mPart IIISymmetry and BondingChapter 4 Vanishing Integrals (零积分)Prof.Dr.XinLu（吕鑫）Email: xinlu@xmu.edu.cnhttp://pcoss.xmu.edu.cn/xlv/index.htmlhttp://pcoss.xmu.edu.cn/xlv/courses/theochem/index.html
Part III Symmetry and Bonding Chapter 4 Vanishing Integrals（零积分） Prof. Dr. Xin Lu （吕鑫） Email: xinlu@xmu.edu.cn http://pcoss.xmu.edu.cn/xlv/index.html http://pcoss.xmu.edu.cn/xlv/courses/theochem/index.html
4. Vanishing integrals.One ofthemost powerful applications of GroupTheory is the ability todecidewhether a particular integral is zero or not without actually evaluating the integral. We will learn how this approach can be used to great advantage in constructingmolecular orbitals and in understanding spectroscopic selection rules.f(x)even·For example, it is clear that the integral ofthe evenfunction ispositive but that of the odd function must be zero. Using Group Theory we can generalise this property of odd andg(x)oddeven functions into a powerful method for deciding whether or notparticularintegralswill bezero
4. Vanishing integrals • One of the most powerful applications of Group Theory is the ability to decide whether a particular integral is zero or not without actually evaluating the integral. • We will learn how this approach can be used to great advantage in constructing molecular orbitals and in understanding spectroscopic selection rules. • For example, it is clear that the integral of the even function is positive but that of the odd function must be zero. • Using Group Theory we can generalise this property of odd and even functions into a powerful method for deciding whether or not particular integrals will be zero
4.1 Symmetry criteria for vanishing integrals. Now consider the integral of a general function y over all space: I= JydtI= Jj+ y(x, y,z)dxdydz (in cartesian coordinates):The integrandmusttransform asthetotallysymmetricIRto maketheintegral non-zero.Ifthe integrand transforms as some otherIR, theintegral is necessarily zeroThe value of the integral I =f ydt is necessarilyzero if ytransforms as anything otherthan the totally symmetric irreducible representation..In other words, if y transforms as a sum of IRs that contain the totally symmetric IR, theintegralisnotnecessarilyzero
4.1 Symmetry criteria for vanishing integrals • Now consider the integral of a general function ψ over all space: I = d • The integrand must transform as the totally symmetric IR to make the integral non-zero. • If the integrand transforms as some other IR, the integral is necessarily zero. I = −∞ +∞ 𝑥, 𝑦, 𝑧 𝑑𝑥𝑑𝑦𝑑𝑧 (𝑖𝑛 𝑐𝑎𝑟𝑡𝑒𝑠𝑖𝑎𝑛 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠) The value of the integral 𝑰 = ψ𝒅𝝉 is necessarily zero if ψ transforms as anything other than the totally symmetric irreducible representation. • In other words, if ψ transforms as a sum of IRs that contain the totally symmetric IR, the integral is not necessarily zero
4.2 Overlap integralsA commonly encountered integral in quantum mechanics, and especially whenconstructing MOs, is the overlap integral between two wavefunctions ; and jJ yrwjdtSij =1.0(H)S0.80.6e0.40.2·InH,theoverlapoftwolsAOs0.02A356centred on differentatoms (ls(A)R/aoand 1s(B)):1s(A) M1s(B)1s(A)×1s(B)
4.2 Overlap integrals • A commonly encountered integral in quantum mechanics, and especially when constructing MOs, is the overlap integral between two wavefunctions ψi and ψj , • In H2 , the overlap of two 1s AOs centred on different atoms (1s(A) and 1s(B)): 𝑺𝒊𝒋 = ψ𝒊 ∗ψ𝒋𝒅𝝉
4.2Overlap integrals· Suppose that y; transforms as the IR [() and y; transforms as [0). Thus y* y; transforms asnro.. For the overlap integral to be non-zero the product r) must be (or at least contains)thetotally symmetricIR.. The only way for the product r to contain the totally symmetric IR is for (i) and 0)to be the same. This leads to a very important conclusion about the overlap integral:The overlap integral Si, = J y'y,dt is non-zero only if w; and yj,transform as thesame IR. (i.e., ‘symmetry compatible'!)
4.2 Overlap integrals • Suppose that ψi transforms as the IR Γ(i) and ψj transforms as Γ(j). Thus ψ𝒊 ∗ ψj transforms as Γ(i)⊗ Γ(j) . • For the overlap integral to be non-zero the product Γ(i)⊗ Γ(j) must be (or at least contains) the totally symmetric IR. • The only way for the product Γ(i)⊗Γ(j) to contain the totally symmetric IR is for Γ(i) and Γ(j) to be the same. This leads to a very important conclusion about the overlap integral: The overlap integral 𝑺𝒊𝒋 = ψ𝒊 ∗ψ𝒋 dτ is non-zero only if ψi and ψj transform as the same IR. (i.e., ‘symmetry compatible’!)