The variational principle in quantum mechanics, lecture 6

1. Lecture 6: density functional theory

In the previous two lectures we discussed several examples of mean-field theory where the variational class is the class of gaussian states or single-particle product states. It turns out that mean-field theory is extremely useful in practice and can give powerful insights into the physics of many strongly interacting systems. This is because it leads to equations which are essentially tractable analytically. Additionally, the solutions often have a direct physical interpretation. Thirdly, the results obtained can often be very good indeed and often match experimentally obtained values to a high precision. Indeed, I’m sure that mean-field theory will continue to be a very useful tool for many years to come, and it should certainly be the first port of call when trying to understand a new system.

In this lecture we turn our attention to another method known as density functional theory (DFT). This can be viewed as, in part, an application of the variational method, although it is not a pure application in the sense of our previous examples.

The pdf version of the lecture notes can be found here.

2. The Hohenberg-Kohn theorem

Throughout the next two lectures we consider many-fermion systems in the second-quantised formalism. Thus we formulate everything in terms of the tuple of quantum field operators {\widehat{\psi}_\alpha(x)}, {\alpha = 0, 1, \ldots, D-1}, obeying the CAR

\displaystyle \{\widehat{\psi}_\alpha(x), \widehat{\psi}^\dag_\beta(y)\} = \delta_{\alpha\beta}\delta(x-y), \ \ \ \ \ (1)

\displaystyle \{\widehat{\psi}_\alpha(x), \widehat{\psi}_\beta(y)\} = 0, \ \ \ \ \ (2)

and

\displaystyle \{\widehat{\psi}^\dag_\alpha(x), \widehat{\psi}^\dag_\beta(y)\} = 0. \ \ \ \ \ (3)

In the case that {D = 2} one can think of each of {\widehat{\psi}_\alpha(x)} as modelling the two spin components of the electron field.

As we’ve seen, a non-relativistic many fermion system is characterised by a second quantised hamiltonian with form

\displaystyle \widehat{H} = \widehat{T} + \widehat{V} + \widehat{W}, \ \ \ \ \ (4)

where

\displaystyle \widehat{T} = -\frac{\hbar^2}{2m} \sum_{\alpha} \int d^3 x \, \widehat{\psi}^\dag_\alpha(x) \nabla^2 \widehat{\psi}_\alpha(x), \ \ \ \ \ (5)

\displaystyle \widehat{V} = \sum_{\alpha} \int d^3 x \, \widehat{\psi}^\dag_\alpha(x) v(x) \widehat{\psi}_\alpha(x), \ \ \ \ \ (6)

and

\displaystyle \widehat{W} = \frac{1}{2} \sum_{\alpha,\beta} \int d^3 xd^3 y \, \widehat{\psi}^\dag_\alpha(x) \widehat{\psi}^\dag_\beta(y) w(x,y) \widehat{\psi}_\beta(y)\widehat{\psi}_\alpha(x), \ \ \ \ \ (7)

where {w(x,y)} is some fixed two-particle interaction. In all the cases we’re interested in {w} is simply the Coulomb interaction.

The fundamental building block of DFT is the Hohenberg-Kohn theorem. At an abstract level this result simply trades one set of variational parameters for another, the local density, via a Legendre transform. The impact of this apparently trivial operation cannot be understated, however, as it unlocks a new way to study electronic systems. One motivation for this transformation is that the many-particle state vector {|\Psi\rangle} is not directly accessible in experiments. Rather, experiments can only access the one-particle density

\displaystyle \rho(x) = \langle \Psi|\widehat{n}(x)|\Psi\rangle = \langle \Psi| \sum_{\alpha} \widehat{\psi}^\dag_\alpha(x)\widehat{\psi}_\alpha(x) |\Psi\rangle \ \ \ \ \ (8)

\displaystyle = N \sum_{\alpha} \int dx_2 dx_3\cdots dx_N |\Psi(x\alpha, x_2, x_3, \ldots, x_N)|^2. \ \ \ \ \ (9)

The first important observation is that essentially the only thing that changes from physical situation to physical situation is the local one-particle potential {\widehat{V}}. Since we are typically interested in the ground-state properties our first step is to construct a special set {\mathcal{V}}, the set of all one-particle potentials with the property that the solution of the eigenvalue equation

\displaystyle \widehat{H} |\Omega\rangle = E_0|\Omega\rangle, \quad \widehat{V}\in \mathcal{V} \ \ \ \ \ (10)

leads to a non-degenerate ground state for a system of {N} fermions. We have thus defined, via solution of the time-independent Schrödinger equation, a map {C} from {\mathcal{V}} to a subset {\mathcal{G}\subset \mathcal{H}} of hilbert space. This map is surjective onto {\mathcal{G}} by construction.

