Basis sets for helium

Cases of three or more electrons were very difficult to treat by the above methods. For instance, for three-electron systems, it is required to have six terms in the expansion of each basis function in order to comply with the antisymmetry criterion, and each term must have factors containing ri2, ri3, r23, etc., if we want to accelerate the convergence. There is, indeed, a real problem with the size of each trial wave function. A symmetrical wavefunction requires that the trial basis set for helium contain two terms to guarantee the permutation of electrons. For an N-electron system, this number grows as N . For a ten-electron system like water, it would be required that each basis set member have more than 3 million terms, and this is in addition to the dependence on 3N variables of each of the terms. These conditions make the Schrodinger equation intractable for systems of even a few electrons. Just the bookkeeping of these terms is practically impossible. 

Exercise 5.4. Formation of a 4-31) split-basis set for helium using the data of Table 5.1 and the calculation of the energy of the helium ground state. 

The basis sets for helium and lithium are more complicated in detail but the principles are the same. In each case the Bethe logarithm comes almost entirely from virtual excitations of the inner Is electron to p-states lying high in the photoionization continuum, and so the basis set must be extended to very short distances for this particle. The outer electrons are to a good approximation just spectators to these virtual excitations. 

By checking the results from Table II of Ref. [19], we see that for the closed-shell system there are discrepancies between the HF and KS-SIC-OEP-KLI methods. Such differences can be attributed to the basis set used for the HF method, or to the methodology used in the KS approach. The aim of this work is twofold. First, we recomputed the Table II from Ref. [19] by using an optimized basis set for the HF calculation. Second, we compute the lowest triplet state of the confined helium atom by the HF method and its results are compared with those obtained by the KS approach. In this way a reliable comparison between HF and KS is made for the lowest singlet and triplet states of the confined helium atom. 

A very accurate determination of the interaction-induced polarizability of He2 at the experimental internuclear separation of 5.6 ao was reported by Jaszunski et al The authors used a very large Ils8p6d5f4g3h basis set for He and high-precision explicitly correlated R12 methods. Their most accurate results for the mean and the anisotropy polarizability were calculated at the CCSD(T)-R12 level of theory and are are aint= 0.00104 and Aa = 0.06179 e ao Eh These values represent reference estimates of the interaction-induced dipole polarizability of two helium atoms. 

Table 6 First Four Augmented Correlation Consistent Valence Basis Sets for Hydrogen, Helium, and the First-row Atoms Boron through Neon...

The polarized core-valence correlation-consistent basis sets for first-row atoms are listed in Table 8.13. We note that the same pattern is followed for the core-valence functions of the first-row atoms as for the valence functions of helium in Table 8.11. The number of ftinctions in the core-valence sets may therefore be calculated from the cardinal number as... 

Table 8.17 The correlation energies recovered by the analytical and numerical correlation-consistent basis sets for the ground-state helium atom. The exact correlation energy is —42.044 mEh...

The two preceding applications showed that our hydrogenic model fits well with the helium atom and the dihydrogen molecule for the determination of the polarization functions except that their exponent ( is different from Co which is the exponent of the genuine basis set It is obvious that the hydrogenic model will fit less and... 

The denominator shift of 1/2 was chosen as a compromise between the situation for hydrogen and helium (where n = 1 + 1 for the cc-pVnZ basis set) and main-group elements (where n = 1). As is immediately obvious upon series expansion, there is considerable coupling between the denominator shift and the exponent. As a result, the three-point extrapolation generally leads to exponents well in excess of three [34],... 

Complete Basis Set Models for Chemical Reactivity from the Helium Atom to Enzyme Kinetics... 

Calculations of IIq(O) are very sensitive to the basis set. The venerable Clementi-Roetti wavefunctions [234], often considered to be of Hartree-Fock quality, get the sign of IIq(O) wrong for the sihcon atom. Purely numerical, basis-set-free, calculations [232,235] have been performed to establish Hartree-Fock limits for the MacLaurin expansion coefficients of IIo(p). The effects of electron correlation on IIo(O), and in a few cases IIq(O), have been examined for the helium atom [236], the hydride anion [236], the isoelectronic series of the lithium [237], beryllium [238], and neon [239] atoms, the second-period atoms from boron to fluorine [127], the atoms from helium to neon [240], and the neon and argon atoms [241]. Electron correlation has only moderate effects on IIo(O). 

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