Basis sets computation accuracy

In the previous section, the Roothaan method was introduced as a method for speeding up the Hartree-Fock calculation by expanding molecular orbitals with a basis set. The accuracy and computational time of the Roothaan calculations depend on the quality and number of basis functions, respectively. Therefore, it is necessary for reproducing accurate chemical reactions and properties to choose basis functions that give highly accurate molecular orbitals with a minimum number of functions. 

The LMTO method [58, 79] can be considered to be the linear version of the KKR teclmique. According to official LMTO historians, the method has now reached its third generation [79] the first starting with Andersen in 1975 [58], the second connnonly known as TB-LMTO. In the LMTO approach, the wavefimction is expanded in a basis of so-called muffin-tin orbitals. These orbitals are adapted to the potential by constmcting them from solutions of the radial Scln-ddinger equation so as to fomi a minimal basis set. Interstitial properties are represented by Hankel fiinctions, which means that, in contrast to the LAPW teclmique, the orbitals are localized in real space. The small basis set makes the method fast computationally, yet at the same time it restricts the accuracy. The localization of the basis fiinctions diminishes the quality of the description of the wavefimction in die interstitial region. 

Several basis schemes are used for very-high-accuracy calculations. The highest-accuracy HF calculations use numerical basis sets, usually a cubic spline method. For high-accuracy correlated calculations with an optimal amount of computing effort, correlation-consistent basis sets have mostly replaced ANO... 

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... 

Single point energy calculations can be performed at any level of theory and with small or large basis sets. The ones we ll do in this chapter will be at the Hartree-Fock level with medium-sized basis sets, but keep in mind that high accuracy energy computations are set up and interpreted in very much the same way. 

Each cell in the chart defines a model chemistry. The columns correspond to differcni theoretical methods and the rows to different basis sets. The level of correlation increases as you move to the right across any row, with the Hartree-Fock method jI the extreme left (including no correlation), and the Full Configuration Interaction method at the right (which fuUy accounts for electron correlation). In general, computational cost and accuracy increase as you move to the right as well. The relative costs of different model chemistries for various job types is discussed in... 

Choosing a model chemistry almost always involves a trade-off between accuracy and computational cost. More accurate methods and larger basis sets make jobs run longer. We ll provide some specific examples of these effects throughout the chapters in this part of the book. 

Determine the effect of basis set on the predicted chemical shifts for benzene. Compute the NMR properties for both compounds at the B3LYP/6-31G(d) geometries we computed previously. Use the HF method for your NMR calculations, with whatever form(s) of the 6-31G basis set you deem appropriate. Compare your results to those of the HF/6-311+G(2d,p) job we ran in the earlier exercise. How does the basis set effect the accuracy of the computed chemical shift for benzene ... 

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. 

A variety of compound methods have been developed in an attempt to accurately model the thermochemical quantities we have been considering. These method-s attempt to achieve high accuracy by combining the results of several different calculations as an approximation to a single, very high level computation which is much too expensive to be practical. We will consider two families of methods the Gaussian-n methods and the Complete Basis Set (CBS) methods. 

In order to calculate total energies with a chemical accuracy of 1 kcal/mol, it is necessary to use sophisticated methods for including electron correlation and large basis sets, which is only computationally feasible for small systems. Instead the focus is usually on calculating relative energies, trying to make the errors as constant as possible. 

However, the direct dynamics calculations are computationally expensive, and cannot employ particularly high levels of electron correlation or large basis sets. If certain regions of the potential cannot be treated to within the required accuracy using a computationally affordable level of theory, the results may have unacceptably large errors. Nevertheless, direct dynamics calculations have played and will play a critical role in the discovery and analysis of competing pathways in chemical reactions. 

Here, n corresponds to the principal quantum number, the orbital exponent is termed and Ylm are the usual spherical harmonics that describe the angular part of the function. In fact as a rule of thumb one usually needs about three times as many GTO than STO functions to achieve a certain accuracy. Unfortunately, many-center integrals such as described in equations (7-16) and (7-18) are notoriously difficult to compute with STO basis sets since no analytical techniques are available and one has to resort to numerical methods. This explains why these functions, which were used in the early days of computational quantum chemistry, do not play any role in modem wave function based quantum chemical programs. Rather, in an attempt to have the cake and eat it too, one usually employs the so-called contracted GTO basis sets, in which several primitive Gaussian functions (typically between three and six and only seldom more than ten) as in equation (7-19) are combined in a fixed linear combination to give one contracted Gaussian function (CGF),... 

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