JOURNAL OF CHEMICAL PHYSICS
VOLUME 112, NUMBER 13
1 APRIL 2000
On the importance of exchange effects in three-body interactions:
The lowest quartet state of Na3
J. Higgins, T. Hollebeek, J. Reho, T.-S. Ho, K. K. Lehmann, H. Rabitz, and G. Scoles
Department of Chemistry, Princeton University, Princeton, New Jersey 08544-1009
Maciej Gutowski
Materials Resources, Pacific Northwest National Laboratory, Richland, Washington 99352
and Department of Chemistry, University of Gdańsk, 80-952 Gdańsk, Poland
共Received 8 November 1999; accepted 10 January 2000兲
Three-body interactions in a homonuclear van der Waals bound trimer 共the 1 4 A 2⬘ state of Na3) are
studied spectroscopically for the first time using laser induced emission spectroscopy on a liquid
helium nanodroplet coupled with ab initio calculations. The van der Waals bound, spin polarized
sodium trimers are prepared via pickup by, and selective survival in, a beam of helium clusters.
Laser excitation from the 1 4 A ⬘2 to the 2 4 E ⬘ state, followed by dispersion of the fluorescence
emission, allows for the resolution of the structure due to the vibrational levels of the lower state and
for the gathering of precise information on the three-body interatomic potential. From previous
experiments on Na2 we know that the presence of the liquid helium perturbs the spectra by a very
small amount 关see J. Higgins et al., J. Phys. Chem. 102, 4952 共1998兲兴. Ab initio potential energy
calculations are carried out at 42 geometries of the lowest quartet state using the coupled cluster
method at the single, double, and noniterative triple excitations level 关CCSD共T兲兴. The full potential
energy surface is obtained from the ab initio points using an interpolation procedure based on a
Reproducing Kernel Hilbert Space 共RKHS兲 methodology. This surface is compared to a second,
constructed using an analytical model function for both the two-body interaction and the
nonadditivity correction. The latter is calculated as the difference between the CCSD共T兲 points and
the sum of the two-body interactions. The bound vibrational states are calculated using the two
potential energy surfaces and are compared to the experimentally determined levels. The calculated
bound levels are combined with an intensity calculation of the ␯ 2⬙ mode of E ⬘ symmetry derived
from a Jahn–Teller analysis of the excited electronic state. The calculated frequencies of ␯ 1⬙ and ␯ 2⬙
are found to be 37.1 cm⫺1 and 44.7 cm⫺1, respectively, using the RKHS potential surface while
values of 37.1 cm⫺1 and 40.8 cm⫺1 are obtained from the analytical potential. These values are
found to be in good to fair agreement with those obtained from the emission spectrum and to be
significantly different from any values calculated from additive potential energy surfaces. The 1 4 A ⬘2
Na3 potential energy surface is characterized by a D 3h symmetry minimum of ⫺850 cm⫺1 共relative
to the three 3 2 S Na atom dissociation limit兲 with a bond distance of 4.406 Å. This bond distance
differs by about 0.8 Å from the value of 5.2 Å found for the sodium triplet dimer. This means that
approximately 80% of the binding energy at the potential minimum is due to three-body effects.
This strong nonadditivity is overwhelmingly due to the deformability of the valence electron density
of the Na atoms which leads to a significant decrease of the exchange overlap energy in the trimer.
© 2000 American Institute of Physics. 关S0021-9606共00兲01813-4兴
and later by Wells and Wilson7 and by Chalasiński and
collaborators8 and confirmed, again theoretically, by the accurate quantum Monte Carlo calculations of Mohan and
Anderson on He3. 9 Recently, highly accurate calculations by
Lotrich and Szalewicz of the binding energy of solid argon
showed that, if agreement with experiment was to be
reached, nonadditive exchange contributions had to be
included.10,11 A recent, rather comprehensive, review on
many-body effects in intermolecular forces has been published by Elrod and Saykally.12
Three-body interactions in the noble gases are not easily
quantified because of two main reasons. The first is their
small size which is due, in part, to cancellations arising from
I. INTRODUCTION
Since the pioneering work of John Barker,1 the independent knowledge of realistic two-body interactions in the rare
gases has allowed an estimate of the contribution of the
three-body forces to the properties of these substances in the
condensed state. While in the earlier period it was believed
that the triple dipole dispersion term 共the so-called Axilrod–
Teller and Muto term兲 was sufficient,2 later high pressure
research led to the realization that many-body effects could
alter the exchange repulsion in an appreciable way3,4 in
agreement with an early suggestion of Jansen.5 A convincing
analysis of the importance of the three-body component of
the exchange repulsion was provided theoretically by Bulski6
0021-9606/2000/112(13)/5751/11/$17.00
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© 2000 American Institute of Physics
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J. Chem. Phys., Vol. 112, No. 13, 1 April 2000
the different signs of the main components. The total threebody effect summed over all terms is indeed reduced to be
about 5% of the total binding energy, which is why cancellation of errors have marked the evolution of this field until
not long ago.12 The second reason is of an experimental nature. It is in fact well known that the most precise information on the two-body intermolecular potential for argon has
been obtained by measuring the v.u.v. spectrum of the argon
dimer.13 However, the v.u.v. spectrum of an Ar trimer has
not yet been obtained and, if obtained, would be quite difficult to analyze. Spectra of nonsymmetric trimers such as
Ar2 – HF or Ar2 – Kr have instead been obtained in easier
regions of the electromagnetic spectrum 共IR and microwave,
respectively兲 and have been used to study three-body
forces.14–16 However, the three-body terms in nonsymmetric
trimers are much more numerous which makes their study
substantially more complicated.17
Atoms with a low ionization potential such as those that
fill the first two columns of the periodic table offer two main
advantages for these types of studies. In the first place they
are much more compressible than the noble gases, showing,
therefore, much larger exchange effects. Furthermore, their
spectra are located in a very convenient region of the electromagnetic spectrum. Not by chance the first theoretical evidence on the existence of three body exchange effects was
gathered by Bulski on Be3. 6 On the other hand, while Be3
and Mg3 have at least partially closed electronic shells, Li3 ,
Na3, etc. can be considered van der Waals systems only in
their lowest quartet 共spin polarized兲 states which are normally next to impossible to prepare even in a molecular
beam environment. However, quite recently, in extending the
spectroscopic use of large helium clusters18,19 to alkali atoms, we have shown how this new kind of matrix spectroscopy introduces very small perturbations and can be used to
prepare and characterize very unstable species such as alkali
triplet dimers20 and quartet trimers.21 In the latter paper,
upon excitation of the 2 4 E ⬘ electronic state of Na3, laser
induced fluorescence excitation and emission spectra were
obtained, confirming that, in this highly deformable, one valence electron atom, nonadditivity effects are very important.
