Contraction, basis set

Combining the full set of basis functions, known as the primitive GTOs (PGTOs), into a smaller set of functions by forming fixed linear combinations is known as basis set contraction, and the resulting functions are called contracted GTOs (CGTOs). 

To some other experts the meaning of the term ab initio is rather clear cut. Their response is that "ab initio" simply means that all atomic/molecular integrals are computed analytically, without recourse to empirical parametrization. They insist that it does not mean that the method is exact nor that the basis set contraction coefficients were obtained without recourse to parametrization. Yet others point out that even the integrals need not be evaluated exactly for a method to be called ab initio, given that, for instance, Gaussian employs several asymptotic and other cutoffs to approximate integral evaluation. 

In order to form the set of 8 atomic orbitals to be used in the VB calculations, we proceed as follows. We associate to each atom a (8s lp) atomic centered gaussian basis set contracted to [2slp] (Table 1). Each orbital is ex-... 

In our calculations we associate to each Li atom a (10s2p) atomic centered gaussian basis set contracted to [4s,2p] (Table 3). We treat only the valence electrons at the VB level and keep the inner shell electrons (LiIs) in a core obtained by HF calculations. Therefore we are neglecting the core-core and core-valence correlation effects, which are small for these small lithium clusters (Fig.5). 

Latajka and Scheiner attempted to follow up on this same basic idea in designing larger basis sets for use with H-bonded systems. As a starting point, they took the 6-31G basis set. Contraction coefficients and scaling factors of... 

If spin-orbit effects are considered in ECP calculations, additional complications for the choice of the valence basis sets arise, especially when the radial shape of the / -f-1/2- and / — 1/2-spinors differs significantly. A noticeable influence of spin-orbit interaction on the radial shape may even be present in medium-heavy elements as 53I, as it is seen from Fig. 21. In many computational schemes the orbitals used in correlated calculations are generated in scalar-relativistic calculations, spin-orbit terms being included at the Cl step [244] or even after the Cl step [245,246]. It therefore appears reasonable to determine also the basis set contraction coefficients in scalar-relativistic calculations. Table 9 probes the performance of such basis sets for the fine structure splitting of the 531 P ground state in Kramers-restricted Hartree-Fock [247] and subsequent MRCI calculations [248-250], which allow the largest flexibility of... 

Basis sets contraction coefficients from scalar-relativistic ground state calculation of the neutral atom (basis set A) or the anion (basis set B). Different contractions for and from... 

The REP and AREP employed for the present calculations are those generated by Christiansen et al. with 7 valence electrons for the halogen atoms, and we optimized basis sets to be of pVTZ quality for the halogen atoms [71]. For the description of the anionic character of the halogen atoms one diffuse basis function is added to each basis set to yield (7s7p3d2f) basis sets contracted to... 

More than one function may be used to represent a particular atomic orbital. This is obviously a well-understood tactic when using Gaussian functions, but the use of basis set contractions also applies to the Slater type orbitals and the numerical basis sets. For a numerical basis set the contraction can be derived from two functions, one corresponding to the neutral atom and the other to a positive ion. 

The geometries of ABN and DMABN were optimized at the CASSCF level of approximation [62]. ANO-S basis sets contracted as C, N[3s2plc/]/H[2s] were used. The cavity radius was optimized to minimize the absolute energy of the ground state at the CASSCF level of theory. This gave a = 13.0ao for ABN in diethylether, and 14.2ao, 13.8fl(, and 13.5flo, for DMABN in cyclohexane, n-butylchloride, and acetonitrile, respectively. 

Y. Ishikawa, H. Sekino, and R, Binning Jr., Chem. Phys. Lett., 165, 237 (1990). Effects of Basis Set Contraction in Relativistic Calculations on Neon, Argon, and Germanium. 

ABSTRACT. Recent advances in electronic structure theory and the availability of high speed vector processors have substantially increased the accuracy of ab initio potential energy surfaces. The recently developed atomic natural orbital approach for basis set contraction has reduced both the basis set incompleteness and superposition errors in molecular calculations. Furthermore, full Cl calculations can often be used to calibrate a CASSCF/MRCI approach that quantitatively accounts for the valence correlation energy. These computational advances also provide a vehicle for systematically improving the calculations and for estimating the residual error in the calculations. Cdculations on selected diatomic and triatomic systems will be used to illustrate the accuracy that currently can be achieved for molecular systems. In particular, the F-I-H2 - HF+H potential energy hypersurface is used to illustrate the impact of these computational advances on the calculation of potential energy surfaces. 

The discussion above applies to uncontracted basis sets. Contracted basis sets present a few further problems. To properly represent the spin-orbit splitting, the two spin-orbit components should be contracted separately. The contraction is now j -dependent, rather than f-dependent, and can only be represented directly in a 2-spinor basis. The problem is not now confined to the small component. If the large-component scalar basis set includes contractions for both spin-orbit components, the product of the contracted basis functions for each spin-orbit component with the spin functions generates a representation for both spin-orbit components. Thus there is a duplication of the basis set that is close to linearly dependent, and some kind of scheme to project out linearly dependent components, either numerically or by conversion to a 2-spinor basis, is mandatory. The same applies to the small component. For example, the contracted p sets for the large-component and d sets both span the same space, but because of the contraction the (i-generated set cannot be made a subset of the -generated set, even if a dual family basis set is used. 

To complete the definition of the truncated basis set, we consider the allowed values of A", the body-fixed projection quantum number. In principle K = 0,..., J for even J - P and 1,..., J for odd J A P- With a finite basis for the Jacobi angle, however, K can not exceed min(J, 2N>il — 1). We have found that for threaction probabilities considered in the present Chapter, convergence is reached with Kmax = 2, in accord with the basis set contraction results of Zhang [35]. This rapid convergence with respect to K ax facilitates exact calculations with very modest increases in CPU time as J increases, and is one of the many useful aspects of the body-fixed representation. 

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