Basis set universal

Large basis sets of Gaussian-type orbitals (up to 35s21p2ld9g6h4i) were used, taken from the universal basis set ofMalli etal. [55]. Many electrons were correlated (5 in B,... 

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. 

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 use of a systematic sequence of basis sets can, of course, be usefully combined with the use of a universal basis set(52). In Figure 8, we display the results of Hartree-Fock calculations on the radial beryllium-like ions Li", B+, C2+, N +, Ol++, F5+, Ne6+, using the basis set given by Schmidt and Ruedenberg(56) specifi-ically for the beryllium atom. It can be seen that for the positive ions the basis sets give a uniform convergence rate. In... 

Universal Basis Sets.—Historically, it has been necessary to restrict the size of basis sets employed in molecular calculations to a reasonably small number of functions in order to keep the computation tractable. However, to achieve high accuracy, moderately large basis sets are ultimately required, especially if a significant fraction of the molecular correlation energy is to be recovered. Since... 

Let us comment at this point on the use of the even-tempered method for generating orbital exponents. A universal basis set clearly need not be an even-tempered one. However, even-tempered basis functions do span the one-electron space in a fairly uniform fashion. Unlike a set of exponents determined by energy optimization for a particular system, we expect that the even-tempered functions will be more suitable for transferring between systems. Even-tempered orbital exponents are defined through the recursion... 

It should be noted that the concept of a universal basis set could prove useful in almost all molecular studies in which finite basis sets are employed, e.g. electron-molecule scattering. Such basis sets lead to accurate results because they are moderately large. Furthermore, they afford a certain degree of uniformity in that they are not optimized for one particular property. 

Comparison of finite difference Hartree-Fock total energies for some fourteen electron diatomic molecules with matrix Hartree-Fock energies obtained by using a universal basis set.a... 

For many-electron systems the C02 has been taken as a prototype(80). Using a basis set containing higher harmonics on the nuclear centres, together with bond centred functions originally developed for JV2 and later applied to CO as a universal basis set, a molecular basis set was developed for CO2 which may be designated... 

Taken from the work of D.M. Silver, S. Wilson and W.C. Nieuwpoort, Universal basis sets and transferability of integrals , Int. J. Quantum Chem., 1978,14,635. 

Today, universal basis sets are employed in a variety of calculations175-189 at both the matrix Hartree-Fock level and including correlation effects, in non-relativistic and relativistic studies. 

Universal Basis Sets and Direct ccMBPT. - Early many-body perturbation theory calculations carried out within the algebraic approximation quickly led to the realization that basis set truncation is the dominant source of error in correlation studies seeking high precision when carried out with respect to an apprpriately chosen reference function. In more recent years, the importance of basis set truncation error control has been more widely recognized. We have described the concept of the universal basis set in Section 2.4.4 which provides a general approach to basis set truncation error reduction. 

The components of the universal basis set and the tmncated universal basis set are summarized in Table 8. In this table, denotes the number of functions of symmetry / on centre X. The total number of primitive Gaussian basis functions, N x ), in the truncated universal basis set is given as a function of the parameter t . 

Table 8 Components of the universal basis set and the truncated universal basis set...

It should be emphasized that the computational characteristics of the methods and basis sets used in this study are such that applications to larger systems, and, in particular, tartaric acid and its derivatives, would be feasible with moderate computer resources. More sophisticated correlation treatments, for example, higher order perturbation theory or cluster expansions [6] [7], and basis sets capable of supporting a higher accuracy, for example, the universal basis sets[8] [9] or the correlation consistent basis sets developed by Dunning and his coworkers[10]-[15], could be used for the 2-... 

The generator coordinate method (GCM), as initially formulated in nuclear physics, is briefly described. Emphasis is then given to mathematical aspects and applications to atomic systems. The hydrogen atom Schrodinger equation with a Gaussian trial function is used as a model for former and new analytical, formal and numerical derivations. The discretization technique for the solution of the Hill-Wheeler equation is presented and the generator coordinate Hartree-Fock method and its applications for atoms, molecules, natural orbitals and universal basis sets are reviewed. A connection between the GCM and density functional theory is commented and some initial applications are presented. 

Diagrammatic many-body perturbation theory calculations of the correlation energy of various diatomic molecules in their ground states using universal basis sets of even-tempered exponential-type functions. Comparison with other approaches. ... 

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