Basics of Electronic-Structure Theory

Transcription

1 A guideline through the tutorial For each exercise in this tutorial, a separate directory should be created in your home directory in which the input files are generated and the calculation is started, like e.g. mkdir tutorial1/n2 mkdir tutorial1/si3... For every exercise, a directory is specified at the top of the description which contains reference files for all required input files as well as a reference output to check your results. However, we strongly recommend to use the provided control.in only in case of time problems and instead to try to generate the input files on your own. All references files are highlighted in italics throughout this script. Note: To avoid confusion, do not touch the directories including the reference files! Instead do all the calculations in your home directory with the created subdirectories as described above. In case you need the reference files and edit them, copy them into your home directory. Several scripts are required for differerent tasks. All scripts that are needed for this tutorial can be found in - /usr/local/aimsfiles/scripts Additional programs are installed to visualize results, like the basic geometry of a structure, the vibrational modes, charge density plots, binding curve etc. These programs are molden vmd jmol gdis As plotting tools, you can use xmgrace gnuplot Several editors are provided, such as emacs vi kwrite Basics of Electronic-Structure Theory How to use FHI-aims In this section, the most fundamental aspects of how to run an FHI-aims calculation are covered. For a detailed description please refer to the manual provided with the conference material. Furthermore, we present the most important tools that will be used throughout the tutorials as well as a guideline for the present exercises. Some fundamental aspects about FHI-aims For the tutorials a binary of FHI-aims will be provided on your workstation. A typical calculation is done in a separate directory which contains the two mandatory input files control.in and geometry.in. FHI-aims is then simply called in this directory by - mpirun -np 2 aims.workshop.mpi.x tee calculation.out which starts an mpi calculation on two processors and pipes the main output to the file calculation.out. This file contains the basic information such as the total energy, atomic forces, etc.. Additional output files might be generated according to the specified settings. The basic input files geometry.in contains the initial system geometry, and any information which is specifically tied to certain atoms. All other runtime settings are included in control.in. We note that the input units in both files for energies and lengths are generally ev and Å, respectively. Internally, these are converted to atomic units (Hartree for energies and bohr radii for lengths), using the 2002 National Institute of Standards CODATA recommended values. The order of lines is irrelevant, execept that information specific to a given atom must follow after the line specifying that atom and before any following atom is specified. The control.in typically consists of a general part, where, again, the particular order of lines is unimportant. In addition, this file contains species subtags that are referenced by geometry.in. Within the description of a given species, the order of lines is again unimportant, but all information concerning the same species must follow the initial species tag in one block. Lines beginning with a # symbol are treated as comments, and empty lines are ignored. FHI-aims provides preconstructed default definitions for the important subkeywords associated with different species (chemical elements) from Z=1-102 (H-Md). These can be found in the directory - /usr/local/aimsfiles/species defaults and are built for inclusion into a control.in file after the general part by simple copy-paste. We provide two default levels of accuracy. tight Tightly converged settings, the basis set might still need to be increased for light elements, though. light For fast prerelaxations. Any of these settings can always be further tightened by modifying them by hand in the species section. A summary of the most important convergence settings can be found in the manual in chapter 2.3.

