HMolecularOrbital TheoryThe schrodinger equation for H,+ can be solved exactly usingconfocal elliptical coordinates(xi) =(r,+rp)/Rn (eta) = (ra-rp)/Rx@is arotationaround z> R≤(r,+rp) < 80Ap7.-R≤(r-rb) ≤R0≤Φ≤2元;1 ≤≤8;Rra=(=+n)R/2 r, =(=-n)R/2-1≤n≤1RH(r,R)μ(r,R)= E(R)μ(r,RH(r)y(r)= Ey(r)fixed12position of theelectron!Yet very TEDIOUS!
R fixed Molecular Orbital Theory H2 Hr R r R E R r R e , , , ˆ 1 1 1 The schrödinger equation for H2 + can be solved exactly using confocal elliptical coordinates: (xi) = (ra+rb )/R (eta) = (ra -rb )/R is a rotation around z ra rb z R z x Yet very TEDIOUS! r a ( )R / 2 r b ( )R / 2 position of the electron! R (ra+rb ) < -R (ra -rb ) R 0 2; 1 ; -1 1 1 1 1 H r r E r ˆ
MolecularOrbital TheoryMolecularorbital (MO) of H,Yelec = F(5, n)[(2元)-1/2 imd(m=0, ±1, ±2, ±3,...)eRadial partAngularpart2 =|ml--orbital angular momentum quantum numberEach electronic level with 0 is doubly degenerate, with m = amh or m (in a.u.) -- the z-component of orbital angular momentum.The one-electron wavefunction (MO) is no longer the eigenfunctionof the operator L?, but is the eigenfunction of L,[L, H] + 0;[L,H]=0Types of molecular orbitals are defined by the value of a (=|m)1203入4Type of MO(bond)8中letter12元Y
(m=0, ±1, ±2, ±3,.) • =|m|-orbital angular momentum quantum number. ( , )[(2 ) ] 1/2 i m elec F e Radial part Angular part [ ˆ , ˆ ] 0; [ ˆ , ˆ ] 0 2 L H Lz H Molecular orbital (MO) of H2 + • Each electronic level with 0 is doubly degenerate, with m = ||. • The one-electron wavefunction (MO) is no longer the eigenfunction of the operator L2 , but is the eigenfunction of Lz . • mħ or m (in a.u.) - the z-component of orbital angular momentum. • Types of molecular orbitals are defined by the value of (=|m|). 0 1 2 3 4 letter Type of MO (bond) Molecular Orbital Theory
For diatomics.For atoms,Pele = F(5, n)(2元)-1/2 eimYele = Rn,(r)O1,m, (0)Dmd2 =ml (m=0, ±1, ±2, ±3,...Quantum numbers: n, l, m,3321201a=|m044SdfletterdletterS元pgaYQuantumNumberof OrbitalangularmomentumAtom: = 0, I, 2... and the atomic orbitals are called: s, p, d, etc& each sublevel contains degenerate AOs with m, = l, ..., -l.Diatomics: = 0,1,2, ... and the molecular orbitals are: , 元, , etc& each level contains degenerate MOs with m = ±aQuestion: Supposing MO's are composed of AO's, what is the12relationship between a (MO) and I (AO), or m (MO) and m,(AO)?
=|m| (m=0, ±1, ±2, ±3,.) =|m| 0 1 2 3 4 letter Quantum Number of Orbital angular momentum • Atom: l = 0, 1, 2,. and the atomic orbitals are called: s, p, d, etc. & each sublevel contains degenerate AOs with ml = l, ., -l. For diatomics, For atoms, ( ) ( ) ( ) , , ml ml elec n l l R r Quantum numbers: n, l, ml l 0 1 2 3 4 letter s p d f g Question: Supposing MOs are composed of AOs, what is the relationship between (MO) and l (AO), or m (MO) and ml (AO)? im elec F e 1/2 ( , )(2 ) • Diatomics: = 0,1,2, . and the molecular orbitals are: , , , etc. & each level contains degenerate MOs with m =
XSymmetry of MO1,)eimpfHAdelec2元Z5 = (ra +r)/R1n = (r.-r)/RA'Ri)Inversion:,=,Φ=Φ+A'(, -n, Φ+元)A(E, n, Φ)F(,-n) =BF(,n), B= +1 or -l;im =i[AF(5,n)eimg J = AF(E,-n)eim(s+z)|=Beimam =B'甲,mmYm is an eigenfunction of inversion with B'= +1 or-1!B'= 1, parity(even), (denoted g);B' =-1, disparity (odd), (denoted u);12Notation valid onlyforhomonuclear diatomics!
i m elec F( , )e 2 1 i) Inversion: [ ( , ) ] ˆ ˆ i m m i i AF e ra rb x R z ra rb Symmetry of MO m is an eigenfunction of inversion with B = +1 or -1 ! Notation valid only for homonuclear diatomics! F(,-) = BF(,), B= +1 or -1; ( ) ( , ) i m AF e m m im e B' B = (ra + rb )/R = (ra rb )/R A(, , ) A(, -, +) i A A ( , , ) ' ' ra rb rb ra • B = 1, parity (even), (denoted g); • B =-1, disparity (odd), (denoted u);
TaSymmetryofMOwavefunctionX7F(5,n)eimo4elec2元Ad = (ra+r)/RZn = (ra-r)/Rii)Reflectionby thexz-plane.(operator xz)(ra=ra, r =, =-)A'(, n, -Φ)A(, n, Φ), = AF(E,n)eim(-) =[AF(5,n)e-im ] = m-mXzi.e.When mO, the molecular orbital wavefunction Y itselfis not an eigenfunction of o12
i m elec F( , )e 2 1 ii) Reflection by the xz-plane. m i m i m xz m AF e AF e ( , ) [ ( , ) ] ( ) ra rb x R z Symmetry of MO wavefunction i.e. When m 0, the molecular orbital wavefunctionm itself is not an eigenfunction of xz! = (ra + rb )/R = (ra rb )/R A(, , ) A(, , -) xz ( , , ) ' ' ra ra rb rb (operator xz)