Abstract
We discuss practical considerations and computational strategies in conducting computational nanocluster research in the context of local research environment. Our emphasis is to explain workable computational strategy for locating the ground state energies of nanocluster, a primary knowledge from which other physical and chemical characteristics can be derived. The strategy in computing ground state structures of nanocluster involve combin- ing global minimum (GM) search algorithms, such as genetic algorithm and basin-hoping, with total energy calculating programs, such as DFT, DFTB and MD softwares. How to practically implement the strategy using readily available software packages is discussed.
Keywords: Nanocluster, Ground state structures, DFT, DFTB, MD, GA, BH, global minimisation search
1. Introduction
Theoretical studies of nanaparticles or nanoclusters is a fundamental part of nanoscience and nanotechnology. It aims to study, understand, manipulate and predict the physical properties of finite (as opposed to periodic) atomic system comprising of a few up to several million atoms. These are nano- objects with a length scale of a few up to a few hundreds nanometer. The best known example being the carbon-60 fullerene which assumes the shape of a 3-D cage-like buckyball. Some nanoclusters have high degree of symmetry, while some appear in the form of amorphous lump. It has been well known from well established experimental and theoretical studies that the chem- ical and physical properties of nanoclusters could be vastly diferent than their bulk counterparts. Furthermore their physical and chemical properties are strongly dependent on the geometry, size and composition they assume. Many nanoclusters exhibit surprising properties otherwise unexpected. For example, metalic clusters are well known to have a size-dependent melting temperatures that could be drastically lower than their bulk counterpart, as was reported by the ground-breaking finding in [1]. Many atoms know to be non-magnetic, such as Pd and Rh, display large magnetic dipole mo- ments when they are aggregated in cluster form due to accidental symmetry enhancement [2]. The magic polyicosahedral core-shell in metallic nanoal- loy systems Ag-Ni and Ag-Cu gives rise to a remarkable stability in their thermodynamic and electronic properties, resulting in melting points higher than their bulk counterparts [3]. There is an intriguing possibility to en- gineer ordinary atoms to form "superatoms", stable nanoclusters that have distinctive chemical properties. Another example is the discovery of mag- netic superatom, in which non-magnetic cluster consisting of sodium atoms can be made magnetic by adding a vanadium atom, forming what is known as magnetic superatom [4]. For a review on superatom, see ref. [5].
Nanoclusters research is now a multidisciplinary research front hotly pursued by materials scientists, physical chemists, condensed matter physicists and medical biologists due to their promising potential, ranging from ad- vanced functional materials (nano-magnets), optics (quantum dots), bio- imaging (nano-magnets), drug delivery (carbon nanotubes) and catalysis (fuel cells). New discoveries, either from theoretical or experimental front on exotic properties of nanoclusters are being reported at an astonishing rate across a wide range of research journals in materials science, physical chemistry to condensed matter physics. The wide range of applications is made possible due to the enormous variety of properties that arises from the many diferent structures of sizes and made of diferent type of atoms. For a comprehensive overview on nanocluster, refer ref. [6].
2. The research tasks in computational nanocluster
The main purpose in the theoretical study of nanocluster is to understand, characterize, and eventually take advantage of the novelty of the properties of nanoluster. The properties of nanoparticle are completely determined by the structures they assume. The complexity in the structures is a consequence of the interplay between the finite size and composition effects, and the extra dependencies that finiteness of size introduces between the different proper- ties. A nanocluster is made up of atoms which interact among themselves via interactions that is electromagnetic in nature and governed by the laws of quantum mechanics. Based on very general argument, all physical characteristics of an atomistic system is completely determined once their total energy as a function of the system's parameters are known. For example, we could derive the vibrational spectra of a nanocluster, its magnetic moment, ionisation energy or thermal properties if we know how the total energy varies with the configuration of the atoms. As such the ability to perform reliable and accurate calculation of the total energy and how it depends on the system's parameters is of utmost importance.
The total energy of an atomistic system depends on the interactions acting among the atoms, the electrons and atoms, and the electrons. An atomic system like the nanocluster is in principle a many-body quantum system. Although theoretically the quantum mechanical equations governing the behaviors of this many-body system can be solved exactly, for example via advanced numerical techniques such as quantum Monte Carlo, this is rarely pragmatic in practice as such approach is extremely expensive in terms of computational resources. To this end, many powerful computational meth- ods have been developed to provide accurate and reliable approximation to obtain the interactions.
2.1. Density functional theory (DFT)
Nanoclusters research is now a multidisciplinary research front hotly pursued by materials scientists, physical chemists, condensed matter physicists and medical biologists due to their promising potential, ranging from ad- vanced functional materials (nano-magnets), optics (quantum dots), bio- imaging (nano-magnets), drug delivery (carbon nanotubes) and catalysis (fuel cells). New discoveries, either from theoretical or experimental front on exotic properties of nanoclusters are being reported at an astonishing rate across a wide range of research journals in materials science, physical chemistry to condensed matter physics. The wide range of applications is made possible due to the enormous variety of properties that arises from the many diferent structures of sizes and made of diferent type of atoms. For a comprehensive overview on nanocluster, refer ref. [6].
2. The research tasks in computational nanocluster
The main purpose in the theoretical study of nanocluster is to understand, characterize, and eventually take advantage of the novelty of the properties of nanoluster. The properties of nanoparticle are completely determined by the structures they assume. The complexity in the structures is a consequence of the interplay between the finite size and composition effects, and the extra dependencies that finiteness of size introduces between the different proper- ties. A nanocluster is made up of atoms which interact among themselves via interactions that is electromagnetic in nature and governed by the laws of quantum mechanics. Based on very general argument, all physical characteristics of an atomistic system is completely determined once their total energy as a function of the system's parameters are known. For example, we could derive the vibrational spectra of a nanocluster, its magnetic moment, ionisation energy or thermal properties if we know how the total energy varies with the configuration of the atoms. As such the ability to perform reliable and accurate calculation of the total energy and how it depends on the system's parameters is of utmost importance.
