6. Force field
AMMP uses a molecular mechanics potential or force field. This is a classical potential and as such is limited in its physical significance. However, it is possible to relate such a classical potential to a power series expansion about a stationary solution of a quantum problem. This relationship is derived via the Feynmann-Hellman theorem which relates the expectation value of the derivative of an operator with its stationary approximation.
We can write Schrodingers equation as:
E<p|p> = <p| H |p>
where E is the energy, H the Hamiltonian operator and the wave function. The molecular mechanics approximation to the energy is given by the Taylor expansion about a given solution:
E(a) = E0 + dE/da (Delta a) +1/2 d2/da2 (Delta a)2.
where is an arbitrary parameter such as the x coordinate value.
The derivatives of E are required for this expansion. Formally:
dE<p|p>/da = d<p| H |p> /da.
This can be expanded as:
dE<p|p>/da = <dp/da | H |p> +<p|d H/da, |p> +<p| H |dp/da>
However, the assumption of a stationary point is simply dp/da = 0 so that the resulting expansion is:
dE/da = <p|d H/da, |>/<p|p>
This expression can be continued, ad infinitum, to produce the necessary derivatives. This Taylor series is NOT infinitely convergent.
Large differences in positions will violate the assumption of a stationary wave function. Electronic transitions are not stationary by definition, and this expansion cannot treat them (nor does it treat non-differentiable problems like spin). However, in the limit of small displacements along bond and geometry, this expansion can be surprisingly good. Interactions between distant atoms, such as Van der Waals and electrostatic interactions are well treated by this approximation because these are relatively weak interactions and the stationary approximation for the wave function is valid.