Section I ab initio electronic structure methods Part

Section I: ab initio electronic structure methods Part II: introduction to the computational methodology Instructor: Marco Buongiorno Nardelli 516 C Cox - tel. 513 -0514 - e-mail: mbnardelli@ncsu. edu Office hours: T, Th, 2: 00 -3: 00 PM http: //ermes. physics. ncsu. edu Suggested readings: • R. M. Martin, Electronic Structure, Cambridge, 2004, http: //www. electronicstructure. org • W. E. Pickett, Pseudopotential methods in condensed matter applications, Comp. Phys. Rep. 9, 115 (1989) • J. I. Gersten, F. W. Smith, The Physics and Chemistry of Materials, Wiley, 2001 • http: //www. pwscf. org Modeling from the nanoscale to the macroscale - Fall 2004

Electronic ground state • Stable structure of solids are classified on the basis of their electronic ground state, which determines the minimum energy equilibrium structure, and thus the characteristics of the bonding between the nuclei • • Closed-shell systems: rare gases and molecular crystals. They remain atom-like and tend to form close-packed solids Ionic systems: compound formed by elements of different electronegativity. Charge transfer between the elements thus stabilizes structures via the strong Coulomb (electrical) interaction between ions Covalent bonding: involves a complete change of the electronic states of the atoms with pair of electrons forming directional bonds Metals: itinerant conduction electrons spread among the ion cores. Electron “gas” as electronic glue of the system Modeling from the nanoscale to the macroscale - Fall 2004

The many-body problem • • • How do we solve for the electronic ground state? Solve a many-body problem: the study of the effects of interaction between bodies, and the behavior of a many-body system The collection of nuclei and electrons in a piece of a material is a formidable many-body problem, because of the intricate motion of the particles in the manybody system: Electronic structure methods deal with solving this formidable problem starting from the fundamental equation for a system of electrons ({ri}) and nuclei ({RI}) Modeling from the nanoscale to the macroscale - Fall 2004

Electronic structure methods • In independent electron approximations, the electronic structure problem involves the solution of a Schroedinger-like equation for each of the electrons in the system • In this formalism, the ground state energy is found populating the lowest eigenstates according to the Pauli exclusion principle Central equation in electronic structure theory. Depending on the level of approximation we find this equation all over: • Semi-empirical methods (empirical pseudopotentials, tight-binding) • Density Functional Theory • Hartree-Fock and beyond Mathematically speaking, we need to solve a generalized eigenvalue problem using efficient numerical algorithms • • Modeling from the nanoscale to the macroscale - Fall 2004

Towards Density Functional Theory • The fundamental tenet of Density Functional Theory is that the complicated many-body electronic wavefunction can be substituted by a much simpler quantity, that is the electronic density • This means that a scalar function of position, n(r), determines all the information in the many-body wavefunction for the ground state and in principle, for all excited states n(r) is a simple non-negative function subject to the particle conservation sum rule • where N is the total number of electrons in the system Modeling from the nanoscale to the macroscale - Fall 2004

Kohn and Sham ansatz • • • H-K theory is in principle exact (there are no approximations, only two elegant theorems) but impractical for any useful purposes Kohn-Sham ansatz: replace a problem with another, that is the original manybody problem with an auxiliary independent-particle model Ansatz: K-S assume that the ground state density of the original interacting system is equal to that of some chosen non-interacting system that is exactly soluble, with all the difficult part (exchange and correlation) included in some approximate functional of the density. Key steps: • Definition of the non-interacting auxiliary system • The auxiliary Hamiltonian contains the usual kinetic energy term and a local effective potential acting on the electrons Actual calculations are performed on this auxiliary Hamiltonian through the solution of the corresponding Schroedinger equation for N independent electrons Modeling from the nanoscale to the macroscale - Fall 2004

Kohn and Sham ansatz Non-interacting auxiliary particles in an effective potential Interacting electrons + real potential • The density of this auxiliary system is then: • The kinetic energy is the one for the independent particle system: • We define the classic electronic Coulomb energy (Hartree energy) as usual: Modeling from the nanoscale to the macroscale - Fall 2004

Kohn and Sham equations • Finally, we can rewrite the full H-K functional as • All many body effects of exchange and correlation are included in Exc • So far theory is still exact, provided we can find an “exact” expression for the exchange and correlation term The minimization of this functional under the particle conservation constraint leads to a set of Schroedinger-like equations • • With an explicit effective potential Modeling from the nanoscale to the macroscale - Fall 2004

