Basis minimal

A minimal basis in which the number of STO or GOTO orbitals is equal to the number of core and valence AOs in tlie atom. 

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. 

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. 

To calculate the matrix elements for H2 in the minimal basis set, we approximate the Slater Is orbital with a Gaussian function. That is, we replace the Is radial wave function... 

To obtain the force constant for constructing the equation of motion of the nuclear motion in the second-order perturbation, we need to know about the excited states, too. With the minimal basis set, the only excited-state spatial orbital for one electron is... 

As mentioned above, HMO theory is not used much any more except to illustrate the principles involved in MO theory. However, a variation of HMO theory, extended Huckel theory (EHT), was introduced by Roald Hof nann in 1963 [10]. EHT is a one-electron theory just Hke HMO theory. It is, however, three-dimensional. The AOs used now correspond to a minimal basis set (the minimum number of AOs necessary to accommodate the electrons of the neutral atom and retain spherical symmetry) for the valence shell of the element. This means, for instance, for carbon a 2s-, and three 2p-orbitals (2p, 2p, 2p ). Because EHT deals with three-dimensional structures, we need better approximations for the Huckel matrix than... 

Even with the minimal basis set of atomic orbitals used m most sem i-empirical calculatitm s. the n urn ber of molecii lar orbitals resulting from an SCFcalciilation exceeds the num ber of occupied molecular orbitals by a factor of about two. The n um ber of virtual orbitals in an ah initio calculation depends on the basis set used in this calculation. 

HyperChem performs an empirical Hiickel calculation to produce th e MO coefficien ts for a minimal basis set and th en projects th ese coefficien ts to the real basis set used in an cife calculation. Th e projected Htickel guess can be applied to rn olecular system s with an atom ic n um ber less th an or equal to 54 (Xe). 

Most simple empirical or semi-empirical molecular orbital methods. including all ofthose ii sed in IlyperCh em, neglect inner sh ell orbitals and electrons and use a minimal basis se.i r>f valence Slater orbitals. 

Minimal basis sets in which 3, 4 etc, Gaussian functions are used to represent the atomic orbitals on an atom... 

Application of the variational self-consistent field method to the Haitiee-Fock equations with a linear combination of atomic orbitals leads to the Roothaan-Hall equation set published contemporaneously and independently by Roothaan and Hall in 1951. For a minimal basis set, there are as many matr ix elements as there are atoms, but there may be many more elements if the basis set is not minimal. 

Semiempirical calculations are set up with the same general structure as a HF calculation in that they have a Hamiltonian and a wave function. Within this framework, certain pieces of information are approximated or completely omitted. Usually, the core electrons are not included in the calculation and only a minimal basis set is used. Also, some of the two-electron integrals are omitted. In order to correct for the errors introduced by omitting part of the calculation, the method is parameterized. Parameters to estimate the omitted values are obtained by fitting the results to experimental data or ah initio calculations. Often, these parameters replace some of the integrals that are excluded. 

The complete neglect of differential overlap (CNDO) method is the simplest of the neglect of differential overlap (NDO) methods. This method models valence orbitals only using a minimal basis set of Slater type orbitals. The CNDO method has proven useful for some hydrocarbon results but little else. CNDO is still sometimes used to generate the initial guess for ah initio calculations on hydrocarbons. 

The smallest basis sets are called minimal basis sets. The most popular minimal basis set is the STO—3G set. This notation indicates that the basis set approximates the shape of a STO orbital by using a single contraction of three GTO orbitals. One such contraction would then be used for each orbital, which is the dehnition of a minimal basis. Minimal basis sets are used for very large molecules, qualitative results, and in certain cases quantitative results. There are STO—nG basis sets for n — 2—6. Another popular minimal basis set is the MINI set described below. 

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. 

MINI—i i = 1—4) These four sets have different numbers of primitives per contraction, mostly three or four. These are minimal basis sets with one contraction per orbital. Available for Li through Rn. 

Some of the basis sets discussed here are used more often than others. The STO—3G set is the most widely used minimal basis set. The Pople sets, particularly, 3—21G, 6—31G, and 6—311G, with the extra functions described previously are widely used for quantitative results, particularly for organic molecules. The correlation consistent sets have been most widely used in recent years for high-accuracy calculations. The CBS and G2 methods are becoming popular for very-high-accuracy results. The Wachters and Hay sets are popular for transition metals. The core potential sets, particularly Hay-Wadt, LANL2DZ, Dolg, and SBKJC, are used for heavy elements, Rb and heavier. 

For gaining a detailed understanding of orbital interactions, use the Mul-liken analysis with a minimal basis set. 

The contracted basis in Figure 28.3 is called a minimal basis set because there is one contraction per occupied orbital. The valence region, and thus chemical bonding, could be described better if an additional primitive were added to each of the valence orbitals. This is almost always done using the even-tempered method. This method comes from the observation that energy-optimized exponents tend to nearly follow an exponential pattern given by... 

J Chem. Phys., 52, 431 (1970)] is a relatively inexpensive one and can be used for calculations on quite large molecules. It is minimal in the sense of having the smallest number of functions per atom required to describe the occupied atomic orbitals of that atom. This is not exactly true, since one usually considers Is, 2s, and 2p, i.e., five functions, to construct a minimal basis set for Li and Be, for example, even though the 2p orbital is not occupied in these atoms. The 2sp (2s and 2p), 3sp, 4sp, 3d,. .., etc. orbitals are always lumped together as a shell , however. The minimal basis set thus consists of 1 function for H and He, 5 functions for Li to Ne, 9 functions for Na to Ar, 13 functions for Kand Ca, 18 functions for Sc to Kr,. .., etc. Because the minimal basis set is so small, it generally can not lead to quantitatively accurate results. It does, however, contain the essentials of chemical bonding and many useful qualitative results can be obtained. 

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