Basis set - minimal

Three basis sets (minimal s-p, extended s-p and minimal s-p with d functions on the second row atoms) are used to calculate geometries and binding energies of 24 molecules containing second row atoms, d functions are found to be essential in the description of both properties for hypervalent molecules and to be important in the calculations of two-heavy-atom bond lengths even for molecules of normal valence. 

It is known that ab initio methods are not accurate in reproducing or predicting molecular dipole moments. For example, a typical basis set minimization with no additional keywords was carried out, and the results show that the computed magnitude of the dipole moment is not particularly accurate when compared with experimental values. For alcohols, MP2 has a root mean squared deviation of 0.146 Debye, while HF had a deviation of 0.0734 Debye when measured against the experimental values. 

In general, BSSE is reduced as the basis set becomes larger and more flexible. However, there is no strict correspondence, and the BSSE can in fact become larger with certain additions to the basis set. Minimal basis sets are particularly prone to large BSSE, as are those like 3-21G with a poor description of the inner shells. 

END theory treats the collisional system in a Cartesian laboratory frame and each level of approximation [3] is defined by a choice of system wave function characterized by a set of time-dependent parameters and choice of basis set. Minimal END employs a system state vector of the form [3]... 

A few ab initio SCF-MO calculations showed that i is very sensitive to changes in basis set. Minimal basis sets of Slater and of Gaussian lobe functions gave i = +0.656 [2, 3] and +0.506... 

The representation of trial fiinctions as linear combinations of fixed basis fiinctions is perhaps the most connnon approach used in variational calculations optimization of the coefficients is often said to be an application of tire linear variational principle. Altliough some very accurate work on small atoms (notably helium and lithium) has been based on complicated trial functions with several nonlinear parameters, attempts to extend tliese calculations to larger atoms and molecules quickly runs into fonnidable difficulties (not the least of which is how to choose the fomi of the trial fiinction). Basis set expansions like that given by equation (A1.1.113) are much simpler to design, and the procedures required to obtain the coefficients that minimize are all easily carried out by computers. 

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. 

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

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... 

Split valence basis sets generally give much better results than minimal ones, but at a cost. Remember that the number of two-electron integrals is proportional to kf , where W is the number of basis functions. Whereas STO-3G has only live ba.sis functions for carbon, 6-31G has nine, resulting in more than a tenfold increase in the size of the calculation,... 

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. 

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). 

The success of simple theoretical models m determining the properties of stable molecules may not carry over into reaction pathways. Therefore, ah initio calcii lation s with larger basis sets ni ay be more successful in locatin g transition structures th an semi-empir-ical methods, or even methods using minimal or small basis sets. 

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

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