of the top panel is postponed until Sec. IV.A). It can be seen that
$E({\cal Z}_{0})$ in both cases displays two local minima. In one case (i.e.
for the sligtly less attractive potential in Fig. 1) a “surface” state for the
Mg atom is energetically preferred, while in the other case the impurity
prefers to sit in the interior to the droplet (although not exactly at its
center). These results are in agreement with the DMC calculations of Ref.
Mel05, , where it has been shown that a bimodal distributions for the Mg
radial probability density function with respect to the center-of-mass of the
helium moiety appears for $N\leq 30$.Mel05  More recent DMC calculations
carried out up to $N\sim 200$ dropsElh08  seem to point out that Mg is always
in a surface state, although somewhat beneath the drop surface. Our DFT
calculations yield that Mg is already solvated for $N=200$.

For both pair potentials, the bottom panel of Fig. 3 shows that the two local
minima in $E({\cal Z}_{0})$ are separated by an energy barrier of about 1 K
height, allowing the impurity to temporarily visit, even at the experimental
temperature $T\sim 0.4$ K, the less energetically favored site. This causes
changes in the total energy of the system by less that 1 %, but has a large
effect on the value of the atomic shift. We will address this important issue,
and its consequencies on the computed spectral properties of Mg@HeN in the
next Sections.

Figure 4 shows the helium configurations for the four stationary points
displayed in Fig. 3, namely those corresponding to ${\cal Z}_{0}=0$, 2, 4, and
6 Å. Although the preference for the surface or the solvated state depends on
the He-Mg pair potential used in the calculations, similarly to the case of
other alkali atoms,Her07  this does not seem to be the case for the stationary
points at ${\cal Z}_{0}=2$ and 4 Å that are present in both curves. We have
compared in Fig. 5 the density profiles along the $z$ axis for the ${\cal
Z}_{0}=2(4)$ Å configuration with the profile of the pure 4He drop, finding
that the appearence of these stationary points is related to the position of
the density peak in the first helium solvation shell with respect to a
maximum(minimum) of the density of the pure drop. We see that a
minimum(maximum) in the energy is associated with a constructive(destructive)
interference in the oscillation pattern of the He density. We are lead to
conclude that the interplay between the density oscillations already present
in pure drops, and the solvation shells generated by the impurity plays an
important role in the solvation properties of the Mg atom. This effect is also
present, although to a lesser extent, in Ca-doped 4He nanodroplets.Her08

Eventually, for larger drops Mg becomes fully solvated. We have found that
this is the case whichever of these two potentials we use (see for instance
the bottom panel of Fig. 6). For this reason, the results we discuss in the
following have been obtained with the Mg-He pair potential of Ref. Hin03, ,
unless differently stated.

When the Mg atom is fully solvated, e.g. for $N=1000$, we have found that
$E({\cal Z}_{0})$ grows monotonously as ${\cal Z}_{0}$ increases (i.e. as the
Mg atom approaches the droplet surface). This is shown in the bottom panel of
Fig. 6. To better understand how $E({\cal Z}_{0})$ depends on $N$, we have
plotted in the top panel of Fig. 6 the energy of the $N=1000$ and 2000 doped
drops, referred to their equilibrium value, as a function of the distance from
the dividing surface, i.e. the radius $R_{1/2}$ at which the density of the
pure drop equals $\rho_{b}/2$, $\rho_{b}$ being the liquid density value
($R_{1/2}=r_{0}N^{1/3}$, with $r_{0}=$ 2.22 Å). These radii are 22.2 and 28.0
Å for $N=1000$ and 2000, respectively.

The top panel of Fig. 6 shows that most of the change in $\Delta E$ takes
place in the 15 Å outer region of the drop, irrespective of its size. As a
consequence of the flatness of $E({\cal Z}_{0})$, the magnesium atom is very
delocalized in the radial direction even in relatively small drops. This
delocalization might affect the absorption spectrum of the attached Mg atom,
and also must be explicitly considered in the interpretation of the electron-
impact yield experiments.Ren07  We address this issue in Sec. IV.

##  III Excitation spectrum of a Mg atom attached to a 4He drop

Lax methodLax52  offers a realistic way to study the absorption spectrum of a
foreign atom embedded in liquid drops. It makes use of the Franck-Condon
principle within a semiclassical approach, and it has been widely employed to
study the absorption spectrum of atomic dopants attached to 4He drops, see
e.g. Ref. Her08,  and references therein. The method is usually applied in
conjunction with the diatomics-in-molecules theory,Ell63  which means that the
atom-drop complex is treated as a diatomic molecule, where the helium moiety
plays a role of the other atom.

In the original formalism, to obtain the line shape one has to carry out an
average on the possible initial states of the system that may be thermally
populated. Usually, this average is not needed for helium drops, as their
temperature, about $0.4$ K,Toe04  is much smaller than the vibrational
excitation energies of the Mg atom in the mean field represented by the second
of Eqs. (6). In small helium droplets thermal effects can show up in the Mg
absorption spectrum due to the high mobility of the atom. For large drops,
thermal motion plays a minor role, as the Mg atom hardly gets close enough to
the drop surface to have some effect on the line shape. In this case, however,
dynamical deformations of the cavity around the impurity may be relevant.Ler93
; Kin96

The line shape for electronic transitions from the ground state $(gs)$ to the
excited state $(ex)$ in a condensed phase system can be written as

$I(\omega)\propto\int\mathrm{d}t~{}\mathrm{e}^{-i\omega
t}\langle\Psi^{gs}|D_{ge}^{\dagger}~{}\mathrm{e}^{\frac{it}{\hbar}\mathcal{H}_{ex}}~{}D_{ge}~{}\mathrm{e}^{-\frac{it}{\hbar}\mathcal{H}_{gs}}|\Psi^{gs}\rangle\;,$
(8)

where $D_{\mathrm{ge}}$ is the matrix element of the electric dipole operator,
$|\Psi^{gs}\rangle$ is the ground state of the system, and $\mathcal{H}_{gs}$
and $\mathcal{H}_{ex}$ are the Hamiltonians that describe the ground and
excited states of the system respectively.

$I(\omega)$ is evaluated using the Born-Oppenheimer approximation and the
Franck-Condon principle, whereby the heavy nuclei do not change their
positions or momenta during the electronic transition. If the excited electron
belongs to the impurity, the helium cluster remains frozen, so that the
relevant coordinate is the relative position $\mathbf{r}$ between the cluster
and the impurity. This principle amounts to assuming that $D_{\mathrm{ge}}$ is
independent of the nuclear coordinates. Taking into account that
$\mathrm{e}^{-\frac{it}{\hbar}\mathcal{H}_{gs}}|\Psi^{gs}\rangle=\mathrm{e}^{-it\omega_{gs}}|\Psi^{gs}\rangle$
and projecting on eigenstates of the orbital angular momentum of the excited
electron $|m\rangle$ one obtains

$I(\omega)\propto|D_{\mathrm{ge}}|^{2}\sum_{m}\int\mathrm{d}t~{}\mathrm{e}^{-i(\omega+\omega^{\mathrm{gs}}_{X})t}\int\mathrm{d}^{n}[\alpha]\int\mathrm{d}^{3}\mathbf{r}~{}{\psi_{X}^{\mathrm{gs}}(\mathbf{r},[\alpha])}^{*}~{}\mathrm{e}^{\frac{it}{\hbar}H^{\mathrm{ex}}_{m}(\mathbf{r},[\alpha])}~{}\psi_{X}^{\mathrm{gs}}(\mathbf{r},[\alpha])\;,$
(9)

where $\hbar\omega^{\mathrm{gs}}_{X}$ and
$\psi_{X}^{\mathrm{gs}}(\mathbf{r},[\alpha])$ are the energy and the wave
function of the ro-vibrational ground state of the frozen helium-impurity
system, and $H^{\mathrm{ex}}_{m}(\mathbf{r},[\alpha])$ is the ro-vibrational
excited Hamiltonian with potential energy
$V^{\mathrm{ex}}_{m}(\mathbf{r},[\alpha])$ determined by the electronic energy
eigenvalue, as obtained for a $p\leftarrow s$ transition. At this point, we
have introduced the variables $[\alpha]$ to represent the degrees of freedom
needed to describe possible deformations of the system, corresponding to the
zero point oscillations of the helium bubble around the impurity. If this
effect is neglected, the deformation parameters $[\alpha]$ are dropped and the
ground state wave function $\psi_{X}^{\mathrm{gs}}$ coincides with the Mg
wavefunction $\Phi$ found by solving Eq. (5).

If the relevant excited states for the transition have large quantum numbers,
they can be treated as approximately classical using the averaged energy
$\hbar\omega^{m}_{\nu}\approx V^{\mathrm{ex}}_{m}(\mathbf{r},[\alpha])$ which
is independent of $\nu$. In this case we obtain the expression

$\displaystyle I(\omega)$ $\displaystyle\propto$
$\displaystyle\sum_{m}\int\mathrm{d}^{n}[\alpha]\int\mathrm{d}^{3}\mathbf{r}~{}|\psi_{X}^{\mathrm{gs}}(\mathbf{r},[\alpha])|^{2}\delta(\omega+\omega^{\mathrm{gs}}_{X}-V^{\mathrm{ex}}_{m}(\mathbf{r},[\alpha])/\hbar)$
(10) $\displaystyle=$
$\displaystyle\hbar\int\mathrm{d}^{n}[\alpha]\int_{\Omega_{m}(\omega)}\;\mathrm{d}^{2}\mathbf{r}~{}\frac{|\psi_{X}^{\mathrm{gs}}(\mathbf{r},[\alpha])|^{2}}{|\mathbf{\nabla}V^{\mathrm{ex}}_{m}(\mathbf{r},[\alpha])|}\,,$

where $\Omega_{m}(\omega)$ is the surface defined by the equation
$\omega+\omega^{\mathrm{gs}}_{X}-V^{\mathrm{ex}}_{m}(\mathbf{r},[\alpha])/\hbar=0$.
If the atom is in bulk liquid helium, or at the center of the drop, the
problem has spherical symmetry and the above equation reduces to

$\displaystyle I(\omega)$ $\displaystyle\propto$ $\displaystyle
4\pi\sum_{m}\int\mathrm{d}^{n}[\alpha]\int\mathrm{d}r~{}|r~{}\psi_{X}^{\mathrm{gs}}(r,[\alpha])|^{2}\delta(\omega+\omega^{\mathrm{gs}}_{X}-V^{\mathrm{ex}}_{m}(r,[\alpha])/\hbar)$
(11) $\displaystyle=$ $\displaystyle
4\pi\hbar\sum_{m}\int\mathrm{d}^{n}[\alpha]\left|\frac{[r~{}\psi_{X}^{\mathrm{gs}}(r,[\alpha])]^{2}}{d{V}^{~{}\mathrm{ex}}_{m}(r,[\alpha])/dr}\right|_{r=r_{m}(\omega)}\;,$

where $r_{m}(\omega)$ is the root of the equation
$\omega+\omega^{\mathrm{gs}}_{X}-V^{\mathrm{ex}}_{m}(r,[\alpha])/\hbar=0$. In
the non-spherical case, we have evaluated $I(\omega)$ using the first
expression in Eq. (10).

The potential energy surfaces $V^{\mathrm{ex}}_{m}(\mathbf{r},[\alpha])$
needed to carry out the calculation of the atomic shifts have been obtained in
the pairwise sum approximation, using the $V_{\Pi}(r)$ and $V_{\Sigma}(r)$ Mg-
He adiabatic potentials from Ref. Mel05, . In cartesian coordinates, and
assuming that the He-impurity spin-orbit interaction is negligible for
magnesium, the eigenvalues of the symmetric matrix

$U_{ij}(\mathbf{r},[\alpha])=\int\mathrm{d}^{3}\mathbf{r^{\prime}}\rho(\mathbf{r^{\prime}}+\mathbf{r},[\alpha])\left\\{V_{\Pi}(r^{\prime})\delta_{ij}+[V_{\Sigma}(r^{\prime})-V_{\Pi}(r^{\prime})]\frac{x^{\prime
i}~{}x^{\prime j}}{r^{\prime 2}}\right\\}$ (12)

are the $V^{\mathrm{ex}}_{m}(\mathbf{r},[\alpha])$ potentials which define the
potential energy surfaces (PES) as a function of the distance between the
center-of-mass of the droplet and that of the impurity.