In the present paper, the data obtained in the experiment
described in Ref. 21 are presented in a more extensive way
and the potential used in their analysis is improved by increasing the number of points and geometries at which high
level ab initio calculations are performed, and by using a
general, multidimensional, reproducing kernel interpolation
procedure. In this way very precise information on the threebody interactions is obtained, making this symmetric trimer a
benchmark system in the study of nonadditivity effects.
II. THE 2 4 E ⬘ \1 4 A 2⬘ EMISSION SPECTRUM OF Na3
The experimental apparatus used to obtain spectroscopic
information on the lowest quartet state of Na3 has been described before.21,22 Here we shall only briefly sketch its main
characteristics. Helium clusters ranging in average size up to
104 atoms per cluster are prepared in an expansion of helium
gas through a 10 ␮m nozzle at a temperature of 17.5 K and
a stagnation pressure of 5.4 MPa. A short distance after the
Higgins et al.
FIG. 1. The solid line is the emission spectrum of the 2 4 E ⬘ →1 4 A 2⬘ transition of Na3 on a helium cluster surface. The dashed line is the excitation
spectrum obtained by collecting the total fluorescence. The excited vibronic
levels are marked with arrows. Experimental conditions were 53.4 bar He
stagnation pressure and 17.5 K nozzle temperature 共11 000 atoms/cluster
before pickup兲.
beam is collimated by a 400 ␮m skimmer the clusters cross
a pickup cell that is connected to an alkali reservoir. A few
centimeters downstream, the doped clusters cross a tunable
continuous wave dye laser beam at the center of a two mirror
laser induced fluorescence 共LIF兲 collector. Emission spectra
were obtained using a monochromator coupled to a liquid
nitrogen cooled charge coupled detector. To obtain the emission spectrum of the 2 4 E ⬘ ←1 4 A 2⬘ transition the dye laser
frequency was positioned at 15 817.8 cm⫺1 on the strongest
band of the excitation spectrum21 which is shown as a dashed
line in Fig. 1.
The resulting emission spectrum is shown as a continuous line in Fig. 1. The strongest fluorescence occurs at about
15 792 cm⫺1, which is 35 cm⫺1 to the red of the excitation.
Since the laser is exciting the 0-0 band of a pure electronic
transition of Na3, the red shift must result from the inelastic
energy transfer to the elementary excitations of the helium
cluster and to the solvent reorganization. The full width at
half maximum 共FWHM兲 of the 0-0 emission band 共approximately 10 cm⫺1兲 is a consequence of the surface location of
the trimer on the helium cluster as electronic chromophores
excited inside the liquid droplets have typically much
broader excitation spectra.
Emission spectra were also acquired at excitation energies above the 0-0 band. Figure 2 displays the resonant emission measured upon excitation of the second, third, fourth,
and sixth bands of the excitation spectrum. Jahn–Teller
analysis of the excited state allows precise quantum numbers
to be assigned to these transitions 共see Fig. 1兲. As can be
seen from the spectra, the emission is in all cases dominated
by transitions arising from the lowest vibronic level of the
excited state which produces a spectrum similar to that obtained upon excitation to the 0-0 level. Indeed, the 0-0 band
near 15 791 cm⫺1 is the most prominent feature in all the
spectra. Some emission from the v ⫽1 – 3 vibronic levels in
the excited state is observed at higher frequencies than the
0-0 transition, but this only constitutes a minor component of
J. Chem. Phys., Vol. 112, No. 13, 1 April 2000
Exchange effects in three-body interactions
5753
TABLE I. Peak positions of the resonant fluorescence channel of the
2 4 E ⬘ →1 4 A ⬘2 transition of Na3 excited at different ( v , j) vibronic levels.
Peak number
in spectrum
1
2
3
4
5
6
7
8
9
10
11
Position
共cm⫺1兲
Excitation at first band 共0,0兲
15 791.39
15 750.74
15 710.30
15 667.45
15 627.44
15 593.68
15 561.78
15 522.30
15 490.69
15 450.73
15 421.95
Difference
共cm⫺1兲
40.65
40.44
42.85
40.01
33.76
31.90
39.48
31.61
39.96
28.78
Excitation at second band 共1,1/2兲 共15 866.0 cm⫺1兲
1
15 868.84
50.83
2
15 818.01
23.97
3
15 794.05
40.66
4
15 753.39
41.33
5
15 712.06
41.99
6
15 670.07
FIG. 2. Resonant emission spectra obtained by exciting different vibronic
levels of the 2 4 E ⬘ →1 4 A ⬘2 transition of Na3. The quantum numbers v , j of
the excited 2 4 E ⬘ state are given in the upper right hand corner of each
panel. The laser excitation frequency is marked by an asterisk. The labeling
of the peaks in the spectra are given as the excited v of the 2 4 E ⬘ state and
the vibrational levels of the lower 1 4 A 2⬘ state.
the emission. These spectra reveal the presence of vibrational
cooling in the excited quartet state similarly to what occurs
in the spectroscopy of the alkali dimers on the surface of
helium clusters.20 Vibrational cooling must occur on a relatively fast time scale since most of the population in the
excited state relaxes to the lowest vibronic state within the
fluorescence lifetime of the excited quartet state which has
been determined to be less than 1.5 ns using time resolved
spectroscopy.23 The positions of the emission spectra bands
are given in Table I.
The 1 4 A ⬘2 state of Na3 has three fundamental vibrational
modes, a symmetric stretch of A 1⬘ symmetry and a doubly
degenerate mode of E ⬘ symmetry. Without vibronic coupling, fluorescence emission from levels in the 2 4 E ⬘ state
could lead into any vibrational level of the symmetric stretch
of the 1 4 A ⬘2 state, but only into even numbered quanta of the
degenerate mode. Due to Jahn–Teller interaction in the
2 4 E ⬘ state, decay into all vibrational states becomes allowed.
