Jahn-Teller Spectral Fingerprint in Molecular Photoemission: C$_{60}$
Nicola Manini
%Gas-phase photoemission off a stable molecule with a Jahn-Teller unstable %final hole state is formulated emphasizing the importance of symmetry %selection, and is demonstrated for C$_{60}$ as a case study. % The $h_u$ hole spectral intensity for C$_{60} \to$~C$_{60}^+$ molecular photoemission is calculated at finite temperature by a parameter-free Lanczos diagonalization of the electron-vibration Hamiltonian, including the full 8 $H_g$, 6 $G_g$, and 2 $A_g$ mode couplings. % %The gigantic vibrational Hilbert space size called for at high temperatures %is handled by means of a specially devised stochastic technique. % The computed spectrum at 800~K is in striking agreement with gas-phase data. The energy separation of the first main shoulder from the main photoemission peak, 230~meV in C$_{60}$, is shown to measure directly and rather generally the strength of the final-state Jahn-Teller coupling. %
pesC60_6.tex
% version 06 -  Handled by Erio 23 Jul and shrunk to 4 pg 24-28 Jul - submitted
% version 05 -  Nick rewriteup + tuning 22 Jul 03
% version 04 -  Nick 16 Jul 03
% version 03 -  Handled by Erio 18+21+22 Jun 03
% version 02 -  Erio's intro + Nick's data and discussion - 13 Jun 03
% version 01 -  first cookup by Nick - 06 Mar 03

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\begin{document}

\title{
Jahn-Teller Spectral Fingerprint in Molecular Photoemission: C$_{60}$
}

\author{Nicola Manini}\email{nicola.manini@mi.infm.it}
\author{Paolo Gattari}
\affiliation{Dipartimento di Fisica, Universit\`a di Milano,
Via Celoria 16, 20133 Milano, Italy}
\affiliation{INFM, Unit\`a di Milano, Milano, Italy}
\author{Erio Tosatti}
\affiliation{International School for Advanced Studies (SISSA), Via Beirut
4, 34014 Trieste, Italy}
\affiliation{INFM Democritos National Simulation Center, and INFM, Unit\`a
Trieste, Italy}
\affiliation{International Centre for Theoretical Physics (ICTP), P.O. Box
586, 34014 Trieste, Italy}


\date{\today}

\begin{abstract}

%Gas-phase photoemission off a stable molecule with a Jahn-Teller unstable
%final hole state is formulated emphasizing the importance of symmetry 
%selection, and is demonstrated for C$_{60}$ as a case study. 
%
The $h_u$ hole spectral intensity for C$_{60} \to$~C$_{60}^+$
molecular photoemission is
calculated at finite temperature by a parameter-free Lanczos
diagonalization of the electron-vibration Hamiltonian, including the full 8
$H_g$, 6 $G_g$, and 2 $A_g$ mode couplings.
%
%The gigantic vibrational Hilbert space size called for at high temperatures
%is handled by means of a specially devised stochastic technique.
%
The computed spectrum at 800~K is in striking agreement with gas-phase
data.  The energy separation of the first main shoulder from the main
photoemission peak, 230~meV in C$_{60}$, is shown to measure directly and
rather generally the strength of the final-state Jahn-Teller coupling.
%
\end{abstract}

\pacs{  33.60.-q,% Photoelectron spectra (molecules)
	33.20.Wr,%vibronic rovibronic and rotation-electronic-spin interactions
	71.20.Tx,% fullerene and related materials (electronic structure)
	36.40.Cg % electronic & magnetic properties of clusters
 }

\maketitle


%-------------------------------------------------------------------------
%\section{Introduction}
%-------------------------------------------------------------------------

Photoemission (PE) from a closed-shell high-symmetry molecule, involving
Jahn-Teller (JT) effect in the final state, is accompanied by
characteristic vibronic structures in the measured hole spectrum
\cite{Lindner96etal}.
% DIRE MEGLIO:
Although accurate spectral calculations have recently appeared in the
chemical literature, e.g.\ for benzene \cite{Koppel02etal}, there still is
a strong need for a qualitative understanding of a more general nature,
possibly system independent, and informative on the nature and the strength
of the JT coupled problem.
%
A recent case in point is that of gas-phase C$_{60}$, where an important
side peak was reported 230~meV above the main PE peak
\cite{Bruhwiler97etal,Canton02etal} (see Fig.~\ref{allmodesT800:fig}).
%
As this excitation energy is in fact larger than all vibrational
frequencies of C$_{60}$ (32-197~meV), its nature could not be
straightforwardly interpreted, leading to a
debate \cite{ManiniCommAndReply03}.


\begin{figure}[b!]
\centerline{
\epsfig{file=allmodesT800.eps,width=83mm,clip=}
}
\caption{\label{allmodesT800:fig}
Experimental PE spectra by Canton {\it et al.}~\cite{Canton02etal} and by
Br\"uhwiler {\it et al.}~\cite{Bruhwiler97etal} compared with the
calculation based on the {\it ab initio} parameters of
Ref.~\cite{Manini01etal} ($T=800$~K, $N_{\rm tier}=31$, $N_{\rm max}=10^5$,
$N_{\rm sample}=245$).
%
Intensities are normalized to unity.
%
%If JT was turned off, the spectrum would be represented by a single sharp
%peak at 112~meV.
%
Inset: $T=800$~K spectra including the single $H_g^{(1)}$ mode, and
$H_g^{(7)}$ and $H_g^{(8)}$.
%
All spectra are shifted to move the main peak to zero energy and
broadened with a 10~meV HWHM Gaussian.
%
The errorbar gives the largest statistical error introduced by random
sampling of the initial states.
}
\end{figure}


