Chapter 7. Symmetry Labeling of Molecular Energies Notes: • Most of the material presented in this chapter is taken from Bunker and Jensen (1998), Chap. 6, and Bunker and Jensen (2005), Chap. 7. 7.1 Hamiltonian Symmetry Operations We have seen in section 6.1.4 of the previous chapter that any elements of the CNPI group, and therefore any elements of the MS group, associated with a given molecule commute with the corresponding molecular Hamiltonian. That is, given a symmetry operator R we have ! ˆ 0 # "H , R$ = 0. (7.1) We also saw that this implies that if a wave function ! of the Hamiltonian is transformed by R such that ! R = R! , then this transformed function is also a wave function of the Hamiltonian with the same energy level as the original function, since ˆ 0 R ˆ 0 ˆ 0 0 0 R H ! n = H R! n = RH ! n = REn! n = En! n , (7.2) 0 where En is the energy associated with the wave function ! n . For a non-degenerate energy level, this in turn implies that R ! n = c! n , (7.3) with c some constant. As we will now see, the determination of the different values that this constant can take is central to the use of symmetry labels, associated with each irreducible representation of the MS group, for identifying the energy levels of a molecule. 7.1.1 Non-degenerate Energy Levels Consider again the case of a MS group operator R acting on a wave function ! n of non- 0 degenerate energy level En . However, because of the group axioms that states that both Rq , with q and integer, and the identity E must be part of the group, it follows that for some integer m we must have Rm = E. (7.4) We therefore find from equation (7.3) that m m R ! n = c ! n = E! n = ! n , (7.5) 119 or alternatively c = m 1. (7.6) For example, for the permutation (12) we find that c = ±1. Example Consider the following three fictitious wave functions for the water molecule (the protons are labeled 1 and 2) ! 1 = sin(X1 " X2 ) ! 2 = cos(X1 " X2 ) (7.7) ! 3 = sin(X1 + X2 ). Determine the constants associated with each operator of the MS group for this molecule, and use the corresponding character table to determine the irreducible representation generated by the wave functions. Solution The MS group for this molecule is C2v (M) and its character table is given in Table 7-1. Applying the different operators to the wave functions we have E ! 1 = sin(X1 " X2 ) = ! 1 (12) ! 1 = sin(X2 " X1 ) = "! 1 (7.8) E# ! 1 = sin(X2 " X1 ) = "! 1 (12)# ! 1 = sin(X1 " X2 ) = ! 1, and similarly E (12) E" (12)" ! 2 = ! 2 = ! 2 = ! 2 = ! 2 (7.9) E (12) E" (12)" ! 3 = ! 3 = #! 3 = #! 3 = ! 3. Table 7-1 – The character table of the C2v (M) group. ! ! C2v (M) : E (12) E (12) A1 : 1 1 1 1 A2 : 1 1 !1 !1 B1 : 1 !1 !1 1 B2 : 1 !1 1 !1 120 It is therefore apparent that the characters of the different wave functions when acted R upon with the MS group operators are such that if we collect the different constants ci , with i = 1,2,3 for the wave functions, we have cE (12) E! (12)! i ci ci ci ! 1 : 1 !1 !1 1 = B1 ! 2 : 1 1 1 1 = A1 ! 3 : 1 1 !1 !1 = A2 And, as indicated in the last column, ! 1, ! 2 , and ! 3 have the symmetry of B1, A1 , and A2 , respectively. Because the energy levels are different for the three wave functions (they are non-degenerate), then we find that we can label the energy levels using the different irreducible representations of the MS group. 7.1.2 Degenerate Energy Levels When an energy level E 0 is l-fold degenerate, there are l wave functions , , n ! n1 … ! nl that transform into linear combinations of themselves when acted upon by an operator R of the MS group. This is expressed mathematically as follows l R D R , (7.10) ! ni = " [ ]ij ! nj j =1 where D[R] is the representation matrix associated to R . Now imagine that we are forming another set of l-fold degenerate functions !nk using an orthogonal matrix A with l (7.11) !nk = # Aki" ni . i=1 Evidently these new functions are also wave functions of the Hamiltonian with the same 0 energy En as the original wave functions ! ni . We would like to find out what representation of the MS group the wave functions !nk