This is an important check of the numerical basis of the new Lagrangian method, a check made possible by the fact that the hydrogenic wave functions are analytical. It is not easy to develop the numerical method, there are certain things that have to be coded up. This has been done perfectly by co author Horst Eckardt. Having completed this check it is now possible to proceed with confidence to the helium atom, which has two electrons and two protons, together with neutrons in the nucleus. The hope is that the Lagrangian method will be of general utility in computational quantum chemistry and can be added to code packages and libraries. I was invited by Clementi in 1988 to be the lead writer for IBM MOTECC. I was an IBM professor with Clement Roothaan of the Roothaan and Roothaan Hall equations of computational quantum chemistry. We helped to pioneer parallel processing using the IBM 3090-6S supercomputer and LCAP system (linear combination of array processors). We also helped to pioneer computer animation. I hope to make available a digitized video tape of one of the first computer animations at IBM Kingston in the mid eighties. This is one of the earliest computer animations ever made of molecular dynamics computer simulation, based on my development of the SERC CCP5 Code TETRA. The code is on www.aias.us and was also used for the Evans Pelkie prize winning animation now kindly put on youtube by AIAS Fellow Michael Jackson. That was made at Cornell Theory Center and took about a year to make. The animation was by Chris Pelkie from my code for the optically active molecule chlorobromofluromethane written at the EDCL in Aberystwyth and extended to double precision at IBM Kingston, New York State. I was the first to use computer simulation at UCW Aberystwyth, and the first to use a microcomputer, a Research Machines microcomputer which we interfaced with the interferometer. The interface was carried out by Irfon Williams and the AIAS Co President, Gareth Evans. For about a decade, I have been working with UPITEC President Horst Eckardt, who uses a desktop and Maxima code. This is almost as powerful as the IBM 3084 mainframe of the mid eighties.

To: EMyrone@aol.com

Sent: 06/03/2017 17:31:04 GMT Standard Time

Subj: Re: 372(2): Analytical SolutionsI have taken the initial conditions for the numerical solution from the analytical solution:

R(r0) = R(r0) [analyt.]

and

dR(r0)/dr = dR(r0)/dr [analyt.]for r0 = 0.001. The analytical solutions were taken from our earlier work, see protocol. In atomic units we have a_0 = 1. Then the numerical solution gives exactly the same as the analytical solution, see attached figure for the 3p radial function. The curves lie on top of each other.

Horst

Am 06.03.2017 um 11:41 schrieb EMyrone:

The analytical solutions of the Lagrangian quantum mechanics of H, (Eq (1) of the attached) are related to the associated Laguerre polynomials as in Eq. (16). Maxima can probably work out all the analytical radial wavefunctions of atomic H for any valid n and l. This has been done many times by Horst Eckardt in previous UFT papers using the hydrogenic wavefunctions. This note checks that the Lagrangian and Hamiltonian quantum mechanics give the same result for H. The Lagrangian method may have computational advantages over the well known Hamiltonian method. That is the purpose of this work. To check the Runge Kutta integration of Eq. (1) with Maxima use the analytical results. The purpose of this is to check that the code works for H, then apply and develop the code for other atoms and molecules. Eq. (1) also holds for hydrogen like atoms, a term used in computational quantum chemistry (for example the alkali metals sodium, potassium and so on). I have used some interpretative material from Atkins, and I hope that he has not made an error. His calculations can be checked by computer algebra.