In this Letter we present first of all a detailed understanding of 
the PE structures that are accurately reproduced for
C$_{60} \to$~C$_{60}^+$ (Fig.~\ref{allmodesT800:fig}) 
through calculations that fully include temperature and that 
contain no adjustable parameters. % whatsoever.
%
By taking further
this calculation to pieces, we
identify the important ingredients that determine the spectrum.
%
The first is symmetry selection: the initial orbital symmetry dictated by 
the suddenly injected bare hole extends to the whole spectrum. 
The second is a property of weakly coupled % excited
JT vibronic 
multiplets that requires the first excitation
energy of the same symmetry as the ground state (GS) to rise above the bare
vibration energy by an amount  strictly
proportional to the squared JT coupling. 
%
Specifically in C$_{60}^+$, symmetry requires the 230~meV peak to
correspond to an $h_u$ vibronic excitation above the $h_u$ GS.
%
JT transforms the highest $H_g^{(8)}$ vibration ($\hbar\omega<200$~meV) of
C$_{60}$ into an $h_u$ vibron near 230~meV.
%
Interestingly, the 20\% magnitude ot this upward shift fingerprints quite
accurately the medium-sized overall JT coupling of C$_{60}$.
%
As both properties above are apparently general, and since the upward
shift tracks the coupling intensity well beyond the weak-coupling limit,
this kind of analysis appears of much more general value.
%
In particular, the first vibronic excitation energy as seen in PE of other
weak to medium JT-coupled molecules should and does gauge their respective
couplings as well.

%____________________________________________________________________
%
%Formulation
%
%______________________________________________________________________
We adopt for C$_{60}$ the simplest model \cite{CeulemansII,hbyh,Moate96etal} 
describing the JT coupling of the orbitally degenerate $h_u$ hole 
with the molecular vibrations.
%
The linear JT coupling is nonzero for two nondegenerate $A_g$, six
fourfold-degenerate $G_g$ and eight fivefold-degenerate $H_g$ vibration
modes, or 66 vibrations altogether \cite{CeulemansII,hbyh,Lueders02etal}. 
%
The resulting $h \otimes (A + G + H)$ JT Hamiltonian thus consists of 
the fivefold $h_u$ highest occupied molecular orbital (HOMO),
%a fivefold $h_u$ electronic level,
of 66 harmonic oscillators, and of the
electron-vibration (e-v) coupling term \cite{Manini01etal,hbyh}:
%
\begin{eqnarray}
\label{modelhamiltonian}
\hat{H} &=& \hat{H}_0 + \hat{H}_{\rm vib} + 
\hat{H}_{\rm e-v} \,,\\
\hat{H}_0     &=& \epsilon_{\rm HOMO} \, \sum_{m \sigma}  
\hat{c}_{m \sigma}^\dagger \hat{c}_{m \sigma} \,, \\
\label{vib-hamiltonian}
\hat{H}_{\rm vib} &=& \frac 12 \sum_{\Lambda j \mu} \hbar \omega_{\Lambda j}
\left(\hat{P}_{\Lambda j \mu}^2+\hat{Q}_{\Lambda j \mu}^2\right) \,, \\
%\hat{b}_{\Lambda j \mu}^\dagger \hat{b}_{\Lambda j \mu} , \\
\label{JT-hamiltonian}
\hat{H}_{\rm e-v} &=& 
%\frac 12
\sum_{{r\,\Lambda j \mu}\atop{\sigma m m'}}
k^\Lambda g^{(r)}_{\Lambda j} \hbar \omega_{\Lambda j}
%\sum_{\sigma m m' \mu} 
C^{r \Lambda \mu}_{m \; -m'} \,
\hat{Q}_{i\Lambda \mu}
\hat{c}^\dagger_{m\sigma} \hat{c}_{m' \sigma } \,.
\end{eqnarray}
%
%$\epsilon$ represents the energy position  of the level where electrons are
%emitted from.
%
Here $m$, $\mu$ label components within the degenerate multiplets
%, for example according to the $C_5$ character in the $I_h\supset D_5\supset C_5$
%group chain 
\cite{hbyh,Butler81},
%
$j$ counts modes of symmetry $\Lambda$,
$C^{r \Lambda \mu}_{m m'}$ are Clebsch-Gordan coefficients \cite{Butler81}
of the icosahedral group $I_h$ that couple the $h_u$ fermion
operators $\hat{c}_{m \sigma}^\dagger$ to a $\Lambda$ vibration.
%
$\hat{Q}_{i\Lambda \mu}
%=\left(\hat{b}_{\Lambda j \mu}^\dagger + \hat{b}_{\Lambda j \mu}\right)/\sqrt 2
$
are the dimensionless %molecular
normal-mode vibration coordinates
%(measured from the %adiabatic
%equilibrium configuration of neutral C$_{60}$,
in units of %the length scale 
$x_0(\omega_{\Lambda j})=\sqrt{\hbar/
(\omega_{\Lambda j} \, m_{\rm C})}$ 
%associated to each harmonic oscillator,
where $m_{\rm C}$ is the mass of the C atom), and $\hat{P}_{i\Lambda \mu}$
the corresponding conjugate momenta.
%
The additional multiplicity $r=1,2$, needed for $H_g$ vibrations only,
labels the two separate kinds of $H_g$ coupling allowed under the {\em
same} symmetry \cite{Manini01etal,Butler81}.

In the present calculation we adopt the fully {\it ab initio} numerical
values of the e-v coupling parameters
$g^{(r)}_{\Lambda j}$
%, listed in Table~\ref{e-v-params:table},
obtained by a Density Functional (DF) calculation in Ref.~\cite{Manini01etal}
\footnote{Another DF calculation \cite{Saito02} based on a different functional
yielded couplings roughly in the same range.
%
However we could not re-compute the PE spectrum with these couplings due to
their lack of distinction between coupling contributions of type $r=1$ and
$r=2$.}.
%(favoring energetically $D_{5d}$ and $D_{3h}$ distortions
%respectively) makes it impossible to run a PE calculation based
%on the couplings of Ref.~\cite{Saito02}.
%
We also use the calculated vibrational frequencies of
Ref.~\cite{Manini01etal}, that are close to the experimental frequencies of
the $A_g$ and $H_g$ modes, but also cover the experimentally unavailable or
unreliable $G_g$ modes.  E-v couplings are dimensionless, each of them
normalized
%as Eq.~\eqref{JT-hamiltonian} refers
%them to the corresponding 
to its respective vibration quantum, so as to reflect directly the
relative coupling strength.
%
%In this notation, modes (e.g.\ the second $G_g$ mode) with
%$g^{(r)}_{\Lambda j}\ll 1$ are only weakly coupled, while, on the other hand
%modes (e.g.\ the lowest $H_g$ mode) with a large $g^{(r)}_{\Lambda j} > 1$ 
%are strongly e-v coupled.
%
Numerical factors % note the factor 2 inclueded here for brevity
$k^{A_g}=\frac {5^{\frac 12}}2$,
$k^{G_g}=\frac {5^{\frac 12}}4$,
$k^{H_g}=\frac 12$,
%which could be reabsorbed into the definition of $g^{(r)}_{\Lambda j}$, 
are included to make contact 
%explicitly for compatibility 
with % the notation of
Ref.~\cite{Manini01etal}.  


