Density-Functional Theory
2. DENSITY-FUNCTIONAL THEORY
2.1. The External Potential and the Electron Density
The reader familiar with the historical development of quantum mechanics can be forgiven if they greet with skepticism the notion that one can extract all the infor-mation contained in a ground-state electronic wave function from the probability distribution function for observing an electron at the point r (that is, the ground-state electron density). The first hint that such a construction might be possible follows directly from the form of the molecular Hamiltonian, Eq. (2). Consider that the form of the kinetic energy operator
and the electron–electron repulsion energy operator ˆVee¼XN
are determined by the fact that we are interested in an N-electron system. The only parts of the Hamiltonian that change when the electronic system changes are the number of electrons, N, and the potential the electrons feel due to their‘‘external’’
environment—that is, those particles/fields that are not due to other electrons. For a system of N electrons bound by an inhomogeneous and non-isotropic electrostatic potential, v(r), one may write
Note what we have gained: because the Hamiltonian operator determines the wave function for the system from the variational principle, it follows that any property, Q, of the ground state of an electronic system may be written as a function of the number of electrons, N, and a functional of a real-valued trivariate function, v(r), which we call the external potential. We denote this functional Q[v(r); N ]. Unfortunately, no expression for Q[v(r); N] with computational utility comparable to Eq. (1) is known.
However, the fact that properties of a system can be expressed as a function of N and a functional of a single trivariate function, v(r), does suggest that there might be a computational useful theory in terms of a trivariate function.
To motivate the subsequent development, recall that the wave function of a system possesses no direct physical significance. Rather, the most informative observ-able property of the wave function is its complex square,jC(r1, s1,. . .,rN, sN)j2, which, according to the Born postulate, represents the probability that an electron has spin s1and is located at r1, another electron has spin s2and is located at r2, etc. Thus
qNðr1; r2; . . . rNÞuX
si
C rð 1; s1; . . . ; rN; sNÞ
j j2 ð12Þ
is the probability distribution function for the electrons in the system. A related tri-variate function is the probability of observing an electron at the point r; this defines the electron density, q(r), at any point in space. From the fact that the electron density at the point r is the sum of the probabilities of any of the N electrons being at r (the other N1 electrons can be anywhere in space),
q rð ÞuXN
Insofar as the electron density is the probability of observing an electron at a point, it is clearly nonnegative; it is also observable experimentally. It is also clear from Eq. (13) that
N½ uq
R
q rð Þdr; ð14Þ
that is, the number of electrons is a functional of the electron density.
2.2. The Ground-State Electron Density as a Descriptor of Electronic Systems
The sweeping theorem of Hohenberg and Kohn is that, like the wave function,‘‘the ground state’s electron density determines all the properties of an electronic system’’
[1]. The result is proved in three steps. First, one recalls that the number of electrons is determined from the electron density using Eq. (14). Next, one demonstrates that the external potential can be determined from the ground-state electron density.
From N and v(r), we may determine the electronic Hamiltonian and solve Schro¨-dinger’s equation for the wave function, subsequently determining all observable properties of the system.
A key to this development is the assertion that the ground-state electron density determines the electronic external potential. Stated mathematically, we must demon-strate that the external potential is a functional of the electron density. Just as f is a function of x if and only if no argument of f, x0, corresponds to more than one value of f(which is to say that f(x) is single-valued for all x in the domain of f ), the external potential, v(r) is a functional of the ground-state electron density, q(r), if and only if no two external potentials correspond to the same ground-state electron density.
That this is true follows directly from the variational principle for the wave function. Consider two different N-electron systems; these systems have different ground-state wave functions, C0and C1, and external potentials, v0(r) and v1(r), which differ by more than an additive constant. From the variational principle for the energy, Eq. (6),
Adding these equations [with substitution of the defining Eq. (11)] yields C0 CˆF 0
From the definition of the electron density, Eq. (13), we obtain the key relation
R
ðq0ð Þ qr 1ð Þr Þ vð 0ð Þ vr 1ð Þr Þdr < 0; ð18Þ where q0(r) and q1(r) are the ground-state densities for the N-electron systems with external potentials v0(r) and v1(r), respectively. Since v0(r) p v1(r) by assumption, Eq. (18) implies that q0(r) p q1(r). Thus no two external potentials correspond to the same ground-state electron density. This result, first obtained by Hohenberg and Kohn [1] in 1964, is generally called the (first) Hohenberg–Kohn theorem. The present proof is modeled after the more general considerations of Levy [2] and Englisch and Englisch [3].2.3. The Variational Principle for the Ground-State Electron Density
The first Hohenberg–Kohn theorem is an existence theorem: it indicates that we can, in principle, determine the ground-state wave function, C[q], the ‘‘purely electronic’’
contribution to the total energy, F [ q]=hC[q]jFˆ jC[q]X, the total energy E[q], and all other properties of an electronic system directly from the ground-state electron density. Leveraging the Hohenberg–Kohn theorem to practical applications requires a method for accurately determining the ground-state electron density. Recalling the utility of the variational principle for determining the wave function, we now wish to derive a variational principle for the ground-state electron density.
