ground state calculations of atoms using gaussian functions
TRANSCRIPT
Atomic Physics
Ground State Calculations of Atoms 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 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?
• 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 atom
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• Quantized Energy and Wave-Particle Duality
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.
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energy level.
• The final level, EF, and the initial level, E
i, can not
be equal else there is no transition, ∆E.
• Each electronic transition has finite energy
associated.
∆E = EF
- Ei
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:
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Spatial Atomic
Wave FunctionSuperposition Principle:
Title: Relativistic corrections to the non-Born-
Oppenheimer energies of the lowest singlet
Rydberg states of He-3 and He-4
Authors: Stanke M, Kedziera D, Bubin S,
Adamowicz L
References
Source: JOURNAL OF CHEMICAL PHYSICS
Volume: 126
Issue: 19 Article
Number: 194312
Published: MAY 21 2007
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ReferencesThe 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
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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 present-
day 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
Basis Functions
Spatial Atomic
Wave Function
Basis Functions Gaussian Functions
ΑΧΣ ΑΧΣ ΑΧΣ ΑΧΣ −−−−βτβτβτβτApril 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. same point in space.
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Work 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, 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.of He.
• Corrected basis functions to describe better the
electron correlations.
• Superposition Principle and Variational
Theorem
• Solved the characteristic equationApril 15th, 2010
Future 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, 2010
Acknowledgements
Dr. Ludwik Adamowicz
Dr. Idar Gabitov
The University of ArizonaThe University of Arizona
Department of Chemistry & Biochemistry
Department of Mathematics
April 15th, 2010