Quantum Chemistry / Quantum Mechanics Many-Particle Problems

Angular Momentum p r

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

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

Fermions and bosons Enrico Fermi (1901- 1954)

Fermions and bosons Satyendra Nath Bose (1894-1974) Enrico Fermi (1901- 1954) Satyendra Nath Bose (1894-1974)

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

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

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

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

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

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

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

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

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

“N4 catastrophe”

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

Basis set ladder STO-3G, “minimal” 6-31G, “split valence” 6-31G*, 6-31G**, “polarization” 6-311++G**, “diffuse” aug-cc-pVTZ aug-cc-pVQZ ....

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