Types of MolecularOrbitalsforH,elec = F(E,n) (2元)-1/2 eims2013480入 =|mlletter元aVParity of molecular orbital (upon inversion): (g ~ even, u ~ odd)Cgdgi) 入=0(a)Bonding1lc!o-type(d)Real!Antibonding(b)ii) 2 = 1 Originally in complex form, but can be expressed in real form!元-typeAfter beingtransformed(f)into real form:(e)BondingAntibonding
0 1 2 3 4 letter elec = F(,) (2) -1/2 e im =|m| -type Types of Molecular Orbitals for H2 + • Parity of molecular orbital (upon inversion): (g ~ even, u ~ odd) Bonding i) =0 Real! Antibonding ii) = 1 -type Originally in complex form, but can be expressed in real form! Bonding Antibonding After being transformed into real form:
Questions1. When we deal with a many-electron diatomic molecule, whatproblem will we encounter?2. What will we encounter when dealing with a many-electronmany-atom molecule?712(i)+H=ZR+EZZ(2r(i)i=1i=NN#MIt is implausible to attain direct solution of the Schrodingerequation of such many-electron system!Mean-field approximation(independent electron approx.)-variationtheorem&LCAO-MO&HF-SCF12
Questions 1. When we deal with a many-electron diatomic molecule, what problem will we encounter? 2. What will we encounter when dealing with a many-electron many-atom molecule? N M NM i j i j n i N N n i r i R r H i 1 1 1 2 1 1 1 2 ) ( ) ( ) ( ˆ It is implausible to attain direct solution of the Schrödinger equation of such many-electron system! Mean-field approximation (independent electron approx.) variation theorem & LCAO-MO & HF-SCF
2. The Variation TheoremGiven a system whose Hamiltonian operator H is time-independent and whose lowest-energy eigenvalue is Ei,if isany normalized, well-behaved function of coordinates of thesystem's particles that satisfies the boundary conditions of theproblem, then<E>=/Φ*Hddt ≥E( d*ddt = 1)The variation theorem allows us to calculate the upper bondfor the system's ground-state energy?
2. The Variation Theorem Given a system whose Hamiltonian operator Ĥ is timeindependent and whose lowest-energy eigenvalue is E1 , if is any normalized, well-behaved function of coordinates of the system’s particles that satisfies the boundary conditions of the problem, then ( 1) ˆ 1 E *Hd E *d The variation theorem allows us to calculate the upper bond for the system’s ground-state energy
To prove the variation theorem, @ is supposed to be expanded interms of the complete, orthonormal set of eigenfunctions ( y of theHamiltonian operator H, i.e.,p=-EaykkwhereHyk=EkVk,dt=，E≥E,(k≥l)i) In case @ is normalized, we have=1 (k= j)1=Jo*ddtJ(Eavi)(Eaw,)dt=0 (k±j)EEaia,Jyiwdt=Eaia,ow=ElakkjK:<E>=[o*Hadt=J(Eayi)H(Ea,V,)dtkZaia,E,=aE≥E,(=ZaE,)12kkjk
E *Hd a H a d j j j k k k ( ) ˆ ( ) ˆ * * , , (k 1) 1 H E ψ ψj dτ δkj Ek E * k k k k ˆ To prove the variation theorem, is supposed to be expanded in terms of the complete, orthonormal set of eigenfunctions {k } of the Hamiltonian operator Ĥ, i.e., k ak k where i) In case is normalized, we have k k k j k j kj k j k j k j j j j k k k a a d a a a * d a a d 2 * * * * * 1 ( )( ) 0 (k j) 1 (k j) kj k k k k j ak aj Ej kj a E 2 * k E ( ak E ) 1 2 1
@= NThen we haveii) In case @ is not normalized, let1=[o*pdt=NfEayEay,)dt<E >=[*Hpdt=a =[*dt=Za=1/ ?=[μ*Hddt / [*ddt ≥E----- a trialvariation function (normalized)<E>=[*Hddt ≥Evariational integral The lower the value of the variational integral, the closer the trialvariational function to the real eigenfunction of ground stateTo arrive at a good approximation to the ground-state energy E,we try many trial variational functions and look for the one thatgives the lowest value of the variational integral.This offers an approximation to approach the solution for a12complex system!
variational integral • The lower the value of the variational integral, the closer the trial variational function to the real eigenfunction of ground state. • To arrive at a good approximation to the ground-state energy E1 , we try many trial variational functions and look for the one that gives the lowest value of the variational integral. This offers an approximation to approach the solution for a complex system! 1 E *H ˆ d E - a trial variation function (normalized) ii) In case is not normalized, let . Then we have 1 / ˆ ˆ *H d * d E E *H d N 2 2 2 2 2 * * 1/ 1 ( )( ) N a * d a N * d N a a d k k k k j j j k k k