As mentioned above, the emission spectra are dominated
by fluorescence originating from the v ⫽0 level of the excited state due to vibrational cooling. Hence, the analysis
will focus only on the data obtained by exciting the v ⫽0
level of the 2 4 E ⬘ state. The line positions provide accurate
spectroscopic data on the vibrational levels of the 1 4 A 2⬘ state
of Na3 which will be used to test the accuracy of ab initio
potential energy surfaces calculated for the lowest quartet
Excitation at third band 共2,5/2兲 共15 932.6 cm⫺1兲
1
15 897.52
32.26
2
15 865.26
47.25
3
15 818.01
23.08
4
15 794.93
41.55
5
15 753.39
37.82
6
15 715.56
1
2
3
Excitation at fourth band 共2,1/2兲 15 955.2 cm⫺1
15 865.26
47.25
15 818.01
23.97
15 794.05
1
2
3
4
Excitation at sixth band 共3,1/2兲 16 038.0 cm⫺1
15 906.51
40.35
15 866.16
47.26
15 818.90
23.10
15 795.82
state of the sodium trimer assessing the importance of the
nonadditive terms. The bound states of the potential energy
surfaces will be calculated and compared to the vibrational
levels obtained from the emission spectrum.
III. THE POTENTIAL ENERGY SURFACE OF Na3 IN
THE LOWEST QUARTET STATE „1 4 A 2⬘ …
In this section, two global potential energy surfaces for
the lowest quartet state 1 4 A 2⬘ of the Na3 trimer are introduced. One is based on a general multidimensional reproducing kernel 共RK兲 interpolation procedure,24–28 while the other
is based on a fit to a Hatree–Fock plus Damped Dispersion
共HFD兲 analytic form for the two-body interaction and to a
damped triple exponential for the nonadditive three-body interaction. Both employ a standard many-body expansion ansatz based on high level ab initio calculations at the coupled
cluster level of theory with single, double, and noniterative
triple excitations 关CCSD共T兲兴.29 While the three-body
CCSD共T兲 interaction has been obtained as described in the
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Higgins et al.
J. Chem. Phys., Vol. 112, No. 13, 1 April 2000
TABLE II. Calculated CCSD共T兲 potential of the 共a兲 1 3 ⌺ ⫹
u state of Na2.
R 共Å兲
V(R) 共cm⫺1兲
R 共Å兲
V(R) 共cm⫺1兲
2.0
2.5
3.0
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
16 304.56
7292.56
2978.40
1289.37
1019.36
793.37
604.68
447.59
317.40
210.11
122.37
51.22
⫺6.0
⫺51.57
⫺87.52
⫺115.51
⫺136.89
4.8
4.9
5.0
5.0911
5.2
5.3
5.4
5.7
6.1
6.6
7.1
8.1
9.1
10.1
11.1
12.1
⫺152.73
⫺163.88
⫺171.09
⫺174.76
⫺176.17
⫺175.15
⫺172.38
⫺157.21
⫺130.14
⫺96.87
⫺69.61
⫺34.47
⫺17.18
⫺8.95
⫺4.90
⫺2.84
with this basis set differ by less than 0.03 Å and 3 cm⫺1,
respectively, from our most accurate results reported for the
dimer in Ref. 30.
Our ab initio calculations for the trimer, shown in Table
III, were performed at 42 carefully selected geometries
which cover the weakly attractive region, the van der Waals
minimum, and the repulsive region of the potential energy
surface. We thoroughly covered the equilateral triangle geometry 共18 D 3h points兲, but we also included numerous bent
structures 共16 C 2 v points兲, linear structures 共4 D ⬁h points兲,
and low symmetry structures 共4 C s points兲.
B. Reproducing Kernel Hilbert Space „RKHS…
interpolation surface
This interpolation procedure is based on a recently formulated one-dimensional 共1-D兲 distancelike reproducing kernel that possesses a correct asymptotic inverse power form:
⫺m⫺1
q n,m 共 x,x ⬘ 兲 ⫽n 2 B 共 m⫹1,n 兲 x ⬎
2F 1
next section, the two-body lowest triplet Na–Na interaction,
based on a larger basis set, has recently been published by
one of us30 and is reported here in Table II.
A. Ab initio calculation of the three-body interaction
The full three-body interaction was calculated at the
coupled cluster level of theory with single, double, and noniterative triple excitations 关CCSD共T兲兴.29,31 All electrons were
correlated with the exception of those which occupy the
atomic 1s levels. First, the total interaction energy was determined for the sodium trimer by subtracting atomic energies E at from the trimer energy E trim
E int共 r 12 ,r 23 ,r 31兲 ⫽E trim共 r 12 ,r 23 ,r 31兲
兺 E iat共 r 12 ,r 23 ,r 31兲 ,
i⫽1
共4兲
where n and m are integers with n⭓1 and m⭓0, x ⬎ , x ⬍ are,
respectively, the larger and smaller of x and x ⬘ , B is the Beta
function, and 2 F 1 is the Gauss’ hypergeometric function. It is
instructive to note that the reproducing kernel q n,m (x ⬍ ,x ⬎ )
⫺(m⫹k⫹1)
is a finite sum of inverse powers x ⬎
, k⫽0,...,(n
⫺1). This property enables one to correctly mimic the
asymptotic forms of both the two- and three-body interactions as described below.
The total potential energy surface V(r 12 ,r 23 ,r 31) of the
lowest quartet state 1 4 A 2⬘ of Na3 can be written as a manybody expansion:
共1兲
where r i j is the interatomic distance between the ith and jth
atoms and E at depends on r i j ’s through the function counterpoise procedure, which is applied to ensure basis set consistency between the trimer and monomer calculations.32–38
Next, the three-body term ⑀ was extracted from E int by subtracting the pairwise interactions ⑀ i j
3
⑀ ⫽E int⫺ 兺 ⑀ i j ,
共2兲
ij
⑀ i j ⫽E dim
⫺E iat⫺E atj
共3兲
i⬍ j
冊
x⬍
,
x⬎
V 共 r 12 ,r 23 ,r 31兲 ⫽V 1 ⫹V 2 共 r 12兲 ⫹V 2 共 r 23兲 ⫹V 2 共 r 31兲
3
⫺
冉
⫻ ⫺n⫹1,m⫹1;n⫹m⫹1;
with every term on the right-hand side of Eq. 共3兲 being also
calculated using the basis set of the trimer.
