Basis sets balance

Another aspect ol basis set balance is the occasional use of mixed basis sets, tor me optinuzation ot oasis lunction exponents ts an example or a nigniy non-imear... 

Balance among different basis sets is as important in the divide-and-conquer method as the basis set balance in the conventional ab initio calculations. Unbalanced basis sets will lead to nonphysical charge shifts. Take N2 molecule as an example. Putting more basis functions on one N atom than on the other will result in a nonzero dipole in this molecule. This artifact exists in both the divide-and-conquer method and the conventional methods. The experiences of balancing basis set from conventional methods can be used directly in the divide-and-conquer calculations. 

The solution to this dilemma is to recognize that the nucleus has a finite size, and that this should be accounted for. Ishikawa and coworkers showed that the use of a finite nucleus instead of a point nucleus allowed for more compact basis sets [12] and also eliminated problems with basis set balance close to the nucleus [13]. Visser et al. [14] performed a full relativistic optimization of exponents for the one-electron atoms Sn and U with and without a finite nucleus, showing that the use of a finite nuclear radius significantly decreased the maximum exponent. 

The next step in iin proving a basis set could be to go to triple zeta, quadruple zeta, etc. Ifone goes in this direction rather than adding functions of higher angular quantum number, the basis set would not be well balanced. With a large number of s and p functions only, one finds, for example, that the equilibrium geometry of am monia actually becomes planar. The next step beyond double z.ela n sit ally in voices addin g polarization fn n ciion s, i.e.. addin g d-... 

The various basis sets used in a calculation of the H and S integrals for a system are attempts to obtain a basis set that is as close as possible to a complete set but to stay within practical limits set by the speed and memory of contemporary computers. One immediately notices that the enterprise is directly dependent on the capabilities of available computers, which have become more powerful over the past several decades. The size and complexity of basis sets in common use have increased accordingly. Whatever basis set we choose, however, we are attempting to strike a balance. If the basis set is too small, it is inaeeurate if it is too large, it exceeds the capabilities of our computer. Whether our basis set is large or small, if we attempt to calculate all the H and S integrals in the secular matrix without any infusion of empirical information, the procedure is described as ab initio. 

Except for very small systems it is impractical to saturate the basis set so that the absolute error in the energy is reduced below chemical accuracy, for example 1 kcal/ mol. The important point in choosing a balanced basis set is to keep the error as constant... 

The presence of the momentum operator means that the small component basis set must contain functions which are derivatives of the large basis set. The use of kinetic balance ensures that the relativistie solution smoothly reduees to the non-relativistic wave function as c is increased. 

In general, the basis set should be in balance with the computational method A highly sophisticated method [e.g., CCSD(T)] in combination with a small basis or a low-level method [e.g., Hartree-Fock (HE)] in combination with a very large basis may be useful only in very specific cases. Consequently, increasing the basis set should be done while increasing the quality of the post-HF approach for a better representation of the electron correlation. 

The goodness of the PP representation can be checked by comparing the all-electron and PP orbital energies and relative stability of atomic states. The comparison is shown in Table 4, and is seen to be very satisfying. For a balanced treatment, also the carbon and oxygen atoms were treated by a PP, as described in previous work5.3d functions were not introduced in the sulphur basis set, mainly because they were not deemed necessary for the illustrative purposes of this chapter. Also, the derivation of a PP representation for polarization functions is not a straightforward matter. The next section is devoted to the discussion of this point. 

The Brueckner-reference method discussed in Section 5.2 and the cc-pvqz basis set without g functions were applied to the vertical ionization energies of ozone [27]. Errors in the results of Table IV lie between 0.07 and 0.17 eV pole strengths (P) displayed beside the ionization energies are approximately equal to 0.9. Examination of cluster amplitudes amd elements of U vectors for each ionization energy reveals the reasons for the success of the present calculations. The cluster operator amplitude for the double excitation to 2bj from la is approximately 0.19. For each final state, the most important operator pertains to an occupied spin-orbital in the reference determinant, but there are significant coefficients for 2h-p operators. For the A2 case, a balanced description of ground state correlation requires inclusion of a 2p-h operator as well. The 2bi orbital s creation or annihilation operator is present in each of the 2h-p and 2p-h operators listed in Table IV. Pole strengths are approximately equal to the square of the principal h operator coefiScient and contributions by other h operators are relatively small. 

A reaction dataset, such as the LLNL database, provides an alternative method for balancing reactions. Such a database contains reactions to form a number of aqueous species, minerals, and gases, together with the corresponding equilibrium constants. Reactions are written in terms of a generic basis set B, which probably does not correspond to set B, our choice of species to appear in the reaction. 

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