Zero-point energy calculations

Zero-point energies calculated from unsealed vibrational frequencies at the HF/6-31G level. 

One additional system, namely LiH, is discussed in this paper. The heat of formation and electron affinity of LiH are taken from the G3/99 (50) data set. The zero-point-exclusive atomization energy (De) was obtained from the heat of formation using the method described elsewhere 10). The electron affinity is converted into a zero-point exclusive electron affinity by removing the neutral and anionic zero-point energies calculated at the mPWlPW91/MG3 level and scaled (18) by 0.9758. 

After transforming to Cartesian coordinates, the position and velocities must be corrected for anharmonicities in the potential surface so that the desired energy is obtained. This procedure can be used, for example, to include the effects of zero-point energy into a classical calculation. 

The second, third, and fourth corrections to [MPd/b-Jl lG(d,p)] are analogous to A (- -). The zero point energy has been discussed in detail (scale factor 0.8929 see Scott and Radom, 1996), leaving only HLC, called the higher level correction, a purely empirical correction added to make up for the practical necessity of basis set and Cl truncation. In effect, thermodynamic variables are calculated by methods described immediately below and HLC is adjusted to give the best fit to a selected group of experimental results presumed to be reliable. 

B synchronously moving away from and toward H the H atom does not move (if A and B are of equal mass). If H does not move in a vibration, its replacement with D will not alter (he vibrational frequency. Therefore, there will be no zero-point energy difference between the H and D transition states, so the difference in activation energies is equal to the difference in initial state zero-point energies, just as calculated with Eq. (6-88). The kinetic isotope effect will therefore have its maximal value for this location of the proton in the transition state. 

Frequencies computed with methods other than Hartree-Fock are also scaled to similarly eliminate known systematic errors in calculated frequencies. The followng table lists the recommended scale factors for frequencies and for zero-point energies and for use in computing thermal energy corrections (the latter two items are discussed later in this chapter), for several important calculation types ... 

When comparing calculated results to thermodynamic quantities extrapolated to zero Kelvin, the zero point energy needs to be added to the total energy. As with the frequencies themselves, this predicted quantity is scaled to eliminate known systematic errors in frequency calculations. Accordingly, if you have not specified a scale factor via input to the Reodlsotopes option, you will need to multiply the values in the output by the appropriate scale factor (see page 64). 

We will provide you with the difference between the HF/6-31G(d) zero-point energy corrections for the two isomers, so you will not need to run frequency calculations ... 

You ll need to run five calculations at each model chemistry oxygen atom, chlorine atom, O2, CIO and ozone (but don t forget that you can obtain lower-level energies from a higher-level calculation). Use the experimental geometries for the various molecules and the following scaled zero-point energy corrections ... 

Some researchers prefer to use HF/6-31G(d) zero-point energy and therma corrections. For some very large systems, performing a Hartree-Focl optimization and frequency calculation followed by a B3LYP/6-31G(d ... 

Step 1. Produce an initial equilibrium structure at the Hartree-Fock level using the 6-31G(d) basis set. Verify that it is a minimum with a frequency calculation and predict the zero-point energy (ZPE). This quantity is scaled by 0.8929. 

CBS-4 is the less expensive of these two methods. It begins with a FlF/3-21G(d) geometry optimization the zero-point energy is computed at the same level. It then uses a large basis set SCF calculation as a base energy, and an MP2/6-31+Gt calculation with a CBS extrapolation to correct the energy through second order. A... 

Optimizations of the reactants and products, followed by frequency calculations and high level energy calculations (to produce zero-point energies and high quality total energies, respectively). 

A frequency calculation (yields the zero-point energy). 

The results of the frequency calculation confirm that the optimized structure is a transition structure, producing one imaginary frequency. The predicted zero-point energy is 0.01774 (after scaling), yielding a total energy of-113.67578 hartrees. 

Make a plot of the relative energies of the various systems, indicating any known paths between them. You ll be able to provide the zero point energy for the transition state from your calculations. 

A frequency calculation to verify the transition structure, compute its zero point energy, and prepare for the IRC (the optimized structure is given in the input file for this exercise). 

The frequency calculation of the given transition structure does produce one imaginary frequency, as required for a transition structure. The computed zero point energy is 0.03062 hartrees. When scaled and added to the MP4 total energy, it produces a relative energy of 0.63 kcal moP compared to the starting reactants. 

Compute AH for each reaction, using the B3LYP/6-31G(d) model chemistry for structures and zero-point energies and the B3LYP/6-311+G(3df,2p) model chemistry for the final energy calculations. 

Two frequency calculations to find their zero point energies. 

The predicted transition structure is at the left. The frequency calculation confirms that it is a transition structure, as well as providing its zero-point energy. 

The molecular mechanics calculations discussed so far have been concerned with predictions of the possible equilibrium geometries of molecules in vacuo and at OK. Because of the classical treatment, there is no zero-point energy (which is a pure quantum-mechanical effect), and so the molecules are completely at rest at 0 K. There are therefore two problems that I have carefully avoided. First of all, I have not treated dynamical processes. Neither have I mentioned the effect of temperature, and for that matter, how do molecules know the temperature Secondly, very few scientists are interested in isolated molecules in the gas phase. Chemical reactions usually take place in solution and so we should ask how to tackle the solvent. We will pick up these problems in future chapters. 

The geometry is optimized at the HF/6-3 lG(d) level, and the vibrational frequencies are calculated. To correct for the known deficiencies at the HF level, these are scaled by 0.893 to produce zero-point energies. 

By assuming additivity in the style of the G2 procedure (Section 5.5), the CCSD(T)/ 6-31G(d,p) results may be combined with the changes due to basis set enlargement to 6-31 lG(2df,2pd) at the MP2 level and the zero-point energy corrections calculated at the MP2/6-31G(d,p) level. The results are shown in Table 11.31. From the observed accuracy of 2 kcal/mol for structures 2-8, the energetics of the species 9-11 may be assumed to be reliable to the same level of accuracy. 

There is an evolution with time the older calculations correspond to isolated molecules in the gas phase without any corrections, the more recent ones include solvent effects, with different approximations, and also some corrections, like ZPE (zero-point energy correction). The contributions of some authors to the understanding of tautomerism have been significant. Some of their contributions are collected in Table II. 

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