Norbornane: An investigation into its valence electronic structure using electron momentum spectroscopy, and density functional and Green's function theories

Article abstract of DOI:10.1063/1.1799014

We report on the results of an exhaustive study of the valence electronic structure of norbornane (C₇H₁₂), up to binding energies of 29 eV. Experimental electron momentum spectroscopy and theoretical Green's function and density functional theory approaches were all utilized in this investigation. A stringent comparison between the electron momentum spectroscopy and theoretical orbital momentum distributions found that, among all the tested models, the combination of the Becke-Perdew functional and a polarized valence basis set of triple-ζ quality provides the best representation of the electron momentum distributions for all of the 20 valence orbitals of norbornane. This experimentally validated quantum chemistry model was then used to extract some chemically important properties of norbornane. When these calculated properties are compared to corresponding results from other independent measurements, generally good agreement is found. Green's function calculations with the aid of the third-order algebraic diagrammatic construction scheme indicate that the orbital picture of ionization breaks down at binding energies larger than 22.5 eV. Despite this complication, they enable insights within 0.2 eV accuracy into the available ultraviolet photoemission and newly presented (e,2e) ionization spectra, except for the band associated with the 1a₂^-1 one-hole state, which is probably subject to rather significant vibronic coupling effects, and a band at ～25 eV characterized by a momentum distribution of "s-type" symmetry, which Green's function calculations fail to reproduce. We note the vicinity of the vertical double ionization threshold at ～26 eV.

Full text of DOI:10.1063/1.1799014

Norbornane: An investigation into its valence electronic structure using
electron momentum spectroscopy, and density functional and Green’s
function theories
S. Knippenberg, K. L. Nixon, M. J. Brunger, T. Maddern, L. Campbell et al.
Citation: J. Chem. Phys. 121, 10525 (2004); doi: 10.1063/1.1799014
View online: http://dx.doi.org/10.1063/1.1799014
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 121, NUMBER 21
1 DECEMBER 2004
Norbornane: An investigation into its valence electronic structure
using electron momentum spectroscopy, and density functional
and Green’s function theories
S. Knippenberg
Department SBG, Limburgs Universitair Centrum, Gebouw D, B-3590 Diepenbeek, Belgium
K. L. Nixon, M. J. Brunger,a) T. Maddern, L. Campbell, and N. Trout
School of Chemistry, Physics and Earth Sciences, Flinders University, GPO Box 2100,
Adelaide SA 5001, Australia
F. Wang
Centre for Molecular Simulation, Swinburne University of Technology, PO Box 218, Hawthorn,
Vic 3122, Australia
W. R. Newell
Department of Physics and Astronomy, University College, Gower Street, London, United Kingdom
and School of Chemistry, Physics and Earth Sciences, Flinders University, GPO Box 2100,
Adelaide SA 5001, Australia
M. S. Deleuze and J.-P. Francois
Department SBG, Limburgs Universitair Centrum, Gebouw D, B-3590 Diepenbeek, Belgium
D. A. Winkler
CSIRO Molecular Science, Private Bag 10, Clayton South MDC, Vic 3169, Australia
͑Received 19 May 2004; accepted 3 August 2004͒
We report on the results of an exhaustive study of the valence electronic structure of norbornane
(C7H12), up to binding energies of 29 eV. Experimental electron momentum spectroscopy and
theoretical Green’s function and density functional theory approaches were all utilized in this
investigation. A stringent comparison between the electron momentum spectroscopy and theoretical
orbital momentum distributions found that, among all the tested models, the combination of the
Becke-Perdew functional and a polarized valence basis set of triple-␨ quality provides the best
representation of the electron momentum distributions for all of the 20 valence orbitals of
norbornane. This experimentally validated quantum chemistry model was then used to extract some
chemically important properties of norbornane. When these calculated properties are compared to
corresponding results from other independent measurements, generally good agreement is found.
Green’s function calculations with the aid of the third-order algebraic diagrammatic construction
scheme indicate that the orbital picture of ionization breaks down at binding energies larger than
22.5 eV. Despite this complication, they enable insights within 0.2 eV accuracy into the available
ultraviolet photoemission and newly presented (e,2e) ionization spectra, except for the band
associated with the 1aϪ2 1 one-hole state, which is probably subject to rather significant vibronic
coupling effects, and a band at ϳ25 eV characterized by a momentum distribution of ‘‘s-type’’
symmetry, which Green’s function calculations fail to reproduce. We note the vicinity of the vertical
double ionization threshold at ϳ26 eV. © 2004 American Institute of Physics.
͓DOI: 10.1063/1.1799014͔
I. INTRODUCTION
the low dipole moment, as well as difficulties in preparing
isotopically enriched samples. There have been a number of
structural studies by electron diffraction4 but the norbornane
molecule is problematic due to strong correlations between
parameters used to determine the similar carbon-carbon bond
lengths in the molecule. The use of x-ray crystallography to
determine an unambiguous structure was complicated by the
fact that norbornane, like many globular molecules, is orien-
tationally disordered at ambient temperatures, transforming
from cubic to hexagonal at 306 K. Single crystals of norbor-
nane have not been available, and Fitch and Jobic5 only re-
cently solved the structure by powder x-ray diffraction meth-
In spite of the importance of norbornane ͑NBA͒ to
chemical and pharmaceutical research,1,2 the experimental
determination of its structure has been problematic. The mol-
ecule has an extremely small dipole moment ͓ϳ0.09 D ͑Ref.
3͔͒, making structural determination by microwave spectros-
copy very difficult. Choplin3 studied the microwave response
of norbornane but was unable to determine its structure be-
cause of the weak intensity of rotational transitions, due to
a͒
Author to whom correspondence should be addressed. Electronic mail:
Michael.Brunger@flinders.edu.au
0021-9606/2004/121(21)/10525/17/$22.00
10525
© 2004 American Institute of Physics
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10526
J. Chem. Phys., Vol. 121, No. 21, 1 December 2004
Knippenberg et al.
ods using a synchrotron radiation source. However this
structure was for solid norbornane, and structures from x-ray
diffraction are subject to substantial deformation because of
crystal lattice interactions. Consequently, a precise gas phase
structure of norbornane has not been determined experimen-
tally and computational approaches have been valuable in
interpreting the available experimental data in a consensus
fashion. van Alsenoy and co-workers6 employed ab initio
Hartree–Fock ͑HF͒ calculations to assist in the interpretation
of the microwave structure model,6 and Allinger’s group7
used molecular mechanics methods to analyze the x-ray dif-
fraction and electron diffraction data to give a consistent
structure for norbornane.
An experimental calibration of the model employed ͑i.e.,
theoretical approach and basis set͒ using electron momentum
spectroscopy ͑EMS͒ provides a way to select a wave func-
tion which is reliable enough for accurately predicting the
molecular structure of norbornane, as well as calculating
other important molecular properties such as the dipole mo-
ment, bond orders, charge distributions, nuclear magnetic
resonance ͑NMR͒, and vibrational spectra. Previous
studies7,8 have used a variety of molecular mechanics and
molecular orbital approaches to determine structural and
electronic properties of norbornane. Here we use the unique
orbital imaging capability9,10 of EMS to determine which of
the employed density functional theory ͑DFT͒ exchange cor-
relation functionals and basis sets best describes the experi-
mental momentum distributions. This optimum basis and ex-
change correlation functional is then used to derive the
structure and molecular properties of norbornane. These data
are next compared with independent experimentally deter-
mined values, and those from other molecular orbital ͑MO͒
calculations, to determine how well the optimum model was
able to reproduce norbornane’s molecular properties.
of ionization, we may probe the influence of substantial cy-
clic strains on chemical bonds. In this respect we note our
preliminary study17 on all three molecules I–III, a paper
which arose from an invited presentation at the Sagamore
14th meeting.
In the following section of this paper we briefly discuss
our EMS measurements, including our ionization spectra.
Details of our HF, DFT, and 1p-GF calculations, and some of
the electronic structure information we can extract from them
are presented in Secs. III and IV, while in Sec. V we compare
and discuss the experimental and theoretical momentum dis-
tributions associated to all bands in the EMS ionization spec-
tra. In Sec. VI the molecular property information derived
from our optimum basis set and exchange correlation func-
tional is detailed, while in Sec. VII some of the conclusions
drawn from the current study are presented.
II. EXPERIMENTAL DETAILS
AND PRELIMINARY ANALYSIS
A sample of high-purity norbornane was synthesized ‘‘in
house’’ using commercially purchased ͑Aldrich Chemical
Company͒ norbornene in the following manner. To a thick-
walled flask we added norbornene ͑5 g, 52 mmol͒, AR
methanol ͑100 ml͒, and a spatula amount of 10% Pd on
carbon. The resulting mixture was hydrogenated under 40 psi
of H2 for 12 h with rocking. There was an instantaneous
uptake of H2 . More H2 was introduced and left overnight.
Water was added and then extracted with CFCl3 ͑2ϫ20 ml͒.
The bottom organic layer was collected and allowed to
evaporate at room temperature. The crude norbornane ͑ϳ1 g͒
was pure according to gas chromatographic ͑GC͒ and 13C
and 1H NMR analysis agreed with previously reported
data.18 This material was then distilled into a U tube im-
mersed in liquid nitrogen and under vacuum and then trans-
ferred into the reaction vessel. The reaction vessel was in
turn connected to the gas handling system of the EMS spec-
trometer. In addition, it was degassed in situ by repeated
freeze-pump-thaw cycles before being introduced into the
interaction region. Comparing our ␾ϭ0°ϩ10° ionization
spectrum with the PES result of Bischof et al.11 shows that
the level of qualitative agreement between them is very
good. This gives further evidence for the purity of our NBA
sample, an important consideration given the high sensitivity
of EMS to the presence of any impurities.
