The basis for the refinement is the calculation of the volume of a
cross peak
between spins i and j, 
, from the atomic coordinates
by means
of the relaxation matrix R
(Macura and Ernst 1980; Ernst et al. 1987; Keepers and
James 1984):
where 
is the mixing time. The relaxation matrix R is a
function
of the transition rates 
which are determined by spectral densities and dipolar coupling strengths (Solomon 1955):
and
is the gyromagnetic ratio of the proton and 
the distance between spins i and j. At present, only protons can be
used in the refinement.
describes the non-NOE magnetization losses
from the lattice.
In the simplest model, it is assumed that a single
isotropic correlation time 
is
sufficient to describe the spectral densities 
(Solomon 1955):
A step beyond this simple model is the ``model-free" approach of
Lipari and Szabo (1982), where the internal motion is described by two
parameters,
an effective correlation time 
and an order parameter 
:
X-PLOR uses an approximation of this equation that
assumes that the internal
motion is much faster than the overall rotation of the molecule
(i.e., 
), such that
the second term in the equation becomes negligible. In order to
take into account the different motional behavior of
different parts of
the molecule, different correlation time and order parameters can be
entered for different proton-proton vectors.
Groups of protons whose resonances are degenerate due to motion
(in general, mostly methyl groups) are
treated roughly as in
CORMA, version 1.5 (Keepers and James 1984).
(Note that cross peaks which are ambiguous due to overlap should be
dealt with in a different way; see the example input file
in Section 21.7.)
Each such group
is represented by
one spin, whose intensity is scaled by the number of protons in the group,
and the distance to
the group
is calculated as the 
or 
average over
the protons in the group (Eq. 21.4).
In addition, a diagonal leakage rate is added for each group
of protons.
Protons can be removed from the spin system (exchangeable protons in
D
O spectra, deuterium-labeled molecules), or their appropriate
occupancy can be specified
(exchangeable protons in H
O spectra).
