Interaction–Induced Spectroscopy of H2 in the
Fullerenes
John Courtenay Lewis1 and Roger M. Herman2
1
Department of Physics and Physical Oceanography, Memorial University of Newfoundland, St. John’s
NL, Canada A1B 3X7
2
Department of Physics, Pennsylvania State University, University Park PA 16802
Abstract. Carbon nanostructures of various sorts have been the subject of intensive research
since their discoveries in the latter part of the 20th century [1]. Much of this research has been
motivated by the intrinsic interest of these structures, though their potential as hydrogen storage
media has also attracted attention [2]. It was realized [3,4] that the carbon–hydrogen interactions
in these media would induce dipole moments which might lead to observable absorption of
infrared spectra, and this work will be reviewed and extended in this paper. The fundamental
vibration–rotation spectrum, of H 2 in a fcc C 60 lattice (fullerite) at room temperature was first
observed by S. A. FitzGerald and coworkers [3], who have subsequently extended their
observations to near liquid nitrogen temperatures. Herman and Lewis have discussed the
theoretical aspects of H 2 in carbon nanotube bundles and in fullerite [4,5,6]. We have
developed a detailed theory for the spectrum of H 2 in fullerite. This theory assumes that the
H 2  C potential can be accurately approximated by an exp–6 potential, the parameters of
which are then obtained by fitting the line frequencies in FitzGerald's spectra. We have also
obtained a model for the H 2 induced dipole moment based on the calculations of Frommhold
and coworkers [7] on the induced dipole in H 2  He . With one adjustable parameter this model
gives a good account of the observed intensities.
In work to date the line width has been taken as an empirical parameter. However, the line
width is in principal determinable from the H 2  C potential and induced dipole moment,
together with the known properties of the phonon modes in fullerite. We conclude this paper
with a discussion of the line width problem for H 2 in fullerite.
Keywords: interaction-induced spectroscopy; buckminsterfullerene;
PACS: 68.43.Pq, 78.30.Na, 33.20.Ea
INTRODUCTION
Since the discovery [1] of C60 by Kroto, Smalley, and coworkers in the mid-80s
and the discovery of carbon tubules or nanotubes by S. Iijima in 1991 the chemical
and physical properties of these all-carbon structures have been extensively studied by
a wide range of methods.
Small molecules adsorbed in nanotubes and fullerenes are interesting for a variety
of reasons, ranging from the advancement of our understanding of condensed phases
(does H2 adsorbed in nanotubes form a weakly bound anisotropic liquid at very low
temperatures, for instance?) to achieving immediately practical applications. Preeminent among the latter is hydrogen storage. The search for an efficient low-pressure
high-density storage system for molecular hydrogen has, in fact, motivated much of
the work on hydrogen-adsorbing nanostructures.
The interaction-induced spectroscopy of the hydrogens in the gas, liquid, and solid
phases has been discussed at previous ICSLSs. In this talk we discuss work from the
last five years on the very interesting spectroscopy of H2 in fullerene lattices. The IR
spectra have been taken by S. A. FitzGerald of Oberlin College and his associates, and
have been analyzed theoretically by T. Yildirim and associates at NIST, and by the
present authors.
Buckminsterfullerenes
It is useful to review some of the properties of buckminsterfullerenes, and in
particular properties of C60. References [1] and [8] are particularly useful in this
regard.
Buckminsterfullerene molecules are polyhedra of carbon atoms where each face is
either a hexagon or a pentagon. They therefore have the same basic structure as
“geodesic domes’’, albeit on a much smaller scale. Essential to their geometries is
Euler’s theorem for polyhedra, which asserts that
F V  E  2 ,
where F  the number of faces, V  the number of vertices, and E  the number of
edges.
If we let h  the number of hexagons and p  the number of pentagons then
F  p  h, 2E  5 p  6h, 3V  5 p  6h
so that
6 F  V  E   12
 6  p  h   2 5 p  6h   35 p  6h 
 p.
Hence there are exactly 12 pentagons in a fullerene molecule, regardless of its size.
The smallest possible fullerene is C20, known as the fullerane C20H20, which contains
no hexagons. However, it is a rule that the presence of adjacent pentagons tends to
