Quantum Chemistry / Quantum Mechanics Many-Particle Problems

Prezentace na téma: "Quantum Chemistry / Quantum Mechanics Many-Particle Problems"— Transkript prezentace:

1 Quantum Chemistry / Quantum Mechanics Many-Particle Problems

2 Angular Momentum p r

3 Operator commutation properties:
z l lz do not commute y x knowing length and one projection is possible

4 Spin of electron - relativistic property
in non-relativistic formalism: equivalent to a coordinate (x,y,z,sz) - also magnetic moment

5 Fermions and bosons Enrico Fermi ( )

6 Fermions and bosons Satyendra Nath Bose (1894-1974)
Enrico Fermi ( ) Satyendra Nath Bose ( )

7 Many particle wavefunction for electrons:
Slater determinant: N = 2:

8 The Born-Oppenheimer Approximation
.. nuclear SE .. electronic SE

9 Atomic units me = 1 ħ = 1 1/(40) = 1 e = -1

10 The Hartree-Fock approximation
orbitals HF equations: non-linear “one-particle”

11 The Hartree-Fock approximation
nuclear attraction Exchange Coulomb (electron repulsion) kinetic

12 Two-electron integrals
Coulomb Exchange - only for the same spins

13 Linear combination of atomic orbitals, LCAO
spin part of the wavefunction spinorbital “molecular” orbital “atomic orbital”, AO MO expansion coefficient

14 Roothan Equations HF equations in “AO” basis not linear - matrix form
solution: iteration electrons in “self-consistent field” (SCF) of the other Clemens C. J. Roothaan, * 1918

15 Orbital terminology Molecular/atomic, MO, AO closed shell, open shell
lowest-unoccupied molecular orbital (LUMO) highest occupied molecular orbital (LUMO)

16

17 “N4 catastrophe”

18 AO basis sets Slater type orbitals, STO Gaussian type orbitals, GTO
“contraction” “primitive functions”

19 Basis set ladder STO-3G, “minimal” 6-31G, “split valence”
6-31G*, 6-31G**, “polarization” G**, “diffuse” aug-cc-pVTZ aug-cc-pVQZ ....

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