The next step is to calculate the ground-state density {n(x)} for each state in {\mathcal{G}}, this gives a map {D:\mathcal{G}\rightarrow \mathcal{N}} to the set of densities via

\displaystyle D[|\psi\rangle] = \langle \psi|\widehat{n}(x)|\psi\rangle. \ \ \ \ \ (11)

This map is, again, surjective by construction.

The first statement of the Hohenberg-Kohn theorem is then that the maps {C} and {D}, and hence {D\circ C}, are injective, and hence bijective. To prove that {C} is injective one needs to show that for any two potentials {\widehat{V} \in \mathcal{V}} and {\widehat{V}' \in \mathcal{V}} always lead to different ground states whenever they differ by more than a constant, i.e.,

\displaystyle \widehat{V} - \widehat{V}' \not= \mbox{const.} \ \ \ \ \ (12)

This is because potentials differing by a constant are physically equivalent. The Schrödinger equation then yields

\displaystyle (\widehat{T} + \widehat{V} + \widehat{W})|\psi\rangle = E_0|\psi\rangle, \ \ \ \ \ (13)

and

\displaystyle (\widehat{T} + \widehat{V}' + \widehat{W})|\psi'\rangle = E_0'|\psi'\rangle, \ \ \ \ \ (14)

so that the assumption {|\psi\rangle = |\psi'\rangle} gives.

\displaystyle (\widehat{V} - \widehat{V}')|\psi\rangle = (E_{0} - E_{0}')|\psi\rangle. \ \ \ \ \ (15)

Since {\widehat{V}} acts via multiplication (in the position basis) we obtain, assuming that the wavefunction is nowhere vanishing, that

\displaystyle \widehat{V} - \widehat{V}' = E_0-E_0', \ \ \ \ \ (16)

contradicting our assumption. (This can be made rigourous, see e.g., Lieb (1982)).

To show that the map {D} is injective one needs to show that for two ground states {|\psi\rangle \not= |\psi'\rangle} then {n(x) \not= n'(x)}. Note that the variational characterisation of the ground eigenvalue implies that

\displaystyle E_0 = \langle \psi|\widehat{H}|\psi\rangle < \langle \psi'|\widehat{H}|\psi'\rangle. \ \ \ \ \ (17)

Also, we have that

\displaystyle \langle \psi'|\widehat{H}|\psi'\rangle = \langle \psi'|\widehat{H}' + \widehat{V} - \widehat{V}' |\psi'\rangle = E_0' + \int d^3\, n'(x)(v(x)-v'(x)). \ \ \ \ \ (18)

Similarly, repeating the argument for {E_0'} gives us

\displaystyle E_0' < E_0 + \int d^3\, n(x)(v'(x)-v(x)). \ \ \ \ \ (19)

Adding these two inequalities and supposing that {n(x) = n'(x)} we conclude that

\displaystyle E_0' + E_0 < E_0 + E'_0, \ \ \ \ \ (20)

which is a contradiction.

Thus we’ve learnt that the map {D} is invertible, meaning that any density {n(x)\in\mathcal{N}} defines a unique state {|\psi[n]\rangle} via

\displaystyle D^{-1}:n(x) \rightarrow |\psi[n]\rangle. \ \ \ \ \ (21)

Hence we obtain the first statement of the Hohenberg-Kohn theorem: the ground-state expectation value of any observable {\widehat{O}} is a unique functional of the ground-state energy density:

\displaystyle O[n]\equiv \langle\psi[n]|\widehat{O}|\psi[n]\rangle. \ \ \ \ \ (22)

The full inverse of the map {D\circ C} shows us that the ground-state density completely determines the external potential of the system (up to a constant):

\displaystyle (D\circ C)^{-1}: n(x) \rightarrow v(x). \ \ \ \ \ (23)

Since the kinetic energy and the interparticle interaction is already specified, the inverse map completely specifies the hamiltonian.