All the 20 occupied MO’s of the complete valence re-
gion of NBA, namely the 3a2 , 5b2 , 7a1 , 5b1 , 6a1 , 4b2 ,
2a2 , 4b1 , 3b2 , 3b1 , 5a1 , 2b1 , 4a1 , 2b2 , 3a1 , 1a2 , 2a1 ,
1b2 , 1b1 , and 1a1 MO’s, were then investigated in several
experimental runs using the Flinders symmetric noncoplanar
EMS spectrometer.9 Details of this coincidence spectrometer
and the method of taking the data can be found in the work
by Brunger and Adcock,10 and Weigold and McCarthy,9 and
so we do not repeat them again here.
The high-purity NBA is admitted into the target chamber
through a capillary tube, the flow rate being controlled by a
variable leak value. Possible clustering, due to supersonic
expansion, was avoided by maintaining a low NBA driving
pressure throughout data collection. The collision region is
differentially pumped by a 700 l sϪ1 diffusion pump. Aper-
While conducting our study, it became quite clear that
existing investigations into the outer and inner valence elec-
tronic structure of norbornane are rather scarce. Previous
photoelectron spectroscopy ͑PES͒ studies include the He͑I͒
measurements from Bischof et al.11 and Getzlaff and
Schonhense12 and the He͑II͒ measurement from Bieri et al.13
¨
Theoretical interpretation of these spectra has been even
more limited with only the modified intermediate neglect of
differential overlap, version 2 ͑MINDO/2͒, result from
Bodor et al.14 currently being available in the literature.
Hence the present HF, DFT, and one-particle Green’s func-
tion ͑1p-GF͒ calculations significantly expand the available
theoretical knowledge of the electronic structure of norbor-
nane. In addition, we believe that the present EMS measure-
ments are the first to be made on this molecule, thus further
expanding our understanding of its electronic structure
through our original momentum space images of its MO’s.
Finally, we note that norbornane is the second molecule
in the chemically similar series norbornadiene͑I͒,15,16 nor-
bornene͑II͒, and norbornane͑III͒, which we have studied us-
ing EMS, HF, and DFT techniques. In going from I to III the
CϭC double bonds in these highly strained bicyclic hydro-
carbons are progressively saturated. It is our thesis that by
unraveling the electronic structure of norbornane using EMS
in conjunction with DFT calculations and the 1p-GF theory
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J. Chem. Phys., Vol. 121, No. 21, 1 December 2004
Norbornane electronic momentum spectroscopy
10527
tures and slits are cut in the collision chamber for the inci-
dent electron beam and the scattered and ejected electrons.
The differentially pumped collision region makes it possible
to increase the target gas density by a factor ϳ3 while keep-
ing the background pressure below 10Ϫ5 Torr. This was im-
portant as it enabled us to maintain workable coincidence
count rates, even with the smaller electron beam current out-
put from the (e,2e) monochromator ͑typically 30 ␮A in this
work͒ compared to that of a normal electron gun.19 The co-
incident energy resolution of the present measurements was
ϳ0.55 eV full width at half maximum ͑FWHM͒ as deter-
mined from measurements of the binding-energy (⑀f) spec-
trum of helium. Note that the profile of the helium spectrum
was found to be well represented by a Gaussian function.
However, due to the natural and vibrational linewidths
͑sometimes also known as the Franck–Condon widths͒ of
the various electronic transitions and a quite strong disper-
sion of the ionization intensity into many-electron processes
at the bottom of the carbon-2s region, the fitted resolutions
of the spectral peaks for NBA varied from ϳ0.88 to 2.31 eV
͑FWHM͒. It is precisely this limitation which forces us to
combine our measured highest occupied molecular orbital
͑HOMO͒ and next highest occupied molecular orbital
͑NHOMO͒ (3a2 and 5b2) momentum distributions ͑MD’s͒,
5b1 and 6a1 orbital MD’s, 4b2 , 2a2 , and 4b1 orbital MD’s,
3b2 and 3b1 orbital MD’s, 5a1 and 2b1 orbital MD’s, and
2a1 , 1b2 , and 1b1 orbital MD’s, respectively. While there is
no doubt one loses some physical information in combining
these MD’s, to not do so would have raised serious question
as to the uniqueness of the MD’s derived in the fits to our
binding energy spectra ͑see below͒. The angular resolution,
which determines the momentum resolution ͓see Eq. ͑1͔͒
was typically 1.2° ͑FWHM͒, as determined from the electron
optics and apertures and from a consideration of the argon
3p angular correlation.
available PES data.11–13 A summary of the available orbital
binding energies from PES data, the present EMS binding
energies and our tentative orbital assignments are given in
Table I. The fact that we use only 13 Gaussians to analyze
spectra containing 20 valence MO’s simply reflects our ear-
lier point that our energy resolution was insufficient to
uniquely deconvolve all the orbitals, so that some were com-
bined ͑summed͒. Notwithstanding this it is clear from Fig. 1
that the fits to the measured binding-energy spectra are ex-
cellent. The least-squares-fit deconvolution technique used in
the analysis of these spectra is based on the work of Beving-
ton and Robinson,20 to whom readers are referred for more
detail. Above ⑀fϳ23 eV there are no PES data available to
guide us in our fitting of the binding-energy spectra. Under
these circumstances the positions and widths of the Gaussian
peaks and the number of Gaussians used in the spectral de-
convolution were simply determined by their utility in best
fitting the observed data for all ␾. The fact that the inner
valence 2a1 , 1b2 , 1b1 , and 1a1 orbitals need three very
broad Gaussians ͑peaks 11–13͒ to incorporate the measured
coincidence intensity into the fit, is undoubtedly indicative of
a severe dispersion of ionization intensity over many satellite
states, an observation which led us to undertake thorough
1p-GF calculations of the valence one-electron and shake-up
ionization spectrum of norbornane ͑see Sec. IV͒.
The EMS ionization spectra of Fig. 1 clearly reflect the
respective symmetries9 of the valence orbitals of norbornane.
For instance, the unresolved HOMO and NHOMO ͑peak 1͒
show significantly more intensity at ␾ϭ10° compared to that
at ␾ϭ0°. This is consistent with the ‘‘p-type’’ symmetry of
these orbitals. On the other hand the 4a1 orbital ͑peak 7͒ has
a much greater intensity at ␾ϭ0° compared to that found at
␾ϭ10°, an angular dependence which corroborates its ‘‘s-
type’’ symmetry. On the basis of the symmetry indicated by
the EMS binding-energy spectra and the results of our cal-
culations in Table II ͑see Secs. III and IV for more details͒
tentative orbital assignments were made and are given in
Table I. In general these orbital assignments are consistent
with those found from our 1p-GF calculations, with the ex-
ception of band 12 in the inner valence region. The angular
dependence of the EMS cross sections indicates that bands
12 and 13 have similar s-type MD’s so that both bands at
first glance could be ascribed to originating from the 1a1
orbital. Our 1p-GF calculations support the notion that band
13 relates essentially to satellites originating from ionization
of the 1a1 orbital. In addition, the EMS and 1p-GF interpre-
tations of band 11 are largely consistent in assigning that flux
as mainly being due to a set of lines related to ionization of
the 2a1 , 1b2 , and 1b1 orbitals. Band 12, however, appears
to be a far more complicated issue than was originally an-
ticipated ͑see Sec. IV͒.
In the present study, noncoplanar symmetric kinematics
were employed; that is, the outgoing electron energies EA
and EB were equal ͑ϭ750 eV͒ and the scattered ͑A͒ and
ejected ͑B͒ electrons made equal polar angles, ␪ϭ45°, with
respect to the direction of the incident electrons. The total
energy E (EϭE0Ϫ⑀fϭEAϩEB) was 1500 eV. The beam
energy is E0 . The binding-energy range of interest (⑀f
ϭ7–29 eV) is stepped through sequentially at each of a cho-
sen set of angles ␾ using a binning mode19 through the entire
set of azimuthal angles ͑␾ϭ0°–30°͒. Scanning through a
range of ␾ is equivalent to sampling different target electron
momenta p as9
1/2
␾
pϭ 2p cos ␪Ϫp ͒2ϩ4p2 sin2 ␪ sin2
.
͑1͒
͑
ͫ
ͩ ͪ
ͬ
A
0
A
2
For zero binding energy (⑀fϭ0 eV), ␾ϭ0° corresponds to
pϭ0 a.u., and for the present binding energies, angular reso-
lution, and kinematics, ␾ϭ0° corresponds to pϷ0.03 a.u.
Note that 1 a.u.ϵ1a0Ϫ1, where a0 is the Bohr radius.
III. THEORETICAL ANALYSIS
OF EMS CROSS SECTIONS
Ionization spectra of norbornane measured at represen-
tative angles ␾ in the region 7–29 eV and at Eϭ1500 eV are
displayed in Fig. 1. The solid curve in each panel represents
the envelope of the 13 fitted Gaussians ͑various dashed
curves͒ whose positions below ⑀fϳ23 eV are taken from the
The plane wave impulse approximation21 ͑PWIA͒ is
used to analyze the measured cross sections for high-
momentum transfer (e,2e) collisions. Using the Born–
Oppenheimer approximation for the target and ion wave
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10528
J. Chem. Phys., Vol. 121, No. 21, 1 December 2004
Knippenberg et al.
