Ensembles¶
Canonical ensemble¶

class
mchammer.ensembles.
CanonicalEnsemble
(structure, calculator, temperature, user_tag=None, boltzmann_constant=8.617330337217213e05, data_container=None, random_seed=None, data_container_write_period=inf, ensemble_data_write_interval=None, trajectory_write_interval=None, sublattice_probabilities=None)[source]¶ Instances of this class allow one to simulate systems in the canonical ensemble (\(N_iVT\)), i.e. at constant temperature (\(T\)), number of atoms of each species (\(N_i\)), and volume (\(V\)).
The probability for a particular state in the canonical ensemble is proportional to the wellknown Boltzmann factor,
\[\rho_{\text{C}} \propto \exp [  E / k_B T ].\]Since the concentrations or equivalently the number of atoms of each species is held fixed in the canonical ensemble, a trial step must conserve the concentrations. This is accomplished by randomly picking two unlike atoms and swapping their identities. The swap is accepted with probability
\[P = \min \{ 1, \, \exp [  \Delta E / k_B T ] \},\]where \(\Delta E\) is the change in potential energy caused by the swap.
The canonical ensemble provides an ideal framework for studying the properties of a system at a specific concentration. Properties such as potential energy or phenomena such as chemical ordering at a specific temperature can conveniently be studied by simulating at that temperature. The canonical ensemble is also a convenient tool for “optimizing” a system, i.e., finding its lowest energy chemical ordering. In practice, this is usually achieved by simulated annealing, i.e. the system is equilibrated at a high temperature, after which the temperature is continuously lowered until the acceptance probability is almost zero. In a wellbehaved system, the chemical ordering at that point corresponds to a lowenergy structure, possibly the global minimum at that particular concentration.
Parameters:  structure (
Atoms
) – atomic configuration to be used in the Monte Carlo simulation; also defines the initial occupation vector  calculator (
BaseCalculator
) – calculator to be used for calculating the potential changes that enter the evaluation of the Metropolis criterion  temperature (float) – temperature \(T\) in appropriate units [commonly Kelvin]
 boltzmann_constant (float) – Boltzmann constant \(k_B\) in appropriate units, i.e. units that are consistent with the underlying cluster expansion and the temperature units [default: eV/K]
 user_tag (str) – humanreadable tag for ensemble [default: None]
 data_container (str) – name of file the data container associated with the ensemble will be written to; if the file exists it will be read, the data container will be appended, and the file will be updated/overwritten
 random_seed (int) – seed for the random number generator used in the Monte Carlo simulation
 ensemble_data_write_interval (int) – interval at which data is written to the data container; this includes for example the current value of the calculator (i.e. usually the energy) as well as ensembles specific fields such as temperature or the number of atoms of different species
 data_container_write_period (float) – period in units of seconds at which the data container is written to file; writing periodically to file provides both a way to examine the progress of the simulation and to back up the data [default: np.inf]
 trajectory_write_interval (int) – interval at which the current occupation vector of the atomic configuration is written to the data container.
 sublattice_probabilities (List[float]) – probability for picking a sublattice when doing a random swap. This should be as long as the number of sublattices and should sum up to 1.
Example
The following snippet illustrate how to carry out a simple Monte Carlo simulation in the canonical ensemble. Here, the parameters of the cluster expansion are set to emulate a simple Ising model in order to obtain an example that can be run without modification. In practice, one should of course use a proper cluster expansion:
from ase.build import bulk from icet import ClusterExpansion, ClusterSpace from mchammer.calculators import ClusterExpansionCalculator from mchammer.ensembles import CanonicalEnsemble # prepare cluster expansion # the setup emulates a second nearestneighbor (NN) Ising model # (zerolet and singlet ECIs are zero; only first and second neighbor # pairs are included) prim = bulk('Au') cs = ClusterSpace(prim, cutoffs=[4.3], chemical_symbols=['Ag', 'Au']) ce = ClusterExpansion(cs, [0, 0, 0.1, 0.02]) # prepare initial configuration structure = prim.repeat(3) for k in range(5): structure[k].symbol = 'Ag' # set up and run MC simulation calc = ClusterExpansionCalculator(structure, ce) mc = CanonicalEnsemble(structure=structure, calculator=calc, temperature=600, data_container='myrun_canonical.dc') mc.run(100) # carry out 100 trial swaps

attach_observer
(observer, tag=None)¶ Attaches an observer to the ensemble.
Parameters:  observer (
BaseObserver
) – observer instance to attach  tag – name used in data container
 observer (

boltzmann_constant
¶ Boltzmann constant \(k_B\) (see parameters section above)
Return type: float

calculator
¶ calculator attached to the ensemble
Return type: BaseCalculator

data_container
¶ data container associated with ensemble
Return type: DataContainer

do_canonical_swap
(sublattice_index, allowed_species=None)¶ Carries out one Monte Carlo trial step.
Parameters:  sublattice_index (
int
) – the sublattice the swap will act on  allowed_species (
Optional
[List
[int
]]) – list of atomic numbers for allowed species
Returns: Return type: Returns 1 or 0 depending on if trial move was accepted or rejected
 sublattice_index (

do_sgc_flip
(chemical_potentials, sublattice_index, allowed_species=None)¶ Carries out one Monte Carlo trial step.
Parameters:  chemical_potentials (
Dict
[int
,float
]) – chemical potentials used to calculate the potential
 difference
 sublattice_index (
int
) – the sublattice the flip will act on  allowed_species (
Optional
[List
[int
]]) – list of atomic numbers for allowed species
Returns: Return type: Returns 1 or 0 depending on if trial move was accepted or rejected
 chemical_potentials (

