《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