Basis sets numerical convergence

This condition is satisfied by for example the sequences n or although these exponents would lead to basis sets with slow convergence and tremendous numerical problems. As for STOs in Section 6.S.6, these criteria are not very helpful in practice since they do not guide us towards basis sets that converge rapidly (since the conditions do not depend in any way on the atoms or orbitals to be expanded). They do tell us, however, that it is possible to construct complete basis sets of the simple form (6.6.15). 

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 /)... 

So far, the only approximation in our description of the FMS method has been the use of a finite basis set. When we test for numerical convergence (small model systems and empirical PESs), we often do not make any other approximations but for large systems and/or ab //i/Y/o-determined PESs (AIMS), additional approximations have to be made. These approximations are discussed in this subsection in chronological order (i.e., we begin with the initial basis set and proceed with propagation and analysis of the results). 

When the FMS method was first introduced, a series of test calculations were performed using analytical PESs. These calculations tested the numerical convergence with respect to the parameters that define the nuclear basis set (number of basis functions and their width) and the spawning algorithm (e.g., Xo and MULTISPAWN). These studies were used to validate the method, and therefore we refrained from making any approximations beyond the use of a... 

Similarly, improvement in the accuracy of the nuclear dynamics would be fruitful. While in this review we have shown that, in the absence of any approximations beyond the use of a finite basis set, the multiple spawning treatment of the nuclear dynamics can border on numerically exact for model systems with up to 24 degrees of freedom, we certainly do not claim this for the ab initio applications presented here. In principle, we can carry out sequences of calculations with larger and larger nuclear basis sets in order to demonstrate that experimentally observable quantities have converged. In the context of AIMS, the cost of the electronic structure calculations precludes systematic studies of this convergence behavior for molecules with more than a few atoms. A similar situation obtains in time-independent quantum chemistry—the only reliable way to determine the accuracy of a particular calculation is to perform a sequence of... 

Note that the NAOs, NBOs, and associated occupancies are in principle uniquely determined by y, and thus by itself. In practice, the NBO-based quantities are found to converge rapidly to well-defined numerical limits, independently of the numerical basis set or other arbitrary details of approximating present work, the level of describing tf will be taken to be sufficiently high that we can generally ignore the small differences that distinguish numerically determined NBOs from the infinite-basis limit. 

The main conclusion of this section is that the matrix elements of all terms in the collision Hamiltonian in the fully uncoupled space-fixed representation can be reduced to simple products of integrals of the type (8.46). Such matrix elements are very easy to evaluate numerically. The fiilly uncoupled representation is therefore very convenient for the development of the coupled channel codes for collision problems involving open-shell molecules with many angular momenta that need to be accounted for. The price for simplicity is a very large number of basis states that need to be included in the expansion of the eigenstates of the full Hamiltonian to achieve full basis set convergence (see Section 8.3.4). 

In the MCHF approach a number of superposed configurations are chosen and the mixing coefficients (weights of the configurations) and also the radial parts of the wave functions are varied. This method does not depend on choice of the basis set and both analytical and numerical wave functions may be used. However, MCHF calculations for complex electronic configurations would require variation of a large number of parameters, which needs powerful computers. Problems may also occur with the convergence of the procedure [45]. 

Numerical inspection shows that the identification of the maximum of F J2 with the real part of E , as well as that of the half width of Fen( 2 with the imaginary part of E , remains very reasonable estimates also when the Coulomb potential is included. This analysis explains the much improved basis set convergence with complex scaling discussed in connection with Figure 5.2. 

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