Even-tempered basis sets

Even-tempered basis sets [40] consist of GTOs in which tlie orbital exponents belonging to series of... 

The basis sets that we have considered thus far are sufficient for most calculations. However, for some high-level calculations a basis set that effectively enables the basis set limit to be achieved is required. The even-tempered basis set is designed to achieve this each function m this basis set is the product of a spherical harmonic and a Gaussian function multiplied... 

The even-tempered basis set consists of the following sequence of functions Is, 2p, 3d, 4f,. .., which correspond to increasing values oi k. The advantage of this basis set is that ii relatively easy to optimise the exponents for a large sequence of basis functions. 

Even-tempered basis sets (M. W. Schmidt and K. Ruedenberg, J. Chem. Phys. 71, 3961 (1979)) consist of GTOs in which the orbital exponents ak belonging to series of orbitals consist of geometrical progressions ak = a, where a and P characterize the particular set of GTOs. 

The constants a,a b and b depend only on the atom type and the type of function. Even-tempered basis sets have the advantage that it is easy to generate a sequence of basis sets which are guaranteed to converge towards a complete basis. This is useful if... 

Even-tempered basis sets have the same ratio between exponents over the whole range. From chemical considerations it is usually preferable to cover the valence region better than the core region. This may be achieved by well-tempered basis sets. The idea is similar to the even-tempered basis sets, tire exponents are generated by a suitable formula containing only a few parameters to be optimized. The exponents in a well-tempered basis of size M are generated as ... 

The rate of convergence of expansions in the basis (1.2) has received little attention except for purely numerical studies [3,7,8,9,16] which indicated that the convergence is at least (unlike for bais set of type) not frustratingly slow. Rather detailed studies were performed for the even-tempered basis set, i.e. for exponents constructed from two parameters and /di (for each /)... 

Esgri, K. (2004) Even tempered basis sets for four-component relativistic quantum chemistry. Chemical Physics, 311,... 

Btiilding on atomic studies using even-tempered basis sets, universal basis sets and systematic sequences of even-tempered basis sets, recent work has shown that molecular basis sets can be systematically developed until the error associated with basis set truncation is less that some required tolerance. The approach has been applied first to diatomic molecules within the Hartree-Fock formalism[12] [13] [14] [15] [16] [17] where finite difference[18] [19] [20] [21] and finite element[22] [23] [24] [25] calculations provide benchmarks against which the results of finite basis set studies can be measured and then to polyatomic molecules and in calculations which take account of electron correlation effects by means of second order perturbation theory. The basis sets employed in these calculations are even-tempered and distributed, that is they contain functions centred not only on the atomic nuclei but also on the midpoints of the line segments between these nuclei and at other points. Functions centred on the bond centres were found to be very effective in approaching the Hartree-Fock limit but somewhat less effective in recovering correlation effects. 

Conversely, if it is assumed that the exponents do form a geometric progression and the parameters a and [3 are optimized for atoms then there is found to be little lost in accuracy. Basis sets developed in this way are termed even-tempered basis sets (for a discussion see [2]) and open up the possibility of constmcting the large and flexible basis sets that are inevitably required for calculations of high precision. 

Using the F ion as a prototype, the convergence of the many-body perturbation theory second-order energy component for negative ions is studied when a systematic procedure for the construction of even-tempered btisis sets of primitive Gaussian type functions is employed. Calculations are reported for sequences of even-tempered basis sets originally developed for neutral atoms and for basis sets containing supplementary diffuse functions. 

Schmidt and Ruedenberg[29] optimized the constants at, a t,bt and hj, for individual atomic species. However, for the large and flmdble even-tempered basis sets required for calculations of high precision it has been recognized that the precise values of these constants is not critical and, building on the concept of a universal basis set[30]-[36], a universal systematic sequence of even-tempered Gaussian basis sets has been investigated[37] [38]. 

The parameters employed in generating sequences of even-tempered basis sets for the neutral F and Ne atoms were taken from the work of Schmidt and Ruedenberg[29] -... 

The Hartree-Fock ground state of the F anion is described by orbitals of s Emd of p symmetry. In the first part of this study, attention was restricted to the convergence of the second order many-body perturbation theory component of the correlation energy for stematically constructed even-tempered basis sets of primitive Gaussian-typ>e functions of s and p symmetry. 

The second order correlation energy component, E Ne [IVe]) calculated for the ground state of the neon atom using systematic sequences of even-tempered basis sets of Gaussian functions designed for the Ne atom and designated [2nsnp] with n = 3,4,..., 13 are also collected in Table 1. 

The accurate description of correlation effects requires the inclusion of functions of higher symmetry than those required for the matrix Hartree-Fock model. The most important of these functions for the F anion are functions of d-type. In this section, the convergence of the total energy through second order and the second order correlation energy component for a systematic sequence of even-tempered basis sets of Gaussian functions of s-, p-and d-type is investigated. 

Most contemporary schemes for designing atomic and molecular Gaussian basis sets e.g. [4]- [7]) exploit, in one form or another, the idea of even-tempered basis sets [8], [9] (see also [4]), which have exponents, (p, defined by the geometric series ... 

See [11] for a recent review of applications of even-tempered basis set to the calculation of accurate molecular polarizabilities and hyperpolarizabilities within the matrix Hartree-Fock approximation. In [11] the results finite basis set Hartree-Fock calculations are compared with finite difference Hartree-Fock calculations. 

EVEN-TEMPERED BASIS SETS FOR EXCITED STATES... 

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