do_vcsgc_flip
(phis, kappa, sublattice_index, allowed_species=None)¶ Carries out one Monte Carlo trial step.
Parameters:  phis (
Dict
[int
,float
]) – average constraint parameters  kappa (
float
) – parameter that constrains the variance of the concentration  sublattice_index (
int
) – the sublattice the flip will act on  allowed_species (
Optional
[List
[int
]]) – list of atomic numbers for allowed species
Returns: Return type: Returns 1 or 0 depending on if trial move was accepted or rejected
 phis (

ensemble_parameters
¶ Returns parameters associated with the ensemble.
Return type: dict

get_random_sublattice_index
(probability_distribution)¶ Returns a random sublattice index based on the weights of the sublattice.
Parameters: probability_distribution – probability distributions for the sublattices Return type: int

observer_interval
¶ minimum number of steps to run Monte Carlo simulation without interruption for observation
Return type: int

observers
¶ Return type: Dict
[str
,BaseObserver
]

random_seed
¶ seed used to initialize random number generator
Return type: int

run
(number_of_trial_steps)¶ Samples the ensemble for the given number of trial steps.
Parameters:  number_of_trial_steps (
int
) – number of MC trial steps to run in total  reset_step – if True the MC trial step counter and the data container will be reset to zero and empty, respectively.
 number_of_trial_steps (

step
¶ current configuration (copy)
Return type: int

structure
¶ current configuration (copy)
Return type: Atoms

sublattices
¶ sublattices for the configuration being sampled
Return type: Sublattices

temperature
¶ Current temperature
Return type: float

update_occupations
(sites, species)¶ Updates the occupation vector of the configuration being sampled. This will change the state of the configuration in both the calculator and the configuration manager.
Parameters:  sites (
List
[int
]) – indices of sites of the configuration to change  species (
List
[int
]) – new occupations (species) by atomic number
Raises: ValueError
– if input lists are not of the same length sites (

user_tag
¶ tag used for labeling the ensemble
Return type: str

write_data_container
(outfile)¶ Updates last state of the Monte Carlo simulation and writes DataContainer to file.
Parameters: outfile ( Union
[str
,Binaryio
,Textio
]) – file to which to write
 structure (
Canonical annealing¶

class
mchammer.ensembles.
CanonicalAnnealing
(structure, calculator, T_start, T_stop, n_steps, cooling_function='exponential', user_tag=None, boltzmann_constant=8.617330337217213e05, data_container=None, random_seed=None, data_container_write_period=inf, ensemble_data_write_interval=None, trajectory_write_interval=None, sublattice_probabilities=None)[source]¶ Instances of this class allow one to carry out simulated annealing in the canonical ensemble, i.e. the temperature is varied in predefined fashion while the composition is kept fixed. See
mchammer.ensembles.CanonicalEnsemble
for more information about the standard canonical ensemble.The canonical annealing ensemble can be useful, for example, for finding ground states or generating low energy configurations.
The temperature control scheme is selected via the
cooling_function
keyword argument, while the initial and final temperature are set via theT_start
andT_stop
arguments. Several predefined temperature control schemes are available including linear and exponential. In the latter case the temperature varies logarithmatically as a function of the MC step, emulating the exponential temperature dependence of the atomic exchange rate encountered in many materials. It is also possible to provide a user defined cooling function via the keyword argument. This function must comply with the following function header:def cooling_function(step, T_start, T_stop, n_steps): T = ... # compute temperature return T
Here
step
refers to the current MC trial step.Parameters:  structure (
Atoms
) – atomic configuration to be used in the Monte Carlo simulation; also defines the initial occupation vector  calculator (
BaseCalculator
) – calculator to be used for calculating the potential changes that enter the evaluation of the Metropolis criterion  T_start (float) – temperature from which the annealing is started
 T_stop (float) – final temperature for annealing
 n_steps (int) – number of steps to take in the annealing simulation
 cooling_function (str/function) – to use the predefined cooling functions provide a string linear or exponential, otherwise provide a function
 boltzmann_constant (float) – Boltzmann constant \(k_B\) in appropriate units, i.e. units that are consistent with the underlying cluster expansion and the temperature units [default: eV/K]
 user_tag (str) – humanreadable tag for ensemble [default: None]
 data_container (str) – name of file the data container associated with the ensemble will be written to; if the file exists it will be read, the data container will be appended, and the file will be updated/overwritten
 random_seed (int) – seed for the random number generator used in the Monte Carlo simulation
 ensemble_data_write_interval (int) – interval at which data is written to the data container; this includes for example the current value of the calculator (i.e. usually the energy) as well as ensembles specific fields such as temperature or the number of atoms of different species
 data_container_write_period (float) – period in units of seconds at which the data container is written to file; writing periodically to file provides both a way to examine the progress of the simulation and to back up the data [default: np.inf]
 trajectory_write_interval (int) – interval at which the current occupation vector of the atomic configuration is written to the data container.
 sublattice_probabilities (List[float]) – probability for picking a sublattice when doing a random swap. This should be as long as the number of sublattices and should sum up to 1.

