Computational Physics

276 Quantum molecular dynamics

with

t=

(α+β)(γ+δ)

α+β+γ+δ

(PQ)^2 , (9.44)

withRPas given above and

RQ=

γRC+δRD

γ+δ

, (9.45)

and

ρ= 2

√

(α+β)(γ+δ)

π(α+β+γ+δ)

. (9.46)

From this form it follows directly that

d

dX

〈α,A;γ,C|g|β,B;δ,D〉=ρ

(

d

dX

Sα,A;β,B

)

Sγ,C;δ,DF 0 (t)

+ρSα,A;β,B

(

d

dX

Sγ,C;δ,D

)

F 0 (t)

+ρSα,A;β,BSγ,C;δ,DF 0 ′(t)

(α+β)(γ+δ)

α+β+γ+δ

2 (PQ)^2

X

(9.47)

where we have used the fact that(PQ)is proportional toXin order to obtain the
last term on the right hand side.
Using these matrix elements, it is possible to construct the derivatives of the
Fock matrix and of the overlap matrix with respect toX, and this gives the force on
Xwhich is needed in the Verlet algorithm. Note that the nuclear kinetic energy is
given by

Ekin,nucl=

Mn

2

[(

X ̇

2

) 2

+

( ̇

X

2

) 2 ]

=

Mn

4

X ̇^2. (9.48)

Therefore, in the equation of motion forX, half the proton mass (that is, the reduced
mass of the two nuclei) has to be used.
Only the ratio of the masses occurring in the electronic and nuclear kinetic energy
is relevant – changing the time stephcorresponds to an overall rescaling of the
masses. In fact, because the mass occurs in the equation of motion in combination
with an acceleration (or, in the kinetic energy, with a velocity squared), rescaling
the mass by a factorband time with a factor

√

bdoes not change the calculated
motion.

programming exercise

Extend the program of the previous subsection to include the nuclear motion.