Vibrational correction

The total energy Eo of a molecular system in its vibrational ground state can be written as the sum of the electronic energy Ee at the equilibrium geometry and the zero-point vibrational energy (ZPVE), denoted as Ezpv, 

The ZPVE may be partitioned into harmonic and anharmonic contributions 

The harmonic frequencies and the anharmonic constants may be obtained from experimental vibrational spectra, although their determination becomes difficult as the size of the system increases. In Table 1.10, we have listed experimental harmonic and anharmonic contributions to the AEs. These contributions may also be obtained from electronic-structure calculations of quadratic force fields (for harmonic frequencies) and cubic and quartic force fields (for anharmonic constants). For some of the larger molecules in Table 1.11, we have used ZPVEs calculated at the CCSD(T)/cc-pVTZ level or higher, see Ref. 12. In some cases, both experimental and theoretical ZPVEs are available and agree to within 0.3 kJ/mol [12, 57], 

Although the harmonic ZPVE must always be taken into account in the calculation of AEs, the anharmonic contribution is much smaller (but oppositely directed) and may sometimes be neglected. However, for molecules such as H2O, NH3, and CH4, the anharmonic corrections to the AEs amount to 0.9, 1.5, and 2.3 kJ/mol and thus cannot be neglected in high-precision calculations of thermochemical data. Comparing the harmonic and anharmonic contributions, it is clear that a treatment that goes beyond second order in perturbation theory is not necessary as it would give contributions that are small compared with the errors in the electronic-structure calculations. 

---

There was less agreement between calculated and experimental energy values. The use of 6-3IG, the best procedure in energy calculations of three-membered rings, yielded a value too low by more than 40 kJ moF in the case of diazirine bond separation energy was calculated as -45 kJ moF the experimental value is +0.4 kJ moF . Vibrational correction and extrapolation to 0 K would reduce this difference by several kJ moF . 

The CCSD model gives for static and frequency-dependent hyperpolarizabilities usually results close to the experimental values, provided that the effects of vibrational averaging and the pure vibrational contributions have been accounted for. Zero point vibrational corrections for the static and the electric field induced second harmonic generation (ESHG) hyperpolarizability of methane have recently been calculated by Bishop and Sauer using SCF and MCSCF wavefunctions [51]. 

Combining [52] the zero point vibrational corrections of Ref. [51] with the CCSD results obtained in the t-aug-cc-pVTZ and d-aug-cc-pVQZ basis sets we obtained the estimates for the ZPV corrected 70, A and B coefficients listed in Table 4. An experimental estimate for 70, A, and B has been derived by Shelton by fitting the results of ESHG measurements to the expresssion = To(l + A0JI2 + This... 

The pointwise given MCSCF results for zero point vibrational corrections for the ESHG hyperpolarizability were added to the CCSD results obtained from the dispersion coefficients and than fitted to a fourth-order polynomial in... 

One of the simplest chemical reactions involving a barrier, H2 + H —> [H—H—H] —> II + H2, has been investigated in some detail in a number of publications. The theoretical description of this hydrogen abstraction sequence turns out to be quite involved for post-Hartree-Fock methods and is anything but a trivial task for density functional theory approaches. Table 13-7 shows results reported by Johnson et al., 1994, and Csonka and Johnson, 1998, for computed classical barrier heights (without consideration of zero-point vibrational corrections or tunneling effects) obtained with various methods. The CCSD(T) result of 9.9 kcal/mol is probably very accurate and serves as a reference (the experimental barrier, which of course includes zero-point energy contributions, amounts to 9.7 kcal/mol). 

The prerequisites for high accuracy are coupled-cluster calculations with the inclusion of connected triples [e.g., CCSD(T)], either in conjunction with R12 theory or with correlation-consistent basis sets of at least quadruple-zeta quality followed by extrapolation. In addition, harmonic vibrational corrections must always be included. For small molecules, such as those contained in Table 1.11, such calculations have errors of the order of a few kJ/mol. To reduce the error below 1 kJ/mol, connected quadruples must be taken into account, together with anhar-monic vibrational and first-order relativistic corrections. In practice, the approximate treatment of connected triples in the CCSD(T) model introduces an error (relative to CCSDT) that often tends to cancel the... 

Vibration corrected bond lengths based on the riding model for C(2)—N(1), rigid body libration for other C—N bonds, libration plus riding motion for N—H bonds. 

The molecular reorientation is found to be correlated with NH2 internal motions. The non-planar nature of the molecules is shown by both the uncorrected and the vibrationally corrected data. 

The conventional approach used to describe the response of a molecule to a static electric field is either to perform pure electronic BO calculations or to perform calculations where the BO values are corrected for vibrational and rotational (thermal) motion of the nuclei. The vibrationally corrected polarizabilities usually do an excellent job of correcting the errors inherent in the pure electronic BO values. Bishop has written several excellent reviews on this topic [78-80]. 

For the vibrational ground state with quantum number u = 0 the averaged polarizabilities are often expressed as e sum of the polarizability at an equilibrium geometry, i e, and a zero-point-vibrational correction (ZPVC)... 

Table 2. LiH dipole and quadrupole polarizability (in atomic units) for the vibrational ground state u = 0 calculated with different response theory methods. P(Pe) is the value at the minimum of the potential energy curve, Pq o is the value in the vibrational ground state and ZPVC = Pq o P(Pe) is the corresponding zero-point-vibrational correction...

In Table 7 we compare the ZPVCs for the dipole and quadrupole polarizabilities of HF. In the same way as for LiH, we have calculated the vibrational averages for each method with two different wavefunctions - one obtained from the PEC of the same or related method as used in the calculation of the property curve and the other obtained from the loo CAS PEC. Compared with the equivalent results for LiH we observe significant differences between the calculations on the two molecules. Eirst of all the vibrational corrections are smaller than in LiH but roughly in the same ratio as the polarizabilities. The influence of the PEC is larger than in LiH. 

Experimental value for the u = 0 state 0.741599 [72] minus a zero-point-vibrational correction... 

Using perturbation dependent atomic orbitals (rotational London orbitals [61]) as basis functions. Experimental values for the u = 0 state gx =0.5654 + 0.0007 and gn =0.5024 + 0.0005 [74] minus a zero-point-vibrational correction Agx = —0.0135 and Agn = —0.0062 calculated with a... 

If the geometry is fully optimized in all the conformations, the 3N-5 neglected vibrations remain at E = 0 during the complete torsion instead of lying at the zero vibrational. For this reason, a zero vibrational correction Vzero can be added to the potential. For each selected conformation the simplest correction is ... 

Finally, the fifth set of levels on Table 1 (SET V) is obtained adding the zero vibrational correction (equation 13) which is neglected in the remaining calculations. 

Finally, the zero point vibration corrections (SET V) use to be much larger than the pseudopotential corrections. In the present case, these zero point corrections seems to give rise to unrrealistic values, probably because of the harmonic approximation used in the calculations. The torsion mode as well as its interactions with the remaining modes are indeed very anharmonic. 

---