We have used Sadlej’s medium-size polarized basis set,
which was originally designed for calculations of molecular
electric properties,39 modifying it to make it more suitable
for calculations of this weak intermolecular potential. First,
we uncontracted the more diffuse set of d functions. Second,
we added one f and one g diffuse functions with exponents of
0.0641 and 0.079 36 a.u., respectively, which were obtained
30
by optimizing the dispersion interaction of the 3 ⌺ ⫹
u dimer.
The CCSD共T兲 values of R e and D e for the dimer obtained
⫹V 3 共 r 12 ,r 23 ,r 31兲 ,
共5兲
where V 1 , V 2 , and V 3 are, respectively, the one-, two-, and
three-body terms. For simplicity, the one-body term, which
may be assigned the value of the dissociation energy in the
three atom limit Na⫹Na⫹Na is taken to be zero. The twobody terms, V 2 , where R⫽r 12 , r 23 , or r 31 , may be expressed as:
N
V 2共 R 兲 ⫽
兺i ␣ 共i 2 兲q n,m共 ␳ i , ␳ 兲
共6兲
in terms of the 1-D distancelike reproducing kernel
q n,m ( ␳ , ␳ ⬘ ). Here the reduced internuclear distance ␳ is defined as
␳⫽
冉冊
R
S
2
共7兲
with S being an arbitrary positive scaling factor chosen to
keep the expansion coefficients ␣ i small. The index i runs
over the ab initio data points. The two-body form has the
following asymptotic form at long distances:
n⫺1
lim V 2 共 R 兲 ⫽⫺
R→⬁
兺
k⫽0
C 2 共 k⫹m⫹1 兲
,
R 2 共 k⫹m⫹1 兲
共8兲
J. Chem. Phys., Vol. 112, No. 13, 1 April 2000
Exchange effects in three-body interactions
5755
TABLE III. The CCSD共T兲 ab initio values of the three-body and total potential for the lowest quartet state
1 4 A ⬘2 of the trimer Na3.
Geometry
r 12 共Å兲
r 23 共Å兲
r 31 共Å兲
V tot 共cm⫺1兲
D 3h
2.0
2.5
3.0
3.6
3.9
4.2
4.3
4.4
4.5
4.6
4.8
5.0911
5.7
6.1
6.6
7.1
7.6
8.1
2.0
2.5
3.0
3.6
3.9
4.2
4.3
4.4
4.5
4.6
4.8
5.0911
5.7
6.1
6.6
7.1
7.6
8.1
2.0
2.5
3.0
3.6
3.9
4.2
4.3
4.4
4.5
4.6
4.8
5.0911
5.7
6.1
6.6
7.1
7.6
8.1
⫺8876.46
⫺6718.70
⫺4510.49
⫺2328.16
⫺1545.93
⫺972.61
⫺824.24
⫺694.64
⫺582.46
⫺485.86
⫺333.47
⫺186.20
⫺47.59
⫺16.69
⫺2.99
0.50
1.12
0.92
41550.46
15675.49
4631.05
113.57
⫺564.26
⫺800.20
⫺826.20
⫺835.87
⫺833.59
⫺822.36
⫺783.08
⫺703.10
⫺516.13
⫺406.53
⫺293.92
⫺208.18
⫺145.43
⫺101.23
C 2v
2.4625
4.6281
6.5254
3.0098
5.6565
7.9755
3.4825
6.5450
9.2282
3.8990
7.3278
10.3319
4.1726
7.8420
4.51
8.4848
3.6
3.6
3.6
4.4
4.4
4.4
5.0911
5.0911
5.0911
5.7
5.7
5.7
6.1
6.1
6.6
6.6
3.6
3.6
3.6
4.4
4.4
4.4
5.0911
5.0911
5.0911
5.7
5.7
5.7
6.1
6.1
6.6
6.6
⫺3481.83
⫺1384.35
⫺253.02
⫺1330.77
⫺300.46
⫺67.29
⫺464.16
⫺60.38
⫺21.51
⫺155.99
⫺10.46
⫺7.45
⫺69.44
⫺1.91
⫺21.50
0.95
6102.36
126.20
1272.74
1570.54
⫺551.99
⫺199.67
286.91
⫺505.29
⫺381.94
⫺138.36
⫺382.50
⫺327.67
⫺253.06
⫺302.88
⫺303.51
⫺219.04
D ⬁h
7.2
8.8
10.1822
11.4
3.6
4.4
5.0911
5.7
3.6
4.4
5.0911
5.7
⫺66.85
⫺62.29
⫺27.24
⫺10.97
1493.05
⫺178.78
⫺380.32
⫺327.40
4.558 784
4.538 180
4.677 665
4.818 342
4.323 550
4.402 865
4.411 442
4.425 702
⫺691.04
⫺691.04
⫺680.59
⫺663.29
⫺816.67
⫺815.42
⫺753.73
⫺648.48
Cs
4.322 079
4.263 256
4.128 087
3.994 653
M
where the C coefficients take on the form:
V 3 共 r兲 ⫽
n 2 B 共 m⫹1,n 兲共 ⫺n⫹1 兲 k 共 m⫹1 兲 k
C 2 共 k⫹m⫹1 兲 ⫽⫺
共 n⫹m⫹1 兲 k k!
2 共 m⫹1 兲
兺 ␣ 共i 2 兲R 2ki ,
i⫽1
兺
j⫽1
␣ 共j 3 兲
再
1
P j Q n,m 共 r j ,r兲
3! 兵 123其 兵 123其
兺
冎
共10兲
in terms of a 3-D reproducing kernel defined as a tensor
product:
N
⫻S
V 3 共cm⫺1兲
共9兲
Q n,m 共 r,r⬘ 兲 ⫽q n,m 共 x,x ⬘ 兲 q n,m 共 y,y ⬘ 兲 q n,m 共 z,z ⬘ 兲
共11兲
with reduced bond distances
where (a) k is the Pochammer’s symbol. In the numerical
implementation, we set n⫽2 and m⫽2 to yield a smooth
interpolation through the ab initio data points and to correctly account for the two leading long-range dispersion interactions ⫺C 6 /R 6 and ⫺C 8 /R 8 .