The total energy of an atomistic system depends on the interactions acting among the atoms, the electrons and atoms, and the electrons. An atomic system like the nanocluster is in principle a many-body quantum system. Although theoretically the quantum mechanical equations governing the behaviors of this many-body system can be solved exactly, for example via advanced numerical techniques such as quantum Monte Carlo, this is rarely pragmatic in practice as such approach is extremely expensive in terms of computational resources. To this end, many powerful computational meth- ods have been developed to provide accurate and reliable approximation to obtain the interactions.
2.1. Density functional theory (DFT)
The most known method is the Nobel prize-winning Density Functional Theory (DFT) [7, 8], a very powerful quantum mechanical approximation that allows physicists to calculate the total energy of a configuration of atoms in either finite or periodic environment. DFT is known as ab initio calculation, meaning that it calculates the total energy of atomistic systems based on first-principles, i.e., quantum mechanics.
In general, the total energy of a atomic system can be divided into electronic and ionic parts. The electronic part is governed entirely by quantum mechanics, whereas the ionic portion, due to the ion's heavy mass, can be well treated classically using molecular dynamics. An accurate calculation of the total energy of a nanocluster requires both the electronic and ionic contributions be taken into account to a sufficient level. DFT is the default theory when it comes to calculating electronic ground states of a many-body atomic system. However, the quantum mechanical calculation of the electronic part is a computational demanding task. In a typical DFT total energy calculation of finite system we specify the atomic positions in 3-D space and the types of atoms at these positions as the initial input. The interactions (both quantum and classical) in the system are automatically handled by the DFT theories and sophisticated computational methodology encoded in the computer codes. Computational expansive it maybe, DFT usually gives very accurate results (up to the percentage level) and is generally acknowledged by the research community to be the benchmark in atomistic calculations. However, it is to be noted that DFT calculations only give the total energy of a system at zero temperature, hence no finite temperature effect is taken into account.
In practice DFT calculation is performed using standard computer program packages, such as VASP, Gaussian, ABINIT, WIEN2k and CRYSTAL. For optimum performance one usually runs the DFT packages in parallel mode.
2.2. Density functional tight-biding (DFTB)
To ease the computational cost on the electronic contribution, Density Functional Tight-Biding (DFTB) [9] is gaining popularity as a compromise between accuracy and computational cost. DFTB is a hybrid approach that combines both ab initio theory and empirical methods that allows electronic contribution to be taken into account, at a cheaper computational cost, via appropriate handling of the electronic interaction energies. This approach involves fitting of the electronic interaction energies using empirical data or results obtained from other ab initio calculations. The results of such fitting are all lumped into what are known as SK files, a table containing a list of numerical values for the DFTB parameters that reproduces the quantitative physical characteristics which are used to fit them. As an example, in order to run DFTB calculation on SiC system, the SK files for Si-Si, Si-C and C-C must be available. Unfortunately, the availability of SK files for an atomic species, or between two types of atom are not always guaranteed. If they do, they are no guarantee to be suitable (it depends on how they are fitted, and in what kinds of chemical or physical environment). One possibility to overcome this problem is to parametrise it oneself. This is by no means a formidable task but most likely a technically tedious one. To DIY a SK file may likely cost as much as a M.Sc project. In practice one usually just makes use of whatever SK file that are available in the literature, which are relatively rare in number and combination as compared to the number of available MD force fields (see below).
In practice DFTB calculation is performed using standard computer pro- gram pacakges, such as DFTB+ [10]. For optimum performance one usually run these packages in parallel mode. The DFTB+ package can also per- form tight-binding molecular dynamics (TBMD) simulation, in which the electronic interactions among the atoms are generated based on the SK files. Temperature effects can be included in a TBMD calculation 1. We cite [11] as an real-life example where DFTB is being applied extensively for investigating the temperature-dependent electronic as well as structural properties of CdS nanocluster. We also cite [12] as an example where TBMD has been applied to calculate the melting of KCl and NaCl.
2.3. Molecular Dynamics (MD)
At a coarser level, the total energy of a nanocluster can also be calculated via non quantum mechanical approaches, such as molecular dynamics (MD). In MD the interactions among the atoms are parametrised into the forms of 'force fields'. Parametrisation of force fields is itself a highly specialised research topic, in which the empirical data that contain the information of how the atoms interact among themselves are encoded into highly specific functional form. The force field of a specific atom-atom interaction is characterised by a set of fix-valued parameters determined from the fitting procedure. MD is able to handle very large system (up to many hundreds or even thousands of atoms). The level of accuracy and reliability of a MD simulation rely heavily on the quality the force fields used. Tremendous advances in MD researches and the high demand from the computational materials science have produced many powerful and accurate force fields for a wide spectrum of systems. Some advanced force fields, such as those 'charge-transferable' or 'first-principles force field' (to cite two such examples, electron force field (eFF) [13] and reactive force field (ReaxFF) [14]), could even allow electronic contribution to be included to a certain degree, hence greatly extending the scope of applicability beyond that achievable by conventional force fields.
