Geometry Optimisation with CASTEP

Transcription

1 Next: Recipe Geometry Optimisation with CASTEP Chris Eames This document (which is available for download as a pdf file) is a basic introduction to the methodology of performing geometry optimisation calculations with the CASTEP program. It is not intended as a review of the ideas behind that methodology. Before approaching this guide the reader is strongly recommended to read a relevent review article (such as [1] or [2]) if they do not have the background knowledge. The aim of this guide is to answer three questions; What structures can be geometry optimised and what types of geometry optimisation are there? What are the basic inputs to a calculation and how are they set up? What are the possible outputs of a calculation and how should they be interpreted? We begin with a discussion of what occurs during a calculation and an outline of what the main considerations should be for the user. Some example calculations are presented at the end to show how it all fits together. Recipe Inputs The.cell file and Pseudopotential libraries The Size and Shape of the Supercell The Initial Configuration of the Atoms Within the Supercell The k-point sampling grid Cell symmetry Constraints on the cell size/shape Constraints on the movement of ions

2 Atomic masses The location of the pseudopotential libraries External pressure that is to be applied to the supercell The.param file General Job Parameters Basis Set Parameters Variable Cell Calculations Electronic Minimisation Parameters Example - Bulk Silicon Outputs The.castep file The.geom file The.check file Other output files Checking the absolute convergence of our Bulk Silicon Calculation A Variable Cell calculation for Bulk Silicon. Example - A Bulk Terminated Silicon Surface References About this document

3 Next: Inputs Up: Geometry Optimisation with CASTEP Previous: Geometry Optimisation with CASTEP Recipe The essence of the calculation is for the ions and electrons in the supercell to be moved around stepwise until the forces on the atoms and the change in total energy between steps fall below some predefined convergence tolerance. The ionic positions are optimised using quasi-newton methods. For each configuration of the ionic positions the electronic configuration is optimised using the method of conjugated gradients. The flow of the calculation is thus; 1. Move ions into new positions using geometry optimisation algorithm 2. Optimise electronic configuration using conjugated gradients method 3. Compare total energy with previous configurations and check if forces within tolerance limits 4. If structure not optimised start at (1) and generate new set of ionic positions This cycle is performed until the forces fall within the tolerance limit and the energy should then be a local minimum. What types of system can this be applied to? In theory CASTEP can geometry optimise most bulk systems in a wide range of materials such as semiconductors, ceramics, metals, minerals and zeolites. It can also look at defects such as surfaces. The main restriction is the amount of computational resources available to the user. Calculations scale with the cube of the number of atoms included. More about this in the discussion of supercells. Next: Inputs Up: Geometry Optimisation with CASTEP Previous: Geometry Optimisation with CASTEP

4 Next: The.cell file and Up: Geometry Optimisation with CASTEP Previous: Recipe Inputs There are three principal inputs; the <seed>.cell file, the <seed>.param file and the pseudopotential library. Subsections The.cell file and Pseudopotential libraries The Size and Shape of the Supercell The Initial Configuration of the Atoms Within the Supercell The k-point sampling grid Cell symmetry Constraints on the cell size/shape Constraints on the movement of ions Atomic masses The location of the pseudopotential libraries External pressure that is to be applied to the supercell The.param file General Job Parameters Basis Set Parameters Variable Cell Calculations Electronic Minimisation Parameters

5 Next: The Size and Shape Up: Inputs Previous: Inputs The.cell file and Pseudopotential libraries Together these contain all of the information relevent to the material in the supercell. The supercell size and shape and the ionic positions within the supercell must be specified. The other cell parameters can be omitted and defaults will be applied in their place. Beware; it is advisable to specify most cell parameters. For example, the default Monkhorst-Pack (MP) k-point grid is unlikely to be suitable for most geometry optimisation jobs. Subsections The Size and Shape of the Supercell The Initial Configuration of the Atoms Within the Supercell The k-point sampling grid Cell symmetry Constraints on the cell size/shape Constraints on the movement of ions Atomic masses The location of the pseudopotential libraries External pressure that is to be applied to the supercell

6 Next: The Initial Configuration of Up: The.cell file and Previous: The.cell file and The Size and Shape of the Supercell There are two ways to specify this. Cartesian coordinates can be used %BLOCK LATTICE_CART units %ENDBLOCK LATTICE_CART Where, for example, is the x-component of the second lattice vector and is the z-component of the third lattice vector and so on. The supercell can also be defined in terms of the magnitudes of the lattice vectors %BLOCK LATTICE_ABC units %ENDBLOCK LATTICE_ABC, and are the magnitudes of, and and, and are the values of, and.