generate. If we denote the new representation matrix associated to R with D[R], then l R D R . (7.12) !nk = " [ ]kr !nr r=1 But using equations (7.11) and (7.10) we can also write 121 l R!nk = # Aki R" ni i=1 (7.13) l l A D R , = # ki # [ ]ij " nj i=1 j =1 and we substitute for ! nj using the inverse of equation (7.11) to get l l l R A D R A"1 !nk = # ki # [ ]ij # jr !nr i=1 j =1 r=1 = A D R A"1 ! (7.14) ##( ki [ ]ij jr ) nr r i, j l = AD R A"1 ! . #( [ ] )kr nr r=1 Comparing this result with equation (7.12) reveals that the new representation D[R] is related to the original representation D[R] via a similarity transformation. That is, D[R] = AD[R]A!1. (7.15) Since we already know that the character of a matrix is invariant under a similarity transformation (see equation (6.50) of the previous chapter), then we find the important result that the character of a representation generated by a set of degenerated wave functions of the Hamiltonian is unique and can be used to reduce it to its irreducible components (see equations (6.59) and (6.60) of Chapter 6). The corresponding degenerated energy level can thus be labeled with the irreducible representations of the MS group. This result is an extension of what was previously found for non-degenerate energy levels. 7.2 Projection Operators Let us again consider a set of wave functions ! js that generate the irreducible j representations ! of dimension (i.e., degeneracy) l j , with s = 1, …,l j . Any other set (of similar size) of functions !n generating a reducible representation ! can expressed as a linear combination of the ! js (because they form a basis) with l j (7.16) !n = ## An, jk" jk . j k =1 (Note: The notation of equation (7.16) is potentially confusing, but it is understood that An, jk is the element of a matrix A belonging to row n and column jk . That is, the indices j and k taken together form a single index. The implication is that the column 122 vector representation of ! jk is such that its first l1 elements belong to j = 1 and k 1, ,l , the next l elements belong to j 2 and k 1, ,l , and so on.) Equation = … 1 2 = = … 2 (7.16) can also be inverted to give "1 (7.17) ! jk = $ Ajk,n#n . n Now consider the following operator l " P!i i D!i R R (7.18) mm = # [ ]mm h R !i where D [R] is the matrix associated to the operator R in the !i irreducible representation of the corresponding symmetry group of order h . We now apply this operator to a function of the !n set l # P!i i D!i R R mm"n = $ [ ]mm "n h R l j l # & ) i D!i R R A = $ [ ]mm ( $$ n, jk% jk + h R ' j k =1 * l j l # A i D!i R R (7.19) = $$ n, jk $ [ ]mm ( % jk ) j k =1 h R l j l j l # & ! ) A i D!i R D j R = $$ n, jk $ [ ]mm ( $ [ ]ks % js + j k =1 h R ' s=1 * l j l j & l # ! ) A i D!i R D j R , = $$ n, jk $% js ( $ [ ]mm [ ]ks + j k =1 s=1 ' h R * where equations (7.16) and (7.10) (suitably rewritten) were used. We can now make use of the GOT (i.e., equation (6.54) of Chapter 6) to find that l j l j !i Pmm"n = %% An, jk %# js ($ij$ mk$ ms ) j k =1 s=1 (7.20) = An,im# im . !i We therefore find that the projection operator Pmm when applied to !n produces the part of !n that belongs to the m th row of the !i irreducible representation. In other words, given the matrix irreducible representations D!i [R] of a group we can use the projection operator to break down an arbitrary set of functions !n into the irreducible 123 representations-generating wave functions ! im that compose it. For example, if the functions !n do not contain the !i irreducible representation, then equation (7.20) will yield !i Pmm"n = 0. (7.21) It is important to emphasize the fact we need to know the matrices D!i [R] in order to use this projection operator. On the other hand, is possible to define a simpler (and less powerful) projection operator from equation (7.18) with l $ !i !i i !i (7.22) P = "Pmm = " # [R] R m h R For example, we can calculate from equation (7.20) that !i (7.23) P "n = $ An,im# im . m So, when applied to !n this projection operator will yield a function that belongs to !i , but not belonging to any particular row m .