Femtosecond study of Cu(H(2)O) dynamics.

Abstract

The short-time nuclear dynamics of Cu(H(2)O) is investigated using femtosecond photodetachment-photoionization spectroscopy and time-dependent quantum wave packet calculations. The Cu(H(2)O) dynamics is initiated in the electronic ground state of the complex by electron photodetachment from the Cu(-)(H(2)O) complex, where hydrogen atoms are oriented toward Cu. Several time-resolved resonant multiphoton ionization schemes are used to probe the ensuing reorientation and dissociation. Immediately following photodetachment, the neutral complex is far from its minimum energy geometry and possesses an internal energy comparable to the Cu-H(2)O dissociation energy and undergoes both large-amplitude H(2)O motion and dissociation. Dissociation is observed to occur on three distinct time scales: 0.6, 8, and 100 ps. These results are compared to the results of time-dependent J=0 wave packet calculations, propagating the initial anion vibrational wave functions on the ground-state potential of the neutral complex. An excellent agreement is obtained between the experimental results and the ionization signals derived from the calculated probability amplitudes. Related experiments and calculations are carried out on the Cu(D(2)O) complex, with results very similar to those of Cu(H(2)O).

Full text

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Femtosecond study of Cu„H2O… dynamics
Felician Muntean,a)Mark S. Taylor, Anne B. McCoy,b)and W. Carl Lineberger
JILA and Department of Chemistry and Biochemistry, University of Colorado, Boulder, Colorado 80309
?Received 3 May 2004; accepted 21 June 2004?
The short-time nuclear dynamics of Cu(H2O) is investigated using femtosecond photo-
detachment-photoionization spectroscopy and time-dependent quantum wave packet calculations.
The Cu(H2O) dynamics is initiated in the electronic ground state of the complex by electron
photodetachment from the Cu?(H2O) complex, where hydrogen atoms are oriented toward Cu.
Several time-resolved resonant multiphoton ionization schemes are used to probe the ensuing
reorientation and dissociation. Immediately following photodetachment, the neutral complex is far
from its minimum energy geometry and possesses an internal energy comparable to the Cu-H2O
dissociation energy and undergoes both large-amplitude H2O motion and dissociation. Dissociation
is observed to occur on three distinct time scales: 0.6, 8, and 100 ps. These results are compared to
the results of time-dependent J?0 wave packet calculations, propagating the initial anion
vibrational wave functions on the ground-state potential of the neutral complex. An excellent
agreement is obtained between the experimental results and the ionization signals derived from the
calculated probability amplitudes. Related experiments and calculations are carried out on the
Cu(D2O) complex, with results very similar to those of Cu(H2O). © 2004 American Institute of
Physics. ?DOI: 10.1063/1.1782176?
I. INTRODUCTION
Both intermolecular energy transfer and solvation dy-
namics play pivotal roles in chemical reactions in the con-
densed phase.1,2Dynamic solvent-solute interactions have
been shown to be important in governing the rates of both
electron transfer reactions3and reactions involving bond
making and breaking.4Femtosecond pump-probe laser spec-
troscopy is a powerful technique for examining intermolecu-
lar energy transfer and solvation dynamics in both the con-
densed phase5–7and within neutral or ionic clusters.8–12In
this study, we employ femtosecond pump-probe laser spec-
troscopy and quantum dynamical calculations to examine the
H2O reorientation and dissociation dynamics within a
Cu(H2O) complex that is produced by photodetachment of
Cu?(H2O).
In related studies, femtosecond pump-probe spectros-
copy has been used to investigate solvation dynamics in both
neutral8,9and anionic10–15gas-phase complexes. The femto-
second photodetachment-photoionization experiments re-
ported here exploit the fact that a polar molecule in a binary
complex with an atomic metal anion will undergo large-
amplitude reorientation and dissociation following electron
photodetachment. Elementary chemical principles and high-
level ab initio calculations concur that Cu?(H2O) is planar
with the hydrogen atoms oriented toward the copper anion,
while neutral Cu(H2O) has the oxygen oriented toward cop-
per and a diminished Cu-OH2separation ?Fig. 1? compared
to the anion. Electron photodetachment of Cu?(H2O) pro-
duces neutral Cu(H2O) initially at the anion geometry with
an internal energy that is near the Cu-H2O dissociation en-
ergy. Immediately after the birth of neutral Cu(H2O), the
water molecule will begin to rotate and vibrate with respect
to Cu, and a fraction of the complexes will dissociate. We
expect the dissociation to have both direct and delayed com-
ponents, the latter arising from coupling between the internal
rotation of H2O and the Cu-H2O stretching vibration.
