Computational Chemistry

136

542

8 11 9

(^)
(^)
¼

136

542

5 þ 37 þ 44 þ 5

(^)
(^)
¼

136

542

574

(^)
(^)
þ

136

542

345

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(^)

4.3.4 The Simple Huckel Method – Theory€

The derivation of the H€uckel method (SHM, or simple H€uckel theory, SHT; also
called H€uckel molecular orbital method, HMO method) given here is not rigorous
and has been strongly criticized [30a]; nevertheless it has the advantage of showing
how with simple arguments one can use the Schr€odinger equation to develop, more
by a plausibility argument than a proof, a method that gives useful results and which
can be extended to more powerful methods with the retention of many useful
concepts from the simple approach.
The Schr€odinger equation (Section 4.2.6, Eq.4.29)

r^2 cþ

8 p^2 m

h^2

ðE"VÞc¼ 0

can after very simple algebraic manipulation be rewritten

"

h^2

8 p^2 m

r^2 þV



c¼Ec (4.35)

This can be abbreviated to the seductively simple-looking form

H^c¼Ec '(4:36)

where

H^¼ h

2

8 p^2 m

r^2 þV



'(4:37)

The symbolHˆ(“H hat” or “H peak”) is an operator (Section 4.3.3): it specifies that
an operation is to be performed onc, and Eq.4.36says that the result of the
operation will beEmultiplied byc. The operation to be performed onc(i.e.c(x,y,
z)) is “differentiate it twice with respect tox, toy and toz, add the partial
derivatives, and multiply the sum by"h^2 /8p^2 m; then add this result toVtimesc”
(now you can see why symbols replaced words in mathematical discourse). The
notationHˆcmeansHˆofc, notHˆtimesc.
Equation4.36says that an operator (Hˆ) acting on a function (c) equals a constant
(E) times the function (“H hat of psi equals E psi”). Such an equation

118 4 Introduction to Quantum Mechanics in Computational Chemistry