LasttimeHydrogen atomAngular & radial componentVariational principleBasisfunctionsSecularequationsElectron spin (can compute by setting spin multiplicity)Calculationrelated:Optimization
Hydrogen atom • Angular & radial component • Variational principle • Basis functions • Secular equations • Electron spin (can compute by setting spin multiplicity) • Calculation related • Optimization Last time 2
ContentsThe Hartree-Fock method1.The problemof manyelectrons2.Hartree atom;3.Self-consistent field approach4.Pauli principle5.Slater determinants6.Coulomb integral7.Exchange integral8.Hartree-Fock equation9.Local densityapproximation10.CorrelationJensen,chp3m
Contents 3 The Hartree-Fock method 1. The problem of many electrons 2. Hartree atom; 3. Self-consistent field approach 4. Pauli principle 5. Slater determinants 6. Coulomb integral 7. Exchange integral 8. Hartree-Fock equation 9. Local density approximation 10. Correlation Jensen, chp 3
TheproblemofmanyelectronsHelium: next simplest atom after hydrogen- 2e in the configuration of (1s)2-isthisthe"same"1sasinH?- No...nuclear charge and shielding give different wavefunction(different distribution ofelectrons)and different energies. To model: again nucleus at origin and stationaryHe atom Schrodinger equatione212e21AY(r,r)=EY(r,r)2m4元64元2-Summationrunsoverbothelectrons"One-electron"terms: kinetic energy and electrostaticattraction to 2+ nuclear charge of each electron;-One"two-electron"term:the electron-electron repulsion;- It's a (many-body) problem4
The problem of many electrons • Helium: next simplest atom after hydrogen – 2e in the configuration of (1s)2 – is this the “same” 1s as in H? 4 – No.nuclear charge and shielding give different wavefunction (different distribution of electrons) and different energies. • To model: again nucleus at origin and stationary • He atom Schrödinger equation • Summation runs over both electrons – “One-electron” terms: kinetic energy and electrostatic attraction to 2+ nuclear charge of each electron; – One “two-electron” term: the electron-electron repulsion; – It’s a (many-body) problem
n-electronatom· Many-electron wavefunction in atomic units- First summation overall electrons;- Second set goes over all electron pairs;- Extending to molecule will only involve elaboration of oneelectron parttoincludemultiplenuclei.h+22(r...r)=EY(r...r)i=l i=i-Z2rSolutions are many-dimensional functions of the coordinatesofalltheelectrons-Cannot solve this analytically!-Mostcommonapproximateapproachesreintroducesingleelectron wavefunction (orbitals)5
n-electron atom • Many-electron wavefunction in atomic units – First summation over all electrons; – Second set goes over all electron pairs; – Extending to molecule will only involve elaboration of oneelectron part to include multiple nuclei. 5 • Solutions are many-dimensional functions of the coordinates of all the electrons – Cannot solve this analytically! – Most common approximate approaches reintroduce single electron wavefunction (orbitals)
TheHartreeatom· Simplest approach: a“Hartree"productAkindofseparationofvariablesY(r..,r,)=y,(r),(r.)山，arefictitious“atomic(spin)orbitals"-Describeseparatelythemotions oftheindividual eThis separation is not physical-Themotionsofthetwoelectronsarenotcorrelated-AmeanfieldmodelHowtobuildthisproduct?-Usevariational principleLagrangemultipliers6
The Hartree atom • Simplest approach: a “Hartree” product – A kind of separation of variables 6 • ψi are fictitious “atomic (spin) orbitals” – Describe separately the motions of the individual e; • This separation is not physical – The motions of the two electrons are not correlated; – A mean field model • How to build this product? – Use variational principle; – Lagrange multipliers