Fundamentals of
Density Functional Theory
Walter Kohn
Physics-Chemistry
University of California, Santa Barbara
Santa Barbara, CA 93106
kohn@physics.ucsb.edu
Introduction
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
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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 nondegenerate or degenerate (In the latter case any of the
possible ground state densities uniquely determines the
potential v(r) ) .
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Because of the key importance of this observation, the simple proof, for nondegenerate 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
vʹ : Hʹ, nʹ(r), N, ɸʹ, Eʹ
(5)
where vʹ≠v + C, and hence ϕʹ≠ϕ. Then, by the Rayleigh Ritz variational principle
(6)
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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(r1 . . .rN) 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,vɸ) + (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)
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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 E0. 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[nʹ(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 nʹ(r) of one of the many
ground states.
By the conventional Rayleigh-Ritz principle we have
(15)
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9
(19)
where
(20)
and
(21)
The total energy is
(22)
In principle, if the exact Exc[n] were used in these equations, the resulting selfconsistent 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
Exc. The simplest of these—which has become a benchmark for all others—is
the local density approximation, LDA:
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(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 vxc 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.)
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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 Exc[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
n1(r), . . . nM(r). Nuclei (= protons + neutrons); electron-hole droplets in
semiconductors.
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d) Free Energy for Finite Temperature
n(r; T). Plasmas
e) Excited States
, where nj(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.
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