3.4 Full Range Process Potentials
103
of convergence of the Taylor expansions around at least one ab initio point. The w i
weight functions are now chosen so as to switch on whenever the ab initio points are
reasonably close to the geometry being considered and the potential to be given by the
weighted average of Taylor expansion estimates from the nearby points. To the end
of smoothly interpolating between adjacent ab initio points, Collins and coworkers
[30] choose as internal coordinates z = 1/r so that a single ab initio point used in
Eq. (3.48) describes the asymptotic behavior of the isolated diatomic potential quite
accurately (Taylor expansions in inverse coordinates have a much larger domain of
convergence than the coordinates themselves).
The w i weight functions are formulated as inverse powers p of the sum of all the
(z k − z k (i))
2
+ k
2 terms with the power parameter determining the drop off of the
weight function and the parameter k avoiding singularities while ensuring sufficient
sharpness near each data point.
An interesting evolution of the local methods is represented by the local mobile
(LM) LS ones in the local basis functions are modulated as a function of the molecular geometry of interest [25]. Because of this, lower order polynomial functions
are needed though the coefficients of the basis functions are now varying with the
geometry and set a heavier computational demand. As a matter of fact in the LM-LS
scheme [31] the value V at point r is represented by a linear combination of linearly
independent basis functions f k (r)( j = 1, . . . , n), as follows:
V (r) = c
T
(r)f(r) = f
T
(r)c
T
(r) =
L
l=1
c l (r) f l (r),
(3.49)
where the coefficients c 1 (r), c 2 (r), . . . , c L (r) depend on the coordinates r.
Being as before the coordinates and energy values to be interpolated r(i) and
v(i)(i = 1, 2, . . . , i max ) with i max the number of data points, the error functional is
formulated as
i max
i=1
w i (r)[V (r) − v(i)]
2
.
(3.50)
This provides fresh ground for the use of previously proposed formulations of the
PESs. Among them is the Diatomic In Molecule (DIM) [32] method that is a simple
method to deal with theoretical studies of electronically nonadiabatic transitions.
The LM-LS scheme fuels also new interest in the formulation of the PES in terms
of many-body expansions (MBE) defined as follows:
V (r) =
pairs
V
(2)
(r
(2)
) +
tri ples
V
(3)
(r
(3)
) + · · · +
Nuples
V
(N )
(r
(N )
).
(3.51)
and that allows to modulate the various each many-body component as the process
progresses. In particular, for a three-body system this means three two-body terms
and one three-body term of the type
103
of convergence of the Taylor expansions around at least one ab initio point. The w i
weight functions are now chosen so as to switch on whenever the ab initio points are
reasonably close to the geometry being considered and the potential to be given by the
weighted average of Taylor expansion estimates from the nearby points. To the end
of smoothly interpolating between adjacent ab initio points, Collins and coworkers
[30] choose as internal coordinates z = 1/r so that a single ab initio point used in
Eq. (3.48) describes the asymptotic behavior of the isolated diatomic potential quite
accurately (Taylor expansions in inverse coordinates have a much larger domain of
convergence than the coordinates themselves).
The w i weight functions are formulated as inverse powers p of the sum of all the
(z k − z k (i))
2
+ k
2 terms with the power parameter determining the drop off of the
weight function and the parameter k avoiding singularities while ensuring sufficient
sharpness near each data point.
An interesting evolution of the local methods is represented by the local mobile
(LM) LS ones in the local basis functions are modulated as a function of the molecular geometry of interest [25]. Because of this, lower order polynomial functions
are needed though the coefficients of the basis functions are now varying with the
geometry and set a heavier computational demand. As a matter of fact in the LM-LS
scheme [31] the value V at point r is represented by a linear combination of linearly
independent basis functions f k (r)( j = 1, . . . , n), as follows:
V (r) = c
T
(r)f(r) = f
T
(r)c
T
(r) =
L
l=1
c l (r) f l (r),
(3.49)
where the coefficients c 1 (r), c 2 (r), . . . , c L (r) depend on the coordinates r.
Being as before the coordinates and energy values to be interpolated r(i) and
v(i)(i = 1, 2, . . . , i max ) with i max the number of data points, the error functional is
formulated as
i max
i=1
w i (r)[V (r) − v(i)]
2
.
(3.50)
This provides fresh ground for the use of previously proposed formulations of the
PESs. Among them is the Diatomic In Molecule (DIM) [32] method that is a simple
method to deal with theoretical studies of electronically nonadiabatic transitions.
The LM-LS scheme fuels also new interest in the formulation of the PES in terms
of many-body expansions (MBE) defined as follows:
V (r) =
pairs
V
(2)
(r
(2)
) +
tri ples
V
(3)
(r
(3)
) + · · · +
Nuples
V
(N )
(r
(N )
).
(3.51)
and that allows to modulate the various each many-body component as the process
progresses. In particular, for a three-body system this means three two-body terms
and one three-body term of the type