Piecewise polynomial configuration interaction natural orbital study of 1 s2 helium

Abstract

We report here an analysis of extensive configuration interaction (CI) wave functions for the 1s2 ground state of the helium atom using piecewise polynomial basis functions. Large numbers of natural radial orbitals (NROs) with l ranging from 0 to 11 have been treated accurately and analyzed systematically. The contribution of each NRO to the total energy is found to follow the formula ΔE∼−0.42(l+1/2) × (n−1/2)−6a.u., where n is the principal quantum number, and the expansion coefficient of the NRO configuration is found to follow the formula, c∼−0.35(l+1/2)2/3 (n−1/2)−4. (The constants for l=0 are a little different). From an examination of the tails of the NROs, we are able to suggest an l-dependent universal asymptotic formula, rβ−l−1+δe−ζr, where β is a constant, where δ=δl0, and where ζ2/2 is the ionization potential. The nodes of the NROs are also found to behave in a systematic way that yields valuable information on the choice of basis functions. So-called ’’L’’-limit energies EL, more accurate than any we could find in the literature, have been determined for L=0 through 11, with a final CI energy calculated here of −2.90370 a.u. from a wave function containing 118 NROs. The increments EL−EL−1 are found to follow the formula −0.0740(L+1/2)−4−0.031 (L+1/2)−5 a.u., which is useful for extrapolation and extension of the CI energy by two significant digits. The n- and l-dependent formulas for the energy contributions make it possible to estimate the size of a CI calculation required for a given accuracy. The high accuracy of these CI calculations is made possible by the flexibility and numerical stability of piecewise polynomial basis functions.

Rate of Convergence of the Configuration Interaction Model for the Helium Ground State

“…This behavior was demonstrated numerically for the ground state of helium by Carroll, Silverstone, and Metzger [5].…”

“…It is notable that this expansion is also slowly convergent and, due to the increasing number of terms, more slowly convergent as increases. While its rate of convergence has been discussed [17,3,5], there are no rigorous results, which, when combined with the results of this paper, would give a rate of convergence in terms of one-electron orbitals. We note however that the expansion of the radial part in r < := min{r 1 , r 2 } and r > := max{r 1 , r 2 } converges rapidly [4,21].…”

Theory and application of explicitly correlated Gaussians

“…The general properties of the CI expansion have been known since the work of Schwartz (1962), which provided the underlying foundation for later investigations (Carroll, Silverstone, and Metzger, 1979;Hill, 1985;Ottschofski and Kutzelnigg, 1997;Bromley and Mitroy, 2007a). The orbital CI expansion is slowly convergent with respect to L max , the maximum angular momentum of any orbital included.…”

Efficient Temperature-Dependent Green’s Function Methods for Realistic Systems: Using Cubic Spline Interpolation to Approximate Matsubara Green’s Functions

“…In chemical physics, cubic spline interpolation has been applied as a basis to solve a complex differential and integral Schrödinger [13][14][15], Dirac [16], and Sham-Schlüter equations [17], Thomas-Fermi model [18], in calculations of vibrational and rotational spectra [19] and many other cases [20][21][22][23][24][25][26][27][28][29][30][31][32]. Since the derivatives of third and higher order polynomials are discontinuous, cubic spline interpolation is limited to applications that are not sensitive to the smoothness of derivatives higher than second order.…”