Numerical Solution of the Quantum Hydrodynamic Equations 337
21.4.3 DYNAMICAL APPROXIMATIONS FOR TREATING QUANTUM MANY-BODY SYSTEMS
The quantum many-body problem is very difficult if not impossible to treat from
first principles due to the well-known exponential scaling of the computational
cost associated with standard quantum mechanical methods. The intuitive trajec-
tory based de Broglie–Bohm formulation of quantum mechanics offers an appealing
new approach to the quantum many-body problem which may result in more efficient
computational methods. However, even within the quantum hydrodynamic formu-
lation, “dynamical approximations” are needed in order make the calculations more
feasible. As mentioned in the introduction (see Section 21.1), dynamical approxi-
mations involve a modification of the original (exact) equations of motion. These
modifications are usually based on the idea that some coupling terms are smaller or
less important than others. Thus, the goal is to identify and partition these coupling
terms into those which are ignored and those which are kept. Unfortunately, due to
the coupled non-linear nature of the quantum hydrodynamic equations, this goal is
difficult to achieve in practice. Intuition and numerical tests are at present the modus
operandi.
Several approximate methods have been developed in recent years, such as: “Lin-
earized Quantum Force,” [17,18] the “Derivative Propagation Method,” [19] and the
VDS [16]. In this section, we will review the VDS. The idea behind the VDS is to
take advantage of a unique property of the de Broglie–Bohm formulation of quantum
mechanics. The property of interest is the well-known cancellation of the classical
and quantum forces for a bound (stationary) vibrational degree of freedom (q
l
). That
is, the total force (classical plus quantum) along this degree of freedom is zero:
f
q
l
c
+ f
q
l
q
= 0. This condition is exact in the asymptotic reactant channel and it
holds for any vibrational interaction potential (i.e., a harmonic oscillator assumption
is not required). Setting the total force zero along the bound vibrational degrees of
freedom decouples the quantum hydrodynamic equations into a set of uncoupled one-
dimensional problems. Thus, the method scales linearly with respect to the number of
vibrational degrees of freedom and can be easily parallelized on a computer by assign-
ing each one-dimensional problem to a separate processor. During the course of the
dynamics, some of the vibrational degrees of freedom will be coupled more strongly
to the reaction path than others. For these degrees of freedom, additional couplings
will need to be included or an exact treatment may be required. These couplings could
be introduced in an iterative fashion or included directly during the propagation. For
the vibrational degrees of freedom which are uncoupled to the reaction path (i.e., they
are “spectators”), the VDS approximation is exact. A practical implementation would
require the following steps:
(1) propagate the uncoupled one-dimensional solutions using the VDS every-
where;
(2) estimate the couplings between the one-dimensional solutions;
(3) re-propagate only the one-dimensional solutions that experience significant
couplings and include these couplings;
(4) repeat steps (2) and (3) until the solutions converge.