destabilize the fullerene; the smallest fullerene with no adjacent pentagons is C60, the
celebrated “Buckyball”, which is the most stable and easily produced of this class of
molecule. The next largest fullerene satisfying the “no-adjacent-pentagons” rule is
C70, which is thermodynamically the most stable of the fullerenes. As of July 2006,
C60 of 99.5% purity is available commercially at US$45 per gm and C70 of 99.0%
purity at US$525 per gm.
C60 has also been detected occurring naturally in rock in Russia and in Colorado.
C60
The basic properties of C60 are as follows:
 the exterior diameter of a C60 is about 10.34 Å;
 all 60 C are equivalent;
 there are two types of bond, a “single” bond of length 1.46 Å, which is
significantly shorter than the C-C bond in alkanes or in diamond; and a
“double” bond of length 1.39 Å;
 there is little resonance and an absence of aromatic characteristics;
 According to a standard reference on fullerene chemistry [8], “C60 is a fairly
electronegative system. It behaves like an electron-poor conjugated polyolefin.”
 Considerable charge separation is evident in C60, with the “single” bonds
positive and the “double” bonds negative [9]. The resultant electrostatic forces
among C60 molecules result in rotational ordering at low temperatures.
 At room temperature its stable phase is fcc. This is known as “fullerite”.
C70
The second most common fullerene, C70, includes among its properties that it has
five distinct C sites and eight distinct C-C bonds; it looks like a rugby football; and its
chemistry is similar to that of C60. As for C60, the stable room-temperature phase of
C70 is fcc [9].
Figure 1. DRIFT spectrum from ref. [3] of the fundamental band of H 2 adsorbed to saturation in a
fullerite lattice approximately at room temperature.
THE INDUCED FUNDAMENTAL BAND OF H2 AT ROOM
TEMPERATURE
When H2 is adsorbed into a fullerite lattice, the interaction of the hydrogen
molecules with the surrounding C60 lattice results in induced dipole moments and
dipole transition moments. Observing the resultant absorption spectra is not
straightforward, because macroscopic crystals of fullerite have not been grown, and
samples are available only as powders. The fullerite lattice is, however, nearly
transparent in the region of the hydrogen fundamental band, and the induced spectrum
can be obtained [3] by the technique of Diffuse Reflectance Fourier Transform
Spectroscopy (DRIFTS). A spectrum taken at room temperature is shown in Fig. 1.
The major features of the spectrum can be labeled in standard notation as follows:
nz  nz 1
Q J R : J  J ,
Q J P : J  J ,
nz  nz 1
S J P : J  J  2 ,
nz  nz 1
S J R : J  J  2 ,
O J R : J  J  2 ,
O J P : J  J  2 ,
nz  nz 1
nz  nz 1
nz  nz 1
Here, “ nz  nz  1 ” is shorthand for “ nz  nz  1 and nx  0 and ny  0 or
nx  nx  1 and ny  0 and nz  0 or ny  ny  1 and nz  0 and nx  0 ”.
If a shift of approximately 54 cm1 from the corresponding free molecule
vibration-rotation energies resulting from the differential perturbation of the
v  0, 1 levels is taken into account, the bands can readily be identified [C,F]. The
larger O(2) R intensity will be obscured by the Q(J )P peaks, for all J, at 3950 cm1 .
The tallest peak, at 4200 cm1 arises from the Q(J )R transitions, for all J. This peak
also contains, in its higher frequency wing, the S(0)P feature. The peak at 4550 cm1
is mostly S(1)P , also supplemented by the weaker S(0) R . The tall peak at 4800 cm1
is mostly S(1) R , and obscures the S(2)P feature. The small peak at 5000 cm1 is
mostly S(2)R , but has the S(3)P in its high frequency wing. The weak peak at
5200 cm1 is almost entirely S(3) R . The reason for the predominance of the S(1) over
the S(0) features comes not only from the larger statistical weight of the J  1 states
over J  0 , but by virtue of the spin statistics, the S  1 state that accompanies odd-J
states being three times as frequently encountered, all other things equal, as the singlet
spin state accompanying the even-J states.
In addition, there are three small relatively sharp features at approximately
4100 cm1 , 4450 cm1 , and 4650 cm1 . These “sharks-teeth” almost certainly reflect a
breakdown of the octahedral symmetry of the fullerite, perhaps from defects or
perhaps from partial rotational ordering of the C60 molecules. The various possibilities
are discussed in ref. [6].
INTERMOLECULAR POTENTIALS AND ENERGY LEVELS
The C-H2 Potential
Our work is based upon the C-H2 potential, which we take to be of two forms:
   12    6 
U r   4      
Lennard-Jones
 r 
 r 