T_start
¶ Starting temperature
Return type: float

T_stop
¶ Starting temperature
Return type: float

attach_observer
(observer, tag=None)¶ Attaches an observer to the ensemble.
Parameters:  observer (
BaseObserver
) – observer instance to attach  tag – name used in data container
 observer (

boltzmann_constant
¶ Boltzmann constant \(k_B\) (see parameters section above)
Return type: float

calculator
¶ calculator attached to the ensemble
Return type: BaseCalculator

data_container
¶ data container associated with ensemble
Return type: DataContainer

do_canonical_swap
(sublattice_index, allowed_species=None)¶ Carries out one Monte Carlo trial step.
Parameters:  sublattice_index (
int
) – the sublattice the swap will act on  allowed_species (
Optional
[List
[int
]]) – list of atomic numbers for allowed species
Returns: Return type: Returns 1 or 0 depending on if trial move was accepted or rejected
 sublattice_index (

do_sgc_flip
(chemical_potentials, sublattice_index, allowed_species=None)¶ Carries out one Monte Carlo trial step.
Parameters:  chemical_potentials (
Dict
[int
,float
]) – chemical potentials used to calculate the potential
 difference
 sublattice_index (
int
) – the sublattice the flip will act on  allowed_species (
Optional
[List
[int
]]) – list of atomic numbers for allowed species
Returns: Return type: Returns 1 or 0 depending on if trial move was accepted or rejected
 chemical_potentials (

do_vcsgc_flip
(phis, kappa, sublattice_index, allowed_species=None)¶ Carries out one Monte Carlo trial step.
Parameters:  phis (
Dict
[int
,float
]) – average constraint parameters  kappa (
float
) – parameter that constrains the variance of the concentration  sublattice_index (
int
) – the sublattice the flip will act on  allowed_species (
Optional
[List
[int
]]) – list of atomic numbers for allowed species
Returns: Return type: Returns 1 or 0 depending on if trial move was accepted or rejected
 phis (

ensemble_parameters
¶ Returns parameters associated with the ensemble.
Return type: dict

estimated_ground_state
¶ Structure with lowest observed potential during run

estimated_ground_state_potential
¶ Lowest observed potential during run

get_random_sublattice_index
(probability_distribution)¶ Returns a random sublattice index based on the weights of the sublattice.
Parameters: probability_distribution – probability distributions for the sublattices Return type: int

n_steps
¶ Number of steps to carry out
Return type: int

observer_interval
¶ minimum number of steps to run Monte Carlo simulation without interruption for observation
Return type: int

observers
¶ Return type: Dict
[str
,BaseObserver
]

random_seed
¶ seed used to initialize random number generator
Return type: int

step
¶ current configuration (copy)
Return type: int

structure
¶ current configuration (copy)
Return type: Atoms

sublattices
¶ sublattices for the configuration being sampled
Return type: Sublattices

temperature
¶ Current temperature
Return type: float

update_occupations
(sites, species)¶ Updates the occupation vector of the configuration being sampled. This will change the state of the configuration in both the calculator and the configuration manager.
Parameters:  sites (
List
[int
]) – indices of sites of the configuration to change  species (
List
[int
]) – new occupations (species) by atomic number
Raises: ValueError
– if input lists are not of the same length sites (

user_tag
¶ tag used for labeling the ensemble
Return type: str

write_data_container
(outfile)¶ Updates last state of the Monte Carlo simulation and writes DataContainer to file.
Parameters: outfile ( Union
[str
,Binaryio
,Textio
]) – file to which to write
 structure (
Semigrand canonical ensemble¶