In the sudden approximation, the angle-integrated PE spectrum $I(E)$ is
given by Fermi's golden rule \cite{Koppel02etal}
\begin{eqnarray}
\label{thermalaverage}
I(E) &=& \frac 1{10} \sum_{m\sigma i} I_{im\sigma}(\omega) P(i) \,,\\
\label{FermiGR}
I_{im\sigma}(E) &=& % \frac{2\pi}{\hbar}
\sum_f \left|\langle f| \hat{c}_{m \sigma}^\dagger | i \rangle \right|^2
\delta\left(E - E_f+E_i\right) \,,
\end{eqnarray}
where $| i \rangle $ represents the starting vibrational eigenstate of
neutral C$_{60}$, of energy $E_i$, and $| f \rangle$ represents a vibronic
JT eigenstate of C$_{60}^+$, of energy $E_f$.
%
%$\hat{c}_{m \sigma}^\dagger$ creates a hole in the orbital labeled $m$ of the
%fivefold degenerate $h_u$ HOMO.
%
$\hat{c}_{m \sigma}^\dagger | i \rangle$ is the initial bare hole, where
the spin-$\sigma$ electron has been ejected from one orbital $m$ of the
fivefold degenerate $h_u$ HOMO, but
the molecule is still unrelaxed.
%
The factor 1/10 and the sum over $m$ and $\sigma$ signifies averaging over
the 5 orbital and 2 spin states of the hole.
%
Assuming thermal equilibrium we generate the
%harmonic oscillator
boson numbers $\{v_{\Lambda j\mu}\}$ of the initial neutral molecule
by randomly sampling the probability distribution
$
%\begin{equation}
%\label{thermaldistribution}
 P(i)= Z^{-1}\exp\left(-E_i / k_{\rm B}T\right) ,
% P(i)= Z^{-1}\exp\left(-\frac{E_i} {k_{\rm B}T}\right) ,
%\end{equation}
$
where $Z=\sum_i\exp\left(-E_i / k_{\rm B}T\right)$ and
$E_i = \sum \hbar \omega_{\Lambda j} v_{\Lambda j\mu}$.
%
In the present medium-coupling regime we compute the final-state vibronic
energies and matrix elements in \eqref{FermiGR} by numerical Lanczos
diagonalization of the Hamiltonian \eqref{modelhamiltonian} on the product
basis of the five $h_u$ states times the harmonic oscillator ladders.
Since these contain an infinite number of states, some truncation is
necessary.
%
%For initial excitations containing only $v_{\Lambda j}\lesssim 2$ quanta, a
%standard truncation including all the states below some maximum number of
%quanta works quite well.
%
%However, 
As we generally want to address high temperatures of the molecular beam
($T=800$~K used for C$_{60}$, $k_{\rm B} T\gtrsim 2\hbar\omega$ of the
strongly coupled $H_g^{(1)}$ mode)
%for which significantly excited initial states are therefore prevalent):
%here a converged 
a standard truncation would involve a far too large Hilbert space size.
%
We generate a smarter basis starting from the initial excitation
$\hat{c}_{m \sigma}^\dagger | i \rangle$, and iteratively
adding sets of states (``tiers'') directly coupled to those of the previous
tier by matrix elements of $\hat{H}_{\rm e-v}$, in a scheme inspired by
Ref.~\cite{Stuchebrukhov93}.
%
This procedure is iterated $N_{\rm tier}$ times so that states relevant at
up to $(2 N_{\rm tier})$-th order in perturbation theory are included
\footnote{
The tier size increases quickly, but we cut it off by including up
to some given number $N_{\rm max}$ of states per tier, according to their
perturbative weight $w_b=\sum_{a} f\left(\langle b|\hat{H}_{\rm
e-v}|a\rangle[E_a-E_b]^{-1}\right) w_a$ where
$f(x)=(1+x^{-2})^{-\frac 12}$,
%$f(x)=\sqrt{1+x^{-2}}$,
the sum extends over all the states $|a\rangle$ in the previous tier,
and the weight of
$\hat{c}_{m \sigma}^\dagger | i \rangle$
%the initial excitation
is taken as unity.  }.
%
The application of about 350 Lanczos steps \cite{Prelovsek00}
generates a tridiagonal matrix, which provides a well converged
spectrum.


The basis for the calculation of Fig.~\ref{allmodesT800:fig}
includes about 720000 states, in particular all the states generated by up
to three applications of $\hat{H}_{\rm e-v}$ to $\hat{c}_{m \sigma}^\dagger
| i \rangle$: the convergence of the truncated basis is quite good.
%
We repeat the whole procedure of basis generation / spectrum calculation for
each of the $N_{\rm sample}$ initial states, and average the obtained
spectra.
%
The above procedure is carried out including all $H_g$ and $G_g$ modes. The
$A_g$ modes separate out, as they can be solved analytically as a simple
displaced oscillators, and are included exactly by convolution.


%_____________________________________________________________________________
%
% Results
%
%_____________________________________________________________________________

Figure~\ref{allmodesT800:fig} compares the theoretical PE %$h_u$ hole
spectrum calculated at $T=800$~K with the
measured spectra by two separate groups.
%
All experimental features, including the characteristically broad and
asymmetric main PE peak, the strongest satellite peak around 230~meV, a
second weaker peak near 400~meV, and a slow decay up to 600~meV are
remarkably reproduced, based purely on {\it ab initio} parameters.
%
The model and its ingredients thus appear to describe quite accurately the
PE spectrum of C$_{60}$. We can now take it apart, so as to understand the
underlying physics to satisfaction.