In general, the total energy of a atomic system can be divided into electronic and ionic parts. The electronic part is governed entirely by quantum mechanics, whereas the ionic portion, due to the ion's heavy mass, can be well treated classically using molecular dynamics. An accurate calculation of the total energy of a nanocluster requires both the electronic and ionic contributions be taken into account to a sufficient level. DFT is the default theory when it comes to calculating electronic ground states of a many-body atomic system. However, the quantum mechanical calculation of the electronic part is a computational demanding task. In a typical DFT total energy calculation of finite system we specify the atomic positions in 3-D space and the types of atoms at these positions as the initial input. The interactions (both quantum and classical) in the system are automatically handled by the DFT theories and sophisticated computational methodology encoded in the computer codes. Computational expansive it maybe, DFT usually gives very accurate results (up to the percentage level) and is generally acknowledged by the research community to be the benchmark in atomistic calculations. However, it is to be noted that DFT calculations only give the total energy of a system at zero temperature, hence no finite temperature effect is taken into account.
In practice DFT calculation is performed using standard computer program packages, such as VASP, Gaussian, ABINIT, WIEN2k and CRYSTAL. For optimum performance one usually runs the DFT packages in parallel mode.
2.2. Density functional tight-biding (DFTB)
To ease the computational cost on the electronic contribution, Density Functional Tight-Biding (DFTB) [9] is gaining popularity as a compromise between accuracy and computational cost. DFTB is a hybrid approach that combines both ab initio theory and empirical methods that allows electronic contribution to be taken into account, at a cheaper computational cost, via appropriate handling of the electronic interaction energies. This approach involves fitting of the electronic interaction energies using empirical data or results obtained from other ab initio calculations. The results of such fitting are all lumped into what are known as SK files, a table containing a list of numerical values for the DFTB parameters that reproduces the quantitative physical characteristics which are used to fit them. As an example, in order to run DFTB calculation on SiC system, the SK files for Si-Si, Si-C and C-C must be available. Unfortunately, the availability of SK files for an atomic species, or between two types of atom are not always guaranteed. If they do, they are no guarantee to be suitable (it depends on how they are fitted, and in what kinds of chemical or physical environment). One possibility to overcome this problem is to parametrise it oneself. This is by no means a formidable task but most likely a technically tedious one. To DIY a SK file may likely cost as much as a M.Sc project. In practice one usually just makes use of whatever SK file that are available in the literature, which are relatively rare in number and combination as compared to the number of available MD force fields (see below).
In practice DFTB calculation is performed using standard computer pro- gram pacakges, such as DFTB+ [10]. For optimum performance one usually run these packages in parallel mode. The DFTB+ package can also per- form tight-binding molecular dynamics (TBMD) simulation, in which the electronic interactions among the atoms are generated based on the SK files. Temperature effects can be included in a TBMD calculation 1. We cite [11] as an real-life example where DFTB is being applied extensively for investigating the temperature-dependent electronic as well as structural properties of CdS nanocluster. We also cite [12] as an example where TBMD has been applied to calculate the melting of KCl and NaCl.
2.3. Molecular Dynamics (MD)
At a coarser level, the total energy of a nanocluster can also be calculated via non quantum mechanical approaches, such as molecular dynamics (MD). In MD the interactions among the atoms are parametrised into the forms of 'force fields'. Parametrisation of force fields is itself a highly specialised research topic, in which the empirical data that contain the information of how the atoms interact among themselves are encoded into highly specific functional form. The force field of a specific atom-atom interaction is characterised by a set of fix-valued parameters determined from the fitting procedure. MD is able to handle very large system (up to many hundreds or even thousands of atoms). The level of accuracy and reliability of a MD simulation rely heavily on the quality the force fields used. Tremendous advances in MD researches and the high demand from the computational materials science have produced many powerful and accurate force fields for a wide spectrum of systems. Some advanced force fields, such as those 'charge-transferable' or 'first-principles force field' (to cite two such examples, electron force field (eFF) [13] and reactive force field (ReaxFF) [14]), could even allow electronic contribution to be included to a certain degree, hence greatly extending the scope of applicability beyond that achievable by conventional force fields.
In practice MD calculation is performed using standard computer program packages, such as LAMMPS [15, 16]. For optimum performance one usually run the LAMMPS package in parallel mode. We also note that temperature efects can be included in a MD calculation.
2.4. Thermal properties of nanocluster using MD and TBMD
One advantage MD and TBMD over DFT is that the former can be used to calculate thermal properties of nanocluster. This includes, among others, heat capacity and melting behavior. To see how a nanocluster melt, we be- gin from the GS structures at zero temperature T = 0 K. The temperature is then raised to a larger temperature T = T + ∆T at a fixed rate. The system is then equilibrated at T for a sufficient amount of steps. The dynamical statistics of the system, including Lindemann index, time and pair correlation functions, fluctuations in potential energy, total energy, velocity and position of the atoms are sampled during the equilibration at T . The system is then heated to the next temperature and the statistical sampling is repeated. The evolution of the atoms in configuration space at each fixed temperature can also be visualised, providing direct insight as to how the nanocluster responses to temperature variation. By this way the dynamical properties and structural evolution of the nanocluster can be monitored in the temperature range simulated. In particular, melting behavior of the nanocluster can be probed in great detail. The indication of cluster melting appears in the form of irregular evolution of the dynamical averages as a function of temperature. In principle TBMD can be employed to investigate temperature dependence of magnetic properties of nanoclusters. In contrast MD cannot be used for this purpose as it does not have capabilities to calculate electronic properties.
We cite, among many literature on cluster melting using MD, the fol- lowing papers as examples: [17, 18, 19, 20, 21, 22, 23]. For a theoretical discussion on the thermodynamics of cluster melting, we refer to the review by ref. [24]. ref.[25] is another recent review useful for the theory on the melting of nanosystem. The review [26] inclines towards the computation aspect of nanocluster melting.
