Derivation of the secular equations. The Nequations can be conveniently expressed in matrix form (Nis the number ofbasis functions):HiNSi1SINS12S13H11H12H13G/0)H21S21H2NS22S23S2NH22H2330H31S31S3NH32H33H3NS32S33-E一00:::...::::::0CNHNISNIHN2HN3HNNSN2SN3SNNN2 H,-type integrals and N2 S,type integrals to be computed!!!These are called the secular equations (久期方程)and in general their solution will leadto N different values of E, each corresponding to a MO: By substituting the corresponding value of the energy E back into the secular equations, thecoefficients ic, corresponding to a particular MO can be found
Derivation of the secular equations • The N equations can be conveniently expressed in matrix form (N is the number of basis functions): • These are called the secular equations (久期方程) and in general their solution will lead to N different values of E , each corresponding to a MO. • By substituting the corresponding value of the energy E back into the secular equations, the coefficients {ci } corresponding to a particular MO can be found. N2 Hij-type integrals and N2 Sij-type integrals to be computed!!!！ 0 0 0 ⋮ 0 = 0 𝑐1 𝑐2 𝑐3 ⋮ 𝑐𝑁 E
6.1.2 The Hiickel approximations.The Hickel approximations1) set S,= J ΦiΦ,dt = 0 (i) or 1 (i-)Then the secular equations look simpler,H12HiNHi1H130001C10H210H22H2N10H23C2H31H320H3NH3300C3= 0-E......::::...:HN1000HN2HN3HNNCNSecularmatrixand can be rewritten as（久期矩阵）H12Hii - EH13HINc1These equations can be solved byH21H23H2NH22 - EC2firstly settingthe determinant of theH31H3NH32H33-EC3=0.secularmatrixnamelythe secular...:.....::determinant(久期行列式),to be zero.HN2HN1HN3HNN-ECN
6.1.2 The Hückel approximations • The Hückel approximations: 1) set Sij= 𝝓𝒊𝝓𝒋𝒅𝝉 = 0 (ij) or 1 (ij) Then the secular equations look simpler, and can be rewritten as Secular matrix (久期矩阵) These equations can be solved by firstly setting the determinant of the secular matrix, namely the secular determinant (久期行列式), to be zero
6.1.2 The Hickel approximations2) Calculating the actual values of the matrix elements H, is itself a formidable task, sowe sidestep this by simply leaving them as parameters,Hi = Φ;HΦ;dt = αi(approx. as the energy of the AO )HickelapproximationsHj = JΦ;Hjdt = βj (resonance integral)β, is zero unless the two orbitals are on adjacent atoms, i.e., directly overlapping!.Some of the β, terms can be: Accordingly, the secular equations becomezero case by case!βiNQ-Eβ13β12C1The values of α, β, can beβ21β23β2Nα2-EC2determined semi-empirically!β31β3203-Eβ3NC3= 0.Quiteeasyfordealingwith元-..............·conjugation systems!βN1βBN2βN3QN-ECNQl:howtodetermineα.&β.?Q2: For an allylic r system, write out the secular equation!
6.1.2 The Hückel approximations 2) Calculating the actual values of the matrix elements Hij is itself a formidable task, so we sidestep this by simply leaving them as parameters, 𝑯𝒊𝒊 = 𝝓𝒊𝑯 𝝓𝒊𝒅𝝉 = 𝜶𝒊 𝑯𝒊𝒋 = 𝝓𝒊𝑯 𝝓𝒋𝒅𝝉 = 𝜷𝒊𝒋 (resonance integral) （approx. as the energy of the AO i ) ij is zero unless the two orbitals are on adjacent atoms, i.e., directly overlapping! • Accordingly, the secular equations become Hückel approximations • Some of the ij terms can be zero case by case! • The values of i , ij can be determined semi-empirically! • Quite easy for dealing with - conjugation systems! • Q1: how to determine c & cc? Q2: For an allylic 𝜋3 𝑥 system, write out the secular equation!
6.1.3 The allyl system. The allyl fragment. the Tt-type MOs formed from these P, orbitals, y = ciΦ, + C2Φ2 + C3Φ3The secular eqs. areβ12β13/α1-E/C1)β21β23α2-EC2=0frameoutofplane2porbitalscofatomβ32α3-EC3β31Can the eqs. be further simplified?!· These are C 2p orbitals. Set α,= α,= α3= α, βi2= β2, = β23 = β32 = β (Hickel approx.),Theseculareqs.thusbecome10(α-E)/βC111(α - E)/β=0C201(C3(α -E)/βNowsetx=(α-E)/B!
6.1.3 The allyl system • The allyl fragment: the π-type MOs formed from these p orbitals, 𝛼1 − 𝐸 𝛽12 0 𝛽21 𝛼2 − 𝐸 𝛽23 0 𝛽32 𝛼3 − 𝐸 𝑐1 𝑐2 𝑐3 = 0 • These are C 2p orbitals. Set 1= 2= 3= , 12= 21 = 23 = 32 = (Hückel approx.). The secular eqs. thus become The secular eqs. are 31 13 Can the Can the eqs eqs. . bbe further simplified?! e further simplified?! 1 2 3 = c11 + c22 + c33 Now set x = (-E)/! (𝛼 − 𝐸)/𝛽 1 0 1 (𝛼 − 𝐸)/𝛽 1 0 1 (𝛼 − 𝐸)/𝛽 𝑐1 𝑐2 𝑐3 = 0
6.1.3 The allyl system10)x/C1). Now we have the simplified secular equations as11C2=0x(with x = (α-E)/β)10(C3)1x: As usual, set the corresponding secular determinant to zero:x(x2 -1)-1× (x- 0)+0 ×(1- 0)= 00)1(xx(x2 - 1) - x = 0det=011xx(x2 - 2) = 010*x=021xE1 = α + V2β,E3=α-V2βE2 = α,. Let us start with x = -V2 that gives E, = α + v2β and the secular equations as1/-V20-V2c1 + C2 = 0[A](C1)Three eqs. are-V2C2=0 C1 - V2c2 + C3 = 0[B]11notindependent!(C3/-V2)[C]0C2 - V2c3 = 01
6.1.3 The allyl system • Now we have the simplified secular equations as (with x = (-E)/) 𝑥(𝑥 2 − 1) − 1 × (𝑥 − 0) + 0 × (1 − 0) = 0 𝑥(𝑥 2 − 1) − 𝑥 = 0 𝑥(𝑥 2 − 2) = 0 𝑑𝑒𝑡 𝑥 1 0 1 𝑥 1 0 1 𝑥 = 0 𝑥 1 0 1 𝑥 1 0 1 𝑥 𝑐1 𝑐2 𝑐3 = 0 • As usual, set the corresponding secular determinant to zero: x = 0, 𝟐 𝐸1 = 𝛼 + 2𝛽, 𝐸2 = 𝛼, 𝐸3 = 𝛼 − 2𝛽 • Let us start with 𝑥 = − 2 that gives 𝐸1 = 𝛼 + 2𝛽 and the secular equations as − 2 1 0 1 − 2 1 0 1 − 2 𝑐1 𝑐2 𝑐3 = 0 − 𝟐𝒄𝟏 + 𝒄𝟐 = 𝟎 [𝑨] 𝒄𝟏 − 𝟐𝒄𝟐 + 𝒄𝟑 = 𝟎 [𝑩] 𝒄𝟐 − 𝟐𝒄𝟑 = 𝟎 [𝑪] Three eqs. are not independent!