DFT – Practice Simple Molecules & Solids [based on Chapters 5 & 2, Sholl & Steckel] Input files Supercells Molecules Solids

The DFT prescription for the total energy (including geometry optimization) Guess ψ ik (r) for all the electrons Is new n(r) close to old n(r) ? Calculate total energy E(a 1,a 2,a 3,R 1,R 2,…R M ) = E elec (n(r); {a 1,a 2,a 3,R 1,R 2,…R M }) + E nucl Yes No Solve! Calculate forces on each atom, and stress in unit cell Move atoms; change unit cell shape/size Yes DONE!!! No Are forces and stresses zero? Self-consistent field (SCF) loop Geometry optimization loop

Approximations Approximation 1: finite number of k-points Approximation 2: representation of wave functions Approximation 3: pseudopotentials Approximation 4: exchange-correlation

Required input in typical DFT calculations Initial guesses for the unit cell vectors (a 1, a 2, a 3 ) and positions of all atoms (R 1, R 2, …, R M ) k-point mesh to “sample” the Brillouin zone Pseudopotential for each atom type Basis function information (e.g., plane wave cut-off energy, E cut ) Level of theory (e.g., LDA, GGA, etc.) Other details (e.g., type of optimization and algorithms, precision, whether spins have to be explicitly treated, etc.) POSCAR KPOINTS POTCAR INCAR VASP input files

The general “supercell” Initial geometry specified by the periodically repeating unit  “Supercell”, specified by 3 vectors {a 1, a 2, a 3 } –Each supercell vector specified by 3 numbers Atoms within the supercell specified by coordinates {R 1, R 2, …, R M } Supercells allow for a unified treatment of isolated atoms & molecules, defect-free & defective solids (i.e., point defects, surfaces/interfaces, and 0-d, 1-d, 2-d & 3-d systems); How? a 1 = a 1x i + a 1y j + a 1z k a 2 = a 2x i + a 2y j + a 2z k a 3 = a 3x i + a 3y j + a 3z k

More on supercells (in 2-d) Primitive cell Wigner-Seitz cell

The simple cubic “supercell” Applicable to real simple cubic systems, and molecules May be specified in terms of the lattice parameter a a 1 = ai a 2 = aj a 3 = ak

The FCC “supercell” The primitive lattice vectors are not orthogonal In the case of simple metallic systems, e.g., Cu  only one atom per primitive unit cell Again, in terms of the lattice parameter a a 1 = 0.5a(i + j) a 2 = 0.5a(j + k) a 3 = 0.5(i + k)

Diatomic molecule, A-B Only one degree of freedom  R AB If we new E(R AB ), then we can determine potential energy surface (PES) Energy R AB R0R0 Bond energy Curvature = m  2 m = m A m B /(m A +m B ) Note: slope = -force Thus, IF E(R AB ) is known, then we can trivially determine equilibrium bond length, bond energy and vibrational frequency!

Isolated diatomic molecules Lets consider a homonuclear diatomic molecule a >> d (i.e., a ~ 10 A) We need to compute energy versus bond length Note: vibrational frequency computations requires accuracy … as frequency is related to curvature in the “harmonic” part of the E vs. d curve [Chapter 5] What about k-point sampling? a 1 = ai a 2 = aj a 3 = ak d

Example – Cl 2 Energy versus bond length of Cl 2 Experiment Curvature needs to be evaluated accurately to yield frequency [Chapter 5, Table 5.1]

Bulk cubic material Only one degree of freedom  lattice parameter a, or Volume V (= a 3 ) If we new E(V), then we can determine potential energy surface (PES) Energy V V0V0 Cohesive energy Curvature = B/V 0 B = bulk modulus Note: slope = stress Thus, IF E(V) is known [equation of state], then we can trivially determine equilibrium lattice parameter, cohesive energy and bulk modulus!

FCC solids We will consider primitive cells of simple FCC systems (Cu) We will determine the equilibrium lattice parameter of Cu and bulk modulus What about k-point sampling? The proper number of k-points in each direction needs to be determined  convergence

Example – FCC Cu k-point sampling Remember: Integration over the Brillouin zone (BZ) replaced by a summation The Monkhorst- Pack scheme requires specification of 3 integers, representing the number of k-points along each of the 3 axis Converged!

Equation of State (EOS) The Harmonic Solid A harmonic or “Hooke” solid is characterized by the “equation of state” (Note: For an ideal gas: PV = NRT) Combined with the thermodynamic relations … … we get the equilibrium bulk modulus, B 0 = V 0 *d 2 E/dV 2 | V0 In other words, fitting the DFT energy vs volume results to the above EOS will yield E 0, V 0 and B 0 (as long as we are close to the minimum  harmonic limit)

A better 2 nd order EOS (“pseudoharmonic” approximation) Note that the bulk modulus is not a constant at all volumes or pressures for a harmonic solid, as B = V*d 2 E/dV 2 | V0 If we require that the bulk modulus remains a constant at all pressures or volumes, we have

The Murnaghan 3 rd order EOS If we allow for the (known) fact that the bulk modulus does depend on pressure, but that the first pressure derivative is a constant By fitting the DFT E vs. V data to the above equation, we can determine E 0, V 0, B 0 and B 0 ’

Comparison of various EOS forms Relative energy per volume vs. scaled volume, for B 0 = 1, B 0 ’ = 3.5 For accurate determination of B 0 and B 0 ’, one will need to use Murnaghan EOS –Fitting DFT energy vs. volume will yield B0 and B0’ For most practical purposes, using the harmonic fit will be sufficient (as long as we are sufficiently close to the minimum), as we will do …

Example – FCC Cu Energy versus lattice parameter Perform quadratic fit: xa 2 + ya + z Minimum in energy at a 0 = –y/(2x) = 3.64 Å (expt: 3.61 Å) Bulk modulus, B = - (16/9)*(x 2 /y) = 146 GPa (expt: 140 GPa) –Why? B = V 0 *d 2 E/dV 2 | V0 = (4/9a 0 )*d 2 E/da 2 | a0 = -(16/9)*(x 2 /y) Experiment

Bulk Silicon [Yin and Cohen, PRB 26, 5668 (1982)] Slope is transformation pressure Why? G = E + PV –TS  G =  E + P  V (  S = 0) Transformation at  G = 0  P = -  E/  V

Bulk Iron DFT-LDA gives the wrong description of bulk iron  it predicts that the non-magnetic FCC structure is the most stable state! One has to use DFT-GGA to get things right DFT-GGA also does better with magnetic moment Expt value Note: spin-polarized calculations help address magnetic phenomena Note: FM  ferro-magnetic; PM  para-magnetic (or non-magnetic)

Anomaly in metallic crystal structure – Polonium (vis a vis the Kepler conjecture!)