reasonably good approximation taking into account the fact, that vibrational
amplitudes are rather low, at least for first two energy levels) one concludes that
Bohr condition, i.e., DE ¼ hm, where m is the frequency of radiation, is fulfilled when
m = m 0 , i.e., when radiation frequency is equal to the classical vibrational frequency
of a molecule. This is why we say that the band on a spectrum corresponds to a given
vibrational frequency. It should be remembered, however, that the band corresponds
to transitions between various energy levels, e.g., t = 0 ! t = 1 in the case of
fundamentals. Thus, in order to find the position of a spectral line on a (pure)
vibrational spectrum of a diatomic molecule, one has to calculate m 0 . This means that
at a given computational level one has to find r e (geometry optimization), next
calculate f as a second energy derivative at r e , and finally apply Eq. (2.10).
Consider the simple case of HCl molecule. It is well known that the fundamental
(experimental) vibrational frequency in gas phase is 2886 cm
−1
. In fact, this is the
wavenumber where the Q-branch (absent on the rotation–vibration spectrum) would
occur. The band’s position calculated according to Eq. (2.10) using CCSD/
aug-cc-pVTZ computational level affords the value 3014 cm
−1 (experimental harmonic frequency based on f ¼ 516
N
m [2] is 2989 cm
−1 ; in addition, r
CCSD
e
¼ 1:2766 Å
and r
expt
e
¼ 1:2746 Å [2]), which is nearly 130 cm
−1 higher as compared with
experimental fundamental. This overestimation is obvious. Parabolic PEC is only an
approximation, and the real PEC of HCl is anharmonic. Apparently, harmonic
approximation overestimated the observed vibrational frequency, and therefore, more
sophisticated treatment is needed to obtain better agreement with the experimental
values. This can be accomplished by means of perturbation or variation approaches.
After neglecting fifth and higher-order terms in Eq. (2.1), the Hamiltonian for a
vibrating molecule is
^
H ¼ À
h
2
2l
d
2
dr 2 þ
1
2
f r À r e
ð
Þ
2 þ
1
6
f
3
ð Þ r À r e
ð
Þ
3 þ
1
24
f
4
ð Þ r À r e
ð
Þ
4
ð2:11Þ
where f
3
ð Þ and f
4
ð Þ are cubic and quartic FCs, respectively, i.e., third and fourth
energy derivatives calculated at r e , cf. Equations (2.1)–(2.3). In the case of perturbation approach terms
^
H
1
ð Þ ¼
1
6
f
3
ð Þ r À r e
ð
Þ
3 and ^
H
2
ð Þ ¼
1
24
f
4
ð Þ r À r e
ð
Þ
4
ð2:12Þ
are first- and second-order perturbations to the unperturbed (harmonic oscillator)
Hamiltonian (2.8), for which exact solutions, w
h
p and E
h
p , are known (symbol “h”
stands for “harmonic”). One can easily calculate f
3
ð Þ and f
4
ð Þ by numerical energy
differentiation. Using central differences on energy method, we obtained f
3
ð Þ
¼
À0:985 a.u. (step h ¼ 0:01 ˚
A) and f
4
ð Þ
¼ 2:51 a.u. (step h ¼ 0:02 ˚
A; in view of
differentiating energy, this value may not be very accurate, sufficient for our purpose, though). These values can be used in standard perturbation theory formulas
for first-order correction to the wavefunction and first- and second-order corrections
56
O. Bąk and P. Borowski
amplitudes are rather low, at least for first two energy levels) one concludes that
Bohr condition, i.e., DE ¼ hm, where m is the frequency of radiation, is fulfilled when
m = m 0 , i.e., when radiation frequency is equal to the classical vibrational frequency
of a molecule. This is why we say that the band on a spectrum corresponds to a given
vibrational frequency. It should be remembered, however, that the band corresponds
to transitions between various energy levels, e.g., t = 0 ! t = 1 in the case of
fundamentals. Thus, in order to find the position of a spectral line on a (pure)
vibrational spectrum of a diatomic molecule, one has to calculate m 0 . This means that
at a given computational level one has to find r e (geometry optimization), next
calculate f as a second energy derivative at r e , and finally apply Eq. (2.10).
Consider the simple case of HCl molecule. It is well known that the fundamental
(experimental) vibrational frequency in gas phase is 2886 cm
−1
. In fact, this is the
wavenumber where the Q-branch (absent on the rotation–vibration spectrum) would
occur. The band’s position calculated according to Eq. (2.10) using CCSD/
aug-cc-pVTZ computational level affords the value 3014 cm
−1 (experimental harmonic frequency based on f ¼ 516
N
m [2] is 2989 cm
−1 ; in addition, r
CCSD
e
¼ 1:2766 Å
and r
expt
e
¼ 1:2746 Å [2]), which is nearly 130 cm
−1 higher as compared with
experimental fundamental. This overestimation is obvious. Parabolic PEC is only an
approximation, and the real PEC of HCl is anharmonic. Apparently, harmonic
approximation overestimated the observed vibrational frequency, and therefore, more
sophisticated treatment is needed to obtain better agreement with the experimental
values. This can be accomplished by means of perturbation or variation approaches.
After neglecting fifth and higher-order terms in Eq. (2.1), the Hamiltonian for a
vibrating molecule is
^
H ¼ À
h
2
2l
d
2
dr 2 þ
1
2
f r À r e
ð
Þ
2 þ
1
6
f
3
ð Þ r À r e
ð
Þ
3 þ
1
24
f
4
ð Þ r À r e
ð
Þ
4
ð2:11Þ
where f
3
ð Þ and f
4
ð Þ are cubic and quartic FCs, respectively, i.e., third and fourth
energy derivatives calculated at r e , cf. Equations (2.1)–(2.3). In the case of perturbation approach terms
^
H
1
ð Þ ¼
1
6
f
3
ð Þ r À r e
ð
Þ
3 and ^
H
2
ð Þ ¼
1
24
f
4
ð Þ r À r e
ð
Þ
4
ð2:12Þ
are first- and second-order perturbations to the unperturbed (harmonic oscillator)
Hamiltonian (2.8), for which exact solutions, w
h
p and E
h
p , are known (symbol “h”
stands for “harmonic”). One can easily calculate f
3
ð Þ and f
4
ð Þ by numerical energy
differentiation. Using central differences on energy method, we obtained f
3
ð Þ
¼
À0:985 a.u. (step h ¼ 0:01 ˚
A) and f
4
ð Þ
¼ 2:51 a.u. (step h ¼ 0:02 ˚
A; in view of
differentiating energy, this value may not be very accurate, sufficient for our purpose, though). These values can be used in standard perturbation theory formulas
for first-order correction to the wavefunction and first- and second-order corrections
56
O. Bąk and P. Borowski