Helium ground state H total 2 2 2 − 2 2 2 e 2 e e = ∇1 + ∇ 22 ) − − + ( 2m 4πε 0 r1 4πε 0 r2 4πε 0 r1 − r2 Approximate antisymmetrized wave function (neglecting electronelectron interactions) Ψ (r1 , r2 ) = Energy neglecting e-e interactions Approximate ground state evaluated with the total Hamiltonian ψ 1s (r1 )ψ 1s (r2 ) 2 ( ↑↓ − ↓↑ ) −13.6 Z 2 E = 2× = − 108.8 eV 2 n E= Ψ H total Ψ Ψ Ψ = −74.83 eV Helium excited states One electron in 1s and one in 2s, E= He Ψ H red Ψ Ψ Ψ 13.6* 22 13.6* 22 =− − = −68 eV 2 2 1 2 The antisymmetric solution Ψ = 0 for r1 = r2 . Helium excited states E = hf = hc/λ Names refer to approximate solutions http://physics.nist.gov/PhysRefData/Handbook/Tables/heliumtable5.htm Exchange (Austauschwechselwirking) 1 ψ A ( r1 , r2 ) = (φ1 (r1 )φ2 (r2 ) − φ1 (r2 )φ2 (r1 ) ) 2 1 ψ A H ψ A = [ φ1 (r1 )φ2 (r2 ) H φ1 (r1 )φ2 (r2 ) − φ1 (r1 )φ2 (r2 ) H φ1 (r2 )φ2 (r1 ) 2 − φ1 (r2 )φ2 (r1 ) H φ1 (r1 )φ2 (r2 ) + φ1 (r2 )φ2 ( r1 ) H φ1 (r2 )φ2 (r1 ) ] 1 ψ S ( r1 , r2 ) = (φ1 (r1 )φ2 (r2 ) + φ1 (r2 )φ2 (r1 ) ) 2 1 ψ S H ψ S = [ φ1 (r1 )φ2 (r2 ) H φ1 (r1 )φ2 (r2 ) + φ1 (r1 )φ2 (r2 ) H φ1 (r2 )φ2 (r1 ) 2 + φ1 (r2 )φ2 (r1 ) H φ1 (r1 )φ2 (r2 ) + φ1 (r2 )φ2 (r1 ) H φ1 (r2 )φ2 (r1 ) ] The difference in energy between the ψA and ψS is twice the exchange energy. Exchange The exchange energy can only be defined when you speak of multi-electron wavefunctions. It is the difference in energy between the symmetric solution and the antisymmetric solution. There is only a difference when the electronelectron term is included. Coulomb repulsion determines the exchange energy. In ferromagnets, the antisymmetric state has a lower energy. Thus the state with parallel spins has lower energy. In antiferromagnets, the symmetric state has a lower energy. Neighboring spins are antiparallel. Many electrons Consider a gold atom (79 electrons) 2 ⎛ d2 ∂ψ =− ⎜ 2 i ∂t 2m ⎝ dx1 3 x 79 = 237 terms Ψ( x1, y1, z1, d2 ⎞ 79e2 + 2 ⎟Ψ − Ψ dz79 ⎠ 4πε 0rj 79 terms + e2 4πε 0rij Ψ 79 x 78 = 3081 terms 2 x79 , y79 , z79 , t ) is a complex function in 237 dimensions Ψ(r1, rN ) 2 is the joint probability of finding an electron at position r1, r2, ... rN. Numerical solution of the Schrödinger equation for one electron ∂Ψ − e 2 = ∇ Ψ− Ψ i ∂t 2m 4πε 0 r 2 2 z Discretize Ψ to solve numerically. For one electron ~ 106 elements are needed. y x Numerical solution for many electrons For a numerical solution, divide Hilbert space along each axis into 100 divisions. z1 y1 100237 = 10474 x1 y2 x2 There are 1068 atoms in the Milky Way galaxy There are ~ 1080 atoms in the observable universe Intractable problem We know the equation that has to be solved. We know how to solve it but we don't have the computer resources to calculate the answer. A Matlab that will calculate the time evolution of an n-electron atom http://lamp.tu-graz.ac.at/~hadley/ss1/studentpresentations/2011/n_electrons.m Quantum computation Sometimes it is possible to map one intractable problem onto another. If you map an intractable problem onto a system of interacting electrons and then measure the energy levels of the electron system, you can find solutions to the intractable problem. Many-electron systems In such a quantum system, the repeated interactions between particles create quantum correlations, or