3 Results and Discussion
3.1 Negative Core Ion NO
−
3 HNO 3
ð
Þ 2
Fifteen different geometries were obtained in the geometry searching of
NO
−
3 HNO 3
ð
Þ 2 . We classified them into two groups, α and β, according to the
number of oxygen atom(s) on NO
−
3 making HBs with two HNO 3 ’s. In the group α
containing three kinds of conformers having a different angle between molecular
planes of NO
−
3 and HNO 3 , one oxygen atom on NO
−
3 has two HBs with two
HNO 3 ’s. In the group β containing twelve kinds of conformers, two different
oxygen atoms on NO
−
3 have one HBs with different HNO 3 ’s. The stable geometries
of all conformers and their relative energies at 0 K (ΔE ZPE ) are shown in Fig. 2,
where the most stable structure at 0 K is the structure α1. Drenck and coworkers
reported the stable structure similar to the structure β9 at B3LYP/6-31++G** level
of DFT calculations [1]. At MP2/6-31++G** level of ab initio calculations,
however, the total energy of the structure β9 is 1.6 kcal/mol higher than that of the
most stable structure α1.
The present ab initio calculations suggested that the negative core ion has the
structure of α1 as the most stable structure at 0 K. It is, however, strongly expected
that the relative abundance of these conformers depends on a thermal condition,
because most of the relative energies at 0 K are within a few kcal/mol. The temperature dependence of relative abundance of conformers is shown in Fig. 3, where
the relative abundance of ith conformer (p i ) at the temperature T is estimated in the
standard way as
p i ðTÞ = exp − ΔG i ̸ k B T
f
g̸ Z, Z = ∑
all
j
exp − ΔG j ̸ k B T
È
É
,
ð1Þ
where ΔG i is the relative Gibbs energy including the enthalpic (H: the sum of the
electronic energy, ZPE, thermal corrections, etc.) and the entropic terms (−TS). In
Fig. 3, the temperature dependences of p i values for all conformers are shown in the
upper figure (a), and those of the sum of p i values over conformers in each group
are given in the lower figure (b). Figure 3 clearly indicates that the relative abundance of NO
−
3 HNO 3
ð
Þ 2 conformers depends on the temperature, and the inversion
of the total abundance of the groups α and β is found at around 250 K. We note
here that the most stable structure α1 at 0 K is a dominant conformer in the low
temperature region below 90 K, while the structure of β9 has the largest relative
abundance in high temperature region above 350 K due to the large entropic
contribution from the lowest twisting mode of two HNO 3 fragments (the harmonic
vibrational frequency ω e = 10 cm
−1 ). Such temperature dependence of the relative
abundances as well as the inversion of the abundance is arising from the entropic
196
A. Ueda et al.
3.1 Negative Core Ion NO
−
3 HNO 3
ð
Þ 2
Fifteen different geometries were obtained in the geometry searching of
NO
−
3 HNO 3
ð
Þ 2 . We classified them into two groups, α and β, according to the
number of oxygen atom(s) on NO
−
3 making HBs with two HNO 3 ’s. In the group α
containing three kinds of conformers having a different angle between molecular
planes of NO
−
3 and HNO 3 , one oxygen atom on NO
−
3 has two HBs with two
HNO 3 ’s. In the group β containing twelve kinds of conformers, two different
oxygen atoms on NO
−
3 have one HBs with different HNO 3 ’s. The stable geometries
of all conformers and their relative energies at 0 K (ΔE ZPE ) are shown in Fig. 2,
where the most stable structure at 0 K is the structure α1. Drenck and coworkers
reported the stable structure similar to the structure β9 at B3LYP/6-31++G** level
of DFT calculations [1]. At MP2/6-31++G** level of ab initio calculations,
however, the total energy of the structure β9 is 1.6 kcal/mol higher than that of the
most stable structure α1.
The present ab initio calculations suggested that the negative core ion has the
structure of α1 as the most stable structure at 0 K. It is, however, strongly expected
that the relative abundance of these conformers depends on a thermal condition,
because most of the relative energies at 0 K are within a few kcal/mol. The temperature dependence of relative abundance of conformers is shown in Fig. 3, where
the relative abundance of ith conformer (p i ) at the temperature T is estimated in the
standard way as
p i ðTÞ = exp − ΔG i ̸ k B T
f
g̸ Z, Z = ∑
all
j
exp − ΔG j ̸ k B T
È
É
,
ð1Þ
where ΔG i is the relative Gibbs energy including the enthalpic (H: the sum of the
electronic energy, ZPE, thermal corrections, etc.) and the entropic terms (−TS). In
Fig. 3, the temperature dependences of p i values for all conformers are shown in the
upper figure (a), and those of the sum of p i values over conformers in each group
are given in the lower figure (b). Figure 3 clearly indicates that the relative abundance of NO
−
3 HNO 3
ð
Þ 2 conformers depends on the temperature, and the inversion
of the total abundance of the groups α and β is found at around 250 K. We note
here that the most stable structure α1 at 0 K is a dominant conformer in the low
temperature region below 90 K, while the structure of β9 has the largest relative
abundance in high temperature region above 350 K due to the large entropic
contribution from the lowest twisting mode of two HNO 3 fragments (the harmonic
vibrational frequency ω e = 10 cm
−1 ). Such temperature dependence of the relative
abundances as well as the inversion of the abundance is arising from the entropic
196
A. Ueda et al.