The ultimate tool to simulate temperature evolution of nanocluster with (almost) full quantum mechanics taken into account is ab initio molecular dynamics, such as Car-Parrinello molecular dynamics (CPMD) [27]. CPMD [28] is a software that implements it. Other ab initio MD is Bohn-Oppenheimer molecular dynamics (BOMD), which computer program implementation can be found in VASP, CPMD and CP2K [29]. BOMD and CPMD are different in the technical details of how the timestep is updated. Ab initio MD is a computationally much expensive than DFT. We have not yet any intention to venture into ab initio MD at the present stage.
DFT, DFTB and MD (including their variants) are the three major total energy calculating programs used by computational materials scientists to probe nanosystems for information that may not be accessible to experimentalists. Herein lies the unique strength of computational approach. The possibility offered by computational nanocluster is unfathomable due to so many possible combination to compose a nanocluster using all types of atoms. MD, TBMD or DFTB simulations can be performed on any type of atom as long as the force fields or SK files are available.
One advantage MD and TBMD over DFT is that the former can be used to calculate thermal properties of nanocluster. This includes, among others, heat capacity and melting behavior. To see how a nanocluster melt, we be- gin from the GS structures at zero temperature T = 0 K. The temperature is then raised to a larger temperature T = T + ∆T at a fixed rate. The system is then equilibrated at T for a sufficient amount of steps. The dynamical statistics of the system, including Lindemann index, time and pair correlation functions, fluctuations in potential energy, total energy, velocity and position of the atoms are sampled during the equilibration at T . The system is then heated to the next temperature and the statistical sampling is repeated. The evolution of the atoms in configuration space at each fixed temperature can also be visualised, providing direct insight as to how the nanocluster responses to temperature variation. By this way the dynamical properties and structural evolution of the nanocluster can be monitored in the temperature range simulated. In particular, melting behavior of the nanocluster can be probed in great detail. The indication of cluster melting appears in the form of irregular evolution of the dynamical averages as a function of temperature. In principle TBMD can be employed to investigate temperature dependence of magnetic properties of nanoclusters. In contrast MD cannot be used for this purpose as it does not have capabilities to calculate electronic properties.
We cite, among many literature on cluster melting using MD, the fol- lowing papers as examples: [17, 18, 19, 20, 21, 22, 23]. For a theoretical discussion on the thermodynamics of cluster melting, we refer to the review by ref. [24]. ref.[25] is another recent review useful for the theory on the melting of nanosystem. The review [26] inclines towards the computation aspect of nanocluster melting.
The ultimate tool to simulate temperature evolution of nanocluster with (almost) full quantum mechanics taken into account is ab initio molecular dynamics, such as Car-Parrinello molecular dynamics (CPMD) [27]. CPMD [28] is a software that implements it. Other ab initio MD is Bohn-Oppenheimer molecular dynamics (BOMD), which computer program implementation can be found in VASP, CPMD and CP2K [29]. BOMD and CPMD are different in the technical details of how the timestep is updated. Ab initio MD is a computationally much expensive than DFT. We have not yet any intention to venture into ab initio MD at the present stage.
DFT, DFTB and MD (including their variants) are the three major total energy calculating programs used by computational materials scientists to probe nanosystems for information that may not be accessible to experimentalists. Herein lies the unique strength of computational approach. The possibility offered by computational nanocluster is unfathomable due to so many possible combination to compose a nanocluster using all types of atoms. MD, TBMD or DFTB simulations can be performed on any type of atom as long as the force fields or SK files are available.
3. Global minimum (GM) search algorithms - Genetic algorithm (GA) and basin-hoping (BH)
To investigate the physical and chemical properties of a nanocluster requires the understanding on the conditions under which one structure is more probable than another. The search for the most probable structures, known as the ground state (GS) structures, involves a strong interplay of experiment with theory and numerical simulation. By definition, a GS is the state with lowest minimal energy, or known as the global minimum (GB) in the potential energy surface (PES). The search for GS of a nanocluster is by no means a trivial task as there are practically infinite possible ways a n-atom cluster comprised of m types of atom could organise themselves. Given the interactions among the atoms are known (which in practice is provided by a total energy calculating program), the search for a structural configuration of the atoms in 3-D space that minimises the total energy is a highly non-trivial task. In practice, one could use experimental data as the initial input. Lowest energy structures are then obtained by optimising the input structure using a local minima search algorithm such as BFGS or conjugate gradient (CG) algorithms which are built in by default in many atomistic computation codes (such a local minimisation is also known as 'relaxation'). Quantum mechanical or molecular dynamics calculations for the their thermal, electronic, optical and chemical properties are then performed based on these relaxed structures.
However, in most cases, experimental data of a cluster's initial configuration are not available. So it is not known a priori what are the lowest energy geometry for a given set of atoms in a cluster. To this end, intelligent search algorithms, which are usually implemented in the forms of parallel programming codes, will have to be deployed to search the configuration space for the lowest possible energy configurations. Many GM search schemes have been developed for such purpose. Genetic algorithm (GA) [30] and basin-hoping [31] are among the most powerful state-of-the-arts search algorithms. Application of these GM search algorithms to obtain GS of cluster is very common. Refs. [32, 33, 34] are a few examples of these. These GM search algorithms could take in an initial random configuration as its input to return as an output the lowest energy configuration that could possibly be found. Given sufficient computational hardware resources, these search algorithms provide a convenient and reliable means to locate the ground state geometry of nanoclusters, which is the primarily piece of information needed to understand them further. Deriving electronic and other physical properties is what come next once the GS structures of a nanocluster is at our disposal. In practice most of the first-principles related total energy codes such as ABINIT, Gaussian, DFTB+ and VASP are used to derive desired electronic structure information from the GS of a nanocluster.
