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An in-depth exploration of N-dimensional cyclic systems in the context of Inorganic Chemistry II. It covers the polynomial derivation of Hückel determinants, the standing wave derivation, and the determination of molecular orbital energies using Schrödinger's equation and the LCAO method. The document also includes the evaluation of integrals and the calculation of eigenvalues and eigenfunctions.
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5.04, Principles of Inorganic Chemistry II
Lecture 9: N-Dimensional Cyclic Systems
Polynomial Derivation
0 N-
where N is the total number of orbitals
The Hückel determinant is given by,
(^3 ) 1
N-
x 1 1 x 1 1 x
D (^) N (x) =
1
= 0 where x = ^ - E 1 x 1 1 x
From a Laplace expansion one finds,
D (^) N (x) = xD (^) n-1 (x) – D (^) N-2 (x)
where
D 1 (x) = x
x 1 D 2 (x) = = x^2 1 1 x
with these defined, the polynomial form of DN (x) for any value of N can be obtained,
D 3 (x) = xD 2 (x) - D 1 (x) = x(x^2 -1) – x = x(x^2 –2) D 4 (x) = xD 3 (x) – D 2 (x) = x^2 (x^2 -2) – (x^2 -1)
and so on
5.04, Principles of Inorganic Chemistry II Lecture 9
MIT Department of Chemistry
The expansion of DN (x) has as its solution,
2 x = 2cos j (j = 0,1,2,3...N 1) N
and substituting for x,
E = + 2 cos 2 j (j = 0,1,2,3...N 1) N
Standing Wave Derivation
An alternative approach to solving this problem is to express the wave function directly in an angular coordinate,
m+1 m m-
N- N-
For a standing wave of about the perimeter of a circle of circumference c,
= sin c j
The solution to the wave function must be single valued a single solution must be obtained for at every 2n…
= sin c^ ( + 2 ) = sin c = sin c^ cos c^2 + sin c^2 cos c^ = sin c
must go to 1 must go to 0
if c
2 = 2 j (j = 0, 1, 2... N 1)
condition for an integral number of ’s about the circumference of a circle
c
= j
5.04, Principles of Inorganic Chemistry II Lecture 9
Evaluating the appropriate integrals,
Substituting for Cjm,
sin 2 m^ j + sin 2 (m^ +^ 1)^ j + sin 2 (m^ ^ 1)^ j = E^2 m j sin^ j N (^) N N (^) N
2 m Dividing by sin j, N
(^2) (m + 1) 2 (m 1) sin j + sin j + ^ N^ N^ = E j sin 2 mj N
Making the simplifying
E (^) j= + sin m
(^) sin m cos + sin cos m + sin m cos sin cos m Ej = + sin^ m
Ej = + 2cos
j =^ ^ +^2 ^ cos^ j^ (j^ =^ 0,1,2...N^ ^ 1) N
5.04, Principles of Inorganic Chemistry II Lecture 9
Let’s look at the simplest cyclic system, N = 3
j N
E 0 = + 2
E = + 2 cos 2 = 1 3 4 E 2 = + 2 cos = 3
Continuing with our approach (LCAO) and using Ej to solve for the eigenfunction, we find…
(^) N ± for N even
m (^) ± (N^ ^ 1) for N odd (^2)
where ^ ~
2i 2
Using the general expression for j , the eigenfunctions are:
i(0) 2 ^ i(0)^4 = e^ i(0)0 + e 3 + e 3 0 1 2 3 i(1)^2 ^ i(1)^4 = e^ i(1)0 + e 3 + e 3
Obtaining real components, and normalizing,
5.04, Principles of Inorganic Chemistry II Lecture 9