6

 
U r   4  exp   r      
exp-6
 r 


with
  11.2912 /  .
The exp-6 potential with this parametrization for  has the same well depth  , the
same zero crossing radius  and the same C6 coefficient as the Lennard-Jones
potential.
The C60-H2 Potential
If the 60 C atoms of a C60 molecule are smeared uniformly over a sphere of radius
a then the above C-H2 potentials yield the following potentials for the interaction of
an H2 molecule with a C60:
 2     10 15    4 15    4 
UC60 -H2 R   4
 3
  
  
 
Ra   R  a 
2 R  a
2 R  a  
from the Lennard-Jones potential and
 2  30 1   R  a    Ra   15    4 15    4 
UC60 -H 2 R   4 
e
 

  
 
Ra 
2  R  a
2 R  a  
 2
for the exp-6 potential. R is the displacement of the H2 molecule from the center of a
C60 molecule.
If the centers of the six C60 molecules bounding an octahedral cell enclosing an H2
are located at d, 0, 0  , 0,  d, 0 , and 0, 0,  d  then the total potential of the H2,
located at x, y, z  relative to the equilibrium position of 0, 0, 0  is
V x, y, z   UC60 -H 2 

d  x 2  y2  z 2   UC
60 -H 2


d  x 2  y2  z 2 
2
2
UC60 -H 2  x 2  d  y   z 2   UC60 -H 2  x 2  d  y   z 2 




2
2
UC60 -H 2  x 2  y 2  d  z    UC60 -H 2  x 2  y 2  d  z   .




Determination of the Energy Levels
We have found that the energy levels can be calculated approximately using a
separable potential which is the sum of the three one-dimensional potentials across the
faces of the fcc cell. If V x, y, z  is the exact nonseparable potential then
V x, y, z  ; V1D x   V1D y   V1D z 
where
1
V1D x   V x, 0, 0   V 0, 0, 0 .
3
The energy levels were determined in two ways. The first was a Discrete-Variable
Representation (DVR) method in which a basis set of products of sines was used, the
matrices of the position operators x̂, ŷ, ẑ were obtained and diagonalized, and their
direct product was formed. The matrix of the potential V x̂, ŷ, ẑ  is diagonal in this
representation and is easily calculated. Details are given in the Appendix. The result is
then added to the matrix for the kinetic energy, and the whole is diagonalized to yield
the energies. This procedure can be used equally for a separable and a non-separable
potential.
The second method was to find the one-dimensional energies using the potential
V1D z  by a shoot-and-match procedure, and use these to obtain the energies for the
separable three-dimensional potential. These energies served as a check on the DVR
procedure, but also provided one-dimensional wave functions in useable form, from
which the band intensities were calculated.
Table 1 compares energies for the separable potential with those for the nonseparable potential to which it approximates. It is evident that the approximation of
separability is satisfactory for the low-lying levels of greatest spectroscopic interest.
o
TABLE 1. exp-6 potential,  = 3.22 A ,  = 3.25 meV
Energy
DegenEnergy for
Energy for
level
eracy
sum potential
V x, y, z 
cm-1
cm-1
1
2
3
4
5
6
7
1
3
3
1
2
1
3
-926.7
-830.9
-739.4
-714.7
-705.6
-652.3
-624.7
-932.1
-836.4
-740.8
-713.1
-713.1
-645.1
-617.5
Principal states
n , n , n 
x
y
z
(000)
( 0 0 1 ) and perms.
( 0 1 1 ) and perms.
( 0 0 2 )* and perms.
same as above
(111)
( 0 1 2 ) and perms.
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
3
3
3
1
2
3
3
3
1
2
1
2
3
3
3
3
1
3
3
-617.5
-566.2
-538.3
-513.1
-501.8
-489.7
-474.2
-427.2
-419.7
-408.3
-407.8
-394.1
-374.5
-368.6
-325.0
-324.6
-318.8
-308.2
-284.5
-617.5
-569.9
-521.8
-494.2
-494.2
-474.3
-474.3
-409.8
-398.5
-398.5
-378.6
-378.6
-351.0
-351.0
-314.2
-314.2
-275.3
-255.3
-255.3
same as above
( 0 0 3 ) and perms.
( 1 1 2 ) and perms.
( 0 2 2 )* and perms.
same as above
( 0 1 3 ) and perms.
same as above
( 0 0 4 ) and perms.
( 1 2 2 )* and perms.
same as above
( 1 1 3 )* and perms.
same as above
( 0 2 3 ) and perms.
same as above
( 0 1 4 ) and perms.
same as above
(222)
( 1 2 3 ) and perms.
same as above
The separation states are linear combinations of the states in the fifth column. (*)
indicates that specific linear combinations are required as first order approximations to
the eigenstates for the nonseparable potential. Thus for the doubly degenerate level 5,
the eigenfunctions for V(x,y,z) can be represented to first order as
1
1
2 0 0 2  2 0 0  0 2 0  and  2 0 0  0 2 0 

