Is it possible to solve the Schrödinger equation of general atoms and molecules with simple mathematics?

Prof. Hiroshi Nakatsuji (Quantum Chemistry Research Institute, Kyodai Katsura Venture Plaza, Goryo Oohara 1-36, Nishikyo-ku, Kyoto 615-8245, Japan)
csü., 2008-06-05 15:00
Kémia épület 063

Schrödinger equation (SE), Hψ=Eψ provides a governing principle of chemistry, biology and physics and has an accurate and powerful predictive power [1]. For over 80 years after its birth, however, this equation has been thought not to be soluble except for a few special cases. Since 2000, we are formulating a general method for solving the SE of atoms and molecules in an analytical expansion form [2-13]. First, we clarified the mathematical structure of the exact wave function and proposed a method, called iterative complement (or configuration) interaction (ICI) method, that gives a series of functions converging to the exact wave function [2,3]. This has been confirmed by applying the method to model harmonic oscillator [4] and to finite-basis expansion method based on the second-quantized Hamiltonian [5]. However, this method includes the integrals of higher powers of Hamiltonian, which diverge when the Hamiltonian involves singular operators like Coulomb potential [6,7]. This problem, called singularity problem, always occurs when we apply the method to atoms and molecules, so this was really a severe problem. However, this difficulty was solved by introducing the scaled Schrödinger equation (SSE) [7]. The ICI method based on the SSE gives the exact wave function at convergence without encountering the singularity problem. Further, the free ICI method was proposed based on this method. It is more easily handled and converges more rapidly than the original ICI method [7,10]. Combined with the variation principle, this method produced the most accurate solutions of the SE for H2 [11,13,14], He [12,15], and others. In particular, the applications to He [12] showed that with the free ICI method we can calculate the solutions of the SE to desired accuracy.
A problem arose here however that the analytical integration is not always possible for the complement functions of general atoms and molecules. This difficulty, called integration difficulty, has been solved by introducing the local SE (LSE) method [13]. It is based on the fact that the free ICI wave function is potentially exact when the order (iteration number) n is large, so that we can use the SE locally. Actually, the local energy relation was confirmed to hold in high accuracy for H2 and He calculated above. With the LSE method, we could calculate the analytical wave functions and energies of several two to five electron atoms and molecules in high accuracy without doing analytical integrations. This free ICI LSE method is general and applicable, in principle, to any atoms and molecules, so that it may be said that now the 80 years’ dogma that the SE can not be solved has been broken. The problems left are to develop efficient algorithms and computing. We note that the free ICI LSE method is suitable for super-parallel computers developed and being developed in recent years. This is different from other accurate quantum chemistry codes. Note that this methodology can also be applied to solve the relativistic Dirac-Coulomb equations [8].
By making the present methodology efficient, we hope we would be able to construct truly accurate and predictive quantum chemistry based on the solutions of the SE. If we can further extend it to simulation methodology, we would be able to reproduce ‘nature’ itself from the principles of quantum mechanics alone. It is indeed our goal.
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