Fundamentals of Density Functional Theory Santa Barbara, CA Walter Kohn Physics-Chemistry University of California, Santa Barbara

Introduction 2 Density functional theory (DFT) is primarily a theory of the electronic structure of atoms, molecules and solids in their ground states, in which the electronic density distribution n(r) plays the central role. Traditionally electronic structure problems were attacked by finding approximate solutions of the Schrödinger equation Where

Basic Density Functional Theory DFT in its modern form originated from the observation that, for an interacting N-electron system in a general external potential v(r), a knowledge of the ground state density n(r) uniquely determines v(r) (to within a physically irrelevant additive constant): This is true whether the ground state is non- degenerate or degenerate (In the latter case any of the possible ground state densities uniquely determines the potential v(r) ). 3

Because of the key importance of this observation, the simple proof, for non- degenerate ground states, will now be presented: : Let v(r) be the external potential of the system, with associated ground state density n(r), total number of particles N = ʃ n(r)dr, Hamiltonian H and ground state and energy, ϕ and E, v : H, N, n(r), ɸ, E(4) Similarly, consider a second system of N particles with v ʹ : H ʹ, n ʹ (r), N, ɸʹ, E ʹ (5) where v ʹ ≠v + C, and hence ϕʹ ≠ ϕ. Then, by the Rayleigh Ritz variational principle (6) 4

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The simple observation (3) has far-reaching consequences. Since n(r) determines v(r) and, trivially, also N, it determines H; hence, implicitly, also all properties derivable from H such as the many electron ground state wave function ɸ (r 1...r N ) and energy E, excited state wave functions and energies, Green’s functions etc. The total ground state energy of a system can be written as E = ( ɸ,v ɸ ) + ( ɸ, (T + U) ɸ ),(9) where the terms on the right hand side are the expectation values of the external potential-, kinetic energy- and interaction energy operators. Clearly (10) While the quantity F[n] ≡ ( ɸ, (T + U) ɸ )(11) is a functional of n(r) (through Ψ[n], defined for any physical n(r) corresponding to some v(r). (Such densities are called v-representable.) Thus, (12) 6

Using the Rayleigh Ritz principle for the ground state energy leads rather simply to the conclusion that, for a given v(r), the expression (12) is a minimum for the correct ground state density n(r). Here is the original proof, given for a non-degenerate ground state ɸʹ with energy E 0. Let ɸʹ be a trial state. Then, by the conventional Rayleigh-Ritz principle, (13) or (14) The equality sign holds only if ɸʹ = ɸ. This is the Hohenberg-Kohn energy variational principle. Note that, so far, F[n ʹ (r)] is defined only abstractly through equation (11) but not yet constructively. The proof can be extended to degenerate ground states, leading again to equation (14). The equality is now obtained for any n ʹ (r) of one of the many ground states. By the conventional Rayleigh-Ritz principle we have (15) 7

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(19) where (20) and (21) The total energy is (22) In principle, if the exact E xc [n] were used in these equations, the resulting self- consistent density n(r) and energy, E, would be exact, including all many body effects. Of course, in real life one must content oneself with approximation for E xc. The simplest of these—which has become a benchmark for all others—is the local density approximation, LDA: 10

(23) Where (n) is the exchange-correlation energy per electron of a uniform electron gas of density n, which is known to great accuracy (~0.1%) from independent studies of the uniform electron gas. The corresponding v xc is, by Eq. (20), given by (24) Refinements of the LDA include the so-called generalized gradient approximations, GGA, (25) Judicious choice of the form of, a function of two variables, typically reduces the error of the LDA by a factor of 2 to 10. This is an area of active research. (The most obvious gradient correction to the LDA is a systematic gradient expansion (26) but this is generally of little quantitative value.) 11

Generalizations (See Parr and Yang (1989 and Dreizler and Gross (1990) The principle of DFT, considering the density n(r) as the basic variable and representing many body effects in a density functional such as E xc [n(r)], has been generalized in many directions. Here we list the most important ones, with indications of the relevant densities. a) Degenerate Ground States b) Spin Density Functional Theory Separate spin up and spin down densities n ↑ (r), n ↓ (r) are introduced. Paramagnetism, cooperative magnetism. c) Multicomponent Systems n 1 (r),... n M (r). Nuclei (= protons + neutrons); electron-hole droplets in semiconductors. 12

d) Free Energy for Finite Temperature n(r; T). Plasmas e) Excited States, where n j (r) is the density of the j, the excited state (see also i) below) f) Orbital Magnetism n(r) (normal density) and ñ(r) (superconducting density) Diagmagnetism, quantum Hall effect g) Superconductors n(r) (normal density) and ñ(r) (superconducting density) h) Relativistic Electrons i) Time-Dependent Phenomena v(r, t), n(r, t) Applications to first order response theory for given frequency ω. Poles of response function give excited state energies. 13