entanglement. As a consequence, the wave function of the system is a complicated object holding a large amount of information, which usually makes analytical calculations impractical. In fact, many-body theoretical physics ranks among the most computationally intensive fields of science. http://en.wikipedia.org/wiki/Many-body_problem The Central Dilemma of Solid State Physics The Schrödinger explains everything but can explain nothing. From a fundamental point of view it is impossible to describe electrons in a metal correctly - Ashcroft and Mermin Neglect the e-e interactions Consider a gold atom (79 electrons) ⎛ d2 − ⎜ 2 2m ⎝ dx1 2 3 x 79 = 237 terms d2 ⎞ 79e2 + 2 ⎟Ψ − Ψ dz79 ⎠ 4πε 0rj 79 terms + e2 4πε 0rij Ψ = EΨ 79 x 78 = 3081 terms 2 Out of desperation: We simplify the model for a solid until the Schrödinger can be solved. If the 3081 electron - electron terms are neglected, the equation can be solved exactly and the total wave function is a product of atomic orbitals. Antisymmetrized product wave functions Ψ (r1 , r2 , , r79 ) = φ179s ↑ (r1 ), φ179s ↓ (r2 ),… , φ679s ↑ (r79 ) The standard first approximation for the many-electron wave-function of any atom is an antisymmetrized product of atomic orbitals. 1s2 2s22p63s23p63d104s24p64d104f145s25p65d106s1 Electron configurations http://lamp.tu-graz.ac.at/~hadley/ss1/molecules/atoms/review3.php Filling of electron shells Ni: 3d84s2 Cu: 3d104s1 Why isn't Ni 3d94s1 or 3d10? You can evaluate the energy of any electron configuration. Ψ (r1 , r2 , , r28 ) = φ128s ↑ (r1 ), φ128s ↓ (r2 ),… , φ328d ↑ (r27 )φ428s ↑ (r29 ) E= Ψ H Ψ Ψ Ψ Lithium The antisymmetric 3 electron wavefunction can be written φ1Lis ↑ (r1 ) φ1Lis ↑ (r2 ) φ1Lis ↑ (r3 ) 1 Li Li Li Li Ψ(r1, r2 , r3 ) = φ1s ↑,φ1s ↓,φ2s ↑ = φ1s ↓ (r1 ) φ1Lis ↓ (r2 ) φ1Lis ↓ (r3 ) 3! Li φ2s ↑ (r1 ) φ2Lis ↑ (r2 ) φ2Lis ↑ (r3 ) There are two possible configurations φ ↑,φ ↓,φ ↑ Li 1s Li 1s Li 2s and φ1Lis ↑,φ1Lis ↓,φ2Lis ↓ Construct the Hamiltonian matrix to see which has the lowest energy Berylium The antisymmetric 4 electron wave function can be written, Ψ(r1, r2 , r3 , r4 ) = φ1Bes ↑,φ1Bes ↓,φ2Bes ↑,φ2Bes ↓ φ1Bes ↑ (r1 ) Be 1 φ1s ↓ (r1 ) = 4! φ2Bes ↑ (r1 ) φ2Bes ↓ (r1 ) φ1Bes ↑ (r2 ) φ1Bes ↓ (r2 ) φ2Bes ↑ (r2 ) φ2Bes ↓ (r2 ) φ1Bes ↑ (r3 ) φ1Bes ↓ (r3 ) φ2Bes ↑ (r3 ) φ2Bes ↓ (r3 ) φ1Bes ↑ (r4 ) φ1Bes ↓ (r4 ) φ2Bes ↑ (r4 ) φ2Bes ↓ (r4 ) Gold The antisymmetric 79 electron wave function can be written, Ψ (r1 , r2 , , r79 ) = φ179s ↑ (r1 ), φ179s ↓ ( r2 ),… , φ679s ↑ (r79 ) The determinant of an N×N matrix has N! terms. 79! = 8.95 × 10116 (more than the atoms in the observable universe) Start with the valence electrons. Stop adding electrons when the matrix gets too big. Pauli exclusion The sign of the wave function must change when two electrons are exchanged 1 ψ1 (r1 ) ψ 2 (r1 ) 1 Ψ(r1, r2 ) = = (ψ1(r1 )ψ 2 (r2 ) −ψ1(r2 )ψ 2 (r1)) 2 ψ1 (r2 ) ψ 2 (r2 ) 2 If two columns are exchanged, the wave function changes sign. If two columns are the same, the determinant is zero. The Pauli exclusion principle only holds in the noninteracting electron approximation when the many electron wave function can be written as a product of single electron wave functions, only one electron can occupy each single electron state.
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