For spherical geometries, Eq. (12) is diagonal with matrix elements (in
spherical coordinates)

$\displaystyle\lambda_{i}(r,[\alpha])$ $\displaystyle\equiv$ $\displaystyle
U_{ii}(r,[\alpha])=2\pi\int\int r^{\prime
2}\sin\theta^{\prime}\mathrm{d}\theta^{\prime}\mathrm{d}r^{\prime}\rho(\sqrt{r^{\prime
2}+r^{2}+2r^{\prime}r\cos\theta^{\prime}},[\alpha])$ (13)
$\displaystyle\times$
$\displaystyle\left\\{V_{\Pi}(r^{\prime})+[V_{\Sigma}(r^{\prime})-V_{\Pi}(r^{\prime})]\left[\frac{1}{2}(\delta_{i1}+\delta_{i2})\sin^{2}\theta^{\prime}+\delta_{i3}\cos^{2}\theta^{\prime}\right]\right\\}$

In this case, two of the PES are degenerate, namely
$\lambda_{1}(r,[\alpha])=\lambda_{2}(r,[\alpha])\neq\lambda_{3}(r,[\alpha])$.notaCa
This holds true for $r\neq 0$, and it is relevant when we take into account
the delocalization of the impurity inside the bubble due to its quantum
motion. Otherwise, since at $r=0$ all the $\lambda_{i}$ coincide, they are
threefold degenerate.

##  IV Results for the absorption spectrum of magnesium atoms

The problem of obtaining the atomic shift of magnesium in a helium drop has
been thus reduced to that of the dopant in the 3D trapping potentials
corresponding to the ground state and $P$ excited states. We consider first
the $[\alpha]=0$ case (i.e. no zero-point deformations of the helium cavity
hosting the impurity). The general situation, in particular the homogeneous
width calculation, is presented later on in this Section.

If $[\alpha]=0$, the model is expected to yield at most the energies of the
atomic transitions, but not the line shapes since the impurity-droplet
excitation interactions as well as inhomogeneous broadening resulting from
droplet size distributions, laser line width and similar effects are not
included. These limitations are often overcome by introducing line shape
functions or convoluting the calculated lines with some effective line
profiles.Sti96 ; Bue07  We discuss here two illustrative examples, namely the
atomic shift of magnesium in $N=50$ and 1000 nanodroplets. The homogeneous
width is calculated in Subsection B for the $N=1000$ droplet.

For Mg@4He50, the calculated shifts at ${\cal Z}_{0}=0$ (spherical
configuration), 2 and 6 Å are 500, 450 and 281 cm-1, respectively (281.3,
281.7 and 283.0 nm wavelengths). No experimental information is available for
such a small drop. Contrarily, the absorption spectrum of the $3s3p$
1P${}_{1}\leftarrow 3s^{2}$ 1S0 transition of Mg atoms attached to large
helium drops has been measured,Reh00  displaying a broad peak strongly blue-
shifted from its position in the gas-phase. This spectrum is remarkably close
to the one obtained by Moriwaki and MoritaMor99  in bulk liquid helium, and
hence it has been concluded that Mg is in the interior of the 4He droplet. We
want to point out that, while the absorption line in liquid helium was
attributed to a single broad peak of energy 281.5 nm (35 524 cm-1), i.e., a
shift of 474 cm-1,Mor99  in large drops a similar line profileNote1  was
fitted by two Gaussians centered at 35 358 and 35 693 cm-1 (282.8 and 280.2 nm
wavelengths) respectively, i.e., shifted 307 cm-1 and 642 cm-1 from the gas-
phase line.Reh00  The origin of the two peaks was attributed to the splitting
of the degenerate $\Pi$ state by dynamical quadrupole deformations of the
cavity surrounding the dopant, since this argument had qualitatively explained
similar doubly-shaped $D_{2}$ excitation spectra of Rb and Cs atoms in liquid
4He due to a quadrupole oscillation of the helium bubble (dynamic Jahn-Teller
effect).Kin96

LIF experiments on the heavier alkaline earth Ca and Sr in large 4He
dropletsSti97  have disclosed the existence of strong blue-shifted, broad
peaks with no apparent structure, although it cannot be discarded that this
broad line could be a superposition of unresolved peaks. The same happens for
Ba.Sti99  The surface location of Ca, Sr, and Ba in these drops has been
further confirmed by DFT calculations.Her07 ; Her08  It is also interesting to
recall that LIF experiments on Ca atoms in liquid 4He and 3He have found a
broad line in the region of the $4s4p$ 1P${}_{1}\leftarrow 4s^{2}$ 1S0
transition with no apparent splitting,Mor05  contrarily to the case of Mg.

Since Mg is fully solvated in the $N=1000$ drop, the calculated atomic shift
$\Delta\omega$ may be sensibly compared with the experimental data where drops
with $N$ in the $10^{3}-10^{4}$ range are studied. We have obtained
$\Delta\omega=659.0$ cm-1 (280.0 nm wavelength); this peak nearly corresponds
to the Gaussian that takes most of the intensity of the absorption line (about
87 %).Reh00  We have carried out a detailed analysis for this drop,
determining the equilibrium structure of Mg@4He1000 as a function of ${\cal
Z}_{0}$, and have used it to evaluate $\Delta\omega$. The results are
displayed in Table 1, showing the actual sensitivity of the absorption
spectrum to the Mg atom environment.

The impurity-drop excitations will determine the homogeneous width of the
spectral line, and the population of excited states may be relevant given the
limit temperature attained by the droplets.Sti06 ; Toe04 ; Bri90  In this
context, the relevant excitation modes of the helium bubble are radial
oscillations of monopole type (breathing modes), and multipole shape
oscillations about the equilibrium configuration, as well as displacements of
the helium bubble inside the droplet. We will address these issues in the
following Subsections.

###  IV.1 Thermal motion and angular momentum effects

To describe the displacement of the helium bubble inside the droplet, we have
fitted the $E({\cal Z}_{0})$ curve of the Mg@4He1000 system to a parabola, and
have obtained the excitation energy $\hbar\omega$ for this 3D isotropic
harmonic oscillator. The hydrodynamic mass of the impurity atom has been
estimated by its bulk liquid helium value, $M^{*}\sim 40$ a.u., obtained by
the method outlined in Appendix A, Eq. (52). We find $\hbar\omega=0.1$ K,
indicating that thermal motion, i.e., the population of the excited states of
the “mean field” $E({\cal Z}_{0})$ is important at the experimental
temperature $T=0.4$ K, and may produce observable effects in the absorption
spectrum and the electron-impact ionization yield.

To describe in more detail the delocalization of the Mg atom inside the drop,
we have used an effective Hamiltonian where we interpret ${\cal Z}_{0}$ as the
radial distance $R$ between the impurity and center of mass of the helium
moiety, and $E({\cal Z}_{0})$ as the “potential energy” $V(R)$ associated with
this new degree of freedom of the impurity in the drop. Namely,

${\cal
H}=\frac{\hat{P}^{2}}{2M^{*}}+V(R)=\frac{\hat{P}_{R}^{2}}{2M^{*}}+\frac{\hat{L}^{2}}{2M^{*}R^{2}}+V(R)\;,$
(14)

where $M^{*}$ is the Mg hydrodynamic mass. In the canonical ensemble, the
total probability distribution $W$ as a function of $R$ can be written as

$\begin{array}[]{rl}\displaystyle
W(R)&\displaystyle=Q^{-1}\int_{0}^{R}\mathrm{d}R^{\prime}R^{\prime
2}\int\mathrm{d}\Omega^{\prime}\sum_{n\ell m}\langle\psi_{n\ell
m}|\mathbf{R}^{\prime}\rangle e^{-{\cal
H}(R^{\prime})/k_{B}T}\langle\mathbf{R}^{\prime}|\psi_{n\ell m}\rangle\\\
&\displaystyle=Q^{-1}\int_{0}^{R}R^{\prime
2}\mathrm{d}R^{\prime}\sum_{n\ell}(2\ell+1)e^{-E_{n\ell}/k_{B}T}|\phi_{n\ell}(R^{\prime})|^{2}\;,\end{array}$
(15)

where the partition function is defined as $Q=Tr\left(e^{-{\cal
H}/k_{B}T}\right)=\sum_{n\ell}(2\ell+1)e^{-E_{n\ell}/k_{B}T}$, being $k_{B}$
the Boltzmann constant. The radial probability density $w$ is

$w(R)=\frac{\mathrm{d}W}{\mathrm{d}R}=Q^{-1}R^{2}\sum_{n\ell}(2\ell+1)e^{-E_{n\ell}/k_{B}T}|\phi_{n\ell}(R)|^{2}\;.$
(16)

This expression has been evaluated for $N=50$ and 1000 by solving the
Schrödinger equation for the Hamiltonian Eq. (14) to obtain the orbitals
$\phi_{n\ell}$ and eigenenergies $E_{n\ell}$. For larger $N$ values, we have
used the semiclassical approximation
$E_{n\ell}\rightarrow\frac{\hat{p}_{R}^{2}}{2M^{*}}+V_{\mathrm{eff}}(R)$,
where the effective potential is

$V_{\mathrm{eff}}(R)=\frac{\hbar^{2}\ell(\ell+1)}{2M^{*}R^{2}}+V(R)\;.$ (17)

Integrating $p_{R}$ in phase space, we obtain for the probability density

$w(R)=Q^{-1}R^{2}\exp\left(-\frac{V(R)}{k_{B}T}\right)\sum_{\ell}(2\ell+1)\exp\left[-\frac{1}{k_{B}T}\frac{\hbar^{2}\ell(\ell+1)}{2M^{*}R^{2}}\right]$
(18)

with the normalization
$Q=\int_{0}^{\infty}dRR^{2}\exp\left(-\frac{V(R)}{k_{B}T}\right)\sum_{\ell}(2\ell+1)\exp\left[-\frac{1}{k_{B}T}\frac{\hbar^{2}\ell(\ell+1)}{2M^{*}R^{2}}\right]$.

Lacking a better choice, we have weighted any possible angular momentum value
with a Boltzmann energy factor. It has been shownLehm04  that some of the
angular momentum deposited in the droplet during the pickup process may be
kept in the impurity atom, resulting in a different angular momentum
distribution than the Boltzmann one. This could yield that some Mg atoms are
actually closer to the drop surface.

If the angular momentum associated with the motion of the magnesium atom
-whose “radius” is $\sim 5$ Å, see Fig. 1\- is such that Mg can be some 10 Å
beneath the drop surface, the shift of the absorption line would be hardly
distinguishable from that of the totally solvated case -as seen in Table 1. At
the same time, the electron-energy dependence of the Mg+ yield observed in
electron impact ionization experimentsRen07  (and which was considered as an
evidence of a surface location of Mg atoms on 4He droplets) could indeed be
due to Penning ionization of the impurity in a collision with a metastable He∗
atom that occupies a surface bubble state in the drop, instead of being due to
the transfer of a positive hole (He+) to the Mg atom, which is the primary
ionization mechanism when the impurity is very attractive and resides in the
deep bulk of the droplet.

We show in the top panel of Fig. 3 the probability densities $w(R)$ at $T=0.4$
K corresponding to the configurations displayed in the bottom panel.notem
Similarly, the top panel of Fig. 6 shows that for Mg@4He1000 and Mg@4He2000,
if thermal motion is taken into account and the impurity retains some of the
pick-up angular momentum, the maximum density probability of Mg is at $\sim
15$ Å beneath the drop surface in both cases. To obtain it, we have taken for
$M^{*}$ the bulk value 40 a.u. As seen from Table 1, the absorption line shift
changes by a small 2% with respect to the $R=0$ configuration. The values of
the angular momentum corresponding to these maximum density probabilities are
$\langle L^{2}\rangle^{1/2}\sim 9\hbar$ and $\sim 10\hbar$, respectively.NoteL
For a $N=10\,000$ drop,Ren07  whose radius is $R_{1/2}=47.8$ Å, we have
extrapolated inwards the $\Delta E$ curves of the calculated $N=1000$ and 2000
drops, and have obtained from it the probability distribution displayed in
Fig. 6. Its maximum is at $\sim 18.5$ Å beneath the surface, with $\langle
L^{2}\rangle^{1/2}\sim 28\hbar$.

