Quantum Chemistry Quantum Mechanics ManyParticle Problems Angular Momentum

Quantum Chemistry / Quantum Mechanics Many-Particle Problems

Angular Momentum p r

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

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 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 kinetic nuclear attraction Coulomb (electron repulsion) Exchange

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

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

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)

“N 4 catastrophe”

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

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

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