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For the present purposes, we define the modern electronic structure problem as finding the ground-state energy of nonrelativistic electrons for arbitrary positions of nuclei within the Born-Oppenheimer approximation.1 If this can be done sufficiently accurately and rapidly on a modern computer, many properties can be predicted, such as bond energies and bond lengths of molecules, and lattice structures and parameters of solids.
Consider a diatomic molecule, whose binding energy curve is illustrated in Figure 1. The binding energy is given by
where E0(R) is the ground-state energy of the electrons with nuclei separated by R, and EA and ZA are the atomic energy and charge of atom A and similarly for B. The minimum tells us the bond length (R0) and the well-depth (De), corrected by zero-point energy ( ), gives us the dissociation energy (D0).
The Hamiltonian for the N electrons is
where the kinetic energy operator is
the electron–electron repulsion operator is
and the one-body operator is
For instance, in a diatomic molecule, v(r) = −ZA/r − ZB/|r − R|. We use atomic units unless otherwise stated, setting , so energies are in Hartrees (1 Ha = 27.2 eV or 628 kcal/mol) and distances in Bohr radii (1 a0 = 0.529 Å). The ground-state energy satisfies the variational principle:
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) approximation, in which Ψ is approximated by a single Slater determinant (an antisymmetrized product) of orbitals (single-particle wavefunctions)2 and the energy is minimized.3 These include configuration interaction, coupled cluster, and Møller-Plesset perturbation theory, and are mostly used for finite systems, such as molecules in the gas phase.4 Other approaches use reduced descriptions, such as the density matrix or Green's function, but leading to an infinite set of coupled equations that must somehow be truncated, and these are more common in applications to solids.5
More accurate methods usually require more sophisticated calculation, which takes longer on a computer. Thus, there is a compelling need to solve ground-state electronic structure problems reasonably accurately, but with a cost in computer time that does not become prohibitive as the number of atoms (and therefore electrons) becomes large.
The electronic density n(r) is defined by the requirement that n(r) d3r is the probability of finding any electron in the volume d3r around r. For a single electron with wavefunction ϕ(r), it is simply |ϕ(r)|2. In density functional theory (DFT), we write the ground-state energy in terms of n(r) instead of Ψ. The first DFT was formulated by Thomas6 and Fermi.7 The kinetic energy density at any point is approximated by that of a uniform electron gas of noninteracting electrons of density n(r), which for a spin-unpolarized system is:†
The interelectron repulsion is approximated by the classical electrostatic self-energy of the charge density, called the Hartree energy:
Because the one-body potential couples only to the density,
The sum of these three energies is then minimized, subject to the physical constraints:
This absurdly crude theory gives roughly correct energies (errors about 10% for many systems) but is not nearly good enough for most properties of interest (for example, molecules do not bind8). For same-spin, noninteracting fermions in 1d, the corresponding kinetic energy is
and makes only a 25% error on the density of a single particle in a box. Hours of endless fun and many good and bad properties of functional approximations can be understood by applying Eq. (11) to standard text book problems in quantum mechanics,‡ and noting what happens, especially for more than one particle.§
But modern DFT began with the proof that the solution of the many-body problem can be found, in principle, from a density functional. To see this, we break the minimization of Eq. (6) into two steps. First minimize over all wavefunctions yielding a certain density, and then minimize over all densities. Because the one-body potential energy depends only on the density, we can define separately9, 10
where the minimization is over all antisymmetric wavefunctions yielding a given density n(r).¶ This is transparently a functional of the density, meaning it assigns a number to each density, as was first proven by Hohenberg and Kohn.**11 Then
where the minimization is over all reasonable densities satisfying Eq. (10). Hohenberg and Kohn proved (i) that all properties are determined by n(r), i.e., they are functionals of n(r),†† (ii) F[n] is a universal functional, independent of v(r), and (iii) the exact density satisfies
where δF/δn(r) is the functional derivative of F with respect to n(r). In fact, these days we use spin DFT,12 in which all quantities are considered functionals of the up, n↑(r), and down, n↓(r), spin densities separately. This makes approximations more accurate for odd electron systems and allows treatment of collinear magnetic fields. All functionals written without spin dependence, such as the ones discussed thus far, are assumed to be referring to a spin-unpolarized system.