Finally, using the effective potential of Eq. (17) we have determined that for
$N=1000(2000)$, a Mg atom with $\ell\sim 24\hbar(44\hbar)$ is in an
“equilibrium position” some 10 Å beneath the drop surface. For the $N=10\,000$
drop, this value is $\ell\sim 148\hbar$. These values look reasonable, and Mg
atoms holding this angular momentum or larger might thus be the origin of the
primary electron-collision ionization yield by the Penning process, without
questioning the conclusion drawn from LIF experiments that magnesium is fully
solvated in 4He drops.

###  IV.2 Homogeneous width from shape deformations of the helium bubble

We have shown that for large drops, the Mg atom is fully solvated and its
thermal motion only produces small changes in the absorption shift. This
allows us to decouple the effect of the translational motion of the helium
bubble on the absorption line, from that of its shape fluctuation. Moreover,
we can address shape fluctuations in the much simpler spherically symetric
ground state, when magnesium is located at the center of the drop.

To quantify the effect of these fluctuations, we have first used the spherical
cap modelAnc95  to estimate the excitation energies of the helium bubble
around the impurity in liquid helium. To this end, we have fitted the
$V_{\text{Mg-He}}$ potential to a Lennard-Jones potential with depth
$\varepsilon=7$ K at a minimum distance $r_{min}=5$ Å. Minimizing the total
energy within this model yields a configuration with an equilibrium radius of
$R_{0}=0.97\times 2^{-1/6}r_{min}$, that we approximate by
$R_{0}=2^{-1/6}r_{min}$ to obtain the excitation energies.

Deformations of the 4He around the Mg atom are modeled asWil64 ; Rin80

$R(\Omega)=R_{\alpha}\left[1+\alpha_{0}+\sum_{\lambda=2}^{\infty}\sum_{\mu=-\lambda}^{\lambda}\alpha_{\lambda\mu}Z_{\lambda\mu}(\Omega)\right]\;,$
(19)

where $R$ is the radius of the bubble cavity hosting the solvated Mg atom,
$\Omega$ represents the solid angle variables $(\theta,\phi)$,
$Z_{\lambda\mu}(\Omega)$ is a real spherical harmonic, and
$\alpha_{\lambda\mu}$ is the amplitude of the $\lambda\mu-$multipole
deformation. The condition
$R_{\alpha}=R_{0}(1-\frac{1}{4\pi}\sum_{\lambda\mu}|\alpha_{\lambda\mu}|^{2})$
ensures the conservation of the number of particles up to second order in
$\alpha_{\lambda\mu}$. The dipole mode amplitude $\alpha_{1\mu}$ is absent
since, for an incompressible fluid, it corresponds to a translation of the
bubble, and this has been considered in the previous Subsection.

If $S[R(\Omega)]$ represents the bubble surface and $\sigma$ the surface
tension, the energy for a large drop can be written as

$\displaystyle E$ $\displaystyle=$ $\displaystyle\sigma
S[R(\Omega)]+\rho_{b}\,\varepsilon\int_{R(\Omega)}^{\infty}\mathrm{d}^{3}\mathbf{r}\left[\left(\frac{r_{min}}{r}\right)^{12}-2\left(\frac{r_{min}}{r}\right)^{6}\right]$
(20) $\displaystyle\simeq$ $\displaystyle
E_{eq}+\frac{1}{2}C_{0}|\alpha_{0}|^{2}+\frac{1}{2}\sum_{\lambda=2}^{\infty}C_{\lambda}\sum_{\mu=-\lambda}^{\lambda}|\alpha_{\lambda\mu}|^{2}\;,$

where $C_{0}=8\pi\sigma R_{0}^{2}\left(1+12\lambda_{A}\right)$ and
$C_{\lambda}=\sigma R_{0}^{2}[(\lambda-1)(\lambda+2)+6\lambda_{A}]$ are the
stiffness parameters, and $\lambda_{A}=\rho_{b}\varepsilon
2^{-1/6}r_{min}\sigma^{-1}$ is the impurity-He solvation parameter.Anc95  The
mass parameters are $B_{0}=4\pi\rho_{b}m_{He}R_{0}^{5}$ and
$B_{\lambda}=\rho_{b}m_{He}R_{0}^{5}/(\lambda+1)$,Fow68  and the excitation
energies are determined from
$\hbar\omega_{\lambda}=\hbar\sqrt{C_{\lambda}/B_{\lambda}}$, yielding
$\hbar\omega_{0}=10.2$ K for the breathing mode, and $\hbar\omega_{2}=9.8$ K
for the quadrupole mode. Given the droplet temperature of 0.4 K, we conclude
that only the ground state is populated. The mean amplitude of the shape
oscillations is estimated from the variance
$\gamma_{\lambda}=\hbar^{1/2}\left(B_{\lambda}C_{\lambda}\right)^{-1/4}/2$,
giving $\gamma_{0}=0.03$ and $\gamma_{2}=0.15$. This model thus yields that
the bubble can experience monopole oscillations of $\sim$3% amplitude, and
quadrupole deformations of $\gamma_{2}\sqrt{3/4\pi}\sim 8$% amplitude.
Amplitudes of this order have been determined within the atomic bubble model
for Cesium atoms in liquid helium.Kin96  Since their effect in the absorption
spectrum is expected to be relevant, we have undertanken a more refined
calculation within DFT taking Mg@4He1000 as a case study.

For helium droplets, we have described bubble deformations in a way similar as
in Refs. Wil64, ; Rin80, ; Kin96, ; Fow68, , namely, if $\rho_{0}(r)$ is the
helium spherical ground state density, deformations are introduced as
$\rho(\mathbf{r},t)=\rho_{0}[R(\mathbf{r},t)]\,{\cal K}^{-1}$, with

$R(\mathbf{r},t)=r+\alpha_{0}(t)+\sum_{\lambda=2}^{\infty}\sum_{\mu=-\lambda}^{\lambda}\alpha_{\lambda\mu}(t)Z_{\lambda\mu}(\widehat{r})\;,$
(21)

where the real spherical harmonics are normalized as $\langle
Z_{\lambda\mu}|Z_{\lambda^{\prime}\mu^{\prime}}\rangle=\frac{4\pi}{2\lambda+1}\delta_{\lambda\lambda^{\prime}}\delta_{\mu\mu^{\prime}}$
for convenience, and the normalization ${\cal
K}=N^{-1}\int\mathrm{d}^{3}\mathbf{r}\rho_{0}[R(\mathbf{r},t)]$ ensures
particle number conservation. If the Mg wave function follows adiabatically
the helium density deformation, it can be shown that to second order in
$\alpha_{\lambda\mu}$, the total energy of the system can be written as

$\displaystyle E([\dot{\alpha}],[\alpha])$ $\displaystyle\simeq$
$\displaystyle E_{gs}+\frac{1}{2}M^{*}_{0}\dot{\alpha}_{0}^{2}+2\pi
E^{(2)}_{0}\alpha_{0}^{2}$ (22)
$\displaystyle+\sum_{\lambda=2}^{\infty}\sum_{\mu=-\lambda}^{\lambda}\left\\{\frac{1}{2}M^{*}_{\lambda}\dot{\alpha}_{\lambda\mu}^{2}+\frac{2\pi}{2\lambda+1}E^{(2)}_{\lambda}\alpha_{\lambda\mu}^{2}\right\\}\;,$

being $E_{gs}$ the ground state energy, $M^{*}_{\lambda}$ the hydrodynamic
mass asociated with the $\lambda$ mode, and $E^{(2)}_{\lambda}$ the second
derivative of the total energy with respect to $\alpha_{\lambda\mu}$. This
equation represents the Hamiltonian of a set of uncoupled harmonic
oscillators, whose quantization yields a ground state to whose energy each
mode contributes with
$\varepsilon_{\lambda\mu}=\frac{1}{2}\hbar\omega_{\lambda}$, with
$\omega_{\lambda}=\sqrt{\frac{4\pi
E^{(2)}_{\lambda}}{(2\lambda+1)M^{*}_{\lambda}}}$, and a ground state wave
function

$\psi([\alpha])=\left(\frac{M^{*}_{0}\omega_{0}}{\pi\hbar}\right)^{1/4}e^{-\frac{M^{*}_{0}\omega_{0}}{2\hbar}\alpha_{0}^{2}}\prod_{\lambda=2}^{\infty}\left(\frac{M^{*}_{\lambda}\omega_{\lambda}}{\pi\hbar}\right)^{1/4}e^{-\frac{M^{*}_{\lambda}\omega_{\lambda}}{2\hbar}\alpha_{\lambda}^{2}}\;,$
(23)

where
$\alpha_{\lambda}^{2}\equiv\sum_{\mu=-\lambda}^{\lambda}\alpha_{\lambda\mu}^{2}$.
Details are given in Appendix A.

We have computed the hydrodynamic masses assuming that the drop is large
enough to use Eq. (52). We have obtained $M^{*}_{0}=15.0m_{He}+0.28m_{Mg}\sim
66.7$ a.u. and $M^{*}_{2}=1.9m_{He}+0.56m_{Mg}\sim 21.0$ a.u. In actual
calculations, instead of using Eq. (39), the energies $E^{(2)}_{0}$ and
$E^{(2)}_{2}$ have been numerically obtained by computing the total energy of
the system for different small values of $\alpha_{0}$ and $\alpha_{2}$. This
has been carried out by numerically introducing the desired deformation
parameter into the ground state density and renormalizating it, solving next
the Schrödinger equation for the Mg atom [Eq. (5)] to determine the ground
state of the impurity, and computing the total energy of the system from Eq.
(1). Fitting these curves to a parabola, we have obtained $E^{(2)}_{0}=49.7$
KÅ-2 and $E^{(2)}_{2}=16.8$ KÅ-2. We have then calculated the ground state
energies $\hbar\omega_{\lambda}/2$ and deformation mean amplitudes
$\gamma_{\lambda}$, obtaining $\hbar\omega_{0}/2=10.6$ K and
$\hbar\omega_{2}/2=6.3$ K, with mean amplitudes $\gamma_{0}=0.18$ Å
($\sim$3.7%) and $\gamma_{2}=0.42$ Å ($\sim$8.5%).

To quantitatively determine the effect of these deformations on the absorption
spectrum, we have developed Eq. (10) to first order in the deformation
parameters, and have explicitly shown that to this order, only the breathing
and quadrupole modes affect the dipole absorption spectrum. The details are
given in Appendix B, where we show that the breathing mode affects the shift
and shape of the line, whereas quadrupole modes only affect the shift.

Consequently, we restrict in Eq. (11) the deformation parameters needed to
properly describe the homogeneous broadening of the absorption dipole line,
namely
$\mathrm{d}[\alpha]\rightarrow\mathrm{d}\alpha_{0}\,\mathrm{d}^{5}\alpha_{2}$
and
$\psi_{X}^{\mathrm{gs}}(r,[\alpha])\rightarrow\psi(\alpha_{0},\alpha_{2})\Phi(r,\alpha_{0})$,
being $\Phi(r,\alpha_{0})$ the wave function of the Mg atom for a given
$\alpha_{0}$ value, and compute the spectrum as

$\displaystyle I(\omega)$ $\displaystyle\propto$ $\displaystyle
4\pi\int\mathrm{d}\alpha_{0}\,\mathrm{d}^{5}\alpha_{2}|\psi(\alpha_{0},\alpha_{2})|^{2}\sum_{m}\int\mathrm{d}r~{}|r~{}\Phi(r,\alpha_{0})|^{2}$
(24)
$\displaystyle\times\delta\left[\omega+\omega^{\mathrm{gs}}_{X}(\alpha_{0})-\frac{1}{\hbar}V^{\mathrm{ex}}_{m}(r,\alpha_{0},[\alpha_{2}])\right]$
$\displaystyle=$ $\displaystyle
4\pi\int\mathrm{d}\alpha_{0}\,\mathrm{d}^{5}\alpha_{2}|\psi(\alpha_{0},\alpha_{2})|^{2}\hbar\sum_{m}\left|\frac{[r~{}\Phi(r,\alpha_{0})]^{2}}{d{V}^{\mathrm{ex}}_{m}(r,\alpha_{0},[\alpha_{2}])/dr}\right|_{r=r_{m}(\omega)}\;,$

where
$[\alpha_{2}]=\\{\alpha_{2-2},\alpha_{2-1},\alpha_{20},\alpha_{21},\alpha_{22}\\}$,
$V^{\mathrm{ex}}_{m}(r,\alpha_{0},[\alpha_{2}])$ are the eigenvalues
$\lambda_{m}(r,[\alpha])$ of the excited potential matrix of Eq. (80), and
$r_{m}(\omega)$ is the root of the equation
$\omega+\omega^{\mathrm{gs}}_{X}(\alpha_{0})-V^{\mathrm{ex}}_{m}(r,\alpha_{0},[\alpha_{2}])/\hbar=0$.