Classical16and quantum17dynamics calculations have pre-
dicted that similar behavior will be exhibited by the com-
plexes Cl(H2O) and Br(H2O) prepared by electron photode-
tachment of their anionic precursors. Analogous energy
transfer dynamics resulting in direct and delayed dissociation
has been observed within the Hg-N2complex.18Using pico-
second laser pump-probe spectroscopy, Soep and co-workers
observed18both direct and delayed dissociation following
electronic excitation of Hg-N2. They concluded that energy
flow from internal rotation of N2into the Hg-N2stretching
vibration was responsible for the delayed dissociation. The
present joint time-resolved experimental and theoretical
study extends such investigations to solvation dynamics oc-
curring on the electronic ground state of a van der Waals
complex. The work described here is also closely related to
the elegant dissociative photodetachment experiments car-
ried out by Continetti.19,20These experiments obtain a
complementary view of dissociation dynamics by employing
photoelectron-photofragment coincidence spectroscopy to
obtain detailed information concerning the dissociation dy-
namics. Several comprehensive reviews of this approach
have recently appeared.21,22Of particular relevance to the
experiments reported here are studies of solvated O?, OH?,
and O2
In this work, electrons are photodetached from a mass-
selected beam of Cu?(H2O) by a 400 nm, 120 fs laser pulse,
?anions.23–27
a?Permanent address: Varian Corporation, Walnut Creek, CA.
b?Permanent address: Department of Chemistry, The Ohio State University,
Columbus, OH.
JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 1222 SEPTEMBER 2004
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preparing an ensemble of time-evolving Cu(H2O) com-
plexes. Delayed resonance enhanced multiphoton ionization
?REMPI? is used to follow the time evolution of the neutral
complexes and their dissociated fragments. This experimen-
tal approach offers advantages over other femtosecond
pump-probe techniques used for studying the dynamics of
gas-phase neutral molecules. Starting with a negative ion al-
lows mass selection and enables the formation of a neutral
complex with a well-defined initial nuclear configuration.
Photodetachment of negative ions initiates the dynamics on
the electronic ground state of the neutrals, while avoiding the
complexity inherent in coherent nonlinear techniques used to
initiate ground state dynamics.28,29Probing by time-delayed
REMPI allows for simultaneous detection of the parent com-
plex and dissociated products by secondary mass analysis.
Wo ¨ste and co-workers developed time-resolved photo-
detachment-photoionization spectroscopy30and used it to
study the rearrangement dynamics of neutral Ag3. Their ap-
proach was modified in our laboratory by replacing the slow
ion beam and quadrupole ion trap with a pulsed supersonic
expansion ion source and fast ion beam, and it was also used
to investigate the rearrangement dynamics of Ag3.31These
developments allowed production of colder negative ions
and simultaneous detection of both neutral and positive ion
photoproducts. The work from our laboratory31and subse-
quent theoretical studies32–34emphasized that the time evo-
lution of the photodetachment-photoionization signals is
caused not only by nuclear dynamics on the ground-state
potential energy surface but also by the involvement of in-
termediate electronically excited states in the REMPI
process.
We choose to study the H2O reorientation dynamics and
energy redistribution in the Cu(H2O) complex for several
reasons. First, both simple chemical principles and ab initio
calculations35,36predict the equilibrium geometries of the an-
ion and neutral complexes to differ significantly ?Fig. 1?.
Negative ion photoelectron spectroscopy37,38and calcula-
tions35,36agree that Cu?(H2O) is well described as a hy-
drated copper atomic anion and that it has a vertical detach-
ment energy of about 1.6 eV. Electron photodetachment us-
ing the second harmonic of the Ti:sapphire laser ?398 nm,
3.12 eV? will produce neutral Cu(H2O) primarily in its
ground electronic state. The ground state of copper has an
alkali-like electronic structure,2S1/2(3d104s1), which gives
rise to two strong2P1/2,3/2←2S1/2transitions at 327 and 325
nm, respectively. Both calculations35and related spectro-
scopic studies39–42suggest that these transition energies will
shift markedly with the H2O orientation and Cu-H2O sepa-
ration. Ionizing the complex using varying excitation ener-
gies will thus probe the cluster in different transient nuclear
configurations.35The approach we take has two components.