7 In both cases if [units] are not specified default units of Å are automatically used. However, angles must be specified in degrees

8 Next: The k-point sampling grid Up: The.cell file and Previous: The Size and Shape The Initial Configuration of the Atoms Within the Supercell Having defined the supercell we can now specify the ionic positions within it. We have two choices of how to do this. We can either specify the Cartesian coordinates of each individual atom %BLOCK POSITIONS_ABS units %ENDBLOCK POSITIONS_ABS Each ion occupies its own individual line. For ion number n the first item on the line is the relevant element symbol or atomic number which is followed by a string of three numbers which are the x, y and z Cartesian coordinates of the ion. The second way is to use fractions of, and for the three coordinates. %BLOCK POSITIONS_FRAC %ENDBLOCK POSITIONS_FRAC

9 After the atomic symbol or the atomic number there are three entries which represent the coordinates of the ion in terms of fractions, and of, and respectively. Next: The k-point sampling grid Up: The.cell file and Previous: The Size and Shape

10 Next: Cell symmetry Up: The.cell file and Previous: The Initial Configuration of The k-point sampling grid The points in k-space at which the Brillouin zone is to be sampled can be defined in one of three ways. As a list %BLOCK KPOINTS_LIST units %BLOCK KPOINTS_LIST The first three numbers on a line are the co-ordinates of a k-point as fractions of the relevent reciprocal space lattice vector. The final number is a weight for the k-point. The weights of all n k-points must add up to 1. Another way is to use a Monkhurst-Pack grid and there are two ways of doing this. The first way is to specify the grid dimensions in the three directions of the reciprocal space lattice vectors KPOINTS_MP_GRID So, for example, we could have a 2x2x2 grid or a 3x4x1 grid. The choice depends upon computational resources and whether a more detailed sampling of the electronic wave function is required in a given reciprocal space direction. A second way of requesting a Monkhurst-Pack grid is to specify a minimum grid density.

11 KPOINTS_MP_SPACING [units] Where is the maximum distance between any two k-points in the grid. The default [units] of the k- point spacing are Å. Another feature of the Monkhurst-Pack grid that the user can specify is the alignment of the grid with respect to the origin of the Brillouin zone. By including KPOINT_MP_OFFSET the Monkhurst-Pack grid can be offset from the origin of the Brillouin zone. The entries are the fractions of the reciprocal space lattice vector that give the value of the offset in the three directions. Next: Cell symmetry Up: The.cell file and Previous: The Initial Configuration of

12 Next: Constraints on the cell Up: The.cell file and Previous: The k-point sampling grid Cell symmetry If the unit cell is invariant under certain symmetry operations these can be specified in the.cell file %BLOCK SYMMETRY_OPS %ENDBLOCK SYMMETRY_OPS Each symmetry operation occupies four lines. The first three lines make a 3x3 array that represent a symmetry rotation and the next line is the translation associated with this rotation. Another way is to allow CASTEP to automatically find the highest symmetry group that applies to the structure. SYMMETRY_GENERATE Note of caution; with a non-cubic cell or with a cell in which the c vector is not along the z direction the automatic symmetry generation can sometimes fail to find all of the symmetry operations that apply to a supercell.

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14 Next: Constraints on the movement Up: The.cell file and Previous: Cell symmetry Constraints on the cell size/shape The default is to allow all of the supercell lattice vectors to vary is size and orientation. To fix the size and shape of the supercell we must include the keyword FIX_ALL_CELL : TRUE There is also the possibility of allowing only some of the supercell lattice vectors/angles to vary while keeping others fixed The BLOCK format is %BLOCK CELL_CONSTRAINTS %ENDBLOCK CELL_CONSTRAINTS The first line contains constraint information for the lattice vectors and the second line contains constraint information for the angles. First, if any entry is zero that quantity stays fixed. If two entries on a line are the same integer then they must both have the same value whatever that might be

15 Next: Atomic masses Up: The.cell file and Previous: Constraints on the cell Constraints on the movement of ions (This page was written by D. Quigley and is taken from the Molecular Dynamics in CASTEP user guide). Any combination of linear constraints can be specified in the cell file. We assign the index to run over all species present in the cell, the index to run over the ions in each species, and the index to run over the three spatial co-ordinates of an ion. The coordinate of the 2nd ion in the 3rd species is then. Associated with each of these degrees of freedom is a co-efficient. The n linear constraint can then be specified as where the constant is determined by the initial conditions. Specification of the n linear constraint therefore reduces to specification of the co-efficients. These are input into the cell file in the following fashion. %BLOCK IONIC_CONSTRAINTS 1 S j 1 S j

16 2 O j 2 O j 2 O j %ENDBLOCK IONIC_CONSTRAINTS For two constraints, one involving two sulfur ions, and another involving three oxygen ions. All coefficients not specified are assumed to be zero. The first column in the block gives a unique number to the constraint specified. The second specifies the species by either atomic symbol (S or O in the above example) or atomic number. The third column is the index within a species current constraint are then specified.. The co-efficients of the three spatial co-ordinates for this ion under the As an example, let us consider the case of restricting a single ion to move along along a plane parallel to. The normal to this plane is and our constraint is that the dot product of this normal with the position vector of the ion in question is zero. If this is the second ion in the fourth species then In this trivial example, we can see that we need (if the fourth species is say, sulfur) %BLOCK IONIC_CONSTRAINTS 1 S %ENDBLOCK IONIC_CONSTRAINTS to satisfy the above equation. A variety of constraints can be specified in this fashion. For simplicity, a special cell file keyword is provided to fix the centre of mass in the calculation. fix_com = true

