Computational Chemistry

associated with the spin operatorSˆzaccording toSˆza¼½(h/2p)aandSˆzb¼
#½(h/2p)b. Unlike most other functions, then, aand beach have only one
eigenvalue, ½(h/2p) and#½(h/2p), respectively. A spin function has the peculiar
property that it is zero unlessx¼½(aspin function) orx¼#½(bspin function).
A function that is zero everywhere except at one value of its variable, where it
spikes sharply, is adelta function(usually ascribed to Dirac – Section 4.2.3). Since
the spin functionc(spinaorb) describing an electron exists only when the spin
variablex¼)½, these two values can be considered the allowed values of the spin
quantum numbermsmentioned in Section 4.2.6.Sometimes an electron with spin
quantum number ½(“an electron with spin ½”) is called anaelectron, and said to
haveupspin, and an electron with spin#½is called abelectron, and said to have
downspin. Up and down electrons are often denoted by arrows"and#, respec-
tively. A nice, brief treatment of the delta function and of the mathematical
treatment of the spin functions is given by Levine [ 9 ].
The Slater wavefunction differs from the Hartree function not only in being
composed of spin orbitals rather than just spatial orbitals, but also in the fact that it
is not a simple product of one-electron functions, but rather adeterminant(Sec-
tion 4.3.3) whose elements are these functions. To construct a Slater wavefunction
(Slater determinant) for a closed-shell species (the only kind we consider in any
detail here), we use each of the occupied spatial orbitals to make two spin orbitals,
by multiplying the spatial orbital byaand, separately, byb. The spin orbitals are
then filled with the available electrons. An example should make the procedure
clear (Fig.5.2). Suppose we wish to write a Slater determinant for a four-electron

used with "electron 1" to make row 1

y 1 (1)a(1) y 1 (1)b(1) y 2 (1)a(1) y 2 (1)b(1)

used with "electron 2" to make row 2

y 1 (2)a(2) y 1 (2)b(2) y 2 (2)a(2) y 2 (2)b(2)

used with "electron 3" to make row 3

y 1 (3)a(3) y 1 (3)b(3) y 2 (3)a(3) y 2 (3)b(3)

used with "electron 4" to make row 4

y 1 (4)a(4) y 1 (4)b(4) y 2 (4)a(4) y 2 (4)b(4)

y 1

y 2 y 2 a

y 1 b

y 2 b

y 1 a

y 3

y 4

y 5

energy

Fig. 5.2 A Slater determinant is made from spin orbitals derived from the occupied spatial
molecular orbitals and two spin functions,aandb

182 5 Ab initio Calculations