The next crucial step in developing the modern theory came from (re)-introducing orbitals. Kohn and Sham13 vastly improved the accuracy of DFT by imagining a fictitious set of noninteracting electrons that are defined to have the same density as the interacting problem. They are still spin- fermions obeying the Pauli principle, so like in HF theory, their wavefunction is (usually) a Slater determinant, an antisymmetrized product of orbitals of each spin, ϕjσ(r), j = 1,…,Nσ, σ = ↑, ↓. These KS electrons satisfy a noninteracting Schrödinger equation:
and the εjσ are called KS eigenvalues and ϕjσ(r) are KS orbitals.‡‡ By evaluating Eq. (3) on the KS Slater determinant, the KS kinetic energy is the sum of the orbital contributions:§§
If we write the energy in terms of KS quantities:
EXC is defined by Eq. (17) and called the exchange-correlation (XC) energy.¶¶ KS showed that one could extract the unknown KS potential if one only knew how the terms depend on the density. Writing the Hartree potential as
This is a formally exact scheme for finding the ground-state energy and density for any electronic problem.*** In Figure 2, we emphasize the exactness of the KS scheme by plotting the exact KS potential for a He atom (which is trivial to find, once the exact density is known from an accurate many-body calculation14):††† Two noninteracting electrons, doubly occupying the 1s orbital of this potential, have a density that matches that of the interacting system exactly.‡‡‡ But in practical calculations, we always use approximations to EXC and hence to vXC(r).
Traditionally, EXC is broken up into exchange (X) and correlation contributions:§§§
The exchange energy is V̂ee evaluated on the KS Slater determinant minus the Hartree energy, and typically dominates.¶¶¶ In terms of the orbitals:
This is precisely the same orbital expression given in HF, though it makes use of the KS orbitals (which are implicit functionals of the density15 rather than the HF orbitals.**** Then the correlation energy is everything else, i.e., defined to make Eq. (17) exact.††††
Practical calculations use some simple approximation to EXC[n↑,n↓]. The KS equations are started with some initial guess for the density, yielding a KS potential via Eq. (20). The KS equations are then solved and a new density is found. This cycle is repeated until changes become negligible, i.e., this a self-consistent field calculation.
The standard approximations are very simple. The local (spin) density approximation (often just called LDA) is13:
where e(n↑,n↓) is the XC energy density of a uniform gas of spin densities n↑ and n↓. The X contribution was first written by Dirac,16 and is found by inserting plane waves (the KS orbitals of a uniform gas) into Eq. (22).‡‡‡‡ In the unpolarized case, the result is§§§§
while e(n↑,n↓) has been calculated and accurately parametrized.17 More accurate energetics are usually obtained with a generalized gradient approximation (GGA), which includes dependence on the gradient of the density
where now the energy density is given by some approximate form. The third standard form of approximation is a hybrid, which mixes in a fixed fraction of the exact exchange energy
where a is a universal parameter (often about 1/4) and EX is the exact exchange defined in Eq. (22).
Although LDA is uniquely defined, there are two basic flavors of the more sophisticated functionals. There are those that are derived without fitting to reference data on atoms and molecules, using information from only the slowly varying electron and known exact conditions on the functional. The standard GGA of this type is PBE,18 while the hybrid is PBE0, mixing 25% exact exchange.19 Of the empirical type, B88 is the standard GGA for exchange,20 LYP for correlation,21 combining to form BLYP. The most commonly used functional today is a hybrid called B3LYP.22
The original LDA became a standard tool in solid-state physics, yielding excellent lattice parameters, and fairly good bulk moduli. But LDA typically overbinds by about 1 eV/bond, which is too large an error to be useful in quantum chemistry. GGAs reduce this error to about 0.3 eV/bond, and hybrids reduce it another factor of 2. Table 1 shows typical results. Empirical parametrized functionals are usually about a factor of 2 better than nonempirical ones, but with less systematic errors.25 Although hybrids are popular in chemistry, where HF codes have been including exact exchange for decades, they are much less popular for solids, due to the singularities in HF for metals.
Table 1. Differences in calculated [using Turbomole (http://www.turbomole.com/)23] versus experimental values (from Ref. 24) for well depth (De) in kcal/mol and equilibrium distance (R0) in pm of N2.
Δ = calc−expt
In contrast, many limitations of these functionals have been identified. Perhaps the most well-known is the gap problem. The fundamental gap is I − A, the difference between the ionization potential and the electron affinity of a system. This is usually larger than the KS gap, the difference between the KS HOMO and LUMO energy eigenvalues. Calculations with LDA and GGA yield fundamental gaps of insulating solids close to the KS gap, and so are too small (by about a factor of 2). This is related to self-interaction error (that is, the functionals are incorrect for one electron). Hybrid functionals often do better. Other well known failures of these approximations include the lack of asymptotic van der Waals forces, which can be important for soft matter.26
There are many suggestions on how to improve these standards. Presently, meta-GGAs, which use KS kinetic energy densities to approximate EXC, can produce accurate energetics without using HF exchange.27 Alternatively, including only the short-ranged contribution to exchange can also avoid the difficulties of applying HF to solids.28 But so far, none of these is even close to replacing the standard functionals in amount of use.