The three-body term V 3 (r), where r⫽ 兵 r 12 ,r 23 ,r 31其 , can
be expressed as
x⫽r 12 /S, y⫽r 23 /S, z⫽r 31 /S.
共12兲
The index j runs over the three-body ab initio data.
P 兵j 123其 is the permutation operator with respect to an arbitrary
permutation of the three integers 1, 2, and 3. The summation
over the subscript 兵123其 is performed over all 3! symmetry
permutations of the trimer bond lengths associated with the
5756
Higgins et al.
J. Chem. Phys., Vol. 112, No. 13, 1 April 2000
FIG. 3. RKHS potential energy curve V 2 (R) of the Na2 共a兲 1 3 ⌺ ⫹
u state,
obtained by interpolating, using the reproducing kernel expansion in Eq. 共6兲,
high-level 共basis 1兲 CCSD共T兲 ab initio data 共see Table II兲. The potential can
be characterized by: R e ⫽5.20 Å, D e ⫽176.17 cm⫺1, ␴ ⫽4.29 Å, C 6
⫽7.6477⫻106 Å 6 , and C 8 ⫽1.8527⫻108 cm⫺1 Å 8 .
jth data point of the potential surface. V 3 is then explicitly
symmetric with respect to the exchange of any two sodium
atoms. In the calculation, n is set to 2 for a smooth interpolation and m⫽2 is chosen such that the leading three-body
term asymptotically behaves as:
V 3 共 r 12 ,r 23 ,r 31兲 ⬇
1
共13兲
3 3 3
r 12
r 23r 31
in line with the behavior of the lowest-order long-range triple
and ␣ (3)
dipole interaction. The expansion coefficients ␣ (2)
i
j
can be obtained by solving the following linear algebraic
equations:
N
V 2共 R k 兲 ⫽
兺i ␣ 共i 2 兲q n,m共 ␳ i , ␳ k 兲 ,
M
V 3 共 rk 兲 ⫽
兺 ␣ 共j 3 兲
j⫽1
再
共14兲
冎
1
P j Q n,m 共 r j ,rk 兲 ,
3! 兵 123其 兵 123其
兺
共15兲
where V 2 (R k ) and V 3 (rk ) are the two- and three-body
CCSD共T兲 ab initio data tabulated in Tables II and III.
The two-body interaction V 2 (R) of the Na2 a 3 ⌺ ⫹
u state
obtained from interpolating the CCSD共T兲 ab initio calculations 共Table II兲 using Eq. 共6兲 is reported in Fig. 3. This new
potential energy curve possesses a well depth D e
⫽176.17 cm⫺1 at the equilibrium position R e ⫽5.20 Å and
crosses the zero value at the distance ␴ ⫽4.29 Å. Moreover,
the calculated C 6 ⫽7.65⫻106 cm⫺1 Å 6 and C 8 ⫽1.85
⫻108 cm⫺1 Å 8 关cf., Eq. 共9兲兴 agree well with the values derived in Ref. 40 fitting the outer part of the RKR results of
7.89⫻106 cm⫺1 Å 6 and 2.12⫻108 cm⫺1 Å 8 , respectively.
The three-body interaction V 3 (r) for the lowest quartet
state (1 4 A ⬘2 ) of Na3 was obtained from interpolating the
FIG. 4. A view of the nonadditivity of the interaction energy in the lowest
quartet state 1 4 A 2⬘ of Na3, as a function of bond length in the D 3h geometries. The solid curve is the nonadditivity of the RKHS potential while the
dashed curve is calculated at the HF-SCF level.
CCSD共T兲 ab initio calculations 共Table III兲 using Eq. 共10兲. In
Fig. 4 the nonadditivity is displayed as a function of the bond
length R at the D 3h geometry showing a very rapid drop in
magnitude toward zero beyond R⫽7.0 Å. The CCSD共T兲 calculations indicate that there may be a very small barrier at
R⫽7.6 Å 共see Table III兲 which is expected, as the longrange Axilrod–Teller–Muto 共ATM兲 triple dipole
nonadditivity41,42 in the equilateral geometry is repulsive
while the dominating term at shorter distance is strongly attractive.
Figure 5 shows the contour plot of the total RKHS potential V(r 12 ,r 23 ,r 31) of the lowest quartet state 1 4 A ⬘2 of the
Na3 trimer as function of two bond lengths r 12 and r 31 , with
the angle between these two bond lengths fixed at 60 deg.43
The resulting Na3 potential energy surface has a global minimum value of ⫺849.37 cm⫺1 at the D 3h geometry with the
bond distance R⫽4.406 Å, which is much smaller than the
potential minimum position R e ⫽5.2 Å of the corresponding
Na2 dimer potential curve 共see Fig. 3兲. A detailed comparison of the two- and three-body contributions to the total
potential energy surface V(r), Eq. 共5兲, in the D 3h geometry
is given in Fig. 6. It is found that about 80% of the total
potential energy around the potential minimum (r 12⫽r 23
⫽r 31⫽4.4 Å) comes from three-body contributions. In general, the two-body contribution dominates at geometries with
small 共e.g., ⬍3.6 Å兲 and large 共e.g., ⬎4.8 Å兲 interatomic
distances, whereas the three-body term is more important in
the intermediate region that includes the potential minimum.
C. The Hartree–Fock Damped Dispersion „HFD… plus
CCSD„T… surface
In our second potential energy surface the total interaction energy for the trimer was represented analytically as a
sum of the two-body and three-body terms. The two-body
J. Chem. Phys., Vol. 112, No. 13, 1 April 2000
Exchange effects in three-body interactions
⌬E CORR共 R 兲 ⫽⫺
冋兺
n
册
C ng n共 ␳ R 兲
f 共 ␳R 兲,
Rn
2 R 2 / 冑n 兲 n
兴 ,
f 共 ␳ R 兲 ⫽1⫺ ␳ R 1.68e ⫺0.78␳ R .
FIG. 5. Contour plot of the total RKHS potential energy surface V(r) of the
lowest quartet state 1 4 A 2⬘ of Na3, as a function of the two bond distances
with included angle equal to 60 degrees. The corresponding
HFD⫹CCSD共T兲 potential energy surface is not shown here because it only
slightly differs from its RKHS counterpart. The total RKHS potential energy
surface possesses a minimum value of ⫺849.37 cm⫺1 at r 12⫽r 23⫽r 31
⫽4.406 Å. The contours start at the dotted line 共corresponding to ⫺100
cm⫺1兲 at the upper right corner, then decrease by 100 cm⫺1 per line until the
innermost circle. Solid contour lines start at zero 共right next to the outer
most dotted line兲 with an increment of 100 cm⫺1 per line.
