170
5 Fast Nonadiabatic Dynamics
wavepackets Θ k is still computationally very demanding. In fact, the computational
burden involved in a brute-force numerical integration grows exponentially with the
number s of vibrational degrees of freedom. For example, we could represent Θ k on
a grid of points, obtained dividing in n parts the interval of values of Q α of interest,
for all α, ending up with a grid of n
s points.
The method that best combines efficiency and accuracy in tackling this difficult
problem is the multiconfigurational time-dependent Hartree (MCTDH) [19], in which
the wavepackets are expanded in terms of “configurations” Φ I
Θ k (Q, t) =
I
A I,k (t)Φ I (Q, t) .
(5.92)
Each configuration is built as a Hartree product (i.e., a simple product, as opposed
to the antisymmetrized products that are necessary for electrons) of “single-particle
functions” (SPFs), in the form
Φ I (Q, t) =
α
ξ I,α (Q α , t) .
(5.93)
In a fast nonadiabatic dynamics the nuclear degrees of freedom are normally strongly
correlated, so that it is mandatory to adopt a multiconfigurational approach (i.e., to
consider more than one configuration per wavepacket). Using more than one configuration allows to represent correctly the splitting of the wavepacket into different
pathways, for instance those leading to distinct photodissociation products. If enough
configurations are added, the numerical time-dependent wavefunction converges to
the exact solution. In the standard version of MCTDH, the SPFs are expanded on a
time-independent basis set of one-dimensional functions with time-dependent coefficients. Each SPF is therefore a contraction of the time-independent basis, suitable
to represent the wavepacket: the number of SPFs needed to reach convergence is
then expected to be small, at least if compared with the number of time-independent
basis functions.
The equations of motions for the coefficients A I,k and the SPFs are obtained using
the Dirac–Frenkel time-dependent variational principle
δΨ
i
∂
∂t
− ˆ
H
Ψ
= 0
(5.94)
where δΨ is an infinitesimal variation of the wavefunction, obtained by changing the
parameters it contains (i.e., the coefficients A I,k and those determining the SPFs, in
the present case). The resulting equation of motions are complex, but their number is
much smaller if compared to the set of equations obtained by using directly the timeindependent basis set. We sketch here a derivation of the time-dependent variational
principle (5.94), following Raab [20]. We assume that the wavefunction Ψ has a
given analytic form and depends on a set of parameters, collected in the vector a.
The time derivative of Ψ is then
5 Fast Nonadiabatic Dynamics
wavepackets Θ k is still computationally very demanding. In fact, the computational
burden involved in a brute-force numerical integration grows exponentially with the
number s of vibrational degrees of freedom. For example, we could represent Θ k on
a grid of points, obtained dividing in n parts the interval of values of Q α of interest,
for all α, ending up with a grid of n
s points.
The method that best combines efficiency and accuracy in tackling this difficult
problem is the multiconfigurational time-dependent Hartree (MCTDH) [19], in which
the wavepackets are expanded in terms of “configurations” Φ I
Θ k (Q, t) =
I
A I,k (t)Φ I (Q, t) .
(5.92)
Each configuration is built as a Hartree product (i.e., a simple product, as opposed
to the antisymmetrized products that are necessary for electrons) of “single-particle
functions” (SPFs), in the form
Φ I (Q, t) =
α
ξ I,α (Q α , t) .
(5.93)
In a fast nonadiabatic dynamics the nuclear degrees of freedom are normally strongly
correlated, so that it is mandatory to adopt a multiconfigurational approach (i.e., to
consider more than one configuration per wavepacket). Using more than one configuration allows to represent correctly the splitting of the wavepacket into different
pathways, for instance those leading to distinct photodissociation products. If enough
configurations are added, the numerical time-dependent wavefunction converges to
the exact solution. In the standard version of MCTDH, the SPFs are expanded on a
time-independent basis set of one-dimensional functions with time-dependent coefficients. Each SPF is therefore a contraction of the time-independent basis, suitable
to represent the wavepacket: the number of SPFs needed to reach convergence is
then expected to be small, at least if compared with the number of time-independent
basis functions.
The equations of motions for the coefficients A I,k and the SPFs are obtained using
the Dirac–Frenkel time-dependent variational principle
δΨ
i
∂
∂t
− ˆ
H
Ψ
= 0
(5.94)
where δΨ is an infinitesimal variation of the wavefunction, obtained by changing the
parameters it contains (i.e., the coefficients A I,k and those determining the SPFs, in
the present case). The resulting equation of motions are complex, but their number is
much smaller if compared to the set of equations obtained by using directly the timeindependent basis set. We sketch here a derivation of the time-dependent variational
principle (5.94), following Raab [20]. We assume that the wavefunction Ψ has a
given analytic form and depends on a set of parameters, collected in the vector a.
The time derivative of Ψ is then