In order to evaluate changes of F up to second order with respect to small deviations from the orientations of the magnetization at sites i and j relative to the (ferromagnetic) ground-state orientation (given with respect to a chosen uniform direction), one has to remember that according to Eq. (4.108) the orientational dependence of the single-site t matrix corresponds to a similarity transformation that rotates the z axis of the reference system (the artificial z axis inherent to LDFT),

ti(z) −1 ≡ mi(z) = Rim0i (z)R+i , Ri ≡ R(ϑi, ϕi) . (7.3)

Changes of mi(z) up to second order in ϑi and ϕi,

∆m(1)i (z) = m ϑ i (z)dϑi +m

∆m(2)i (z) = 1

2 mϑϑi (z)dϑidϑi +m

2 mϕϕi (z)dϕidϕi , (7.5)

can therefore easily be expressed by means of the below derivatives of the rotation matrices Ri,

mϑi (z) ≡ ∂mi(z) ∂ϑi

= ∂Ri ∂ϑi

, (7.6)

mϕi (z) ≡ ∂mi(z) ∂ϕi

= ∂Ri ∂ϕi

, (7.7)

mϑϑi (z) ≡ ∂2mi(z) ∂ϑi∂ϑi


+ 2 ∂Ri ∂ϑi


mϕϑi (z) ≡ ∂2mi(z) ∂ϕi∂ϑi

= ∂2Ri ∂ϕi∂ϑi


+ ∂Ri ∂ϑi


mϕϕi (z) ≡ ∂2mi(z) ∂ϕi∂ϕi


+ 2 ∂Ri ∂ϕi