266 Quantum molecular dynamics
The (classical) force on nucleusnis given as the negative gradient∇nof the energy
with respect to theRn:
Fn=−∇nE(S)=−∇n[
〈ψG|H(S)|ψG〉
〈ψG|ψG〉]
. (9.2)
It should be noted that there is not only the explicit S-dependence in the
Hamiltonian, but the ground state is evaluated for the Hamiltonian with a particular
configurationS. Therefore the ground state also depends onS.
TheHellmann–Feynman theorem[ 2 , 3 ], which we discussed for the single-
parameter case inSection 5.3, states that we can neglect this dependence: ifψGis
an eigenstate of the HamiltonianH(S),wehave
(〈ψG|ψG〉)^2 ∇nE
=[
〈(∇nψG)|H|ψG〉+〈ψG|(∇nH)|ψG〉+〈ψG|H|(∇nψG)〉]
〈ψG|ψG〉
−〈ψG|H|ψG〉[
〈(∇nψG)|ψG〉+〈ψG|(∇nψG)〉]
, (9.3)
where we have omitted theS-dependence of the Hamiltonian. Except for the term
including〈ψG|(∇nH)|ψG〉, all the terms on the right hand side cancel: this follows
directly from the fact thatHψG=EGψGand from the fact thatHis Hermitian. So
we are left with
∇nE=
〈ψG|(∇nH)|ψG〉
〈ψG|ψG〉. (9.4)
In practice we do not know the exact ground state, but we have only a variational
approximation to it. Therefore, in actual calculations, the Hellmann–Feynman
theorem does not predict the actual forces exactly and the variation of the (approxim-
ated) ground state wave function should be taken into account as well. Nevertheless,
the Hellmann–Feynman theorem is used quite often for predicting ground state
configurations, because the inclusion of other contributions is cumbersome.
9.2 The molecular dynamics method
In principle, all the ingredients for a molecular dynamics simulation using forces
calculated from the quantum electronic structure are at our disposal. However, at
each step in the MD simulation, a full electronic structure calculation is required, so
the method consumes a lot of computer time. In 1985, Car and Parrinello proposed
a method in which not only the nuclear positions, but also the electronic states are
calculated using MD algorithms. This results in a description of the system in which
the electronic structure does not, in general, relax completely to the ground state of
the actual configuration of nuclei; however, the calculated electronic structure will
follow the exact one rather closely. We start the description of the Car–Parrinello