And basis sets

The orthonormality of the (2n) e"i flmetions will be demonstrated explieitly later. Before doing so however, it is useful to review both eomplex numbers and basis sets. 

There are a number of other technical details associated with HF and other ah initio methods that are discussed in other chapters. Basis sets and basis set superposition error are discussed in more detail in Chapters 10 and 28. For open-shell systems, additional issues exist spin polarization, symmetry breaking, and spin contamination. These are discussed in Chapter 27. Size-consistency and size-extensivity are discussed in Chapter 26. 

A basis set is a set of functions used to describe the shape of the orbitals in an atom. Molecular orbitals and entire wave functions are created by taking linear combinations of basis functions and angular functions. Most semiempirical methods use a predehned basis set. When ah initio or density functional theory calculations are done, a basis set must be specihed. Although it is possible to create a basis set from scratch, most calculations are done using existing basis sets. The type of calculation performed and basis set chosen are the two biggest factors in determining the accuracy of results. This chapter discusses these standard basis sets and how to choose an appropriate one. 

For many projects, a basis set cannot be chosen based purely on the general rules of thumb listed above. There are a number of places to obtain a much more quantitative comparison of basis sets. The paper in which a basis set is published often contains the results of test calculations that give an indication of the accuracy of results. Several books, listed in the references below, contain extensive tabulations of results for various methods and basis sets. Every year, a bibliography of all computational chemistry papers published in the previous... 

Table 12.1 gives the partial charges for the atoms in acetic acid computed with a number of dilferent methods and basis sets. All calculations use the molecular... 

It is a well-known fact that the Hartree-Fock model does not describe bond dissociation correctly. For example, the H2 molecule will dissociate to an H+ and an atom rather than two H atoms as the bond length is increased. Other methods will dissociate to the correct products however, the difference in energy between the molecule and its dissociated parts will not be correct. There are several different reasons for these problems size-consistency, size-extensivity, wave function construction, and basis set superposition error. 

Modeling the lighter main group inorganic compounds is similar to modeling organic compounds. Thus, the choice of method and basis set is nearly identical. The second-row compounds (i.e., sulfur) do have unfilled d orbitals, making it often necessary to use basis sets with d functions. 

Model chemistries are characterized by the combination of theoretical procedure and basis set. Every calculation performed with Gaussian must specify the desired theoretical model chemistry in addition to specifying the molecular system to consider and which results to compute for it. 

The route section of a Gaussian input file specifies the kind of job you want to run as well as the specific theoretical method and basis set which should be used. All of these items are specified via keywords. Recall that the first line of the route section begins with a sign (or T to request terse output). 

A frequency job must use the same theoretical model and basis set as produced the optimized geometry. Frequencies computed with a different basis set or procedure have no validity. We U be using the 6-31G(d) basis set for all of the examples and exercises in this chapter. This is the smallest basis set that gives satisfactory results for frequency calculations. 

Predict the zero point or thermal energy by running a frequency job at the optimized geometry, using the same method and basis set. (Note that these two steps maybe run via a single Gaussian job via the Opt Freq keyword.)... 

Seiufion Here are the results for a variety of theoretical methods and basis sets ... 

In the last two chapters, we discussed the ways that computational accuracy varies theoretical method and basis set. We ve examined both the successes and failures o variety of model chemistries. In this chapter, we turn our attention to mod designed for modeling the energies of molecular processes very accurately. 

Atoms defined in this way can be treated as quantum-mechanically distinct systems, and their properties may be computed by integrating over these atomic basins. The resulting properties are well-defined and are based on physical observables. This approach also contrasts with traditional methods for population analysis in that it is independent of calculation method and basis set. 

The amount of spacing between items is not significant in Gaussian input. In the route section, commas or slashes may be substituted for spaces if desired (except within parenthesized options, where slashes don t work). For example, the previous route section used a slash to separate the procedure and basis set, spaces to separate other keywords, and commas to separate the options to the Unite keyword. 

The route section specifies the procedure and basis set we want to use for this calculation ... 

To correct for electron correlation beyond QCISD(T) and basis set limitations, an empirical correction is added to the total energy. 

The Parameterized Configuration Interaction (PCI-X) method simply takes the correlation energy and scales it by a constant factor X (typical value 1.2), i.e. it is assumed that the given combination of method and basis set recovers a constant fraction of the correlation energy. 

All of these properties ean be ealeulated at different levels of sophistieation (eleetron eorrelation and basis sets). It should be noted that dynamic properties, where one or more of the external electrie and/or magnetie fields are time-dependent, may involve different frequeneies. 

In this chapter we will illustrate some of the methods described in the previous sections. It is of course impossible to cover all types of bonding and geometries, but for highlighting the features we will look at the H2O molecule. This is small enough that we can employ the full spectrum of methods and basis sets. 

Table 11.12 H2O dipole moment (Debye) as a function of DFT functional and basis set the ...

The starting point to obtain a PP and basis set for sulphur was an accurate double-zeta STO atomic calculation4. A 24 GTO and 16 GTO expansion for core s and p orbitals, respectively, was used. For the valence functions, the STO combination resulting from the atomic calculation was contracted and re-expanded to 3G. The radial PP representation was then calculated and fitted to six gaussians, serving both for s and p valence electrons, although in principle the two expansions should be different. Table 3 gives the numerical details of all these functions. 

The Bethe Sum Rule and Basis Set Selection in the Calculation of Generalized Oscillator Strengths... 

Notable in the series of homoleptic polynitrogen systems is the absence of the N ring. The structure of hexaazabenzene strongly depends on the choice of theoretical model and basis set [97] [112], van der Waals type structure of two... 

Six-dimensional, numerically accurate four-atom wave packet calculations were pioneered by Zhang and Zhang [17] and Neuhauser [18]. While numerous details of the present RWP implementation differ from these earlier approaches, it should be noted that many of the general ideas remain the same. In applications, finite-sized grids and basis sets are introduced to describe the wave packet, and... 

Comparing the last two entries in Figure 4.7, the all-electron Douglas-Kroll coupled cluster result for A te is in perfect agreement with the RPPA [156, 157]. Figure 4.8 shows the relativistic effects in dissociation energies. Here, relativistic effects are very sensitive to the level of electron correlation and basis sets used. RPPA... 

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