Solving the Schrödinger equation of helium and its isoelectronic ions with the exponential integral (Ei) function in the free iterative complement interaction method

Abstract

We introduce here the exponential integral (Ei) function for variationally solving the Schrödinger equation of helium and its isoelectronic ions with the free iterative complement interaction (ICI) method. In our previous study [J. Chem. Phys., 2007, 127, 224104], we could calculate very accurate energies of these atoms by using the logarithmic function as the starting function of the free ICI calculation. The Ei function has a weak singularity at the origin, similarly to the logarithmic function, which is important for accurately describing the three-particle coalescence region. The logarithmic function, however, has a node and a maximum along the radial coordinate which may be physically meaningless. In contrast, the Ei function does not have such unphysical behaviors and so would provide an improvement over the logarithmic function. Actually, using the Ei function, instead of the logarithmic function, we obtained the energy, E= -2.903 724 377 034 119 598 311 159 245 194 404 446 696 924 865 a.u. for the helium ground state with 21 035 functions, which is a slight improvement over our previous result (the bold face shows the digits that are believed to have converged). This result supports the suggestion that the Ei function is better than the logarithmic function for describing the three-particle coalescence region.

How does the free complement wave function become accurate and exact finally for the hydrogen atom starting from the Slater and Gaussian initial functions and for the helium atom on the cusp conditions?

“…In the FC method, such explicit r 12 dependence is automatically generated in the ICI step and, therefore, the FC wave function shows quite excellent convergence to the exact wave function. This has actually been shown previously for the helium atom, giving highly accurate energy, wave function, and properties 19, 21, 23, 24. More recently 25, we have further examined the local energy, H‐square error, and energy upper and lower bounds using the FC wave function of the helium atom and shown the highly accurate nature of the FC wave function.…”

“…Since 1999, one of the authors has been involved in this difficult task and has published a series of articles to formulate a general method of solving the SE of atoms and molecules in an analytical expansion form 7–25. First, he clarified the mathematical structure of the exact wave function and proposed a method, called the iterative complement (or configuration) interaction (ICI) method that gives a series of functions converging to the exact wave function 7, 8.…”

“…Combined with the variation principle, this method produced the most accurate solutions of the SE for H 2 15, 20, He 19, 23, 24, and others. In particular, the applications to He 19, 24 showed numerically that with the free ICI method we can calculate the solutions of the SE to any desired accuracy.…”

Can correlation bring electrons closer together?

“…The exponents l i , m i , and n i are nonnegative integers and, along with the nonlinear parameter and the linear parameters c i , these are optimized using the Variation Theorem. The wavefunction that we have used has N ¼ 204 and was developed by Koga et al [10], who showed that it reproduces the exact ground-state energy of the helium atom [11] to within 3 nanohartrees. The intracule of the 204-term Hylleraas wavefunction (7) reduces [12] to …”

Rydberg states of the helium atom

“…Recently, they have applied their approach to compute the ground-state energy of the helium atom to an astonishing 43 decimal digits [12] and also to obtain spectacularly accurate energies [13] for its singlet and triplet 1s ns states, for n = 2, 3, . .…”