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> iandi<j,thus half needstoberemoved(E)=e,-J(r),(r)drdr,17e-j>i8
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