%__________________________________________________________________
%
% Understanding
%
%__________________________________________________________________

We address the two main features of the spectrum, namely a) the large
broadening and b) the 230~meV peak.
%
By eliminating $G_g$ and $A_g$ modes from the calculation there is no
large change in the spectrum. Thus only the $H_g$ modes are important.
%
Among $H_g$ modes, the largest couplings belong to the low-frequency
$H_g^{(1)}$ 32~meV quadrupolar radial mode, and to the two high-frequency
C-C stretch modes $H_g^{(7)}$ and $H_g^{(8)}$ at $\hbar\omega=181$ and
197~meV).
%
When we include $H_g^{(1)}$ alone we find (inset) that the thermally
broadened asymmetric main peak is well reproduced.
%
Therefore $H_g^{(1)}$ is the main responsible for the large
broadening.
%
%The harmonic wiggles separated by about $\hbar\omega_{H_g 1}\simeq $32$~meV
%are washed out by many-modes interference.

When only $H_g^{(1)}$ is included, all satellite structures above 200~meV
are absent, and do not reappear even if we arbitrarily increase the
$H_g^{(1)}$ coupling to larger and larger values.
%
This rules out the possibility to attribute the 230~meV structure to an
``electronic'' splitting \cite{ManiniCommAndReply03}.
%
As it turns out, electronic JT splittings become visible in experimental PE
spectra, such as those of Fe(CO)$_5$ \cite{Hubbard81}, and in theoretical
spectra, like those of Martinelli {\it et al.}~\cite{Martinelli91etal} only
for very strong e-v coupling, whereas in C$_{60}$ coupling is intermediate
at most.
%
That suggests that the 230~meV peak in C$_{60}$ should rather be associated
to the high-frequency modes $H_g^{(7)}$ and $H_g^{(8)}$. However, the
connection between their frequencies $<200$~meV and the 230~meV
peak position is far from obvious.
%
To clarify this, we calculate the spectrum by including $H_g^{(7)}$ and
$H_g^{(8)}$ alone (inset of Fig.~\ref{allmodesT800:fig}).
%
The result is a much narrower spectrum ($H_g^{(1)}$ is now omitted),
showing a high-energy satellite at a significantly (10\%) higher energy
than $\hbar\omega_{H_g8}$.
%216~meV
This increase in vibronic excitation energy above the purely vibrational
$\hbar\omega$ for the $v$=1-phonon excitation of a trivial displaced
oscillator is in fact a characteristic signature of the dynamic JT effect,
and can be understood on the basis of symmetry.
%
For $H_g^{(7)}$ and $H_g^{(8)}$, $\hbar\omega$ is almost three times the
thermal energy: the main initial-state contribution to the spectrum comes
from $|i\rangle$ = the neutral molecule GS, of symmetry $A_g$.
%
The bare hole $\hat{c}_{m \sigma}^\dagger | i \rangle$ therefore carries
the orbital $h_u$ symmetry of the HOMO.
%
This state has nonzero overlap strictly only with final states $|f\rangle$
of the same symmetry $h_u$.
%
This {\em single symmetry channel}\ thus represents the only contributor to the
low-temperature PE spectrum according to \eqref{FermiGR}
\footnote{
This, incidentally, rules out the possibility to observe tunnel split states
\cite{Canton02etal} in PE, as they necessarily possess
{\em different} symmetries.
}.
%
The blue shift of the vibronic peak relative to the
bare vibration is now due to two general properties (in fact common
to all dynamical JT systems): (i) the coupling independence of the {\em
average} excitation energy in any vibronic multiplet at weak coupling (the
leading change is of order $g^4$) \cite{BersukerWeakCoupl}; (ii) the
increasing ``repulsion'' that the GS level exerts for increasing $g$ on all
excited states of the same symmetry, which pushes them upward.


\begin{figure}
\centerline{
\epsfig{file=alpha0.5.eps,width=85mm,clip=}
}
\caption{\label{alpha0.5:fig}
Vibronic excitation energies for a single $H_g$ mode coupled to the $h_u$
level, as a function of coupling, for a ratio $g^{(2)}/g^{(1)}$
characteristic of the $H_g^{(7/8)}$ modes.
%
The {\it average} refers to the multiplet originating from the 1-phonon states.
Bold: $h_u$ states, the only visible states in $T=0$ PE.
%as thick solid lines.
}
\end{figure}


Both concepts are illustrated in Fig.~\ref{alpha0.5:fig}, depicting the
vibronic excitation energies (with special attention to the $h_u$ states)
for the simplest case of a single $H_g$ mode, as a function of the e-v
coupling strength $g^2$.
%
The approximate coupling independence of the {\em average} excitation
energy of the multiplet of states derived from the $v$=1-phonon manifold is
apparent.
%
Within this multiplet (made of $h_u\times H_g$ = $a_u + t_{1u} + t_{2u} + 2
g_u + 2 h_u$), the $h_u$ vibronic states are the sole that have another
state of the same symmetry (the GS itself) lower in energy ``pushing''
them upward.
%
As a result, for small but increasing coupling, the $h_u$ excited vibronic
states, move necessarily upward in energy above $\hbar\omega_{H_g}$.
%
Moreover the amount of shift is at weak coupling proportional to $g^2$
and reflects the strength of the coupling.
%
None of these results seems specific to the $h\otimes H$ JT model of
C$_{60}$. For example, identical conclusions apply to the $e\otimes
E$ problem \cite{BersukerWeakCoupl}.  Indeed a 15\% blue shift of the first
vibronic satellite is clearly observable in the PE spectra of benzene
\cite{Baltzer97etal}.

