Computational Chemistry

can be written as a Slater determinant of the “component” spatial wavefunctionsc
(by including spin functions), and in principle anyway, any property of a molecule
can be calculated fromC. The component wavefunctionscand their energy levels
eare extremely useful, as chemists rely heavily on concepts like the shape and
energies of, for example, the HOMO and LUMO of a molecule (MO concepts are
reviewed inChapter 4). The energy levels enable (with a correction term) the total
energy of a molecule to be calculated, and so the energies of molecules can be
compared and reaction energies and activation energies can be calculated. The
Roothaan–Hall equations, then, are a cornerstone of modern ab initio calculations,
and the procedure for solving them is outlined next. These ideas are summarized
pictorially in Fig.5.5.
The fact that the Roothaan–Hall equations Eqs.5.56are actually a total ofm'm
equations suggests that they might be expressible as a single matrix equation, since
the single matrix equationAB¼ 0 , whereAandBarem'mmatrices, represents
m'm“simple” equations, one for each element of the product matrixAB(work it
out for two 2'2 matrices). A single matrix equation would be easier to work with
thanm^2 equations and might allow us to invoke matrix diagonalization as in
the case of the simple and extended H€uckel methods (Sections 4.3.4 and 4.4.1).
To subsume the sets of equations5.54-1–5.54-m, i.e. Eqs.5.56, into one matrix

weighted sum

MO # energy

Using, e.g., a set of 4 basis functions:

y 4 e 4

y 3 e 3

y (^2) e
2
y 1 e 1
{ f 1 , f 2 , f 3 , f 4 } (the weighting factors are the MO coefficients c)
y 1 (1)a(1) y 1 (1)b(1) y 2 (1)a(1) y 2 (1)b(1)
y 1 (4)a(4) y 1 (4)b(4) y 2 (4)a(4) y 2 (4)b(4)
y 1 (3)a(3) y 1 (3)b(3) y 2 (3)a(3) y 2 (3)b(3)
y 1 (2)a(2) y 1 (2)b(2) y 2 (2)a(2) y 2 (2)b(2)
If there are 4 electrons in the molecule, then y 1 and y 2
are occupied (and y 3 and y 4 are virtual orbitals). The
occupied orbitals are used to construct the total wavefunction,
as a Slater determinant of spin orbitals.
ψ =
Fig. 5.5 Pictorial representation of basis functions, MO’s, total wavefunction, and energy levels
5.2 The Basic Principles of the ab initio Method 201