Kohn and Sham equations • • The great advantage of recasting the H-K functional in the K-S form is that separating the independent particle kinetic energy and the long range Hartree terms, the remaining exchange and correlation functional can be reasonably approximated as a local or nearly local functionals of the electron density Local Density Approximation (LDA): Exc[n] is a sum of contribution from each point in space depending only upon the density at each point independent on other points where • is the exchange and correlation energy per electron. is a universal functional of the density, so must be the same as for a homogeneous electron gas of given density n The theory of the homogeneous electron gas is well established and there are exact expression (analytical or numerical) for both exchange and correlation terms Exchange as • Correlation from exact Monte Carlo calculations (Ceperley, Alder, 1980) • • Modeling from the nanoscale to the macroscale - Fall 2004

Kohn and Sham equations • • • Finally, the set of K-S equations with LDA for exchange and correlation give us a formidable theoretical tool to study ground state properties of electronic systems Set of self-consistent equations that have to be solved simultaneously until convergence is achieved Note: K-S eigenvalues and energies are interpreted as true electronic wavefunction and electronic energies (electronic states in molecules or bands in solids) Note: K-S theory is a ground-state theory and as such is supposed to work well for ground state properties or small perturbations upon them Extremely successful in predicting materials properties - golden standard in research and industry Modeling from the nanoscale to the macroscale - Fall 2004

Overview of solid-state concepts • • • In DFT calculations, one solves iteratively the Kohn-Sham equations for the electron density in the external potential of the ions We need to frame the K-S equations in a crystalline (or molecular or atomic) environment and be able to formulate the problem in a computationally tractable manner (a crystalline set-up is the most general for our purposes and for most available computer codes) A crystal is an ordered state of matter in which the position of the nuclei are repeated periodically in space Completely specified by the shape of one repeat unit (primitive unit cell) and type and position of nuclei in that unit (basis) The unit cell is repeated infinitely in space through a set of rules that describe the repetition (translations) - the Bravais lattice Crystal Points or translation vectors Modeling from the nanoscale to the macroscale - Fall 2004 Basis

Overview of solid-state concepts • Simple Bravais lattice (cubic cell): primitive cell with one atom basis: Note: A primitive cell is defined by 3 unit vectors directed along the edges of the periodic “box” a 3 a 2 • a 1 The mimimal primitive cell that can generate the whole periodic lattice is called the Wigner-Seitz cell, and depends on the particular symmetry of the crystal (lattice + basis) a 2 a 1 Wigner-Seitz Cell • Translations are defined as vectors T multiple of the primitive cell vectors: T=la 1+ma 2+na 3 Modeling from the nanoscale to the macroscale - Fall 2004

Overview of solid-state concepts • Any function defined in a crystal has to satisfy the periodicity of the system: • • This applies to all the quantities that we have introduced so far: Hamiltonians, wavefunctions, electron densities, etc. It is well known that periodic functions can be represented by Fourier transform in terms of Fourier components at specific wavevectors q. We define a Fourier transform: • For periodic functions the above expression can be written as: • In a periodic crystal, Fourier components are restricted to those that are periodic within a large volume of crystal made of Ncell: • The set of Fourier components that satisfy the above condition form the “reciprocal crystal lattice”. The reciprocal lattice is itself a Bravais lattice whose axis bi are defined through the relation Modeling from the nanoscale to the macroscale - Fall 2004

Overview of solid-state concepts • Since the reciprocal lattice is a Bravais lattice, it has a primitive cell. The minimal primitive cell of the reciprocal lattice is called “Brillouin zone” and its principal axis are given by a 2 b 2 a 1 Wigner-Seitz Cell • b 2 a 2 b 1 a 1 b 1 Brillouin Zone The reciprocal space is comprised of a lattice of points defined by the reciprocal lattice vectors G=lb 1+mb 2+nb 3 so that the Fourier transform of the periodic function can be written as Modeling from the nanoscale to the macroscale - Fall 2004