In C$_{60}$ we must consider the two high-frequency modes $H_g^{(7)}$
and $H_g^{(8)}$ at $\hbar\omega=181$ and 197~meV).
%
The couplings for these two modes alone accounts for a 10\%, or roughly
20~meV shift of the main $H_g^{(7,8)}$ vibronic satellite above
$\hbar\omega$.
%
This is about half of the total observed and calculated shift of $\sim
40$~meV of the 230~meV satellite relative to $\hbar\omega$.
%
The remainder is a collective effect of all other modes simultaneously
interacting with the HOMO, and pushing their respective $v=1$ satellites
(weak and thus invisible at large temperature) upward.
%
All goes as if effectively the coupling of the $H_g^{(7/8)}$ modes were
larger, i.e.\ effectively moving to the right of the vertical dashed line
in Fig.~\ref{alpha0.5:fig}.


\begin{figure}
\centerline{
\epsfig{file=allmodesTdependence.eps,height=85mm,clip=}
}
\caption{\label{allmodesTdependence:fig}
Temperature dependence of the PE, computed including all modes of C$_{60}$
for $T=0$, $T=300$~K and $T=800$~K.
%
Labels ${\Lambda^{(j)}}_v$ indicate the leading component in the vibronic
states.
%
Inset: computed spectra ($T=800$~K) based on the same DF coupling parameters
$g_{\Lambda j}^r$ divided by $\sqrt 2$, original, and multiplied by
$\sqrt 2$, compared to experiment~\cite{Bruhwiler97etal}.
%All calculations convolved with a 2~meV HWHM Gaussian, $N_{\rm tier}=29$,
%$N_{\rm max}=4\cdot 10^4$, $N_{\rm sample}\simeq 1700$.
}
\end{figure}

We have thus attained the following understanding of the 800~K PE spectrum
of C$_{60}$: (i) the strong 230~meV satellite (and the weak 400~meV too) is
due to $h_u$-symmetry vibrons derived from $H_g^{(7/8)}$ modes, blue
shifted by repulsive coupling to the $h_u$ GS; (ii) the large
spectral broadening is due mainly to $H_g^{(1)}$.
%
Clearly, this broadening of the 800~K spectrum prevents direct extraction
of further detailed information about the actual e-v couplings of the
fullerene cation.
%
Our calculations, when repeated for lower temperatures, indicate a wealth
of structures that should emerge. % in that case.
%
Figure~\ref{allmodesTdependence:fig} shows that already a measurement
carried out at 300~K would provide enough detail to get precise indications
about the couplings of individual modes.
%
For example, the intensity ratios between the main peak and the subsequent
peaks at 35 and 65~meV is a direct measure of $g_{H_g1}$.
%
A cool molecular beam experiment \cite{Miller88} would be extremely useful.
% for this purpose: we could readily repeat our calculations for a given
% experimental condition and compare it to the measured spectrum.
%
Also in view of future 
cool-beam
data it is meaningful to address the question
of how accurate the calculated DF e-v couplings used here really are.
%
For $t_{1u}$ electrons coupled to $H_g$ vibrations, the calculated DF
couplings were argued for example to be too weak by a substantial amount,
by comparison to the larger couplings needed in order to fit C$_{60}^-$ PE
\cite{Gunnarsson95etal}.
%because DF and Hartree Fock (HF) are the two main {\it ab initio}
%approaches to calculate these couplings, and the results differ hugely (HF
%giving much larger couplings than LDA). 
%
%While keeping in mind that the C$_{60}^-$ problem -- where JT is present in
%the {\em initial} state -- is totally different from ours, our bias was to
%expect a similar failure of LDA couplings in the present HOMO e-v
%couplings.
%
In our HOMO case, the excellent agreement of calculations with experiment
%even with the limited resolution permitted by the temperature of 800~K,readily
suggests that there is no such failure.
%
Figure~\ref{allmodesTdependence:fig} (inset) shows several PE calculations
with rigidly rescaled e-v parameters. Comparison with experiment indicates
that the DF couplings are, on the whole, quite close (if slightly on the
weak side) to the actual couplings of C$_{60}^+$.
%
Much larger JT coupling values like those obtained using different approaches
\cite{Bendale92etal} are ruled out.
%
That is very good news, for it gives further confidence in the future
possibility to rely solidly on DF for a good description of the
intra-fullerene chemical bond, and of its perturbations.


%
%________________________________________________________________________
%
%Conclusions
%
%_________________________________________________________________________

In conclusion, we present here a detailed calculation of the PE spectrum of a
high-symmetry molecule, C$_{60}$, which is JT active in its one-hole state.
%
We find excellent agreement with experiment, confirming for the first
time the accuracy of the e-v couplings as extracted from DF calculations.
%
Perhaps more importantly, we reach a fresh understanding of the PE
spectrum, with its width, its vibronic peak, and the reason for its blue
shift relative to the bare vibrations.
%
The blue shift of the one-vibron satellite is argued to represent a more
general characteristic fingerprint of JT systems in the
weak-to-intermediate coupling regime, and is found for example in the
$e\otimes E$ JT of benzene as well.% \cite{Baltzer97etal}.

%\section*{Acknowledgments}

We are indebted to G.P. Brivio, P.\ Br\"uhwiler, A.\ Del Monte, D.\ Galli,
M.\ Lueders, L.\ Molinari, G.\ Onida, F.\ Parmigiani, A.\ Parola, and G.E.\
Santoro, for useful discussion.
%
This work was supported by the European Union, contracts ERBFMRXCT970155
(TMR FULPROP) as well as by MIUR COFIN01, and FIRB RBAU017S8R
operated by INFM.


%&&&&&&&&&&&&&&&&&&&&&&&&&&&&&FINE&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&


%%%%%%%%%%%%%%%%%%%%%%%%%%%% REFERENCES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\bibliographystyle{apsrev}
%\bibliography{biblio}

\begin{thebibliography}{21}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname bibnamefont\endcsname\relax
  \def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
  \def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

\bibitem[{\citenamefont{{ R.\ Lindner {\it et al.}, Science {\bf 271}, 1698
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\bibitem[{\citenamefont{{ H.\ K\"oppel {\it et al.}, J.\ Chem.\ Phys. {\bf
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\bibitem[{\citenamefont{{ P.\ Br\"uhwiler {\it et al.}, Chem.\ Phys. Lett.\
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\bibitem[{\citenamefont{{ S.\ E.\ Canton {\it et al.}, Phys.\ Rev.\ Lett.\ {\bf
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\bibitem[{\citenamefont{{ N.\ Manini and E.\ Tosatti, Phys.\ Rev. Lett.\ {\bf
  90}, 249601 (2003); S.\ E.\ Canton {\it et al.}, ibid,
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\bibitem[{\citenamefont{{ N.\ Manini {\it et al.}, Philos.\ Mag. B {\bf 81},
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\bibitem[{\citenamefont{{ A.\ Ceulemans, and P.\ W.\ Fowler, J.\ Chem.\ Phys.\
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\bibinfo{author}{\bibnamefont{{ A.\ Ceulemans, and P.\ W.\ Fowler, J.\ Chem.\
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\bibitem[{\citenamefont{{ N.\ Manini and P.\ De Los Rios, Phys.\ Rev. B {\bf
  62}, 29 (2000)}}()}]{hbyh}
\bibinfo{author}{\bibnamefont{{ N.\ Manini and P.\ De Los Rios, Phys.\ Rev. B
  {\bf 62}, 29 (2000)}}}.

