《结构化学 Structural Chemistry》课程教学资源（课件讲稿）Chapter 4 Structures of Diatomic Molecules

Substitute E, into the secular equations(1)(α-E)c+(β-ES)c,=0(2)(β-ES)c+(α-E)C, = 0α+βBα+S)c, = 0(α+761+ S1+S(αS-β)c.+(β-αS)c, =0C。- C,= 0=Ca= Cb.d=CaVa+CV,=Ca(ya+b)Yet, c, remains unknown! However, the wavefunction should benormalized, i.e.,[d'ddt=113

( ) ( ) ( ) ( ) ( ) ( ) 0 2 0 1         a b a b ES c E c E c ES c     0 1 1          a b S c S c S ( ) ( )       Yet, ca remains unknown! However, the wavefunction should be normalized, i.e., Substitute E1 into the secular equations,  (S   )ca  (  S )cb  0 a b a b  c  c  0  c  c ( ) a a b b a a b 1  c   c  c   1 1  1    d *

nomalization condition: [d'odt = 1[(ca(ya+y,)"ca(ya+y,)dt=1= Ic? +2cyay, +c*w, dt =1→2c。(1+ S )=1=c。=1/ /2(1+ S )=→d =(ya+V)/ /2(1+S Similarly, substituting E, into the secular equations, we haveCa+ Cβ = 0= Ca = -Cbd2=caVa+C,V=Ca(y。-Vh)Normalizationd =(y。-Vr)/ /2(1- S)12

1 1 1 1                c c d nomalization condition d a a b a a b ( ( )) ( ) : * * a b a b c  c  0  c  c Similarly, substituting E2 into the secular equations, we have ( ) / ( ) [ ] c S c S c c c d a a a a a a b a b            2 1 1 1 2 1 2 1 2 2 2 2 2 2      ( )/ ( S ) 1   a  b 2 1 ( )/ ( S ) 2   a b 2 1 ( ) a a b b a a b 2  c   c  c   Normalization

Now we have20*E8α-a-yhE2D1-S[2(1- S)H1sH1SBYa+Whα+E,?1+S/2(1 + S)10E(E, <E2)Can we simplifytheprocess by usingthemolecularsymmetry?H,+ has an inversion center. The bonding and antibonding orbitalsshould be symmetric and asymmetric, respectively, upon inversion, i.e.c and c'm=c(Va+,);Φasym=c'(a-bsvmnormalization12*Ho.*HoE.dt.EdtsymasmasymsymCSU

2(1 ) ( ) , 1 2(1 ) ( ) , 1 1 1 2 2 S S E S S E a b a b                       + + - Can we simplify the process by using the molecular symmetry? E1 E2 H2 + has an inversion center. The bonding and antibonding orbitals should be symmetric and asymmetric, respectively, upon inversion, i.e., Now we have ( E1 < E2 ) ( ); '( ) sym a b asym a b   c     c   normalization c and c E H d E φ *Hφ dτ sym   sym sym asym   asym asym ˆ , * ˆ   

OverlapSab = [ yaedtRab = o, Sah =0;Rab =0, Sb =1integral11CoulombicHa - J y, Hyadt;一integralR2arbInternuclearHa= v(-↓v?.repulsion2RrarJva(-=v?-w.dt7wRh[=dt)=E+](J ~ 5.5%E HEHα=H..=Eμ+J~EThe attractiveenergyofelectronElectrostaticinteractionof H, by theGround-stateexerted by the nucleasnucleas of Hp12energy of H, atomof H to H, atom

Overlap integral a a H H a H H b H a b a a a a a a a a b a a a a b a a a a H E J E d E J J E R r E d r d R d r d r r R H r r R H H d H                                                       ) ( 5.5% ) 1 1 ( 1 1 ) 1 2 1 ( ) 1 1 1 2 1 ( 1 1 1 2 1 ˆ ; ˆ 2 * 2 * * * 2 2 Coulombic * integral  S    d ab a b * Rab = , Sab =0; Rab =0, Sab =1 Ground-state energy of Ha atom The attractive energy of electron of Ha by the nucleas of Hb . Electrostatic interaction exerted by the nucleas of Hb to Ha atom. Internuclear repulsion

resonance integral交换积分H=R2Hab = fya. Hy,dtrra[ (--→wRaEWdt+山RSabESV,dt)= EHSab +K= βabRSThe stabilization of chemicalWadt=Kbonding (Sab >0) upon the nucleas ofR9H,approaching H, atomnegative0

                                              d E S K R r S E S d r d R E d d r R d r H H d a b H a b a a b H a b a b a a b b a b b a b a b a a b a b ) 1 ( 1 1 ) 1 1 ) ( 1 2 1 ( ˆ * * * * * 2 * * resonance integral The stabilization of chemical bonding (Sab >0) upon the nucleas of Hb approaching Ha atom. d K R r S a b a a b       1 * negative 交换积分 r r R H a b 1 1 1 2 1 2       ˆ