共18兲
共19兲
n⫽3, 6, 8, 10, . . . ,
g n 共 ␳ R 兲 ⫽ 关 1⫺e 共 ⫺2.1␳ R/n 兲 ⫺ 共 0.109␳
5757
共20兲
共21兲
The universal damping functions f and g scale with ␳,
which was determined by using the ionization potential of
Na. The long-range dispersion coefficients (C6, C8, C10) calculated by Marinescu et al.45 for Na(3 2 S)⫹Na(3 2 S) were
used. The HF-SCF energy was calculated with the GAUSSIAN
46
92 suite of programs using the 6-311G** basis set. No
further fitting was performed as in this way, the resulting
potential energy curve is very similar to both the RKR curve
and the ab initio CCSD共T兲 potential.
The HFD two-body functional form was used to define
the nonadditive potential surface V 3 via the relationship
total
V 3 共 r 12 ,r 23 ,r 31兲 ⫽V CCSD共T兲
共 r 12 ,r 23 ,r 31兲
⫺ 兵 V 2 共 r 12兲 ⫹V 2 共 r 23兲 ⫹V 2 共 r 31兲 其 . 共22兲
The so-obtained ab initio three-body data were fit to an
empirical formula47,48 that approaches zero at long range:
V 3 共 r 12 ,r 23 ,r 31兲 ⫽A 3 f 共 r 12兲 e ⫺ ␥ r 12 f 共 r 23兲 e ⫺ ␥ r 23 f 共 r 31兲 e ⫺ ␥ r 31,
共23兲
where
interaction energy between two sodium atoms (V 2 ) with parallel spins was represented by the Hartree–Fock Damped
Dispersion 共HFD兲 model suggested by Douketis et al.44 According to this model, the total interaction energy between
the two atoms (V 2 (r)) is split into a contribution calculated
at the Hartree–Fock self-consistent field level (⌬E SCF ,
which ignores electron correlation兲 and a damped multipole
expansion which accounts for dispersion interactions
(⌬E CORR).
V 2 共 R 兲 ⫽⌬E SCF共 R 兲 ⫹⌬E CORR共 R 兲 ,
⌬E SCF共 R 兲 ⫽A 1
R ␣ 1 e ⫺ ␤ 1 R ⫹A
2
R ␣2e ⫺␤2R,
共16兲
共17兲
FIG. 6. Comparison of the total RKHS potential energy surface of the 1 4 A ⬘2
state of Na3 with its two-body contribution in the D 3h geometry. Solid
curve: the total potential surface, dotted curve: the two-body contribution,
and dashed curve: the potential energy curve of the 共a兲 1 3 ⌺ ⫹
u state of Na2.
f 共 r 兲 ⫽1⫺r ␩ e ⫺␭r
共24兲
is a damping function used to reproduce the correct form of
the nonadditivity at short internuclear separations. A 3 , ␥, ␩,
and ␭ are adjustable parameters. The final HFD⫹CCSD共T兲
surface is obtained by summing the fitted two-body HFD
terms 关cf., Eq. 共16兲兴 and the fitted three-body surface of Eq.
共23兲. It is found that two fitted HFD⫹CCSD共T兲 surfaces using 共a兲 only the D 3h data and 共b兲 using all of the available
data differ only slightly from each other. The result based on
the D 3h data is a potential whose binding energy, relative to
three free ground state Na atoms, is ⫺835 cm⫺1 at an internuclear separation of 4.4 Å. The parameters of this potential
surface of Na3 are listed in Table IV. The HFD⫹CCSD共T兲
potential surface along the symmetric stretch coordinate behaves very similarly to the RKHS surface which is displayed
in Fig. 5.
We note that the HFD⫹CCSD共T兲 surface does not contain a three-body long-range component of the ATM type.
The fact that, as we will see later, this approximation leads to
reasonably good agreement with experiment is due to the
fact that the dominating three-body nonadditivity is already
present at the self-consistent field 共SCF兲 level of theory
where no electron correlation is included 共see Fig. 4兲. The
nonadditivity of the potential energy is the result of a decrease in the overlap repulsion in the trimer as the third sodium atom 共with a parallel electron spin兲 approaches the triplet dimer. Since the valence electron clouds of the sodium
atoms are highly deformable, the deformation induced by the
presence of a third sodium atom is significant. This deformation allows the contraction of the bond distance in the trimer
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Higgins et al.
J. Chem. Phys., Vol. 112, No. 13, 1 April 2000
TABLE IV. Parameters of the HFD potential energy surface of the 共a兲
4
1 3⌺ ⫹
u state of Na2 and of the HFD⫹CCSD共T兲 potential of the 1 A ⬘
2 state of
Na3. Values are given for r in Å and V in cm⫺1.
Na2HFD
Parameter
Value
A1
␤1
␣1
A2
␤2
␣2
C6
C8
C 10
␳
De
Re
␴
1.010⫻106
5.7348
4.3736
9.6005⫻104
1.7342
1.7861
7.0789⫻106
1.509 85⫻108
4.181 65⫻109
0.5261
⫺178.13
5.078
4.192
Na3 HFD⫹CCSD共T兲
⫺1.3082⫻106
0.5869
25.575
9.8821
A3
␥
␩
␭
and is accompanied by the consequent increase in binding
energy. This increase is responsible for the relatively large
difference between the additive two-body contribution to the
binding energy of the trimer and the actual total binding
energy including nonadditive effects. The RKHS method
produces a potential surface that differs slightly from the
HFD⫹CCSD共T兲 surface and can better describe the surface
over a wider range of geometries.