\bibitem[{\citenamefont{{ C.\ P.\ Moate {\it et al.}, Phys.\ Rev.\ Lett.\ {\bf
  77}, 4362 (1996)}}()}]{Moate96etal}
\bibinfo{author}{\bibnamefont{{ C.\ P.\ Moate {\it et al.}, Phys.\ Rev.\ Lett.\
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\bibitem[{\citenamefont{{ M.\ L\"uders {\it et al.}, Philos.\ Mag. B {\bf 82},
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\bibitem[{\citenamefont{{ P.\ H.\ Butler, {\it Point Group Symmetry
  Applications} (Ple\-num, New York, 1981)}}()}]{Butler81}
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\bibitem[{\citenamefont{{ A.\ A.\ Stuchebrukhov and R.\ A.\ Marcus, J.\ Chem.\
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\bibinfo{author}{\bibnamefont{{ A.\ A.\ Stuchebrukhov and R.\ A.\ Marcus, J.\
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\bibitem[{\citenamefont{{ J.\ Jaklic and P.\ Prelovsek, Adv.\ Phys.\ {\bf 49},
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\bibitem[{\citenamefont{{ J.\ L.\ Hubbard and D.\ L.\ Lichtenberger, J.\ Chem.\
  Phys.\ {\bf 75}, 2560 (1981)}}()}]{Hubbard81}
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\bibitem[{\citenamefont{{ L.\ Martinelli {\it et al.}, Phys.\ Rev.\ B {\bf 43},
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\bibitem[{\citenamefont{{ I.\ B.\ Bersuker and V.\ Z.\ Polinger, {\it Vibronic
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  Sect.\ 4.1}}()}]{BersukerWeakCoupl}
\bibinfo{author}{\bibnamefont{{ I.\ B.\ Bersuker and V.\ Z.\ Polinger, {\it
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  1989), Sect.\ 4.1}}}.

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\end{thebibliography}


\end{document}


% trailing stuff -- ideas for recycling:

As was explored in Ref.~\cite{Martinelli91} in a similar context, the
Lanczos tridiagonalization method allows one to express the dynamical
response \eqref{FermiGR} in terms of a continued fraction, with extremely
attractive convergence properties.
%
The idea is that a Lanczos chain $\psi_0, ... \psi_K$ is started from the
vector $\psi_0 =\hat{c}_{m \sigma}^\dagger | i \rangle$, and the sequence
of Lanczos coefficients $b_i= ...$


Here we assume that the orbitals are basically unchanged upon distortion,
%
We also assume no change of the phonon frequencies $\omega_{\Lambda j}$ and
couplings $g^{(r)}_{\Lambda j}$ upon charging.


The present calculation was carried out in the {\em linear} e-v coupling
approximation.
%
As the coupling and thus the distortions are fairly large, quadratic and
higher-order (in ${\bf Q}$) e-v interactions and vibration anharmonicity
could be relevant.
%
Unfortunately, no estimate for those higher-order couplings is available yet.


%NICOLA IL RESTO SOTTO IO LO TOGLIEREI

The basic idea is that sudden PE of an electron generates the C$_{60}^+$
ion in the initially perfectly icosahedral molecular geometry,
with a hole in the fivefold degenerate $h_u$ HOMO.

The ensuing final-state JT molecular distortion leads to a splitting of
this degeneracy, with a lowering of the total energy.

This process is however not adiabatic, and a finite probability of leaving
behind an excited C$_{60}$ ion gives rise to a rich hole spectrum,
comprising strong shake-up satellites.

Because the electronic and vibrational excitations are mutually entangled
in the dynamic JT state, the ion final-state excitations are not just
vibrations (whose energies would be independent of couplings) or just
electronic excitations, but rather vibronic -- entangled
electron-vibrational -- states whose energies may resemble either
vibrations or electronic excitations, but will differ in fact
quantitatively from both in a coupling-dependent manner.
%
That makes these vibronic spectral features generally hard to interpret,
sometimes even at the qualitative level \cite{ManiniCommAndReply03}.
%
The correct quantitative theory of these non adiabatic spectra and of their
vibronic satellite positions permits, in direct comparison with experiment,
both to learn how to interpret %qualitatively
these spectra and to obtain quantitative information about the couplings.
%
These, in turn, are necessary ingredients to the understanding not only of
the PE spectrum of pure C$_{60}$ \cite{Rudolf99}, but also the properties
of C$_{60}$ cation-based compounds, which are attracting increasing
interest \cite{Gasyna,Datars95,Reed00,Panich02,Panich03}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

NICOLA: LA ROBA SOTTO NIENTE...\

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Peak structures will arise in the spectrum due to the summation over the final
vibronic eigenstates $| f \rangle$, while the $i$-summation over the
initial thermally excited vibrational states will generally produce thermal
broadening.

The initial neutral molecule states are unaffected by JT
and are therefore simple superpositions of eigenstates 
of the 66 harmonic oscillators in $\hat{H}_{\rm
vib}$, of energy $E_i = \sum \hbar \omega_{\Lambda j} v_{\Lambda j\mu}$.

The calculation of the PES of C$_{60}$ requires therefore a complete
knowledge of the spectra of the neutral and singly-ionized states.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%NICOLA: LA STORIA DI ALFA QUI SOTTO IO LA TOGLIEREI DEL TUTTO...