2 Check the output of the atomic calculation. Is Hund s rule fulfilled? The multiplicity (J = 2S + 1) you obtain should be 4. (N.pw-lda.out.ref) Plot the binding energy as a function of the bond distance (using e.g. xmgrace). Redo the upper exercise with the pbe xc functional. Compare the (roughly) obtained equilibrium bond distance and binding energies to the experimental value of 9.8 ev and 1.1 Å [4]. Do you get the expected result for pw-lda and pbe? (N.pbe.out.ref) Now let s check the most important parameter to converge: the size of the basis set. Take the equilibrium bond distance (which should be 1.1 Å) and recalculate the binding energy for the basis sets minimal, tier1, tier2, tier3 using the pw-lda-functional (still using light settings for all other parameters). This is done by simply uncommenting/commenting basis functions near the end of the species defaults. Recalculate the binding energy with the tight settings. Have a closer look at the differences in the parameters, like the grid settings for instance. How much do the results change? To separate the effect of the basis set, compare the results with the light settings using the same basis set as specified in the tight settings. The nitrogen dimer Problem I: Getting started - N 2 Directory: /usr/local/aimsfiles/tutorial1/n2 /usr/local/aimsfiles/tutorial1/n2/reference output Motivation: In this exercise, the basics of FHI-aims will be conveyed, that are the generation of the input files the discussion of the output file the different functionals available in FHI-aims basis set convergence technical parameters to be aware of Generate a simple geometry.in file by hand for a nitrogen dimer with an initial (too short) bond distance of 0.8 Å(geometry.in.N2.ref). Generate a control.in file for nitrogen using light species defaults and the following settings for the physical model and the smearing (control.in.pw-lda.light.ref). For the general part, you can use the provided template (control.in.basic). xc pw-lda spin none occupation type gaussian 0.01 KS method lapack empty states 5 The self-consistency accuracy can be chosen throughout the whole tutorial as sc accuracy rho 1E-5 sc accuracy eev 1E-3 sc accuracy etot 1E-6 sc iter limit 200 Paste the species defaults in the control.in file. /usr/local/aimsfiles/species defaults/light/07 N default Now, run FHI-aims for this bond distance and have a look at the final total energy after self-consistency is achieved. Redo the calculation for the bond distances (1.0, 1.1, 1.2, 1.5 Å) and plot the binding curve (e.g. using xmgrace). Note: Since you are dealing with clusters, you will always consider the total energy and not the back-extrapolated total energy for T->0! To get the binding energy, you need the atomic reference energy. Therefore create a geometry.in filefortheatom.(geometry.in.atom.ref) The atom needs to be calculated spin-polarized. Therefore, add to the general part of the control.in file which does an unconstrained spin calculation. Furthermore, the cutoff radius needs to be enhanced for a single atom since there are no overlapping basis functions of the neighbouring atoms. cut pot

3 Now, open the Si3.prerelax.irc file with molden. molden -m Si3.prerelax.irc Open the file Si3.prerelax.out and copy the final atomic structure into a new geometry.in file as initial geometry for the ensuing post-relaxation. (geometry.in.si3.prerelaxed) Generate a control.in file with the upper specifications but with tight species defaults for Si. (control.in.si3.pbe.tight, control.in.si3.pw-lda.tight) Post-relax the structure for both the pw-lda and the pbe functional and have a look at the corresponding structures using molden. (Si3.postrelax.pbe.out, Si3.postrelax.pw-lda.out) How much does the structure change going from light to tight settings? What are the differences in the structure between the two functionals? estimated cpu time: 5 min Problem III: Vibrational analysis (harmonic) Directory: /usr/local/aimsfiles/tutorial1/si3/vib pw-lda, /usr/local/aimsfiles/tutorial1/si3/vib pbe The harmonic vibrational analysis is done using finite differences by the provided script aims.vibrations.pl. This script has to be called in the working directory which contains the control.in and geometry.in files. Create subdirectories (e.g. vib pw-lda and vib pbe) for the corresponding calculations and copy the corresponding control.in files that you have used for the post-relaxation. (control.in.si3.pbe.tight, control.in.si3.pw-lda.tight) Create the corresponding geometry.in files like in the procedure above by copying and pasting the finalized atomic structure from the output files of the post-relaxation. (geometry.in.si3.postrelaxed.pw-lda, geometry.in.si3.postrelaxed.pbe) The same control.in files from the post-relaxationscan be used. (control.in.si3.pbe.postrelax, control.in.si3.pw-lda.postrelax) In case of a vibrational analysis, the script does always a preceding local structural relaxation to ensure the