Expression (24) has been integrated using a Monte Carlo method. We have sorted
$M=10^{6}$ sets of values
$[\alpha]^{i}=\\{\alpha_{0}^{i},\alpha_{2-2}^{i},\alpha_{2-1}^{i},\alpha_{20}^{i},\alpha_{21}^{i},\alpha_{22}^{i}\\}$
using the square of the wave function of Eq. (23) as probability density.
Next, for each set we have found the eigenvalues
$V^{\mathrm{ex}}_{m}(r,\alpha_{0}^{i},[\alpha_{2}]^{i})$ of the $U_{i,j}$
matrix that define the potential energy surfaces and have used a trapezoidal
rule to evaluate the integrals

$I_{m}(\omega,[\alpha]^{i})=4\pi\int\mathrm{d}r~{}|r~{}\Phi(r,\alpha_{0}^{i})|^{2}\delta\left[\omega+\omega^{\mathrm{gs}}_{X}(\alpha_{0}^{i})-\frac{1}{\hbar}V^{\mathrm{ex}}_{m}(r,\alpha_{0}^{i},[\alpha_{2}]^{i})\right]$
(25)

using a discretized representation of the delta function.Her08  Finally, we
have obtained the spectrum as

$I(\omega)\propto\frac{1}{M}\,\sum_{i=1}^{M}\sum_{m}I_{m}(\omega,[\alpha]^{i}).$
(26)

Figure 7 shows the absorption spectrum of one Mg atom attached to 4He1000 in
the vicinity of the $3s3p$ 1P${}_{1}\leftarrow 3s^{2}$ 1S0 transition when
homegeneous broadening is considered. We have decomposed the absorption line
into its three components, the higher frequency component being the $\Sigma$
one. The starred vertical line represents the gas-phase transition, and the
experimental curve, adapted from Ref. Reh00, , has been vertically offset for
clarity. Also shown is the absorption spectrum obtained by neglecting
homogeneous broadening (hatched region). This figure shows that the both the
energy and width of the absorption peak that takes most of the experimental
intensity are correctly described by our calculations.

##  V Two magnesium atoms attached to a 4He drop

The attachment of magnesium atoms in 4He droplets has been recently addressed
using resonant two-photon-ionization.Prz07  In particular, the authors of Ref.
Prz07,  have obtained the absorption spectrum for drops doped with different,
selected numbers of Mg atoms. From their measurements it appears that two main
features contribute to the observed line shapes, one peaked at about 279 nm,
and another at about 282 nm. This is in agreement with the results of Refs.
Mor99, ; Reh00,  (we recall that actually, the two peaks were not resolved by
the authors of the bulk liquid experiment). The structure at 282 nm, however,
only appears if the droplet contains more than one Mg atom. Thus the two-peak
structure cannot be due to the splitting of the absorption line due to
dynamical quadrupole deformations of the helium bubble around the impurity, as
previously believed. We have indeed shown in the previous Section that this
coupling only produces a broad peak, in good agreement with the results of
Ref. Prz07,  for helium drops containing just one Mg atom.

Another interesting observation reported in Ref. Prz07,  is that their
experimental results for multi-atom doped 4He droplets are not consistent with
the formation of compact, metallic Mg clusters inside the 4He droplet. The
magnesium atoms in the droplet appear instead to be relatively isolated from
each other, showing only a weak interaction and leading to the 282 nm shift in
the observation.

To confirm this scenario and find an explanation for the origin of the low
energy component in the absorption peak, we have carried out DFT calculations
to determine the structure of a two-magnesium doped 4He drop. Our goal is to
verify whether the helium density oscillation around a magnesium atom may
result in an energy barrier preventing the Mg atoms from merging into a Mg2
dimer, as suggested by Przystawik et al.Prz07  To obtain the structure of two
Mg atom in a 4He drop, we have minimized the energy of the system written as

$\displaystyle E[\Psi,\Phi_{1},\Phi_{2}]$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2\,m_{He}}\int\mathrm{d}^{3}\mathbf{r}\,|\nabla\Psi(\mathbf{r}\,)|^{2}+\int\mathrm{d}^{3}\mathbf{r}\,{\cal
E}(\rho)$ (27) $\displaystyle+$
$\displaystyle\frac{\hbar^{2}}{2\,m_{Mg}}\int\mathrm{d}^{3}\mathbf{r}\,\\{|\nabla\Phi_{1}(\mathbf{r}\,)|^{2}+|\nabla\Phi_{2}(\mathbf{r}\,)|^{2}\\}$
$\displaystyle+$
$\displaystyle\int\int\mathrm{d}^{3}\mathbf{r}\,\mathrm{d}^{3}\mathbf{r^{\prime}}\,\\{|\Phi_{1}(\mathbf{r})|^{2}+|\Phi_{2}(\mathbf{r})|^{2}\\}\,V_{Mg-
He}(|\mathbf{r}-\mathbf{r^{\prime}}|)\,\rho(\mathbf{r^{\prime}})$
$\displaystyle+$
$\displaystyle\int\int\mathrm{d}^{3}\mathbf{r}\,\mathrm{d}^{3}\mathbf{r^{\prime}}\,|\Phi_{1}(\mathbf{r})|^{2}\,V_{Mg-
Mg}(|\mathbf{r}-\mathbf{r^{\prime}}|)\,|\Phi_{2}(\mathbf{r^{\prime}})|^{2}\;,$

where $V_{Mg-Mg}(|\mathbf{r}-\mathbf{r^{\prime}}|)$ is the Mg-Mg pair
potential of Ref. Gue92, , and the other ingredients have the same meaning as
in Eq. (1).

There are at least two additional effects which are not considered when
modeling the Mg-Mg interaction via the pair-potential in vacuum, as implied in
the above expression. The first is due to three-body (and higher) correlation
effects involving the 4He atoms surrounding the Mg pair: these should exert an
additional, albeit small, screening effect due to He polarization, which is
expected to reduce the absolute value of the dispersion coefficients in the
long-range part of the Mg-Mg interaction. The second is a possible reduction
of the Mg atom polarizability due to the presence of the surrounding 4He
cavity which, because of the repulsive character of the electron-He
interaction, should make the electronic distribution of the impurity atom
slightly “stiffer”, thus reducing further the values of the dispersion
coefficients in the Mg-Mg pair interactions. Although in principle these
effects might reduce the net interaction between a pair of Mg atoms embedded
in liquid 4He, in practice in the present system they are indeed very small.
The correction to the leading term of the long-range dispersion interaction,
$-C_{6}/r^{6}$, due to three-body correlation effects can be written to first
orderrenne  as $-C_{6}(1-2\pi n\alpha/3)/r^{6}$, $\alpha$ being the static
polarizability of the host fluid ($\alpha_{He}=1.39\,a_{0}^{3}$). Such
correction is of the order of only 1% in our case. To estimate the change in
the Mg atomic polarizability due to the surrounding He, we computed, using ab-
initio pseudopotential calculations, the (static) polarizability of a Mg atom
in the presence of an effective (mainly repulsive) potential acting on the Mg
valence electrons due to the presence of the surrounding He. The effective
interaction is derived from the equilibrium shape of the 4He bubble hosting
the Mg atom, as predicted by our DFT calculations, and assuming a (local)
electron-He density-dependent interaction which was proposed by Cole et
al.Che94  We find a very small change in the static atomic polarizability
$\alpha$ of Mg. Assuming that, roughly, $C_{6}\propto\alpha^{2}$ we find a
reduction of the $C_{6}$ coefficient of about 1-2%.

The minimisation of the total energy functional written above under the
constraint of a given number of helium atoms and normalized Mg ground state
wave functions should in principle yield the equilibrium configuration of the
system. In practice, depending on the initial configuration, we have found
several local minima, whose origin is again the “interference” of the He
solvation shells around the Mg atoms. We have found three such metastable
configurations for (Mg+Mg)@4He1000 if we start the minimization procedure with
one Mg atom near the center of the droplet, and the other placed off center,
at some distance from the first. They are displayed in Fig. 8. The energy
difference between the innermost (Mg-Mg distance $d=9.3$ Å) and the outermost
($d=18.5$ Å) configurations is 12.5 K. The energy of the $d=9.3$ Å
configuration is sensibly that of the configuration specularly symmetric about
the $z=0$ plane (-5581.4 K) also shown in the figure. It is worth noticing
that, since $R_{1/2}=22.2$ Å and the “radius” of Mg is $\sim 5$ Å, only the
upper left corner configuration has the Mg impurity in a surface state.

Figure 9 shows two density profiles of the symmetric configuration obtained
along the $z$-axis (solid line) and the $x$\- or $y$-axis (dashed line). It
shows a relatively high density helium ring around the two-bubble waist,
clearly visible in Fig. 8, where the local density is almost three times the
bulk liquid 4He density, and that prevents the collapse of the two-bubble
configuration. The Mg wave functions are peaked at $\sim\pm 4.75$ Å, and very
narrow. This justifies a posteriori the assumptions we have made to write the
total energy of the system, Eq. (27).

Our results confirm the existence of the energy barrier suggested in Ref.
Prz07, , that prevents the two Mg atoms from coming closer than some 9 Å, and
thus hindering, at least temporarily, the formation of the Mg2 dimer. This
barrier is shown in Fig. 10 as a function of the Mg-Mg distance $d$, which is
kept fixed in a constrained energy minimization. Note that the energy of the
Mg+Mg system increases as $d$ does because the two Mg atoms in a drop form a
state more bound than that of a pure drop with the impurities well apart.

Since the height of the barrier is larger than the experimental temperatures,
$T\sim 0.4\,K$, two solvated Mg atoms will not easily merge into a dimer, but
rather form of a metastable weakly bound state. Based on these finding, we
suggest that several Mg atoms solvated inside 4He drops might form a sparse,
weakly-bound “foam”-like aggregate rather than coalesce into a more tightly
bound metallic cluster. Partial coagulation of impurities was already invoked
by Toennies and coworkers to explain their experimental findings for the
successive capture of foreing atoms and molecules in helium clustersLew95
(see also Ref. Gor04,  for the case of bulk liquid helium). Very recently, a
kind of “quantum gel” has been predicted to be formed in 4He drops doped with
neon atoms.Elo08  Although some degree of mutual isolation between foreign
atoms is expected in the case of strongly attractive impurities (like those
studied in the two cases mentioned above), where they are kept apart by the
presence a solid 4He layer coating the impurity,Gor04  our calculations show
that this effect is possible even for relatively weakly attractive impurities
like Mg, where such solid-like 4He layer is absent.

One may estimate the mean life of the metastable state as

$\tau=2\pi\sqrt{\frac{\mu^{*}_{Mg}}{U^{\prime\prime}(d_{eq})}}\,\exp{[\Delta
U/(k_{B}T)]}\;,$ (28)

where $\mu^{*}_{Mg}=M^{*}_{Mg}/2\sim 20$ a.u. is the hydrodynamic reduced mass
of the Mg+Mg system and $\Delta U$ is the barrier height. From Fig. 10 we have
that $U^{\prime\prime}(d_{eq})\sim 40$ K Å-2. This yields a mean life of a few
nanoseconds, which is about five to six orders of magnitude smaller than the
time needed for its experimental detection.Prz07 ; Notabarrier  The mean life
becomes increasingly large as the relative angular momentum $L$ deposited into
the two Mg system increases. Writing

$\Delta E=\Delta
E(L=0)+\frac{\hbar^{2}}{2\mu^{*}_{Mg}}\frac{L(L+1)}{d^{2}}\;,$ (29)

one obtains the $L-$dependent energy barriers displayed in Fig. 10. For $L=30$
we have $\tau\sim 0.6$ $\mu$s, and for $L=40$, $\tau\sim 0.1$ milliseconds.
Thus, there is an angular momentum window that may yield mean lifes compatible
with the experimental findings. Increasing $L$ much further would produce too
a distant Mg+Mg system which would correspond to a two independent Mg
impurities in a drop.