It is clear that in order to embark on computational nanocluster research, the availability of sufficient hardware resources, computational tools and the hands-on know-how to handle them are of pragmatic relevance. In practical terms, these requirements mean lots of processor cores, cluster computer systems (usually operated in Linux OS), knowledge to maintain Linux systems, hands-on knowledge to install, maintain and use the DFT, DFTB, MD software in parallel mode, availability of the global optimisation codes (BH, GA), the knowledge to coupled the optimisation codes to the total energy calculating software, and paying the tremendous electric bills.
4. Coupling global minimisation algorithms to total energy calculating methods
GM search algorithms and the total energy calculation codes are two integral pieces of tools for computational nanocluster research. For the purpose of performing GM search, we need a GA or a BH code that could be coupled to the total energy packages. Purpose-specific GA and BH algorithms are being refined for better performance on a regular basis, providing faster and more accurate method to search for ground state geometries in the PES of nanocluster. For example, GASP [35] is a GA program (coded in Java) developed by a Cornell group, and has been incorporated into LAMMPS.
A novel GM search code, known as Parallel Tempering Multicanonical Basin Hoping Genetic Algorithm (PTMBHGA), developed by a research group from National Central University (NCU), Taiwan, was reported in [36]. In PTMBHGA, both GA and BH algorithms are combined to pro- vide a more superior search quality. PTMBHGA can be used in GA-only, BH-only or the full PTMBHGA mode (where GA and BH are combined). The GA in the original PTMBHGA code, as reported in ref. [36], contains 5 genetic operators. The weight of each operator in generating a child genera- tion has been optimised for the purpose to obtain lowest energy nanocluster structures. Recently the NCU group has coupled the PTMBHGA code into LAMMPS and DFTB+. In addition, the original PTMBHGA code has been improved to include a new 2-point crossover operator, known as the cut-and- splice operator, based on that proposed by ref. [34]. The genetic operators in the PTMBHGA now include
1. Inversion
2. Arithmetic mean
3. Geometric mean
4. N-point crossover
5. 2-point crossover
6. Cut-and-splice
There are basically three modes of operations: (a) operators 1 - 5 are switched on while operator 6 is of. In this mode, the weight of these oper- ators (excluding operator 6) during the production of new child generations have been optimally tuned for best performance [36]. GA using these 5 operators is presumably most suited for running metal-only atoms, where the directionality of the bondings is less important. (b) Only operator 6 is switched on. This mode is presumably most suited for running atomic species which directionality in the boding is important (e.g. non-metallic atoms such as C or Si atoms). When coupled with the LAMMPS MD pack- age and using BH + GA in this mode the well-known C-60 fullerene structure using the empirical Brenner force field was successfully reproduced [37]. (c) Switch on operator 1-4, and also operator 6. This mode is presumably suited for arbitrary combination of atomic species in the cluster when mode (a) or (b) fails to give satisfactory result. The additional difculty to use the GA in this mode is that the weight of each operators will have to be manually optimised.
5. Computation schemes based on cluster size and atomic composition
5.1. Atomic composition
To investigate the physical and chemical properties of a nanocluster requires the understanding on the conditions under which one structure is more probable than another. The search for the most probable structures, known as the ground state (GS) structures, involves a strong interplay of experiment with theory and numerical simulation. By definition, a GS is the state with lowest minimal energy, or known as the global minimum (GB) in the potential energy surface (PES). The search for GS of a nanocluster is by no means a trivial task as there are practically infinite possible ways a n-atom cluster comprised of m types of atom could organise themselves. Given the interactions among the atoms are known (which in practice is provided by a total energy calculating program), the search for a structural configuration of the atoms in 3-D space that minimises the total energy is a highly non-trivial task. In practice, one could use experimental data as the initial input. Lowest energy structures are then obtained by optimising the input structure using a local minima search algorithm such as BFGS or conjugate gradient (CG) algorithms which are built in by default in many atomistic computation codes (such a local minimisation is also known as 'relaxation'). Quantum mechanical or molecular dynamics calculations for the their thermal, electronic, optical and chemical properties are then performed based on these relaxed structures.
However, in most cases, experimental data of a cluster's initial configuration are not available. So it is not known a priori what are the lowest energy geometry for a given set of atoms in a cluster. To this end, intelligent search algorithms, which are usually implemented in the forms of parallel programming codes, will have to be deployed to search the configuration space for the lowest possible energy configurations. Many GM search schemes have been developed for such purpose. Genetic algorithm (GA) [30] and basin-hoping [31] are among the most powerful state-of-the-arts search algorithms. Application of these GM search algorithms to obtain GS of cluster is very common. Refs. [32, 33, 34] are a few examples of these. These GM search algorithms could take in an initial random configuration as its input to return as an output the lowest energy configuration that could possibly be found. Given sufficient computational hardware resources, these search algorithms provide a convenient and reliable means to locate the ground state geometry of nanoclusters, which is the primarily piece of information needed to understand them further. Deriving electronic and other physical properties is what come next once the GS structures of a nanocluster is at our disposal. In practice most of the first-principles related total energy codes such as ABINIT, Gaussian, DFTB+ and VASP are used to derive desired electronic structure information from the GS of a nanocluster.