So far we referred just to the $H_g^{(7,8)}$ coupling strength as the main
parameter. As is well known, the $H\otimes h$ problem is characterized by a
second parameter, namely the value of the mixing angle $\alpha$
\cite{hbyh,Manini01etal} fixing the ratio $g_{H_gj}^2/g_{H_gj}^1 =
\tan \alpha_j$.

In C$_{60}$ 
this parameter, often ignored, is quite important. In static JT, it
determines the symmetry of the distortion ($D_{5d}$ for $\alpha =0$,
but  $D_{3d}$ for $\alpha=\pi/2$). In dynamic JT, it again determines 
the GS symmetry, which is  $h_u$ for $\alpha\approx 0$, but which would 
be $a_u$ for  $\alpha\approx \pi/2$ \cite{hbyh}.

By repeating PE calculations for variable $\alpha$ we find that the
spectrum does change although not dramatically.

In PE spectroscopy, it is crucial in determining the precise splitting
pattern of the 1-vibron multiplet, even though the general feature that the
$h_u$ states shift upwards holds for any value of $\alpha$.


%
\rem{
In particular, we will demonstrate in C$_{60}$ how the energy separation
between the ``zero-vibron'' main PE %hole
line and the first vibron
satellite differs from the bare vibration energy by an amount which depends
sensitively (and thus gauges delicately) on the magnitude and even the
phase of the relevant e-v coupling.
%
As it turns out, two strongly coupled vibrations $H_g^{(7)}$ at 181~meV and
$H_g^{(8)}$ at 197~meV are the main responsible for the 230~meV PE
satellite. The large blue shift relative to the bare vibrational energies
is proportional to the JT coupling magnitudes, and in fact does require
couplings whose magnitude and phase must coincide with an error of order
10\% with the values derived from DF electronic-structure
calculations by Manini {\it et al}.~\cite{Manini01etal}
% ERIO: e` piuttosto ottimista che sappiamo veramente gli accoppiamenti al 10%:
% se qualche modo di quelli debolmente accoppiati fosse accoppiato magari
% il doppio o la meta`, lo spettro non cambierebbe visibilmente.  Uno statement
% piu` fair sarebbe che i modi piu` accoppiati hanno un accoppiamento nel
% giusto range.  Ma capisco che a livello di intro non si puo`
% sottiglizzare troppo.
%
The basic physical considerations here are a) low-excitation probability
($\hbar\omega>k_{\rm B}T$) for the high-energy phonons $H_g^{(7)}$ and
$H_g^{(8)}$, b) symmetry, and c) tracelessness of the JT-induced splittings
in the weak-coupling limit.
%
At $T=800$~K, most spectral intensity is associated to initial states $| i
\rangle$ where the $H_g^{(7)}$ and $H_g^{(8)}$ high-frequency modes are in
their GS.
%
These $H_g^{(7)}$ and $H_g^{(8)}$ phonon symmetry is therefore trivially
$A_g$.
%
The initial unrelaxed pure-hole state $\hat{c}_{m \sigma}^\dagger | i
\rangle$ has therefore the same symmetry $h_u$ as the HOMO, and thus it
generally retains, in a dynamic JT problem, precisely the same overall
symmetry $\Gamma$.\cite{ham}\footnote{
%ERIO: quello che dici qui e` vero ma misleading. Non importa quale sia la
%simmetria del GS vibronico.  Comunque, lo stato iniziale con la buca
%appena creata nell'HOMO, prima che i vibroni comincino a rilassare ha
%simmetria h_u.  Anche se i parametri fossero tali che il GS fosse di
%simmetria a_u, comunque nello spettro PES uno vedrebbe solo gli stati h_u
%e null'altro.  Tra l'altro questo mostra che non c'e` speranza di vedere
%il tunnel splitting, come dicevamo l'altro giorno parlandone...
In fact, the $h\otimes H$ problem might for certain couplings lead to a
hole of $a_u$ symmetry instead of $h_u$ symmetry as in Ham's theorem (see
Ref.~\cite{Delos96,hbyh}). However, realistic couplings have been shown to
lead to the standard $h_u$ hole GS symmetry.
%
}
only overlaps to vibronic states of the same symmetry $h_u$.

It is a general property 
the first excitation of same symmetry $\Gamma$ above the $\Gamma$
GS -- leading to the first important shake-up peak -- must rise
in energy above all other excitations of symmetry $\Gamma'$ different from  $\Gamma$.
In that manner the cumulative average energy of the whole first excitation 
multiplet $\Gamma + \Gamma'$ can remain coupling independent, as required
by a null trace. This rise leads to an energy of the first shake-up vibronic peak that
most be {\em larger} than the bare vibrational quantum $\hbar\omega$ by an
amount precisely proportional to the total JT coupling, at least so long
as that coupling is not too large.
%
%Focusing on C$_{60}$ as our working example, we will show that 
%even in such
%a complicated intermediate-coupling case 
%one can use the measured 230~meV energy difference of the first
%vibronic shake-up satellite as a fingerprint to gauge the magnitude of
%the JT actual couplings.
%
} % end of \rem


% BUONA INTRO AL MODELLO:
PE in C$_{60}$ suddenly creates a $h_u$  hole in the fivefold orbitally 
degenerate $h_u$ highest occupied molecular orbital (HOMO). Due to JT
coupling, a bare hole in the frozen molecule is not an eigenstate of the 
coupled system. It must therefore be expanded into vibron eigenstates,
clearly restricting to those of $h_u$ symmetry. The (zero-temperature)
excitation spectrum corresponds precisely to this expansion.


