Multi-Scale Modeling from First Principles

Transcription

1 m mm Multi-Scale Modeling from First Principles μm nm m mm μm nm space space Predictive modeling and simulations must address all time and Continuum Equations, densityfunctional space scales Rate Equations and Finite Element theory Modeling (and beyond) (Introduction to Electronic Structure Theory) ab initio ab initio kinetic Monte Carlo Molecular ElectronicDynamics Structure Theory One long MD run is not (always) useful; time after a few hops the memory about the past is f s lost. p Many s MD n s runs with μ s different m s starting s hours years conditions give a better impression about possibly relevant processes. densityfunctional theory (and beyond) ab initio Molecular ElectronicDynamics Structure Theory Predictive modeling and simulations must address all time and Continuum Equations, space scales Rate Equations and Finite Element Modeling ab initio kinetic Monte Carlo time +Friday f s p s n s μ s m s s hours years Now! the base 1

2 Modeling Materials and Bio-Molecular Properties and Functions: The Many-Body Schrödinger Equation With: 1,??? We know the operators and the inter- actions. We can write them down. No open question here! Born-Oppenheimer Approximation Where Φ ν are solutions of the electronic Hamiltonian : ({r k }) ({r k }) = ({r k }) frequently (commonly) applied approximations: neglect non-adiabatic coupling (terms of order m/m I ) keep onlyλ 0 the dynamics of electrons and nuclei decouple 2

3 Some Limits of the Born-Oppenheimer Approximation Itd does not account tfor correlated ltddynamics of fions and electrons. For example: - polaron-induced superconductivity - dynamical Jahn-Teller effect - some phenomena of diffusion in solids - non-adiabaticity in molecule-surface scattering - etc. Wave-Function Theories <Φ H e Φ> Restrict the study to a selected subclass of functions Φ (Hartree and Hartree-Fock theory); or Quantum Monte Carlo. 3

4 The Hohenberg-Kohn Theorem (1964) n(r) = n[φ] N = <Φ δ(r r i ) Φ> i The set of non-degenerate ground state wave functions Φ of arbitrary N-electron Hamiltonians. The set of particle densities n(r) belonging to nondegenerate ground states of the N-electron problem. The dashed arrow is not possible Comparison of Wave-Function and Density-Functional theory 4

5 Comparison of Wave-Function and Density-Functional theory Summary of Hohenberg-Kohn Density-Functional Theory (DFT) The many-body Hamiltonian determines everything. (standard quantum mechanics) -- There is a one-to-one correspondence between the ground-state wave function and the many-body Hamiltonian [or the nuclear (or ionic) potential, υ(r)]. (standard quantum mechanics) -- There is a one-to-one correspondence between the ground-state electron-density and the groundstate wave function. (Hohenberg and Kohn) 5

6 The Kohn-Sham Ansatz -- Kohn-Sham (1965) Replace the original manybody problem with an independent electron problem that can be solved! -- Only the ground state density and the ground state energy are required to be the same as in the original many-body problem. -- Maybe the exact tee xc [n] functional cannot be written as a closed mathematical expression. Maybe there is a detour similar to that taken for T s [n]? The challenge is to find useful approximate xc functionals. T s, E Hartree, and E xc are all universal functionals in n(r), i.e., they are independent of the special system studied. (general theory: see the work by Levy and Lieb) ϵ xc-jellium (n) E xc-lda = ϵ xc-jellium (n) n(r) d 3 r Ceperley and Alder (1980) neglecting is the localdensity approximation n 6

7 The Exchange-Correlation Hole Hatree Hartree- Fock DFT (exact) Comparison of Hartree, Hartree-Fock, and density-functional theories for jellium For non-jellium systems and the LDA (or the GGA) the shape of n xc (r, r ) is incorrect. However, only its spherical average enters: n xc (r, r ) E xc [n] = Exchange-Correlation Hole in Silicon R. Q. Hood, M. Y. Chou, A. J. Williamson, G. Rajagopal, and R. J. Needs, PRB 57, 8972 (1998) The spherically averaged exchange-correlation hole in variational Monte Carlo (VMC) and DFT-LDA with (a) one electron fixed at the bond center, (b) one electron fixed at the tetrahedral interstitial site, and (c) plots (a) and (b) superimposed with the same scale. 7

8 Most-Cited Papers in APS Journals 11 papers published in APS journals since 1893 with >1000 citations in APS journals (~5 times as many references in all science journals) From Physics Today, June, 2005 Certainties about Density Functional Theory 1. DFT in principle: It is exact; a universal E xc [n] functional exists. 2. DFT in practice: It is probably not possible to write down E xc [n] as a closed mathematical expression. We need approximations. The success of DFT proves that simple approximations The success of DFT proves that simple approximations to the exchange-correlation functional can provide good results if one knows what one is doing. 8

9 Some remarks about excited states Bulk and Surface Defects in Semiconductors; Example: The P Vacancy at InP (110) In I In II [110] Ph. Ebert et al., Phys. Rev. Lett. 84 (2000) [001] calculated l electronic structure [110] Calculated geometry... the systematic error for the calculated energies of the charge transfer levels is too large to identify the symmetry of the vacancy on the position of the defect level only. Use Kohn-Sham Density-Functional Theory 1a level of neutral P vacancy at InP(110) 9

