Energy correction

Pack R T and Hirschfelder J O 1970 Energy corrections to the Born-Oppenheimer approximation. The best adiabatic approximation J. Chem. Phys. 52 521-34... 

The first-order energy correction with respect to the unperturbed problem is then... 

The zeroth-order energy level is twofold degenerate. The corresponding vibronic basis functions are ur K+2 0 0 —) = 11) and luj- A"—2 0 0 +) = 2). The first-order energy correction is... 

The second-order energy corrections have the form (B.8) with... 

The second-order energy correction is expressed as follows ... 

When this result is used in the earlier expression for the second-order energy correction, one obtains ... 

An essential thing to stress concerning the above development of so-called Rayleigh-Schrodinger perturbation theory (RSPT) is that each of the energy corrections... 

The second-order energy correction can be evaluated in like fashion by noting that... 

There is also a local MP2 (LMP2) method. LMP2 calculations require less CPU time than MP2 calculations. LMP2 is also less susceptible to basis set superposition error. The price of these improvements is that about 98% of the MP2 energy correction is recovered by LMP2. 

The more recently developed methods define an energy expression for the combined calculation and then use that expression to compute gradients for a geometry optimization. Some of the earlier methods would use a simpler level of theory for the geometry optimization and then add additional energy corrections to a final single point calculation. The current generation is considered to be the superior technique. 

It is possible to make a method approximately size-extensive by adding a correction to the final energy. This has been most widely used for correcting CISD energies. This is a valuable technique because a simple energy correction formula is easier to work with than full Cl calculations, which require an immense amount of computational resources. The most widely used correction is the Davidson correction ... 

There are several ways to include relativity in ah initio calculations more efficiently at the expense of a bit of accuracy. One popular technique is the Dirac-Hartree-Fock technique, which includes the one-electron relativistic terms. Another option is computing energy corrections to the nonrelativistic wave function without changing that wave function. 

The Cannon-Fenske viscometer (Fig. 24b) is excellent for general use. A long capillary and small upper reservoir result in a small kinetic energy correction the large diameter of the lower reservoir minimises head errors. Because the upper and lower bulbs He on the same vertical axis, variations in the head are minimal even if the viscometer is used in positions that are not perfecdy vertical. A reverse-flow Cannon-Fen ske viscometer is used for opaque hquids. In this type of viscometer the Hquid flows upward past the timing marks, rather than downward as in the normal direct-flow instmment. Thus the position of the meniscus is not obscured by the film of Hquid on the glass wall. 

Some orifice viscometers, such as the Shell dip cup and the European ISO cup, which resembles a Eord cup with a capillary, have long capillaries. These cups need smaller kinetic energy corrections and give better precision than the corresponding short-capiHary viscometers. However, they are stiU not precision instmments, and should be used only for control purposes. 

The tetramethylammonium salt [Me4N][NSO] is obtained by cation exchange between M[NSO] (M = Rb, Cs) and tetramethylammonium chloride in liquid ammonia. An X-ray structural determination reveals approximately equal bond lengths of 1.43 and 1.44 A for the S-N and S-O bonds, respectively, and a bond angle characteristic bands in the IR spectrum at ca. 1270-1280, 985-1000 and 505-530 cm , corresponding to o(S-N), o(S-O) and (5(NSO), respectively. Ab initio molecular orbital calculations, including a correlation energy correction, indicate that the [NSO] anion is more stable than the isomer [SNO] by at least 9.1 kcal mol . ... 

That is, the sum of the electronic energy and nuclear repulsion energy of the molecule at the specified nuclear configuration. This quantity is commonly referred to as the total energy. However, more complete and accurate energy predictions require a thermal or zero-point energy correction (see Chapter 4, p. 68). 

To compute zero-point vibration and thermal energy corrections to total energies as well as other thermodynamic quantities of interest such and the enthalpy and entropy of the system. 

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 ... 

You should be aware that the optimal scaling factors vary by basis set. For example, Bauschlicher and Partridge computed the B3LYP/6-311+G(3df,2p) ZPE/thermal energy correction scaling factor to be 0.989. Additional scaling factors have also been computed by Wong and by Scott and Radom. 

Gaussian predicts various important thermodynamic quantities at the specified temperature and pressure, including the thermal energy correction, heat capacity and entropy. These items are broken down into their source components in the output ... 

The scale factor is optional. If Included, it says to scale the frequencies before performing the thermochemicai analysis. Note that including the factor affects the thermochemistry output only (including the ZPE) the frequencies printed earlier in the output remain unsealed. This parameter is the means by which scale foctors are applied to thermal energy corrections. 

In order to predict the energy of a system at some higher temperature, a thermal energy correction must be added to the total energy, which includes the effects of molecular translation, rotation and vibration at the specified temperature and pressure. Note that the thermal energy includes the zero-point energy automatically do not add both of them to an energy value. 

When comparing energy results to experiments performed at particular temperatures, the thermal energy correction given in the output should be added to the total energy (this sum is also given in the output). In order to apply the appropriate scale factor to a thermal energy correction, you must specify a scale factoi via input to the Readlsotopes option. The quantity reported in the output cannot simply be multiplied by the scale factor itself as it is composed of several terms, only some of which should be scaled. 

Here is how the zero-point and thermal energy-corrected properties appear in the output from a frequency calculation ... 

The raw zero-point energy and thermal energy corrections are listed first, followed by the predicted energy of the system taking them into account. The output also includes corrections to and the final predicted values for the enthalpy and Gibbs free energy. All values are in Hartrees. 

Zero-point and thermal energy corrections are usually computed with the same model chemistry as the geometry optimization. However, you may also choose to follow the common practice of always using the HF/6-31G(d) model chemistry for predicting zero-point and thermal energies (see page 149). Of course, such frequency calculations must follow a HF/6-31G(d) geometry optimization. 

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 ... 

How well do your results agree with the experimental value of about 3.4 kcal-mol i Since this observation is very approximate, we will not worry about zero-point energy corrections in this exercise. 

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