Ehf X Dab ha 2 X Da Dgd XaXAXXe Kn 199

Differentiation (using l as a general geometrical displacement of a nucleus) yields eq. (10.100).

___ Drs + Daß^r \((XaXr\XßiXs)- (XaXrlXeXß»+ (10.100)

1 Mbasis d dV

The third and fourth terms are identical and may be collected to cancel the factor of V2. Rearranging the terms gives eq. (10.101).

T7L- X Daß-^L + T X DaßDrSTT(XaXrlXßX^-<XaXrlXdXß))

Mbasis dD Mbasis ^D

+ X haß + X "di^A«(XaXrlXßX^ - (XaXrlXdXß»

dVnn di

The first two terms involve products of the density matrix with derivatives of the atomic integrals, while the two next terms can be recognized as derivatives of the density matrix times the Fock matrix (eq. (3.52)).

Mbasis ai,

dh 1 Mbasis d

^^ " + ^X DaßDJS 37 ((XaXrlXßXd) - (XaXrlXdXß ))


ab di ab

The derivative in eq. (10.102) of the nuclear repulsion (third term) is trivial since it does not involve electron coordinates. The one-electron derivatives are given in eq. (10.103).

di di

The central term is recognized as the Hellmann-Feynman force. The two-electron derivatives in eq. (10.102) become eq. (10.104).

di i dg

Xb dXd di

The central term is again the Hellmann-Feynman force, which vanishes since the two-electron operator g is independent of the nuclear positions.

The last term in eq. (10.102) involves a change in the density matrix, i.e. the MO coefficients.

N elec

0 0

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