10 Kohn-Sham eigenvalues are not excitation energies. However, they may serve as a starting point. What About Kohn-Sham Eigenvalues? The ionization energy is: n(r) = f i φ i (r) 2 i = 1 f i are occupation numbers (Well defined for the highest occupied state. Otherwise, this only holds approximately.) Here we assume that the positions of the atoms don t change upon ionization, or that they change with some delay (Franck-Condon principle). Using the mean-value theorem of integral algebra gives: This is the Slater-Janak transition state. It is the DFT analog of Koopmans theorem. 10

11 Kohn-Sham Electron Bands Kohn-Sham band gap: Δ KS = є CB - є VB of the N-particle system The measured (optical) band gap is something else: Silicon removal addition conduction band (empty states) } the KS gap Δ valence band (filled states) E gap = Δ KS + Δ xc There is a discontinuity in V L Γ X W xc at integer values of occupation numbers. We don t know how to determine Δ xc within DFT What is measured and what is calculated? Photoemission Inverse Photoemission Energies of direct or inverse photoemission are given by totalenergy differences or, equivalently, by the Green function. This can be interpreted as a propagator: For t > t it describes a process in which an extra electron is added to the system at time t. A Fourier analysis yields the energy spectrum (for j = (N+1)... ) Likewise, for t > t the Green function describes the propagation of an extra hole between t and t, yielding the energies (for j = 1... N ) 11

12 Insert a complete set of eigenstates between the field operators and Fourier transform to the frequency axis. This gives: The poles of the Green function correspond directly to the quasiparticle energies These are the true band structure. The highest occupied state equals the exact ionization potential and the lowest unoccupied state the exact electron affinity. Many-body perturbation theory provides a convenient way to analyze both defect states and charge-transition levels. Many-Body Perturbation Theory; The Self-Energy Occupied and unoccupied states and their energies are obtained from the quasiparticle equation The self-energy Σ xc incorporates all contributions from exchange and correlation processes. In contrast to the xc-potential of DFT, it is nonlocal, energy-dependent and has a finite imaginary part. We evaluate Σ xc in the G 0 W 0 approximation: i W 0 is the screened Coulomb interaction, and G 0 is the zeroth order Green function, i.e. that calculated from Kohn-Sham wave functions 12

13 Exploiting the formal similarity between the self-energy and the Kohn-Sham equations by evaluating the quasiparticle energies within first-order perturbation theory gives This treatment (first-order perturbation theory) is justified if the eigenvalues of the underlying Kohn-Sham system are already sufficiently close to the expected quasiparticle band structure. Using Σ xc = i G 0 W 0 (i.e. refraining from a self-consistent evaluation of G and W) implies that the quasi-particle spectrum depends on the input Green s function, G 0. Still, the approach provides a good account of the discontinuity as well as other shortcomings of the employed xc functional. Exploiting the formal similarity between the self-energy and the Kohn-Sham equations by evaluating the quasiparticle energies within first-order perturbation theory gives This treatment (first-order perturbation theory) is justified if the eigenvalues of the underlying Kohn-Sham system are already sufficiently close to the expected quasiparticle band structure. Using Σ xc = i G 0 W 0 (i.e. refraining from a self-consistent evaluation of G and W) implies that the quasi-particle spectrum might depend on the input Green s function, G 0. Still, the approach provides a good account of the discontinuity as well as other shortcomings of the employed xc functional. 13

14 Band Gaps of II-VI Semiconductors and of GaN P. Rinke, A. Qteish, J. Neugebauer, C. Frey- soldt, and M.S., Combining GW calcula-tions with exact-exchange density-functional theory: an analysis of valence-band photoemission for compound semiconductors. New J. Phys. 7, 126 (2005). How to evaluate charge-transition levels? Example: The (+/0) level of V P at InP(110) CB 0 + VB This is a total-energy difference and well defined within ihi DFT....? Not true for the known xc functionals. As different charge states occur, the Δ xc term is needed. -- We rewrite the equation as lattice relaxation energy electron affinity level of the of the neutral charge state positive charge state Charge-transition levels for the P vacancy at InP(110) in ev LDA G 0 W experiment 0.75±0.1 M. Hedström, et al. PRL 97 (2006). 14

15 Summary -- Interacting electrons determine the properties and function of real materials and bio molecules. -- Approximate xc functionals have been very successful, but for highly correlated situations and for excited states there are problems. Important arenas for future theoretical work: -- Correlated systems, e.g. f-electron chemistry -- Thermodynamic phase transitions, e.g. melting. -- Surfaces, interfaces, nanostructures in realistic environments. -- Modeling the kinetics, e.g. of catalysts or crystal growth (self-assembly and self-organization) -- Molecules and clusters in solvents, electrochemistry, fuel cells, external fields, transport. -- Biological problems The challenges: -- Find practical ways to correct the xc approximation. -- Develop methods for bridging the length and time scales. 15