Zero point vibrational correction

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). 

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...

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... 

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. 

Density functional theory (DFT) calculations have been carried out to elucidate the structure and energetics of the various isomeric (Si-N) rings as well as the corresponding anions and dianions. The local minima on the potential energy surfaces were verified by computation of the eigenvalues of the respective Hessian matrices. From these, harmonic vibrational frequencies and the zero-point vibration corrected energetics were calculated. The calculations were carried out for isolated molecules in the gas phase. The theoretical results are expected to be reliable for molecules in non-polar or weakly aprotic polar solvents. 

We have carried out DFT (B3LYP/6-31G(d)) calculations (the basis set comprises 312 cGTOs) in order to establish the energetic order of the different possible isomers of (Me2Si-NH)4, OMCTS (Fig. 15). At the local minima on the potential energy surfaces, the Hessian matrices were computed. Harmonic vibrational frequencies were used to calculate the zero-point vibration-corrected energetics. (Results are collected in Table I and Fig. 16.)... 

The most comprehensive near-HF calculation on H2O was carried out by Dunning, Pitzer, and Aung,468 who examined the use of a variety of GTO and STO basis sets in a study of a large number of one-electron properties of the ground state. About 70% of the dissociation energy was obtained, with the energy 0.003 0.002 hartree above the HF limit. The computed VIP s were 1—1.5 eV too low, and the force constants were in error by 15—20%. Kern and co-workers have made an extensive study469 472 of zero-point vibrational corrections to one-electron properties. 

J. Kongsted, K. Ruud, Solvent effects on zero-point vibrational corrections to optical rotations and nuclear magnetic resonance shielding constants, Chem. Phys. Lett. 451 (2008) 226. 

FIGURE 3. Schematic representation of calculated relative energies including calculated gas-phase proton affinities (PA) for vinylamines and related systems (MP4/G-311 + G together with zero-point vibrational corrections). Reprinted with permission from J. Am. Chem. Soc., 114, 36 (1992). Copyright (1992) American Chemical Society... 

Dimitrova and Peyerimhoff focused their work on geometry I and helped provide a more accurate assessment of its interaction energy. Their highest level of theory stopped at MP2 but incorporated a 6-311 -t- H-G(2d,2p) basis set. The SCF part of the interaction energy is —4.79 kcal/mol, reduced to —4.04 when corrected for BSSE. The superposition-corrected contribution from MP2 correlation is —0.97 kcal/mol, adding up to a value of AE jec = 5.0 kcal/mol, quite similar to the best estimates for the water dimer. When zero-point vibrational corrections are added, this quantity lowers in magnitude to —3.3 kcal/mol. 

Table 1. The activation barrier AE [kcal mol ] incorporating the zero-point vibrational correction for the decomposition HN5 -> HN3 -1- N2, calculated with several methods.

MP2/6-31G level. In the last column, B3LYP/6-31G //MP2/6-31G energies are reported. Their (S ) values are always about 0.78 and 0.75 before and after projection, respectively. The reported zero point vibrational corrections (ZPE), thermal energies (TE) and imaginary frequencies (in cm ) at the transition states are calculated with the HF/6-31G method. TS stands for the transition state, and the OH-xylene are the hydroxycyclohexadienyl radicals with the OH added to the indicated carbon atom of the ring. Roman numbers indicated in parentheses refer to formulas shown in the text... 

Table 2.3 Reaction energies for the reactions considered in Eqs. (1-3) total electonic energy differences [kcal moT ] including zero-point vibrational corrections for the reactions of THVS and GOTHS with the silsesquioxanes.

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