class
mchammer.ensembles.
SemiGrandCanonicalEnsemble
(structure, calculator, temperature, chemical_potentials, user_tag=None, data_container=None, random_seed=None, data_container_write_period=inf, ensemble_data_write_interval=None, trajectory_write_interval=None, boltzmann_constant=8.617330337217213e05, sublattice_probabilities=None)[source]¶ Instances of this class allow one to simulate systems in the semigrand canonical (SGC) ensemble (\(N\Delta\mu_i VT\)), i.e. at constant temperature (\(T\)), total number of sites (\(N=\sum_i N_i\)), relative chemical potentials (\(\Delta\mu_i=\mu_i  \mu_1\), where \(i\) denotes the species), and volume (\(V\)).
The probability for a particular state in the SGC ensemble for a \(m\)component system can be written
\[\rho_{\text{SGC}} \propto \exp\Big[  \big( E + \sum_{i>1}^m \Delta\mu_i N_i \big) \big / k_B T \Big]\]with the relative chemical potentials \(\Delta\mu_i = \mu_i  \mu_1\) and species counts \(N_i\). Unlike the canonical ensemble, the number of the respective species (or, equivalently, the concentrations) are allowed to vary in the SGC ensemble. A trial step thus consists of randomly picking an atom and changing its identity with probability
\[P = \min \Big\{ 1, \, \exp \big[  \big( \Delta E + \sum_i \Delta \mu_i \Delta N_i \big) \big / k_B T \big] \Big\},\]where \(\Delta E\) is the change in potential energy caused by the swap.
There exists a simple relation between the differences in chemical potential and the canonical free energy \(F\). In a binary system, this relationship reads
\[\Delta \mu =  \frac{1}{N} \frac{\partial F}{\partial c} ( N, V, T, \langle c \rangle).\]Here \(c\) denotes concentration (\(c=N_i/N\)) and \(\langle c \rangle\) the average concentration observed in the simulation. By recording \(\langle c \rangle\) while gradually changing \(\Delta \mu\), one can thus in principle calculate the difference in canonical free energy between the pure phases (\(c=0\) or \(1\)) and any concentration by integrating \(\Delta \mu\) over that concentration range. In practice this requires that the average recorded concentration \(\langle c \rangle\) varies continuously with \(\Delta \mu\). This is not the case for materials with multiphase regions (such as miscibility gaps), because in such regions \(\Delta \mu\) maps to multiple concentrations. In a Monte Carlo simulation, this is typically manifested by discontinuous jumps in concentration. Such jumps mark the phase boundaries of a multiphase region and can thus be used to construct the phase diagram. To recover the free energy, however, such systems require sampling in other ensembles, such as the varianceconstrained semigrand canonical ensemble.
Parameters:  structure (
Atoms
) – atomic configuration to be used in the Monte Carlo simulation; also defines the initial occupation vector  calculator (
BaseCalculator
) – calculator to be used for calculating the potential changes that enter the evaluation of the Metropolis criterion  temperature (float) – temperature \(T\) in appropriate units [commonly Kelvin]
 chemical_potentials (Dict[str, float]) – chemical potential for each species \(\mu_i\); the key denotes the species, the value specifies the chemical potential in units that are consistent with the underlying cluster expansion
 boltzmann_constant (float) – Boltzmann constant \(k_B\) in appropriate units, i.e. units that are consistent with the underlying cluster expansion and the temperature units [default: eV/K]
 user_tag (str) – humanreadable tag for ensemble [default: None]
 data_container (str) – name of file the data container associated with the ensemble will be written to; if the file exists it will be read, the data container will be appended, and the file will be updated/overwritten
 random_seed (int) – seed for the random number generator used in the Monte Carlo simulation
 ensemble_data_write_interval (int) – interval at which data is written to the data container; this includes for example the current value of the calculator (i.e. usually the energy) as well as ensembles specific fields such as temperature or the number of atoms of different species
 data_container_write_period (float) – period in units of seconds at which the data container is written to file; writing periodically to file provides both a way to examine the progress of the simulation and to back up the data [default: np.inf]
 trajectory_write_interval (int) – interval at which the current occupation vector of the atomic configuration is written to the data container.
 sublattice_probabilities (List[float]) – probability for picking a sublattice when doing a random flip. This should be as long as the number of sublattices and should sum up to 1.
Example
The following snippet illustrate how to carry out a simple Monte Carlo simulation in the semicanonical ensemble. Here, the parameters of the cluster expansion are set to emulate a simple Ising model in order to obtain an example that can be run without modification. In practice, one should of course use a proper cluster expansion:
from ase.build import bulk from icet import ClusterExpansion, ClusterSpace from mchammer.calculators import ClusterExpansionCalculator from mchammer.ensembles import SemiGrandCanonicalEnsemble # prepare cluster expansion # the setup emulates a second nearestneighbor (NN) Ising model # (zerolet and singlet ECIs are zero; only first and second neighbor # pairs are included) prim = bulk('Au') cs = ClusterSpace(prim, cutoffs=[4.3], chemical_symbols=['Ag', 'Au']) ce = ClusterExpansion(cs, [0, 0, 0.1, 0.02]) # set up and run MC simulation (T=600 K, delta_mu=0.8 eV/atom) structure = prim.repeat(3) calc = ClusterExpansionCalculator(structure, ce) mc = SemiGrandCanonicalEnsemble(structure=structure, calculator=calc, temperature=600, data_container='myrun_sgc.dc', chemical_potentials={'Ag': 0, 'Au': 0.8}) mc.run(100) # carry out 100 trial swaps
Todo
 add check that chemical symbols in chemical potentials are allowed

attach_observer
(observer, tag=None)¶ Attaches an observer to the ensemble.
Parameters:  observer (
BaseObserver
) – observer instance to attach  tag – name used in data container
 observer (

boltzmann_constant
¶ Boltzmann constant \(k_B\) (see parameters section above)
Return type: float

calculator
¶ calculator attached to the ensemble
Return type: BaseCalculator

chemical_potentials
¶ chemical potentials \(\mu_i\) (see parameters section above)
Return type: Dict
[int
,float
]

data_container
¶ data container associated with ensemble
Return type: DataContainer

do_canonical_swap
(sublattice_index, allowed_species=None)¶ Carries out one Monte Carlo trial step.
Parameters:  sublattice_index (
int
) – the sublattice the swap will act on  allowed_species (
Optional
[List
[int
]]) – list of atomic numbers for allowed species
Returns: Return type: Returns 1 or 0 depending on if trial move was accepted or rejected
 sublattice_index (

do_sgc_flip
(chemical_potentials, sublattice_index, allowed_species=None)¶ Carries out one Monte Carlo trial step.
Parameters:  chemical_potentials (
Dict
[int
,float
]) – chemical potentials used to calculate the potential
 difference
 sublattice_index (
int
) – the sublattice the flip will act on  allowed_species (
Optional
[List
[int
]]) – list of atomic numbers for allowed species
Returns: Return type: Returns 1 or 0 depending on if trial move was accepted or rejected
 chemical_potentials (

do_vcsgc_flip
(phis, kappa, sublattice_index, allowed_species=None)¶ Carries out one Monte Carlo trial step.
Parameters:  phis (
Dict
[int
,float
]) – average constraint parameters  kappa (
float
) – parameter that constrains the variance of the concentration  sublattice_index (
int
) – the sublattice the flip will act on  allowed_species (
Optional
[List
[int
]]) – list of atomic numbers for allowed species
Returns: Return type: Returns 1 or 0 depending on if trial move was accepted or rejected
 phis (

ensemble_parameters
¶ Returns parameters associated with the ensemble.
Return type: dict

