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XtalOpt is a free and truly open source evolutionary algorithm designed for a priori crystal structure prediction.

A detailed user guide for the XtalOpt setup and usage:

User Guide

Full description of the implemented methodology:

Methodology

A quick overview of the XtalOpt code features:

Graphical User Interface Multi-Objective Search Evolutionary Operations

Graphical User Interface

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Structure Limits

Optimization Settings

Search Settings

Multiobjective Search

Progress

• As XtalOpt performs the search, the progress table continuously updates, providing information about each structure.
• In the figure we see individuals in various stages of completion:
• One structure XtalOpt displays is optimized (in blue), another structure has been automatically marked as a duplicate (dark green) and removed from the breeding pool, yet another structure is currently undergoing a local optimization (light green), while another structure is waiting to be optimized (light blue). For a multi-objective run, while the objectives are being calculated, the structure is marked as calculating objectives (yellow). If these objective calculations fail, then the structure will be marked as objective fail (gray).
• Other useful information is displayed about each structure, such as the time spent in optimization, the optimized enthalpy, the cell volume, spacegroup, and each structure’s ancestry (i.e. parent(s) and parameters for the evolutionary operator that generated it).
• A status bar on the bottom of the window shows the number of structures that are optimized, running, and failing at any given time.
• This information is visible regardless of which tab is currently being viewed.
• An additional feature of the progress table is the ability to immediately visualize any of the individuals in the Avogadro main window – simply clicking on a row in this table will display the three-dimensional structure in Avogadro, where it can be visualized, modified, or exported.
• If the user would like to add a bit of “intelligent design” to the evolutionary process, a structure can be modified and then resubmitted using a context (right-click) menu from the progress table.
• The context menu provides tools to (un)kill a structure, resubmit for local optimization at an arbitrary optimization step, or replace a problematic structure with a new, random individual.

The "Progress" tab

Plot

• Another visualization and analysis tool available during the search is the interactive plot.
• Allows investigation of various trends in the search by plotting a point for each individual.
• Each point can be plotted based upon structure number, generation number, enthalpy, energy, PV enthalpy term, lattice parameters, cell volume, or objectives (in a multi-objective search) on either axis.
• Further information is available by labeling the points with the individual’s structure ID, spacegroup number, Hermann-Mauguin spacegroup symbol, enthalpy, energy, PV term, volume, generation, or index number.
• The plot is also interactive; zooming and panning are possible using simple mouse controls.
• Clicking on a structure’s point in the plot will load it in the main Avogadro window, allowing all the same functionality as described above in the progress table section.

Multi-Objective Search

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User-specified objectives

• The user can utilize an arbitrary number of objectives for multi-objective optimization.
• Each objective is corresponding to a property of the system that can be calculated and represented with a single numerical value (enthalpy is a trivial one!).
• After successful local optimization, XtalOpt provides the external code with structural information of the relaxed structure for calculating the desired property.
• To introduce an objective, the user should specify the external code or script that calculates that property and the file name to which the corresponding numerical value is written.
• Performing the multi-objective optimization, requires the user to further specify the optimization type and a weight for each objective to be used in calculating the generalized fitness function.

Generalized fitness function

• XtalOpt uses a scalar generalized fitness function to assess the suitability of each structure in a multi-objective optimization.
• For a set of objectives \(\{X\}\) and \(\{Y\}\) to be minimized and maximized, respectively, the generalized fitness function for \(s^\textrm{th}\) structure is calculated as:

\(f_s= \sum_i w^i_X \left( \frac{X^i_{max} - X^i_{s} }{X^i_{max}-X^i_{min}} \right) + \sum_j w^j_Y \left( \frac{Y^j_{s} - Y^j_{min}}{Y^j_{max}-Y^j_{min}} \right) \)

where \(\{X_s\}\) and \(\{Y_s\}\) are the values of the corresponding objectives calculated for the \(s^\textrm{th}\) structure, and \(\{w\}\) is the set of weights associated with the objectives chosen to reflect their relative significance
• With the total objective weights' equal to 1.0, the fitness function will be a value between 0 and 1.
• Choosing as many structures as the user-specified pool size according to the fitness value of candidate structures, an accumulative probability is being calculated. This probability will be used to chose a structure randomly for procreation by means of evolutionary operations.

Evolutionary Operations

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Strain

• The strain operator allows XtalOpt to mutate the unit cell parameters, but not the fractional positions of the atoms.
• One can multiply the cell row vectors by the symmetric voight matrix, as shown below,

\( \vec{\textrm{v}}_{\textrm{new}} = \vec{\textrm{v}} \begin{bmatrix} 1+\varepsilon_{11} & \frac{\varepsilon_{12}}{2} & \frac{\varepsilon_{13}}{2} \\ \frac{\varepsilon_{12}}{2} & 1+\varepsilon_{22} & \frac{\varepsilon_{32}}{2} \\ \frac{\varepsilon_{13}}{2} & \frac{\varepsilon_{23}}{2} & 1+\varepsilon_{33} \end{bmatrix} \)

where \(\varepsilon_{i,j}\) is a random selected value, taken from a zero-centered normal distribution.

Exchange

• Exchange is the simpliest operator that exchanges two atoms of different types in a cell.

Ripple

• The ripple operator acts as a periodic displacement of the atomic coordinates in a cell.
• A random axis is chosen (for this example the \(z\) axis will be chosen) and the atoms along this axis are shifted by an amount, \(\Delta z\), where the new coordinate along that axis will be \( z_\mathrm{new} = z + \Delta z \).
• The displacement along the axis chosen is dependent on atomic placement in the orthogonal plane (\(x\) and \(y\) axes for this example) following the formula:



\( \Delta z = \rho\cos(2\pi\mu x+\theta_{x})\cos(2\pi\eta y+\theta_{y})~~~~~\mu\in\mathbb{Z},~\eta\in\mathbb{Z} \)

• In this equation:
• \(\rho\) is the maximum displacement of an atom along the axis.
• \(\mu\) and \(\eta\) determines the number of cosine waves in the orthogonal directions.
• \(\theta_{x}\) and \(\theta_{y}\) are randomly chosen values between \(0\) and \(2 \pi\) that specify the strength of the ripple in various areas of the cell.

Crossover

• The crossover, or "cut and splice", operator is the equivalent of breeding in biology of breeding in which two parents are combined to form an offspring.
• This is the only operator in XtalOpt that requires two parents to generate an offspring.
• For XtalOpt's implementation, two parents are selected from the breeding pool.
• The atoms in both parent structures are randomly reflected and rotated to avoid biasing one given orientation.
• Each newly mutated parent is then "cut" to form the two halves of what will be the offspring structure.
• Finally, the two halves are "spliced" together.
• This occurs through first joining the atomic positions in fractional coordinates.
• The cell dimensions are defined through random weighting of each of the parents' cell vectors.