17 It is possible to fix the positions of all ions and hence only conduct dynamics for the cell degrees of freedom using the following. fix_all_ions = true Similarly, it is possible to specify that all cell degrees of freedom remain fixed. fix_all_cell = true Fixing both the cell and ionic degrees of freedom in this way will eliminate all motions and will hence return an error. Non-linear constraints, such as those used to fix the distance between two ions, are not yet supported. Constraints on individual cell degrees of freedom are also not yet supported. Next: Atomic masses Up: The.cell file and Previous: Constraints on the cell

18 Next: The location of the Up: The.cell file and Previous: Constraints on the movement Atomic masses The mass of each species of ion in the calculation may be specified in the cell file. The format for the BLOCK is %BLOCK SPECIES_MASS units %ENDBLOCK SPECIES_MASS Each species is treated on a separate line. The first entry for species n is either the atomic symbol or the atomic number and the second entry is the mass, which has default units of atomic mass units. If the mass of any species is not specified then the default value of the mass will be assigned to that species

19 Next: External pressure that is Up: The.cell file and Previous: Atomic masses The location of the pseudopotential libraries In this BLOCK the user tells CASTEP where the pseudopotentials can be found %BLOCK SPECIES_POT %ENDBLOCK SPECIES_POT The species symbol/atomic number must be consistent with that in the IONIC_POSITIONS block and the SPECIES_MASS block. If the pseudopotential files are contained in a working directory then only the filename is specified. If the user is working on a shared machine such as a cluster in which the pseudopotentials are kept in a library for access by multiple users then the full directory path must be specified

20 Next: The.param file Up: The.cell file and Previous: The location of the External pressure that is to be applied to the supercell A pressure tensor is used to specify any external pressure that is to be applied to the cell %BLOCK EXTERNAL_PRESSURE units %ENDBLOCK EXTERNAL_PRESSURE Here we have, for example - the -component of the pressure and the -component of the pressure and so on. The default units are GPa. If the whole BLOCK is omitted then no external pressure is applied by default. For example we might specify %BLOCK EXTERNAL_PRESSURE atm %ENDBLOCK EXTERNAL_PRESSURE Which will apply a hydrostatic pressure of 1 atmosphere upon the supercell from all three directions

21 Next: General Job Parameters Up: Inputs Previous: External pressure that is The.param file This contains all of the parameters which the user requires within the calculation. The parameters can be specified in any order but only one parameter must be specified per line. The most fundamental parameter is the task (there are other types of CASTEP calculation besides geometry optimisation). A geometry optimisation.param file should always include the line task : GeometryOptimization Note the format keyword : value Most of the parameters (there are very many of them) can be left as defaults. The main parameters that the user may find useful to specify and why are outlined below Subsections General Job Parameters Basis Set Parameters Variable Cell Calculations Electronic Minimisation Parameters

22 Next: Basis Set Parameters Up: The.param file Previous: The.param file General Job Parameters An important first choice is which method to use for optimisation of the ionic positions. The default option is the Broyden-Fletcher-Goldfarb-Shannon (BFGS) method and this is used throughout the rest of this user guide. For fixed cell calculations there are two other options, damped molecular dynamics geom_method : damped_md and delocalised internal coordinates geom_method : delocalized The user is free to learn about these methods from the literature and try them out if they suit a particular problem. Since we will be doing variable cell calculations later on we will stick with BFGS. The verbosity of information in the output files is controlled by iprint : value value can be 0,1,2 or 3 in order of increasing verbosity with a default value of 1, which in most cases will be sufficient to understand how the calculation has progressed. iprint : 3 is for full debugging. For users who wish to know a little bit more about what occurs during a calculation it might be interesting to set iprint : 2. If a calculation is unexpectedly interrupted many hours of work might be lost. CASTEP automatically backs up the calculation after every 5 BFGS steps. The parameter num_backup_iter : value allows the backup interval to be user specified. For very slow jobs it is sensible to back up the results after every BFGS step num_backup_iter : 1

23 When a calculation that was stopped is resumed the parameter continuation : default must be included so that the calculation will read the.check file and use the last completed BFGS step as a starting point for the calculation. This feature is essential when the calculation is run on large shared computational facilities where the run time per job is restricted. It is possible to choose whether the algorithm should favour a speedier calculation at the cost of using more RAM during the calculation or whether to to slow down the calculation and make it less RAM intensive. opt_strategy : value value can be 'MEMORY' or 'SPEED'. The default is for a balance between memory usage and performance. If the calculation is at all likely to be RAM limited then MEMORY should be chosen. On RAM extensive architectures SPEED should be chosen. SPEED should also be used for most clusters, multiple processors or supercomputers because these write data temporarily to disk so as to conserve RAM which significantly hampers performance. Next: Basis Set Parameters Up: The.param file Previous: The.param file

24 Next: Variable Cell Calculations Up: The.param file Previous: General Job Parameters Basis Set Parameters The number of plane waves included in the calculation is controlled by the cutoff energy. cut_off_energy : value [units] Default [units] are ev. Increasing the value of this parameter increases the computational cost of the calculation. However, is a variational parameter and the total energy will convergence towards the variational minimum as is increased. It is crucial to repeat the calculation a few times with a fixed k- point grid but with increasing. The total final energy will asymptotically reduce to the variational minimum and when it levels off the user can be confident that they have the right value for and that the calculation will be properly converged. How do we decide what cutoff energy to use when first approaching a new system? Trial and error is not feasible for large jobs. There is a parameter which will automatically choose a cutoff energy basis_precision : value value can be coarse, medium, fine, precise or extreme. For fixed cell calculations it is suggested that the basis precision should be at least fine and for variable cell calculations it should be at least precise but the user should carefully check in both cases. Next: Variable Cell Calculations Up: The.param file Previous: General Job Parameters