It is clear that in order to embark on computational nanocluster research, the availability of sufficient hardware resources, computational tools and the hands-on know-how to handle them are of pragmatic relevance. In practical terms, these requirements mean lots of processor cores, cluster computer systems (usually operated in Linux OS), knowledge to maintain Linux systems, hands-on knowledge to install, maintain and use the DFT, DFTB, MD software in parallel mode, availability of the global optimisation codes (BH, GA), the knowledge to coupled the optimisation codes to the total energy calculating software, and paying the tremendous electric bills.
4. Coupling global minimisation algorithms to total energy calculating methods
GM search algorithms and the total energy calculation codes are two integral pieces of tools for computational nanocluster research. For the purpose of performing GM search, we need a GA or a BH code that could be coupled to the total energy packages. Purpose-specific GA and BH algorithms are being refined for better performance on a regular basis, providing faster and more accurate method to search for ground state geometries in the PES of nanocluster. For example, GASP [35] is a GA program (coded in Java) developed by a Cornell group, and has been incorporated into LAMMPS.
A novel GM search code, known as Parallel Tempering Multicanonical Basin Hoping Genetic Algorithm (PTMBHGA), developed by a research group from National Central University (NCU), Taiwan, was reported in [36]. In PTMBHGA, both GA and BH algorithms are combined to pro- vide a more superior search quality. PTMBHGA can be used in GA-only, BH-only or the full PTMBHGA mode (where GA and BH are combined). The GA in the original PTMBHGA code, as reported in ref. [36], contains 5 genetic operators. The weight of each operator in generating a child genera- tion has been optimised for the purpose to obtain lowest energy nanocluster structures. Recently the NCU group has coupled the PTMBHGA code into LAMMPS and DFTB+. In addition, the original PTMBHGA code has been improved to include a new 2-point crossover operator, known as the cut-and- splice operator, based on that proposed by ref. [34]. The genetic operators in the PTMBHGA now include
1. Inversion
2. Arithmetic mean
3. Geometric mean
4. N-point crossover
5. 2-point crossover
6. Cut-and-splice
There are basically three modes of operations: (a) operators 1 - 5 are switched on while operator 6 is of. In this mode, the weight of these oper- ators (excluding operator 6) during the production of new child generations have been optimally tuned for best performance [36]. GA using these 5 operators is presumably most suited for running metal-only atoms, where the directionality of the bondings is less important. (b) Only operator 6 is switched on. This mode is presumably most suited for running atomic species which directionality in the boding is important (e.g. non-metallic atoms such as C or Si atoms). When coupled with the LAMMPS MD pack- age and using BH + GA in this mode the well-known C-60 fullerene structure using the empirical Brenner force field was successfully reproduced [37]. (c) Switch on operator 1-4, and also operator 6. This mode is presumably suited for arbitrary combination of atomic species in the cluster when mode (a) or (b) fails to give satisfactory result. The additional difculty to use the GA in this mode is that the weight of each operators will have to be manually optimised.
5. Computation schemes based on cluster size and atomic composition
5.1. Atomic composition
In performing nanocluster computation, atomic composition of a cluster has to be considered apart from number of atoms. However, atomic composition is more a qualitative (e.g., related to the nature of interactions among the atoms) than a quantitative issue (computing cost and time spent). It is possible to envisage the following combination in the cluster make-up:
• Cluster made up of single type of metallic atom, such as Ag, Fe, Cu.
• Cluster made up of single type of non-metallic atom, such as C, Si, N.
• Cluster comprised of more than one type of atom, which are all non-metallic, e.g., C-C, Si-C, N-O, O-C.
• Cluster comprised of more than one type of atom, which are all metallic (nanoalloy cluster), e.g., Au-Cu, Cu-Fe, Ag-Au.
• Cluster comprised of non-metallic and metallic atoms, e.g., Ag-C, Ni-C, C-Fe.
The computing cost for GM search depends very sensitively to the size of the nanocluster. For large size cluster, to involve DFT in the GM search can be impractical. Due to practical limitation of computing resources, we strategically divided the nanocluster calculation into two GS search schemes:
5.2. Scheme A: DFT + GM search for small size cluster
For small size cluster (presumably less than 10 atoms), DFT is coupled to GM search codes to obtain the GS structures of atomic clusters directly. The GS so obtained are supposed to be more reliable and less controversial, but the computational cost could be orders of magnitude higher than using DFTB or MD codes. The advantage is that this scheme does not require any SK file or force field, hence could be used to calculate nanocluster of any types of atoms, including all possible combination as listed in subsection 5.1. Refs. [38, 39, 40] are examples that calculate GS of small cluster via this scheme. The coupling of DFT programs with GM codes is not yet realised by us but we do have a plan to do so.
5.3. Scheme B: DFTB (or MD) + GM + DFT
For large nanocluster we means cluster with a number of total atoms as large as could possibly be handled by our computational resources and patience. DFTB+ (or LAMMPS) coupled with GM search algorithm could handle large cluster but the GS so obtained may not coincide with the GM of the PES at the DFT level.
In this scheme, we first generate a sufficiently large number of candidate GS structures at the DFTB+ or MD level. Out of these, we choose N candidate structures with lowest energy to feed into an ab initio total energy calculating program for further optimisation (via local optimisation scheme built in these DFT software). This is to drive the candidate GS structures into the GM of the DFT PES. The process can be repeated as many time as possible until no structure with lower energy could be found. The lowest energy structure at the end of this two-stage process can then be taken as the GS at the DFT level (but not without ambiguity). Search scheme B is a practical and feasible strategy to supplant scheme A when the latter becomes too expensive and impractical, though there is still a sense of in- definiteness whether the minimum so obtained can truly represent the true global minimum at the DFT level.
