《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

Part IISymmetry and BondingGraphical Method in Hickel Molecular OrbitalsProf.Dr.XinLu（吕鑫）Email: xinlu@xmu.edu.cnhttp://pcossgroup.xmu.edu.cn/old/users/xlu/group/courses/theochem/

Part III Symmetry and Bonding Graphical Method in Hückel Molecular Orbitals Prof. Dr. Xin Lu （吕鑫） Email: xinlu@xmu.edu.cn http://pcossgroup.xmu.edu.cn/old/users/xlu/group/courses/theochem/

6.6.1 General process for linear [nlpolyenesGraphical method to predefine the coefficients of HMOsfor conjugated systems (developed byQianerZhangetal.)必Ciii=1For a linear [nlpolyene, we have n secular equations (x =(α-E)/β) :xCI +C2=0C) + xC2 + C3= 000C11Ci+I + Ci-1 = -xc;C2001xsinA + sinB = 2sin A+BA-BO=0Ci-1 + xc,+ Ci+1 = 0220(cyclic formula)cmif A= (i+l)e, B= (i-l)000CCn-2 + xcn-1+ Cn = 0then x =-2coso& ci =sinioCn-I +xc, = 0

6.6.1 General process for linear [n]polyenes Graphical method to predefine the coefficients of HMOs for conjugated systems (developed by Qianer Zhang et al.) • For a linear [n]polyene, we have n secular equations (x = (-E)/ ) : i 1 2 3 n-1 n 𝑥 1 . 0 0 1 𝑥 . 0 0 . . . . . 0 0 . 𝑥 1 0 0 . 1 𝑥 𝑐1 𝑐2 . 𝑐𝑛−1 𝑐𝑛 = 0 xc1 + c2=0 c1 + xc2 + c3= 0 . ci-1 + xci+ ci+1 = 0 . cn-2 + xcn-1+ cn = 0 cn-1 + xcn = 0 (cyclic formula) ci+1 + ci-1 = xci 𝑠𝑖𝑛𝐴 + 𝑠𝑖𝑛𝐵 = 2𝑠𝑖𝑛 𝐴+𝐵 2 𝑐𝑜𝑠 𝐴−𝐵 2 tℎ𝑒𝑛 𝑥 = 2𝑐𝑜𝑠 𝑖𝑓 𝐴= (i+1), 𝐵= (i1) & 𝑐𝑖 = 𝑠𝑖𝑛𝑖 𝝍 𝝅 = 𝒊=𝟏 𝒏 𝒄𝒊𝝓𝒊

6.6.1 General processfor [nlpolyenesGraphical methodtopredefine thecoefficientsof HMOsfor conjugated systems(developed by Qianer Zhang et al.)n+11-1元ciΦi00c,sinosin29sin39sinesin(n-1)e sinn sin(n +1) = 0For a linear [n]polyene, we have n secular equations (x = (α-E)/β) :xc + c2=0;BoundaryconditionC, = sin2 0setCn+1 = sin(n+ I)0= 0CI +xC2 + c3= 0;C, = sin30x = -2cos0, = kπ(n+l) (k=l,...,n)Ci-I +xc,+ Ci+1 = 0;C,=singE= α+2βcos0C, = sini(cyclic formula)nS=Φ;sin(iok)..; cn-I +xc, = 0Cn = sinngi=1Nowrecall the sinewaverulewelearntinthe1stsemester!(kdefinestheenergylevel!)