FIG. 1. Typical binding-energy spectra from our 1500
eV noncoplanar symmetric EMS investigation into nor-
bornane. The curves show the fits to the spectra at ͑a͒
␾ϭ0° (pϷ0.03 a.u.) and ͑b͒ ␾ϭ10° (pϷ0.92 a.u.) us-
ing the known energy resolution. The peak positions of
the Gaussians used in the fit ͑see also Table I͒ are indi-
cated. Note that indicative error bars are shown on this
figure.
TABLE I. Norbornane—electronic structure ͑experimental͒.
⑀f (eV)
Experimental
Natural
Orbital
number Classification
Present
Present
EMS
width ͑eV͒
͑Refs. 11–13͒
PES ͑Ref. 11͒
ϳ10.2
PES ͑Ref. 12͒ PES ͑Ref. 13͒
1
2
3a2
5b2
7a1
5b1
6a1
4b2
2a2
4b1
3b2
3b1
5a1
2b1
4a1
2b2
3a1
1a2
2a1
1b2
1b1
1a1
ϳ10.3
ϳ10.3
10.3
0.72
ͬ
ͬ
ͬ
ͬ
ͬ
3
4
ϳ10.7
ϳ10.9
ϳ11.6
ϳ10.9
ϳ11.6
10.9
11.6
0.72
0.86
ͬ
ͬ
ͬ
ͬ
ͬ
5
6
ϳ11.4–12.12
ͬ
7
ϳ12.4
ϳ13.6
ϳ15.6
ϳ12.4
ϳ13.5
ϳ15.6
12.4
13.5
15.6
1.20
ͬ
ͬ
ͬ
8
9
ϳ13.4
ϳ15.5
1.14
ͬ
ͬ
ͬ
ͬ
ͬ
ͬ
ͬ
10
11
12
13
14
15
16
17
18
19
20
0.64
ͬ
ͬ
ϳ16.4
ϳ17.5
¯
ϳ16.5
ϳ17.5–17.8
ϳ18.1
ϳ19.4
¯
ϳ16.5
ϳ17.65
ϳ18.1
ϳ19.4
16.5
17.65
18.1
19.4
0.86
0.86
0.72
0.86
¯
¯
¯
¯
ϳ22.62
ϳ22.6
2.25
ͬ
ͬ
ͬ
¯
¯
a
¯
¯
¯
24.9
27.5
1.80
1.80
aThis assignment is controversial. See text.
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J. Chem. Phys., Vol. 121, No. 21, 1 December 2004
Norbornane electronic momentum spectroscopy
10529
TABLE II. Norbornane–electronic structure ͑theory͒.
functions, the EMS differential cross section ␴, for randomly
oriented molecules and unresolved rotational and vibrational
states, is given9 by
⑀f (eV)
Basis sets
Orbital
number
Present
Classification
Present
HF/TZVP
Present HF/
cc-pVDZ
Present DFT
BP/TZVP
N
i
p⌿NϪ1
͉
⌿
͉ ,
͘
͑2͒
2
␴ϭK d⍀
͉
͗
͵
f
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
3a2
5b2
7a1
5b1
6a1
4b2
2a2
4b1
3b2
3b1
5a1
2b1
4a1
2b2
3a1
1a2
2a1
1b2
1b1
1a1
11.332
11.666
12.033
12.585
12.659
13.140
13.439
13.695
14.767
14.887
17.079
17.332
18.474
19.793
20.547
22.372
25.576
26.593
27.135
31.606
11.328
11.641
12.006
12.531
12.607
13.108
13.384
13.634
14.734
14.831
16.999
17.232
18.360
19.772
20.500
22.328
25.542
26.570
27.067
31.532
6.88
7.24
7.54
7.90
7.98
8.19
8.41
8.65
9.41
where K is a kinematical factor which is essentially constant
in the present experimental arrangement. ⌿Nf Ϫ1 and ⌿iN
are the electronic many-body wave functions for the final
(NϪ1) electron͔ ion and target ͓N-electron͔ ground states,
and p is a plane wave representing the ionized electron. The
d⍀ denotes the integral required for averaging the com-
͓
͐
puted (e,2e) cross sections over all gas phase molecular ori-
entations ͑spherical averaging͒. The average over the initial
vibrational state is well approximated by evaluating orbitals
at the equilibrium geometry of the molecule. Final rotational
9.46
11.16
11.43
12.26
12.82
13.42
14.64
16.94
17.68
18.10
21.48
and vibrational states are eliminated by closure.9
The momentum space target-ion overlap p⌿NϪ1
͉
⌿
N
͗
͘
f
i
can be evaluated using configuration interaction descriptions
of the many-body wave functions,22 but usually the weak
coupling approximation19 is made. Here the target-ion over-
lap is replaced by the relevant orbital of, typically, the
Hartree–Fock or Kohn–Sham23 ground state ⌽0 , multiplied
by a spectroscopic amplitude. With these approximations Eq.
͑2͒ reduces to
UniChem.30 The molecular structure of norbornane has been
optimized through energy minimization with various
gradient-corrected functionals and basis sets, employing the
UniChem user interface. Note that a geometry optimization
was performed in DGAUSS with each basis set used. The elec-
tronic structural calculations using restricted Hartree–Fock
͑RHF͒ and second-order Møller–Plesset ͑MP2͒ approaches
along with a polarized valence basis set of triple-␨ ͑TZVP͒
quality are based on GAMESS.31 A subset of our calculated
orbital energies from both our DFT RHF calculations is
given in Table II. Clearly, none of these results give particu-
larly good agreement with the corresponding experimental
values of Table I. Despite Koopmans’ theorem, all HF orbital
energies overestimate the measured ionization energies by
ϳ1 to ϳ3 eV, which indicates that these energies are sub-
stantially influenced by electron-correlation effects, and,
more importantly, electron relaxation effects. On the other
hand, the BLYP- and BP-DFT orbital energies all underesti-
mate the respective experimental binding energies by ϳ3.5–
4.7 eV. Such a result was, however, not entirely unexpected.
It is known ͑Ref. 32 and references therein͒ that XC func-
tionals, whether at LSD or GGA levels, fail to give the cor-
rect dispersion interaction in the large r region. This error in
the asymptotic limit of the XC functionals leads to ionization
energies that underestimate those determined by experiment
by as much as 5 eV.
͑ f ͒
2
ជ
␴ϭKSj
d⍀
͉
␾ p͒
͉
,
͑3͒
͑
͵
j
ជ
where ␾j(p) is the momentum space orbital. Note that the
relaxation of the final state has been neglected in this ap-
proximation. Further, note that the basis of the orbital imag-
ing capability of EMS is immediately apparent from Eq. ͑3͒.
The spectroscopic factor S(jf) is the square of the spectro-
scopic amplitude for orbital j and ion state f. It satisfies the
sum rule
͑ f ͒
S
ϭ1.
͑4͒
͚
j
j
Hence S(jf) may be considered as the probability of finding
the one-hole configuration in the many-body wave function
of the ion.
The Kohn-Sham equation23 of DFT may be considered
as an approximate quasiparticle equation, with the potential
operator approximated by the exchange-correlation
potential.22 Often this is done using the local spin density
͑LSD͒ approximation, although in this study we concentrate
on approximating the exchange-correlation ͑XC͒ functional
with functionals that depend on the electron density and its
gradients24–27 ͓i.e., the generalized gradient approximation
͑GGA͔͒. Specifically, here we employed two different ap-
proximations to the XC energy functional due to Becke and
Perdew24–26 ͑BP͒ and Becke, Lee, Yang, and Parr
͑BLYP͒.24,25,27 To compute the coordinate space Kohn–Sham
orbitals ␺j , we employed DGAUSS, a program package origi-
nally developed at CRAY Research by Andzelm and
co-workers.28,29 It has been known for a number of years30
that HF theory provides momentum distributions of lower
quality than DFT, therefore we do not assess HF momentum
distributions again here. DGAUSS is itself a part of
Information of the molecular structure and the molecular
orbital wave functions for the ground electronic state of
NBA, obtained from the DGAUSS DFT calculations, were
next treated as input to the Flinders-developed program
AMOLD,19 which computes the momentum space spherically
averaged molecular-structure factor21 and the (e,2e) cross
section or MD ͓see Eq. ͑3͔͒. Note that all the theoretical
MD’s we report in this paper have had the experimental an-
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10530
J. Chem. Phys., Vol. 121, No. 21, 1 December 2004
Knippenberg et al.
gular resolution folded in using the method of Frost and
Weigold.33
a Block-Davidson diagonalization procedure45 in the final
diagonalization step. The assumption of frozen core electrons
has been used throughout and symmetry has been exploited
to the extent of the C2v point group. Our results from these
calculations are presented in Table III. For comparison pur-
poses, more specifically to evaluate the sensitivity of the
computed ionization energies to the quality of the basis set, a
few results obtained from outer-valence Green’s function
͓OVGF ͑Refs. 46, 47͔͒ calculations, performed with the
GAUSSIAN 98 package,48 are also presented in Table III. For
these benchmark computations of one-electron ionization en-
ergies, specifically, we will consider basis sets such as Dun-
ning’s correlation consistent polarized valence basis set of
triple-␨ quality ͓cc-pVTZ ͑Ref. 43͔͒, and the cc-pVDZ basis
The comparisons of calculated MD’s with experiment
͑see Sec. V͒ may be viewed as an exceptionally detailed test
of the quality of the XC energy and basis set. From our
previous experience,34,35 the GGA-DFT methods using the
BP and BLYP XC functionals give best agreement with the
experimental MD’s, compared to the LSD method. As a re-
sult, GGA-BP and GGA-BLYP are used in combination with
three basis sets to examine the behavior of the XC function-
als and basis sets. These basis sets are denoted by the acro-
nyms DZVP, DZVP2, and TZVP. The notations DZ and TZ
denote basis sets of double- or triple-␨ quality. V denotes a
calculation in which such a basis is used only for the valence
orbitals and a minimal basis is used for the less chemically
reactive core orbitals. The inclusion of long-range polariza-
tion functions is denoted by P. We note, in particular, that the
basis sets of DGAUSS were specially designed for DFT
calculations.28,36 The TZVP basis set has a contraction
scheme ͓7111/411/1͔ for carbon and ͓3111/1͔ for hydrogen.
