Kramer

Kramer - 1A = N then Ψ 1 = N p zA 1.618 p zB 1.618 p zC p...

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Coefficients of Butadiene Orbitals Using Kramer's Rule Rule: The ratio of the coefficients = ratio of the corresponding minors The secular determinant is: x 1 0 0 1 x 1 0 0 1 x 1 0 0 1 x for Ψ i = c iA p zA + c iB p zB + c iC p zC + c iD p zD i = 1 . .. 4 You can only find the ratio of coefficients. So for convenience find the ratio with respect to c iA . For example, expand the minors for the first row: c iB c iA = 1 1 0 0 x 1 0 1 x x 1 0 1 x 1 0 1 x = –(x 2 -1) x 3 -2x Use for each of the MO's in turn. For the most bonding orbital, Ψ 1 : x = -1.618 c 1B c 1A = -1.618 -1 = 1.618 c iC c iA = 1 x 0 0 1 1 0 0 x x 1 0 1 x 1 0 1 x = x x 3 -2x for Ψ 1 , c 1C c 1A = 1.618 c iD c iA = 1 x 1 0 1 x 0 0 1 x 1 0 1 x 1 0 1 x = –1 x 3 -2x for Ψ 1 c 1D c 1A = 1 For normalization, let c
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Unformatted text preview: 1A = N then Ψ 1 = N ( p zA + 1.618 p zB + 1.618 p zC + p zD ) ⌡ ⌠ Ψ 1 * Ψ 1 d τ = N 2 [ ⌡ ⌠ p zA 2 d τ +1.618 2 ⌡ ⌠ p zB 2 d τ +1.618 2 ⌡ ⌠ p zC 2 d τ + ⌡ ⌠ p zD 2 d τ ] = 1 Because cross terms like ⌡ ⌠ p zA p zB d τ = S AB = 0 in the Hückel approximation. The p orbitals are normalized so ⌡ ⌠ p zA 2 d τ = ⌡ ⌠ p zB 2 d τ = ⌡ ⌠ p zC 2 d τ = ⌡ ⌠ p zD 2 d τ = 1 ⌡ ⌠ Ψ 1 * Ψ 1 d τ = N 2 [ 1+ 1.618 2 + 1.618 2 + 1] = 1 giving N = 0.372 Ψ 1 = 0.372 p zA + 0.602 p zB + 0.602 p zC + 0.372 p zD...
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This document was uploaded on 02/22/2012.

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