To check whether this foam-like structure of the Mg aggregate also appears in
3He drops, we have carried out calculations for (Mg+Mg)@3He1000 using the same
density functional as in Ref. Sti04, , and the method presented in this
Section. Figure 11 shows the energy of the (Mg+Mg)@3He1000 complex as a
function of $d$. For distances smaller than some 8.6 Å we have found that the
system has a tendency to collapse into a dimer -physically unreachable from
our starting point, Eq. (27). We are led to conclude that there is no barrier
in the case of liquid 3He. The configuration corresponding to the closest $d$
we have calculated is shown in Fig. 12.

We are now in the position to determine the effect of these weakly-bound
systems on the LIF and R2PI experiments on 4He droplets containing more than
one Mg atom. Notice that the bottom right panel of Fig. 8 shows that the
helium bubbles have a non-zero static quadrupole moment, whereas they are
spherically symmetric for one single Mg atom in the drop. It is precisely the
existence of this static quadrupole moment that causes an additional
separation between the $\Sigma$ and $\Pi$ spectral components in the
absorption spectra, which results, as a consequence of the broadening of each
line, in an double-peak structure of the computed spectra, in semi-
quantitative agreement with the experimental data. Details of our calculation
are given in Appendix C, see Eq. (82). Figure 13 shows such two-peak structure
corresponding to the specularly symmetric configuration displayed in Fig. 8,
and indicates that the 282 nm structure observed in the experiments may be
attributed to the distortion produced by neighbour Mg bubbles; these bubbles
contribute incoherently to the absorption spectrum, and the relative intensity
of the 282 and 279 nm peaks might reflect the different population of drops
doped with one and two Mg atoms, since those hosting a compact cluster (dimer,
trimer, etc), would not yield an absorption signal in the neighborhood of the
monomer $3s3p$ 1P${}_{1}\leftarrow 3s^{2}$ 1S0 transition. Notice that a large
static distortion of the helium bubble could also arise if the Mg atom were in
a shallow dimple at the drop surface, but Fig. 6 discards this possibility.

We finally note that we have not considered in our work another source for an
additional splitting of the spectral lines of a Mg atom in the field produced
by a neighboring one, i.e. the resonant dipole-dipole interaction occurring
during the electronic excitation. This effect could in principle lead to an
additional (but probably small, compared with the effect discussed here)
splitting of the calculated lines. An accurate determination of the dipole
moment is required for a proper inclusion of this effect, which is beyond the
scope of the present paper.

##  VI Summary

We have obtained, within DFT, the structure of 4He droplets doped with Mg
atoms and have discussed in detail the magnesium solvation properties. In
agreement with previous DMC calculations,Mel05 ; Elh08  we have found that Mg
is not fully solvated in small 4He drops, whereas it becomes fully solvated in
large droplet.

As a consequence of its interaction with the helium environment, it turns out
that magnesium is radially quite delocalized inside the droplets. This large
delocalization provides a way to reconcile two contradictory results on the
solution of one Mg atom in a 4He drop, namely center localization (LIF and
R2PI experimentsReh00 ; Prz07 ), and surface localization (electron-impact
ionization experimentsRen07 ).

We have calculated the absorption spectrum of magnesium in the vicinity of the
$3s3p$ 1P${}_{1}\leftarrow 3s^{2}$ 1S0 transition. For the large Mg@4He1000
droplet, where Mg is fully solvated, we reproduce the more intense component
of the absorption line found by LIF and R2PI experiments in large drops and in
liquid helium. This agreement is only achieved when homegeneous broadening due
to the coupling of the dipole excitation with the quadrupole deformations of
the helium bubble are fully taken into account. This coupling is naturally
included in Quantum Monte Carlo simulations of the absorption spectrum,Oga99 ;
Mel02  whereby one takes advantage of the inherent fluctuations present in
these simulations. These fluctuations are the full quantal equivalent of the
dynamical distortions of the helium bubble we have introduced for the
description of homogeneous broadening. An alternative method to include shape
fluctuations within DFT has been proposed and applied to the case of Cs in
bulk liquid helium.Nak02  It would be interesting to adapt this method to the
drop geometry, since it is not simple to handle dynamical bubble distortions
in a non-spherical environment, or in 3He drops.

To explain the origin of the low energy peak in the absorption line and
confirm the likely existence of soft, “foam”-like structure of Mg aggregates
in 4He drops as proposed by Przystawik et al,Prz07  we have addressed the
properties of two Mg atoms in 4He1000 and have found that indeed, Mg atoms are
kept apart by the presence of helium atoms that prevent the formation of a
compact Mg cluster. We have estimated that the height of the energy barrier
for the formation of the Mg dimer in 4He drops is $\sim 2-3$ K, which should
be enough, at the droplet experimental temperature of $0.4$ K, to guarantee a
relatively long lifetime to these weakly-bound Mg aggregates. We predict that,
contrarily, Mg atoms adsorbed in 3He droplets do not form such metastable
states.

The presence of neighboring Mg atoms in these structures induces a static
quadrupole deformation in the helium bubble accomodating a Mg atom. As a
consequence, the dipole absorption line around the $3s3p$ 1P${}_{1}\leftarrow
3s^{2}$ 1S0 transition splits. We attribute to this static quadrupole moment
the origin of the low energy peak in the absorption line, and confirm the
suggestion made by the Rostock group Prz07  that the splitting of the
absorption line, rather than being due to a dynamical (Jahn-Teller)
deformation of the helium bubble, is due to the presence of more than one
magnesium atom in the same droplet.

Our previous study on Ca doped helium dropsHer08  and the present work show
that DFT is able to quantitatively address the dipole absorption of dopants in
4He drops within the diatomics-in-molecules approach, provided the impurity-
helium pair potentials are accurately determined. However, we want to point
out that, while we have a consistent scenario that explains the results of LIF
and R2PI experiments, the understanding of the electron-impact ionization
experiment reported in Ref. Ren07,  still requires further analysis. Indeed,
since Mg atoms may be in the bulk of the drop or just beneath the drop
surface, the experimental ion yield curve should reflect both possibilities,
whereas apparently it does not (see Fig. 2 of Ref. Ren07, ). One possible
explanation may be that for electron-impact experiments, a $N=10\,000$ drop is
still small, so that the impurity is always close enough to the drop surface
to make the Penning ionization process to prevail on the direct formation of a
He+ ion.

## Acknowledgments

We would like to thank Massimo Mella and Fausto Cargnoni for useful comments
and for providing us with their Mg-He excited pair potentials, and Marius
Lewerenz, Vitaly Kresin, Karl-Heinz Meiwes-Broer, Josef Tiggesbäumker, Andreas
Przystawik, Carlo Callegari and Kevin Lehmann for useful comments and
discussions. This work has been performed under Grants No. FIS2005-01414 from
DGI, Spain (FEDER), No. 2005SGR00343 from Generalitat de Catalunya, and under
the HPC-EUROPA project (RII3-CT-2003-506079), with the support of the European
Community - Research Infrastructure Action under the FP6 “Structuring the
European Research Area” Programme.

## Appendix A

In this Appendix we obtain the energy of the doped drop up to second order in
the deformation parameters and the hydrodynamic mass of the helium bubble. To
this end, the helium order parameter and Mg wave function are expressed as
$\Psi(\mathbf{r},t)=\sqrt{\rho(\mathbf{r},t)}\exp[i\frac{m_{He}}{\hbar}S(\mathbf{r},t)]$
and
$\Phi(\mathbf{r},t)=|\Phi(\mathbf{r},t)|\exp[i\frac{m_{Mg}}{\hbar}\varphi(\mathbf{r},t)]$,
respectively. Neglecting the velocity-dependent terms of the Orsay-Trento
functional that mimic backflow effects,Dal95  the total energy of the system
is written as

$\displaystyle E$
$\displaystyle=\frac{1}{2}m_{He}\int\mathrm{d}^{3}\mathbf{r}\,\rho(\mathbf{r},t)|\nabla
S(\mathbf{r},t)|^{2}+\frac{1}{2}m_{Mg}\int\mathrm{d}^{3}\mathbf{r}\,|\Phi(\mathbf{r},t)|^{2}|\nabla\varphi(\mathbf{r},t)|^{2}$
(30) $\displaystyle+$
$\displaystyle\frac{\hbar^{2}}{2\,m_{He}}\int\mathrm{d}^{3}\mathbf{r}\,|\nabla\sqrt{\rho(\mathbf{r},t)}|^{2}+\frac{\hbar^{2}}{2\,m_{Mg}}\int\mathrm{d}^{3}\mathbf{r}\,\left|\nabla|\Phi(\mathbf{r},t)|\right|^{2}+\int\mathrm{d}^{3}\mathbf{r}\,{\cal
E}(\rho)$ $\displaystyle+$
$\displaystyle\int\int\mathrm{d}^{3}\mathbf{r}\,\mathrm{d}^{3}\mathbf{r^{\prime}}\,|\Phi(\mathbf{r},t)|^{2}\,V_{Mg-
He}(|\mathbf{r}-\mathbf{r^{\prime}}|)\,\rho(\mathbf{r^{\prime}},t)\;,$

where the functions $\rho$($|\Phi|$) and $S$($\varphi$) fulfill the continuity
equations

$\displaystyle-\frac{\partial}{\partial t}\rho(\mathbf{r},t)$ $\displaystyle=$
$\displaystyle\nabla\left[\rho(\mathbf{r},t)\nabla S(\mathbf{r},t)\right]$
(31) $\displaystyle-\frac{\partial}{\partial t}|\Phi(\mathbf{r},t)|^{2}$
$\displaystyle=$
$\displaystyle\nabla\left[|\Phi(\mathbf{r},t)|^{2}\nabla\varphi(\mathbf{r},t)\right]$
(32)

that allow to identify $S(\mathbf{r},t)$ and $\varphi(\mathbf{r},t)$ as
velocity field potentials, and the first two terms in Eq. (30) as a collective
kinetic energy, whose density we denote as $t[\rho,S,|\Phi|,\varphi]$. Thus,

$E=T+V=\int\mathrm{d}^{3}\mathbf{r}\left\\{t[\rho,S,|\Phi|,\varphi]+v[\rho,|\Phi|]\right\\}\;.$
(33)

In the adiabatic approximation, the dynamics of the system requires the
following steps: i) introduce a set of collective variables (or deformation
parameters) $[\alpha(t)])$ that define the helium density,
$\rho(\mathbf{r},t)=\rho(\mathbf{r},[\alpha(t)])$; ii) for each helium
configuration defined by $[\alpha]$, solve the time-independent Schrödinger
equation obeyed by $|\Phi(\mathbf{r},[\alpha])|$; iii) obtain the potential
surface $V[\rho,|\Phi|]$ by computing the static energy for each
configuration; iv) determine the velocity field potentials
$S(\mathbf{r},[\alpha(t)])$ and $\varphi(\mathbf{r},[\alpha(t)])$ by solving
the continuity equations; v) compute the collective kinetic energy to obtain
the hydrodynamic mass, and vi) solve the equation of motion associated with
the effective Hamiltonian written as a function the deformation parameters.

We aim to describe harmonic deformations of a spherical helium bubble created
by an impurity in the ground state, and have to determine the helium density
$\rho(\mathbf{r},[\alpha(t)])$ resulting from a change in the radial distance
to the center of the spherical bubble induced by the $[\alpha(t)]$ parameters:

$r\longrightarrow
r+\sum_{\lambda=0}^{\infty}\sum_{\mu=-\lambda}^{\lambda}\alpha_{\lambda\mu}(t)Z_{\lambda\mu}(\widehat{r})\;,$
(34)

where the $\lambda=1$ deformation is now introduced to allow for displacements
of the bubble. We recall that the real spherical harmonics have been
normalized as $\langle
Z_{\lambda\mu}|Z_{\lambda^{\prime}\mu^{\prime}}\rangle=\frac{4\pi}{2\lambda+1}\delta_{\lambda\lambda^{\prime}}\delta_{\mu\mu^{\prime}}$,
with $Z_{00}(\widehat{r})=1$, $Z_{10}(\widehat{r})=\cos\theta$, etc. The
breathing mode corresponds to $\lambda=0$, an infinitesimal translation to
$\lambda=1$ (provided the fluid is incompressible), and a quadrupolar
deformation to $\lambda=2$. To first order, the density can be written as

$\rho(\mathbf{r},t)\simeq\rho_{0}(r)+\rho_{0}^{\prime}(r)\sum_{\lambda=0}^{\infty}\sum_{\mu=-\lambda}^{\lambda}\alpha_{\lambda\mu}(t)Z_{\lambda\mu}(\widehat{r})\;,$
(35)

where from now on, the prime will denote the derivative of the function with
respect to its argument.