Overview of solid-state concepts • • • Electrons in solids are subject to the external potential of the nuclei that has the periodicity of the crystal Independent electrons (such as the ones of DFT in the K-S approach) in a periodic potentials have a very important property as a general consequence of the periodicity of the potential Bloch’s theorem: the eigenstates of the crystal Hamiltonian (the electrons’ wavefunctions) can be always chosen as the product of a plane wave times a function with the periodicity of the Bravais lattice with k within the primitive cell of the reciprocal lattice Note: for the crystal we have r and T, for the reciprocal space we have k and G All possible eigenstates of the Hamiltonian are specified by k within the primitive cell of the reciprocal lattice. For each k there is a descrete set of eigenvalues that form what are the energy bands of the crystal. Usually they are calculated within the Brillouin zone and along specific directions in reciprocal space For a complete discussion on reciprocal space vectors and periodic boundary conditions see the Ashcroft and Mermin book or any other book on Solid-state theory Modeling from the nanoscale to the macroscale - Fall 2004

Example • Brillouin zone and band structure for a cubic fcc solid: Au z X L G X L K S D K y X W U W X U X From: http: //cst-www. nrl. navy. mil/Electronic. Structure. Database/ Modeling from the nanoscale to the macroscale - Fall 2004

An electronic structure code: PWscf • • PWscf is a state-of-the-art software package that we will use as an introduction to the actual procedure of modeling the electronic structure of a solid http: //www. pwscf. org Modeling from the nanoscale to the macroscale - Fall 2004

An electronic structure code: PWscf Modeling from the nanoscale to the macroscale - Fall 2004

An electronic structure code: PWscf Best first step: READ THE USER MANUAL! Modeling from the nanoscale to the macroscale - Fall 2004

An electronic structure code: PWscf • Three basic preliminary steps. From the User’s manual: • Installation The PWscf package can be downloaded from the http: //www. pwscf. org site. Presently, only source files are provided. Some precompiled executables (binary files) are provided only for the GUI. Providing binaries for PWscf would require too much effort and would work only for a small number of machines anyway. Uncompress and unpack the code in an empty directory of your choice that will become the root directory of the distribution. On Linux machines, you may use: tar -xvzf pw. 2. 0. tgz. On other Unix machines: gzip -dc pw. 2. 0. tgz | tar -xvf Automatic configuration: An experimental automatic configuration, using the GNU "configure" utility, is available (thanks to Gerardo Ballabio, CINECA). From the root directory, type: . /configure The script will examine your hardware and software, generate dependencies needed by the Makefile's, produce suitable configuration files make. sys and make. rules. Presently it is expected to work for Linux PCs, IBM sp machines, SGI Origin, some HP-Compaq Alpha machines. For more details, read the INSTALL file. • • • Modeling from the nanoscale to the macroscale - Fall 2004

An electronic structure code: PWscf • • Compilation There a few adjustable parameters in Modules/parameters. f 90. The present values will work for most cases. All other variables are dynamically allocated: you do not need to recompile your code for a different system. You can compile the following codes: make pw produces PW/pw. x and PW/memory. x. pw. x calculates electronic structure, structural optimization, molecular dynamics, barriers with NEB. memory. x is an auxiliary program that checks the input of pw. x for correctness and yields a rough (under-)estimate of the required memory. make ph produces PH/ph. x calculates phonon frequencies and displacement patterns, dielectric tensors, effective charges (uses data produced by pw. x). make d 3 produces D 3/d 3. x calculates anharmonic phonon lifetimes (thirdorder derivatives of the energy), using data produced by pw. x and ph. x. make gamma produces Gamma/phcg. x is a version of ph. x that calculates phonons at = 0 using conjugate-gradient minimization of the density functional expanded to second-order. Only the ( = 0 ) point is used for Brillouin zone integration. It is faster and takes less memory than ph. x, but does not (yet) support Ultrasoft pseudopotentials. make pp produces a variety of post-processing codes make tools produces utility programs, mostly for phonon calculations Modeling from the nanoscale to the macroscale - Fall 2004

An electronic structure code: PWscf • Running Pwscf for an electronic and ionic structure calculation: • Main information: the input file Modeling from the nanoscale to the macroscale - Fall 2004

PWscf input file • • The input data is organized as several namelists, followed by other fields introduced by keywords. The namelists are • &CONTROL: general variables controlling the run • &SYSTEM: structural information on the system under investigation • &ELECTRONS: electronic variables: self-consistency, smearing • &IONS (optional): ionic variables: relaxation, dynamics • &CELL (optional): variable-cell dynamics • &PHONON (optional): information needed to produce data for phonon calculations Optional namelist may be omitted if the calculation to be performed does not require them. This depends on the value of variable calculation in namelist &CONTROL. Most variables in namelists have default values. Only the following variables in &SYSTEM MUST be specified: • ibrav (integer): bravais-lattice index • celldm (real, dimension 6): crystallographic constants • nat (integer): number of atoms in the unit cell • ntyp (integer): number of types of atoms in the unit cell • ecutwfc (real): kinetic energy cutoff (Ry) for wavefunctions. Description of all the input cards can be found in the file INPUT_PW Modeling from the nanoscale to the macroscale - Fall 2004