get_random_sublattice_index
(probability_distribution)¶ Returns a random sublattice index based on the weights of the sublattice.
Parameters: probability_distribution – probability distributions for the sublattices Return type: int

observer_interval
¶ minimum number of steps to run Monte Carlo simulation without interruption for observation
Return type: int

observers
¶ Return type: Dict
[str
,BaseObserver
]

random_seed
¶ seed used to initialize random number generator
Return type: int

run
(number_of_trial_steps)¶ Samples the ensemble for the given number of trial steps.
Parameters:  number_of_trial_steps (
int
) – number of MC trial steps to run in total  reset_step – if True the MC trial step counter and the data container will be reset to zero and empty, respectively.
 number_of_trial_steps (

step
¶ current configuration (copy)
Return type: int

structure
¶ current configuration (copy)
Return type: Atoms

sublattices
¶ sublattices for the configuration being sampled
Return type: Sublattices

temperature
¶ temperature \(T\) (see parameters section above)
Return type: float

update_occupations
(sites, species)¶ Updates the occupation vector of the configuration being sampled. This will change the state of the configuration in both the calculator and the configuration manager.
Parameters:  sites (
List
[int
]) – indices of sites of the configuration to change  species (
List
[int
]) – new occupations (species) by atomic number
Raises: ValueError
– if input lists are not of the same length sites (

user_tag
¶ tag used for labeling the ensemble
Return type: str

write_data_container
(outfile)¶ Updates last state of the Monte Carlo simulation and writes DataContainer to file.
Parameters: outfile ( Union
[str
,Binaryio
,Textio
]) – file to which to write
 structure (
Varianceconstrained semigrand canonical ensemble¶

class
mchammer.ensembles.
VCSGCEnsemble
(structure, calculator, temperature, phis, kappa, boltzmann_constant=8.617330337217213e05, user_tag=None, data_container=None, random_seed=None, data_container_write_period=inf, ensemble_data_write_interval=None, trajectory_write_interval=None, sublattice_probabilities=None)[source]¶ Instances of this class allow one to simulate systems in the varianceconstrained semigrand canonical (VCSGC) ensemble (\(N\phi\kappa VT\)), i.e. at constant temperature (\(T\)), total number of sites (\(N=\sum_i N_i\)), and two additional dimensionless parameters \(\phi\) and \(\kappa\), which constrain average and variance of the concentration, respectively.
The below examples treat the binary case, but the generalization of to ternaries and higherorder systems is straightforward. The probability for a particular state in the VCSGC ensemble for a \(2\)component system can be written
\[\rho_{\text{VCSGC}} \propto \exp\Big[  E / k_B T + \kappa N ( c_1 + \phi_1 / 2 )^2 \Big],\]where \(c_1\) represents the concentration of species 1, i.e. \(c_1=N_1/N\). (Please note that the quantities \(\kappa\) and \(\phi\) correspond, respectively, to \(\bar{\kappa}\) and \(\bar{\phi}\) in [SadErh12].). The \(\phi\) may refer to any of the two species. If \(\phi\) is specified for species A, an equivalent simulation can be carried out by specifying \(\phi_B\) as \(2\phi_A\). In general, simulations of \(N\)component systems requires the specification of \(\phi\) for \(N1\) elements.
Just like the semigrand canonical ensemble, the VCSGC ensemble allows concentrations to change. A trial step consists of changing the identity of a randomly chosen atom and accepting the change with probability
\[P = \min \{ 1, \, \exp [  \Delta E / k_B T + \kappa N \Delta c_1 (\phi_1 + \Delta c_1 + 2 c_1 ) ] \}.\]Note that for a sufficiently large value of \(\kappa\), say 200, the probability density \(\rho_{\text{VCSGC}}\) is sharply peaked around \(c_1=\phi_1 / 2\). In practice, this means that we can gradually change \(\phi_1\) from (using some margins) \(2.1\) to \(0.1\) and take the system continuously from \(c_1 = 0\) to \(1\). The parameter \(\kappa\) constrains the fluctuations (or the variance) of the concentration at each value of \(\phi_1\), with higher values of \(\kappa\) meaning less fluctuations. Unlike the semigrand canonical ensemble, one value of \(\phi_1\) maps to one and only one concentration also in multiphase regions. Since the derivative of the canonical free energy can be expressed in terms of parameters and observables of the VCSGC ensemble,
\[k_B T \kappa ( \phi_1 + 2 \langle c_1 \rangle ) =  \frac{1}{N} \frac{\partial F}{\partial c_1} (N, V, T, \langle c_1 \rangle ),\]this ensemble allows for thermodynamic integration across multiphase regions. This means that we can construct phase diagrams by directly comparing the free energies of the different phases. This often makes the VCSGC ensemble more convenient than the semigrand canonical ensemble when simulating materials with multiphase regions, such as alloys with miscibility gaps.
When using the VCSGC ensemble, please cite Sadigh, B. and Erhart, P., Phys. Rev. B 86, 134204 (2012) [SadErh12].
Parameters:  structure (
Atoms
) – atomic configuration to be used in the Monte Carlo simulation; also defines the initial occupation vector  calculator (
BaseCalculator
) – calculator to be used for calculating the potential changes that enter the evaluation of the Metropolis criterion  temperature (float) – temperature \(T\) in appropriate units [commonly Kelvin]
 phis (Dict[str, float]) – average constraint parameters \(\phi_i\); the key denotes the species; for a Ncomponent sublattice, there should be N  1 different phi_i (referred to as \(\bar{\phi}\) in [SadErh12])
 kappa (float) – parameter that constrains the variance of the concentration (referred to as \(\bar{\kappa}\) in [SadErh12])
 boltzmann_constant (float) – Boltzmann constant \(k_B\) in appropriate units, i.e. units that are consistent with the underlying cluster expansion and the temperature units [default: eV/K]
 user_tag (str) – humanreadable tag for ensemble [default: None]
 data_container (str) – name of file the data container associated with the ensemble will be written to; if the file exists it will be read, the data container will be appended, and the file will be updated/overwritten
 random_seed (int) – seed for the random number generator used in the Monte Carlo simulation
 ensemble_data_write_interval (int) – interval at which data is written to the data container; this includes for example the current value of the calculator (i.e. usually the energy) as well as ensembles specific fields such as temperature or the number of atoms of different species
 data_container_write_period (float) – period in units of seconds at which the data container is written to file; writing periodically to file provides both a way to examine the progress of the simulation and to back up the data [default: np.inf]
 trajectory_write_interval (int) – interval at which the current occupation vector of the atomic configuration is written to the data container.
 sublattice_probabilities (List[float]) – probability for picking a sublattice when doing a random flip. The list should be as long as the number of sublattices and should sum up to 1.
Example
The following snippet illustrate how to carry out a simple Monte Carlo simulation in the varianceconstrained semicanonical ensemble. Here, the parameters of the cluster expansion are set to emulate a simple Ising model in order to obtain an example that can be run without modification. In practice, one should of course use a proper cluster expansion:
from ase.build import bulk from icet import ClusterExpansion, ClusterSpace from mchammer.calculators import ClusterExpansionCalculator from mchammer.ensembles import VCSGCEnsemble # prepare cluster expansion # the setup emulates a second nearestneighbor (NN) Ising model # (zerolet and singlet ECIs are zero; only first and second neighbor # pairs are included) prim = bulk('Au') cs = ClusterSpace(prim, cutoffs=[4.3], chemical_symbols=['Ag', 'Au']) ce = ClusterExpansion(cs, [0, 0, 0.1, 0.02]) # set up and run MC simulation structure = prim.repeat(3) calc = ClusterExpansionCalculator(structure, ce) phi = 0.6 mc = VCSGCEnsemble(structure=structure, calculator=calc, temperature=600, data_container='myrun_vcsgc.dc', phis={'Au': phi}, kappa=200) mc.run(100) # carry out 100 trial swaps