The auxiliary basis set corresponding to the TZVP basis is
called A1,37 in which the s-, p-, and d-orbital exponents were
determined separately from an optimization that reproduces,
as accurately as possible, the energy from an atomic DFT
calculation. The contraction schemes of the A1 basis sets for
H are ͓4/1͔ and for C ͓8/4/4͔.
The DFT DGAUSS calculations were performed on a Sili-
con Graphics 02 ͑R5200͒ workstation as the UniChem client
and a CRAY J90se/82048 computer as the DFT computa-
tional engine. Further restricted Hartree–Fock ͑RHF͒ and
MP2 calculations using the TZVP basis set and a GAMESS 02
suite of programs,31 were carried out on the Compaq Alpha
Server SC cluster at the Australian Partnership for Advanced
Computing National Facilities.
In light of the marginal agreement between the DFT and
experimental ionization energies, which we described earlier,
further calculations employing more sophisticated Green’s
function techniques were undertaken. These calculations are
all based on geometries that have been optimized using den-
sity functional theory by means of the GAMESS 02 program31
employing the TZVP basis set and the nonlocal hybrid Becke
three-parameter Lee–Yang–Parr functional ͑B3LYP͒.27,38
augmented by a set of diffuse s,p functions on hydrogens,
͕
͖
and a set of diffuse s,p,d functions on carbons ͓aug-cc-
͕
͖
pVDZ ͑Refs. 43, 49͔͒. With the cc-pVDZ, aug-cc-pVDZ and
cc-pVTZ basis sets, 158, 269, and 378 basis functions in
total are incorporated in the OVGF computations on norbor-
nane, respectively.
Because of the complexity of the outermost valence
bands, encompassing the contributions of many and strongly
overlapping ionization lines, it is preferable to resort to the-
oretical simulations for analyzing the available PES mea-
surements. As a guide to the eye, the identified solutions of
the secular ADC͑3͒/cc-pVDZ eigenvalue problem are there-
fore displayed as a spike spectrum and in the form of a
convoluted density of states, along with the ultraviolet photo-
12
¨
ionization spectra by Getzlaff and Schonhense and Bieri
et al.13 ͑see Fig. 2 and Table III͒. The convolution has been
performed using as a spread function a combination of a
Gaussian and a Lorentzian with equal weight, a FWHM pa-
rameter of 0.6 eV, and by simply scaling the line intensities
according to the computed ADC͑3͒ pole ͑spectroscopic͒
strengths. Despite the neglect of cross section effects, the
shape, position and the relative intensities of bands in the
He͑I͒ and He͑II͒ spectra are overall very finely reproduced in
the simulation. In particular, in line with the convoluted
spectrum, three substructures are seen with the outermost
He͑II͒ ionization band, namely, a shoulder at ϳ10.9 eV, and
two maxima at ϳ11.7 and ϳ12.1 eV.
There are several points we would like to highlight from
the results in Table III: First, the current Green’s function
results for ⑀f of each respective orbital are in satisfactory
agreement with those correspondingly found in the previous
PES work11–13 ͑see Fig. 2͒ and present EMS study ͑see Table
I͒, particularly for the outer valence orbitals. Second, our
ADC͑3͒ results predict that the ionization intensity resulting
from the inner valence 2a1 , 1b2 , 1b1 , and 1a1 orbitals is
severely split due to final state electron correlation effects.
For these orbitals, the fractions of intensity recovered under
the form of lines with a spectroscopic strength larger than
0.005 amount to 0.765, 0.697, 0.725, and 0.481, respectively.
This observation is entirely consistent with previous one-
particle Green’s function50–53 or MR-SDCI ͑Ref. 54͒ studies
of the ionization spectra of saturated hydrocarbons larger
than ethane. As has been noted earlier,50,51 the dispersion of
ionization intensity over many shake-up lines at energies
larger than 22 eV correlates well with significant band broad-
IV. THEORETICAL ANALYSIS OF VALENCE
IONIZATION SPECTRA
Vertical ionization spectra have been computed using
one-particle Green’s function ͑1p-GF͒ theory at the level of
the third-order algebraic diagrammatic construction
͓ADC͑3͔͒ scheme,39–42 in conjunction with Dunnings’ corre-
lation consistent polarized valence basis set of double-␨ qual-
ity ͓cc-pVDZ ͑Ref. 43͔͒, and with the original code inter-
faced to the GAMESS 92 package.31 With the 1p-GF/ADC͑3͒
approach, the primary one-hole (1h) and the shake-up two-
hole-one-particle (2h-1p) ionization energies are recovered
through third and first order in correlation, respectively. Con-
stant self-energy diagrams have been computed through
fourth order in correlation, using charge-consistent44 one-
electron densities. A threshold on pole strengths of 0.005 has
been retained for solving the ADC͑3͒ secular equation, using
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J. Chem. Phys., Vol. 121, No. 21, 1 December 2004
Norbornane electronic momentum spectroscopy
10531
TABLE III. Norbornane—electronic structure ͑theory͒. Binding energies are given in eV, along with the OVGF and ADC͑3͒ spectroscopic factors in
parentheses. Results obtained using ͑I͒ B3-YP/TZVP, ͑II͒ B3LYP/cc-pVTZ, and MP2/aug-cc-pVDZ geometries.
⑀f (eV)
Basis sets
Orbital
number
Present
classification
Present ADC͑3͒/
cc-pVDZ ͑I͒
Present OVGF/
cc-pVDZ ͑I͒
Present OVGF/
aug-cc-pVDZ ͑I͒
Present OVGF/
cc-pVTZ ͑I͒
Present OVGF/
cc-pVTZ ͑II͒
Present OVGF/
cc-pVTZ ͑III͒
Symbol
u
t
s
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
3a2
5b2
7a1
5b1
6a1
4b2
2a2
4b1
3b2
3b1
5a1
2b1
4a1
2b2
3a1
1a2
2a1
10.513 ͑0.91͒
10.863 ͑0.91͒
11.189 ͑0.91͒
11.657 ͑0.90͒
11.670 ͑0.91͒
12.102 ͑0.91͒
12.445 ͑0.91͒
12.645 ͑0.90͒
13.657 ͑0.90͒
13.736 ͑0.90͒
15.757 ͑0.89͒
15.948 ͑0.89͒
16.897 ͑0.88͒
17.866 ͑0.86͒
18.473 ͑0.86͒
19.926 ͑0.83͒
10.390 ͑0.91͒
10.746 ͑0.91͒
11.063 ͑0.91͒
11.513 ͑0.91͒
11.529 ͑0.91͒
11.986 ͑0.91͒
12.390 ͑0.91͒
12.545 ͑0.91͒
13.589 ͑0.91͒
13.687 ͑0.91͒
15.587 ͑0.91͒
15.734 ͑0.90͒
16.698 ͑0.90͒
17.817 ͑0.89͒
18.405 ͑0.88͒
19.953 ͑0.87͒
10.467 ͑0.91͒
10.830 ͑0.91͒
11.154 ͑0.91͒
11.607 ͑0.91͒
11.615 ͑0.91͒
12.072 ͑0.91͒
12.453 ͑0.91͒
12.629 ͑0.91͒
13.670 ͑0.91͒
13.762 ͑0.91͒
15.624 ͑0.90͒
15.771 ͑0.90͒
16.740 ͑0.89͒
17.843 ͑0.89͒
18.435 ͑0.88͒
19.979 ͑0.87͒
10.443 ͑0.91͒
10.793 ͑0.91͒
11.121 ͑0.91͒
11.555 ͑0.91͒
11.557 ͑0.91͒
12.043 ͑0.91͒
12.452 ͑0.91͒
12.592 ͑0.91͒
13.650 ͑0.91͒
13.755 ͑0.91͒
15.650 ͑0.90͒
15.784 ͑0.90͒
16.741 ͑0.89͒
17.872 ͑0.89͒
18.449 ͑0.88͒
19.988 ͑0.87͒
22.566 ͑0.85͒
10.392 ͑0.91͒
10.758 ͑0.91͒
11.075 ͑0.91͒
11.534 ͑0.91͒
11.554 ͑0.91͒
11.995 ͑0.91͒
12.406 ͑0.91͒
12.569 ͑0.91͒
13.605 ͑0.91͒
13.706 ͑0.91͒
15.619 ͑0.91͒
15.772 ͑0.90͒
16.746 ͑0.90͒
17.831 ͑0.89͒
18.429 ͑0.88͒
19.980 ͑0.88͒
10.359 ͑0.91͒
