Many Electron Atoms

to index 2011.09.13 previous lecture note LECTURE NOTE 2011.09.15 2011.09.20 next lecture note

Many Electron Atoms
SCF and DFT
The Self Consistent Field method we are now working with is in recent time often replaced by
a method originating from chemistry: Density Functional Theory
In practice they are very similar - DFT in applications is based on finding orbitals iteratively.
1_DFT_Density_Functional_Theory_vs_SCF.png

       1_DFT_Density_Functional_Theory_vs_SCF.png
LINKS FOR DFT: Nobel prize in chemistry http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1998/
                            wikipedia: http://en.wikipedia.org/wiki/Density_functional_theory
                                  functional: http://en.wikipedia.org/wiki/Functional_(mathematics)

Qualitative and Quantitative applications of Hartree type theories

We shall look at systematics of ionization potentials and the "electron configurations"

1b_Ionization_energies_tab.png

1b_Ionization_energies_tab.png

To understand the origin of (1s)² (2s)² (2p)⁶ (3s)² (3p)⁶ filled shell (the set (3d)¹⁰ does not make a shell)
we look at a model of the SCF

First the model: Radial potentials (including the centrifugal barrier: (Screened Coulomb=model of Hartree )
The value of screening parameter alpha must be found empirically (must be Z-dependent )

2__Coulomb_Centrifug_vs_Hartree_Centrifug.png

2__Coulomb_Centrifug_vs_Hartree_Centrifug.png

For Coulomb potential 3s 3p 3d have all the same energy. For other potentials
this is not true
2_Coulomb_Centrifug_vs_Hartree_Centrifug.png

2_Coulomb_Centrifug_vs_Hartree_Centrifug.png

This drawing summarizes this point: the spectra of states in Hartree-type potential have the
structur shown
3_states_n_l_in_Hartree_PERIODIC_TABLE.png

3_states_n_l_in_Hartree_PERIODIC_TABLE.png

Work with formal derivation of Hartree and Hartree-Fock theories

We need to evaluate the expectation value of the total energy in the SLATER DETERMINANT
(antisymmetrized independent particle - product state)

We started with revisiting the Helium excited state (done last time) - here in detail, showing the
difference between single particle operators and pair-particle operators

HELIUM ATOM ENERGY - as an example for Many=2
4_Helium_energy_Slater.png

4_Helium_energy_Slater.png

NEXT: 3 electrons,

LITHIUM ATOM ENERGY - as an example for Many=3
5_Lithium_description.png

5_Lithium_description.png

5_expectation_value_single_particles_and_pairs.png

5b_Lithium_terms_pairs.png

We can generalize the result
(the pair interaction can be discussed in more detail)
9_Energy_Summary.png

       9_Energy_Summary.png

This result will be used to derive the SCF equations next time

Next time (following our 2010 notes - PDF Many_Electrons_Atoms_2010.11.30.pdf   ):
         Pair interaction result in more detail
         Schrödinger equation from variational method
         Variational method - deriving Hartree-Fock Equations
         Hartree-Fock Equations
         Total energy and the selfconsistent orbital energies