structure to be in a local minimum. In your case, this should be done in one geometry step since you use the post-relaxed structure as a starting geometry. Invoke the vibrational analysis by aims.vibrations.pl jobname where you should provide a reasonable jobname (e.g. Si3 pw-lda, or Si3 pbe respectively). In the working directory, the main output is jobname.xyz which can be opened with jmol to visualize the frequencies and the corresponding eigenmodes and IR-frequencies. Alternatively, the output file jobname.mol can be used to visualize the vibrational spectrum with molden. AlltheFHI-aims output-files, that have been created, are summarized in the jobname.tar.gz file, which can be opened with tar -xzvf jobname.tar.gz Regarding the interpretation of the eigenvalues in the output Si3 pbe.vib.out, thevalues of the first six are compared with the largest one and should be zero (the three translational and vibrational degrees of freedom). Since SI-units are used at this point (the actual frequencies are given in wavenumbers), these values are very large (> O(20)). Due to numerical uncertainties, the six smallest ones are not exactly zero but several orders of magnitude smaller than the largest one. What do the frequencies tell you about the stability of the isomer? estimated cpu time: 15 min The silicon trimer Problem II: Local structural relaxation Motivation: In this exercise, the basic procedure for a local structural relaxation is presented: First, pre-relax a given initial structure with light settings using the pw-lda functional. Then, further post-relax the structure using tight settings and the desired exchangecorrelation functional. Directory: /usr/local/aimsfiles/tutorial1/si3 Generate a geometry.in file containing a random structure of three silicon atoms using the provided random structure generator create geometry.x: create geometry.x d min D Si 3 > geometry.in (geometry.in.si3 random) which creates a random structure with each atom having coordinates between { D...D} and not being closer to each other than d min. The origin of the coordinate system is set to the centroid of the cluster. Provide reasonable ranges d min=0.5 Å, D=1.0 Å. create geometry.x Si 3 > geometry.in (geometry.in.si3 random) Note: Type 1.0 and not 1 as d min! Generate a control.in file using the following settings (control.in.si3.prerelax) xc pw-lda occupation type gaussian 0.1 spin mix param 0.2 relax geometry bfgs 1d-2 and the light species defaults for Si. For that, paste /usr/local/aimsfiles/species defaults/light/14 Si default into the control.in file. Throughout this tutorial, the self-consistency accuracy for the forces can be set to sc accuracy forces 1E-4 Invoke the FHI-aims calculation and pipe the output to a file, e.g. mpirun -np 2 aims.workshop.mpi.x tee Si3.prerelax.out Si3.prerelax.out Visualize the local structural relaxation with molden. Therefore, you first have to generate a molden file by the provided script create relax movie.pl create relax movie.pl Si3.prerelax.out > Si3.prerelax.irc Si3.prerelax.irc

4 Figure 2: Isosurface (0.5 Å 3 ) of the charge density distribution for a silicon trimer with pw-lda. X (+) Si 3,X=Sc,Ti,V,Cr Problem V: Predicting the structure of small molecules Directory: /usr/local/aimsfiles/tutorial1/x(+)si3/xsi3.pw-lda, /usr/local/aimsfiles/tutorial1/x(+)si3/xsi3.pbe, /usr/local/aimsfiles/tutorial1/x(+)si3/x+si3.pw-lda, /usr/local/aimsfiles/tutorial1/x(+)si3/x+si3.pbe Motivation: This exercise gives a simple example of how to determine cluster geometries. In this simple approach, atoms are placed randomly (like in the case of Si 3) and locally relaxed. This is done multiple times, thereby generating cluster geometries without putting any chemical intuition into the starting geometry. The identified ground-state geometries are then analyzed and compared. More sophisticated methods that are important for larger systems like Basin-Hopping [1, 2] or Genetic Algorithms [3] are more or less based upon this principle but differ in the way to generate trial geometries. A structure search is done for the four-atomic systems X (+) Si 3,withX=Sc,Ti,Vand Cr. Each system should be treated both cationic and neutral for both the pw-lda and the pbe functional, so all in all sixteen cases. Note: Each group will only take care of ONE of these cases, where three groups will share one system, thus tripling the amount of trial structures for each case. The distribution of the systems will be coordinated by the tutor. Create a new working directory for your testsystem. With the already known tool create geometry.x, five random structures should be generated like - create geometry.x d min D Si 3 Sc 1 > geometry.in.struct1 which for instance generates a four-atomic geometry containing three silicon atoms and one scandium atom with minimum bond distance d min=0.5 ÅandD=1.0 Å. Generate a control.in file using light species defaults for the corresponding species. Use the specific control.in file