25 Next: Electronic Minimisation Parameters Up: The.param file Previous: Basis Set Parameters Variable Cell Calculations A variable cell calculation is performed if we are attempting to geometry optimise a material for which the lattice parameters and atomic positions are not even approximately known. In this situation, placing constraints on the size and shape of the supercell might affect the result. A ground state structure with a large lattice constant might not be found if a small supercell was used which confined the atoms to be too close together. The solution is to perform a variable cell calculation which will allow the supercell size and shape to be optimised along with the atomic positions. A variable cell calculation is the default and will be performed as long as FIX_ALL_CELL : TRUE is not in the.cell file. A finite basis set correction [3] is used to reduce errors associated with changes in the total number of plane waves as the system changes size. As the supercell changes size the basis set associated with each k-point is altered. To adjust for this the cutoff energy could be varied to maintain the number of plane waves as a constant. However, it is generally considered [4] to be more acceptable to keep the cutoff energy constant and vary the number of plane waves. This 'graininess' in the basis set can be smoothed out with a finite basis set correction. This involves the calculation of the variation of the total energy with the logarithm of the cutoff energy, which is used as a smoothing parameter. The principle keyword that activates a finite basis set correction whenever the cell parameters change is finite_basis_corr : n The default value of n is none, meaning no finite basis set correction is performed. The user can specify the value of the smoothing parameter by setting n = manual and then including basis_de_dloge: v where v is the value of the smoothing parameter. There is also the handy option of allowing castep to automatically perform a finite basis set correction. Set finite_basis_corr : auto The correction is made by performing several total energy calculations for a given configuration of atoms but using different cutoff energies in each one. The variation in the total energy with the logarithm of the cutoff energy extrapolated from these singlepoint calculations enables the evaluation of the smoothing parameter. The user can specify.

26 finite_basis_npoints : n The number of different cutoff energies at which to evaluate the total energy and work out basis_de_dloge. The default value is 3. If an auto finite basis set correction has been performed in the first run of a job it need not be repeated at the start of the continuation run. The value of basis_de_dloge is given in the.castep file and this can be included in the.param file along with finite_basis_corr : manual Another neat trick is the use of finite basis set corrections to efficiently converge with respect to the basis set (see this section for more details about convergence with respect to the basis set). The keyword finite_basis_spacing : Allows specification of the spacing, in ev, between the finite_basis_npoints:. These are evenly spaced up to the specified cutoff energy with a default spacing of 5eV. If the user was to specify finite_basis_npoints : 7 With finite_basis_spacing : 50 and a cutoff energy of 350eV then singlepoint calculations would be performed at cutoff energies of 50, 100, 150, 200, 250, 300, 350. This would enable very efficient calculations of the convergence of the total energy with respect to the cutoff energy for a given k-point mesh. Next: Electronic Minimisation Parameters Up: The.param file Previous: Basis Set Parameters

27 Next: Example - Bulk Silicon Up: The.param file Previous: Variable Cell Calculations Electronic Minimisation Parameters The calculation may fail to converge if the electronic minimisation parameters are not suitable. First, the exchange correlation functional. The default choice is the Local Density Approximation and this will probably be suitable in most cases. If the calculation fails to converge the parameter xc_functional : value can be used to try a Generalised Gradient functional. The options are PW91 (Perdew Wang '91), PBE (Perdew Burke Ernzerhof) and RPBE (Revised Perdew Burke Ernzerhof). When satisfied that a calculation has completely converged with the LDA it is probably a good idea to rerun the whole calcualtion using a GGA functional to check that the final result is not drastically different. In general the LDA is better suited to bulk calculations and the GGA to surface calculations. There is also the option of using a non-local xc_functional (for fixed cell calculations only). The options are HF Hartree-Fock SHF Screened Hartree-Fock EXX Exact exchange LDA-X Local Density Approximation for exchange LDA-C Local Density Approximation for correlation Health warning; non-local xc functionals are very computationally expensive. A cap can be placed on the maximum number of allowed SCF cycles. The calculation is stopped if the electronic minimisation is unconverged after this many step. The keyword string is max_scf_cycles : value If a calculation has taken more than 100 SCF cycles then it probably won't converge, although it does

28 depend on the nature of the system and the method used to treat any metallic species in the system metals_method : value For insulators no special treatment is required. Including a parameter fix_occupancy : true defines the system as an insulator and metals method will be set to a default value of NONE. If the user defines fix_occupancy : false then by default a choice will be made by CASTEP for metals_method : EDFT Now with Ensemble Density Functional Theory (EDFT) the calculation is more likely to converge than with the alternative method Density Mixing (DM) and will usually require fewer SCF cycles to minimise the electronic structure for a given ionic structure. However, DM is much faster per SCF cycle than EDFT. For systems containing metallic atoms it is advisable to first try DM with max_scf_cycles : 50 and if this is failing to converge switch to EDFT. The electronic minimisation is considered to have converged when the change in the total energy from one iteration to the next remains below some tolerance value per atom for a few scf steps. The default value of the parameter elec_energy_tol : value is ev per atom and is usually suitable but it might be suitable to reduce the strictness of this tolerance limit if a calculation is consistently failing to converge. The number of iterations for which the change in the total energy must remain below elec_energy_tol is the convergence window and it can be specified by. elec_convergence_win

