Atomic PhysicsGround State Calculations of Atoms Ground State Calculations of Atoms using Gaussian FunctionsKeeper SharkeyDr. Ludwik AdamowiczApril 15th, 2010PurposeVERY VERY ACCURATE calculations for reproducing the electronic spectra of small atoms. Rigorous variational method using explicitly correlated Rigorous variational method using explicitly correlated Gaussian FunctionsGaussian functions are the only functions at present that allow for performing such high accuracy calculations for atoms with more than three electrons.April 15th, 2010What are Atoms?What are Atomic Spectra?What is the Mathematical Model?April 15th, 2010What 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 •Electrons exist at fixed energy levels in space outside and around the nucleus.• The simplest atom is the hydrogen atomApril 15th, 2010• Quantized Energy and Wave-Particle DualityWhat 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.April 15th, 2010energy level.• 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.∆E = EF- EiElectronic Spectra of HydrogenApril 15th, 2010Electronic Spectra of HydrogenElectronic Spectra of HydrogenExperimental data can be found on the NIST website:http://physics.nist.gov/PhysRefData/ASD/index.htmlThe Mathematical ModelCharacteristic Equation: Secular EquationVariation Theorem:April 15th, 2010Spatial Atomic Wave FunctionSuperposition Principle:- The Hamiltonian -Cartesian Coordinate FrameApril 15th, 2010- The Hamiltonian -Cartesian Coordinate FrameApril 15th, 2010Title: Relativistic corrections to the non-Born-Oppenheimer energies of the lowest singlet Rydberg states of He-3 and He-4Authors: Stanke M, Kedziera D, Bubin S, Adamowicz LReferencesSource: JOURNAL OF CHEMICAL PHYSICS Volume: 126 Issue: 19 ArticleNumber: 194312 Published: MAY 21 2007April 15th, 2010ReferencesThe helium atom is a system that has been described incalculations since the very early stages of the development ofquantum mechanics. It is also one of the systems where theexperiment has achieved the highest levels of precision. Recenttheoretical studies of the helium atom that include the worksperformed by Morton et al.,1 Korobov and Yelkhovsky,2 Korobov,3and Pachucki,4–6 have demonstrated that by systematicallyincluding relativistic and QED corrections to the nonrelativisticenergiesofthegroundandexcitedstatesofthissystem,onecanApril 15th, 2010energiesofthegroundandexcitedstatesofthissystem,onecanachieve an accuracy of the predicted ionization and transitionenergies that in some cases exceed the accuracy of the present-day experiment. The recently published summary of the availabletheoretical and experimental results for bound stationary states ofHe by Morton et al.1 demonstrates the high level agreementbetween theory and experiment very well. It also shows that fora few states such as 21P1 and 23PJ there is still some noticeabledisagreement between the theory and the experiment.6,7Basis FunctionsSpatial Atomic Wave FunctionBasis FunctionsGaussian FunctionsΑΧΣ ΑΧΣ ΑΧΣ ΑΧΣ −−−−βτβτβτβτApril 15th, 2010ResultsΑΧΣ ΑΧΣ ΑΧΣ ΑΧΣ −−−−βτβτβτβτApril 15th, 2010Problems• 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. same point in space. April 15th, 2010Work Performed• Derivated overlap and Hamiltonian matrix elements and Energy gradient.• Coded formulas using Fortran90 • Debugged Fortran90 code using Mathematica.Mathematica.• Numerical differentiation to debug the Energy gradient code• Implementation on ICE super computer using MPI protocol• Application to He April 15th, 2010New ResultsΑΧΣ ΑΧΣ ΑΧΣ ΑΧΣ −−−−βτβτβτβτApril 15th, 2010Summary• Built a more representative spatial wave function using exponentially and preexponentially correlated Gaussian basis set.• Effectively calculated the ground state energy of He.of He.• Corrected basis functions to describe better the electron correlations.• Superposition Principle and Variational Theorem• Solved the characteristic equationApril 15th, 2010Future Directions• Excited state calculations• Calculations on larger systems such as Be, Li-, B+...Be, Li, B...• Calculations where both functions have prefactors.ΑΧΣ ΑΧΣ ΑΧΣ ΑΧΣ −−−−βτβτβτβτApril 15th, 2010AcknowledgementsDr. Ludwik AdamowiczDr. Idar GabitovThe University of ArizonaThe University of ArizonaDepartment of Chemistry & BiochemistryDepartment of MathematicsApril 15th, 2010QuestionsQuestionsApril 15th,