D. Bound-state calculations
The vibrational levels of the 1 4 A ⬘2 state were calculated
using the TRIATOM suite of programs.49 This program uses a
variational approach utilizing a finite basis of Morse oscillators. The Hamiltonian for the triatomic system was set up
using body-fixed coordinates (r 1 ,r 2 , ␪ ) where r 1 , r 2 are two
bond lengths of the trimer and ␪ is the included angle. The
angular basis functions are symmetrized rotation matrix elements:
兩 j,k,p 典 ⫽
1
&
关 ⌰ jk 兩 JM K 典 ⫹ 共 ⫺1 兲 p ⌰ jk 兩 JM ⫺K 典 ],
共25兲
where ⌰ is an associated Legendre polynomial and p is the
parity. Motion in the radial coordinate is carried out using a
Morse oscillator basis set
兩 n 典 ⫽ 冑␤ N n ␣ L ␣n 共 y 兲 e 共 ⫺1/2兲 y y 共 ␣ ⫹1兲/2,
共26兲
where y⫽Ae ⫺ ␤ (r⫺r e ) , A⫽4D e / ␤ , ␤ ⫽ ␻ e ( ␮ /2D e ) 1/2, and
␣ ⫽integer(A) are the Morse parameters and N n ␣ L n (y) is an
associated LaGuerre polynomial. The total wave function for
the lth energy level becomes:
⌿ l⫽
d kJljmn 兩 jk 典 兩 m 典 兩 n 典 .
兺k 兺
jmn
共27兲
FIG. 7. Calculated vibrational levels of the ab initio potential energy surfaces of the 1 4 A 2⬘ state of Na3. The top panel displays the experimental
emission spectrum with the energy referenced to the ground vibrational
level of the 1 4 A 2⬘ state. The middle panel is the vibrational levels calculated
using the RKHS potential surface obtained by fitting the CCSD共T兲 ab initio
data with the RKHS method. The lower panel are the levels calculated with
the HFD⫹CCSD共T兲 potential surface constructed of the HFD two-body
interaction and a damped triple exponential fit to the nonadditivity.
A Morse oscillator basis set consisting of 3000 functions
was used and the Hamiltonian was diagonalized to give the
corresponding eigenvalues. The comparison of the energy
levels calculated with the HFD⫹CCSD共T兲 three-body potential surface, the RKHS surface, and the experimental emission spectrum is given in Fig. 7. The first peak in the emission spectrum is the 0-0 band and is set as the zero of energy.
The second prominent band is composed of two vibronic
levels, 1 ␯ 1 and 1 ␯ 2 . In the absence of three-body forces, a
symmetry analysis of the normal modes reveals that the
asymmetric stretch ␯ 2 should be exactly 1/2 of the frequency
of the symmetric stretch ␯ 1 . In the case of quartet Na3, the
three-body forces constitute such a large part of the binding
energy that the frequency of ␯ 2 actually becomes larger than
that of ␯ 1 . The frequency of ␯ 2 therefore becomes a sensitive probe of the magnitude of the three-body forces present
in the molecule. By calculating the energy of the two fundamental modes using a number of different potential energy
surfaces, it is found that there is a direct correlation between
the magnitude of the nonadditive component of the potential
and the frequency of ␯ 2 . As the nonadditivity increases, the
relative ratio of ␯ 2 to ␯ 1 increases. The calculated vibrational
frequencies of the two normal modes are ␯ 1 ⫽37.1 cm⫺1 for
the symmetric stretch and ␯ 2 ⫽40.8 cm⫺1 for the degenerate
asymmetric stretch using the HFD⫹CCSD共T兲 surface, while
the RKHS potential surface yields values of ␯ 1 ⫽37.1 cm⫺1
and ␯ 2 ⫽44.7 cm⫺1.
J. Chem. Phys., Vol. 112, No. 13, 1 April 2000
Exchange effects in three-body interactions
5759
TABLE V. Calculated vibrational levels of the 1 4 A 2⬘ state of Na3. The two
potential surfaces described in the text were utilized as input for the variational calculation of the vibrational levels.
Vibrational
Level
HFD two-body
⫹CC three-body
共cm⫺1兲
Reproducing kernel fit to
CC potential surface
共cm⫺1兲
共0,0兲
共1,0兲
共0,1兲
共2,0兲
共1,1兲
共0,2兲
共3,0兲
共2,1兲
共1,2兲
共0,3兲
共4,0兲
共3,1兲
共2,2兲
共1,3兲
共0,4兲
0.00
37.1
40.8
73.3
76.4
80.3
108.7
111.1
114.6
118.6
143.4
145.9
148.9
153.0
155.1
0.00
37.1
44.7
73.2
80.0
88.1
108.4
114.5
121.8
130.0
143.2
149.0
156.9
162.9
170.9
As the experimental line shape is uncertain we cannot
carry out a quantitative comparison between theory and experiment, however, the vibrational frequencies calculated
from both surfaces appear to be in good agreement with the
measured emission spectrum. Table V lists the calculated
vibrational levels of the 1 4 A 2⬘ state of Na3 using the two
potential surfaces described previously. The notation
( ␯ 1 , ␯ 2 ) is used to denote the quantum numbers of the vibrational levels. All energies are referenced to the 共0, 0兲 level of
the lowest quartet state. Incomplete convergence of the
bound state calculation produces residual splitting of the two
components of the asymmetric E ⬘ mode. The splitting does
not allow accurate assignment of quantum numbers to calculated levels 200 cm⫺1 above the 0-0 level. The experiment
also reveals congestion in the emission spectrum below
15 600 cm⫺1. Below this energy, an average was taken of the
energies of the components of the E mode. Above the 共0, 4兲
level, all calculated vibrational levels are displayed in Fig. 7.
Overall, the spectrum would allow, in principle, the determination the vibrational levels up to 700 cm⫺1 in energy which
accounts for ⬃90% of the binding energy of the quartet trimer 关 D 0 ⬇775 cm⫺1 using the HFD⫹CCSD共T兲 surface, the
RKHS surface gives D 0 ⬇783 cm⫺1]. However, to exploit
this opportunity, a calculation of comparable quality of the
excited state potential energy surface would be necessary,
which would be a far from trivial undertaking.