Computer codes also need some basis set to represent the KS orbitals. Basis functions localized on atoms are used for molecular systems in quantum chemistry. Most codes now use Gaussians, but some use Slater-type orbitals (exponentials centered on atoms) and localized basis sets. In solids and materials, periodic boundary conditions are convenient for extended systems, making plane waves a natural choice. Core electrons are often treated as frozen or by pseudo-potentials, saving calculational effort. A few codes use numerical grids.5
Electronic KS-DFT has been implemented in a huge variety of codes, and is being applied in many areas of science. Modern computers allow routine calculation of hundreds of atoms, far more than with traditional quantum chemical methods, but with a reasonable (not very high) degree of accuracy and reliability. In traditional areas of condensed-matter physics and quantum chemistry, many experimental groups run codes themselves, and many purely computational groups exist. For example, new catalysts have been found by calculation, then demonstrated in the lab, patented, and are being further developed for industrial production.29 But such calculations and calculators are also appearing in materials science, geology, and biochemical departments. Recent uses also include protein science, soil science, and astrophysics. Liquids can be simulated using ab initio molecular dynamics (AIMD).30 There is increasing interest in matter under high pressure and temperature (but not high enough to be pure plasma) and here DFT simulation is making a large impact.31
In its original form, electronic DFT focused purely on the ground-state energy and density. Several approaches exist to tackle excited states, but the most popular is time-dependent density functional theory (TDDFT), based on a formal proof for time-dependent one-body potentials.32 In the linear response regime, this leads to a simple scheme for calculating transition frequencies which has become popular for molecules. The response scheme uses the ground-state KS orbitals as its starting point. In Figure 2, the (doubly occupied) 1s orbital is at −24.57 eV, precisely −I, the ionization potential for the system. The 2p orbital is at −3.429 eV, so the first allowed optical transition of the KS electrons has frequency 21.14 eV.33 TDDFT corrects these transitions to the exact value 20.97 eV,34 but once again requires knowledge of an unknown functional, called the XC kernel, which depends on the perturbing frequency.35 Present applications all use the adiabatic approximation, which ignores the frequency dependence and allows the kernel to be extracted from a ground-state XC approximation. Typical results give moderate accuracy for transition frequencies (0.4 eV errors), but excellent results for the properties of molecules in excited states.36 Dramatic failures of these approximations include double excitations, charge transfer, and a missing contribution to the optical response of solids.37
Tables 2 and 3 give a list of acronyms and notation used in this essay. Perhaps the best pedagogical introduction of recent years is “A Primer in DFT.”38 For physicists with some background in many-body theory, there's nothing better than Dreizler and Gross,39 or the more recent Engel and Dreizler.40 Analogously, if you are a chemist with knowledge of quantum chemical methods, try Parr and Yang.41 A more pragmatic approach is that of Koch and Holthausen42 who, in Part B, discuss the accuracy of different approximate functionals for different molecular properties. For TDDFT, the standard reference is that edited by Marques,37 and there is a new introduction by Ullrich.43
To check you have followed the essay, you should certainly try the questions posed in the text, which are collected and answered in the Supporting Information.
Explain why a vibrational frequency is a property of the ground-state of the electrons in a molecule.
Evaluate the TF kinetic energy of the H atom and deduce the % error. Repeat using spin-DFT.
Calculate the TF kinetic energy for a 1d particle of mass m = 1 in (a) a harmonic well (v(x) = x2/2) and (b) in a delta-well (v(x) = −δ(x)). Give the % errors.
Deduce the exact energy for N same-spin fermions in a flat box of width 1 bohr. Then evaluate the local approximation to the kinetic energy for N = 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 Weizsäcker kinetic energy.
Technically, HK only proved fact this for v-representable densities, i.e. densities which are the ground-state of some external one-body potential.
Again, HK proved this only for v-representable densities.
Is the sum of the KS eigenvalues equal to the total energy?
Apply Eq. (12) to the KS system to define TS without ever mentioning vS(r). Then prove that T ≥ TS always.
Give the signs of E, T, Vee, V, U, EX, and EC for real systems (i.e., atoms, molecules, and solids).
Why is a KS calculation much faster than direct solution of the Schrödinger equation?
Write the formula that extracts vS(r) from n(r) for the helium atom. Can you explain why this does not tell us the vital vXC[n](r) for any spin-unpolarized two-electron systems?
Why is the helium KS potential less deep than the original potential, −2/r?
What is EXC for a one-electron system?
What is the expectation value of the Hamiltonian (Ĥ = T̂ + V̂ + V̂ee) evaluated on the KS Slater determinant? Use this to prove that the DFT definition of correlation energy is never positive.
Derive a formula for EX[n↑,n↓] in terms of EX[n], evaluated on various densities. Does the same formula apply to the KS kinetic energy, but with EX replaced by TS? How about for EC?
This definition differs slightly from that of quantum chemistry.
Show that if electrons repelled via a contact repulsion, δ(r − r′), the exchange is given exactly by LDA, and give its expression, including spin-dependence.
Use dimensional analysis to explain the powers of the density in the local approximations of the kinetic energy in 1d and 3d, as well as the exchange energy in 3d.
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.]