10.734 ͑0.91͒
11.055 ͑0.91͒
11.507 ͑0.91͒
11.507 ͑0.91͒
11.975 ͑0.91͒
12.353 ͑0.91͒
12.518 ͑0.91͒
13.557 ͑0.91͒
13.635 ͑0.91͒
15.552 ͑0.91͒
15.685 ͑0.91͒
16.649 ͑0.90͒
17.741 ͑0.89͒
18.335 ͑0.88͒
19.881 ͑0.87͒
r
q
p
n
m
l
k
j
i
h
g
f
e
d
a
b
b
b
b
21.695 ͑0.02͒
22.088 ͑0.09͒
22.389 ͑0.13͒
22.560 ͑0.85͒
22.588 ͑0.85͒
22.595 ͑0.85͒
22.497 ͑0.85͒
c
d
22.484 ͑0.51͒
23.573 ͑0.01͒
23.961 ͑0.01͒
22.493 ͑0.01͒
22.951 ͑0.39͒
22.960 ͑0.02͒
23.053 ͑0.04͒
23.162 ͑0.07͒
23.235 ͑0.01͒
23.397 ͑0.01͒
23.345 ͑0.08͒
23.448 ͑0.04͒
23.650 ͑0.02͒
23.968 ͑0.01͒
24.042 ͑0.01͒
24.108 ͑0.02͒
22.327 ͑0.01͒
22.555 ͑0.01͒
22.810 ͑0.01͒
23.167 ͑0.02͒
23.190 ͑0.02͒
23.287 ͑0.04͒
23.378 ͑0.08͒
23.444 ͑0.05͒
23.456 ͑0.01͒
23.533 ͑0.01͒
23.597 ͑0.17͒
23.663 ͑0.24͒
23.708 ͑0.01͒
24.091 ͑0.01͒
24.177 ͑0.01͒
24.263 ͑0.01͒
24.452 ͑0.01͒
24.514 ͑0.01͒
25.318 ͑0.01͒
25.410 ͑0.01͒
25.676 ͑0.01͒
26.104 ͑0.01͒
26.350 ͑0.01͒
26.411 ͑0.01͒
26.445 ͑0.01͒
26.459 ͑0.01͒
26.493 ͑0.01͒
b
b
b
b
c
18
1b2
23.288 ͑0.84͒
¯
23.286 ͑0.84͒
23.327 ͑0.84͒
23.256 ͑0.84͒
b
b
b
b
b
19
1b1
23.786 ͑0.84͒
¯
23.782 ͑0.83͒
23.834 ͑0.84͒
23.735 ͑0.84͒
a
20
1a1
¯
¯
¯
¯
¯
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10532
J. Chem. Phys., Vol. 121, No. 21, 1 December 2004
Knippenberg et al.
TABLE III. ͑Continued.͒
⑀f (eV)
Basis sets
Orbital
number
Present
classification
Present ADC͑3͒/
cc-pVDZ ͑I͒
Present OVGF/
cc-pVDZ ͑I͒
Present OVGF/
aug-cc-pVDZ ͑I͒
Present OVGF/
cc-pVTZ ͑I͒
Present OVGF/
cc-pVTZ ͑II͒
Present OVGF/
cc-pVTZ ͑III͒
Symbol
26.581 ͑0.01͒
26.655 ͑0.01͒
26.669 ͑0.01͒
26.685 ͑0.01͒
26.729 ͑0.02͒
26.804 ͑0.03͒
26.917 ͑0.02͒
26.930 ͑0.03͒
27.012 ͑0.01͒
27.099 ͑0.01͒
27.163 ͑0.01͒
27.183 ͑0.02͒
27.208 ͑0.01͒
27.228 ͑0.01͒
27.279 ͑0.04͒
27.287 ͑0.01͒
27.331 ͑0.03͒
27.352 ͑0.01͒
27.368 ͑0.01͒
27.385 ͑0.01͒
27.393 ͑0.01͒
27.402 ͑0.01͒
27.432 ͑0.02͒
27.437 ͑0.01͒
27.469 ͑0.02͒
27.518 ͑0.01͒
27.679 ͑0.01͒
27.784 ͑0.01͒
27.993 ͑0.01͒
a
20
1a1
¯
¯
¯
¯
¯
aDominant electronic configuration: 3a2Ϫ28aϩ1 1 (HOMOϪ2 LUMOϩ1).
bBreakdown of the MO picture of ionization; see J. Chem. Phys. 116, 7012 ͑2002͒.
cDominant electronic configuration: 5b2Ϫ28aϩ1 1 (HOMO-1)Ϫ2 LUMOϩ1
.
͔
͓
dDominant electronic configuration: 3a2Ϫ15b2Ϫ16bϩ1 1
.
ening on the experimental side ͓see the FWHM values re-
ported in Table I for peaks 11–13͔. Finally, the present cal-
culations confirm the empirical rule55 ͑and references
therein͒ that OVGF pole strengths smaller than 0.85 very
consistently foretell a breakdown of the MO picture of ion-
ization at the ADC͑3͒ level. In other words, the quasiparticle
approach that has been somewhat unfortunately referred to
over the last two decades as the OVGF approach, can also be
used for inner-valence states as long as the OVGF spectro-
scopic strengths remain larger than 0.85. Within that part of
the spectrum which can be reliably described by one-hole
states, i.e., up to binding energies of 20 eV, the OVGF and
ADC͑3͒ ionization energies do not differ by more than
ϳ0.13 eV. For the 2a1 orbital the MO picture still holds to
some extent, since among the identified satellites one of
them emerges at 22.5 eV, in the ADC͑3͒ ionization spectrum,
with rather dominant intensity (S(jf)ϭ0.51) and a rather clear
2aϪ1 1 one-hole character. At higher binding energies, how-
ever, the breakdown of the MO picture intensifies and the
OVGF approach can no longer be applied. Note that the
impact of diffuse functions on the one-hole ionization ener-
gies is very limited ͑Ͻ0.1 eV͒—see Table III. Convergence,
within ϳ0.1 eV accuracy, of the OVGF/cc-pVDZ and, by
extension, ADC͑3͒/cc-pVDZ ionization energies ͑with re-
gards to further improvements of the basis set͒ is also con-
firmed by comparison with the OVGF/cc-pVTZ results. Fi-
nally, the last two columns of Table III, obtained using
geometries optimized at the B3LYP/cc-pVDZ and MP2/aug-
cc-pVDZ levels, demonstrate the very limited dependence of
the computed ionization spectra on details of the molecular
structures. All in all, at the ADC͑3͒/cc-pVDZ level, we thus
expect accuracies of Ϯ0.2 eV on the computed vertical one-
electron ionization energies. Indeed, an agreement better
than 0.2 eV is found upon comparing the theoretical one-
electron binding energies reported in Table III with the He͑I͒,
He͑II͒, and EMS experimental values of Table I.
Nonetheless, a discrepancy of ϳ0.6 eV is noticed for the
1aϪ2 1 ionization line. Although one can never exclude some
calibration problems on the experimental side ͓the He͑I͒ and
He͑II͒ ionization energies reported in Ref. 13 can be in error
by approximately Ϯ0.2 eV͔, this unusually large discrepancy
most presumably relates to strong geometry relaxation ef-
fects and vibronic interactions in a molecule characterized by
pronounced cyclic strains. It can in particular be noticed that
the corresponding band in the He͑I͒ and He͑II͒ spectra,12,13
reproduced in Fig. 2, has a very asymmetric shape, which is
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J. Chem. Phys., Vol. 121, No. 21, 1 December 2004
Norbornane electronic momentum spectroscopy
10533
the 1a1 ionization intensity should normally be recovered
under the form of an extremely long correlation tail, extend-
ing from ϳ27 eV up to binding energies of 60 eV, and pos-
sibly beyond.
Upon performing further MP2/cc-pVDZ calculations of
the total energy of norbornane in its neutral and dicationic
ground (1A1) states, including full geometry optimization for
both species, it was found that the vertical and adiabatic
double ionization potentials of norbornane amount to 25.9
and 23.5 eV, respectively. Further studies, based on two-
particle Green’s function calculations of doubly ionized
states, or highly challenging one-particle Green’s function
calculations incorporating very diffuse functions, Coulomb
and distorted plane waves in the basis set might thus be
necessary for identifying with certainty the origin of band
12. Note that, as the 1a1 ionization intensity falls clearly
much above the double ionization threshold, the shake-up
lines which have been identified for that orbital should most
correctly be regarded as discrete ͑bound and excited͒ cationic
states embedded in a continuum of unbound ͑resonance and
shake-off͒ dicationic states.
Finally, we note that all the MP2, OVGF, and ADC͑3͒
calculations described in Sec. IV were carried out on a DEC-
Compaq ES40 workstation at the Limburgs Universitair Cen-
trum in Belgium.
V. COMPARISON BETWEEN EXPERIMENTAL
AND THEORETICAL MOMENTUM DISTRIBUTIONS
FIG. 2. Comparison between the measured ͑a͒ He͑I͒ ͑Ref. 12͒, ͑b͒ He͑II͒
͑Ref. 13͒, and ͑c͒ ADC͑3͒/cc-pVDZ theoretical ionization spectrum of nor-
bornane.
