Density basis sets

With this approximation, the evaluation of the Coulomb term scales as N2M, in contrast to the standard way, which scales as N4 (N and M are the number of primitive functions in the orbital and density basis sets, respectively). The expansion coefficients of the electronic density in Eq. (8) are chosen such as to minimize the error in the Coulomb term arising from the difference between the real density and the fitted density [25],... 

A diagrannnatic approach that can unify the theory underlymg these many spectroscopies is presented. The most complete theoretical treatment is achieved by applying statistical quantum mechanics in the fonn of the time evolution of the light/matter density operator. (It is recoimnended that anyone interested in advanced study of this topic should familiarize themselves with density operator fonnalism [8, 9, 10, H and f2]. Most books on nonlinear optics [13,14, f5,16 and 17] and nonlinear optical spectroscopy [18,19] treat this in much detail.) Once the density operator is known at any time and position within a material, its matrix in the eigenstate basis set of the constituents (usually molecules) can be detennined. The ensemble averaged electrical polarization, P, is then obtained—tlie centrepiece of all spectroscopies based on the electric component of the EM field. 

Frequent approximations made in TB teclmiques in the name of achieving a fast method are the use of a minimal basis set, the lack of a self-consistent charge density, the fitting of matrix elements of the potential. 

Pisani [169] has used the density of states from periodic FIP (see B3.2.2.4) slab calculations to describe the host in which the cluster is embedded, where the applications have been primarily to ionic crystals such as LiE. The original calculation to derive the external Coulomb and exchange fields is usually done on a finite cluster and at a low level of ab initio theory (typically minimum basis set FIP, one electron only per atom treated explicitly). 

Although split valence basis sets give far better results than minimal ones, they still have systematic weaknesses, such as a poor description of three-inembered rings, This results from their inability to polarize the electron density to one side of an atom. Consider the /T-bond shown in Figure 7-23. 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

Basis sets can be further improved by adding new functions, provided that the new functions represent some element of the physics of the actual wave function. Chemical bonds are not centered exactly on nuclei, so polarized functions are added to the basis set leading to an improved basis denoted p, d, or f in such sets as 6-31G(d), etc. Electrons do not have a very high probability density far from the nuclei in a molecule, but the little probability that they do have is important in chemical bonding, hence dijfuse functions, denoted - - as in 6-311 - - G(d), are added in some very high-level basis sets. 

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. 

The energy obtained from a calculation using ECP basis sets is termed valence energy. Also, the virial theorem no longer applies to the calculation. Some molecular properties may no longer be computed accurately if they are dependent on the electron density near the nucleus. 

There have been a few basis sets optimized for use with DFT calculations, but these give little if any increase in efficiency over using EIF optimized basis sets for these calculations. In general, DFT calculations do well with moderate-size HF basis sets and show a significant decrease in accuracy when a minimal basis set is used. Other than this, DFT calculations show only a slight improvement in results when large basis sets are used. This seems to be due to the approximate nature of the density functional limiting accuracy more than the lack of a complete basis set. 

A much less basis set dependent method is to analyze the total electron density. This is called the atoms in molecules (AIM) method. It is designed to examine the small effects due to bonding in the primarily featureless electron density. This is done by examining the gradient and Laplacian of electron density. AIM analysis incorporates a number of graphic analysis techniques as well as population analysis. The population analysis will be discussed here and the graphic techniques in the next chapter. 

Transition structures are more dihicult to describe than equilibrium geometries. As such, lower levels of theory such as semiempirical methods, DFT using a local density approximation (LDA), and ah initio methods with small basis sets do not generally describe transition structures as accurately as they describe equilibrium geometries. There are, of course, exceptions to this, but they must be identihed on a case-by-case basis. As a general rule of thumb, methods that are empirically dehned, such as semiempirical methods or the G1 and G2 methods, describe transition structures more poorly than completely ah initio methods do. 

The most popular of the SCRF methods is the polarized continuum method (PCM) developed by Tomasi and coworkers. This technique uses a numerical integration over the solute charge density. There are several variations, each of which uses a nonspherical cavity. The generally good results and ability to describe the arbitrary solute make this a widely used method. Flowever, it is sensitive to the choice of a basis set. Some software implementations of this method may fail for more complex molecules. 

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