###  A.1 Impurity wave function

To first order, the wave function $|\Phi(\mathbf{r},[\alpha])|$ is written as

$|\Phi(\mathbf{r},t)|\simeq\Phi_{0}(r)+\sum_{\lambda=0}^{\infty}\sum_{\mu=-\lambda}^{\lambda}\alpha_{\lambda\mu}(t)\Phi^{(1)}_{\lambda\mu}(\mathbf{r})\;.$
(36)

The amplitudes $\Phi^{(1)}_{\lambda\mu}(\mathbf{r})$ are determined in first
order perturbation theory from the multipole expansion of the impurity-helium
pair potential

$\displaystyle U^{(1)}_{\lambda\mu}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{3}\mathbf{r}^{\prime}\rho_{0}^{\prime}(r^{\prime})Z_{\lambda\mu}(\widehat{r}^{\prime})V_{X-He}(|\mathbf{r}-\mathbf{r^{\prime}}|)$
(37) $\displaystyle=$
$\displaystyle\int\mathrm{d}^{3}\mathbf{r}^{\prime}\rho_{0}^{\prime}(r^{\prime})Z_{\lambda\mu}(\widehat{r}^{\prime})\sum_{\lambda\mu}V_{X-He}^{\lambda}(r,r^{\prime})Z_{\lambda\mu}(\widehat{r})Z_{\lambda\mu}(\widehat{r}^{\prime})$
$\displaystyle\equiv$ $\displaystyle
U^{(1)}_{\lambda}(r)Z_{\lambda\mu}(\widehat{r})\;,$

which defines $U^{(1)}_{\lambda}(r)$. We obtain

$\displaystyle\Phi^{(1)}_{\lambda\mu}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\sum_{n\ell
m}\frac{\langle\Phi_{0}|U^{(1)}_{\lambda}Z_{\lambda\mu}|\Phi_{n\ell}Z_{\ell
m}\rangle}{\varepsilon_{0}-\varepsilon_{n\ell m}}\Phi_{n\ell}(r)Z_{\ell
m}(\widehat{r})$ (38) $\displaystyle=$
$\displaystyle\left[\sum_{n}\frac{4\pi}{2\lambda+1}\frac{\langle\Phi_{0}|U^{(1)}_{\lambda}|\Phi_{n\lambda}\rangle}{\varepsilon_{0}-\varepsilon_{n\lambda\mu}}\Phi_{n\lambda}(r)\right]Z_{\lambda\mu}(\widehat{r})$
$\displaystyle\equiv$
$\displaystyle\Phi^{(1)}_{\lambda}(r)Z_{\lambda\mu}(\widehat{r})$

that shows that actually, $\Phi^{(1)}_{\lambda\mu}$ is $\mu$ independent.

Once we have obtained the wave function, we can compute the energy surface
$V[\rho,|\Phi|]$. Since we describe deformations around a spherically
symmetric ground state, the first order term vanishes, and the derivative
$g_{\lambda}\equiv\partial^{2}v/\partial\alpha_{\lambda\mu}\partial\alpha_{\lambda^{\prime}\mu^{\prime}}$
is also spherically symmetric. We can evaluate the second order contribution
to the collective potential energy as

$\displaystyle V^{(2)}$ $\displaystyle=$
$\displaystyle\sum_{\lambda\mu}\sum_{\lambda^{\prime}\mu^{\prime}}\frac{1}{2}\int\mathrm{d}^{3}\mathbf{r}\left.\frac{\partial^{2}v}{\partial\alpha_{\lambda\mu}\partial\alpha_{\lambda^{\prime}\mu^{\prime}}}\right|_{\rho_{0},\varphi_{0}}\alpha_{\lambda\mu}Z_{\lambda\mu}(\widehat{r})\alpha_{\lambda^{\prime}\mu^{\prime}}Z_{\lambda^{\prime}\mu^{\prime}}(\widehat{r})$
(39) $\displaystyle=$
$\displaystyle\sum_{\lambda\mu}\frac{2\pi}{2\lambda+1}\int_{0}^{\infty}\mathrm{d}r\,r^{2}g_{\lambda}(r)\alpha_{\lambda\mu}^{2}$
$\displaystyle\equiv$
$\displaystyle\sum_{\lambda=0}^{\infty}\frac{2\pi}{2\lambda+1}E^{(2)}_{\lambda}\sum_{\mu=-\lambda}^{\lambda}\alpha_{\lambda\mu}^{2}\;,$

which defines $E^{(2)}_{\lambda}$. The parameters $[\alpha]$ are the dynamical
variables that describe the evolution of the system.

###  A.2 Velocity field potentials

Introducing the expansion
$S(\mathbf{r},t)\equiv\sum_{\lambda\mu}\dot{\alpha}_{\lambda\mu}(t)\tilde{S}_{\lambda}(r)Z_{\lambda\mu}(\widehat{r})$,
where the dot denotes the time-derivative, the continuity equation for the
liquid helium is, to first order,

$-\sum_{\lambda\mu}\dot{\alpha}_{\lambda\mu}~{}Z_{\lambda\mu}~{}\frac{\mathrm{d}\rho_{0}}{\mathrm{d}r}=\sum_{\lambda\mu}\dot{\alpha}_{\lambda\mu}~{}Z_{\lambda\mu}\left\\{\frac{\mathrm{d}\rho_{0}}{\mathrm{d}r}\frac{\mathrm{d}\tilde{S}_{\lambda}}{\mathrm{d}r}+\rho_{0}\left[\frac{\mathrm{d}^{2}\tilde{S}_{\lambda}}{\mathrm{d}r^{2}}+\frac{2}{r}\frac{\mathrm{d}\tilde{S}_{\lambda}}{\mathrm{d}r}-\frac{\lambda(\lambda+1)}{r^{2}}\tilde{S}_{\lambda}\right]\right\\}\;.$
(40)

Hence,

$-\frac{\mathrm{d}\rho_{0}}{\mathrm{d}r}=\frac{\mathrm{d}\rho_{0}}{\mathrm{d}r}\frac{\mathrm{d}\tilde{S}_{\lambda}}{\mathrm{d}r}+\rho_{0}\left[\frac{\mathrm{d}^{2}\tilde{S}_{\lambda}}{\mathrm{d}r^{2}}+\frac{2}{r}\frac{\mathrm{d}\tilde{S}_{\lambda}}{\mathrm{d}r}-\frac{\lambda(\lambda+1)}{r^{2}}\tilde{S}_{\lambda}\right].$
(41)

When $r\rightarrow\infty$, the density vanishes for a drop, and approaches
$\rho_{b}$ for the liquid. In the later case, Eq. (41) reduces to the radial
part of the Laplace equation

$0=\frac{\mathrm{d}^{2}\tilde{S}_{\lambda}}{\mathrm{d}r^{2}}+\frac{2}{r}\frac{\mathrm{d}\tilde{S}_{\lambda}}{\mathrm{d}r}-\frac{\lambda(\lambda+1)}{r^{2}}\tilde{S}_{\lambda}$
(42)

whose general solution is

$\tilde{S}_{\lambda}(r)=A_{\lambda}~{}r^{\lambda}+\frac{B_{\lambda}}{r^{\lambda+1}}\;.$
(43)

We have solved Eq. (41) adapting the method proposed in Ref. Leh02, . Let
$r_{i}$ be the first point where $\rho_{0}(r)$ is significantly different from
zero $[\rho_{0}(r)=0\,\,{\rm for}\,\,r\leq r_{i}]$. At this point, Eq. (41)
implies that

$\left.-\frac{\mathrm{d}\rho_{0}}{\mathrm{d}r}\right|_{r_{i}}=\left.\frac{\mathrm{d}\rho_{0}}{\mathrm{d}r}\frac{\mathrm{d}\tilde{S}_{\lambda}}{\mathrm{d}r}\right|_{r_{i}},$
(44)

which determines the boundary condition at $r_{i}$:

$\left.\frac{\mathrm{d}\tilde{S}_{\lambda}}{\mathrm{d}r}\right|_{r_{i}}=-1\;.$
(45)

For the liquid, the other boundary condition is that when
$r\rightarrow\infty$, the solution behaves as in Eq. (43) with $A_{\lambda}=0$
to have a physically acceptable solution. If the bubble has a sharp surface of
radius $r_{i}$ and the liquid is uniform, $\tilde{S}_{\lambda}(r)$ is
completely determined by Eq. (43) and the velocity field potential that
fulfills Eq. (45) corresponds to the coefficients

$\displaystyle A_{\lambda}$ $\displaystyle=$ $\displaystyle 0$ $\displaystyle
B_{\lambda}$ $\displaystyle=$
$\displaystyle\frac{r_{i}^{\lambda+2}}{\lambda+1}\;.$ (46)

To find the velocity field potential in a large drop, we have defined a radial
distance $r_{b}$, far from the bubble and from the drop surface at $R_{1/2}$,
around which on one may consider that the density is that of the liquid.
Starting from $r=r_{b}$ with the liquid solution fixed by the coefficients
given in Eq. (46), we have integrated inwards Eq. (41), finding the solutions
$\tilde{S}_{\lambda}^{\mathrm{inh}}(r)$ and
$\tilde{S}_{\lambda}^{\mathrm{h}}(r)$ that correspond, respectively, to the
non-homegeneous and to the homegeneous differential equation that results by
setting to zero the left hand side of Eq. (41). The general solution that
satisfy the boundary condition Eq. (45) is obtained as

$\tilde{S}_{\ell}(r)=\tilde{S}_{\lambda}^{\mathrm{inh}}(r)+C\tilde{S}_{\lambda}^{\mathrm{h}}(r)$
(47)

with

$C=-\frac{1+\left.\frac{\mathrm{d}\tilde{S}_{\lambda}^{\mathrm{inh}}}{\mathrm{d}r}\right|_{r_{i}}}{\left.\frac{\mathrm{d}\tilde{S}_{\lambda}^{\mathrm{h}}}{\mathrm{d}r}\right|_{r_{i}}}\;.$
(48)

The field $\varphi$ is analogously obtained after introducing the expansion
$\varphi(\mathbf{r},t)=\sum_{\lambda\mu}\dot{\alpha}_{\lambda\mu}(t)\tilde{\varphi}_{\lambda}(r)\mathcal{Z}_{\lambda\mu}(\widehat{r})$.
We have assumed that the wave function of the impurity in the ground state is
a Gaussian $\Phi_{0}(r)=A\exp\left(-\beta r^{2}\right)$ whose shape has been
determined by fitting it to the actual wave function, and have introduce a
cutoff distance $r_{g}$ such that safely $\Phi_{0}(r)=0$ if $r\geq r_{g}$. We
then obtain the following differential equation to determine
$\tilde{\varphi}_{\lambda}$:

$4\beta r=-4\beta
r\frac{\mathrm{d}\tilde{\varphi}_{\lambda}}{\mathrm{d}r}+\frac{\mathrm{d}^{2}\tilde{\varphi}_{\lambda}}{\mathrm{d}r^{2}}+\frac{2}{r}\frac{\mathrm{d}\tilde{\varphi}_{\lambda}}{\mathrm{d}r}-\frac{\lambda(\lambda+1)}{r^{2}}\tilde{\varphi}_{\lambda}\;.$
(49)

The appropriate boundary conditions are

$\displaystyle\left.\frac{\mathrm{d}\tilde{\varphi}_{\lambda}}{\mathrm{d}r}\right|_{r_{0}}$
$\displaystyle=$ $\displaystyle-1$ $\displaystyle\tilde{\varphi}_{\lambda}(0)$
$\displaystyle=$ $\displaystyle 0\;,$ (50)

and the solution is analytical but very involved (we have found it by using
the Mathematica packpage). The dipole mode is the only exception; if
$\lambda=1$ we have $\tilde{\varphi}_{1}=-r$.