First, femtosecond photodetachment-photoionization experi-
ments are used to investigate the evolution of Cu(H2O) for
the first 100 ps following photodetachment of Cu?(H2O).
Second, the dynamics of Cu(H2O) are simulated by quantum
wave packet calculations: the Cu?(H2O) vibrational wave
functions populated under the experimental conditions are
propagated quantum mechanically on the neutral Cu(H2O)
surface. Details of the Cu?(H2O) and Cu(H2O) potential
energy surfaces and vibrational wave functions evaluated
from these surfaces are presented in the following paper.35
The remainder of this paper is structured as follows. We
present our experimental methods and apparatus in Sec. II. In
Sec. III, we describe the theoretical and computational meth-
odology used to calculate the time-dependent dynamics of
Cu(H2O). We present the results of our femtosecond
photodetachment-photoionization experiments on Cu(H2O)
in Sec. IV. In Sec. V, the dynamics of Cu(H2O) are discussed
and interpreted in light of the experimental observations and
the time-dependent wave packet calculations.
II. EXPERIMENT
The time-resolved photodetachment-photoionization of
Cu?(H2O) is performed using the charge-reversal instru-
ment described previously,31so only relevant details will be
given here.
A. Ion production, transport, and product analysis
A schematic of the ion-beam part of the apparatus is
presented in Fig. 2. A high-pressure pulsed sputtering dis-
charge ion source31is used to make Cu?(H2O)ncluster ions.
A mixture of 80% Ne and 20% Ar at stagnation pressures
between 6 and 8 atm flows through a small water container at
room temperature and expands into vacuum through a pulsed
?200 Hz? General Valve ?0.8 mm orifice, Parker Hannifin?. A
sputtering discharge is initiated when the gas pulse flows
FIG. 1. Schematic illustration of the Cu(H2O) dynamics upon electron pho-
todetachment of the negative ion.
FIG. 2. Schematic diagram of the ion beam portion of the photodetachment-
photoionization apparatus.
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past a negative 2–3 kV potential between a copper rod cath-
ode and a stainless steel rod at ground potential. The
Cu?(H2O) complexes are produced in the high-pressure re-
gion immediately following the discharge, inside a 3 mm
diameter source channel. They are further cooled and stabi-
lized downstream, inside the cone-shaped expansion chan-
nel, and then allowed to expand into the (0.5–1.0)?10?4
Torr pressure of the source chamber. Along with Cu?(H2O)n
ions,thesourcegenerates
CuOH?(H2O)n, O?(H2O)n, and OH?(H2O)n. In order to
avoid mass contamination of63Cu?(H2O) with65CuO?, we
study the less abundant, but isotopically pure, m/z 83 ion
packet ?65Cu?(H2O)?, and for the deuteration experiments
we employ m/z 85 ?65Cu?(D2O)?. The rotational and vibra-
tional temperatures of the anions made in this source are not
well characterized. Previous studies of large cluster ions43,44
indicated that they both range between 40 and 60 K. For
these smaller clusters, we expect both to be higher, possibly
100–200 K.
At 10 cm below the nozzle, a pulsed transverse electric
field extracts the negative ions into a differentially pumped
Wiley-McLaren45time-of-flight mass spectrometer ?Fig. 2?.
They are further accelerated to ?3 keV energy, and brought
to a spatial and temporal focus at the photodetachment re-
gion 1.8 m downstream. The fast neutrals produced by the
photodetachment pulse are detected by an in-line channeltron
detector. The cation products enter a reflectron mass spec-
trometer and are subsequently counted with an off-axis
microchannel-plate detector. This dual collector arrangement
enables the normalization of the cation signal to the neutral
signal, which is found to be very helpful for reducing the
effects of fluctuations in the negative ion intensity and flight
time.
