1 / 14

The Nuts and Bolts of First-Principles Simulation

The Nuts and Bolts of First-Principles Simulation. 3 : Density Functional Theory. CASTEP Developers’ Group with support from the ESF  k Network. Density functional theory Mike Gillan, University College London. Ground-state energetics of electrons in condensed matter

Download Presentation

The Nuts and Bolts of First-Principles Simulation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. The Nuts and Bolts of First-Principles Simulation 3: Density Functional Theory CASTEP Developers’ Groupwith support from the ESF k Network CASTEP Workshop, Durham University, 6 – 13 December 2001

  2. Density functional theoryMike Gillan, University College London • Ground-state energetics of electrons in condensed matter • Energy as functional of density: the two fundamental theorems • Equivalence of the interacting electron system to a non-interacting system in an effective external potential • Kohn-sham equation • Local-density approximation for exchange-correlation energy CASTEP Workshop, Durham University, 6 – 13 December 2001

  3. The problem • Hamiltonian H for system of interacting electrons acted on by electrostatic field of nuclei: • with T kinetic energy, U mutual interaction energy of electrons, V interaction energy with field of nuclei. • To develop theory, V will be interaction with an arbitrary external field: with ri position of electron i. • Ground-state energy is impossible to calculate exactly, because of electron correlation. DFT includes correlation, but is still tractable because it has the form of a non-interacting electron theory. CASTEP Workshop, Durham University, 6 – 13 December 2001

  4. Energy as functional of density: the first theorem For given external potential v(r), let many-body wavefunction be . Then ground-state energy Eg is: and the electron density n(r) by: Theorem 1: It is impossible that two different potentials give rise to the same ground-state density distribution n(r). Corollary: n(r) uniquely specifies the external potential v(r) and hence the many-body wavefunction . where the density operator is defined as: CASTEP Workshop, Durham University, 6 – 13 December 2001

  5. Convexity of the energy (1) Convexity means: For two external potentials and , go along linear path between them; if is ground-state energy for then: Proof of follows from 2nd-order perturbation theory: with and wavefns of ground and excited states, and their energies, and . Theorem 1 expresses convexity of the energy Egas function of external potential. CASTEP Workshop, Durham University, 6 – 13 December 2001

  6. Convexity of the energy (2) Theorem 1 is equivalent to saying that a change of external potential cannot give a vanishing change of density This follows from convexity. Convexity implies that at is less than at . But , so that: so that: Hence: which demonstrates that , and this is Theorem 1. CASTEP Workshop, Durham University, 6 – 13 December 2001

  7. Since ground-state energy Eg is uniquely specified by n(r), write it as Eg[n(r)]. It’s useful to separate out the interaction with the external field, and write: Where F[n(r)] is ground-state expectation value of H0 when density is n(r). Proof: Let v(r) and v’(r) be two different external potentials, with ground-state energies Eg and Eg’ and ground-state wavefns and . By Rayleigh-Ritz variational principle: DFT variational principlethe second theorem Theorem 2 (variational principle): Ground-state energy for a given v(r) is obtained by minimising Eg[n(r)] with respect to n(r) for fixed v(r), and the n(r) that yields the minimum is the density in the ground state. Where n’(r) is density associated with . This proves the theorem. The usual assumptions of non-degenerate ground state is needed. CASTEP Workshop, Durham University, 6 – 13 December 2001

  8. The Euler equation Write F[n(r)] as: where T[n] is kinetic energy of a system of non-interacting electrons whose density distribution is n(r). Then: Variational principle: subject to constraint: Handle the constant-number constraint by Lagrange undetermined multiplier, and get: with undetermined multiplier the chemical potential. CASTEP Workshop, Durham University, 6 – 13 December 2001

  9. Kohn-Sham equation Rewrite the Euler equation for interacting electrons: by defining , so that: But this is Euler equation for non-interacting electrons in potential veff(r), and must be exactly equivalent to Schroedinger equation: with n(r) given by: Then put n(r) back into G[n(r)] to get total energy: CASTEP Workshop, Durham University, 6 – 13 December 2001

  10. Self consistency How to do DFT in practice??? • We don’t know G[n(r)], and probably never will, but suppose we know an adequate approximation to it. • Make an initial guess at n(r), calculate and hence • for this initial n(r). • Solve the Kohn-Sham equation with this veff(r) to get the KS orbitals • and hence calculate the new n(r): • The output n’(r) is not the same as the input n(r). So iterate to reduce residual: The whole procedure is called ‘searching for self consistency’. CASTEP Workshop, Durham University, 6 – 13 December 2001

  11. Exchange-correlation energy • We have already split the total energy into pieces: • Now separate out the Hartree energy: • Then exchange-correlation energy Exc[n] is defined by: So far, everything is formal and exact. If we knew the exact Exc[n], then we could calculate the exact ground-state energy of any system! CASTEP Workshop, Durham University, 6 – 13 December 2001

  12. Local density approximation • There is one extended system for which Exc is known rather precisely: the uniform electron gas (jellium). For this system, we know exchange-correlation energy per electron as a function of density n. • Local density approximation (LDA): assume the xc energy of an electron at point r is equal to for jellium, using the density n(r) at point r. Then total Exc for the whole system is: • Some kind of justification can be given for LDA (see xxxxxxx). But the main justification is that it works quite well in practice. CASTEP Workshop, Durham University, 6 – 13 December 2001

  13. Kohn-Sham potential in LDA The effective Kohn-Sham effective potential in general is: The Hartree potential is: Exchange-correlation potential in LDA: Where: So in LDA, everything can be straightforwardly calculated! CASTEP Workshop, Durham University, 6 – 13 December 2001

  14. Useful references • Here is a selection of references that contain more detail about DFT: • P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964) • W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965) • N. D. Mermin, Phys. Rev. 137, A1441 (1965) • R. O. Jones and O. Gunnarsson, Rev. Mod. Phys., 61, 689 (1989) • M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias and J. D. Joannopoulos, Re. Mod. Phys., 64, 1045 (1992) CASTEP Workshop, Durham University, 6 – 13 December 2001

More Related