attach_observer
(observer, tag=None)¶ Attaches an observer to the ensemble.
Parameters:  observer (
BaseObserver
) – observer instance to attach  tag – name used in data container
 observer (

boltzmann_constant
¶ Boltzmann constant \(k_B\) (see parameters section above)
Return type: float

calculator
¶ calculator attached to the ensemble
Return type: BaseCalculator

data_container
¶ data container associated with ensemble
Return type: DataContainer

do_canonical_swap
(sublattice_index, allowed_species=None)¶ Carries out one Monte Carlo trial step.
Parameters:  sublattice_index (
int
) – the sublattice the swap will act on  allowed_species (
Optional
[List
[int
]]) – list of atomic numbers for allowed species
Returns: Return type: Returns 1 or 0 depending on if trial move was accepted or rejected
 sublattice_index (

do_sgc_flip
(chemical_potentials, sublattice_index, allowed_species=None)¶ Carries out one Monte Carlo trial step.
Parameters:  chemical_potentials (
Dict
[int
,float
]) – chemical potentials used to calculate the potential
 difference
 sublattice_index (
int
) – the sublattice the flip will act on  allowed_species (
Optional
[List
[int
]]) – list of atomic numbers for allowed species
Returns: Return type: Returns 1 or 0 depending on if trial move was accepted or rejected
 chemical_potentials (

do_vcsgc_flip
(phis, kappa, sublattice_index, allowed_species=None)¶ Carries out one Monte Carlo trial step.
Parameters:  phis (
Dict
[int
,float
]) – average constraint parameters  kappa (
float
) – parameter that constrains the variance of the concentration  sublattice_index (
int
) – the sublattice the flip will act on  allowed_species (
Optional
[List
[int
]]) – list of atomic numbers for allowed species
Returns: Return type: Returns 1 or 0 depending on if trial move was accepted or rejected
 phis (

ensemble_parameters
¶ Returns parameters associated with the ensemble.
Return type: dict

get_random_sublattice_index
(probability_distribution)¶ Returns a random sublattice index based on the weights of the sublattice.
Parameters: probability_distribution – probability distributions for the sublattices Return type: int

kappa
¶ kappa \(\bar{\kappa}\) constrain parameter (see parameters section above)
Return type: float

observer_interval
¶ minimum number of steps to run Monte Carlo simulation without interruption for observation
Return type: int

observers
¶ Return type: Dict
[str
,BaseObserver
]

phis
¶ phis \(\phi_i\), one for each species but their sum must be \(2.0\) (referred to as \(\bar{\phi}\) in [SadErh12])
Return type: Dict
[int
,float
]

random_seed
¶ seed used to initialize random number generator
Return type: int

run
(number_of_trial_steps)¶ Samples the ensemble for the given number of trial steps.
Parameters:  number_of_trial_steps (
int
) – number of MC trial steps to run in total  reset_step – if True the MC trial step counter and the data container will be reset to zero and empty, respectively.
 number_of_trial_steps (