Cun
?,CuO?(H2O)n,
B. Laser system
Most of the femtosecond laser system has been exten-
sively described.31,44A Ti:sapphire oscillator ?Coherent Mira
Basic? pumped by a Nd:VO4laser ?Coherent Verdi V5? pro-
duces ?85 fs pulses at 750–850 nm. The pulses are ampli-
fied by a regenerative, multipass Ti:sapphire amplifier
?Quantronix Titan?, pumped by a Nd:YLF laser ?Quantronix,
model 527 DQ?, to 3 mJ/pulse and 120 fs, at a repetition rate
of 400 Hz. Two-thirds of this infrared light is used to gener-
ate second harmonic ?398 nm? and third harmonic ?265 nm?
?CSK Optronics Supertripler, model 8315A? radiation, while
the remainder pumps an optical parametric amplifier ?OPA,
Light Conversion TOPAS?, whose fourth harmonic output
provides tunable radiation between 300 and 400 nm. Finally,
the OPA and the 398 nm pulses pass through delay stages
and are further combined with the 265 nm pulse in a collin-
ear configuration. All three pulses are focused to a spot of
?1 mm diameter at the point of interaction with the ion
packet, where their typical energies are 100 ?J for 398 nm,
60 ?J for 265 nm, and 10–12 ?J for the OPA.
C. Measurements performed
We investigate the dynamics of the Cu(H2O) complex
through four time-resolved photodetachment-photoionization
measurements. For all measurements reported here, the neu-
tral Cu(H2O) complex is formed by electron photodetach-
ment from Cu?(H2O) using a 398 nm pulse and the cation
signal is recorded as a function of the delay between the
photodetachment and the excitation/ionization pulses ?OPA/
265 nm?. The four ionization schemes are described below
and three are shown schematically in Fig. 3. In this figure,
the various laser pulse sequences are superimposed upon a
schematic, partial potential energy diagram46displaying the
electronic states involved in the experiments.
1. Three-color photodetachment photoionization
of CuÀ
The time resolution of the experimental apparatus is de-
termined in situ, using three-color, three-photon photode-
tachment photoionization of Cu??Fig. 3, scheme 1?,
Cu?
——→
398 nm ?t?0?
Cu
——→
327?265 nm ?t??t?
detect Cu?.
?1?
The 398 nm pulse photodetaches an electron from Cu?and a
sequence of 327 and 265 nm photons resonantly ionize the
2S1/2Cu atom through the2P1/2intermediate state. The delay
between the excitation and the ionization ?265 nm? pulses is
fixed at 0.3 ps. As the photodetachment–resonant photoion-
ization delay time (?t) is varied, the Cu?signal exhibits a
step function increase at ?t?0 ?Fig. 4? and the rise time of
this step gives a direct measurement of the time resolution of
the apparatus. The solid line in Fig. 4 is a fit of the data to the
form A?1?tanh(t/?)? and the derivative of this fit ?dashed
line? has a sech2form. The width ?full width at half maxi-
mum ?FWHM?? of this sech2function provides the effective
time resolution of the experimental apparatus, 260 fs.
FIG. 3. Schematic representation of the measurements performed, as de-
scribed in the text. Dotted lines represent cuts in the Cu-OH2C2vconfigu-
rations, while solid lines show cuts in the Cu-H2O C2vgeometry.
5678J. Chem. Phys., Vol. 121, No. 12, 22 September 2004Muntean et al.
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Page 4

2. Two-color resonant ionization of Cu arising from
dissociative photodetachment of CuÀ(H2O)
The Cu atom fragment resulting from dissociative pho-
todetachment is monitored as a function of the delay time
between the photodetachment photon ?398 nm? and the
2P1/2←2S1/2excitation of the copper atom by a 327 nm pho-
ton ?Fig. 3, scheme 2?,
Cu??H2O?
——→
t??t
398 nm ?t?0?
Cu?H2O?
——→
Cu?H2O ——→
327?265 nm
detect Cu??H2O.
?2?
The delay between the resonant excitation ?327 nm? pulse
and the ionization ?265 nm? pulse is fixed at 0.3 ps. This is
exactly the same measurement process described above, ex-
cept that the laser pulses intersect Cu?(H2O), rather than
Cu?. The major background contribution is from two-color
?327 and 265 nm? photodetachment photoionization of
Cu?(H2O) and is typically 10% of the maximum of the
three-color signal.
3. One-color, resonant multiphoton ionization
of initially bound Cu„H2O…
The bound Cu(H2O) complex is probed, as a function of
time after photodetachment, by one-color multiphoton ion-
ization using a tunable ?OPA? pulse ?Fig. 3, scheme 3?,
Cu??H2O?