2
6
while level 4 to first order has the eigenfunction
1
 0 0 2  2 0 0  0 2 0 .
3
Levels 4 and 5 form one triply degenerate level in the separable-potential
approximation.
The approximate separability of the potential is important, because the DVR
method can certainly be used to calculate intensities, or indeed any other matrix
element of functions of x, y, z, it is complicated to implement such calculations and we
have not as yet been completely successful in doing so.
It should be emphasized that for realistic parameters  ,  the potential V x, y, z  is
neither very much like a box nor like an isotropic harmonic oscillator. In fact for
typical values of these parameters it has a weak maximum at the center. Hence its
approximate separability was not expected.
Our approach to calculating the energies was to vary  ,  by hand, recalculating
the spectrum for each new set of values. This procedure was speeded by the
approximate separability property of the potential V, because it is of course much
faster to obtain one-dimensional eigenvalues by a shoot-and-match procedure than to
diagonalize the large matrix which is required by the DVR procedure. We started with
values of  ,  which were close to those for graphene-hydrogen [10]. The results
were not satisfactory, as can be seen in Fig. 3. The graphene-hydrogen potential of
Wang et al. [11] was similarly unsatisfactory [6]. Agreement was finally obtained, as
shown in Fig. 2, by using a value of  which is about 13% greater and 21% shallower
than the graphene-hydrogen values of ref. [10].
We could also distinguish between the exp-6 potential and the Lennard-Jones 12-6
potential, as can be seen in Fig. 4. The Lennard-Jones potential has been found to be
satisfactory in graphene-adsorbate studies, but for H2 in fullerite it gives a spectrum
which is too “boxy”, i.e. the V(x,y,z) which results is too “hard”. The softer
exponentially dependent repulsion of the exp-6 potential seems to be necessary for
accurate energies.
o
Our final best C-H2 potential was of exp-6 form with   3.22 A,   3.25 meV .
Figure 2. The DRIFTS from Fig. 1 together with our best theoretical spectrum, which was calculated
o
for the exp-6 potential for which   3.22A and   3.25meV . The principal component Lorentzian
1
line shapes have HWHH of 24cm . Other parameters are described in the text.
INDUCED DIPOLE MOMENTS
To calculate the intensities as opposed to the positions of the lines it is necessary to
have some estimate for the dipole moment induced between the C60 molecules and the
H2 molecules. We define s  s,  ,   to be the orientation vector for the H2,
r  r, ,   to be the vector from a C to an H2, and R  R,, to be the vector
from the center of a C60 molecule to the center of an H2 molecule. Then according to
Poll and Hunt the  th spherical tensor component of the dipole moment induced
between a C and an H2 is given by
4
p r,s  
 AL r, s C L1; M   M YM  , YL  M  , .
3 LM
For vibrational 0  1 transitions this gives
p001 r  
4
 BL r C L1; M,M YM  , YL,M , 
3 LM
where p001 r   0 AL r, s  1 is the matrix element of p0 r, s  between the ground
and first vibrational levels.
From the symmetry of the H2 molecule only even  and odd L contribute to the
induced dipole moment. This implies a translational selection rule:
nx  ny  0 and nz  1
and permutations thereof.
It is expected that the most important contributions to p001 will be the B01 term,
mainly from the overlap contribution, and the B23 term, which contains the dominant
long range quadrupole induced contribution. However, we also include the induction
contribution to B01.
It is known that the overlap and induction contributions to the total dipole moment
have an origin similar to the intermolecular force in distortion of the molecular charge
clouds, and a similar functional dependence on the intermolecular separation, albeit
with a somewhat greater range. It would be possible for us to calculate intensities with
such a functional form and adjust parameters so as to obtain agreement with the
intensities. However, we will endeavour to estimate those parameters a priori, or at
any rate on the basis of detailed ab initio calculations of HD-H2 by Borysow,
Frommhold and Mayer [7], and on He-H2 by Schaeffer and Kohler [12], rather than to
interpret fitted parameters a posteriori.
Figure 3. Theoretical spectrum using an exp-6 potential based on the Novaco [10] C-H2 potential in
o
graphene, for which   2.85 A and  4.11meV ; with DRIFTS.
Long Range Behavior of the Induced Dipole Moment
At long range a bond polarizability model is used to relate B 23 R HeH
01
01
to B 23
2
R C60 H 2 . The result is
B 23 R C
01
60 H 2
;
long range
93.3
R4
in atomic units. The long range (dispersion) part of B01 varies as R7 and can be
neglected in first approximation.
Figure 4. Theoretical spectrum using a Lennard-Jones interaction, with the same  and  as used for
the exp-6 potential in Fig. 2.
Short Range Behavior of the Induced Dipole Moment
By graphical analysis of data from the afore-mentioned authors we find that
0.75
01
B01
r HeH2 ; 0.251  Fshort range r HeH2 
with numerical values in atomic units, for values of r around . We also find that
01
01
B23
r  ; 0.24B01
r .
We conjecture that these relations hold for other atoms and small molecules as well
as for He, provided that the bonding is largely physical rather than covalent, and
provided that account is taken of differing charge deformability. The factor of 0.271
by itself cannot, be a general rule for relating the induced dipole moment and the 3/4th
power of the intermolecular force. Both the intermolecular force and the induced
dipole moment depend on the extent to which the atomic/molecular charge clouds are
deformed by proximity of the colliding partner. However, this enters the
intermolecular force as, in some sense, a sum of the deformabilities of the two
collision partners, and the induced dipole moment as a difference of the
deformabilities. Indeed, if the two collision partners are identical atoms, the induced
dipole moment is zero. We conjecture that the reciprocal of the ionization potential I is
an appropriate measure of the charge cloud deformability. Then, in general, instead of
the factor 0.251 obtained for He-H2, we expect that
 1 / IY  1 / IX 
0.251