29 the default value is 3 but it must be at least 2. Next: Example - Bulk Silicon Up: The.param file Previous: Variable Cell Calculations

30 Next: Outputs Up: Geometry Optimisation with CASTEP Previous: Electronic Minimisation Parameters Example - Bulk Silicon At this point we introduce geometry optimisation of bulk Silicon as an example. We will use the known bulk structure as a starting point and check what structure CASTEP suggests and what the bond length is optimised to. Bulk Silicon is shown in figure 1 below. Figure 1:Bulk silicon

31 We must now choose a supercell to enclose as few atoms as possible but still reproduce the whole structure. The following configuration can be used provided we include the relevant symmetry operations in the cell file. Figure 2:The primitive cell which we will take as our supercell The complete.cell file for this supercell is included in the link. The first thing that we notice is that we are using the abc format to specify the size and shape of the lattice. The shape of our unit cell naturally suggests that we do this. The lengths of a,b and c are the same as those in the bulk lattice. Later we will do a variable cell calculation as both an example of how to set one up and to check how accurately the primitive cell shape and contents can be optimised. Note that we are using a 2x2x2 Monkhurst-Pack grid and also the inclusion of symmetry_generate to ensure that castep is aware of the symmetry operations that our supercell satisfies. The parameters file si.param is here Notice that we can include the parameter line Comment : value This will appear in the output as a title (see the next section). Having set up our input files we can now run the job. The job is small and we will execute it on a desktop

32 pc under the command prompt. To do this we type castep <seed> ie castep si2 when we are in the folder in which the castep executable and the input files are located (including the pseudopotential files). In the next section we will use the results of this calculation to discuss the output files. Next: Outputs Up: Geometry Optimisation with CASTEP Previous: Electronic Minimisation Parameters

33 Next: The.castep file Up: Geometry Optimisation with CASTEP Previous: Example - Bulk Silicon Outputs Subsections The.castep file The.geom file The.check file Other output files

34 Next: The.geom file Up: Outputs Previous: Outputs The.castep file The complete si.castep file can be found in the link. The file begins with a header which contains the details of the version of castep which was compiled into the executable. Authors and contributors are acknowledged and copyright and licensing information is summarised. The details of a publication which should be cited in any work arising from use of CASTEP are given. A full summary of all of the parameters (even those left to default) to be used in the calculation are listed in groups; general parameters, Exchange-Correlation parameters, Pseudopotential parameters, Basis Set parameters, Electronic parameters, Electronic Minimization parameters, Density Mixing, Population Analysis parameters, Geometry Optimization parameters. Notice that our request for medium basis set precision has produced the default cutoff energy of 120eV. The cell parameters follow, including a list of the k-point co-ordinates k-points For BZ Sampling MP grid size for SCF calculation is Number of kpoints used = Number Fractional coordinates Weight After symmetry and constraint information the results of the calculation begin. For the initial configuration of the atoms in the supercell the electronic minimisation begins <-- SCF SCF loop Energy Energy gain Timer <-- SCF per atom (sec) <-- SCF <-- SCF Initial E <-- SCF The iterations in the electronic structure continue until 2 consecutive iterations fall within the electron convergence window E E <-- SCF E E <-- SCF E E <-- SCF

35 E E <-- SCF E E <-- SCF E E <-- SCF E E <-- SCF E E <-- SCF <-- SCF Final energy = ev The electronic minisation for this ionic configuration is now complete. A convergence check is performed ******************************** Forces ********************************* * * * Cartesian components (ev/a) * * * * x y z * * * * Si * * Si * * * ************************************************************************* BFGS: finished iteration 0 with enthalpy= E+002 ev <-- BFGS Parameter value tolerance units OK? <-- BFGS <-- BFGS de/ion E E-005 ev No <-- BFGS F max E E-002 ev/a No <-- BFGS dr max E E-003 A Yes <-- BFGS <-- BFGS F max is the maximum allowed force on any atom and dr max is the tolerance for the change in any atomic position between iterations. Since the calculation has not converged a new set of ionic co-ordinates must be generated. A new BFGS iteration is started and a summary of the previous BFGS step is given ================================================================================ Starting BFGS iteration 1... ================================================================================ <-- min BFGS Step lambda F.delta enthalpy <-- min BFGS <-- min BFGS previous <-- min BFGS <-- min BFGS What is meant by lambda and F.delta? F.delta is the search direction in BFGS and this doesn't change during a BFGS iteration. F is the force vector. Ideally, the step length lambda should take the value that makes F.delta = 0 (to ensure conjugacy of the search directions). In practice, the value of F.delta for which the energy is lowest is chosen and so F.delta is not