The second statement of the Hohenberg-Kohn theorem establishes a variational property of the functional

\displaystyle E_{v_0}[n] \equiv \langle\psi[n]|\widehat{T} + \widehat{V}_0 + \widehat{W}|\psi[n]\rangle, \ \ \ \ \ (24)

where {\widehat{V}_0} is the external potential of interest with ground-state density {n_0(x)} and ground-state energy {E_0}. The variational method directly implies that

\displaystyle E_0 < E_{v_0}[n], \quad \mbox{for} \quad n\not=n_0, \ \ \ \ \ (25)

and

\displaystyle E_0 = E_{v_0}[n_0]. \ \ \ \ \ (26)

We learn that the ground-state energy can be found via minimisation of {E_{v_0}[n]} with respect to {n} instead of {|\psi\rangle}:

\displaystyle E_0 = \inf_{n\in\mathcal{N}} E_{v_0}[n]. \ \ \ \ \ (27)

Since the map {D^{-1}} doesn’t depend on {\widehat{V}_0} we have that

\displaystyle E_{v_0}[n] = F_{{HK}}[n] + \int d^3 x \, v_0(x)n(x) \ \ \ \ \ (28)

where

\displaystyle F_{{HK}}[n] = \langle\psi[n]|\widehat{T} + \widehat{W}|\psi[n]\rangle. \ \ \ \ \ (29)

Remarkably {F_{{HK}}[n]} is universal meaning that it doesn’t depend on {\widehat{V}_0}. It is exactly the same whether we talk about atoms, molecules, solids, or gases since {\widehat{W}} is given in all cases by the Coulomb repulsion for electrons.

3. The Kohn-Sham scheme

The Hohenberg-Kohn theorem allows one to determine the ground-state density of a specific many particle system via a direct variation with respect to densities. In practice there are considerable advantages in replacing the direct variation with respect to the density with an intermediate orbital calculation. This results in a self-consistent scheme that is now the cornerstone of the density functional formalism.

To describe the Kohn-Sham scheme we consider an auxiliary system of {N} non-interacting particles with hamiltonian

\displaystyle \widehat{H}_s = \widehat{T} + \widehat{V}_s. \ \ \ \ \ (30)

According to the Hohenberg-Kohn theorem there is a unique energy functional

\displaystyle E_s[n] = T_s[n] + \int d^3 x \, v_s(x)n(x), \ \ \ \ \ (31)

such that the variational extremum given by {\delta E_s[n] = 0} yields the exact ground-state density {n_s(x)}. Here {T_s[n]} denotes the universal kinetic-energy functional for non-interacting particles.

The main assertion of the Kohn-Sham scheme is: for any interacting system there is a local single-particle potential {v_s(x)} such that the exact ground-state density {n(x)} of the interacting system is identical to the ground-state density of the auxiliary problem, i.e.,

\displaystyle n(x) = n_s(x). \ \ \ \ \ (32)

If the ground state of {\widehat{H}_s} is nondegenerate, the ground-state density {n_s(x)} is uniquely represented by

\displaystyle n(x) = n_s(x) = \sum_{j=1}^N |\phi_j(x)|^2 \ \ \ \ \ (33)

where

\displaystyle \left(-\frac{\hbar^2}{2m}\nabla^2 + v_s(x)\right)\phi_j(x) = \epsilon_j \phi_j(x), \quad \epsilon_1 < \epsilon_2 < \cdots. \ \ \ \ \ (34)

The degenerate case is more subtle, and we don’t discuss it here (see, e.g., Dreizler (1991) for the complete case). Having assumed the existence of {v_s(x)} the uniqueness of {v_s(x)} follows immediately follows from the Hohenberg-Kohn theorem. This implies that the single-particle orbitals {\phi_j(x)} are also unique functionals of the density {n(x)}

\displaystyle \phi_j(x) = \phi_j(n; x) \ \ \ \ \ (35)

and hence so is the non-interacting kinetic energy via

\displaystyle T_s[n] = \sum_{j=1}^N \int d^3 x \, \overline{\phi}_j(x)\left(-\frac{\hbar^2}{2m}\nabla^2\right) \phi_j(x). \ \ \ \ \ (36)

Now we are going to show how to obtain the single-particle potential {v_{s,0}(x)} of our auxiliary non-interacting system which generates the ground-state density {n_0(x)} of a given interacting system with potential {v_0(x)}.

