TUTORIAL REVIEW WWW.Q-CHEM.ORG
DFT in a Nutshell Kieron Burke[a,b] and Lucas O. Wagner*[a,b]
The purpose of this short essay is to introduce students and something new outside their area. Important questions other newcomers to the basic ideas and uses of modern varying in difficulty and effort are posed in the text, and are electronic density functional theory, including what kinds of answered in the Supporting Information. VC 2012 Wiley approximations are in current use, and how well they work (or Periodicals, Inc. not). The complete newcomer should find it orients them well, while even longtime users and aficionados might find DOI: 10.1002/qua.24259
1 X 1 Electronic Structure Problem V^ ¼ ; (4) ee 2 r r i6¼j j i jj For the present purposes, we define the modern electronic struc- and the one-body operator is ture problem as finding the ground-state energy of nonrelativistic electrons for arbitrary positions of nuclei within the Born-Oppen- XN [1] heimer approximation. If this can be done sufficiently accurately V^ ¼ vðrjÞ: (5) and rapidly on a modern computer, many properties can be pre- j¼1 dicted, such as bond energies and bond lengths of molecules, For instance, in a diatomic molecule, v(r) ¼ Z /r Z /|r R|. and lattice structures and parameters of solids. A B We use atomic units unless otherwise stated, setting Consider a diatomic molecule, whose binding energy curve e2 ¼ h ¼ m ¼ 1, so energies are in Hartrees (1 Ha ¼ 27.2 eV is illustrated in Figure 1. The binding energy is given by e or 628 kcal/mol) and distances in Bohr radii (1 a0 ¼ 0.529 A˚). The ground-state energy satisfies the variational principle:
E ¼ min hWjH^jWi; (6) W
where the minimization is over all antisymmetric N-particle
wavefunctions. This E was called E0(R) in Eq. (1).* Many traditional approaches to solving this difficult many- body problem begin with the Hartree–Fock (HF) approxima- tion, in which W is approximated by a single Slater determi- nant (an antisymmetrized product) of orbitals (single-particle wavefunctions)[2] and the energy is minimized.[3] These include Figure 1. Generic binding energy curve. For N2, values for R0 and De are configuration interaction, coupled cluster, and Møller-Plesset given in Table 1. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] perturbation theory, and are mostly used for finite systems, such as molecules in the gas phase.[4] Other approaches use ZA ZB reduced descriptions, such as the density matrix or Green’s EbindðRÞ¼E0ðRÞþ EA EB (1) R function, but leading to an infinite set of coupled equations
where E0(R) is the ground-state energy of the electrons with that must somehow be truncated, and these are more com- [5] nuclei separated by R,andEA and ZA are the atomic energy mon in applications to solids. and charge of atom A and similarly for B. The minimum tells us More accurate methods usually require more sophisti-
the bond length (R0) and the well-depth (De), corrected by cated calculation, which takes longer on a computer. Thus, zero-point energy (h x=2), gives us the dissociation energy (D0). there is a compelling need to solve ground-state electronic The Hamiltonian for the N electrons is structure problems reasonably accurately, but with a cost in
H^ ¼ T^ þ V^ee þ V^; (2) [a, b] K. Burke, L. O. Wagner where the kinetic energy operator is Department of Chemistry, University of California, Irvine, California 92697 Department of Physics, University of California, Irvine, California 92697 1 XN T^ ¼ r2; (3) E-mail: [email protected] 2 j j¼1 *Explain why a vibrational frequency is a property of the ground-state of the electrons in a molecule. the electron–electron repulsion operator is VC 2012 Wiley Periodicals, Inc.
96 International Journal of Quantum Chemistry 2013, 113, 96–101 WWW.CHEMISTRYVIEWS.ORG WWW.Q-CHEM.ORG TUTORIAL REVIEW
Kieron Burke is a professor of heretical physical and computational chemistry at the University of California in Irvine. He spent the last two decades of his life in functionals. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
Lucas Wagner began working in 2009 as a graduate student with Kieron Burke starting off curious, plucky, and impressionable in the wide world of DFT. Lucas studies simple (mostly 1d) models and enjoys the combination of analytical and numerical studies involved. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
computer time that does not become prohibitive as the enough for most properties of interest (for example, molecules number of atoms (and therefore electrons) becomes large. do not bind[8]). For same-spin, noninteracting fermions in 1d, the corresponding kinetic energy is Basic DFT Z p2 TTF 1d½n ¼ dx n3ðxÞ (11) The electronic density n(r) is defined by the requirement that 6 n(r) d3r is the probability of finding any electron in the volume d3r around r. For a single electron with wavefunction f(r), it is and makes only a 25% error on the density of a single particle simply |f(r)|2. In density functional theory (DFT), we write the in a box. Hours of endless fun and many good and bad prop- ground-state energy in terms of n(r) instead of W. The first erties of functional approximations can be understood by DFT was formulated by Thomas[6] and Fermi.[7] The kinetic applying Eq. (11) to standard text book problems in quantum energy density at any point is approximated by that of a uni- mechanics,‡ and noting what happens, especially for more form electron gas of noninteracting electrons of density n(r), than one particle.§ which for a spin-unpolarized system is:† But modern DFT began with the proof that the solution of Z the many-body problem can be found, in principle, from a TF 3 5=3 2 2=3 T ¼ aS d rn ðrÞ; aS ¼ 3ð3p Þ =10: (7) density functional. To see this, we break the minimization of Eq. (6) into two steps. First minimize over all wavefunctions The interelectron repulsion is approximated by the classical yielding a certain density, and then minimize over all densities. electrostatic self-energy of the charge density, called the Har- Because the one-body potential energy depends only on the tree energy: density, we can define separately[9,10] Z Z 1 nðrÞ nðr0Þ U ¼ d3r d3r0 : (8) 0 F½n ¼minhWj T^ þ V^ee jWi; (12) 2 jr r j W!n
Because the one-body potential couples only to the density, where the minimization is over all antisymmetric wavefunc- Z tions yielding a given density n(r).¶ This is transparently a V ¼hV^i¼ d3rnðrÞ vðrÞ: (9) functional of the density, meaning it assigns a number to each density, as was first proven by Hohenberg and **[11] The sum of these three energies is then minimized, subject to Kohn. Then the physical constraints: Z nðrÞ 0; d3rnðrÞ¼N: (10) ‡Calculate the TF kinetic energy for a 1d particle of mass m ¼ 1 in (a) a har- monic well (v(x) ¼ x2/2) and (b) in a delta-well (v(x) ¼ d(x)). Give the % errors. This absurdly crude theory gives roughly correct energies §Deduce the exact energy for N same-spin fermions in a flat box of width 1 (errors about 10% for many systems) but is not nearly good bohr. Then evaluate the local approximation to the kinetic energy for N ¼ 1, 2, and 3, and calculate the % error. ¶Derive F[n] for a single electron. It has no electron–electron interaction, and is known as the von Weizs€acker kinetic energy. †Evaluate the TF kinetic energy of the H atom and deduce the % error. Repeat **Technically, HK only proved fact this for v-representable densities, i.e. den- using spin-DFT. sities which are the ground-state of some external one-body potential.
International Journal of Quantum Chemistry 2013, 113, 96–101 97 TUTORIAL REVIEW WWW.Q-CHEM.ORG