How to construct the trial wave function of the Hydrogen atom?
This text is based upon the following presentation:
Quantum Chemical Methods
The exact wave function of atoms are unknown except for Hydrogen. For that we proceed to calculate an approximate one called the trial wave function. To look how to get a trial wave function close to the exact one, we take the Hydrogen atom as a real case to do the job. We will be interested only the radial part of the full wave function because of the spherical symmetry.
We know that it is not possible to solve the Schrodinger equation exactly for a many-body system because of the electron-electron interactions. For that, we use approximate methods which are of two types:
1- Perturbation method: We treat electron-electron interactions as a minor perturbation.
2- Variational method: We guess a suitable form of wave function and optimize it
In this text, we present how to construct the trial function of the Hydrogen atom using the variational method.
Variational Method
The real wave function of the Hydrogen atom
The proposed parameter dependent trial wave function is as follows:
Background of the Variational Theory
To get the background of the variational theory check the pages 24 to 30 of the previous presentation.
Properties of the trial function : for any suitable trial wavefunction:
Construction and Optimization of Trial Wavefunctions
Choosing the basis functions : The gaussian function
How to get the parameters of the new trial function which is a linear combination of gaussian functions?
check the pages 35 to 39.
Quantum Chemical Methods
The exact wave function of atoms are unknown except for Hydrogen. For that we proceed to calculate an approximate one called the trial wave function. To look how to get a trial wave function close to the exact one, we take the Hydrogen atom as a real case to do the job. We will be interested only the radial part of the full wave function because of the spherical symmetry.
We know that it is not possible to solve the Schrodinger equation exactly for a many-body system because of the electron-electron interactions. For that, we use approximate methods which are of two types:
1- Perturbation method: We treat electron-electron interactions as a minor perturbation.
2- Variational method: We guess a suitable form of wave function and optimize it
In this text, we present how to construct the trial function of the Hydrogen atom using the variational method.
Variational Method
The real wave function of the Hydrogen atom
The proposed parameter dependent trial wave function is as follows:
Background of the Variational Theory
To get the background of the variational theory check the pages 24 to 30 of the previous presentation.
Properties of the trial function : for any suitable trial wavefunction:
Construction and Optimization of Trial Wavefunctions
Choosing the basis functions : The gaussian function
How to get the parameters of the new trial function which is a linear combination of gaussian functions?
check the pages 35 to 39.
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