2.1 Molecular Structure
15
3N–6 (or 3N–5) eigenvalues of the Hessian matrix correspond to the force constants
{k i } for molecular normal vibrations regarded as the harmonic vibrations with the
relationship
(2πν i )
2
=
k i
m
(2.3)
where {ν i } stands for the frequencies of the corresponding i-th normal vibrations
and m the reduced mass for the corresponding normal vibration. Thus at the local
minimum all of the force constants k i ’s are positive affording the “real” frequencies
and the optimization process is mostly performed along with the normal-vibration
analysis. Moreover, the thermochemical data such as enthalpy, entropy, and Gibbs
free energy also can be obtained by making up the corresponding partition functions
as well. Figure 2.14, for instance, affords the normal-vibration modes obtained at the
same time of the optimization of formaldehyde molecule.
In order to effectively perform the processes described above, information on the
analytical forms of ∂ E/∂ R i and ∂
2 E/∂ R i ∂ R j are obviously desirable. It is currently
possible to obtain the analytical forms both in the ground and in the first singlet and
the triplet excited states (Foresman et al. 1992 and van Caillie and Amos 1999) and
these prescriptions have been incorporated into most of the computational softwares.
1D polymers shown in Fig. 2.7 include one more freedom, translation length, say
a, being added to all the nuclear coordinates R of the atoms constructing the unit
cell as has already been mentioned. The optimization process of 1D polymers is also
incorporated into the current major computational softwares.
1199.17 cm -1 (1.59)
1274.46 cm -1 (12.62)
1553.98 cm -1 (6.82)
1845.33 cm -1 (96.41)
2901.63 cm -1 (54.29)
2959.64 cm -1 (158.37)
Fig. 2.14 Normal-vibration frequencies of formaldehyde in the S 0 state with the vibration modes
indicated by blue arrows. IR intensities are also indicated in parentheses. These data were obtained
along the optimization at DFT/B3LYP/6-31G**
15
3N–6 (or 3N–5) eigenvalues of the Hessian matrix correspond to the force constants
{k i } for molecular normal vibrations regarded as the harmonic vibrations with the
relationship
(2πν i )
2
=
k i
m
(2.3)
where {ν i } stands for the frequencies of the corresponding i-th normal vibrations
and m the reduced mass for the corresponding normal vibration. Thus at the local
minimum all of the force constants k i ’s are positive affording the “real” frequencies
and the optimization process is mostly performed along with the normal-vibration
analysis. Moreover, the thermochemical data such as enthalpy, entropy, and Gibbs
free energy also can be obtained by making up the corresponding partition functions
as well. Figure 2.14, for instance, affords the normal-vibration modes obtained at the
same time of the optimization of formaldehyde molecule.
In order to effectively perform the processes described above, information on the
analytical forms of ∂ E/∂ R i and ∂
2 E/∂ R i ∂ R j are obviously desirable. It is currently
possible to obtain the analytical forms both in the ground and in the first singlet and
the triplet excited states (Foresman et al. 1992 and van Caillie and Amos 1999) and
these prescriptions have been incorporated into most of the computational softwares.
1D polymers shown in Fig. 2.7 include one more freedom, translation length, say
a, being added to all the nuclear coordinates R of the atoms constructing the unit
cell as has already been mentioned. The optimization process of 1D polymers is also
incorporated into the current major computational softwares.
1199.17 cm -1 (1.59)
1274.46 cm -1 (12.62)
1553.98 cm -1 (6.82)
1845.33 cm -1 (96.41)
2901.63 cm -1 (54.29)
2959.64 cm -1 (158.37)
Fig. 2.14 Normal-vibration frequencies of formaldehyde in the S 0 state with the vibration modes
indicated by blue arrows. IR intensities are also indicated in parentheses. These data were obtained
along the optimization at DFT/B3LYP/6-31G**