Cyclic-N
3
. II. Significant geometric phase effects in the vibrational spectra
Dmitri Babikov
Chemistry Department, Marquette University, Wehr Chemistry Building, Milwaukee, Wisconsin 53201
Brian K. Kendrick
Theoretical Chemistry and Molecular Physics (T-12), Los Alamos National Laboratory, Los Alamos MS
B268, New Mexico 87545
Peng Zhang and Keiji Morokuma
Department of Chemistry, Emory University, Chemistry Building, Atlanta, Georgia 30322
͑Received 30 July 2004; accepted 5 October 2004; published online 11 January 2005͒
An accurate theoretical prediction of the vibrational spectra for a pure nitrogen ring ͑cyclic-N
3
)
molecule is obtained up to the energy of the
2
A
2
/
2
B
1
conical intersection. A coupled-channel
approach using the hyperspherical coordinates and the recently published ab initio potential energy
surface ͓D. Babikov, P. Zhang, and K. Morokuma, J. Chem. Phys. 121, 6743 ͑2004͔͒ is employed.
Two independent sets of calculations are reported: In the first set, the standard Born–Oppenheimer
approximation is used and the geometric phase effects are totally neglected. In the second set, the
generalized Born–Oppenhimer approximation is used and the geometric phase effects due to the
D
3h
conical intersection are accurately treated. All vibrational states are analyzed and assigned in
terms of the normal vibration mode quantum numbers. The magnitude of the geometric phase effect
is determined for each state. One important finding is an unusually large magnitude of the geometric
phase effects in the cyclic-N
3
:itisϳ100 cm
Ϫ1
for the low-lying vibrational states and exceeds 600
cm
Ϫ1
for several upper states. On average, this is almost two orders of magnitude larger than in the
previously reported studies. This unique example suggests a favorable path to experimental
validation. © 2005 American Institute of Physics. ͓DOI: 10.1063/1.1824905͔
I. INTRODUCTION
The previously unknown energetic form of nitrogen has
been recently predicted theoretically
1
and produced
experimentally.
2–5
This is a stable ring-N
3
isomer ͑having the
form of an isosceles triangle͒ called cyclic-N
3
hereafter.
Cyclic-N
3
is metastable with respect to dissociation to the
ground state N(
4
S) ϩN
2
(⌬E ϭϪ1.4 eV), which is spin for-
bidden. Furthermore, recent results
6
show that the doublet-
quartet surface crossings that must be traversed for dissocia-
tion lie about 1 eV above the cyclic-N
3
minimum. Thus,
cyclic-N
3
is very stable and carries a lot of energy; it is an
excellent new candidate for technological applications in en-
ergy storage, high nitrogen explosives, and clean propellants.
It is worth mentioning that the nitrogen resources on our
planet are practically limitless.
The cyclic-N
3
is a new molecule and experimental stud-
ies of it have been somewhat ahead of theory. Valuable the-
oretical guidance for designing experiments and for inter-
preting experimental results has been notably lacking during
the last couple years. This paper is the second one in a series
of theoretical papers we intend to publish which focus on
cyclic-N
3
. In the first paper,
7
we presented an accurate ab
initio potential energy surface ͑PES͒ for cyclic-N
3
. Cyclic-
N
3
is a Jahn–Teller molecule that exhibits a conical intersec-
tion between two of its potential energy surfaces at the D
3h
͑equilateral triangle͒ configuration.
6,7
That conical intersec-
tion causes the equilibrium geometry to distort off the D
3h
geometry. In the present paper, we report calculations of the
vibrational states of cyclic-N
3
with particular emphasis on
the associated geometric phase effects.
The origin of the geometric phase dates back to 1963
when Herzberg and Longuet-Higgins
8
showed that a Born–
Oppenheimer electronic wave function changes sign for any
closed path in the nuclear parameter space which encircles a
conical intersection of two electronic PESs. The geometrical
interpretation of the sign change was first recognized by
Mead and Truhlar
9
in 1979. They showed that the sign
change can be expressed in terms of the ‘‘magnetic flux’’ due
to a pseudomagnetic solenoid centered at the degeneracy
point. Later, Mead
10
called this effect the ‘‘molecular
Aharonov–Bohm’’ effect. In 1984, Berry
11
showed that the
sign change was a special case of a more general geometric
phase factor often referred to as ‘‘Berry’s phase.’’ Due to the
universal nature of this effect, Berry’s influential paper gen-
erated widespread interest which continues to this day.
As noted by Ham,
12
probably the first experimentally
verified example of a geometric phase effect was in crystal
defects with strong Jahn–Teller coupling where the lowest
vibronic state was shown to have E symmetry instead of A
1
or A
2
.
13
The first experimental verification of this ordering
was for Cu
2ϩ
in MgO using electronic paramagnetic reso-
nance ͑EPR͒.
14
This ordering is a direct consequence of the
geometric phase. If it were not properly included, the oppo-
site ordering would be predicted ͑i.e., the lowest vibronic
state would be A
1
or A
2
).
The theoretical treatment of geometric phase effects in
molecular spectra date back to Longuet–Higgins et al.
15
who
THE JOURNAL OF CHEMICAL PHYSICS 122, 044315 ͑2005͒
122, 044315-10021-9606/2005/122(4)/044315/20/$22.50 © 2005 American Institute of Physics