 1 / I H 2  1 / I He 
will be the appropriate proportionality factor for colliding atoms and small molecules
X and Y, in the absence of covalent bonding. Then for C-H2
 1 / I  1 / IC 
0.75
01


B01
F
r
r CH2 ; 0.251 H2


 short range CH2 
 1 / I H  1 / I He  
2
01
should be a good representation of the short-range part of B01
r CH . For the exp-6
2
potential the short-range part then yields
120 d  1   R  a    Ra  
Fshort range R C -H   2
e


60
2
R
 a dR 

which finally gives
 
d  1   R  a    Ra   
e
R C60 -H2  3.55  2 e 
 .
dR 
R
  a



Then the 23  component is given by
93.9
01
01
B23
RC60 -H2  0.24B01
RC60 -H2  4 .
R
01
The dimensionless parameter  which enters both B01 R C -H and, implicitly,
0.75
01
B01
60
01
B23
RC
60 -H 2
2
, is an adjustable parameter representing our imprecise knowledge of
both the short range part and the long range part of the induced dipole moments. The
ratio of the two tallest peaks in the spectrum is sensitive to it; we found that the value
  3.1 gave a good fit.
The calculation of intensities from these induced dipole moments is described in ref.
[6].
Low-Temperature Spectra
S. A. FitzGerald and associates at Oberlin College have recently taken spectra at
80K and 10K [14,15]. We have applied our model to the 80K data. We obtain a good
fit provided that:
 we assume that the lattice contracts by 1.4%;
 we take the temperature to be somewhat higher than 80K.
Figure 5. Observed (DRIFT) and theoretical spectra of H2 in fullerite at approximately 80K. The
DRIFTS was taken by S. A. FitzGerald and associates [14].
LINE WIDTHS AND LINE SHAPES
Whereas both energy levels and intensities are based on a priori estimates of
relevant parameters, the line widths in our room temperature and 80K were chosen
freely, to give the best fits with the experimental data. Our best spectrum, shown in
Fig. 2, uses Lorentzian lines all of the same FWHH. We also used the same
parameters with Gaussian line shapes, the results being shown in Fig. 6. Clearly the
Gaussian line shapes give poorer agreement with the experimental data than do
Lorentian lines, and we can therefore conclude that the broadening is largely
homogeneous.
Given that the H2-fullerite potential and induced dipole moment are now known, it
should be possible to calculate the line broadening from first principles.
C70
It should be relatively straightforward to obtain the induced spectrum of at least the
fundamental band of H2 in a C70 lattice. The stable room-temperature phase is fcc with
complete orientational disorder [9]. Given the spheroidal shape of C70 and the
presence of 5 inequivalent C atoms it is likely that the induced spectrum of H2 in C70
would show considerably more homogeneous broadening than H2 in C60. At low
temperatures, as the rotational motions freeze out, considerable inhomogeneous
broadening would be expected.
Figure 6. Theoretical room temperature spectrum using Gaussian line shapes. The width (and all other)
parameters are the same as for the Lorentzian fit of Fig 2.
CONCLUSIONS
The first conclusion from the studies discussed herein is that the C-H2 interaction in
C60, and probably in other small fullerenes, is substantially different from the C-H2
interaction in graphene. This conclusions is in accord with present knowledge of the
bonding in fullerenes [9] and of fullerene chemistry [8].
A more general conclusion is that DRIFT spectra, supplemented by neutron
scattering spectra [13], contain much information on the C60-H2 interaction and likely
