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2 Actual Potentials of Theoretical Chemistry: What Can Be Obtained
2.3.3 Spin Density
In the open-shell molecule with different numbers of α and β spins, there is in
general deviation of those spins depending on r. This situation can be analyzed by
considering spin density defined as a difference of α-spin density ρ α (r) and β-spin
density ρ β (r) as
ρ(r) spin ≡ ρ α (r) − ρ β (r) =
occ
i
n α,i ψ
∗
α,i (r)ψ α,i (r) −
occ
j
n β, j ψ
∗
β, j (r)ψ β, j (r)
(2.15)
In this expression, positive ρ(r) spin indicates α spin is dominant at r and negative
ρ(r) spin β spin at r. Examples of spin densities are illustrated in Fig. 2.30 showing
appearance of dominant spin distribution depending on atoms. It is seen that α and
β spins are shown up rather alternately, which demonstrates, what is called, spin
polarization as is often seen in the open-shell molecules. Arrangement of the spin
polarization of the molecules in Fig. 2.30 is indicated by arrows.
(a)
(b)
Fig. 2.30 Spin density of a an allyl radical and b phenoxy radical obtained by unrestricted
DFT/B3LYP/6-31G**. Yellow and light blue colors represent the site abundant in α-spin and β-spin
densities, respectively. Upward and downward arrows simply represent these spins