Hückel Theory 2 (Eigenvalues)

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 7: Hückel Theory 2 (Eigenvalues) The energies (eigenvalues) may be determined by using the Hückel approximation. ψ A1g = 16 (φ1 +φ2 +φ3 +φ4 +φ5 +φ6 ) E ψ A1g ⎞⎟⎠ ⎛⎜⎝ψ A1g H ψ A1gdτψ A| H |ψ 1g∫==A1g = 16 (φ1 +φ2 +φ3 +φ4 +φ5 +φ6 ) H 16 (φ1 +φ2 +φ3 +φ4 +φ5 +φ6 ) = 1 ( H + H + H + H + H + H + H + H + H + H + H + H 61112 1314 1516 21 22 23 24 25 26 ↓↓ ↓↓↓↓αβ ββαβ⎞⎟⎟⎠+ H3i(i = 1 − 6) + H4i(i = 1 − 6) + H5i(i = 1 − 6) + H6i(i = 1 − 6) ↓↓ ↓ ↓α+2βα+2βα+2βα+2β= 1 (6)(α+ 2β) =α + 2β 6 E ψ A1g ⎞⎟⎠ ⎞⎟⎠ The energy of the LCAO, ψB2g ⎛⎜⎝ B2g 2g =ψ | H |ψ 2g ⎛⎜⎝ ⎛⎜⎝ E ψ E ψ B B 11 = (φ−φ +φ −φ +φ −φ)| H | (φ−φ +φ −φ +φ −φ)123456 12345666 = 61 ( H11 − H12 + H13 − H14 + H15 − H16 + H2i(i = 1 − 6) + H3i + H4i + H5i + H6i ) ↓↓ ↓↓ ↓↓↓↓ αβ βα–βα–2βα–2β α–2βα–2β = 1 (6)(α− 2β) =α − 2β 6 ⎞⎟⎠B2g 5.04, Principles of Inorganic Chemistry II Lecture 7 Prof. Daniel G. Nocera Page 1 of 4 The energies of the remaining LCAO’s are: ⎞⎟ ⎠⎞⎟ ⎠=ψ ⎟ = ⎜ψ ⎟Note the energies of the E orbitals are degenerate. Constructing the energy level diagram, we set α = 0 and β as the energy parameter (a negative quantity, so an MO whose energy is positive in units of β has an absolute energy that is negative), B2g–2 –1 E2u ⎛⎜⎝⎛ψψ⎜⎝⎛⎜⎜⎝⎛ ⎜⎝αβEE+baE1g E1g ⎞⎞α −βE= E=⎟⎠baE2u E2u E/0 1 E1g 6 π bonding electrons 2 A2u The energy of benzene based on the Hückel approximation is Etotal = 2(2β) + 4(β) = 8β What is the delocalization energy (i.e. π resonance energy)? To determine this, we consider cyclohexatriene, which is a six-membered cyclic ring with 3 localized π bonds; in other terms, cyclohexatriene is the product of three condensed ethylene molecules. For ethylene, Following the procedures outlined above, we find, ψ1(A) = 1(φ1 +φ2)2ψ2(B) = 1(φ1 −φ2)2 5.04, Principles of Inorganic Chemistry II Lecture 7 Prof. Daniel G. Nocera Page 2 of 4 E(ψ1) = 1 (φ +φ)| H |1 (φ+φ) = 1 (2α+ 2β) =β 212 212 2 E(ψ2) = 1 (φ −φ)| H |1 (φ−φ) = 1 (2α− 2β) = −β 212 212 2 The above was determined in the C2 point group. Correlating to D2h point group gives A in C2 → B1u in D2h and B in C2 → B2g in D2h: E/The Hückel energy of ethylene is, Etotal = 2(β) = 2β Therefore, the energy of cyclohexatriene is 3(2β) = 6β. The resonance energy is therefore, Eres(C6H6) = 8β – 6β = 2β ↓↓ Etotal Etotal benzene cyclohexatriene The bond order is given by, coefficients of electron i and electron j in a given bond B.O. = ∑necicjorbital e– occupancy j,i5.04, Principles of Inorganic Chemistry II Lecture 7 Prof. Daniel G. Nocera Page 3 of 4 Consider the B.O. between the C1 and C2 carbons of benzene ⎤ ⎥⎦ ⎡⎣ ⎡⎣ ⎛⎜ ⎜⎛⎜ ⎜⎞⎟ ⎟⎞⎟ ⎟111[ )] ) ⎤)⎥⎦ ψ1(A 2 ⎝==2u 6⎝⎠36⎠⎛⎜ ⎜⎛⎜ ⎜⎞⎟ ⎟⎝⎠⎞⎟ ⎟212 11⎢ ψ3(E1ga 2 ⎝==12 3⎠()0⎛⎜⎜⎝⎞⎟⎟121⎢ ψ4(E1gb 0==2⎠235.04, Principles of Inorganic Chemistry II Lecture 7 Prof. Daniel G. Nocera Page 4 of 4