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Starting by Few slides on Spin (missing in the He part)          - see last year Many electron atoms start at a blue region below (link)             Some slides got lost - replaced by reconstructions and last year versions   ../2014_09_23/index.html

Extra on spin  Pauli matrices  - (up to the hbar/2 ) represent the "spin angular momentum" The same commutation relations as angular momentum    ... see also the vector product of the operator L by itself   <----->    commutation relations Magnetic moment      and      angular momentum     /      spin magnetic moment Addition of two spins                             Addition of two   angular momenta

00010.png        00010.png

Above :   Addition of two spins                             Addition of two   angular momentuma  Under :     Singlet and triplet energies    - Effective  spin - spin interaction

00020.png        00020.png

Singlet and triplet energies    - Effective  spin - spin interaction         since there is different energy for two values of "total spin"  -  S=0  and  S=1        we can try to write the energy as function of the SCALAR PRODUCT                                                                            - i.e the ANGLE between the 2 electron spins       It thus becomes proportianal to the relative orientations of the spins       ....  in this case   parallel   or   anti-parallel

zz_effective_spin_spin_interaction.png        zz_effective_spin_spin_interaction.png

The exchange interaction  -  leads to effective spin - spin alignment  (   minus sign  )

zz_z_effective_spin_spin_ferromagnetic.png        zz_z_effective_spin_spin_ferromagnetic.png

The above comments:  FERROMAGNETISM; Theories  - Heisenberg Ferromagnet model Thus understanding exchange interaction /  effective spin spin   in He   ===>   Nature of Ferromagnetism Also: spin - spin alignment - effective spin - spin  --   SEE HUND'S RULE in the foollowing part

Many Electron Atoms - Part 1

Filling of the shells, Build - up - "Aufbau - Prnciple Hund's Rule - spins want to be parallel ==>  effective spin-spin interaction; cf. Helium triplet (orthohelium)                               Historically, Hund's rules were very important.                               For us they are just an example of the "effecive spin interaction - the Helium triplet states,                               i.e. the Pauli principle - and thus antisymmetry of the space part

00041_Hunds_Rule.png        00041_Hunds_Rule.png

Filling of the shells, Build - up - "Aufbau - Prnciple  - Pauli

00042_Aufbau__building-up.png        00042_Aufbau__building-up.png

Number of states in n-shell n l m quantum numbers                                                             n     l    m

00043_Number_of_states_in_a_shell_N_.png        00043_Number_of_states_in_a_shell_N_.png                                              But the Noble gases are not Z=2, 10, 28, 60;  rather  Z= 2, 10, 18, 36, 54, ....

Experimental Ionization Potentials    (  minus  times  the Binding energies ) Closed shells  -  PEAK    -  i.e. large DIP in Binding energy                             WHY   -   see the sketch in the lower part Closed shells are NOT   closed n - shells    --- except  for  Z=2  and Z=10  Next should be Z=28

00046_Ionization_Energies_and_Binding_E.png        00046_Ionization_Energies_and_Binding_E.png

above - schematic drawing:    Closed shells  -  PEAK    -  i.e. large DIP in Binding energy                                                    WHY   -     - 1/2  Z2 / n2      -   ---        gets smaller and smaller with Z                                                                                                                 until n filled   - JUMP up to    Z2 / (n+1) 2          Towards the SELF-CONSISTEN FIELD THEORY Electron cloud   -   improve the Helium - effective charge by much more flexible "cloud of negative charge" What is the interaction of an electron with such a "cloud" ? interaction of an electron with a "cloud" of charge, the charge density, and the charge distribution from the probability - trying to find which potential energy should be used to describe one independent electron in the atom                                              (Hartree 1926- 1930s)                                                                                                                            Here WE HAVE LOST ORIGINAL - THIS IS A RECONSTRUCTION

00050_two_charges_dq1_dq2.png        00050_two_charges_dq1_dq2.png

The density (rho) coming from the orbitals, but the orbitals are found solving Schrödinger equation with the potential resulting from the density (rho)

00060_Interaction_with_the_CLOUD.png        00060_Interaction_with_the_CLOUD.png

Electron density from probability density  ... from wavefunctions ( orbitals) --> cloud Orbitals from    the potential       < ----    with the CLOUD                                                                                                     ITERATION

00070_Plug_the_W_into_Schroedinger_Eq.png        00070_Plug_the_W_into_Schroedinger_Eq.png

ITERATION                                                                                                               SELF-CONSISTENT criterium

00078_Iteration_selfconsistent_field.png        00078_Iteration_selfconsistent_field.png

xcf_0020_orbitals_in_cloud.png        xcf_0020_orbitals_in_cloud.png ----   COPIED FROM 2014       xcf_0025_ORBITALS.png        xcf_0025_ORBITALS.png   Illustration of the iteration; all necessary orbitals must be found                                                                       ---- COPIED FROM 2014       xcf_0029_ITERATIONS.png        xcf_0029_ITERATIONS.png                                                                                                                                                            ---- COPIED FROM 2014 The iteration in other symbols ....                         xcf_0030_self-Consistent_field.png        xcf_0030_self-Consistent_field.png                               SELF-CONSISTENT criterium

00083_Iteration__Converge-and-stop.png        00083_Iteration__Converge-and-stop.png

The above point concludes the introduction of the SELFCONSISTENT FIELD MODEL                                                                                                                                                            ---- COPIED FROM 2014 Configurations and energies Closed shells do not follow the    n2   rule            1s  2s  2p  3s       3p     3d      4s    4p     4d    4f     5s                                                                                               1s  2s  2p  3s       3p     4s      3d   4p      5s    4d    4f

xcf_0040_Configuration_orbitals.png        xcf_0040_Configuration_orbitals.png

The explanation of this behaviour   -  in terms of the self-consitent field   - NEXT LECTURE

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