Chemistry Reference
In-Depth Information
theorem, each ionization energy
I
E
corresponds to an orbital energy, hence the
photoelectron spectrum of a molecule should be a direct representation of the
MO energy diagram. However, the photoelectron spectra of molecules are rather
complex, as can be appreciated from Fig. 1.1(b). Ionized N
2
exhibits photoemis-
sion lines at 15.6, 16.7-18.0 and at 18.8 eV. The 15.6 eV line corresponds to the
3
σ
−
g
MO, where the
1 superscript represents the removal of one electron. The
group of peaks between 16.7 and 18.0 eV correspond to
−
π
−
1
and finally the line at
u
σ
−
u
. In fact, each MO gives a group of peaks in the spec-
trum because of the possibility of vibrational as well as electronic excitations. Each
resolved peak stands for a single vibrational line and represents a definite number of
quanta of vibrational energy of the molecular ion (Eland, 1984). Ionization is a fast
process, about 10
−
15
s being required for the ejected electron to leave the immediate
neighbourhood of the molecular ion. The time is so short that motions of the atomic
nuclei that make up vibrations and proceed on a much longer time scale of 10
−
13
s
can be considered as frozen during ionization. This results in many accessible
vibrational states and the effect is known as the Franck-Condon principle.
The fact that the number of peaks largely exceeds the number of MOs shows the
limitation of Koopmans' theorem. In the case of the 2
18.8 eV corresponds to 2
σ
g
MOs, at least five features
are associated to them in the photoemission spectra (Krummacher
et al.
, 1980),
which is a consequence of the strong electron correlation in the ionic state. Bond-
ing orbitals should exhibit distinct vibrational structure as compared to non-bonding
orbitals because removal of one electron should strongly perturb the orbitals in-
volved. This is clearly observed for the 16.7-18.0 eV features, corresponding to
bonding
π
u
orbitals, as well as for the features around 15.6 and 18.8 eV, associated
with the non-bonding 3
u
MOs.
Since we are here interested in solid nitrogen, we can ask ourselves which are
the aggregation states of a simple molecule such as N
2
. The answer comes in the
form of a phase diagram, shown in Fig. 1.2. This pressure vs. temperature phase
diagram reveals a certain degree of complexity, including several solid crystallo-
graphic phases or polymorphs (Bini
et al.
, 2000; Gregoryanz
et al.
, 2002). From
the fluid phase, which exists up to
c
. 2 GPa (1 GPa
σ
g
and
σ
=
10 kbar) at RT, nitrogen
solidifies in a disordered hexagonal
-phase, where the N
2
molecular axes are
randomly distributed. This is a common feature of diatomic molecular crystals.
Low-pressure and low-temperature cubic
β
-phases represent two ways of
ordered quadrupolar packing. At higher pressures, other classes of structures with
non-quadrupolar-type ordering exist (
α
- and
γ
).
Because of the weak intermolecular interactions involved, the energy bands
associated with solid N
2
can be generated, as a first approximation, by gently
broadening the N
2
MOs, resulting in small finite bandwidths
W
of a few tenths of
eV. The solid bands would originate from the perturbation of the molecular levels
δ
,
δ
loc
,
ε
,
ζ
