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analytical potentials and the manual process of selecting a good functional form of
the potential are disadvantages of the first case of deriving the non-bonded CG
potentials.
Concerning the second option to generate numerically a tabulated potential that
closely reproduces a given melt structure, the iterative Boltzmann inversion (IBI)
method [
29
,
41
,
51
,
52
] has been developed.
3.1.1
Iterative Boltzmann Inversion Method
The main feature of the IBI method is the automatic and iterative way of determin-
ing the effective bead-bead interactions that match a set of structural quantities
(such as intermolecular RDFs) calculated from a more detailed reference simulation
model (i.e., atomistic). Henderson [
53
] proved that at a given density and tempera-
ture, there is a unique mapping between the RDF and the intermolecular potential.
Thus, a potential that reproduces the target RDF is a fixed point of the iteration and,
if the algorithm converges, a valid solution is obtained for the CG potential. For a
complete polymer model, one assumes that the total potential energy
U
CG
can be
separated into bonded (covalent) and non-bonded contributions:
X
U
CG
b
X
U
CG
nb
U
CG
¼
þ
;
(1)
where
U
CG
b
and
U
CG
nb
represent the bonded and non-bonded part of the potential,
respectively. The bonded interactions are derived such that the conformational
distribution
P
CG
, which is characterized by specific CG bond lengths
r
between
adjacent pairs of CG beads, angles
y
between neighboring triplets of beads, and
between neighboring quadruplet of beads, i.e.,
P
CG
torsions
, in the CG
simulation is reproduced. If one assumes that the different internal CG degrees of
freedom are uncorrelated, then
P
CG
'
ð
r
; y;'Þ
factorizes into independent probability
distributions of bond, angle, and torsional degrees of freedom:
ð
r
; y;'Þ
P
CG
r
P
CG
P
CG
P
CG
ð
; y;'
Þ¼
ð
r
Þ
ðyÞ
ð'Þ:
(2)
To obtain the bonded potentials, the individual distributions
P
CG
,
P
CG
ð
r
Þ
ðyÞ
, and
P
CG
ð'Þ
are first fitted by a suitable sum of Gaussians functions and then Boltzmann
inverted. It should be noted that the bond length and bond angle probability
distributions are normalized by taking into account the corresponding metric,
namely
r
2
for bending angles. It should be noted that
the Boltzmann inversion of a distribution leads to a potential of mean force (PMF),
i.e., a free energy, which is only in certain limiting cases identical to a potential
energy. This means that using a free energy in place of a potential energy is wrong
in a strict statistical-mechanical sense. In the case of bonded interactions, however,
which are rather stiff and energy-dominated and which separate well from the
for bond lengths and sin
ðyÞ
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