LCAO and Hückel Theory 1(Eigenfunctions)

MIT OpenCourseWarehttp://ocw.mit.edu 5.04 Principles of Inorganic Chemistry II �� Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.04, Principles of Inorganic Chemistry II Prof. Daniel G. Nocera Lecture 6: LCAO and Hückel Theory 1 (Eigenfunctions) A common approximation employed in the construction of molecular orbitals (MOs) is the linear combination of atomic orbitals (LCAOs). In the LCAO method, the kth molecular orbital, ψk, is expanded in an atomic orbital basis, ψk = caφa + cbφb + … ciφi where the φis are normalized atomic wavefunctions and ∫ φiφidτ= 1. Solving Schrödinger’s equation and substituting for ψk yields, Ηψ k = Eψ kΗ− E= 0ψ kΗ− Ecaφa + cbφb + ... + ciφi = 0 Left-multiplying by each φi yields a set of i linear homogeneous equations, ca Η− E + cb Η− E + K + ci Η− E = 0φa φa φa φb φa φicaΗ− E + cb Η− E + K + ci Η− E = 0φb φa φb φb φb φi ca Η− E + cb Η− E + K + ci Η− E = 0φi φa φi φb φi φi Solving the secular determinant, aaaa ES–H abab ES–H LL aiai ES–H baba ES–H bbbb ES–H L L bibi ES–H M O M = 0 M O M iaia ES–H ibib ES–H L L iiii ES–H where Hii = ∫ φiΗφidτ ; Sii = ∫ φiφidτ= 1; Hij = ∫ φiΗφjdτ ; Sij = ∫ φiφjdτ 5.04, Principles of Inorganic Chemistry II Lecture 6 Prof. Daniel G. Nocera Page 1 of 6 In the Hückel approximation, Hii = αHij = 0 for φi not adjacent to φj Hij = β for φi not adjacent to φj Sii = 1 Sij = 0 The foregoing approximation is the simplest. Different computational methods treat these integrals differently. Extended Hückel Theory (EHT) includes all valence orbitals in the basis (as opposed to the highest energy atomic orbitals), all Sijs are calculated, the Hiis are estimated from spectroscopic data (as opposed to a constant, α) and Hijs are estimated from a simple function of Sii, Hii and Hij (zero differential overlap approximation). The EHT (and other Hückel methods) are termed semi–empirical because they rely on experimental data for quantification of parameters. Other semi-empirical methods include CNDO, MINDO, INDO, etc. in which more care is taken in evaluating Hij (these methods are based on self-consistent field procedures). Still higher level computational methods calculate the pertinent energies from first principles – ab initio and DFT. Here core potentials must be included and high order basis sets are used for the valence orbitals. As an example of the Hückel method, we will examine the frontier orbitals (i.e. determine eigenfunctions) and their associated orbital energies (i.e. eigenvalues) of benzene. The highest energy atomic orbitals of benzene are the C pπ orbitals. Hence, it is reasonable to begin the analysis by assuming that the frontier MO’s will be composed of LCAO of the C 2pπ orbitals: The matrix representations for this orbital basis in D6h is, ⎡⎡⎤⎤φ1 φ1 φ100000⎥⎥⎥⎥⎥⎥⎥⎥=⎥⎥⎥⎥⎥⎥⎥⎥=⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦φ2 φ2 φ3 4 φ2 φ3 4 100000100000100φ3E•x = 6trace 4 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣⎤⎦⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣⎤⎦⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣1 0 0 φ 0 φφ 0 φ 0 φ 5.04, Principles of Inorganic Chemistry II Lecture 6 Prof. Daniel G. Nocera Page 2 of 6 φ5 00010φ5 φ5 00001φ66 6 ⎡⎤⎤⎡⎤⎤φ1 φ φ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣φ1 φ210 000φ2 φ201 0003 0 0 0 0 1 ⎡ ⎢⎢⎢⎢ φ0⎢ ⎢ ⎢ ⎢⎣⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦φ φ φ φ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦φ3 φ300 1004 0C6 • ==x = trace000104 φ5 6 4 5 φ5000016 φ6 φ100000000001−1φ 1φ⎡⎤1φ⎡⎤⎡⎤⎡⎤ C2 ′ • ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣φ φ φ φ φ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎢⎣000 000 2 3 = 0004 100 −5 010 −6 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣φ φ φ φ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦00 − 010 1 − 2 3 001− 4 5φ000 000 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎥⎦6 