step
¶ current configuration (copy)
Return type: int

structure
¶ current configuration (copy)
Return type: Atoms

sublattices
¶ sublattices for the configuration being sampled
Return type: Sublattices

temperature
¶ temperature \(T\) (see parameters section above)
Return type: float

update_occupations
(sites, species)¶ Updates the occupation vector of the configuration being sampled. This will change the state of the configuration in both the calculator and the configuration manager.
Parameters:  sites (
List
[int
]) – indices of sites of the configuration to change  species (
List
[int
]) – new occupations (species) by atomic number
Raises: ValueError
– if input lists are not of the same length sites (

user_tag
¶ tag used for labeling the ensemble
Return type: str

write_data_container
(outfile)¶ Updates last state of the Monte Carlo simulation and writes DataContainer to file.
Parameters: outfile ( Union
[str
,Binaryio
,Textio
]) – file to which to write
 structure (
Hybrid ensemble¶

class
mchammer.ensembles.
HybridEnsemble
(structure, calculator, temperature, ensemble_specs, probabilities=None, boltzmann_constant=8.617330337217213e05, user_tag=None, data_container=None, random_seed=None, data_container_write_period=inf, ensemble_data_write_interval=None, trajectory_write_interval=None)[source]¶ Instances of this class allows one to combine multiple ensembles. In particular, a dictionary should be provided for each ensemble, which must include the type (ensemble) as well as the index of the sublattice (sublattice_index). In addition, it is possible to provide a list of allowed symbols (allowed_symbols), which must represent a subset of the elements that can occupy the sites on the specified sublattice. Note that additional arguments are required for the SGC and VCSGC ensembles, namely chemical potentials (chemical_potentials) for the former and constraint parameters (phis and kappa) for the latter. For more detailed information regarding the different ensembles, please see
CanonicalEnsemble
,SemiGrandCanonicalEnsemble
, andVCSGCEnsemble
.This class is particularly useful for effectively sampling complex multicomponent systems with several active sublattices, in which case different ensembles can be defined for each of the latter. The fact that it is possible to set the allowed chemical symbols means that one can vary the concentrations of a few selected species, with the help of a VCSGC or semigrand canonical ensemble, while still allowing swaps between any two sites, using a canonical ensemble (see also the below example). It is advisable to carefully consider how to define the ensemble probabilities. By default the ensembles are weighted by the sizes of the corresponding sublattices, which should give suitable probabilities in most cases. As is shown in the example below, it might be prudent to provide different values if allowed symbols are provided as well as for cases where there are several ensembles that are active on different sublattices.
Parameters:  structure (
Atoms
) – atomic configuration to be used in the Monte Carlo simulation; also defines the initial occupation vector  calculator (
BaseCalculator
) – calculator to be used for calculating the potential changes that enter the evaluation of the Metropolis criterion  temperature (float) – temperature \(T\) in appropriate units [commonly Kelvin]
 ensemble_specs (List[Dict]) –
A list of dictionaries, which should contain the following items:
 ’ensemble’, which could be either “vcsgc”; “semigrand” or “canonical”, lowercase and uppercase letters or any combination thereof are accepted (required)
 ’sublattice_index’, index for the sublattice of interest (required)
 ’allowed_symbols’, list of allowed chemical symbols (default: read from ClusterSpace)
 ’chemical_potentials’, a dictionary of chemical potentials for each species \(\mu_i\); the key denotes the species, the value :specifies the chemical potential in units that are :consistent with the underlying cluster expansion (only :applicable and required for SGC ensembles)
 ’phis ‘, dictionary with average constraint parameters
‘\(\phi_i\); the key denotes the species; for a
Ncomponent sublattice, there should be N  1
different phi_i (referred to as
\(\bar{\phi}\) in [SadErh12]; only applicable
and required for VCSGC ensembles, see also
VCSGCEnsemble
)  ’kappa’, parameter that constrains the variance of the ‘concentration (referred to as \(\bar{\kappa}\) in [SadErh12]; only applicable :and required for VCSGC ensembles)
 probabilities (List[float]) – list of floats with the probabilities for choosing a particular ensemble with the same length as ensemble specs. If left unspecified the probabilties are weighted by the sizes of the associated sublattices
 boltzmann_constant (float) – Boltzmann constant \(k_B\) in appropriate units, i.e. units that are consistent with the underlying cluster expansion and the temperature units [default: eV/K]
 user_tag (str) – humanreadable tag for ensemble [default: None]
 data_container (str) – name of file the data container associated with the ensemble will be written to; if the file exists it will be read, the data container will be appended, and the file will be updated/overwritten
 random_seed (int) – seed for the random number generator used in the Monte Carlo simulation
 ensemble_data_write_interval (int) – interval at which data is written to the data container; this includes for example the current value of the calculator (i.e. usually the energy) as well as ensembles specific fields such as temperature or the number of atoms of different species
 data_container_write_period (float) – period in units of seconds at which the data container is written to file; writing periodically to file provides both a way to examine the progress of the simulation and to back up the data [default: np.inf]
 trajectory_write_interval (int) – interval at which the current occupation vector of the atomic configuration is written to the data container.
Example
The following snippet illustrates how to carry out a simple Monte Carlo simulation using a combination of one canonical and one VCSGC ensemble. Specifically, the concentration of one species (Au) is kept constant while the others (Ag and Pd) are varied, while swaps are still allowed. Here, the parameters of the cluster expansion are set to emulate a simple Ising model in order to obtain an example that can be run without modification. In practice, one should of course use a proper cluster expansion:
from ase.build import bulk from icet import ClusterExpansion, ClusterSpace from mchammer.calculators import ClusterExpansionCalculator from mchammer.ensembles import HybridEnsemble # prepare cluster expansion # the setup emulates a second nearestneighbor (NN) Ising model # (zerolet and singlet ECIs are zero; only first and second neighbor # pairs are included) prim = bulk('Au') cs = ClusterSpace(prim, cutoffs=[4.3], chemical_symbols=['Ag', 'Au', 'Pd']) ce = ClusterExpansion( cs, [0, 0, 0, 0.1, 0.1, 0.1, 0.02, 0.02, 0.02]) # define structure object structure = prim.repeat(3) for i, atom in enumerate(structure): if i % 2 == 0: atom.symbol = 'Ag' elif i % 3 == 0: atom.symbol = 'Pd' # the default probabilities for this case would be [0.5, 0.5], but # since the VCSGC ensemble only performs flips on a subset of all # sites on the sublattice, namely those originally occupied by Ag # and Pd atoms, specific values will be provided weights = [len(structure), len([s for s in structure.get_chemical_symbols() if s != 'Au'])] norm = sum(weights) probabilities = [w / norm for w in weights] # set up and run MC simulation calc = ClusterExpansionCalculator(structure, ce) ensemble_specs = [ {'ensemble': 'canonical', 'sublattice_index': 0}, {'ensemble': 'vcsgc', 'sublattice_index': 0, 'phis': {'Ag': 0.2}, 'kappa': 200, 'allowed_symbols':['Ag', 'Pd']}] mc = HybridEnsemble(structure=structure, calculator=calc, ensemble_specs=ensemble_specs, temperature=600, probabilities=probabilities, data_container='myrun_hybrid.dc') mc.run(100) # carry out 100 trial steps