To do this, consider a system consisting of N-electrons, with electron density q0(r), confined by external potential, v1(r), for which q0(r) is not a ground-state electron density. The purely electronic contribution to the energy,
F½ ¼ Cq0 D 0 CˆF 0E
; ð19Þ
does not depend on the external potential; such universal functionals play a key role in density-functional theory. The interaction energy between the external forces on the electrons and the electrons is, from classical electrostatics,
Vext½v1; q0 ¼
R
q0ð Þvr 1ð Þdr;r ð20Þ
and so the total energy of this system can be written as Ev1½ uF qq0 ½ þ0
R
q0ð Þvr 1ð Þdr:r ð21Þ
Here we adopt the standard notation, which explicitly denotes the fact that the external potential, v1(r), is a parameter that is constructed at the beginning of the variational calculation from the molecular geometry and external fields of interest.
The comparison between the energies of the‘‘wrong’’ electron density q0(r) and a ground-state electron density for v1(r) follows directly from Eq. (16) and the defining Eq. (21):
Eg:s:½ ¼ Eq1 v1½ uF qq1 ½ þ1
R
q1ð Þvr 1ð Þdrr
ð22Þ VEv1½ uF qq0 ½ þ0
R
q0ð Þvr 1ð Þdr;r
the equality holds only when q0(r) is an electron density for this system (which, by assumption, it is not). Equation (22), which is often referred to as the second Hohenberg–Kohn theorem [1], is the foundation of all practical procedures for finding the ground-state electron density:
Eg:s:¼ Ev qg:s:
¼ |{z}min
all Nelectron qðrÞ
Ev½ :q ð23Þ
2.4. Recapitulation: Analogies to the Wave Function Theory
It is instructive to compare the Hohenberg–Kohn theorems to their counterparts in conventional wave function–based quantum theory. The wave function provides a
complete quantum mechanical description of any state of any system. The first Hohenberg–Kohn theorem indicates that the ground-state electron density determines all the properties of any electronic system, by which we mean a system whose Hamiltonian operator assumes the specific form of Eq. (11). The practical utility of quantum mechanics depends upon efficient computational methods for determining the wave function; the most fundamental of these is the variational principle, Eq. (6).
Similarly useful in the density-functional theory context is the second Hohenberg–
Kohn theorem, Eq. (22).
Instead of direct implementation of the variational principle, one often seeks to solve the associated Schro¨dinger equation, Eq. (7). A similarly useful equation in density-functional theory is derived from Eq. (23): given the N-electron ground-state electron density, qN(r), for a system, v(r), with a nondegenerate ground state, all other N-electron densities have greater energy, Ev[q]>Ev[qN]. Subject to sufficient smooth-ness in the energy functional, this indicates that the energy functional is stationary with respect to small normalization-preserving perturbations of the ground-state electron density. That is, given a function, D(r), for which mD(r)dr = 0, then
Ev½q0ðrÞ þ eDðrÞ Ev½q0ðrÞ~e2 ð24Þ in the limit of small e. Thus,
dEv½q0ðrÞ þ eDðrÞ de
e¼ 0¼ 0: ð25Þ
The restriction in Eq. (25) to variations for which mD(r) dr = 0 is computa-tionally inconvenient. To avoid this, we introduce the notion of a functional derivative,yEyqðrÞv½ qjq¼ q0. Just as the gradient of a function at a point is defined as that vector,jf(x)jx= x0, which maps small changes in x about x0to the resulting changes in the value of the function according to
[
the functional derivative is defined as the function that maps small changes in the electron density to small changes in the energy according to the equation:[
Choosing D(r) = d(rr0) in Eq. (27), we obtain the computationally useful formula yEv½q
We can now express the density-functional variational principle in terms of functional derivatives, namely,
where qN(r) is the ground-state N-electron density for the external potential v(r) and the Lagrange multiplier, l[qN], constrains the variation to N-electron densities. To find the value of the Lagrange multiplier, consider that the stationary condition holds also for small changes in the electron density associated with changing the number of electron in the system; thus:
B
Ev½q lN½q
BN ¼ 0: ð30Þ
There results l¼ BE
BN
vðrÞ; ð31Þ
allowing us to identify l as the chemical potential for the electrons in this system [4].
The introduction of the chemical potential in the density-functional variational principle, Eq. (30), is analogous to the transformation to the grand canonical ensemble in statistical mechanics [5].
Simplifying Eq. (29) yields a functional differential equation for the ground-state density,
yEv½q
yqðrÞ ¼ l ¼ constant; ð32Þ
that is directly analogous to Schro¨dinger’s partial differential equation for the wave function. This is most apparent when the Schro¨dinger equation is written in the form
ˆHC
C ¼ E ¼ constant: ð33Þ
Equations (32) and (33) possess comparable importance and similar utility in the density-functional and wave-functional approaches to quantum mechanical systems, respectively.
3. FURTHER DEVELOPMENTS IN DENSITY FUNCTIONAL