《无机化学 Chemical Theory》课程教学资源（讲稿）Part II Molecular Spectroscopy Chapter 6 a complementary part regarding the Graphical Representations of HMOs

Frontier MO's of[npolyene店sin20singsin(ne)kTtVk= AZm=1 sinmOk (k=1,2,...,n; A =[2/(N+1)}/2)E,=α+2βcosomn+1LUMOwithk=(n+2)/2a)When n=even,n/2 bonding MOs,HOMO with k = n/2.En元T(n+2)元元1YOHOMo = 2(n+1)OLUMO2 + 2(n+1)22(n+1)2(n+1)：E(n+2)2(n+2)/2in=4l+2EnnNHOMO: C,=C Cr-1 =C2..., symmetricWan2…...LUMO: C, =-C, Cn-, =-C2 .., anti-symmetricasym.sym.E242i)n=4asym.EiHOMO: Ch =-C, Cn-1 = -C2.., anti-symmetricVisym.Odd-numberedMO:coeff.sym...,symmetricLUMO: C, = CiCn-, = C2Even-numberedMOs:coeff.asym

Frontier MOs of [n]polyene k = 𝒌𝝅 𝒏+𝟏 Ek = +2cosm k = 𝑨 𝒎=𝟏 𝒏 𝒔𝒊𝒏𝒎𝜽𝒌 (k=1,2,.,n; A = [2/(N+1)]1/2) sin sin2 sin(n) a) When n=even, n/2 bonding MOs, (n+2)/2 E E1 E2 En/2 E(n+2)/2 . . . 1 2 n/2 n HOMO with k = n/2, HOMO: Cn = C1 LUMO with k = (n+2)/2, HOMO = 𝒏𝝅 𝟐(𝒏+𝟏) = 𝝅 𝟐 − 𝝅 𝟐(𝒏+𝟏) LUMO = (𝒏+𝟐)𝝅 𝟐(𝒏+𝟏) = 𝝅 𝟐 + 𝝅 𝟐(𝒏+𝟏) i) n = 4l+2 Cn-1 = C2 ., symmetric LUMO: Cn = -C1 Cn-1 = -C2 ., anti-symmetric ii) n = 4l HOMO: Cn =-C1 Cn-1 = -C2 ., anti-symmetric LUMO: Cn = C1 Cn-1 = C2 ., symmetric sym. asym. sym. asym. Odd-numbered MO: coeff. sym. Even-numbered MOs: coeff. asym

6.6.2 Symmetry classification:a.[nlpolyenes with n=even(n/2)2(n/2)Symmetric MOs:"C,=C,n0cos=00cos=Q cos=cOS-COSCOSC, = C...222222Cn/2 =C(n/2)&Ck-1 +Ck+1 = 2C, cos0 (Cyclic formula)Let coefficients of central atoms (1 & 1') be cos(0/2)30003000= C, =C,,= 2 coscos0-(coscoscoscoscos222222050030050C, =C3. = 2 coScOsO-cOScoscOSCOScOS222222(n-1)0C(n/2) =C(n/2) = COs→ Boundary condition: cos[(n+1)/2]=022m+1Esym一>0m= α + 2βcos0mn+1 (m=0, ,.., (n-2)/2)m

 Boundary condition: cos[(n+1)/2]=0 6.6.2 Symmetry classification: a. [n]polyenes with n=even   2 1 , cos 2 1 cos Symmetric MOs: &Ck1 Ck1  2Ck cos Let coefficients of central atoms (1 & 1) be  2 3 cos (Cyclic formula)  2 3 cos cos( / 2 ) 1 1 (n/2)  C2  C2'  2 2 (n/2) / ( / )' ' ' ., 2 2 2 2 1 1 Cn C n C C C C     2 5 cos 2 ) cos 2 cos 2 5 (cos 2 cos cos 2 3 2cos 3 3'        C  C        2 ( 1) cos n  2 ( 1) ., cos ( / 2) ( / 2)'     n C n C n 2 2 2    cos cos  cos 2 3 2 2 2 3     (cos  cos ) cos  cos  2 ( 1) cos n   m = 2𝑚+1 𝑛+1  (m=0, 1,2,., (n-2)/2) 𝐸𝑚 𝑠𝑦𝑚 = 𝛼 + 2𝛽𝑐𝑜𝑠m

6.6.2 Symmetry classification:a.nlpolyenes with n=even(n/2)(n/2)AsymmetricMOsn-n-nn2sin(0/2) sir4sin(/sinC;Hsinsin2222 C =-C,C2 =-C2"., &Ck-- +Ck+1 = 2C, cos0Let coefficients for central atoms be -sin(0 /2),sin(0/2)n-1n-lsinand-sinThencoefficientsforterminalatomsare-22> Boundary condition: sin[(n+1)0/2]=02mEasym> 0.(m=1,2,., n/ 2)= α + 2βcos,n+1m

 Boundary condition: sin[(n+1)/2]=0 6.6.2 Symmetry classification: a. [n]polyenes with n=even Asymmetric MOs: (n/2) 2 1 1 2 (n/2)  m = 2𝑚 𝑛+1  (m=1,2,., n/2) 𝐸𝑚 𝑎𝑠𝑦𝑚 = 𝛼 + 2𝛽𝑐𝑜𝑠m  sin( / 2 ), sin( / 2 )   2 3 , sin 2 1 sin     n n   2 1 , sin 2 3 sin n  n  &Ck1 Ck1  2Ck cos Let coefficients for central atoms be Then coefficients for terminal atoms are   2 1 and sin 2 1 sin    n n Ci , ,., 1 1 2 2 - C  C  C  C   sin( / 2 ), sin( / 2 )