PWscf input file • Input file for an electronic structure calculation for a bulk Si crystal • &control: • scf = self-consistent solution of the Kohn-Sham equations (ground state energy, electron densities) from an arbitrary initial density (‘from_scratch’) • prefix = prefix for file names • tstress, tprnfor = compute also forces and stresses in the given geometry • Specify working directories (optional) Modeling from the nanoscale to the macroscale - Fall 2004

Input parameters: &system • The &system namelist is the one where we specify the geometrical parameters of our simulation cell: • ibrav = specifies the Bravais lattice type • celldm = specifies the crystallographic constants (the dimension of the simulation cell) Modeling from the nanoscale to the macroscale - Fall 2004

Input parameters: &system • • nat and ntyp specify the number of atoms in the simulation cell and how many different atomic species are in the system we want to study Together with the information contained under the keywords ATOMIC SPECIES and ATOMIC POSITIONS they determine the basis in the primitive cell • ATOMIC SPECIES: list of all the atomic species in the system with information on the atomic mass and a link to an external file that contains the information necessary to describe that particular atom - pseudopotential file • ATOMIC POSITIONS: list of all the different atoms in the simulation cell with the specification of their atomic coordinates Modeling from the nanoscale to the macroscale - Fall 2004

Input parameters: the pseudopotential Modeling from the nanoscale to the macroscale - Fall 2004

Numerical solution: plane waves • • Kohn-Sham equations are differential equations that have to be solved numerically To be tractable in a computer, the problem needs to be discretized via the introduction of a suitable representation of all the quantities involved Various discretization approeches. Most common are Plane Waves (PW) and real space grids. In periodic solids, plane waves of the form are most appropriate since they reflect the periodicity of the crystal and periodic functions can be expanded in the complete set of Fourier components through orthonormal PWs • In Fourier space, the K-S equations become • We need to compute the matrix elements of the effective Hamiltonian between plane waves Modeling from the nanoscale to the macroscale - Fall 2004

Numerical solution: plane waves • Kinetic energy becomes simply a sum over q • The effective potential is periodic and can be expressed as a sum of Fourier components in terms of reciprocal lattice vectors • • Thus, the matrix elements of the potential are non-zero only if q and q’ differ by a reciprocal lattice vector, or alternatively, q = k+Gm and q’ = k+Gm’ The Kohn-Sham equations can be then written as matrix equations • where: • We have effectively transformed a differential problem into one that we can solve using linear algebra algorithms! Modeling from the nanoscale to the macroscale - Fall 2004

Input parameters: ecutwfc • In this representation both the potentials and the Bloch functions, solution of the K-S problem, are expanded on a set of plane waves • In principle, the plane waves basis set is infinite, since I have an infinite number of reciprocal lattice vectors (or, in other words, Fourier components). In practice, we need to limit ourselves to a finite basis set for the practical solution of the linear equations: approximation! Remember: in Fourier space, that is in reciprocal space in a crystal, small q components describe long-range features (wave-length), while large q components, describe short-range features. Very sharp oscillations, for instance, need to be described by a large number of plane waves with large G vectors Increasing the dimension of the basis (number of plane waves, so larger magnitudes of Gm) allows for a better description of short-range features in either the potential or the density. ecutwfc is the parameter that controls the number of PW’s in the basis, so it affects directly the accuracy of the calculation: convergence parameter |Gmax|2 ecutwfc • • Modeling from the nanoscale to the macroscale - Fall 2004

Input parameters: &electrons • • Kohn-Sham equations are always selfconsistent equations: the effective K-S potential depends on the electron density that is the solution of the K-S equations In reciprocal space the procedure becomes: where and • Iterative solution of self-consistent equations often is a slow process if particular tricks are not used: mixing schemes Modeling from the nanoscale to the macroscale - Fall 2004