We start with the single-particle equations

\displaystyle n_0(x) = \sum_{j=1}^N |\phi_{j,0}(x)|^2 \ \ \ \ \ (37)

where

\displaystyle \left(-\frac{\hbar^2}{2m}\nabla^2 + v_{s,0}(x)\right)\phi_{j,0}(x) = \epsilon_{j,0} \phi_{j,0}(x), \quad \epsilon_{1,0} < \epsilon_{2,0} < \cdots. \ \ \ \ \ (38)

We then rewrite the total energy functional {E_{v_0}[n]} for the interacting system by adding and subtracting {T_s[n]} and a Hartree term:

\displaystyle E_{v_0}[n] = T_s[n] + \int d^3x\, v_0(x)n(x) + \frac12 \int\int d^3xd^3x'\, n(x)w(x,x')n(x') + E_{xc}[n], \ \ \ \ \ (39)

where the exchange-correlation functional {E_{xc}[n]} is defined to be

\displaystyle E_{xc}[n] = F_{HK}[n] - \frac12 \int\int d^3xd^3x'\, n(x)w(x,x')n(x') - T_s[n]. \ \ \ \ \ (40)

Via the Hohenberg-Kohn theorem we know that {E_{v_0}[n]} is stationary with respect to small variations {n_0(x) \mapsto n_0(x) + \delta n(x)}:

\displaystyle 0 = E_{v_0}[n_0 + \delta n] - E_{v_0}[n_0] \ \ \ \ \ (41)

\displaystyle = \delta T_s + \int d^3 x \, \delta n(x) \left[v_0(x) + \int d^3x'\, w(x,x')n_0(x') + v_{xc}(n_0; x)\right]. \ \ \ \ \ (42)

Here {v_{xc}(n_0; x)} denotes the exchange-correlation potential, given by

\displaystyle v_{xc}(n_0; x) = \frac{\delta E_{xc}[n]}{\delta n(x)}\Bigg|_{n_0}. \ \ \ \ \ (43)

There is a subtle point to mention here: not all variations {n_0(x) + \delta n(x)} are allowed, instead, only those which could have arisen from a local potential. Now our main assumption ensures that this variation can be represented with a unique variation {v_{s,0}(x) + \delta v_s(x)} of the local potential of the single-particle problem. This, in turn, leads to a variation {\phi_{j,0}(x) + \delta \phi_j(x)} of the single-particle orbitals. The variation of {T_s[n]} may now be expressed in terms of these orbital variations:

\displaystyle \delta T_s = \sum_{j=1}^N \int d^3 x\, \left[\delta\overline{\phi}_j(x) \left(-\frac{\hbar^2}{2m}\nabla^2\right)\phi_{j,0}(x) + \delta\phi_{j,0}(x)\left(-\frac{\hbar^2}{2m}\nabla^2\right)\overline{\phi}_{j,0}(x)\right], \ \ \ \ \ (44)

which follows from Green’s theorem. Since the orbitals {\phi_{j,0}(x)} obey the Schrödinger equation we obtain, to first order,

\displaystyle \delta T_s = \sum_{j=1}^N \epsilon_j \int d^3 x\, \delta|\phi_j(x)|^2 - \sum_{j=1}^N \int d^3 x\, v_{s,0}(x)\delta|\phi_j(x)|^2. \ \ \ \ \ (45)

The first term vanishes thanks to the normalisation of the wavefunctions. This, combined with (42), leads to

\displaystyle v_{s,0}(x) = v_0(x) + \int d^3x'\, w(x,x') n_0(x') + v_{xc}(n_0; x). \ \ \ \ \ (46)

This is a self-consistent equation which can be solved via iteration.

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