attach_observer
(observer, tag=None)¶ Attaches an observer to the ensemble.
Parameters:  observer (
BaseObserver
) – observer instance to attach  tag – name used in data container
 observer (

boltzmann_constant
¶ Boltzmann constant \(k_B\) (see parameters section above)
Return type: float

calculator
¶ calculator attached to the ensemble
Return type: BaseCalculator

data_container
¶ data container associated with ensemble
Return type: DataContainer

do_canonical_swap
(sublattice_index, allowed_species=None)¶ Carries out one Monte Carlo trial step.
Parameters:  sublattice_index (
int
) – the sublattice the swap will act on  allowed_species (
Optional
[List
[int
]]) – list of atomic numbers for allowed species
Returns: Return type: Returns 1 or 0 depending on if trial move was accepted or rejected
 sublattice_index (

do_sgc_flip
(chemical_potentials, sublattice_index, allowed_species=None)¶ Carries out one Monte Carlo trial step.
Parameters:  chemical_potentials (
Dict
[int
,float
]) – chemical potentials used to calculate the potential
 difference
 sublattice_index (
int
) – the sublattice the flip will act on  allowed_species (
Optional
[List
[int
]]) – list of atomic numbers for allowed species
Returns: Return type: Returns 1 or 0 depending on if trial move was accepted or rejected
 chemical_potentials (

do_vcsgc_flip
(phis, kappa, sublattice_index, allowed_species=None)¶ Carries out one Monte Carlo trial step.
Parameters:  phis (
Dict
[int
,float
]) – average constraint parameters  kappa (
float
) – parameter that constrains the variance of the concentration  sublattice_index (
int
) – the sublattice the flip will act on  allowed_species (
Optional
[List
[int
]]) – list of atomic numbers for allowed species
Returns: Return type: Returns 1 or 0 depending on if trial move was accepted or rejected
 phis (

ensemble_parameters
¶ Returns parameters associated with the ensemble.
Return type: dict

get_random_sublattice_index
(probability_distribution)¶ Returns a random sublattice index based on the weights of the sublattice.
Parameters: probability_distribution – probability distributions for the sublattices Return type: int

observer_interval
¶ minimum number of steps to run Monte Carlo simulation without interruption for observation
Return type: int

observers
¶ Return type: Dict
[str
,BaseObserver
]

probabilities
¶ Ensemble propabilities
Return type: float

random_seed
¶ seed used to initialize random number generator
Return type: int

run
(number_of_trial_steps)¶ Samples the ensemble for the given number of trial steps.
Parameters:  number_of_trial_steps (
int
) – number of MC trial steps to run in total  reset_step – if True the MC trial step counter and the data container will be reset to zero and empty, respectively.
 number_of_trial_steps (

step
¶ current configuration (copy)
Return type: int

structure
¶ current configuration (copy)
Return type: Atoms

sublattices
¶ sublattices for the configuration being sampled
Return type: Sublattices

temperature
¶ Current temperature
Return type: float

trial_steps_per_ensemble
¶ Number of Monte Carlo trial steps for each ensemble
Return type: Dict
[str
,float
]

update_occupations
(sites, species)¶ Updates the occupation vector of the configuration being sampled. This will change the state of the configuration in both the calculator and the configuration manager.
Parameters:  sites (
List
[int
]) – indices of sites of the configuration to change  species (
List
[int
]) – new occupations (species) by atomic number
Raises: ValueError
– if input lists are not of the same length sites (

user_tag
¶ tag used for labeling the ensemble
Return type: str

write_data_container
(outfile)¶ Updates last state of the Monte Carlo simulation and writes DataContainer to file.
Parameters: outfile ( Union
[str
,Binaryio
,Textio
]) – file to which to write
 structure (