36 necessarily equal to zero at the end of each BFGS step. By default a step of lambda = 1 is taken first BFGS: starting iteration 1 with trial guess (lambda= ) The electronic minimisation is performed and the total energy calculated. Now the line minimisation is performed <-- min BFGS Step lambda F.delta enthalpy <-- min BFGS <-- min BFGS previous <-- min BFGS trial step <-- min BFGS <-- min BFGS BFGS: improving iteration 1 with line minimization (lambda= ) Which results in <-- min BFGS Step lambda F.delta enthalpy <-- min BFGS <-- min BFGS previous <-- min BFGS trial step <-- min BFGS line step <-- min BFGS <-- min BFGS BFGS: finished iteration 1 with enthalpy= E+002 ev This is repeated for each BFGS step. The calculation proceeds until the end of BFGS iteration 2, after which the change in the energy falls within the convergence tolerance and BFGS: finished iteration 2 with enthalpy= E+002 ev <-- BFGS Parameter value tolerance units OK? <-- BFGS <-- BFGS de/ion E E-005 ev Yes <-- BFGS F max E E-002 ev/a Yes <-- BFGS dr max E E-003 A Yes <-- BFGS <-- BFGS BFGS: Geometry optimization completed successfully. All that is left to do is to print the final supercell configuration ================================================================================

37 BFGS: Final Configuration: ================================================================================ Cell Contents xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx x Element Atom Fractional coordinates of atoms x x Number u v w x x x x Si x x Si x xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx BFGS: Final Enthalpy = E+002 ev The bonding information is revealing Atomic Populations Species Ion s p d f Total Charge (e) ============================================================== Si Si ============================================================== Bond Population Length (A) ==================================== Si 1--Si ==================================== We can see that the bonds are sp hybridised as we might expect in Si. The bond length is Å. The accepted value for the Si-Si bulk bond length is Å. Our calculation is 8% out. More about this in a later section. Next: The.geom file Up: Outputs Previous: Outputs

38 Next: The.check file Up: Outputs Previous: The.castep file The.geom file The <seedname>.geom file contains a brief record of the main results at the end of each BFGS step. Note that everything in the.geom file is in atomic units. So, for example, the end of BFGS iteration 0 would correspond to the following output in the.geom file E E+000 <-- E E E E+000 <-- h E E E+000 <-- h E E E+000 <-- h Si E E E+000 <-- R Si E E E+000 <-- R Si E E E-003 <-- F Si E E E-003 <-- F The first line contains the iteration number. The next line is the final energy in Hartrees (1eV = Ha). This is repeated twice - why? In a finite pressure calculation it is the free energy, the enthalpy, that is minimised. This is E + PV where E is the energy, P is the pressure and v is the volume. In this case P = 0 so the energy and the enthalpy are the same. The final energy is listed first and then the enthalpy. The next three lines are the unit cell vectors in cartesian coordinates in units of Bohrs. There exist programs such as geom2xyz to convert the.geom file into a list of xyz coordinates. For each atom in the supercell the absolute position is listed and then the forces are listed (both quantities in atomic units)

39 Next: Other output files Up: Outputs Previous: The.geom file The.check file This Fortran binary file is a very detailed store of all of the information gathered during a calculation. The.check file is vital for checkpointing. When a job is resubmitted after an intentional or accidental interruption the information in the.check file is used to resume the calcultion from the end of the last successfully completed BFGS step. The parameters may be changed during the interruption. The line continuation : default must be included in the.param file to specify a restart and not a fresh calculation

40 Next: Checking the absolute convergence Up: Outputs Previous: The.check file Other output files.wvfn.nnnn this file contains a record of the wavefunction. The presence of this file allows a secondary restart mechanism. In the param file set ELEC_RESTORE_FILE To the root name..bands this file contains the electronic eigenvalues related to the individual k-points..err.nnnn this is an error file, written by every MPI process. If a job terminates unexpectedly this file will provide a ready source of diagnostic information. nnnn is the MPI rank. A job run in parallel on, say, 4 processors will return up to 4.err. files depending on which nodes the problem has occured. If the calculation is run in serial without MPI then nnnn =

41 Next: A Variable Cell calculation Up: Geometry Optimisation with CASTEP Previous: Other output files Checking the absolute convergence of our Bulk Silicon Calculation The example calculation with bulk silicon does not produce a very accurate answer. Can it be improved? There are two main sources of error in our calculation; Fundamental errors associated with the approximations used in the calculation such as the use of the LDA. Systematic errors associated with inaccuracies in the wavefunctions introduced by using a finite basis set and wavefunction sampling in reciprocal space. The latter of these must be reduced to a minimum. To show how this might be done I have repeated the calculation several times with increasing precision in the basis set and with various MP grid densities. Figure 3 below shows the total energy as a function of the basis set size for calculations with 1, 10, 28 and 60 K-Points. (See the section on variable cell calculations for a neat trick to efficiently perform many singlepoint energy calculations at a variety of cutoff energies at a fixed k-point density).

