Computational Chemistry

The dot product is also

v 1 +v 2 ¼v 1 xv 2 xþv 1 yv 2 yðwith an obvious extension to 3D spaceÞ

i.e.

cosy¼ðv 1 xv 2 xþv 1 yv 2 yÞ=jv 1 kjv 2

¼



1

p

2



"

1

p

2



þ



1

p

2



1

p

2



=ð 1 Þð 1 Þ¼ 0

and so

y¼ 90 

Likewise, the three columns of the matrixA 2 above represent three mutually
perpendicular, normalized vectors in 3D space. A better name for an orthogonal
matrix would be an orthonormal matrix. Orthogonal matrices are important in
computational chemistry because molecular orbitals can be regarded as orthonor-
mal vectors in a generalizedn-dimensional space (Hilbert space, after the mathe-
matician David Hilbert). We extract information about molecular orbitals from
matrices with the aid ofmatrix diagonalization.

4.3.3.5 Matrix Diagonalization

Modern computer programs use matrix diagonalization to calculate the energies
(eigenvalues) of molecular orbitals and the sets of coefficients (eigenvectors) that
help define their size and shape. We met these terms, and matrix diagonalization,
briefly in Section 2.5; “eigen” means suitable or appropriate, and we want solutions
of the Schr€odinger equation that are appropriate to our particular problem. If a
matrixAcan be writtenA¼PDP"^1 , whereDis a diagonal matrix (you could callP
andP"^1 pre- and postmultiplying matrices), then we say thatAisdiagonalizable
(can be diagonalized). The process of findingPandD(gettingP"^1 fromPis simple
for the matrices of computational chemistry – see below) ismatrix diagonalization.
For example

if A¼

4 " 2

11



thenP¼

12

11



;D¼

20

03



;and P"^1 ¼

" 1

1

2

" 2



Check it out. Linear algebra texts describe an analytical procedure using deter-
minants, but computational chemistry employs a numerical iterative procedure
called Jacobi matrix diagonalization, or some related method, in which the
off-diagonal elements are made to approach zero.
Now, it can be proved that if and only ifAis a symmetric matrix (or more
generally, if we are using complex numbers, a Hermitian matrix – see symmetric
matrices, above), thenPis orthogonal (or more generally, unitary – see orthogonal
matrices, above) and so the inverseP"^1 of the premultiplying matrixPis simply the

4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 115