Crystal Structure Prediction and its Application in Earth and Materials Sciences: Performance and extensions

1.6 Performance and extensions

So far, the USPEX method has been widely successfully applied to different kinds of systems. And the success rate and efficiency is quite encouraging. To predict very large and complex crystal structures, this method has made many progresses (to generate randomly symmetric structures, to make smart variation operators learning about preferable local environments, to include anti-aging technique, .etc). Fig. 2.4 is an illustration showing how USPEX works. According to the first blind test of inorganic crystal structure prediction methods as shown in Table 2.1, it shows that USPEX outperforms other methods in terms of efficiency and reliability.

$\includegraphics[scale=0.6]{chapter2/pdf/uspex.png}$

Figure 2.4: Illustration of USPEX method for crystal structure prediction. An example of a test on the prediction of the crystal structure of MgSiO $_3$ with 80 atoms in the super cell. The lower panel shows that the global-minimum structure was found with 3200 structural relaxations. Taken from Ref.[20]

Table 2.1: First blind test for inorganic crystals

	Ramdom sampling	Simulated annealing	USPEX
Test 1. BaMgAl $_4$ Si $_4$ O $_{16}$ with fixed cubic cell (with force field)
Number of runs	1	10	2
(successful runs)	(1)	(1)	(2)
Minimum energy (eV)	-876.94	-877.99	-877.71
Minimum number of	14794	7330	1465
local optimizaitons
Test 2. Ba $_2$ Mg $_2$ Al $_8$ Si $_8$ O $_{32}$ with fixed cubic cell (with force field)
Number of runs	1	9	2
(successful runs)	(1)	(1)	(1)
Minimum energy (eV)	-1751.57	-1756.03	-1757.14
Minimum number of	14102	2435	3210
local optimizaitons
Test 3. Ba $_2$ Mg $_2$ Al $_8$ Si $_8$ O $_{32}$ with variable cell (with force field)
Number of runs	1	9	1
(successful runs)	(1)	(1)	(1)
Minimum energy (eV)	-1943.46	-1949.10	-1950.53
Minimum number of	13029	685	4610
local optimizaitons
Test 4. Ba $_2$ Mg $_2$ Al $_8$ Si $_8$ O $_{32}$ with variable cell (ab initio)
Number of runs	1	-	1
(successful runs)	(1)	-	(1)
Minimum energy (eV)	-68.82	-	-70.37
Minimum number of	978	-	4071
local optimizaitons

Although the standard framework of evolutionary algorithm is very powerful, new ideas can frequently come out to solve the more tricky concerns which are beyond the traditional CSP problems. Here I just list a few:

Molecular crystals

In case of many organic crystals, assembling the most stable structure from single atom will give a mixture of H $_2$ O, CO $_2$ , and some other simple molecules, for a simple reason that most organic complexs are metastable rather than thermodynamically stable. The prediction of stable complex organic structures can be achieved under the constraint of fixed molecules (or partially fixed as flexible molecules) as building blocks. Chapter 3 and 4 will go into details about this module.

Variable compositions

This is a function to enable simultaneous prediction of all stable stoichiometries and structures. The pioneering study was done by Johnnesson et al ²⁹, who succeeded in predicting stable stoichiometries of alloys within a given structure type. This means that we are dealing with a complex landscape consisting of compositional and structural coordinates. And the optimizing target is no longer one dimensional (only involving energy), but a two dimensional convex hull represented in energy versus composition plot (see Fig. 2.5). Chapter 6 will show an example to study the Mg-O system under high pressure.

$\includegraphics[scale=0.7]{chapter2/pdf/vc.png}$

Figure 2.5: Examples of a variable composition prediction in a binary Lenard-Jones system. Solid circles denote ground states, while open circles denote metastable solutions. Taken from Ref.[20]

Surfaces and clusters

Dimension can be viewed as another type of extreme condition. A lot of new physics and chemistry can be achieved by simply reducing the dimensionality. Low dimensional structure prediction is very similar to crystal structure prediction, but there are also critical differences. Under periodic boundary conditions (the currently most popular model in most widely used codes), we should insert enough ‘vacuum’ to eliminate the interaction with its periodic images ³⁰. For surface, substrate should be also taken into account. Yet, the variation operators should not care about them. There should exists inconsistency between local optimization model and global optimization model. Surface also brings another independent thermodynamical parameter, chemical potential (which is not needed to be under consideration for 3D infinite bulk crystals). Since the stability of surface configuration depends on the chemical potential, the established convex hull for multicomponent system is quite different from bulk ³¹.

New fitness function

As mentioned above, we can also consider landscapes of properties other than the (free) energy. In this case, hybrid optimization needs to be considered - combining local optimization by energy and global optimization with respect to the property. This allows searching for new materials with target properties (for instanace, the superdense ³² and superhard materials ³³). The example of density as the fitness function will be described in chapter 7.