###  A.3 Kinetic energy

Once the velocity fields $S(\mathbf{r},t)$ and $\varphi(\mathbf{r},t)$ have
been determined, the collective kinetic energy can be easily calculated to
second order in the collective parameters:

$\displaystyle T$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{3}\mathbf{r}~{}t[\rho,S,|\Phi|,\varphi]$ (51)
$\displaystyle=$
$\displaystyle\frac{1}{2}m_{He}\int\mathrm{d}^{3}\mathbf{r}\,\rho(\mathbf{r},t)|\nabla
S(\mathbf{r},t)|^{2}+\frac{1}{2}m_{Mg}\int\mathrm{d}^{3}\mathbf{r}\,|\Phi(\mathbf{r},t)|^{2}|\nabla\varphi(\mathbf{r},t)|^{2}$
$\displaystyle\simeq$
$\displaystyle\sum_{\lambda=0}^{\infty}\frac{4\pi}{2\lambda+1}\left\\{\frac{1}{2}m_{He}\int\mathrm{d}r~{}r^{2}~{}\rho_{0}(r)~{}\left[\left(\frac{\mathrm{d}\tilde{S}_{\lambda}}{\mathrm{d}r}\right)^{2}+\frac{\lambda(\lambda+1)}{r^{2}}\tilde{S}_{\lambda}^{2}\right]\right.$
$\displaystyle\left.+\frac{1}{2}m_{Mg}\int\mathrm{d}r~{}r^{2}~{}|\Phi_{0}(r)|^{2}~{}\left[\left(\frac{\mathrm{d}\tilde{\varphi}_{\lambda}}{\mathrm{d}r}\right)^{2}+\frac{\lambda(\lambda+1)}{r^{2}}\tilde{\varphi}_{\lambda}^{2}\right]\right\\}\sum_{\mu=-\lambda}^{\lambda}\dot{\alpha}_{\lambda\mu}^{2}$
$\displaystyle\equiv$
$\displaystyle\frac{1}{2}\sum_{\lambda=0}^{\infty}M_{\lambda}^{*}\sum_{\mu=-\lambda}^{\lambda}\dot{\alpha}_{\lambda\mu}^{2}$

which defines the hydrodynamic mass $M_{\lambda}^{*}$ for each $\lambda$ mode.

Using the continuity equations and the Gauss theorem, one can find an
alternative expression for $M_{\lambda}^{*}$

$M_{\lambda}^{*}=\frac{4\pi}{2\lambda+1}\left\\{m_{He}\int\mathrm{d}r~{}r^{2}~{}\rho^{\prime}_{0}(r)~{}\tilde{S}_{\lambda}(r)+m_{Mg}\int\mathrm{d}r~{}r^{2}~{}2\Phi_{0}(r)\Phi^{\prime}_{0}(r)\tilde{\varphi}_{\lambda}(r)\right\\}\;.$
(52)

We have checked that both expressions yield the same values for
$M_{\lambda}^{*}$, which constitutes a test on the numerical accuracy of the
method. It is easy to see from Eq. (51) that in bulk liquid helium, the
$\lambda=1$ hydrodynamic mass coincides with that given in Ref. Leh02, . Using
that $\tilde{\varphi}_{1}=-r$, it is also easy to check from the above
expressions that the impurity contribution to the $\lambda=1$ hydrodynamic
mass is just the bare mass of the Mg atom.

The sum of Eqs. (39) and (51) represents the Hamiltonian of a set of uncoupled
harmonic oscillators whose frequency only depends on $\lambda$.

## Appendix B

In this Appendix we work out in detail the expressions we have used to
describe the homogeneous broadening of the absorption line. We consider a
spherical ground state defined by a helium density $\rho_{0}(r)$ and impurity
wave function $\Phi_{0}(r)$, both modified by the action of the breathing mode
defined in Eq. (21), namely $\rho(r,\alpha_{0})=\rho_{0}(r+\alpha_{0})\,{\cal
K}^{-1}$, and $\Phi(r,\alpha_{0})$. The computation of the spectra for a given
$\alpha_{0}$ can be carried out starting from Eq. (11)

$I(\omega,\alpha_{0})=4\pi\sum_{i}\int\mathrm{d}r~{}|r~{}\Phi(r,\alpha_{0})|^{2}\delta[\omega+\omega(\alpha_{0})-V^{\mathrm{ex}}_{i}(r,\alpha_{0})/\hbar]\;,$
(53)

where $\hbar\omega(\alpha)=\varepsilon(\alpha_{0})$ is the impurity
eigenenergy and $V^{\mathrm{ex}}_{i}(r,\alpha_{0})$ are the PES defined by Eq.
(13), where $[\alpha]$ reduces to $\alpha_{0}$. Notice that $\alpha_{0}$ is
not introduced perturbatively; it is unnecessary since this mode does not
break the spherical symmetry. Next, we perturbatively introduce the modes with
$\lambda\geq 2$ to first order; Eq. (53) becomes

$\displaystyle I(\omega,[\alpha])$ $\displaystyle\simeq$ $\displaystyle
I(\omega,\alpha_{0})$ (54)
$\displaystyle+\sum_{i}\sum_{\lambda=2}^{\infty}\sum_{\mu=-\lambda}^{\lambda}\left\\{\int\mathrm{d}\mathbf{r}~{}\Phi(r,\alpha_{0})\Phi^{(1)}_{\lambda}(r,\alpha_{0})Z_{\lambda\mu}(\hat{r})\delta[\omega+\omega(\alpha_{0})-V^{\mathrm{ex}}_{i}(r,\alpha_{0})/\hbar]\right.$
$\displaystyle\left.-4\pi\int\mathrm{d}r~{}|r~{}\Phi(r,\alpha_{0})|^{2}\delta^{\prime}[\omega+\omega(\alpha_{0})-V^{\mathrm{ex}}_{i}(r,\alpha_{0})/\hbar]\frac{1}{\hbar}\epsilon_{\lambda\mu}^{i}(r)\right\\}\alpha_{\lambda\mu}\;,$

where $\epsilon_{\lambda\mu}^{i}(r)=\left.\partial
V^{\mathrm{ex}}_{i}(r)/\partial\alpha_{\lambda\mu}\right|_{[\alpha]_{\lambda\geq
2}=0}$. The first integral is zero due to the orthogonality of the spherical
harmonics. To evalue the second integral, we expand
$\epsilon_{\lambda\mu}^{i}(r)$ as a power series of $r$. Taken into account
that the PES have a stationary point at $r=0$ due to the spherical geometry
–the first order term is zero– we can safely stop the expansion at the zeroth
order term, since the wave function $\Phi(r,\alpha_{0})$ is very narrow

$\displaystyle I(\omega,[\alpha])\simeq I(\omega,\alpha_{0})$
$\displaystyle-4\pi\sum_{i}\sum_{\lambda=2}^{\infty}\sum_{\mu=-\lambda}^{\lambda}\int\mathrm{d}r~{}|r~{}\Phi(r,\alpha_{0})|^{2}\delta^{\prime}[\omega-\omega(\alpha_{0})-V^{\mathrm{ex}}_{i}(r,\alpha_{0})/\hbar]\left(\frac{1}{\hbar}\epsilon_{\lambda\mu}^{i}(0)\alpha_{\lambda\mu}\right)\;.$
(55)

This equation may be interpreted as the expansion to first order of a shift in
$\omega$, so it can be written as

$\displaystyle I(\omega,[\alpha])$ $\displaystyle\simeq$ $\displaystyle
4\pi\sum_{i}\int\mathrm{d}r~{}|r~{}\Phi(r,\alpha_{0})|^{2}\delta\left[\omega+\omega(\alpha_{0})-\frac{1}{\hbar}\left(V^{\mathrm{ex}}_{i}(r,\alpha_{0})+\sum_{\lambda=2}^{\infty}\sum_{\mu=-\lambda}^{\lambda}\epsilon_{\lambda\mu}^{i}(0)\alpha_{\lambda\mu}\right)\right]$
(56) $\displaystyle=$
$\displaystyle\sum_{i}I\left[\omega+\sum_{\lambda=2}^{\infty}\sum_{\mu=-\lambda}^{\lambda}\epsilon_{\lambda\mu}^{i}(0)\alpha_{\lambda\mu}/\hbar,\alpha_{0}\right]\;.$

Thus the eigenvalues $\epsilon_{\lambda\mu}^{i}(0)$ are related to the
diagonalization of the expansion of $U_{ij}(\mathbf{r},[\alpha])$ defined in
Eq. (12). Writing this matrix equation as a function of the real spherical
harmonics

$\displaystyle
U(\mathbf{r},[\alpha])=\int\mathrm{d}^{3}\mathbf{r^{\prime}}\rho(\mathbf{r^{\prime}}+\mathbf{r},[\alpha])\left\\{\frac{1}{3}[V_{\Sigma}(r^{\prime})+2V_{\Pi}(r^{\prime})]Z_{00}(\hat{r})\left(\begin{array}[]{ccc}1&0&0\\\
0&1&0\\\ 0&0&1\end{array}\right)\right.$ (60)
$\displaystyle\left.+\frac{1}{\sqrt{3}}[V_{\Sigma}(r^{\prime})-V_{\Pi}(r^{\prime})]\left(\begin{array}[]{ccc}-\frac{1}{\sqrt{3}}Z_{20}(\hat{r})+Z_{22}(\hat{r})&Z_{2-2}(\hat{r})&Z_{21}(\hat{r})\\\
Z_{2-2}(\hat{r})&-\frac{1}{\sqrt{3}}Z_{20}(\hat{r})-Z_{22}(\hat{r})&Z_{2-1}(\hat{r})\\\
Z_{21}(\hat{r})&Z_{2-1}(\hat{r})&\frac{2}{\sqrt{3}}Z_{20}(\hat{r})\end{array}\right)\right\\}\;,$
(64)

expanding $\rho(\mathbf{r^{\prime}}+\mathbf{r},[\alpha])$ to first order in
$[\alpha]_{\lambda\geq 2}$, and evaluating the first order contribution at
$r=0$ we obtain

$\displaystyle U(\mathbf{r},[\alpha])$ $\displaystyle\simeq$ $\displaystyle
U(r,\alpha_{0})+\sum_{\lambda=2}^{\infty}\sum_{\mu=-\lambda}^{\lambda}\frac{4\pi}{2\lambda+1}\int\mathrm{d}r^{\prime}\,r^{\prime
2}\,\rho^{\prime}(r^{\prime},\alpha_{0})\frac{1}{\sqrt{3}}[V_{\Sigma}(r^{\prime})-V_{\Pi}(r^{\prime})]$
(68)
$\displaystyle\times\left(\begin{array}[]{ccc}-\frac{1}{\sqrt{3}}\delta_{2\lambda}\delta_{0\mu}+\delta_{2\lambda}\delta_{2\mu}&\delta_{2\lambda}\delta_{-2\mu}&\delta_{2\lambda}\delta_{1\mu}\\\
\delta_{2\lambda}\delta_{-2\mu}&-\frac{1}{\sqrt{3}}\delta_{2\lambda}\delta_{0\mu}-\delta_{2\lambda}\delta_{2\mu}&\delta_{2\lambda}\delta_{-1\mu}\\\
\delta_{2\lambda}\delta_{1\mu}&\delta_{2\lambda}\delta_{-1\mu}&\frac{2}{\sqrt{3}}\delta_{2\lambda}\delta_{0\mu}\end{array}\right)\alpha_{\lambda\mu}\;,$

This shows that, to first order, only quadrupolar deformations are coupled to
the dipole electronic transition, and that its effect is a shift of the
spectral line, as shown in Eq. (56). At variance with the approach of Ref.
Kin96, , where the above matrix is approximated by its diagonal expression,
implying that only the $\mu=0$ and 2 components of the quadrupole deformation
are considered, our approach incorporates all five components.

