The Zero-Point Vibrational Energy

In a recent benchmark study [57] on the CH2=NH molecule, we explicitly computed a CCSD(T)/VTZ quartic force field at great expense (the low symmetry necessitated the computation of 2241 energy points in Cs symmetry and 460 additional points in Ci symmetry). The resulting anharmonic Ezpv, 24.69 kcal/mol, is only 0.10 kcal/mol above the scaled B3LYP/VTZ estimate, 24.59 kcal/mol. At least for fairly rigid 

If we use B3LYP/VTZ+1 harmonics scaled by 0.985 for the Ezpv rather than the actual anharmonic values, mean absolute error at the W1 level deteriorates from 0.37 to 0.40 kcal/mol, which most users would regard as insignificant. At the W2 level, however, we see a somewhat more noticeable degradation from 0.23 to 0.30 kcal/mol - if kJ/mol accuracy is required, literally every little bit counts . If one is primarily concerned with keeping the maximum absolute error down, rather than getting sub-kJ/mol accuracy for individual molecules, the use of B3LYP/VTZ+1 harmonic values of Ezpv scaled by 0.985 is an acceptable fallback solution . The same would appear to be true for thermochemical properties to which the Ezpv contribution is smaller than for the TAE (e.g. ionization potentials, electron affinities, proton affinities, and the like). 

A reliable assessment of the performance of a method in the kJ/mol accuracy range is, by its very nature, only possible where experimental data are themselves known to this accuracy. 

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At low temperatures nearly all bonds will be in their lowest vibrational level, n = 0, and will, therefore, possess the zero-point vibrational energy, Eq = hvl2. Presuming the molecule behaves as a simple harmonic oscillator, the vibrational frequency is given by... 

The inequality of the C—H bonds in the radical cation implies that all C—H bonds do not have the same force constants. In a simplistic approximation, the zero-point vibrational energy (ZPVE) of a C—H stretching vibration will be proportional to (k/mn), where k is the force constant of the C—H bond and j// is the mass of the hydrogen nucleus. The effect on the ZPVE of replacing one proton by a deuteron will hence depend on the deuteration site, such that the ZPVE will be lowered more if the deuteron occupies a site with a larger fcrce constant, i.e. a shorter bond. This, in general, means a site with low unpaired spin density. 

The molecule has jRT from translational energy, RT from the term pV, RT from the two rotational degrees of freedom, and then the zero-point vibrational energy. The atom has only contributions from translational energy and the PV term ... 

The values were calculated taking into consideration the zero-point vibration energy. Experimental 4.2kJ/mol (506a). cBy NMR data (see Table 3.25). dRef. (505)b. 

In fact, the barrier for proton transfer in the maleate anion appears to lie below the zero-point vibrational energy level (W. M. Westler, private communication). Thus, vibrationally averaged properties of the maleate anion will correspond to a symmetrically bridged Cjv transition-state structure rather than to either of the asymmetrically bridged equilibrium structures in Fig. 5.22. For present purposes this interesting feature of the potential surface can be ignored. 

The inclusion of vibrational terms is dealt with later. We note in passing the fascinating example of solid He, where lattice statics is completely inappropriate. The binding forces are so weak that even at the lowest temperatures solidification occurs only at pressures of at least 2.5 MPa, and it is the zero-point vibrational energy that stabilizes the structure. 

Scheme 11.2 Products of chlorine atom addition to propadiene (la) and heats of formation [kj mol-1] of addition products 2a and 3a [16], a> QCISD(T)/6—311+G(d,p)//QCISD(T)/6—31+G(d,p) MP2/6-31+G(d,p) for calculation of the zero point vibrational energy.

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 zero-point vibrational energy (Ezpv) is obtained from harmonic B3LYP/VTZ+1 frequencies scaled by 0.985 in the case of Wl theory. For W2 theory, anharmonic values of Ezpv from quar-tic force fields at the CCSD(T)/VQZ+1 (or comparable) level are preferred where this is not feasible, the same procedure as for Wl theory is followed as a fallback solution . 

Tables 1 and 2 show the lowest torsional energy levels of hydrogen peroxide and deuterium peroxide which have been determined variationally using as basis functions the rigid rotor solutions. Experimental data are from Camy-Peiret et al [15]. The first set of leval data are from Camy-Peiret et al [15]. The first set of levels (SET I) has been calculated without including the pseudopotential V = 0). The levels corresponding to the other sets (SET II, SET III and SET IV) were obtained including pseudopotentials calculated with different numerical and analytical algorithms. Finally, the zero point vibration energy correction was introduced in the SET V [14],...

Reaction enthalpies were determined at 0 K and corrected by the zero-point vibrational energy. Wavenumbers were scaled with a factor of 0.89. 

Under standard conditions, both molecules exist in their lowest vibrational energy levels. These are known as their zero-point vibrational states, in which the value of the vibrational quantum number is zero. The fact that molecules in their zero-point vibrational states possess vibrational energy is a consequence of the Uncertainty Principle this would be violated if the internuclear distance was unchanging. The dissociation limits for both species are identical the complete separation of the two atoms, which is taken as an arbitrary zero of energy. The difference between the zero of energy and the zero-point vibrational energy in both cases represents the bond dissociation energies, respectively, of H2+ and H2. 

For accurate comparison of relative energies, one must add to the BO-optimized energy the zero-point vibrational energy (ZPVE), which in the harmonic approximation is half the sum of the fundamental frequencies. This correction is most critical for the calculation of activation energies. The contribution of the ZPVE of the mode corresponding most closely to the reaction coordinate is lost completely. Processes that involve breaking of a bond to H are the most seriously affected torsional changes are the least affected. 

Within the harmonic oscillator approximation, the energy of the lowest vibrational level can be determined from Eq. (9.47) as ha>/2 where h is Planck s constant (6.6261 x 10- J s) and a> is the vibrational frequency. The sum of all of these energies over all molecular vibrations defines the zero-point vibrational energy (ZPVE). We may then define the internal energy at 0 K for a molecule as... 

In the limit of a particular vibration going to zero, we see from Eq. (10.1) that it ceases to contribute to the zero-point vibrational energy. However, it is less obvious what happens to... 

Fig. VII-2.—Some vibrational energy levels for an idealised diatomic molecule. The electronic energy curve has been approximated by a parabola, corresponding to a Qooke s-law interaction between the two atoms. The firat five vibrational states are represented. They are separated by the energy difference hv. The lowest vibrational state, with v 0, has the zero-point vibrational energy...

It is seen that the vibrational energy levels are equally spaced, being separated by the energy value hv . The vibrational energy of the lowest vibrational state, with v = 0, is hv even in the lowest state the molecule has this amount of vibrational energy. This quantity is called the zero-point vibrational energy of the molecule (Fig. VII-2). 

Similarly, the symbol D0 is used for the difference in energy of the lowest state, with v = 0 and K = 0, and the separated atoms it is called the dissociation energy of the molecule. It is smaller than Dt by the amount the zero-point vibrational energy. 

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