The `basin-sampling' algorithm as described by Bogdan, Wales and Calvo [2] combines the `basin-hopping' [1] and Wang-Landau sampling techniques [3] to study the thermodynamics of the transformed PES. It provides a direct temperature-independent estimate of the total energy density of states, along with thermodynamic properties such as the free energy and entropy via ensemble averages using samples of local minima, rather than instantaneous configurations.

In the GMIN implementation setting *TEMPERATURE*
to zero and specifying the keyword *HISTOGRAM* invoke a `basin-sampling' run.

In the microcanonical `basin-sampling'
procedure we start from a random configuration and perform a random walk in energy space
by perturbing and minimising the structures. Assuming that a given energy is visited with probability
reciprocal to the true density of states,
we obtain a flat energy distribution. To restrict the search of configuration
space to bound clusters, the structures are confined to a spherical container.
For every visited state the current estimate of the energy density of local minima
is updated by a multiplicative modification factor (*histfac*).
Acting as a convergence parameter, is relatively large at first to allow for
fast accumulation of the histograms over the full energy range,
and over time it is self-consistently reduced towards unity.
The length of one WL iteration, over which the value of the
modification factor stays the same, is regulated by the
`flatness parameter' (*hpercent*)--the percentage by which histogram entries
are allowed to deviate from the mean. As approaches a predefined
value (*histfacmin*), and provided the random walk is unbiased
and all the energy levels are sampled uniformly, the energy density of local
minima converges to its true value. If
the *VISITPROP* keyword is specified, the convergence of one WL iteration is governed by
the number of visits being proportional to
[4].

The energy spectrum in question is bounded from below by the potential energy of the global
minimum *histmin* and is separated into *hbins*
equally spaced energy windows, which constitute histogram bins of width *histint*.

For a random configuration the probability of quenching to a minimum with potential energy
lying in a given bin *i* is
, where is the probability of a random minimum having potential
energy in a given bin, and is the average configuration space volume
of a basin of attraction for minima in this range. In the original
basin-sampling study is approximated by
,
where
is the mean distance of a random starting point to the quenched minimum in the
corresponding bin, and is the number of vibrational degrees of freedom.

During the random walk we accumulate
(the density of local minima in each bin--*hweight*),
and
(the global energy histogram *histvals* and the
local energy histogram *lhistvals*, i.e. the
number of visits in a given energy bin during one WL iteration),
and
(*hdist*), which is updated when a new quenched minimum is added
to the corresponding bin.

The run is started from a uniform energy distribution, , and samples the configuration space with a probability inversely proportional to . At each step all the Cartesian coordinates are displaced by a random number in the range times STEP. The structure obtained after each geometrical perturbation is then minimised using the modified limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm[5]. In the case of an atom leaving the container at any point the coordinates are reset to the starting geometry, and the previous minimum is recounted. If the minimisation is successful, the weights of the bins to which the starting () and the quenched () minima belong are compared . If , the density of states of the -th bin is updated as , its energy histograms are incremented as and , and the distance between the starting and the quenched geometries is used to update . If the attributes of the -th bin are modified instead. If the walk goes outside the defined energy range we recount the structure in the -th bin to avoid boundary effects. After updating the histogram we do not reset the coordinates at each successful step to those of the quenched minimum but allow the geometry to vary continuously. The opposite strategy is generally found to be more effective for global optimisation, but here we must maintain detailed balance.

The energy histograms are periodically checked against the convergence criterion .
Because the energy spectrum is discrete some energy bins may never be visited.
Also, to prevent trapping when is close to
, bins where has fewer than of the average number of entries are
ignored by setting the *ignorebin* flag
to true. When the non-zero parts of the
histogram are considered sufficiently flat,
the modification factor is reduced using
a square root function, the values of
are reset and another WL iteration is started.
The final statistical weights, , can be used to calculate the canonical partition function in
terms of contributions from the catchment basins in each energy range .

The geometries of minima can be saved along the run by specifying *BINSTRUCTURES * keyword. Keyword *EQUILIBRATION*
regulates the starting point and frequency of recording statistics.

The output of the `basin-sampling' run is printed in `BL.Pjnorm.lnGj.Djnm.Djm.VT.his` in the
following format:
*histmin*+(i-1/2)*histint*; *hweight*(i)(*distmin*/*dist(i)*)
normalised to ;
normalized to ;
average *hdist*(i) minimised with respect to rigid body coordinates; unminimised average
*hdist*(i); *histvals*(i).

Calculation of the
vibrational density of states for a given minimum is invoked by
the additional *TETHER* keyword, which
requests a conventional Wang-Landau sampling of the configuration
space restricted to the average volume of the basin of
attraction to which a given minimum belongs.[2]

David Wales 2018-02-19