The relation between the eigenvalues $\lambda_{i}(r,[\alpha])$ of
$U(r,[\alpha])$ and the coefficients $\epsilon_{\lambda\mu}^{i}(0)$ is

$\lambda_{i}(r,[\alpha])\equiv
V^{\mathrm{ex}}_{i}(r,\alpha_{0})+\sum_{\mu=-2}^{2}\epsilon_{2\mu}^{i}(0)\alpha_{2\mu}\;.$
(69)

Finally, the total spectrum is written as

$\displaystyle I(\omega)$ $\displaystyle\propto$ $\displaystyle
4\pi\int\mathrm{d}\alpha_{0}\,\mathrm{d}^{5}\alpha_{2}|\psi(\alpha_{0},\alpha_{2})|^{2}I(\omega,[\alpha])$
(70) $\displaystyle\simeq$ $\displaystyle
4\pi\int\mathrm{d}\alpha_{0}\,\mathrm{d}^{5}\alpha_{2}|\psi(\alpha_{0},\alpha_{2})|^{2}\sum_{m}\int\mathrm{d}r~{}|r~{}\Phi(r,\alpha_{0})|^{2}$
$\displaystyle\times\delta\left[\omega+\omega^{\mathrm{gs}}_{X}(\alpha_{0})-\frac{1}{\hbar}\lambda_{i}(r,[\alpha])\right]\;,$

## Appendix C

In this Appendix we consider that the doped drop is cylindrically symmetric.
We expand the helium density and ground state wave function into spherical
harmonics with $\mu=0$

$\displaystyle\rho(\mathbf{r})$ $\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{\infty}\beta_{\lambda}\tilde{\rho}_{\lambda}(r)Z_{\lambda
0}(\hat{r})$ $\displaystyle\Phi(\mathbf{r})$ $\displaystyle=$
$\displaystyle\sum_{\lambda=0}^{\infty}\gamma_{\lambda}\tilde{\Phi}_{\lambda}(r)Z_{\lambda
0}(\hat{r})\;,$ (71)

where
$\beta_{\lambda}=\frac{2\lambda+1}{4\pi}\int\mathrm{d}^{3}\mathbf{r}\rho(\mathbf{r})Z_{\lambda
0}(\hat{r})$ and
$\tilde{\rho}_{\lambda}(r)=\frac{2\lambda+1}{4\pi\beta_{\lambda}}\int\mathrm{d}\Omega\rho(\mathbf{r})Z_{\lambda
0}(\hat{r})$, with analogous definitions for $\tilde{\Phi}_{\lambda}$ and
$\gamma_{\lambda}$. If
$\beta_{0}\gg\beta_{\lambda>0}\,(\Rightarrow\gamma_{0}\gg\gamma_{\lambda>0})$,
we can compute the line shape to first order in $\beta_{\lambda>0}$
($\gamma_{\lambda>0}$); in analogy with Eq. (54) we write

$\displaystyle I(\omega,[\beta])\simeq I(\omega,\beta_{0})$
$\displaystyle-4\pi\sum_{i}\sum_{\lambda=1}^{\infty}\int\mathrm{d}r~{}|r~{}\gamma_{0}\tilde{\Phi}_{0}(r)|^{2}\delta^{\prime}[\omega-\omega(\alpha_{0})-V^{\mathrm{ex}}_{i}(r,\beta_{0})/\hbar]\frac{1}{\hbar}\epsilon_{\lambda
0}^{i}(0)\beta_{\lambda}\;,$ (72)

with $\epsilon_{\lambda 0}^{i}$ defined now by the potential matrix

$\displaystyle U(\mathbf{r},[\beta])$ $\displaystyle\simeq$ $\displaystyle
U(r,\beta_{0})+\frac{4\pi}{15}\int\mathrm{d}r^{\prime}\,r^{\prime
2}\,\tilde{\rho}_{2}(r^{\prime})[V_{\Sigma}(r^{\prime})-V_{\Pi}(r^{\prime})]\left(\begin{array}[]{ccc}-1&0&0\\\
0&-1&0\\\ 0&0&2\end{array}\right)\beta_{2}\;.$ (76)

Introducing the shape deformations defined in Eq. (34), to first order this
matrix becomes

$\displaystyle U(\mathbf{r},[\beta],[\alpha])$ $\displaystyle\simeq$
$\displaystyle
U(r,\beta_{0},\alpha_{0})+\frac{4\pi}{5\sqrt{3}}\int\mathrm{d}r^{\prime}\,r^{\prime
2}\,\beta_{0}\,\tilde{\rho}^{\prime}_{0}(r^{\prime},\alpha_{0})[V_{\Sigma}(r^{\prime})-V_{\Pi}(r^{\prime})]$
(80)
$\displaystyle\times\left(\begin{array}[]{ccc}-\frac{1}{\sqrt{3}}\tilde{\alpha}_{20}+\alpha_{22}&\alpha_{2-2}&\alpha_{21}\\\
\alpha_{2-2}&-\frac{1}{\sqrt{3}}\tilde{\alpha}_{20}-\alpha_{22}&\alpha_{2-1}\\\
\alpha_{21}&\alpha_{2-1}&\frac{2}{\sqrt{3}}\tilde{\alpha}_{20}\end{array}\right)\;,$

where we have defined $\tilde{\alpha}_{20}\equiv\alpha_{20}+\beta_{2}C$, with

$C=\frac{\displaystyle\int\mathrm{d}r^{\prime}\,r^{\prime
2}\,\tilde{\rho}_{2}(r^{\prime})[V_{\Sigma}(r^{\prime})-V_{\Pi}(r^{\prime})]}{\displaystyle\int\mathrm{d}r^{\prime}\,r^{\prime
2}\,\beta_{0}\,\tilde{\rho}^{\prime}_{0}(r^{\prime},\alpha_{0})[V_{\Sigma}(r^{\prime})-V_{\Pi}(r^{\prime})]}\;.$
(81)

These equations show that the computation of the line shape for this geometry
is as in the spherical case but with a shift in the $\alpha_{20}$ parameter.

The dipole absorption spectrum is finally obtained as

$\displaystyle I(\omega)$ $\displaystyle\propto$ $\displaystyle
4\pi\int\mathrm{d}\alpha_{0}\,\mathrm{d}^{5}\alpha_{2}|\psi(\alpha_{0},\alpha^{\prime}_{2})|^{2}\sum_{m}\int\mathrm{d}r~{}|r~{}\Phi(r,\alpha_{0})|^{2}$
(82)
$\displaystyle\times\delta\left[\omega+\omega^{\mathrm{gs}}_{X}(\alpha_{0})-\frac{1}{\hbar}\lambda_{i}(r,[\alpha])\right]$

with
${\alpha^{\prime}}_{2}^{2}=({\alpha}_{20}-\beta_{2}C)^{2}+\alpha_{22}^{2}+\alpha_{2-2}^{2}+\alpha_{21}^{2}+\alpha_{2-1}^{2}$.

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Figure 1:  (Color online) $X^{1}\Sigma$ Mg-He pair potentials used in this
work: squares connected with a solid line, from Ref. Hin03, ; circles
connected with a dotted line, from Ref. Par01, .

Figure 2:  Energy of the Mg atom as a function of the number of atoms in the
drop, obtained using the Mg-He potential of Ref. Hin03,  (squares). The values
given in Ref. Her07,  are also displayed (dots). The lines are drawn to guide
the eye.

Figure 3: (Color online) Bottom panel: total energy (K) of Mg@4He50 as a
function of ${\cal Z}_{0}$ (Å) obtained using the Mg-He potential of Ref.
Hin03,  (squares) and of Ref. Par01,  (circles). The energies are referred to
their equilibrium values, -157.0 K and -153.8 K, respectively. The vertical
line locates the drop surface at $R_{1/2}=r_{0}N^{1/3}$, with $r_{0}=2.22$ Å.
The horizontal line has been drawn 0.4 K above the equilibrium energy. Top
panel: probability densities for the configurations displayed in the bottom
panel; the single peak distribution corresponds to the Mg-He interaction of
Ref. Par01, .

Figure 4: (Color online) Helium density plots of the Mg@4He50 droplet in the
$y=0$ plane obtained using the Mg-He potential of Ref. Hin03, . From top to
bottom and left to right, the ${\cal Z}_{0}$ values correspond to the
stationary points displayed in Fig. 3, namely 0, 2, 4, and 6 Å, respectively.
The brighter regions are the higher density ones.

Figure 5: (Color online) Helium density profiles of the Mg@4He50 droplet along
the $z$ axis obtained using the Mg-He potential of Ref. Hin03, . The ${\cal
Z}_{0}$ value is indicated in each panel. Dashed lines, pure drops; solid
lines, doped drops.

Figure 6: (Color online) Bottom panel: total energy (K) of Mg@4He1000 as a
function of ${\cal Z}_{0}$ (Å) obtained using the Mg-He potential of Ref.
Hin03,  (squares) and of Ref. Par01,  (circles). The energies are referred to
their equilibrium values, -5482.0 K and -5476.8 K, respectively. The vertical
lines roughly delimit the drop surface region, conventionally defined as the
radial distance between the points where the density equals 0.1$\rho_{b}$ and
0.9$\rho_{b}$, being $\rho_{b}=0.0218$ Å-3 the bulk liquid density. The
horizontal line has been drawn 0.4 K above the equilibrium energy. Top panel:
total energy (K) of Mg@4HeN with $N=1000$ and 2000 (vertical right scale) as a
function of ${\cal Z}_{0}$ (Å) obtained using the Mg-He potential of Ref.
Hin03, . The energies are referred to their equilibrium values, $-5482.0$ K
and $-11\,629.8$ K, respectively, and the distances (horizontal scale) are
referred to the $R_{1/2}$ radius; also shown are the corresponding probability
densities (vertical left scale): dot(dot-dash) line, $N=1000(2000)$. The solid
line represents the probability density of the $N=10\,000$ drop.

Figure 7: (Color online) Total absorption spectrum of one Mg atom attached to
4He1000 in the vicinity of the $3s3p$ 1P${}_{1}\leftarrow 3s^{2}$ 1S0
transition. The line has been decomposed into its two $\Pi$ and one $\Sigma$
components, the former one is the higher frequency transition. The starred
vertical line represents the gas-phase transition. The experimental curve,
adapted for Ref. Reh00, , has been vertically offset for clarity. Also shown
is the absorption spectrum obtained neglecting homogeneous broadening (hatched
region)

Figure 8: (Color online) From top to bottom and left to right, (Mg+Mg)@4He1000
metastable configurations for Mg-Mg interatomic distances $d=18.5$ Å, 12.9 Å,
and 9.3 Å, and total energies -5567.8 K, -5573.9 K, and -5580.3 K,
respectively. The bottom right panel shows the specularly symmetric
configuration at $d=9.5$ Å with total energy $-5581.4$ K. The brigther regions
are the higher density ones.

Figure 9: (Color online) Helium density profiles of the (Mg+Mg)@4He1000
symmetric configuration ($d=9.5$ Å) along the $z$-axis (solid line) and the
$x$\- or $y$-axis (dashed line).

Figure 10: (Color online) Energy (K) of the Mg+Mg@4He1000 system as a function
of the Mg-Mg distance (Å). The energies have been referred to that of the
metastable equilibrium configuration (local minimum) at $L=0$. The lines have
been obtained by a cubic spline of the actual calculations. From bottom to
top, the curves correspond to $L=0$ to $50\hbar$ in $10\hbar$ steps.

Figure 11: (Color online) Energy (K) of the Mg+Mg@3He1000 system as a function
of the Mg-Mg distance (Å). The line has been drawn to guide the eye.

Figure 12: (Color online) Top panel: helium density profiles of the
(Mg+Mg)@3He1000 complex at $d=8.6$ Å along the $z$-axis (solid line) and the
$x$\- or $y$-axis (dashed line). Bottom panel: equidensity lines corresponding
to the same configuration. The brigther regions are the higher density ones.

Figure 13: (Color online) Total absorption spectrum of one Mg atom attached to
4He1000 in the vicinity of the $3s3p$ 1P${}_{1}\leftarrow 3s^{2}$ 1S0
transition in the distorted environment created by the presence of another Mg
atom. The line has been decomposed into its two $\Pi$ and one $\Sigma$
components, the former one is the higher frequency transition. The starred
vertical line represents the gas-phase transition. The experimental curve,
adapted for Ref. Reh00, , has been vertically offset for clarity. Also shown
is the absorption spectrum obtained neglecting homogeneous broadening (hatched
region)

Table 1: Atomic shift $\Delta\omega$ of Mg@4He1000 ($R_{1/2}=22.2$ Å) as a function of the average distance between the magnesium atom and the center of mass of the 4He1000 moiety. Also indicated is the corresponding wavelength $\lambda$. The value of the transition energy in the gas phase is 35 051 cm-1.Mar80  ${\cal Z}_{0}$ (Å) |  $\Delta\omega$ (cm-1) |  $\lambda$ (nm)
---|---|---
$0$ | 659.0 | 280.0
$10$ | 642.1 | 280.2
$14$ | 615.8 | 280.4
$18$ | 520.3 | 281.1
$22$ | 492.5 | 281.4