42 Figure 3:Convergence of the bulk Si calculation with respect to the basis set We can see that gamma point sampling with a single k-point produces a structure with a much higher energy than sampling with other k-point densities. Figure 4 below shows a plot of figure 3 with the 1 k- point data removed for greater clarity. Figure 4:Convergence of the basis set showing only the higher k- point densities The cutoff energy is a variational parameter and as it is increased the energy will converge asymptotically on the ground state from above. However, it is important to remember that the sampling set size is not a variational parameter. It entirely possible to increase the total final energy of the system by increasing the density of the sampling grid. If we know more information about the wavefunction it does not mean that this information will provide us with a lower energy structure. The key point about convergence with respect to the sampling grid density is that the difference in energy from one grid density to the next should be minimised. We can see from figure 4 that increasing the MP grid density from 28 to 60 k-points has a negligible effect on the total final energy. Indeed, the two curves lie on top of one another and cannot be separately resolved. Also, the total energy is converged with respect to the cutoff energy when this is above about 280eV. In summary, our fully converged basis set parameters are; 28 k-points in the reciprocal space sampling grid and a cutoff energy of 280eV or higher. Has the reduction of systematic error helped? The bond length for the calculation with the optimal set of parameters for both accurate wavefunction modelling and conservation of computational resources is Å. This is only 1.9% away from the acepted value of Å. Convergence has resulted in an improvement of 6% on the answer obtained using an

43 arbitrary set of basis set parameters. One final point on sampling grids. An important factor in dictating the required sampling grid density is the size of the supercell. A large real space supercell might require only a very sparse reciprocal space sampling grid. Our small bulk silicon unit cell requires quite a dense MP grid. In the later silicon (100) surface example the supercells contain 20 atoms but require only a 9 k-point sampling grid. Of course, the electronic complexity of the constituent atoms also plays a role in determining the required sampling grid density. Next: A Variable Cell calculation Up: Geometry Optimisation with CASTEP Previous: Other output files

44 Next: Example - A Bulk Up: Geometry Optimisation with CASTEP Previous: Checking the absolute convergence A Variable Cell calculation for Bulk Silicon. In our bulk Silicon calculation we had a rigid supercell shape and the starting atomic positions of the silicon atoms were those known to be correct. What effect will a variable cell calculation have? In the cell file FIX_ALL_CELL : FALSE In the.param file our keywords are; finite_basis_corr : 2 which ensures that an automatic finite basis set correction is made The complete output is in the link. We see that the first three steps are indeed singlepoint energy calculations Calculating finite basis set correction with 3 cut-off energies. Calculating total energy with cut-off of eV <-- SCF SCF loop Energy Energy gain Timer <-- SCF per atom (sec) <-- SCF <-- SCF Initial E <-- SCF E E <-- SCF E E <-- SCF E E <-- SCF E E <-- SCF E E <-- SCF E E <-- SCF E E <-- SCF E E <-- SCF <-- SCF Final energy = ev (not corrected for finite basis set) Calculating total energy with cut-off of eV <-- SCF SCF loop Energy Energy gain Timer <-- SCF per atom (sec) <-- SCF <-- SCF Initial E <-- SCF E E <-- SCF E E <-- SCF

45 E E <-- SCF <-- SCF Final energy = ev (not corrected for finite basis set) Calculating total energy with cut-off of eV <-- SCF SCF loop Energy Energy gain Timer <-- SCF per atom (sec) <-- SCF <-- SCF Initial E <-- SCF E E <-- SCF E E <-- SCF E E <-- SCF <-- SCF Final energy = ev (not corrected for finite basis set) For future reference: finite basis detot/dlog(ecut) = eV Total energy corrected for finite basis set = ev With the finite basis set correction in place the calculation can proceed as normal. The calculation completes and we have a silicon bond length of Å which is 0.17% out and an improvement of about 1.7% on our fixed cell calculation. We get a bonus. The calculation of the finite basis set correction provides information that can be used to estimate the bulk modulus of silicon based upon the energy changes as the cell size changes. BFGS: Final bulk modulus = E+001 GPa The known bulk modulus of silicon is 98.8 GPa at room temperature (Kittel page 59 [5]) and our zero temperature estimation is only 1 % out. A more rigorous way to calculate this quantity would be to plot energy versus cell volume for a variety of cell volumes (perhaps individually done as fixed cell calculations) and to fit a curve to this plot. The bulk modulus B can then be calculated from (1) In summary the user should become more inclined to a finite basis set correction whenever there is uncertainty about whether the final expected configuration of the supercell might be very different from the initial configuration. Next: Example - A Bulk Up: Geometry Optimisation with CASTEP Previous: Checking the absolute convergence

46 Next: References Up: Geometry Optimisation with CASTEP Previous: A Variable Cell calculation Example - A Bulk Terminated Silicon Surface When the supercell does not have full 3D periodicity the calculation must be converged with respect to other supercell parameters. Consider a surface. We will use the Si(100) surface reconstruction as an example. Figure 5 below shows the bulk terminated Si(100) surface. Figure 5:The bulk terminated Si(100) surface In one direction we have a defect; after some point there will be a complete absence of atoms, essentially out to infinity. In the opposite direction we move into the bulk and in principle we should include an infinite number of bulk layers below the top few surface layers. Both of these requirements are computationally impractical. In practice we must include just enough bulk layers so that the inner bulk atoms remain in place during the calculation and only the top surface layers reconstruct.also, only a finite amount of vacuum space above the top surface layer is practical. However, when a supercell is repeated periodically in all three spatial directions the bottom bulk-like region of one supercell borders upon the top vacuum region of the supercell below it (figure 6). If the vacuum region is not thick enough the uncompensated charges on the bottom of the one supercell and the top of the supercell under it will cause an unphysical interaction. Vertically adjacent supercells must not be able to 'see' one another.