Deconvolving the ionization spectra measured at each of
a chosen set of angles ␾ by means of a least-squares-fit
technique20 allows us to derive the MD’s associated to each
of the bands identified in Figs. 1͑a͒ and 1͑b͒. Although the
measured MD’s are not absolute, relative magnitudes for the
different transitions are obtained.19 In the current EMS inves-
tigation of the valence states of NBA, the experimental MD’s
are placed on an absolute scale by summing the experimental
flux for each measured ␾ for the first ten outer valence or-
bitals, and then normalizing this to the corresponding sum
for our PWIA-BP/TZVP calculation.
a quite typical feature for such effects. Further studies of the
Franck–Condon vibrational profiles associated to this one-
electron ionization line would be necessary for quantitatively
clarifying this issue.
The most striking discrepancy between the EMS mea-
surements displayed in Fig. 1 and the ADC͑3͒/cc-pVDZ
spectrum of Fig. 2͑c͒ is the band ͑12͒ seen at 24.9 eV in the
experimental spectrum, which does not correlate to any set
of ionization lines with appreciable enough intensity on the
theoretical side. At this point, it is worth recalling that, be-
cause of the rather weakly correlated nature of wide band-
gap compounds such as saturated hydrocarbons, the expected
accuracies of vertical one-electron and shake-up ionization
energies at the ADC͑3͒/cc-pVDZ level are around 0.2 ͑see
above͒ and 0.6 eV, respectively. On the basis of the angular
dependence of band 12 ͑Fig. 1͒, and of the related MD,
which appear to be very similar to that of band 13 ͑see Sec.
V͒, it would be very tempting to assign both bands to orbital
1a1 . However, upon examining the ADC͑3͒/cc-pVDZ simu-
lation of Fig. 2͑c͒ and the corresponding data in Table III, it
is immediately apparent that the shake-up lines ascribed to
ionization of orbital 1a1 concentrate only around 27.5 eV.
By analogy with a band–Lanczos study52 of the valence ion-
ization spectra of n-alkanes, the missing fraction ͑52%͒ of
The results from this process for the unresolved HOMO
(3a2) and NHOMO (5b2) orbitals are shown in Fig. 3. In
this case we find very good agreement between all the cal-
culated PWIA-XC/DFT momentum distributions and our
corresponding EMS data taken in two independent runs ͑runs
A and B͒. Note that the error bars on all the MD data repre-
sent one standard deviation uncertainty. Further, note that the
experimental MD data from independent runs A and B are in
very good agreement with one another, a feature that is re-
peated for all the measured MD’s. The results in Fig. 3
strongly suggest that the EMS spectroscopic factors for both
the respective 3a2 and 5b2 orbitals are ϳ1. This observation
is entirely consistent with our calculated ADC͑3͒ and OVGF
spectroscopic factors for these orbitals ͑see Table III͒. Al-
though not shown, a similar level of agreement between the
experimental and theoretical MD’s is found for the 7a1 or-
bital. This result implies SE7aMS (⑀fϭ10.9 eV)ϳ1, which is
1
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10534
J. Chem. Phys., Vol. 121, No. 21, 1 December 2004
Knippenberg et al.
FIG. 3. 1500 eV symmetric noncoplanar MD for the 3a2ϩ5b2 orbitals or
norbornane (⑀fϳ10.3 eV). The present data for run A ͑᭹͒ and run B ͑ᮀ͒ are
compared against the results of our PWIA-DFT calculations: ͑- - - -͒ BP/
FIG. 5. 1500 eV symmetric noncoplanar MD for the 4b2ϩ2a2ϩ4b1 orbit-
als of norbornane (⑀fϳ12.4 eV). The legend is the same as that for Fig. 3.
• • • •
DZVP, ͑– –͒ BLYP/DZVP, ͑• – •–͒ BP/DZVP2, ͑
͒ BLYP/DZVP2,
͑
͒ BP/TZVP, and ͑- –͒ BLYP/TZVP. Acronyms are defined in the text.
of the BLYP exchange correlation functional and DZVP ba-
sis set is not providing a very good representation of these
orbitals. While it is a less striking effect, Fig. 4 also appears
to indicate, for momenta in the region 0.1 a.u.рpр0.6 a.u.,
that the PWIA-BLYP/DZVP2 MD somewhat underestimates
the magnitude of the experimental MD. Nonetheless, the
good level of agreement between theory and experiment for
the remaining XC/DFT basis set results indicates that the
EMS spectroscopic factors of both the 5b1 and 6a1 orbitals
are respectively ϳ1. This finding is consistent with the MO
picture of ionization being valid here for these outer-valence
orbitals, a result in good agreement with our ADC͑3͒ and
OVGF calculations of Table III.
also in good accord with our calculated ADC͑3͒ and OVGF
pole strengths ͑see again Table III͒.
In Fig. 4 we show the measured and calculated MD’s for
the 5b1ϩ6a1 orbitals of norbornane. In this case we find
that the momentum distributions calculated at the BLYP/
DZVP level within the Plane Wave Impulse Approximation
significantly overestimates the magnitude of the experimen-
tal cross section for all p. This indicates that the combination
The present MD’s for the 4b2ϩ2a2ϩ4b1 orbitals of
norbornane are shown in Fig. 5. In this case there is a very
interesting trend for momenta in the range 0.1 a.u.рp
р0.55 a.u.. Specifically, in this region all the PWIA-BLYP/
DFT MD’s exhibit a somewhat higher cross section magni-
tude compared to all the corresponding PWIA-BP/DFT
MD’s with the experimental cross sections favoring the
PWIA-BP/DFT results. This is quite unusual in our
experience10,15,16 as typically we have found that our experi-
mental MD’s are more discriminating in terms of the types of
basis sets employed, rather than the type of XC functional
used. We would characterize the overall level of agreement
between our PWIA-BP/DFT momentum distribution results
and the experimental momentum distributions as being good,
suggesting EMS spectroscopic factors for each of these or-
bitals lying somewhere in the range 0.9–1.0. Such EMS
spectroscopic factors for the 4b2 , 2a2 , and 4b1 orbitals are
found again to be in good agreement with the predictions
from our ADC͑3͒ and OVGF calculations, as can be seen in
Table III.
The 4a1 orbital momentum distributions are illustrated
in Fig. 6. In this case we see that all the MD’s are strongly
FIG. 4. 1500 eV symmetric noncoplanar MD for the 5b1ϩ6a1 orbitals of
norbornane (⑀fϳ11.6 eV). The legend is the same as that for Fig. 3.
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J. Chem. Phys., Vol. 121, No. 21, 1 December 2004
Norbornane electronic momentum spectroscopy
10535
FIG. 6. 1500 eV symmetric noncoplanar MD for the 4a1 orbital of norbor-
nane (⑀fϳ16.5 eV). The legend is the same as that for Fig. 3.
peaked ͑large cross section͒ as p→0 a.u., indicating an
s-type symmetry9 which is probably due to strong C(2s)
contributions. For pу0.2 a.u. all the theoretical MD’s are in
good agreement with each other and with the experimental
MD results. For pϽ0.2 a.u., however, only the BP/TZVP,
BLYP/TZVP, and, to a lesser extent, the BP/DZVP models
are providing a good description of the measurements. When
we combine this observation with what we have previously
discussed from Figs. 4 and 5, we start to see a trend emerg-
ing. Namely, in the one-electron ionization part of the spec-
trum, the BP/TZVP model gives overall the most accurate
description for each of the experimental MD’s. Note that this
observation also holds for all the MD’s we do not specifi-
cally plot. Hence, from the results obtained for the one-
electron ionization lines, the BP/TZVP wave function ap-
pears to be one of the best suited wave functions for studying
further structural, vibrational and electronic properties of
norbornane—see Sec. VI.
Let us now consider the most challenging part of the
ionization spectrum, namely, the inner valence region be-
yond the shake-up threshold at ϳ22 eV. In Fig. 7͑a͒ we plot
the experimental MD for the sum of peaks 11–13 of Fig. 1,
and the corresponding theoretical MD’s from the models
considered. Here all the theoretical MD’s do a fair job in
predicting the shape of the experimental result, although they
all underestimate the magnitude of the experimental cross
section across most of the measured momentum range. This
result might reflect a breakdown in the inner valence region
for the PWIA description of the reaction mechanism. There
certainly exists a large body of evidence that shows that for
certain atomic systems9 the PWIA breaks down for inner
valence orbitals. In these cases the (e,2e) ionization process
has to be described within a distorted wave framework.9
The ADC͑3͒ calculation suggests that peak 11 originates
␮
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10536
J. Chem. Phys., Vol. 121, No. 21, 1 December 2004
Knippenberg et al.
FIG. 8. 1500 eV symmetric noncoplanar MD for the shake-up band 13 and
the 1a1 orbital of norbornane (⑀fϳ27.5 eV). The legend is the same as that
*
for Fig. 3, except 0.5 BP/TZVP ͑–͒ is also shown.
supports the idea that there is additional 1a1 flux at ⑀f
Ͼ29 eV. As peak 12 has a similar ͑although by no means
identical͒ MD to that of peak 13 ͑see Fig. 9͒, it is tempting to
conclude that it too might originate from the 1a1 orbital.