——→
t??t
398 nm ?t?0?
Cu?H2O?
——→
Cu?H2O?
——→
2??315–345? nm
detect Cu??H2O?.
?3?
The excitation wavelength is tuned over the range 315–
345 nm. The major background contribution is from one-
color ?OPA? photodetachment-photoionization of Cu?, and
is typically 20% of the maximum of the two-color ?398
OPA? signal.
4. One-color dissociative multiphoton ionization of
initially bound Cu„H2O…
This measurement is the same as the one above, except
that it is a higher order multiphoton process, and the ion
detected is Cu?
Cu??H2O?
——→
t??t
398 nm ?t?0?
Cu?H2O?
——→
Cu?H2O?
——→
3??315–345? nm
detect Cu?
?H2O.
?4?
The one-color ionization used here does not allow effi-
cient multiphoton ionization of the isolated Cu atom, even
when the OPA is tuned to the 327 nm Cu atom resonance
transition. At that color, the energy of two 327 nm photons
?7.58 eV? is insufficient to ionize the isolated Cu atom ?ion-
ization potential (IP)?7.73 eV]. In contrast with the two-
color resonant ionization of the neutral Cu fragment arising
from dissociative photodetachment, the Cu?cation signal in
this case comes from dissociative photoionization of the
Cu(H2O) complex. The Cu?time dependence observed in
this process is essentially the same as that observed for the
Cu?(H2O) produced by one-color multiphoton ionization of
Cu(H2O).
D. Experimental details and data acquisition
Optimal laser-ion spatial overlap is of primary impor-
tance for our two- and three-color experiments. The optimal
overlap is best achieved through the optimization of the pho-
todetachment photoionization of Cu?in a two-step proce-
dure. First, the photodetachment ?398 nm? pulse is posi-
tioned in time and space to maximize the Cu neutral signal.
The resonant excitation ?327 nm? OPAand the ionization 265
nm pulses are introduced and adjusted spatially to maximize
the Cu?cation signal. Special care is also taken to ensure
that the laser modes are as uniform as possible in the far
field, in the region of interaction with the ions. Typically,
5%–10% of the parent negative ion beam is photodetached
by the 398 nm radiation, but a significantly smaller propor-
tion of the photodetached neutrals are themselves ionized by
the subsequent pulses.
The data acquisition procedure consists of scanning a
range of time delays between the 398 nm pulse and the fixed
OPA and 265 nm pulses, while simultaneously measuring the
total ?both intact Cu(H2O) and dissociated neutral products?
neutral signal and the individual Cu?and Cu?(H2O) prod-
uct cation signals. One single scan consists of sweeping the
entire range of time delays while data are accumulated dur-
ing 800 laser shots per time delay. Many back and forth
scans are performed for a total of 20000–40000 laser shots
per time delay. This procedure allows continuous monitoring
of the measured signals at all time delays to check for any
major drift. The final signal ?after appropriate background
corrections? is given by the cation signal divided by the neu-
tral signal. Although we do not continuously monitor the
intensities of the ionizing laser pulses during data acquisi-
tion, we periodically check their intensities and spatial posi-
tions.
FIG. 4. The three-color photodetachment photoionization of Cu?with 398,
327, and 265 nm laser pulses as a function of the time delay between the
photodetachment ?398 nm? and the resonant excitation ?327 nm? pulses. The
solid line represents a tanh fit to the data and the dashed line is the derivative
(sech2) of the fit, an in situ measurement of the effective time resolution of
the experiment.
5679
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Page 5

The maximum Cu?(H2O) and Cu?count rates are about
0.2 and 0.5 counts/laser shot, respectively. The principal
challenge of the experiment arises from maintaining stability
during the several hours of signal optimization and data ac-
quisition. It is especially challenging to maintain a stable
spatial overlap of the three laser beams and the ion packet,
due to the variations of the pulsed valve operation and to the
slight pointing drift of the three laser beams as the room
temperature changes. Background signals due to the indi-
vidual photodetachment or photoionization pulses are typi-
cally recorded at the beginning and end of data acquisition
and are subtracted from the time-dependent signals.
