Self Consistent Cycle

Transcription

1 Self Consistent Cycle

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3 Step 0 : defining your system namelist SYSTEM

4 How to specify the System All periodic systems can be specified by a Bravais Lattice and and atomic basis

5 How to specify the Bravais Lattice / Unit cell Input parameter ibrav - gives the type of Bravais lattice (SC, FCC, HEX,...) Input parameters {celldm(i)} - give the lengths (& directions if needed) of the BL vectors Note that one can choose a non-primitive unit cell (e.g., 4 atom SC cell for FCC structure).

6 Atoms inside the unit cell: How many, where? Input parameter nat - Number of atoms in the unit cell Input parameter ntyp - Number of types of atoms Field ATOMIC_POSITIONS - Initial positions of atoms (may vary when relax done). - Can choose to give in units of lattice vectors ( crystal ) or in Cartesian units ( alat or bohr or angstrom )

7 Step 1 : obtaining V_ext

8 Nuclear potential Electrons experience experience a Coulomb potential due to the nuclei This has a known simple form But this leads to computational problems!

9 Problems for Plane-Wave basis Core wavefunctions: Sharply peaked close to nuclei due to deep Coulomb potential. Valence wavefunctions: Lots of wiggles near nuclei due to orthogonality to core wavefunctions High Fourier component are present i.e. large kinetic energy cutoff needed

10 Solutions for Plane-Wave basis Core wavefunctions: Sharply peaked close to nuclei due to deep Coulomb potential. Valence wavefunctions: Lots of wiggles near nuclei due to orthogonality to core wavefunctions Don't solve for core wavefunction Remove wiggles from valence wavefunctions Replace hard Coulomb potential by smooth PseudoPotentials

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12 Pseudopotentials for Quantum ESPRESSO Go to

13 Pseudopotentials for Quantum ESPRESSO Click on the element for which the PP is desired

14 Pseudopotentials for Quantum ESPRESSO Pseudopotential's name gives Information about -exchange correlation functional -type of pseudopotential

15 Atomic and V_ion info for QE Input card ATOMIC_SPECIES example: ATOMIC_SPECIES Ba Ba.pbe-nsp-van.UPF Ti Ti.pbe-sp-van_ak.UPF O O.pbe-van_ak.UPF NOTE should use the same XC functional for all pseudopentials. ecutwfc, ecutrho depend on type of pseudopotentials used (should test for system & property of interest).

16 Step 2 : initial guess for rho_in

17 Initial choice of rho_in Various possible choices, e.g.,: Superpositions of atomic densities. Converged n(r) from a closely related calculation (e.g., one where ionic positions slightly different). Approximate n(r), e.g., from solving problem in a smaller/different basis. Random numbers.

18 Initial choice of rho_in Various possible choices, e.g.,: Superpositions of atomic densities. Converged n(r) from a closely related calculation (e.g., one where ionic positions slightly different). Approximate n(r), e.g., from solving problem in a smaller/different basis. Random numbers. Initial guess of wfc Input parameter startingwfc 'atomic' 'atomic+random' 'random' 'file'

19 Step 3 : compute VH & Vxc

20 Hartree potential Hartree potential is computed from the Poisson equation which is diagonal in reciprocal space. VH (G) = 4 π e^2 rho(g) / G^2 The divergent G=0 term cancel out (for neutral systems) with analogous terms present in the ion-ion and electronion interactions. In charged systems a compensating uniform background is assumed.

21 Exchange-Correlation potential - Local Density Approximation - Generalized Gradient Approximation

22 Step 4 : diagonalization

23 Kohn Sham equation in a PW basis Eigenvalue equation is Matrix elements are V_ion contains the NON-LOCAL pseudopotential terms

24 Exact diagonalization is expensive find eigenvalues & eigenfunctions of H k+g,k+g Typically, NPW > 100 x number of atoms in unit cell. Expensive to store H matrix: NPW^2 elements to be stored Expensive (CPU time) to diagonalize matrix exactly, ~ NPW^3 operations required. Note, NPW >> Nb = number of bands required = Ne/2 or a little more (for metals). So ok to determine just lowest few eigenvalues.

25 Iterative diagonalization Can recast diagonalization as a minimization problem. Then use well-established techniques for iterative minimization by searching in the space of solutions, e.g., Conjugate Gradient. Another popular iterative diagonalization technique is the Davidson algorithm. Input parameter diagonalization algorithm used for iterative diagonalization Input parameter nbnd how many eigenvalues to compute

26 Iterative diagonalization Explicit storage of the hamiltonian matrix is not needed The only requirement is a routine able to compute H * Psi * Dual space formalism * apply kinetic energy in reciprocal space apply local potentials in real space compute dot products wherever is more effcient move between spaces by FFT

27 Iterative diagonalization Can recast diagonalization as a minimization problem. Then use well-established techniques for iterative minimization by searching in the space of solutions, e.g., Conjugate Gradient. Another popular iterative diagonalization technique is the Davidson algorithm. Input parameter diagonalization algorithm used for iterative diagonalization Input parameter nbnd how many eigenvalues to compute

28 Step 5 : new charge density

29 Brilloun Zone Sums Many quantities (e.g., n, Etot) involve sums over k. - In principle, need infinite number of k s. - In practice, sum over a finite number: BZ Sampling. - Number needed depends on band structure. - Typically need more k s for metals. - Need to test convergence wrt k-point sampling.

30 Types of K-points used Special Points: [Chadi & Cohen] Points designed to give quick convergence for particular crystal structures. Monkhorst-Pack grids: Equally spaced mesh in reciprocal space. May be centred on origin [ non-shifted ] or not [ shifted ] K_POINTS {tpiba crystal automatic gamma} If 'automatic' use M-P grids nk1, nk2, nk3, ik1, ik2, ik shift

31 Step 6 : test for convergence

32 How to decide if converged? Check for self-consistency. Could compare: New and old wavefunctions / charge densities. New and old total energies. Compare with energy estimated using Harris-Foulkes Input parameter conv_thr typically ok to use 1.d-8 Input parameter electron_maxstep maximum number of scf steps performed

33 Step 7 : mixing

34 Mixing Once iteration n of the self-consistent cycle has completed how to get next guess for rho? direct iteration in which rho_out is fed directly in rho_in rho_in(n) rho_outt(n) rho_in(n+1) usually doesn t converge. One needs to mix, take some combination of input and output densities (may include information from several previous iterations). Goal is to achieve self consistency (rho_out=rho_in) in as few iterations as possible.

35 Mixing Simplest prescription: linear mixing rho_in(n+1) = beta * rho_out(n) + (1-beta) rho_in(n). Usually slow but should converge for small enough values of beta There exist more sophisticated prescriptions (Broyden mixing, modified Broyden mixing of various kinds ) based on Quasi Newton Raphson methods. Input parameter mixing_mode plain TF local-tf Input parameter mixing_beta -Typical values between 0.1 & 0.7 (depend on type of system)

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38 The end : convergence achieved

39 Output quantities Converged diagonalization eigenvaules and eigenvectors This allows to compute - Ground state charge density: ρ(r) = Σ Ψ ^2 - Ground state energy: sum of the eigenvalues, corrected for double counting of Hartree and XC term + ion-ion int.... can be used to get structures, heats of formation, adsorption energies, diffusion barriers, activation energies, elastic moduli, vibrational frequencies... to be continued