\rem{
\begin{table}[t]
\centerline{
\begin{tabular}{c|r|rrrr}
\hline
\hline
mode    &    Raman  & \multicolumn{4}{c}{DFT - LDA calculation} \\
%\hline
symmetry&   $\hbar\omega_{\Lambda j}$ & $\hbar\omega_{\Lambda j}$  &  $\hbar\omega_{\Lambda j}$  &   $g_{\Lambda j}$    &  $\alpha_{\Lambda j}$      \\
$\Lambda$ & {\small [cm$^{-1}$]} &  {\small [cm$^{-1}$]}  &  {\small [meV]} &  {\small [dim.less]}    & {\small [rad]}   \\
\hline
$A_g$   &   496     &       500     &       62.0    &       0.0591   & -  \\
\       &  1470     &      1511     &      187.4    &       0.2741   & -  \\
\hline
$G_g$   &    -      &       483     &       59.9    &       0.7567   & -  \\
\       &    -      &       567     &       70.3    &       0.1024   & -  \\
\       &    -      &       772     &       95.7    &       0.8003   & -  \\
\       &    -      &      1111     &      137.8    &       0.6239   & -  \\
\       &    -      &      1322     &      163.9    &       0.2277   & -  \\
\       &    -      &      1519     &      188.4    &       0.4674   & -  \\
\hline
$H_g$   &   271     &       261     &      32.4     &     3.0417     &-0.0015\\
\       &   437     &       429     &      53.2     &     1.2234     &0.5250 \\
\       &   710     &       718     &      89.0     &     0.9950     &1.5605 \\
\       &   774     &       785     &      97.3     &     0.7842     &-0.0394\\
\       &  1099     &      1119     &     138.7     &     0.2212     & 1.3361\\
\       &  1250     &      1275     &     158.0     &     0.5194     & 0.4889\\
\       &  1428     &      1456     &     180.5     &     0.9616     & 0.4905\\
\       &  1575     &      1588     &     196.9     &     0.8686     &-0.5430\\
\hline
\hline
\end{tabular}
}
\caption{\small \label{e-v-params:table}
Experimental \cite{Bethune91,Zhou} and {\it ab initio} \cite{Manini01etal}
vibrational frequencies for the JT active modes.  {\it Ab initio} e-v
linear coupling parameters \cite{Manini01etal} for the $h_u$ HOMO in C$_{60}$.
} 
\end{table}

}


%
\rem{ OLD INSET with convergence test:
\centerline{
\epsfig{file=convergenceT800.eps,height=60mm, angle=270,clip=}
}
Inset: convergence for basis enlargement;
red solid:
$N_{\rm tier}=24$, $N_{\rm max}=3\cdot 10^3$, $N_{\rm sample}=2500$;
green dashed:
$N_{\rm tier}=25$, $N_{\rm max}=1\cdot 10^4$, $N_{\rm sample}=2907$;
blue short dashed:
$N_{\rm tier}=29$, $N_{\rm max}=4\cdot 10^4$, $N_{\rm sample}=2447$;
pink dotted:
$N_{\rm tier}=31$, $N_{\rm max}=1\cdot 10^5$, $N_{\rm sample}=245$.
%
}

\rem{
\begin{figure}
\centerline{
\epsfig{file=onlyh7h8.eps,height=85mm, angle=270,clip=}
}
\caption{\label{onlyh7h8:fig}
Computed spectrum ($T=800$~K, $N_{\rm tier}=29$, $N_{\rm max}=4\cdot 10^4$,
$N_{\rm sample}=1000$) including only the $H_g^{(7)}$ and $H_g^{(8)}$ modes.
}
\end{figure}
}

cond-mat0307696.extracted_bib
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\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
  \def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

\bibitem[{\citenamefont{{ R.\ Lindner {\it et al.}, Science {\bf 271}, 1698
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\bibitem[{\citenamefont{{ H.\ K\"oppel {\it et al.}, J.\ Chem.\ Phys. {\bf
  117}, 2657 (2002)}}()}]{Koppel02etal}
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\bibitem[{\citenamefont{{ S.\ E.\ Canton {\it et al.}, Phys.\ Rev.\ Lett.\ {\bf
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\bibitem[{\citenamefont{{ C.\ P.\ Moate {\it et al.}, Phys.\ Rev.\ Lett.\ {\bf
  77}, 4362 (1996)}}()}]{Moate96etal}
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  {\bf 77}, 4362 (1996)}}}.

\bibitem[{\citenamefont{{ M.\ L\"uders {\it et al.}, Philos.\ Mag. B {\bf 82},
  1611 (2002)}}()}]{Lueders02etal}
\bibinfo{author}{\bibnamefont{{ M.\ L\"uders {\it et al.}, Philos.\ Mag. B {\bf
  82}, 1611 (2002)}}}.

\bibitem[{\citenamefont{{ P.\ H.\ Butler, {\it Point Group Symmetry
  Applications} (Ple\-num, New York, 1981)}}()}]{Butler81}
\bibinfo{author}{\bibnamefont{{ P.\ H.\ Butler, {\it Point Group Symmetry
  Applications} (Ple\-num, New York, 1981)}}}.

\bibitem[{\citenamefont{{ A.\ A.\ Stuchebrukhov and R.\ A.\ Marcus, J.\ Chem.\
  Phys. {\bf 98}, 6044 (1993)}}()}]{Stuchebrukhov93}
\bibinfo{author}{\bibnamefont{{ A.\ A.\ Stuchebrukhov and R.\ A.\ Marcus, J.\
  Chem.\ Phys. {\bf 98}, 6044 (1993)}}}.

\bibitem[{\citenamefont{{ J.\ Jaklic and P.\ Prelovsek, Adv.\ Phys.\ {\bf 49},
  1 (2000)}}()}]{Prelovsek00}
\bibinfo{author}{\bibnamefont{{ J.\ Jaklic and P.\ Prelovsek, Adv.\ Phys.\ {\bf
  49}, 1 (2000)}}}.

\bibitem[{\citenamefont{{ J.\ L.\ Hubbard and D.\ L.\ Lichtenberger, J.\ Chem.\
  Phys.\ {\bf 75}, 2560 (1981)}}()}]{Hubbard81}
\bibinfo{author}{\bibnamefont{{ J.\ L.\ Hubbard and D.\ L.\ Lichtenberger, J.\
  Chem.\ Phys.\ {\bf 75}, 2560 (1981)}}}.

\bibitem[{\citenamefont{{ L.\ Martinelli {\it et al.}, Phys.\ Rev.\ B {\bf 43},
  8395 (1991)}}()}]{Martinelli91etal}
\bibinfo{author}{\bibnamefont{{ L.\ Martinelli {\it et al.}, Phys.\ Rev.\ B
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\end{thebibliography}