Atomic Physics Ground State Calculations of Atoms using Gaussian Functions Keeper Sharkey Dr Ludwik Adamowicz April 15th 2010 Purpose VERY VERY ACCURATE calculations for reproducing the electronic spectra of small atoms Rigorous variational method using explicitly correlated Gaussian Functions Gaussian functions are the only functions at present that allow for performing such high accuracy calculations for atoms with more than three electrons April 15th 2010 What are Atoms What are Atomic Spectra What is the Mathematical Model April 15th 2010 What are Atoms Atoms are composed of protons neutrons and electrons Protons and neutrons exist at the nucleus of the atom Electrons exist at fixed energy levels in space outside and around the nucleus Quantized Energy and Wave Particle Duality The simplest atom is the hydrogen atom April 15th 2010 What is Atomic Spectra Spectroscopy is the study of electronic transitions of an atom An electronic transition is the excitation or relaxation of an electron from an initial energy level to a final energy level E EF Ei The final level EF and the initial level Ei can not be equal else there is no transition E Each electronic transition has finite energy associated April 15th 2010 Electronic Spectra of Hydrogen April 15th 2010 Electronic Spectra of Hydrogen Electronic Spectra of Hydrogen Experimental data can be found on the NIST website http physics nist gov PhysRefData ASD index html The Mathematical Model Characteristic Equation Secular Equation Variation Theorem Spatial Atomic Wave Function Superposition Principle April 15th 2010 The Hamiltonian Cartesian Coordinate Frame April 15th 2010 The Hamiltonian Cartesian Coordinate Frame April 15th 2010 References Title Relativistic corrections to the non BornOppenheimer energies of the lowest singlet Rydberg states of He 3 and He 4 Authors Stanke M Kedziera D Bubin S Adamowicz L Source JOURNAL OF CHEMICAL PHYSICS Volume 126 Issue 19 Article Number 194312 Published MAY 21 2007 April 15th 2010 References The helium atom is a system that has been described in calculations since the very early stages of the development of quantum mechanics It is also one of the systems where the experiment has achieved the highest levels of precision Recent theoretical studies of the helium atom that include the works performed by Morton et al 1 Korobov and Yelkhovsky 2 Korobov 3 and Pachucki 4 6 have demonstrated that by systematically including relativistic and QED corrections to the nonrelativistic energies of the ground and excited states of this system one can achieve an accuracy of the predicted ionization and transition energies that in some cases exceed the accuracy of the presentday experiment The recently published summary of the available theoretical and experimental results for bound stationary states of He by Morton et al 1 demonstrates the high level agreement between theory and experiment very well It also shows that for a few states such as 21P1 and 23PJ there is still some noticeable disagreement between the theory and the experiment 6 7 April 15th 2010 Basis Functions Spatial Atomic Wave Function Gaussian Functions Basis Functions April 15th 2010 Results April 15th 2010 Problems Gaussian Basis have improper short range cusp behavior A too fast decaying long range behavior have a maximum when electron occupy the same point in space April 15th 2010 Work Performed Derivated overlap and Hamiltonian matrix elements and Energy gradient Coded formulas using Fortran90 Debugged Fortran90 code using Mathematica Numerical differentiation to debug the Energy gradient code Implementation on ICE super computer using MPI protocol Application to He April 15th 2010 New Results April 15th 2010 Summary Built a more representative spatial wave function using exponentially and preexponentially correlated Gaussian basis set Effectively calculated the ground state energy of He Corrected basis functions to describe better the electron correlations Superposition Principle and Variational Theorem Solved the characteristic equation April 15th 2010 Future Directions Excited state calculations Calculations on larger systems such as Be Li B Calculations where both functions have prefactors April 15th 2010 Acknowledgements Dr Ludwik Adamowicz Dr Idar Gabitov The University of Arizona Department of Chemistry Biochemistry Department of Mathematics April 15th 2010 Questions April 15th 2010