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An Introduction to Density Functional Theory (DFT) and Derivatives

1.1 The Problem of a N-electron System

The density functional theory (DFT) is first resulted from the work by Hohenberg and Kohn [1], wherein the complicated individual electron orbitals are substituted by the electron density. Namely, the DFT is entirely expressed in terms of the functional of electron density, rather than the many-electron wave functions. In this case, DFT significantly reduces the calculations of the ground state properties of materials. That is why DFT is useful for calculating electronic structures, especially with many electrons. As the foundation of DFT, two theorems are proposed by Hohenberg and Kohn [1]. The first theorem presents that the ground state energy is a functional of electron density. The second theorem shows that the ground state energy can be achieved by minimizing system energy on the basis of electron density.

It should be noted that, although Hohenberg and Kohn point out there are relations between properties and electron density functional, they do not present the exact relationship. But fortunately, soon after the work of Hohenberg and Kohn, Kohn and Sham simplified the many-electron problems into a model of individual electrons in an effective potential [2]. Such a potential contains the external potential and exchange-correlation interactions. For exchange-correlation potential, it is a challenge to describe it rigorously.

The simplest approximation for treating the exchange-correlation interaction is the local density approximation (LDA) [3], wherein the exchange and correlation energies are obtained by the uniform electron gas model and fitting to the uniform electron gas, respectively. LDA can provide a realistic description of the atomic structure, elastic, and vibrational properties for a wide range of systems. Yet, because LDA treats the energy of the true density using the energy of a local constant density, it cannot describe the situations where the density features rapid changes such as in molecules [4, 5]. To address this problem, the generalized gradient approximation (GGA) is proposed [6-8], which depends on both the local density and the spatial variation of the density. And in principle, GGA is as simple to use as LDA. Currently, in the vast majority of DFT calculations for solids, these two approximations are adopted.

By considering the Born-Oppenheimer and non-relativistic approximations, the effective Hamiltonian of a N-electron system in the position representation can be given by,

The first term is kinetic energy operator. The second term is an external potential operator. In systems of interest to us, the external potential is simply the Coulomb interaction of electrons with atomic nuclei:

where the ri is the coordinate of electron i and the charge on the nucleus at Ra is Za. The third term of Eq. (1.1) is the electron-electron operator. The electronic state can be obtained by the Schrödinger equation:

Here, ?(r1, r2, rN) is a wave function in terms of space-spin coordinates. Apparently, the wave function is antisymmetric under exchanging the coordinates. Under Dirac notation, the Eq. (1.1) can be expressed in representation-independent formalism:

In principle, the ground state energy E0 of the N-electron system can be found based on the variational theorem, which is obtained by the minimization:

Here, the search is over all antisymmetric wave functions ?. In this regard, better approximations for ? can readily result in the ground state energy E0 of the N-electron system, but the computational cost would be very high. Therefore, the direct solution is not feasible. To address this issue, DFT is developed, which is based on a reformulation of the variational theorem in terms of electron density.

We know that |?|2 = ?*? represents the probability density of measuring the first electron at r1, the second electron at r2, . and the Nth electron at rN. By integrating |?|2 over the first N?-?1 electrons, the probability density of the Nth electron at rN is determined. Then the probability electron density that defines any of the N electrons at the position r is given by:

And the electron density is normalized to the electron number:

The energy of the system is expressed as:

Here,

1.2 The Thomas-Fermi Theory for Electron Density

Before discussing the Hohenberg-Kohn theorems, we first introduce the Thomas-Fermi theory. The Thomas-Fermi theory is important as it gives the relation between external potential and the density distribution for interacting electrons moving in an external potential:

Here,

and µ is the r independent chemical potential. The second term is Eq. (1.12) is the classical electrostatic potential raised by the density ?(r). Based on the Thomas-Fermi theory, Hohenberg and Kohn build up the connection between electron density and the Schrödinger equation. And in the following, we will introduce the two Hohenberg-Kohn theorems, which lie at the heart of DFT.

1.3 The First Hohenberg-Kohn Theorem

By replacing the external potential vne(r) with an arbitrary external local potential v(r), the corresponding ground state wave function ? can be found by solving the Schrödinger equation. Based on the obtained wave function, the ground state density ?(r) can be computed. And obviously, two different local potentials would give two different wave functions and thus two different electron densities. This gives the map:

Based on the Thomas-Fermi theory, Hohenberg and Kohn demonstrated that the preceding mapping can be inverted, namely, the ground state electron density ?(r) of a bound system of interacting electrons in some external potential v(r) determines the potential uniquely:

This is known as the first Hohenberg-Kohn theorem.

To demonstrate this theorem, we consider two different local potentials v1(r) and v2(r), which differ by more than the constant. These two potentials yield two different ground state wave functions ? and ?´, respectively. And apparently, these two ground state wave functions are different. Assume v1(r) and v2(r) correspond to the same ground state wave function, then

By subtracting Eq. (1.17) from Eq. (1.16), we can obtain:

which can be expressed in position representation,

This suggests that

thus in contradiction with the assumption that v1(r) and v2(r) differ by more than a constant. Accordingly, two different local potentials that differ by more than the constant cannot share the same ground state wave function, which demonstrate the map:

Then, we demonstrate the map:

Let ? and ?´ be the ground state wave functions corresponding to v1(r) and v2(r), respectively. Assuming that ? and ?´ exhibit the same ground state electron density ?(r), then the variational theorem gives the ground state energy as:

By subtracting Eq. (1.23) from Eq. (1.24), we can obtain:

This makes no sense. This finally leads to the conclusion that there cannot exist two local potentials differing by more than an additive constant that has the same ground state density.

1.4 The Second Hohenberg-Kohn Theorem

According to the first Hohenberg-Kohn theorem, the ground state density ?(r) determines the local potential v(r), and in turn determines the Hamiltonian. Therefore, for a given ground state density ?0(r) that is generated by a local potential, it is possible to compute the corresponding ground state wave function ?0. That is to say, ?0 is also a unique functional of...