However, as noted earlier, our ADC͑3͒ calculation does not
support such an assignment. It is possible that peak 12 partly
originates from the 2a1 orbital with some additional 1b1 and
1b2 contributions. Such a scenario is allowed by our
FIG. 7. ͑a͒ 1500 eV symmetric noncoplanar MD for peaks 11–13 in the
ionization spectrum of norbornane. The legend is the same as that for Fig. 3.
͑b͒ 1500 eV symmetric noncoplanar MD for the shake-up band 11 and the
2a1ϩ1b1ϩ1b2 orbitals of norbornane. The legend is the same as that for
Fig. 3.
mainly from the 2a1 , 1b2 , and 1b1 orbitals and the present
EMS experimental MD for this peak supports such a notion.
As can be seen from Fig. 7͑b͒, the experimental MD for
2a1ϩ1b2ϩ1b1 orbitals has very good shape agreement
with the corresponding theoretical MD’s, although as it
might be expected from Fig. 7͑a͒ there is a mismatch in the
magnitude of these cross sections. Nonetheless, the present
experimental momentum profile exhibits clearly a minimum
at pϳ0.2 a.u., in fair agreement with the theoretical predic-
tions for the summed 2a1ϩ1b2ϩ1b1 orbital set, and thus
nicely reflects the fact that band 11 consists of a mixture of
ionization lines with s-type and p-type symmetries.
If we consider the experimental momentum distribution
for peak 13, compared to 0.5ϫ1a1 for PWIA-BP/TZVP ͑see
Fig. 8͒, then we see that the level of agreement between them
is quite good. This is strong evidence that peak 13 largely
originates from the innermost valence 1a1 orbital, a result
which is consistent with our ADC͑3͒ findings. We would like
to recall that the missing experimental flux ͑ϳ50%͒ is ex-
pected to be found at binding energies beyond the range
sampled in the present study. There is evidence in Fig. 1 that
FIG. 9. 1500 eV symmetric noncoplanar MD for band 12 of the EMS
binding energy spectra. The present data for run A ͑᭹͒ and run B ͑ᮀ͒ are
shown.
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J. Chem. Phys., Vol. 121, No. 21, 1 December 2004
Norbornane electronic momentum spectroscopy
10537
ADC͑3͒/cc-pVDZ results ͑Table III͒ which suggest that up to
23.5%, 30.3%, and 27.5% of the 2a1 , 1b1 , and 1b2 fluxes
might reside under peak 12, respectively, in the form of long
correlation tails52 consisting of shake-up lines with a spec-
troscopic strength smaller than 0.005. However, even upon
admitting that this missing fraction of the 2a1 shake-up in-
tensity would be entirely recovered under peak 12, it would
still be far too small to explain the intensity of this peak in
the spectrum recorded at the azimuthal angle ␾ϭ0°, relative
to that of band 11 ͓Fig. 1͑a͔͒. This, the fact that the 1p-GF/
ADC͑3͒ and density functional theories of ionization and
(e,2e) cross sections provide very consistent insights into
the shape, energy location, and into the momentum distribu-
tions characterizing the neighboring peaks 11 and 13, and the
vast experience accumulated over the last 25 years with
1p-GF calculations of the shake-up transitions of saturated
hydrocarbons51–53 and many other molecules ͑see, for in-
stance, Refs. 40, 42, 55–57 and references therein͒, lead us
to believe that band 12 does not belong to the vertical one-
electron and 2h-1p shake-up ionization spectrum of norbor-
nane in its ground electronic state, as described by the
ADC͑3͒ model of ionization. A band-Lanczos study of the
correlation tails in the ionization spectrum of NBA might be,
however, useful to fully confirm this assertion.
FIG. 10. Structural representation of norbornane and the atom numbering.
study,4,6 and 1.578 Å from Fitch and Jobic’s powder x-ray
diffraction study.5 The remaining carbon-carbon bonds in-
volving the bridge or bridgehead carbon atoms are also in
excellent agreement with experiment. The agreement with
experiment is better than for the small basis set ab initio and
semiempirical MO-derived geometries in Table IV.8,58 The
distance between the two single bonds involving the four
methylene carbon atoms (C2-C3 and C5-C6) was particularly
well reproduced with the C2¯C6 distance from BP/TZVP of
2.520 Å compared with the experimental distance of 2.542 Å
from powder x-ray diffraction studies.
Finally, we note that there are still quite a few orbital
MD’s that we have not specifically discussed or plotted in
this section. Plots of these MD’s are available on request to
the corresponding author ͑M.J.B.͒. These MD’s reinforce the
argument for the utility of BP/TZVP that we have made in
this section, but do not add any further insight.
Bond angles were also well reproduced, especially the
bridge and bridgehead angles. The bridge angle ͑e.g.,
ЄC1C7C4) of 94.5° from our DFT calculations compares
well with 93.1° from the x-ray structure and 93.4° from elec-
tron diffraction. The bridgehead angles ͑e.g., ЄC2C1C7)
were calculated to be 101.4° by our DFT calculation, com-
pared with 102.0° from electron diffraction studies, and
99.3° from the x-ray diffraction studies. There was some
evidence of lattice perturbations in the x-ray structure when
compared with the electron diffraction structure and the
structures predicted by MO methods, as illustrated in Table
IV. For example, the bridgehead bond angle ЄC2C1C6 is
substantially larger in the x-ray structure than in the other
experimental and theoretical structures, as is the angle
ЄC1C2C3 , which is approximately 4° larger than in the
other structures.
VI. MOLECULAR PROPERTY INFORMATION
We now use the BP/TZVP model which best described
the experimental MD’s to derive the structure and a selection
of the molecular properties of norbornane. These are com-
pared in detail with independent experimentally determined
values and those from other MO calculations, to determine
how well the BP/TZVP model was able to reproduce these
molecular properties.
A. Molecular geometries
In general, our calculations of molecular geometries us-
ing the BP/TZVP model are in very good agreement with
experimentally determined molecular geometries ͑given the
experimental uncertainties͒, and compare favorably with the
results from other MO calculations. The results are summa-
rized in Table IV. Note that in Table IV we have also in-
cluded relevant data from our B3LYP/cc-pVTZ and MP2/
aug-cc-pVDZ calculations. While these basis sets were not
prevalidated using our EMS MD’s, we have included them
for completeness and in general their results appear to com-
pare well with those from BP/TZVP. Further note that to
assist the reader in the discussion that follows, a structural
representation and atom numbering of the norbornane mol-
ecule is given in Fig. 10.
B. Dipole moment
Like all saturated hydrocarbons, norbornane has a small
dipole moment which has been well reproduced by our BP/
TZVP DFT calculations. We obtain a value of 0.076 D from
our calculations compared with a very accurate value of
0.091͑8͒ D inferred from the Stark effect in the microwave
spectrum of norbornane.3 Wilcox and co-workers had earlier
estimated the dipole moment as 0.03͑2͒ from dielectric
measurements,59 which appears to be too low.
The two single bonds (C2-C3 and C5-C6) involving the
four methylene carbon-carbon have bond distances of 1.571
Å from our calculations, in excellent agreement with the two
experimental values of 1.573 Å from an electron diffraction
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10538
J. Chem. Phys., Vol. 121, No. 21, 1 December 2004
Knippenberg et al.
TABLE V. 13C NMR chemical shifts ͑in ppm͒.
error of 0.09 ppm. Put another way, at the GIAO level the
BP/TZVP approach yields overestimates between 4% and
10%, in the experimental proton shifts.
*
BP/TZVP BP/TZVP HF/6-31ϩG
BP/TZVP
Carbon Experimental LORG
IGLO
GIAO ͑Ref. 66͒ GIAOa
The well-known differences in the chemical shifts be-
tween the endo and exo protons in norbornane are very
nicely reproduced by our BP/TZVP calculations. At this
level, and using the GIAO approach, we calculate a differ-
ence of 0.34 ppm compared with the experimental difference
of 0.31 ppm.
1
2
3
7
6
5
4
36.8
30.1
30.1
38.7
30.1
30.1
36.8
46.5
35.0
35.0
43.5
35.0
35.0
46.5
47.8
36.3
36.3
45.2
36.3
36.3
47.8
33.2
27.4
27.4
34.4
27.4
27.4
33.2
43.0
34.6
34.6
41.9
34.6
34.6
43.0
a
*
Results obtained using a B3LYP/6-31G geometry.
D. Vibrational spectra
The DFT calculations were able to calculate the frequen-
cies of the vibrational modes of norbornane with reasonable
accuracy. Table VII shows the vibrational frequencies calcu-
lated at the BP/TZVP level in the present work. The calcu-
lated intensities of the transitions are also in reasonable
agreement with the observed68 experimental IR spectrum of
norbornane, as Table VII also illustrates. The level of agree-
ment between our ͑unscaled͒ BP/TZVP frequencies and ex-
periment is similar to that of the work of Shaw et al.,69 who
studied the norbornane infrared spectrum using a rescaled
HF/3-21G ab initio force field. The assignment of the nor-
bornane vibrational modes follows from the work of Levin
and Harris.70 For completeness we note that according to the
dipole selection rules for IR spectroscopy, transitions from
the zero-point level to the excited vibrational levels belong-
C. NMR properties
There have been many measurements of the chemical
shifts60–62 of carbon and protons in norbornane, examples
of which are the work of Abraham and co-workers60 and
Lippmaa et al.61 We used the localized orbital/local origin
͑LORG͒,63 individual gauge localized orbitals ͑IGLO͒ ͑Ref.