on the dynamics of the fullerite lattice itself. This information can be obtained by
careful analysis but without heroic extensions either to present experimental technique
or to computational resources. The sensitivity which we have demonstrated of the
theoretical spectra to small changes in parameters will lead to an excellent
understanding of the C60-H2 interaction.
ACKNOWLEDGMENTS
We thank Stephen FitzGerald for permission to use his DRIFTS data in our figures,
and for several extensive discussions about experiment and theory.
We acknowledge useful discussions on graphene and carbon nanostructures with
M. W. Cole, P. C. Eklund, and V. Crespi.
RMH gratefully acknowledges receipt of a Fulbright Fellowship, with which part
of this work was carried out, and thanks the Department of Physics and Physical
Oceanography of the Memorial University of Newfoundland for hospitality during
working visits.
JCL acknowledges support from the Natural Sciences and Engineering Research
Council of Canada and thanks the Department of Physics of the Pennsylvania State
University for its hospitality during several visits.
APPENDIX: A DISCRETE VARIABLE REPRESENTATION
METHOD
We begin with a complete set of one-dimensional wave functions  n x n . For
the H2-fullerite problem either sines (eigenfunctions for infinite rectangular wells), or
simple harmonic oscillator wave functions, are feasible, though early on in our work
we found that the sines seemed to give more rapid convergence than the simple
harmonic wave functions. We denote the matrix element of an operator Ô by

n

Ô  n  . In particular the matrix element of the position operator
x̂
is xnn  n x̂ n . This can be diagonalized by a unitary operator û :
  x    unl  n x 
n
*
*
x%
      x̂      un  n x̂  n  un     un xnn  un   .
nn 
nn 
The matrix element of the one-dimensional kinetic energy in the  basis is given
by


t      tˆ      un*  n tˆ  n  un     un* t nn un   .
nn 
nn 
The operator t is diagonal in the sines basis, though not in the harmonic oscillator
basis, but we make no use of that fact.
If Iˆ is the one-dimensional identity operator then the operators for position and
kinetic energy in three-dimensional space become
ˆ Yˆ  Iˆ  ŷ  I,
ˆ Ẑ  Iˆ  Iˆ  ẑ
X̂  x̂  Iˆ  I,
T̂  tˆx  Iˆ  Iˆ  Iˆ  tˆy  Iˆ  Iˆ  Iˆ  tˆz
and the three-dimensional bases are, with some abuse of notation,

nx ,ny ,nz  N    nx x  ny y  nz z 
x , y , z     x x  y y  z z 
with the unitary transformation Û connecting the two bases:
  U N  N 
N
U N   u n x  x u n y  y u nz  z
In the  basis the matrix elements of X̂ are diagonal:
%
X    x%
              X   
x
x
x
y y
z z
and similarly for Yˆ and Ẑ ; the matrix elements of the kinetic energy are given by
%
%
T  t%
x x  y y  z z   x x t y y  z z   x x  y y t z z .
It is straightforward and indeed almost trivial to calculate the matrix elements of the
potential energy V in the  basis, because the position operators are diagonal; no
integration is required, and little more than a function evaluation is necessary:


%
V   x x  y y  z z V x%
x , y%
y , z%
z   V x%
, y
 , z%
 .
The matrix elements of the three-dimensional translational Hamiltonian are given
by H    T   V  and diagonalization of H   yields translational energy levels
such as those shown in Table 1.
Evaluation of matrix elements of quantities dependent upon position requires backtransformation from the basis in which the Hamiltonian is diagonal, to the  basis.
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