6.6.1 General process for [n]polyenes Graphical method to predefine the coefficients of HMOs for conjugated systems (developed by Qianer Zhang et al.) For a linear [n]polyene, we have n secular equations (x = (-E)/ ) : xc1 + c2=0; c1 + xc2 + c3= 0; . ci-1 + xci+ ci+1 = 0; (cyclic formula) .; cn-1 + xcn = 0 set x = 2cos c1 = sin c3 = sin3 . c2 = sin2 ci = sini . cn = sinn Boundary condition: cn+1 = sin(n+1) = 0 Ek=  + 2 cosk k = k/(n+1) (k=1,.,n) i 1 2 3 n-1 n sin 𝜃 sin 2𝜃 sin 3𝜃 sin(𝑛 − 1)𝜃 sin 𝑛𝜃 0 n+1 ci sin 0 sin(𝑛 + 1)𝜃 = 0 𝝍 𝝅 = 𝒊=𝟏 𝒏 𝒄𝒊𝝓𝒊 (k defines the energy level!) 𝝍𝒌 𝝅 = 𝒊=𝟏 𝒏 𝝓𝒊𝒔𝒊𝒏(𝒊𝜽𝒌) Now recall the sine wave rule we learnt in the 1st semester!

: The method can be used for dealing with more complicated systems: Recent work developed by Prof. Zhenhua Chen can be found as “ Graphical representationof Hickel Molecular Orbitalsin J. Chem.Educ.2020, 97(2),448-456.(https://pubs.acs.org/doi/10.1021/acs.jchemed.9b00687).FYI:"Introduction to Computational Chemistry:TeachingHuckelMolecularOrbital TheoryUsinganExcelWorkbookforMatrixDiagonalizationin J. Chem.Educ.2015,92(2),291-295.(https://pubs.acs.0rg/doi/full/10.1021/ed500376g): after-class assignment 2: Please figure out the trends in the energies and compositionsof LUMO andHOMOforlinear and cyclic [n]ployenes, respectively! (n=4k,4k+1,4k+2,4k+3): After-class assignment 3: Ex. 29

• The method can be used for dealing with more complicated systems. • Recent work developed by Prof. Zhenhua Chen can be found as “Graphical representation of Hückel Molecular Orbitals” in J. Chem. Educ. 2020, 97(2), 448-456. (https://pubs.acs.org/doi/10.1021/acs.jchemed.9b00687) • FYI： “Introduction to Computational Chemistry: Teaching Hückel Molecular Orbital Theory Using an Excel Workbook for Matrix Diagonalization” in J. Chem. Educ. 2015, 92(2), 291-295. (https://pubs.acs.org/doi/full/10.1021/ed500376q) • after-class assignment 2: Please figure out the trends in the energies and compositions of LUMO and HOMO for linear and cyclic [n]ployenes, respectively! (n = 4k,4k+1,4k+2,4k+3) • After-class assignment 3: Ex. 29

Frontier MO's of [nlpolyeneC,sin24sinesin30sin(n)kVk= A Em=1 sin(m0k) (k=1,2,..,n; A =[2 /(n+1)}/2)0E,=α+2βcos(0,)n+1a) When n=odd, SOMO with k = (n+1)/2, soMo = 元/2 , EsoMo = αNon-bonding!E soMo = A( - + Φs -..) with C, = C4i+1 =- C4i+3 = A, C, = C2i = 0Vn：SimplifieddiagramGE(n+1)/2F+(n+1)/2ofSOMO:E(n-1)/2N(n-1)/2coefficientsofAOsinSOMO8n=4+1.n = 4l+1symmetricE2Y2E,Vin=4l+3asymmetric+

a) When n=odd, ( .)  SOMO  A 1 3 5  Non-bonding! Frontier MOs of [n]polyene (n+1)/2 E E1 E2 E(n-1)/2 E(n+1)/2 . . . 1 2 (n-1)/2 n SOMO with k = (n+1)/2,  k= 𝒌𝝅 𝒏+𝟏 Ek = +2cos(k )  k = 𝑨 𝒎=𝟏 𝒏 𝒔𝒊𝒏(𝒎𝒌 ) (k=1,2,.,n; A = [2/(n+1)]1/2) sin sin2 sin(n) with C1 = C4i+1 = – C4i+3 = A, C2 = C2i = 0 coefficients of AOs in SOMO SOMO = /𝟐 , ESOMO =  Simplified diagram of SOMO: n=4l+1 n=4l+3 n = 4l+1 symmetric n = 4l+3 asymmetric Ci sin3