There exist many papers that uses such a scheme to investigate nanocluster, such as [41, 42, 43, 44]. All the types of atomic composition mentioned in subsection 5.1 can be performed using this approach. However, the main constraint of this scheme is that the relevant SK files or force fields must be available.
We note the possibility that once the GS structures of some large clsuters are obtained this way, they can be heated by using thermostat in LAMMPS or DFTB+ so that the thermal properties can be investigated. Specifically, we could calculate the melting properties as the function of the cluster composition.
Despite this scheme has already been used in the literature for GS structures of nanocluster, it is not being widely used in the search for GM of nanocluster. Many authors in the research community prefer to use the ex- pensive ab initio MD (coupled with simulated annealing, another GM search algorithm) to search for the GS of nanocluster. Some resort to experimental data as the initial guess, while others simply fed random guess structures as the initial input into a DFT program to obtain the 'GS', before deriving their physical and electronic properties.
6. Summary
6.1. Software tools
The computational software tools required for computational nanocluster could be divided into two independent aspects: total energy calculating programs and global minimisation search algorithms.
Total energy (TE) calculating programs2 can be further categorised into
1. First-principles methods. Software implementation includes: VASP, ABINIT, WIEN2k, Gaussian, Crystal.
2. Semi-empirical methods. Software implementation includes: DFTB+.
3. Molecular dynamics methods. Software implementation includes: LAMMPS.
There exist many state-of-the-arts Global Minimisation (GM) search algorithms in the research front. We list only the software implementation that are readily accessible to us in USM.
1. Basin hoping (as an independent mode of operation in the PTMBHGA code)
2. Genetic algorithm (as an independent mode of operation in the PTMBHGA code)
3. Full PTMBHGA (mixture of BS and GA)
Furthermore the GA can be operated using combination of operators, as discussed at the end of Section 4.
In additional to the GM as mentioned above, the GASP code is also available for GM search purpose. However, we have only limited experience with this code as compared to the PHMBHGA code.
6.2. Coupling of TE and GM codes
To obtain the GS of a nanocluster, the TE program has to be coupled with the GM search algorithms. We envisage the following coupling to be possible:
Scheme A (Our subgroup has not yet realised this coupling)
1. ABINIT + BH
2. ABINIT + GA
3. VASP + BH
• Cluster made up of single type of metallic atom, such as Ag, Fe, Cu.
• Cluster made up of single type of non-metallic atom, such as C, Si, N.
• Cluster comprised of more than one type of atom, which are all non-metallic, e.g., C-C, Si-C, N-O, O-C.
• Cluster comprised of more than one type of atom, which are all metallic (nanoalloy cluster), e.g., Au-Cu, Cu-Fe, Ag-Au.
• Cluster comprised of non-metallic and metallic atoms, e.g., Ag-C, Ni-C, C-Fe.
The computing cost for GM search depends very sensitively to the size of the nanocluster. For large size cluster, to involve DFT in the GM search can be impractical. Due to practical limitation of computing resources, we strategically divided the nanocluster calculation into two GS search schemes:
5.2. Scheme A: DFT + GM search for small size cluster
For small size cluster (presumably less than 10 atoms), DFT is coupled to GM search codes to obtain the GS structures of atomic clusters directly. The GS so obtained are supposed to be more reliable and less controversial, but the computational cost could be orders of magnitude higher than using DFTB or MD codes. The advantage is that this scheme does not require any SK file or force field, hence could be used to calculate nanocluster of any types of atoms, including all possible combination as listed in subsection 5.1. Refs. [38, 39, 40] are examples that calculate GS of small cluster via this scheme. The coupling of DFT programs with GM codes is not yet realised by us but we do have a plan to do so.
5.3. Scheme B: DFTB (or MD) + GM + DFT
For large nanocluster we means cluster with a number of total atoms as large as could possibly be handled by our computational resources and patience. DFTB+ (or LAMMPS) coupled with GM search algorithm could handle large cluster but the GS so obtained may not coincide with the GM of the PES at the DFT level.
In this scheme, we first generate a sufficiently large number of candidate GS structures at the DFTB+ or MD level. Out of these, we choose N candidate structures with lowest energy to feed into an ab initio total energy calculating program for further optimisation (via local optimisation scheme built in these DFT software). This is to drive the candidate GS structures into the GM of the DFT PES. The process can be repeated as many time as possible until no structure with lower energy could be found. The lowest energy structure at the end of this two-stage process can then be taken as the GS at the DFT level (but not without ambiguity). Search scheme B is a practical and feasible strategy to supplant scheme A when the latter becomes too expensive and impractical, though there is still a sense of in- definiteness whether the minimum so obtained can truly represent the true global minimum at the DFT level.
There exist many papers that uses such a scheme to investigate nanocluster, such as [41, 42, 43, 44]. All the types of atomic composition mentioned in subsection 5.1 can be performed using this approach. However, the main constraint of this scheme is that the relevant SK files or force fields must be available.
We note the possibility that once the GS structures of some large clsuters are obtained this way, they can be heated by using thermostat in LAMMPS or DFTB+ so that the thermal properties can be investigated. Specifically, we could calculate the melting properties as the function of the cluster composition.
Despite this scheme has already been used in the literature for GS structures of nanocluster, it is not being widely used in the search for GM of nanocluster. Many authors in the research community prefer to use the ex- pensive ab initio MD (coupled with simulated annealing, another GM search algorithm) to search for the GS of nanocluster. Some resort to experimental data as the initial guess, while others simply fed random guess structures as the initial input into a DFT program to obtain the 'GS', before deriving their physical and electronic properties.
