Zero-point effects

The para-ortho energy difference increases to 2.5 kcal/mol. Inclusion of correlation effects increases the ortho-para AE and, of course, reverses the stability of the ortho and meta isomers. The zero point effects are not inconsequential for the diaminobenzenes with the ortho isomer having a higher zero point energy than the other two isomers. This lowers the relative energies of the meta and para isomers by 0.4 and 0.5 kcal/mol respectively. 

The QFH potential approximately captures two key quantum effects. When an atom is near a potential minimum, the curvature is positive and thus so is the QFH correction this models the zero-point effect. On the other hand, near potential maxima the curvature is negative, and the QFH potential models tunneling. 

Calculations at the 6-3IG level indicate that in the gas phase, 2//-l,2,3-triazole is more stable than 1//-1,2,3-triazole by about 4.5 kcal moC. In solution, the IH isomer becomes the more stable species because the large difference in dipole moments favors the more polar tautomer. The triazolium ion (75) is predicted to be more stable than (76) by about 13.5 kcal mol <89Mi40i-0i>. 2//-1,2,3-Triazole represents more than 99.9% of the equilibrium mixture in the gas phase. However, the ab initio calculated proton affinity of 1//-benzotriazole is 10.2 kcal mol larger than that of 2//-benzotriazole, which is consistent with ICR measurements (1-methylbenzotriazole is 10 kcal mol more basic than 2-methylbenzotriazole). Measurements of enthalpies of solution, vaporization, sublimation and solvation in water, methanol and DMSO confirm the predominance of the IH tautomer in solution <89JA7348>. The energy difference between the tautomers of 1,2,3-triazole has also been estimated at the 6-31G (MP2)//3-21G level including zero-point effects. The... 

The model with an unprotonated His77 was used. Zero-point effects are not included. Cleavage over the Ni-Fe bridge. Proton goes to Cys495. 

See Ref. [148] for a physically appealing derivation.) Equation (12) yields an efficient, accurate and formally exact scheme to perform quantum dynamics. It involves the action of a handed, sparse and Toeplitz matrix on a vector. The propagation scheme has been shown to accurately represent [147] all quantum dynamical features including zero-point effects, tunneling as well as over-barrier reflections and in this sense differs from standard semi-classical treatments. The approach also substantially differs from other formalisms such as centroid dynamics [97,98,168-171], where the Feynman path centroid is propagated in a classical-like... 

I80. In practice, effects are much smaller than this, largely because bonds are not completely broken at the transition state. With these small effects, the effect of change of mass on molecular moments of inertia cannot be neglected. Such ponderal kinetic isotope effects have a lower dependence on temperature than zero point effects, for which rin( L/ H) = constant (equation 1.14) holds generally, T being the absolute temperature. 

In molecular calculations, the minimum energy is usually reached when the atoms are in their equilibrium positions. Care must be taken in calculations on extended solids, since the relaxation of the interatomic distances in most codes usually does not include zero-point effects. We shall see later that these effects can have appreciable consequences when dealing with light elements, especially hydrogen. 

The nature of the hydrogen bond in 2-fluoroacetaldehyde enol is quite different from that in the alcohol. Among the four structures shown below, the cis-syn structure is the most stable (Table V). The difference in energies of the cis-syn and cis-anti structures gives an estimate of the hydrogen bond energy. At the MP-2 level, this is 3.53 kcal/mol and corrections for zero-point effects yield a value of 3.21 kcal/mol. (The... 

Mu + H2 Muonium (Mu) is an isotope of the hydrogen atom with a positive muon as the nucleus. Since it is 8.8 times lighter than H, tunnelling and zero point effects are enormous, and in fact the Mu +... 

Er has a c-axis cone structure, with a periodicity of almost eight atomic layers, below Tc=18K. Neutron diffraction measurements at 6 K indicated c-axis and basal plane moments of (4.44 0.12)ju,B and (7.80 0.12)/iB, respectively, i.e. a total ionic moment of (8.98 0.18)ju,B and cone angle of 29.6 0.8 . Jensen s (1976) prediction of a 1% reduction of the fully ordered moment by zero point effects is thus within the experimental error. 

In this equation, a allows for the number of equivalent reaction paths, for example, cr = 4 for H + CH4, ( /F), is the per volume partition function for the specified species (z), denotes the transition state species, and is the difference in energy between the zeroth levels in the transition state and in the reactants, sometimes referred to as the vibrationally adiabatic barrier. It should be noted that, because of zero-point effects, the energy of the classical barrier, V, is related to but not the same as AE and that, because the preexponential term on the right-hand-side of eqn (1.14) depends on temperature, ds.E is also not identical with Eaa as defined in eqn (1.8). 

Computed average vibrational free energies for H-transfer (blue) and D-transfer (green) in wild-type DHFR. The results are fitted to an inverse Eckart function, and the locations of the transition state at 5 °C and 45 °C are indicated by the vertical lines. The Boltzmann factor of between H- and D-transfers gives the vibrational free energy contribution (dominantly zero-point effects) to the overall KIE. The reaction coordinate, ARc, is defined as the difference of the distances of the transferring proton from the donor (NADPH) and acceptor (dihydrofolate) atoms. 

The barrier height for reaction (6) is less accurately known from experiment. A recent review of the literature (37) concludes that the threshold for this reaction, including zero point effects, is approximately 84 kcal/mol. Using the frequencies of Frisch et dl. (31) to correct for zero point energies, leads to an estimated experimental of 89.5 kcal/mol. This result is in good... 

Hiraoka etal. [10] discuss the zero point effect of the magnetic dipolar interaction described by Holstein, Primakoff [11] as a possible mechanism of the reduction in the ferromagnetic moment at 0 K (see pp. 214, 216). The estimated high dipolar interaction suggests that it has an appreciable influence on the low temperature spin wave spectrum and in pressure-induced ferromagnetic EuSe, probably also causes a deviation of the NMR-v(T) dependence from the law at T/Tc< 1. This is expected to occur somewhere below 1.7 K [10]. 

Table 2.1 Bulk properties ofAl, Cu, Pd, and Ag, as computed from DFT with the LDA, PBE, TPSS, and PBEO exchange-correlation functionals. All data is taken from Refs. [39, 66], and [67]. Ref [66] is an all-electron study with a Gaussian basis set. Data from Ref [39] has been obtained with the plane-wave plus PAW method and data from Ref [67] has been obtained with the FP-LAPW method. The experimental values in parenthesis have been corrected for zero-point effects. 

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