Basis size

The LMTO method is the fastest among the all-electron methods mentioned here due to the small basis size. The accuracy of the general potential teclmique can be high, but LAPW results remain the gold standard . 

In the present implementation, the unperturbed functions are not subject to any orthogonality constraint nor are required to diagonalize any model hamiltonian. This freedom yields a faster convergence of the variational expansion with the basis size and allows to obtain the phaseshift of the basis states without the analysis of their asymptotic behaviour. 

Expectation Values of the Li-H Internuclear Distance, Its Square, and Electron-Positron Contact Densities Evaluated at 3200-Function Basis Size [122] ... 

Here r is the basis size, N is the electron number, Va the number of spin-up electrons, and 25 + 1 is the multiplicity. The geometries used are the experimental ones from Ref. [20]. The basis set is STO-6G for all systems. 

We also see that the number of basis functions grows rapidly with the number of electrons. In Chapter 16 we will discuss another method for dealing with the escalation of basis size with greater numbers of atoms and electrons. 

Thus the one-particle basis determines the MOs, which in turn determine the JV-particle basis. If the one-paxticle basis were complete, it would at least in principle be possible to form a complete jV-particle basis, and hence to obtain an exact wave function variationally. This wave function is sometimes referred to as the complete Cl wave function. However, a complete one-paxticle basis would be of infinite dimension, so the one-paxticle basis must be truncated in practical applications. In that case, the iV-particle basis will necessarily be incomplete, but if all possible iV-paxticle basis functions axe included we have a full Cl wave function. Unfortunately, the factorial dependence of the iV-paxticle basis size on the one-particle basis size makes most full Cl calculations impracticably large. We must therefore commonly use truncated jV-paxticle spaces that axe constructed from truncated one-paxticle spaces. These two truncations, JV-particle and one-particle, are the most important sources of uncertainty in quantum chemical calculations, and it is with these approximations that we shall be mostly concerned in this course. We conclude this section by pointing out that while the analysis so fax has involved a configuration-interaction approach to solving Eq. 1.2, the same iV-particle and one-particle space truncation problems arise in non-vaxiational methods, as will be discussed in detail in subsequent chapters. 

In principle, Equation (3.5) represents an infinite set of coupled equations. In practice, however, we must truncate the expansion (3.4) at a maximal channel n which turns (3.5) into a finite set that can be numerically solved by several, specially developed algorithms (Thomas et al. 1981). The required basis size depends solely on the particular system. The convergence of the close-coupling approach must be tested for each system and for each total energy by variation of n until the desired cross sections do not change when additional channels are included. Expansion (3.4) should, in principle, include all open channels (k > 0) as well as some of the closed vibrational channels (k% < 0). Note, however, that because of energy conservation the latter cannot be populated asymptotically. 

Further analysis performed in [27] allowed authors to establish the convergence specifically of the correlation energy with respect to the basis size in the form ... 

First, we try the Sturmian method outUned in Section 6.2.1. The computations are performed in a basis of 50 Sturmian states as defined in (6.2.3) for three different choices of the Sturmian label, a = 3,4,5. The resulting ionization probabiUties after 100 cycles of the field, P (IOO), are shown in Figs. 6.8(a) - (c) for the three different Stm-mian labels, respectively. Our first impression is that the results depend strongly on the choice of the Sturmian label. This dependence can in principle be reduced, but only at the cost of increasing the basis size substantially. 

With increase of the basis size the different pair variants appear to converge to the same results. 

The reason for this, and for the extreme sensitivity to basis size seen in Tables VI and VII, is qualitatively explained by Figure... 

TABLE I. Convergence study of the position of the resonance state (l j 2, J M=0) of AT.H2 van der Waals molecule (potential U(III)) with respect to basis size N. Shown here are calculations with two channel blocks (with the same basis size N.. ), and = 9.0 a ,... 

It affects also computational data since, apart from very special models, the eigenstates need be determined by diagonalizing the Hamiltonian in a basis set that is exceedingly large. As the basis size is increased, the very weakest transitions are (as is only to be expected on the basis of Murphy s law and other considerations) the slowest to converge. 

The correction term is usually ignored. However, it is important if < o is not a good approximation to its exact counterpart. Calculations in basis expansions converge much faster with extension of the basis size [22], if the correction term is included. Note, that the correction term is spin-independent, it does not affect the spin-orbit term. 

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