64͒ and gauge-independent atomic orbital ͑GIAO͒ methods65
to calculate 13C chemical shifts from our BP/TZVP calcula-
tions. Chemical shifts were determined by comparisons with
1
the H and 13C isotropic shifts computed for tetramethylsi-
lane at the BP/TZVP level. Our chemical shift values are
compared in Tables V and VI with those determined by
Sauers66 from a GIAO calculation using Hartree–Fock
theory. As in many previous computations of NMR chemical
shifts ͑see Ref. 67 and references therein͒, these HF results
systematically underestimate the experimental values
whereas the opposite is seen with our BP/TZVP results.
The LORG method produced better agreement with the
experimental 1H and 13C chemical shifts than the IGLO
method, particularly for the proton spectrum. However, it
appears that when a correlated wave function is used, the
GIAO approach provides the best agreement with experi-
ment. At this level, the chemical shifts for carbon predicted
by our DFT calculations are overall in good agreement with
the experimental shifts, although the bridgehead ͑methine͒
carbons had a larger error ͑ϳ7 ppm͒ than the other ͑methyl-
ene͒ carbons ͑error ϳ3 ppm͒. The proton chemical shifts
were in excellent agreement with experiment with an average
ing to the a2 irreducible representation of the C2 point
v
group are forbidden by symmetry.71 One of these transitions
is nonetheless detected in the IR spectrum of norbornane, in
the form of an extremely weak line at 542 cmϪ1. This line
must thus be described as a hot band.
VII. CONCLUSIONS
We have reported on the first comprehensive EMS study
into the complete valence electronic structure of norbornane,
in conjunction with DFT calculations of orbital MD’s and
1p-GF ͓OVGF and ADC͑3͔͒ calculations of the one-electron
and shake-up ionization spectrum. Excellent agreement is
generally found between the experimental PES and EMS
binding energies on the one hand and the 1p-GF results on
the other hand. Where a comparison is possible, pole
strengths calculated by our 1p-GF procedures, certainly for
the outer valence orbitals, were found to be largely consistent
with those determined from our EMS MD data. Strong final
state configuration interaction effects are predicted in our
ADC͑3͒ calculation for the inner valence 2a1 , 1b2 , 1b1 ,
and 1a1 orbitals, and this prediction is consistent with the
very significant band broadening observed at binding ener-
gies beyond ϳ22 eV. A striking discrepancy between one-
particle Green’s function theory and experiment has been
noted, however. It takes the form of a very intense band at
ϳ25 eV in the EMS spectrum recorded at an azimuthal angle
␾ϭ0°, which could not be reproduced by the large scale
ADC͑3͒ calculations presented in this work. According to the
related momentum distribution, this band has apparently
s-type symmetry. Further theoretical studies will be needed
to establish whether it relates, for instance, to shake-up tran-
sitions to particularly diffuse bound states, to double ioniza-
TABLE VI. 1H NMR chemical shifts ͑in ppm͒.
*
BP/TZVP BP/TZVP HF/6-31ϩG
Proton Experimental LORG
BP/TZVP
IGLO
GIAO ͑Ref. 66͒ GIAOa
1
2
2
3
3
4
5
5
6
6
7
2.19
1.16
1.47
1.16
1.47
2.19
1.16
1.47
1.16
1.47
1.18
2.36
1.27
1.49
1.27
1.49
2.36
1.27
1.49
1.27
1.49
1.23
4.56
3.32
3.68
3.32
3.68
4.56
3.32
3.68
3.32
3.68
3.08
1.91
2.28
1.28
1.62
1.28
1.62
2.28
1.28
1.62
1.28
1.62
1.25
1.12b
1.37b
1.12b
1.37b
1.92b
1.12b
1.37b
1.12b
1.37b
1.13
Ј
Ј
Ј
Ј
a
*
Results obtained using a B3LYP/6-31G geometry.
bThis work.
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J. Chem. Phys., Vol. 121, No. 21, 1 December 2004
TABLE VII. Infrared vibrational frequencies and intensities.
Norbornane electronic momentum spectroscopy
10539
Experimental spectrum
BP/TZVP spectrum
Mode TZVP ͑cmϪ1 Intensity ͑km molϪ1
͓Levin and Harris ͑Ref. 70͔͒
Symmetry
label
͒
͒
Frequency ͑cmϪ1
͒
Intensity
Assignment
a2
b2
a1
b1
a2
a1
b2
b1
a1
b2
a1
b1
a1
a2
b1
a2
b2
a1
b1
b2
b1
a2
a1
b2
b1
a2
b2
a1
b2
a2
a2
a1
b2
b1
a2
a1
b2
b1
a1
b2
a2
a1
b1
a1
a2
b2
b1
b2
a1
b1
a1
7
8
9
164.82
332.85
392.74
437.64
532.85
738.32
744.02
776.35
797.49
804.12
857.80
873.00
908.14
926.69
927.12
937.65
937.94
0.0
0.2
0.0
0.0
0.0
0.8
0.1
0.4
0.0
2.7
1.4
1.3
1.6
0.0
0.5
0.0
0.6
0.1
0.4
0.2
0.1
0.0
1.0
0.3
3.2
0.0
1.2
1.0
0.0
0.0
0.0
1.8
2.6
0.0
0.0
7.3
2.1
5.9
0.5
59.9
0.0
51.4
95.4
15.6
0.0
10.8
91.4
4.2
1.9
62.8
94.0
344
407
485
542
755
w
w
w
26, 39, 51
v
v
v
v
v
v
v
15
35
14
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
w
v
s
13, 99
v
787
ms
24, 37
v
v
814
874
889
925
s
s
s
s
13
48
11
v
v
v
949
m
36, 47
v
v
958
990
w
w
m
w
w
w
m
m
w
23
10
v
v
v
v
v
v
v
v
973.06
1004.32
1054.78
1092.25
1103.68
1125.17
1136.82
1186.00
1195.51
1225.65
1236.83
1242.20
1253.65
1276.66
1293.44
1293.70
1297.16
1433.85
1439.98
1446.88
1452.03
1475.76
2962.27
2963.18
2964.46
2973.15
2973.44
2996.72
2998.48
3012.16
3012.82
3016.85
3017.78
3022.19
1031
1069
1091
1103
1120
1140
1160
1207
1217
1242
1259
35, 46
v
31
22
34
9
33
m
mw
w
8, 45
v
v
v
v
v
30
32
7
mw
1274
1301
1317
1400
w
m
m
w
44
19
31
v
v
v
1442
1455
1465
m
s
18
v
v
v
6, 30, 43
v
v
5
2866
2912
2928
2954
2964
m
m
m
s
s
v
v
tion processes, or to autoionization via electronically excited
and dissociating states.72 The latter suggestion is in particular
worthy of consideration, in light of the extent of the cyclic
strains in a compound such as norbornane. On the experi-
mental side, further He͑II͒, Penning ionization and XPS
studies of the innermost valence levels of norbornane are
also clearly necessary.
ϩ6a1, 4b2ϩ2a2ϩ4b1 , 3b2ϩ3b1 , 5a1ϩ2b1 , 4a1 , 2b2 ,
3a1 , 1a2 , 2a1ϩ1b2ϩ1b1 , and 1a1 orbitals were measured
and compared against a series of PWIA-based calculations
using DFT DGAUSS basis sets. Our calculations, for each of
the three basis sets ͑DZVP, DZVP2, TZVP͒, were performed
using both BP and BLYP exchange correlation corrections to
the DFT functional. On the basis of this comparison between
the experimental and theoretical MD’s, we found that BP/
Momentum distributions for the 3a2ϩ5b2 , 7a1 , 5b1
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10540
J. Chem. Phys., Vol. 121, No. 21, 1 December 2004
Knippenberg et al.
TZVP provided the most physically reasonable representa-
tion of the NBA wave function. Molecular property informa-
tion derived from this ‘‘optimum’’ BP/TZVP wave function
was seen to be in generally good agreement with the results
from independent measurements. This provides compelling
evidence for the pedigree of EMS in a priori evaluation of a
quantum chemical wave function. For a molecule such as
NBA, where unambiguous molecular geometry information
is not readily available from traditional methods, this can be
particularly useful.
Our next major study will concentrate on the valence
electronic structure of norbornene (C7H10 , NBN). We pro-
pose this investigation in order to probe how the electronic
structure of the chemically similar nonbonnadione ͑C7H8,
NBD, NBN, and NBA molecules changes as the double
bonds of NBD are progressively saturated. That study will
search for any discernible trends, particularly in the momen-
tum distributions, and if so can we quantify them in a logical
manner.
Finally, the present work highlights the need for imple-
menting more efficient diagonalization approaches that pre-
serve the total spectral moments for exhaustively studying
with larger basis sets the innermost correlation tails in the
1p-GP/ADC͑3͒ ionization spectra. Also, we note that an im-
provement in the (e,2e) reaction mechanism description,
particularly for the inner valence and core orbitals, by the
development of a distorted wave framework9 for multicen-
tred targets ͑i.e., molecules͒ is still desirable. While this is a
very difficult task, a clear need for its implementation exists.
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This work was supported in part by the Australian Re-
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while another ͑K.L.N.͒ thanks the Ferry Trust for her schol-
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S.K., and J.P.F. acknowledge financial support from the
Bijzonder Onderzoeks Fonds ͑BOF͒ of the Limburgs Univer-
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