Mixing schemes • • • Key problem: updating the potential and/or the density between successive iteration steps (loop in the solution of the K-S equations) A direct approach, where we start from an arbitrary density and use the solution of the KS loop as input of the next will not work instabilities in the solution of the minimization problem - the numerical procedure does not converges into the minimum Simplest approach to resolve the issue is linear mixing: estimate an improved density input nini+1 at step I+1 as the linear combinantion of input and output densities nini and nouti at step i • mixing_mode = particular algorithm to mix input and output density/potentials • mixing_beta = numerical value of • conv_thr = convergence threshold for selfconsistency, estimated energy error < conv_thr Modeling from the nanoscale to the macroscale - Fall 2004

Mixing schemes • • is the parameter that controls the rate of convergence • Large : output density from previous iteration weighs most - fast convergence (large density updates from one step to the other). Works well for strongly bound, rigid systems (insulators or semiconductors, regions around the ionic cores) • Small : output density weighs lest - slower convergence (smaller density updates from one step to the other) but more stable. Works well for “soft” systems such as metals, alloys, surfaces or open systems (lots of empty space). Various mixing scheme beyond linear mixing have been developed and are available in PWscf (Anderson, Broyden, etc. ) Modeling from the nanoscale to the macroscale - Fall 2004

Symmetry and special points • • • Crystals are very symmetric systems whose geometrical properties are best described by the specification of their space group Space group of a crystal is the group composed by the whole set of translations and point symmetry operations (rotations, inversions, reflections and combinations of the above) that leave the system invariant (including a particular set of operations that combines a rotation with a non-integer translation, or glide, of a fraction of a crystal translation vector, called non-symmorphic operations) Any function that has the full symmetry of the crystal is invariant upon any operation of the space group: Sng(r) = g(Snr) In particular, since the Hamiltonian is invariant upon any symmetry operation, any operation Sn leads to a new equation with r Snr and k Snk The new solution of the transformed equation is still an eigenfunction of the Hamiltonian with the same eigenvalue: • A “high-symmetry” k point is defined by the identity relation: Snk k. Helpful in the classification of electronic states. • One can define the Irreducible Brillouin Zone (IBZ), which is the smallest fraction of the Brillouin Zone that is sufficient to determine all the information on the electronic structure of the crystal. All the properties for k outside the IBZ are obtainable via symmetry operations Modeling from the nanoscale to the macroscale - Fall 2004

Symmetry and special points • • • This concept becomes particularly significant when we have to compute properties that require an integration over the Brillouin Zone, such as energy or electron density. In general an integral over the BZ becomes a sum over a discrete set of states corresponding to different k points Two important issues: • To have an accurate numerical integration the discrete set of k values has to be dense enough to have a sufficient number of points in regions where the integrand varies rapidly. Crucial difference between metals and insulators. Since insulators have only filled bands, integrals can be computed using a few well-chosen points in the BZ: special points • Symmetry must be used to reduce the calculations to ones that involve only k-points comprised in the IBZ Modeling from the nanoscale to the macroscale - Fall 2004

Input parameters: K-POINTS • • Set of special points in the BZ can be chosen for an efficient integration of smooth periodic functions Most general method has been proposed by Monkhorst and Pack (implemented in PWscf). Uniform sets of points are chosen in reciprocal space according to the formula: where bi are the reciprocal space basis vectors and Ni are the parameters that determine the mesh size (ni = 1, 2, …, Ni) • • They form a uniform mesh in reciprocal space Can be specified in an automatic way by the program: nk 1, 2, 3 = N 1, 2, 3 and k 1, 2, 3 = 0 or 1 0 = centered grid, 1 = shifted grid Modeling from the nanoscale to the macroscale - Fall 2004

Input parameters: K-POINTS • • A sum over a uniform set of special k-points integrates exactly a periodic function that has Fourier components that extend only to Ni. Ti in each direction This logic can be easily understood in one dimension: the value of the following integral is given by the value of the integrand function f 2(k)= sin(k) at the mid-point k = , f 2(k= )= sin( )=0 If one has two integrand functions, f 2(k)=A 1 sin(k) +A 2 sin(2 k), the integral is given by the sum over two points For functions with the symmetry of the full crystal symmetry group, the density of the special k-point mesh gives a direct measure of the accuracy with which we compute the integrals: convergence parameter Modeling from the nanoscale to the macroscale - Fall 2004