settings: (control.in.prerelax.ref) charge 1.0 (for cationic systems) charge 0.0 (for neutral systems) spin mix param 1.0 relax geometry bfgs 1d-2 Problem IV: Charge density plots Figure 1: IR-spectra for the silicon trimer using tight. Directory: /usr/local/aimsfiles/tutorial1/si3/vib pw-lda, /usr/local/aimsfiles/tutorial1/si3/vib pbe The cube files for the charge density plots as well as for the HOMO can be generated as post-processing in the same sub-directories where the vibrational analysis has been done. The restart.dat files can be used to reinitialize the densities by adding to the control.in file. (control.in.si3.pw-lda.cube.out, control.in.si3.pbe.cube.out) restart read only restart.dat output cube total density cube origin cube edge cube edge cube edge output cube eigenstate density i max cube state σ where i max is the index of the HOMO to plot the eigenstate density of the HOMO that you can get by checking the eigenvalue spectrum of the post-relaxed structures. σ denotes the corresponding spin channel (1 ˆ=spin-up, 2 ˆ=spin-down). total density plots the total charge density distribution of the system. Running the calculation again will generate the cube files for the total density and for the HOMO state. Depending upon which spin-channel contains the HOMO, your cube-file of interest is eigenstate density i max spin σ.cube Visualize the cube files with molden or jmol. estimated cpu time: 2 min

5 References [1] D.J. Wales and H.A. Scheraga, Science 285, 1368 (1999). [2] D.J. Wales, J.P.K. Doye, M.A. Miller, P.N. Mortenson, and T.R. Walsh, Adv. Chem. Phys. 115, 1 (2000). [3] D.M. Deaven and K.M. Ho, Phys. Rev. Lett. 75, 288 (1995). [4] K. Huber and G. Herzberg, Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules (Van Nostrand, Princeton, 1979). occupation type gaussian 0.2 restart write only restart.dat Give the control.in file an appendix like control.in.prerelax mv control.in control.in.prerelax The provided script combi run.pl which is simply called in the working directory runs all combinations of control.in and geometry.in files and pipes the output to aims.struct1.prerelax for instance. The corresponding restart.dat file is copied to restart.struct1.prerelax.dat so that it can later be used for post-processing. Generate all the relaxation files with create relax movie.pl and have a look at the geometries. Compare results with the partner groups and generate a small table with the energy differences between the different isomers you found. Generate the total density plot and the HOMO eigenstate density plot for the post-relaxed ground-state structure using the restart file. Additionally, have a look at the density of states. Use the following control.in settings (control.in.cube.out.ref) occupation type gaussian 0.05 restart read only restartfile output dos output cube total density cube origin cube edge cube edge cube edge output cube eigenstate density i max cube state σ where again i max corresponds to the index of the HOMO eigenstate and σ to the spin channel (1 ˆ=spin-up, 2 ˆ=spin-down). Use as restartfile the corresponding file created in the prerelaxation, e.g. restart read only restart.struct1.prerelax.dat The density of states is printed to the ASCII-files KS DOS total raw.dat and KS DOS total.dat. The second file contains the dos shifted with the Fermi energy as reference energy while the first one contains the original dos. Printed is the energy range ev using 5000 points and a gaussian smearing of 0.01 ev to obtain sharp peaks for the Kohn-Sham eigenvalues. The electronic smearing is now reduced to 0.05 ev since you start from a reasonable density which facilitates the scf-convergence. Note: In the reference directories of the systems, all structure files structurei.xyz are given in the energetic order of the isomers. Additionally, the corresponding geometry.in.i files are provided that can be used for any kind of post-processing. For the identified ground-state structure, the dos, a cube file of the total density as well as of the HOMO are provided. The energetic differences and spin states are summarized in isomers.dat. Compare your results with those based upon the other xc functional. What are the differences in the geometries and energetic differences? Compare your results with those with a different charge state. Differences? Do the structural motifs or the energetic order change due to the different charge? Plot the dos for both spin channels using the shifted dos. Compare the different results obtained for the individual cases. Try to understand where the overall peaks come from and why they change going to the right in the row of the 3d-metals. Postrelax the ground-state structure with tight settings and a tighter force convergence criterium. (control.in.tight.ref) relax geometry bfgs 1.d-3 Does anything change significantly? Check the stability of the isomer explicitly by doing a vibrational analysis as explained in Problem III. estimated cpu time: 30 min