6. Summary
6.1. Software tools
The computational software tools required for computational nanocluster could be divided into two independent aspects: total energy calculating programs and global minimisation search algorithms.
Total energy (TE) calculating programs2 can be further categorised into
1. First-principles methods. Software implementation includes: VASP, ABINIT, WIEN2k, Gaussian, Crystal.
2. Semi-empirical methods. Software implementation includes: DFTB+.
3. Molecular dynamics methods. Software implementation includes: LAMMPS.
There exist many state-of-the-arts Global Minimisation (GM) search algorithms in the research front. We list only the software implementation that are readily accessible to us in USM.
1. Basin hoping (as an independent mode of operation in the PTMBHGA code)
2. Genetic algorithm (as an independent mode of operation in the PTMBHGA code)
3. Full PTMBHGA (mixture of BS and GA)
Furthermore the GA can be operated using combination of operators, as discussed at the end of Section 4.
In additional to the GM as mentioned above, the GASP code is also available for GM search purpose. However, we have only limited experience with this code as compared to the PHMBHGA code.
6.2. Coupling of TE and GM codes
To obtain the GS of a nanocluster, the TE program has to be coupled with the GM search algorithms. We envisage the following coupling to be possible:
Scheme A (Our subgroup has not yet realised this coupling)
1. ABINIT + BH
2. ABINIT + GA
3. VASP + BH
4. VASP + GA
5. ABINIT + PTMBHGA
6. VASP + PTMBHGA
Scheme B (Our subgroup has realised this coupling)
1. (LAMMPS + BH) + DFT
2. (LAMMPS + GA) + DFT
3. (LAMMPS + PTMBHGA) + DFT
4. (DFTB+ + BH) + DFT
6. VASP + PTMBHGA
Scheme B (Our subgroup has realised this coupling)
1. (LAMMPS + BH) + DFT
2. (LAMMPS + GA) + DFT
3. (LAMMPS + PTMBHGA) + DFT
4. (DFTB+ + BH) + DFT
5. (DFTB+ + GA) + DFT
6. (DFTB+ + PTMBHGA) + DFT
7. (GASP + LAMMPS)3 + DFT
6.3. Computing cost
The actually implementation of computation route depends on how large a nanocluster is. Scheme A is most suited to locate GS for a system with small number of atoms (presumably less than ∼ 10). This scheme has the great advantage of being accurate up to the DFT benchmark with minimal ambiguity. In addition it does not suffer from the problem of unavailability of SK files or force fields. In principle nanocluster of any type or combination of atoms can be calculated, as long as the computing time is within the limit we can tolerate.
Scheme B is most suited for larger system. This scheme is a feasible strategy to circumvent the computing time bottleneck suffered by Scheme A. GS structures of large cluster can be found this way economically, effectively at relative small computing cost.
6.4. Physical, electronic and thermal properties
6. (DFTB+ + PTMBHGA) + DFT
7. (GASP + LAMMPS)3 + DFT
6.3. Computing cost
The actually implementation of computation route depends on how large a nanocluster is. Scheme A is most suited to locate GS for a system with small number of atoms (presumably less than ∼ 10). This scheme has the great advantage of being accurate up to the DFT benchmark with minimal ambiguity. In addition it does not suffer from the problem of unavailability of SK files or force fields. In principle nanocluster of any type or combination of atoms can be calculated, as long as the computing time is within the limit we can tolerate.
Scheme B is most suited for larger system. This scheme is a feasible strategy to circumvent the computing time bottleneck suffered by Scheme A. GS structures of large cluster can be found this way economically, effectively at relative small computing cost.
6.4. Physical, electronic and thermal properties
With the GS structures of the nanocluster at our disposal, the physical and electronic properties can be evaluated using DFT software. Most of the DFT software comes with built-in program to derive many electronic properties by reading in GS structures an input. The properties include, e.g., magnetic dipole moment, phonon vibrational mode, DOS, band structure, HOMO-LUMO gap, configuration of the chemical bonding, etc. Temperature- dependent physical properties, notably the melting behaviors of nanocluster and the temperature-dependence of magnetic properties can be also evaluated at affordable computing cost. By varying the sizes of the nanocluster, the computation schemes as discussed here allow us to calculate the size dependence of the physical, electronic and thermal properties at greater detail.
6.5. Conclusion
To quote Francis Crick, "If you want to study function, study structure". A working global-miminisation algorithm that can be incorporated into total energy calculation programs in parallel mode is the key weapon to dissect nanoclusters. The GS structures of nanoclusters can be probed this way despite the absence of experimental hint. Discovery of totally new properties and unexpected surprises in nanocluster could be found if the computational schemes discussed here can be realised.
Now, what remains next is: what is your pet cluster to calculate?
Footnote:
1. At the present stage our research group has not any practical experience in running a TBMD calculation. We plan to learn running TBMD using the DFTB+ package.
2. We list only the software that are readily accessible and technically familiar to us in the computational physics subgroup in the School of Physics, USM, Malaysia)
3. We have only limited experience to run computation using this combination.
References
[1] M. Schmidt, R. Kusche, B. von Issendorf, H. Haberland, Irregular variations in the melting point of size-selected atomic clusters, Nature 393 (1998) 238-240.
Now, what remains next is: what is your pet cluster to calculate?
Footnote:
1. At the present stage our research group has not any practical experience in running a TBMD calculation. We plan to learn running TBMD using the DFTB+ package.
2. We list only the software that are readily accessible and technically familiar to us in the computational physics subgroup in the School of Physics, USM, Malaysia)
3. We have only limited experience to run computation using this combination.
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