47 Figure 6:A finite vacuum gap may cause an interaction between neighbouring atomic layers in vertically adjacent supercells Other effects are caused by uncompensated charges. The bottom layer of the supercell, which should be bulk-like, may reconstruct. To prevent this we can constrain these atoms to remain in their bulk positions. Another effect is an interaction between the top surface layer uncompensated charges and the bottom layer uncompensated charges through the intermediate layers. The supercell must contain sufficient numbers of layers of atoms to space the top and bottom layers out so as to minimise the internal interactions in the supercell. A neat trick is to use hydrogen passivation. In this the dangling bonds of the bottom bulk like layers are compensated by bonding these atoms to hydrogen atoms. This introduces extra atoms into the calculation but reduces the overall number of atoms needed because the number of spacer layers in the supercell is reduced (also, hydrogen is computationally cheap). The thickness of the vacuum gap can also be reduced when hydrogen passivation is used (remember empty space is a computational expense too). The Si(100) surface is known to have a simple 2x1 dimerised structure. The small in-plane unit cell of the reconstructed surface means that we can use a small supercell and we do not need many atoms. The following initial supercell will be sufficient (figure 7). Notice the hydrogen passivation.

48 Figure 7:A hydrogen passivated Si(100) supercell with a 7 Åvacuum gap and 9 layers of silicon. This is a crucial feature of performing geometry optimisation on surfaces. The initial surface unit cell must be large enough to enclose any expected reconstruction. A Si(111)-1x1 supercell will not reconstruct into the Si(111)-7x7 Takayanagi reconstruction! If no suggestions have been provided as to possible reconstructed periodicities by previous theoretical or experimental work then the user might have to begin with a large supercell and perform a variable cell calculation (in case of lateral relaxations). Now we must perform the abovementioned supercell convergence checks. We will use a 9 k-point MP grid with a cutoff energy of 260eV for this (we will converge our calculation with respect to basis set parameters later). Let us start with a large vacuum gap of 15Å (I know that this is entirely sufficient from experience). We will now be able to perform several calculations with 7, 8, 9, 10 and 11 layers of silicon and not have to worry about the vacuum gap - one problem at a time. The supercell lattice matrix and the atomic coordinate list in the.cell file for 7 layers of silicon looks like %BLOCK LATTICE_CART

49 %ENDBLOCK LATTICE_CART %BLOCK POSITIONS_FRAC H H Si Si Si Si Si Si Si Si Si Si Si Si Si Si %ENDBLOCK POSITIONS_FRAC Notice the hydrogen atoms. Now for the constraints on the bottom layer silicon atoms and the hydrogen atoms (the H atoms must also be constrained because the have a tendency to move into awkward formations). The list of constraints whose format was specified before looks like %BLOCK IONIC_CONSTRAINTS 1 H H H H H H Si Si Si Si Si Si %ENDBLOCK IONIC_CONSTRAINTS So now we have our supercells ready to run. The calculations were performed on 9 nodes of a Beowulf cluster. The total final energy per supercell atom as a function of the number of layers of silicon is shown

50 in figure 8 below Figure 8:The total final energy per supercell atom as a function of the number of layers of silicon. We can see that the energy per atom is decreasing almost linearly with the number of layers. We would need to add very many layers for the energy to not vary by the inclusion of more layers. We will use 9 layers of silicon in future calculations. With 9 silicon layers the vacuum gap was varied from 3 Å to 15 Å in 2 Å steps. The total final energy as a function of vacuum gap separation is shown below. Figure 9:The total final energy as a function of the vacuum gap

51 thickness We can see that the total energy is converged with respect to the vacuum gap when this has a thickness greater than about 7 Å (there is a slight upwards kink after this). We will use a 9 Å vacuum gap in all future calculations. Now it remains to converge the basis set paramters. Cutoff energies of 140, 180, 230, 260, 290, 320, 350, and 370eV were used with MP grids containing 4, 8 and 9 k-points. The calculations were performed on a Beowulf cluster using 8 nodes for the 4 and 8 k-point calculations and 9 nodes for the 9 k-point calculation. The convergence of the total energy with basis set parameters is shown below. Figure 10:Convergence of the total energy with respect to the basis set parameters We can see that using 9 layers of silicon, a 9 Å vacuum gap, 9 k-points and a 370 ev cutoff energy will ensure minimisation of systematic errors with respect to both supercell aperiodicity and the basis set repectively. So what results do we get? Click on the movie below to see the evolution of the atomic positions as the calculation progresses. Please use the 'back' button to close the movie.

52 We can see that the result is an initial vertical relaxation followed by asymmetric dimerisation of the top layer silicon atoms. Asymmetric dimerisation has been seen experimentally [6] and has been confirmed by detailed theoretical calculations [7]. The bond lengths are shown in figure 11 below (the numbers in brackets are taken from reference [7]). Figure 11:Principle bond lengths compared with the results of other theoretical work (in brackets).