Input parameters: K-POINTS • • • Integrals over the full Brillouin Zone (BZ) can be replaced by integrals over the first Brillouin Zone (IBZ). In this way we need to perform summations on a subset of the full k-point mesh in the BZ. NOTE: All the k points of the full BZ can be obtained from the k points in the IBZ via the symmetry operations of the crystal space group. If we define the weight factor wk as the total number of distinguishable k points related by symmetry to a k point in the IBZ divided by the total number of points Nk. With this proviso, any sum over the BZ can be written as a sum over the IBZ that includes the appropriate weight for each point: Modeling from the nanoscale to the macroscale - Fall 2004

A few notes on “convergence” • • • The accuracy of an electronic structure calculation depends on various approximation factors, both “physical” and numerical: Physical approximations: • description of nuclei/ions: all-electron vs. pseudopotential, type of pseudopotential used • choice of exchange and correlation functional (LDA vs. GGA. vs. hybrid functionals or mixed schemes (this is usually done in the pseudopotential input file) Numerical approximations: • The accuracy of the basis set: • In a plane wave basis, the extension of the reciprocal space mesh (Fourier basis) • In a real space calculation, the density of the grid in real space • The accuracy of the integrals in reciprocal space: density of the special kpoint mesh. Particularly important for metallic systems, where large sets of k -points have to be used for a consistent description of the system • The accuracy of the self-consistent solution of the K-S equations: mixing schemes and convergence threshold Modeling from the nanoscale to the macroscale - Fall 2004

Results • Calculation for a crystal of Si • Diamond structure • Equilibrium lattice parameter • FCC cell with two atoms in the basis - 48 point symmetry operation with non -symmorphic translations a 3 a 2 a 1 Modeling from the nanoscale to the macroscale - Fall 2004

Results pw. x < si. scf. in > out • Preliminaries • First part: geometry • Warning on the existence of non-symmorphic point group operations Modeling from the nanoscale to the macroscale - Fall 2004

Results • More on geometry and computational parameters: lattice parameter and volume of the cell, atoms and species, kinetic energy cutoff (ecutwfc), convergence threshold and mixing parameters, exchange and correlation functional, real and reciprocal space basis vectors Modeling from the nanoscale to the macroscale - Fall 2004

Results • Pseudopotential parameters • Symmetry operations and atomic coordinates • k-mesh for BZ integration Modeling from the nanoscale to the macroscale - Fall 2004

Results • Iterate! Modeling from the nanoscale to the macroscale - Fall 2004

Results • • • Convergence! Electronic energies (eigenvalues of the K-S equation) for the occupied bands of the solid Total energy = ground state energy = minimum of the K-S functional Individual contributions to the ground state energy (given in an alternative way, that groups together various contributions and uses the sum of the eigenvalues of the K-S equations): • band energy • 1 -electron contribution • Hartree energy • x-c energy • Ewald (ionic) energy Implicitly, we clearly know the ground state electron density Modeling from the nanoscale to the macroscale - Fall 2004

Results • • • From the knowledge of the electron density, we can compute all the ground state properties of interest Forces (in this case are zero because of symmetry) Stress (zero, or negligible, since the calculation has been done at the equilibrium lattice parameter) Modeling from the nanoscale to the macroscale - Fall 2004

Results • Collect run-time statistics and finish the run Modeling from the nanoscale to the macroscale - Fall 2004

Project • 1. 2. 3. 4. Using the input file given on your home directory on pharos. physics. ncsu. edu do the following exercises: Study the variation of the total ground state energy as a function of the convergence parameters: 1. ecutwfc = 2, 5, 8, 12, 18, 24, 32 Ry with the given k-point mesh. Discuss the results and find the converged value for Etot. 2. k-point mesh: using the converged value for ecutwfc from above, use the option {automatic} and check convergence for various grid sizes both centered and not: 1 1 1 0 0 0, 1 1 1, 2 2 2 0 0 0, 2 2 2 1 1 1, etc. Discuss the results Displace one of the atoms in the cell by a small amount along the diagonal of the cell and repeat the study of point 1 above looking at the values of the forces and stresses. Discuss the results Using the ideal geometry, do a study of total ground state energy versus lattice parameter (volume). The minimum of the curve will give you theoretical lattice parameter for Si corresponding to the pseudopotential used (GGA-PBE: Si. pbe-rrkj. UPF) Repeat the study of point 3 above using the LDA pseudopotential (Si. pzvbc. UPF). Discuss the results. Modeling from the nanoscale to the macroscale - Fall 2004