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
Hartreeproduct2h+2之minimize the expectation(E)=(V....Y../=1/=/+1/value of the energy8(E)SW(v/y.)=8,maintain constraint that L=(E)-ZZe, (yV))-8)areorthonormalSL=0
Hartree product 7 maintain constraint that ψi are orthonormal minimize the expectation value of the energy
Hartree equation and energy one-e Hartree equation[h+23/v(t)=eW(r),(r)=Rm(r)Ym(e),g)j,=Jw;(r)dr,Repulsionfromall othere'sHave to solvethis for all n electrons of the atom/moleculeTo avoid double-counting of electron-electron repulsions- ijincludesj> iandii8
Hartree equation and energy • To avoid double-counting of electron-electron repulsions – i ≠j includes j > i and i <j, thus half needs to be removed 8 Have to solve this for all n electrons of the atom/molecule • one-e Hartree equation Repulsion from all other e’s
Thechicken-or-eggdilemmaPresentsanobviousdifficulty-Needtoknow山,aheadoftime- H and 出 are not separable[6+i/v(r)=ew(r)j,=Jv,(r,)二dr,1-Howto constructHartreeequationtosolvethisdifferentialequationfor 中,?
The chicken-or-egg dilemma 9 • Presents an obvious difficulty – Need to know ψj ahead of time – H and ψ are not separable – How to construct Hartree equation to solve this differential equation for ψi ?
Self-consistentfield(SCF)approach[h+Z3/(r)=ew(c)Hartree&father1930j,= Jlv,(r,)二d,1. Guess initial set of 山2.Construct Hartree potential for eachorbital i3.Solve differential equations for new 山4.Check to see whether new and old are sufficiently close(basedon E, E,orother criteria)ifyes, done!5. If no, return to 2 using new ,and repeatFor He, for instance, we'd arrive at Y =i,(r;)α(1)is(r,)β(2)Quantitatively incorrect!Setsthegroundworkfor mostapproachesthatfollow.10
Self-consistent field (SCF) approach 1. Guess initial set of ψi 2. Construct Hartree potential for each orbital i 3. Solve differential equations for new ψi 4. Check to see whether new and old are sufficiently close (based on E, εi , or other criteria) If yes, done! 5. If no, return to 2 using new ψi and repeat 10 For He, for instance, we’d arrive at Hartree & father 1930 Quantitatively incorrect! Sets the groundwork for most approaches that follow
Pauliprincipleandanti-symmetrizationKey problem with the Hartree model-Itdistinguishesbetweenelectrons-Inrealityelectronsareindistinguishable!Pauliprinciple-FundamentalpostulateofQM-"Thewavefunction of a multi-particle systemmust beanti-symmetricto coordinate exchangeif the particles arefermions and symmetrictoexchange if the particles are bosons."Y(x,X2)=-4(x2,x)-xdescribesspaceandspincoordinatesofanelectronElectrons have half-integer spin and are thus fermions11
Pauli principle and anti-symmetrization • Key problem with the Hartree model – It distinguishes between electrons – In reality electrons are indistinguishable! • Pauli principle – Fundamental postulate of QM – “The wavefunction of a multi-particle system must be anti-symmetric to coordinate exchange if the particles are fermions and symmetric to exchange if the particles are bosons.” – x describes space and spin coordinates of an electron • Electrons have half-integer spin and are thus fermions 11