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 )

6.6.2 Symmetry classification:a.[nlpolyenes with n=even2m+1H(m=0, 1,2,.., <n/2)n+1学受鸟学学Sym.Em=α+2βcos0m2m元 (m=1,.., /2)[["Asym0n+1E..Thus the lowest n/2 MOs with 0., < 元/2 are bonding MOsE(n+2)/2—(n+2)/2En/2 n2n=4k,n/2=2kn=4k+2,n/2=2k+l:(n+2)/2= 2k+1(n+2)/2= 2k+2E4Asym.4n+2LUMOn+2n+2EsmAsym..mSym..V3Sym.mm2(n+1)2(n+1)No.(n+2)/24E2V2Asym.nn-HOMOm=Sym.Asym.m=, 0m=2(n+1)2(n+1)E,4 SymNo. n/2Yi福

6.6.2 Symmetry classification: a. [n]polyenes with n=even Asym. m = 2𝑚 𝑛+1  (m=1,2,., n/2) Thus the lowest n/2 MOs with m < /2 are bonding MOs. Sym. –[ 1 2 ] [ 1 2 –[ ] 3 2 –[ ] 𝑛−3 2 –[ ] 𝑛−1 2 ] [ 3 2 ] [ 𝑛−1 2 [ ] 𝑛−3 2 ] ( 1 2 ) ( 3 2 ) ( 𝑛−3 2 ) ( 𝑛−1 2 ( ) 𝑛−1 2 ) ( 𝑛−3 2 ) ( 1 2 ( ) 3 2 ) m = 2𝑚+1 𝑛+1  (m=0, 1,2,., <n/2) 𝐸𝑚 = 𝛼 + 2𝛽𝑐𝑜𝑠m Sym. Asym. Asym. Sym. (n+2)/2 E E1 E2 En/2 E(n+2)/2 . . . 1 2 E3 E4 3 4 n/2 LUMO No.(n+2)/2 HOMO No. n/2 n= 4k, n= 4k+2, Asym. Sym., Asym., Sym., n/2= 2k n/2= 2k+1 (n+2)/2= 2k+1 (n+2)/2= 2k+2 m= 𝑛 4 , m = 𝑛+2 2(𝑛+1)  m= 𝑛 4 , m = 𝑛 2(𝑛+1)  m= 𝑛+2 4 , m = 𝑛+2 2(𝑛+1)  m= 𝑛−2 4 , m = 𝑛 2(𝑛+1) 

b. [n]polyenes with n=odd[(n-1)/2][(n-1)/2]'Symmetric MO's:n-ln-3n-3ncos , 1, cos0AcosHcoscoscos2: Ck-1 +Ck+1 = 2Ck Cos0222&C。 = ,C, = C.., C(n-1)/2} = (= C, = C, = cos0((n-1)/2/)n-3h= C, = 2C, cos0-1= cos20UC(n-1)/2=Cos=→ .., C(n-3)/2 = cOS22Boundary conditions:2m+1n+1n+12m+1A=0 (m=0, 1,2,..., (n-1)/2)cos元222n+1Esym=α+2βcos0mm

b. [n]polyenes with n=odd Symmetric MOs:  2 n  3 cos    2 2m 1 2 1 0 2 1      n  n cos Boundary conditions: Ck1 Ck1  2Ck cos [(n-1)/2] 1 0 1 [(n-1)/2] ' [( )/ ] [( )/ ]' & , ,., C0 1 C1  C1 C n1 2  C n1 2  C1  C1'  cos  C2  2C1 cos 1 cos 2  2 1 1 2     n C n cos ( ) / 1  2 n 1 cos  2 3 3 2     n C n ., cos ( ) /  2 n  3  cos 2 n 1 cos cos, 1, cos  m = 2𝑚+1 𝑛+1  (m=0, 1,2,., (n-1)/2) 𝐸𝑚 𝑠𝑦𝑚 = 𝛼 + 2𝛽𝑐𝑜𝑠m