Computational Chemistry

Before moving on to H€uckel theory we take a look at matrices, since matrix
algebra is the simplest and most elegant way to handle the linear equations that arise
when MO theory is applied to chemistry.

4.3.3 Matrices and Determinants

Matrix algebra was invented by Cayley^22 as a systematic way of dealing with
systems of linear equations. The single equation in one unknown

ax¼b

has the solution x¼a"^1 b

C C+ CC

C C

C C + C C

C C

pi bond

sigma bond sp

(^5) /sp (^5) bond
sp^5 /sp^5 bond
Approximately
But more like
Fig. 4.10The model of a C/C double bond as as/pbond is at bottom really equivalent to the
sp^5 /sp^5 þsp^5 /sp^5 model: both result in the same electron distribution, which is the physically
significant thing. There are no gaps in electron density between the carbons: as the contribution to
density from thesbond (or one of thesp^5 /sp^5 bonds) falls off, the contribution from thepbond (or
the othersp^5 /sp^5 bond) increases. The electron density falls off smoothly with distance from the
C/C axis. For some purposes one of the models,s/por bent (banana) bonds, may be more useful
(^22) Arthur Cayley, lawyer and mathematician, born Richmond, England, 1821. Graduated Cam-
bridge. Professor, Cambridge. After Euler and Gauss, history’s most prolific author of mathemati-
cal papers. Died Cambridge, 1895.
108 4 Introduction to Quantum Mechanics in Computational Chemistry