III. THEORY
From a theoretical standpoint, the photodetachment-
photoionization dynamics of the Cu(H2O) complex can be
considered in three parts. The first part involves the charac-
terization of Cu?(H2O) and is described in the following
paper.35In the second part, the photodetachment process will
be modeled assuming a constant transition moment and an
infinitely sharp laser pulse. As such, in this step of the cal-
culations, we take a set of vibrational wave functions for the
anion and propagate each of them on the neutral surface,
using standard time-dependent approaches.47–49Finally, the
ionization process needs to be considered. We use ab initio
calculations of the electronic excitation and ionization ener-
gies of Cu(H2O), as described in the following paper,35to
determine the configurations for which the system is most
likely to be ionized and analyze the results of the simulations
according to this model. Excitation energies are determined
for transitions between the ground electronic state and those
excited electronic states that correlate the2P1/2,3/2states of
Cu and ground-state H2O.
A. Coordinates and Hamiltonian
As the intramolecular vibrations of water are fast com-
pared to the intermolecular vibrational motions of Cu(H2O)
in all of the states that are probed by the experiment, we will
consider the potential and the dynamics in terms of the three
intermolecular coordinates only. These three coordinates are
the distance between the copper atom and the center of mass
of the water, R, and the two orientation angles of the water
molecule relative to the vector along R, ?, and ? ?see fol-
lowing paper for details?. We take as our reference geometry
the configuration in which all four atoms lie in a plane and R
lies along the C2axis of the water molecule, i.e., ??0° and
??90°. Motion along ? corresponds to rotation of the water
molecule out of the plane of the complex, while motion
along ? corresponds to rotation in the plane of the complex.
In this representation of the copper-water complex, the
Hamiltonian is given by
Hˆ??
?2
2?R
?2
?R2?2?c
?
?bxjˆx
2?byjˆy
2?bzjˆz
2?
??Jˆ?jˆ?2
2?RR2?V?R,?,??
??
?2
2?R
?2
?R2?2?c
??
bx?by
2
?jˆ2?jˆz
2?
?bx?by
4
?jˆ?
2?jˆ?
2??bzjˆz
2???Jˆ?jˆ?2
2?RR2?V?R,?,??.
?5?
Here, ?Rrepresents the reduced mass of the copper-water
complex, while the b?give the vibrationally averaged rota-
tional constants of water. A mass of 62.929599 amu was
used for Cu, 15.994915 amu for O, and 1.007825 amu for H
?2.014102 amu for D?. In the present coordinate system, bx
?27.88, by?14.51, and bz?9.28 cm?1for H2O and bx
?15.25, by?7.30, and bz?4.94 cm?1for D2O.35Finally, V
represents the intermolecular potential for this system, de-
scribed in detail in the following paper.35
B. Dynamics
In order to investigate the dynamics of the copper-water
complex, we start the system in one of the vibrational eigen-
states of the anion. We assume a constant electron photode-
tachment transition moment, and solve the time-dependent
Schro ¨dinger equation for each of the ten lowest energy vi-
brational states of Cu?(H2O) and Cu?(D2O). In all cases,
the system is propagated to 9.6 ps, through a series of 8000,
50 a.u. ?1.2 fs? time steps. In order to evaluate
??R,?,?,t??t??e?iHˆ?t/???R,?,?,t?,
?6?
we express the Hamiltonian matrix in terms of a low-order
Lanczos recursion, consisting of 50 vectors that are initiated
using ?(R,?,?,t). We solve for the eigenvalues and eigen-
vectors of this system and use them to evaluate Eq. ?6?.50,51
For these calculations, the wave functions are expressed in
an evenly spaced grid of 500 points in R, ranging from 1.03
to 12.99 Å, while the angular dependence is expanded in
spherical harmonics with j?10 for Cu(H2O) and j?15 for
Cu(D2O).
Since as much as 65% of the wave packet will reach the
asymptotic region of the potential in 10 ps, we multiply the
part of the wave function for which R?Rabs?10.34 Å by a
damping factor:52
a?R??exp??2e?2(Rmax?Rabs)/(R?Rabs)?
?7?
at each time step, where Rmax?12 Å. This choice ensures
that there will not be any reflection of the wave packet off of
the edge of the grid. As our analysis does not involve the
parts of the wave packets that have amplitude at R?8 Å, we
do not require detailed information about this portion of the
wave functions.
5680J. Chem. Phys., Vol. 121, No. 12, 22 September 2004Muntean et al.
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