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣φ φ φ φ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦6 5φ 2−= x = trace 4 3 2 The only orbitals that contribute to the trace are those that transform into +1 or –1 themselves (i.e. in phase or with opposite phase, respectively). Thus the trace of the remaining characters of the pπ basis may be determined by inspection: E 2C6 2C3 C2 3C2’3C2” i 2S3 2S6 σh 3σv 3σdD6h 6 0 0 0 –2 0 0 0 0 –6 2 0Γpπ The Γpπ representation is a reducible basis that must be decomposed into irreducible representations. Decomposition of reducible representations may be accomplished with the following relation: no. of members in the class ⎛⎜⎝⎞⎟⎠1RR ⋅ xi ⋅ CR∑the number of times ai = xha Γirr contributes to Γred R character of Γirr order character of Γred under operation R under operation Returning to the above example, a=1 [ 6 ⋅ 1 ⋅ 1 + 0 ⋅ 0 ⋅ 0 + (−2)(1)(3) + 0 + 0 + 0 + 0 + (−6)(1)(1) + 2 ⋅ 1⋅ 3 + 0 ] = 0 A1g 24thus A1g does not contribute to ΓpπR 5.04, Principles of Inorganic Chemistry II Lecture 6 Prof. Daniel G. Nocera Page 3 of 6 How about aA2u? a = 1 [ 6 ⋅1⋅1 + 0 ⋅ 0 ⋅ 0 + (−2)(−1)(3) + 0 + 0 + 0 + 0 + (−6)(1)(−1) + 2 ⋅ 1⋅ 3 + 0 ] = 1A2u 24 Continuing the procedure, one finds, Γpπ = A2u + B2g + E1g + E2u these are the symmetries of the MO’s formed by the LCAO of pπ orbitals in benzene. With symmetries established, LCAOs may be constructed by “projecting out” the appropriate linear combination. A projection operator, P(i), allows the linear combination of the ith irreducible representation to be determined, dimension of Γi operator [ x (R)](i) l (i)i h P•R∑R =⎤⎦ character of Γi under operator Rorder A drawback of projecting out of the D6h point group is the large number of operators. The problem can be simplified by dropping to the pure rotational subgroup, C6. In this point group, the full extent of mixing among φ1 through φ6 is maintained; however the inversion center, and hence u and g symmetry labels are lost. Thus in the final analysis, the Γis in C6 will have to be correlated to those in D6h. Reformulating in C6, ⎥ E C6 C3 C2 C32 C35C66 0 0 0 0 0Γpπ Γpπ = A + B + E1 + E2 A2u B2g E1g E2u in D6h The projection of the SALC that from φ1 transforms as A is, 1⎡⎣1 ⋅ E ⋅ ⎢ 2345(A)φ1 φ1 ⋅φ1 φ1 φ1 φ1 φ1P1C1 ⋅ C1 ⋅ C1 ⋅ C1 ⋅ C+++++=⋅⋅⋅⋅⋅6 6 6 6 66≅φ1 +φ2 +φ3 +φ4 +φ5 +φ6 drop constant since LCAO will be normalized 5.04, Principles of Inorganic Chemistry II Lecture 6 Prof. Daniel G. Nocera Page 4 of 6 Continuing, P(B)φ1 =φ1 −φ2 +φ3 −φ4 +φ5 −φ6 P(E1a)φ1 =φ1 + εφ2 −ε *φ3 −φ4 − εφ5 +ε *φ6 P(E1b)φ1 =φ1 +ε *φ2 − εφ3 −φ4 −ε *φ5 + εφ6 P(E2a)φ1 =φ1 −ε *φ2 − εφ3 +φ4 −ε *φ5 − εφ6 P(E2b)φ1 =φ1 − εφ2 −ε *φ3 +φ4 − εφ5 −ε *φ6 The projections contain imaginary components; the real component of the linear combination may be realized by taking ± linear combinations: For ψ(E1a) SALC’s: ψ3’(E1a) + ψ4’(E1b) = 2φ1 + (ε + ε*)φ2 – (ε + ε*)φ3 -2φ4 – (ε + ε*)φ5 + (ε + ε*)φ6 ψ3’(E1a) – ψ4’(E1b) = (ε – ε*)φ2 + (ε – ε*)φ3 + (ε* – ε)φ5 + (ε* – ε)φ6 where in the C6 point group, ⎛⎜⎜⎞⎟⎟2π2π 2πε=i = cos − isinexp ⎝666 2π 2π 2π 2π 2π ⎠∴ ε+ε * =− isin isin 2cos 1++==cos cos 6 666 6 *2π 2π 2π 2π 2πε −ε= − cos + isin − cos + isin = 2isin = 6 666 6 3i 2π 2π2π 2π⎞⎟ ⎠⎟=−2isin 2π6−⎛⎜ ⎝⎜cosε −ε * =cos 3i−=−isin isin+6666∴ the E1a LCAO’s reduce to (again ignoring the constant prefactor), ′′ψ()=ψ(E )+ψ (E )= φ+φ−φ− 2φ −φ +φE 231 31a 41b 123 456 ′′ψ()E =ψ (E )−ψ (E )=φ +φ −φ−φ41 31a 41b 2356 Similarly for the ψ5(E2) and ψ6(E2) LCAO’s… normalizing the SALC’s ψ (A) = 1 (φ+φ+φ+φ+φ+φ) ψ (B) = 1 (φ−φ +φ −φ +φ+φ)1 6123456 2 6123456 ψ (E ) = 1 (2φ+φ −φ − 2φ −φ +φ) ψ (E ) = 1 (φ +φ−φ−φ)31 12123 456 4122356 ψ5(E2) = 1 (2 φ1 − φ2 − φ3 + 2 φ4 − φ5 − φ6 ) ψ 6(E2) = 21 (φ2 − φ3 + φ5 − φ6 )125.04, Principles of Inorganic Chemistry II Lecture 6 Prof. Daniel G. Nocera Page 5 of 6 The pictorial representation of the SALC’s are, 5.04, Principles of Inorganic Chemistry II Lecture 6Prof. Daniel G. Nocera Page 6 of 6