The intensities of the transitions into the E ⬘ mode of the
lowest 1 4 A 2⬘ quartet state were calculated through a Jahn–
Teller analysis of the excited 2 4 E electronic state. The overlap integrals were calculated in the harmonic approximation
and the resulting intensities are shown in Fig. 8, normalized
to the 0-0 level. A complete modeling of the intensities of all
the levels could not be accomplished since the frequency of
␯ 1⬘ could not be determined from the excitation spectrum. As
stated above, only a full ab initio calculation of the excited
state surface would provide such information. Good agreement is obtained between the experimentally determined and
calculated emission intensity of the E ⬘ progression. The tran-
FIG. 8. Comparison of the experimental emission spectrum and the calculated intensities of the transitions into the E ⬘ vibrational levels of the 1 4 A 2⬘
state of Na3. Panel 共A兲 displays the calculated intensities based on the level
positions of the RKHS potential energy surface. Panel 共B兲 uses the level
positions of the HFD⫹CCSD共T兲 surface.
sitions group into bands for the first five prominent lines that
appear in the emission spectrum due to the similar frequencies of the ␯ 1⬙ and ␯ 2⬙ modes. The first five bands of the
emission spectrum beginning with the 0-0 band are composed of one, two, three, four, and five individual lines, respectively. The intensities could only be calculated for the
transitions into the E ⬘ mode. For this reason, the intensity of
only a single line of each clump involving an overtone level
of the ␯ 2⬙ mode could be calculated. The intensities calculated using the line positions of the HFD⫹CCSD共T兲 surface
appear to better match the experimental spectrum since the
transition frequencies tend to clump together near the center
of the bands as can be seen in Fig. 7, although better agreement with the absolute positions of the lines is obtained using the values calculated with the RKHS potential energy
surface. This is demonstrated in panel A of Fig. 8 where it
can be seen that the height of the line near 15 750 cm⫺1
matches the intensity of the shoulder seen in the experimental spectrum.
IV. DISCUSSION
Several comparisons were made in order to assess the
strengths and weaknesses of the RKHS method when applied
to the lowest quartet state of Na3. Two versions of the RKHS
surface and the HFD⫹CCSD共T兲 surface were made. In each
case, the first one 共not shown兲 was based only on the D 3h
configurations, while the second one 共cf., Fig. 5兲 was based
on all the available ab initio data. This allows effects due to
symmetric and asymmetric configurations to be compared.
In general, a better agreement is qualitatively observed
between the experiment and the levels calculated with the
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Higgins et al.
J. Chem. Phys., Vol. 112, No. 13, 1 April 2000
reproducing kernel interpolation of the CCSD共T兲 potential
energy surface. When only D 3h ab initio data are used, the
HFD⫹CCSD共T兲 and RKHS results are very similar. The
HFD⫹CCSD共T兲 has the advantage of giving a simple intuitive form for the short-range interaction as well, while the
RKHS form uses a more general but less physically motivated form which has the advantage of correctly reproducing
all the ab initio calculations.
When the non-D 3h configurations are added, the difference between the two is noticeably larger, and the RKHS
method appears to give better qualitative agreement 共nearly
quantitative, although comparison is difficult without information about peak shape or amplitudes兲 with the experimental spectrum. Since the HFD⫹CCSD共T兲 form does not include any three-body terms which attempt to model non-D 3h
configurations, the HFD⫹CCSD共T兲 result only changes
slightly when the new points are added to the fit. Better
agreement would require the addition of new terms to give
the function more flexibility. Doing this without sacrificing
the intuitive nature of the HFD⫹CCSD共T兲 form would require careful insight into the nature of the interactions being
modeled; simply adding ad hoc terms with less transparent
physical justification to the HFD⫹CCSD共T兲 potential would
sacrifice the intuitiveness that is the strength of the
HFD⫹CCSD共T兲 method probably without achieving the accuracy of the RKHS surface. On the other hand, the RKHS
method is able to incorporate the new data without any additional effort, after which the agreement with experiment is
clearly improved. The reproducing kernel method of interpolation appears to be a good method to produce a complete
potential surface from individual points since it possesses the
correct asymptotic form at long range and has enough flexibility for incorporating data from non-D 3h geometries. This
allows a larger portion of the surface to be calculated even
though theoretical data may not be available for those particular geometries.
V. CONCLUSION
Using laser excitation and emission spectroscopy, the
1 4 A 2⬘ lowest quartet state of Na3 residing on the surface of a
liquid He nanodroplet was identified for the first time and its
vibrational levels were subsequently obtained. As the perturbations due to the liquid He droplet have been shown to be
quite small20 we believe that the observed vibrational structure reflects quite closely the vibrational structure of those
species in the gas phase. The ab initio potential energy surface of the 1 4 A 2⬘ state of Na3 displays strong nonadditivity
effects. In particular, the experimentally determined asymmetric stretch frequency in the lowest quartet state provides a
sensitive measure of the three-body contributions to the potential energy. An increased nonadditive component causes
the frequency of ␯ 2 to increase from its expected value of
1/2⫻ ␯ 1 .
With the aid of ab initio calculations, a potential surface
has been obtained for the lowest quartet state of Na3. The
validity of the surface was confirmed by comparing the calculated vibrational states of the theoretical potential energy
surface with the experimental levels. The nonadditive contri-
butions to the interaction energy of this homonuclear triatomic system have been found to be large. In fact, the equilibrium distance of the trimer is about 15% smaller than the
bond length of the dimer. This large three-body contribution
is caused by a decreased overlap repulsion of the electrons in
the trimer which is due to the highly deformable valence
electron shells of the interacting sodium atoms.
The lowest quartet state of alkali trimers represents
somewhat of an extreme case of atomic deformability in
which three-body effects are large and almost exclusively
due to exchange overlap effects. This contrasts the case of
the noble gases in which the relatively small ATM dispersion
contributions are instead larger than those due to electron
overlap. For instance, the equilibrium distance calculated for
Ar3 is the same as the bond length in the dimer.50 As the rest
of the atoms in the periodic table have ionization potentials,
and therefore valence electron distribution deformability,
which are intermediate between the two groups mentioned
above, we now have a more precise feeling for what to expect when three-body effects in homonuclear systems are
considered.
ACKNOWLEDGMENTS
J.H., J.R., K.K.L, and G.S. were supported by AFOSR
共HEDM program兲. T.H.T.-S.H., and H.R. were supported by
the Department of Energy. M. G. acknowledges support of
this work by the Division of Chemical Sciences and the Division of Geosciences and Engineering both of the Office of
Science of the US Department of Energy. His work was
performed in part under auspices of the U.S. Department of
Energy, under Contract No. DE-AC06-76RLO 1830, with
Battelle Memorial Institute, which operates the Pacific
Northwest National Laboratory.
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