case in the electronic structure theory), and wavefunctions w
trial
0 and w
trial
1 as (nearly)
exact solutions. Since w
trial
1 is orthogonal to w
trial
0 , the second lowest eigenvalue is an
upper bound to the first excited state energy (t = 1). The two lowest eigenvalues
correspond to E
trial
0 $ 1498, and E
trial
1 $ 4434 cm
−1 , and thus, the energy gap corresponds to 2936 cm
−1 . The difference between theoretical and experimental
transition frequency is therefore 50 cm
−1 , which is again much lower than within
the harmonic oscillator framework (130 cm
−1 ). Since, as mentioned above, the
calculated wavefunctions are (nearly) exact for Hamiltonian (2.11), and assuming
that the overall picture will not change upon inclusion of fifth and higher-order
terms, the difference as large as 50 cm
−1 should be regarded as following from the
shortcomings in the treatment of correlation effects at the adopted computational
level.
1 In view of this discussion, the better performance of the (approximate)
second-order perturbation approach (yielding the difference 24 cm
−1 ) is due to
fortuitous cancelation of errors.
The relative error between the calculated harmonic and the experimental fundamental frequency is quite significant. In the case described above, it is equal to
4.4%. The values obtained within the perturbation or variation frameworks are also
not perfect. The reported differences of 24 and 50 cm
−1 follow from the approximate nature of the calculations. First, we considered merely cubic and quartic terms
in the expansion of PEC. In addition, we used the second-order correction within
the perturbation framework. In the case of variation procedure, we used expansion
(2.14) which is long enough to assure (nearly) exact results for a given computational level and the assumed form of Hamiltonian (2.11). However, the quadratic,
cubic, and quartic FCs are approximate (recall that the CCSD/aug-cc-pVTZ method
was used), which also results in the above-mentioned deviations.
The energy levels of anharmonic oscillator are no longer equidistant. They
approach each other, which means that harmonic approximation overestimates the
observed fundamentals (in an overwhelming majority of cases). In addition, the
approximate nature of calculations (neglect of a part of correlation effects, and basis
set incompleteness) leads to errors in the predicted geometry and/or curvature of
PEC around the equilibrium position, which result in the error in the predicted FC,
and, consequently in the predicted harmonic frequency. It should be emphasized
that even in the case of highly accurate geometries (in the case considered above,
the deviation between the computed and experimental bond length is 0.16%), there
is no guarantee that the calculated curvature is correct. For these reasons, additional
endeavors are needed for the harmonic approximation to be conclusive when
applied to various physicochemical problems. One of them is scaling techniques
discussed in the present chapter.
1
In fact, this is not the case. Inclusion of fifth-order term to Hamiltonian (2.11) with the estimated
value of f
(5) = -5.2 a.u. lowers the predicted by variation method transition frequency from
2936 cm
−1 down to 2911 cm
−1 (which is mostly due to substantial lowering of energy for t= 1).
This value is only 25 cm
−1 higher as compared with the fundamental (25 cm
−1 is also the difference between the calculated and experimental harmonic frequencies).
58
O. Bąk and P. Borowski
trial
0 and w
trial
1 as (nearly)
exact solutions. Since w
trial
1 is orthogonal to w
trial
0 , the second lowest eigenvalue is an
upper bound to the first excited state energy (t = 1). The two lowest eigenvalues
correspond to E
trial
0 $ 1498, and E
trial
1 $ 4434 cm
−1 , and thus, the energy gap corresponds to 2936 cm
−1 . The difference between theoretical and experimental
transition frequency is therefore 50 cm
−1 , which is again much lower than within
the harmonic oscillator framework (130 cm
−1 ). Since, as mentioned above, the
calculated wavefunctions are (nearly) exact for Hamiltonian (2.11), and assuming
that the overall picture will not change upon inclusion of fifth and higher-order
terms, the difference as large as 50 cm
−1 should be regarded as following from the
shortcomings in the treatment of correlation effects at the adopted computational
level.
1 In view of this discussion, the better performance of the (approximate)
second-order perturbation approach (yielding the difference 24 cm
−1 ) is due to
fortuitous cancelation of errors.
The relative error between the calculated harmonic and the experimental fundamental frequency is quite significant. In the case described above, it is equal to
4.4%. The values obtained within the perturbation or variation frameworks are also
not perfect. The reported differences of 24 and 50 cm
−1 follow from the approximate nature of the calculations. First, we considered merely cubic and quartic terms
in the expansion of PEC. In addition, we used the second-order correction within
the perturbation framework. In the case of variation procedure, we used expansion
(2.14) which is long enough to assure (nearly) exact results for a given computational level and the assumed form of Hamiltonian (2.11). However, the quadratic,
cubic, and quartic FCs are approximate (recall that the CCSD/aug-cc-pVTZ method
was used), which also results in the above-mentioned deviations.
The energy levels of anharmonic oscillator are no longer equidistant. They
approach each other, which means that harmonic approximation overestimates the
observed fundamentals (in an overwhelming majority of cases). In addition, the
approximate nature of calculations (neglect of a part of correlation effects, and basis
set incompleteness) leads to errors in the predicted geometry and/or curvature of
PEC around the equilibrium position, which result in the error in the predicted FC,
and, consequently in the predicted harmonic frequency. It should be emphasized
that even in the case of highly accurate geometries (in the case considered above,
the deviation between the computed and experimental bond length is 0.16%), there
is no guarantee that the calculated curvature is correct. For these reasons, additional
endeavors are needed for the harmonic approximation to be conclusive when
applied to various physicochemical problems. One of them is scaling techniques
discussed in the present chapter.
1
In fact, this is not the case. Inclusion of fifth-order term to Hamiltonian (2.11) with the estimated
value of f
(5) = -5.2 a.u. lowers the predicted by variation method transition frequency from
2936 cm
−1 down to 2911 cm
−1 (which is mostly due to substantial lowering of energy for t= 1).
This value is only 25 cm
−1 higher as compared with the fundamental (25 cm
−1 is also the difference between the calculated and experimental harmonic frequencies).
58
O. Bąk and P. Borowski