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.

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What are Atoms?

What are Atomic Spectra?

What is the Mathematical

Model?

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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

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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:

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Spatial Atomic

Wave FunctionSuperposition Principle:

- The Hamiltonian -Cartesian Coordinate Frame

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- The Hamiltonian -Cartesian Coordinate Frame

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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

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Results

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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

New Results

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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.

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Acknowledgements

Dr. Ludwik Adamowicz

Dr. Idar Gabitov

The University of ArizonaThe University of Arizona

Department of Chemistry & Biochemistry

Department of Mathematics

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QuestionsQuestions

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