Density Functional Theory

In the previous section, we have shown that the ``simulated annealing'' technique can be used to minimize the electronic energy of a system by optimizing the electronic degrees of freedom, the coefficients $c_k^i$, or to minimize the atomic potential energy by optimizing the atomic degrees of freedom, the positions ${\bf R}_\alpha$. But we have not yet introduced the quantum mechanical relation between the energy and the electronic coefficients and atomic positions, to obtain the forces on the atoms and coefficients (eq. 2.30). The reason is, that in principle various quantum mechanical (or even semi-empirical methods) can be used, such as the Hartree-Fock method or the configurational interaction method. Hartree-Fock, however, lacks for many practical systems the desired chemical accuracy while on the other hand the configurational interaction method, although it can be highly accurate, is unpractical due to its dramatic scaling of the computational expense with the number of electrons. In practice, therefore, the energy and the forces are obtained from an electronic structure calculation using density functional theory (DFT), which is both accurate and has a favorable scaling.

Density functional theory is based on the notion that for a many-electron system there is a one-to-one mapping between the external potential and the electron density: $v({\bf r}) \leftrightarrow \rho({\bf r})$. In other words, the density is uniquely determined given a potential, and vice versa. All properties are therefore a functional of the density, because the density determines the potential, which determines the Hamiltonian, which determines the energy ($E[\rho]$) and the wave function ($\Psi[\rho]$), from which all physical properties can be determined. This theorem was proven in 1964 by Hohenberg and Kohn, using the variational principle for systems with a non-degenerate ground-state. Hohenberg and Kohn also showed that for a given potential $v({\bf r})$, which corresponds with a ground-state $\Psi_0[\rho]$ and ground-state energy $E_0[\rho]$, the energy functional $E_v[\rho]$ has its minimum equal to $E_0[\rho]$ at the ground-state density (second Hohenberg-Kohn theorem), or:

$$\left< \Psi[\rho] \left \vert {\hat {\cal H}} \right\vert \P... ...r}) + T[\rho] + V_\mathrm{ee}[\rho] = E_v[\rho] \ge E_0[\rho]$$ (36)

with $T[\rho]$ the electronic kinetic energy functional and $V_\mathrm{ee}[\rho]$ the electron-electron interaction energy functional.

A prescription to obtain the energy from the $N$-electron density was given by Kohn and Sham in a one-electron formalism, posing that there exists an auxiliary system of $N$ non-interacting electrons, feeling only a local potential $v_s({\bf r})$ which yields exactly the same density as the system of interacting electrons with potential $v({\bf r})$. The electron density is represented by the sum of the densities of the $N/2$ doubly occupied single-particle spatial orbitals,

$$\rho({\bf r}) = 2 \sum_i^{occ} \left\vert \psi_i({\bf r}) \right\vert^2$$ (37)

and the Kohn-Sham (KS) expression for the electronic energy functional is given by:

$$E[\rho] = T_s[\rho] + V_\mathrm{N}[\rho] + J_\mathrm{ee}[\rho] + E_\mathrm{xc}[\rho]$$ (38)

Here, $T_s[\rho]$ is the kinetic energy of non-interacting electrons, $V_\mathrm{N}[\rho]$ describes the electron-nuclei interaction and the so-called Hartree term, $J_\mathrm{ee}[\rho]$, is the Coulombic interaction among the electrons:

$$T_s(\rho) = -\frac{\hbar^2}{2m}\sum^N_i \int\! \mathrm{d}{\bf r}\, \psi_i^*({\bf r}) \nabla^2 \psi_i({\bf r})$$ (39)

$$V_\mathrm{N}[\rho] = \int\! \mathrm{d}{\bf r}\, v^{\rm ext}({\bf r})\, \rho({\bf r})$$ (40)

$$J_\mathrm{ee}[\rho] = \frac{e^2}{2} \int\! \mathrm{d}{\bf r}\!\int\! \mathrm{d}{\bf r}^... ...rac{\rho({\bf r})\rho({\bf r}^\prime)}{\left\vert {\bf r-r^\prime} \right\vert}$$ (41)

The last term in equation 2.37 is the exchange correlation energy functional $E_\mathrm{xc}[\rho]$ which compensates for the electron-electron interaction $V_\mathrm{ee}$ being described only by the Coulomb interaction $J_\mathrm{ee}$ (thereby neglecting the exchange interaction and the electron correlation interaction) and for the kinetic energy functional which describes the kinetic energy for non-interacting electrons:

$$E_\mathrm{xc}[\rho] = V_\mathrm{ee} - J_\mathrm{ee}[\rho] + T[\rho] - T_s[\rho]$$ (42)

The exchange correlation energy functional also has to correct for a spurious self interaction arising from $J_\mathrm{ee}$, which is in ab initio methods canceled exactly by the exchange term, but in DFT actually is the largest contribution to $E_\mathrm{xc}[\rho]$.

The electronic ground-state density is found by minimizing the Kohn-Sham energy functional, which is achieved by solving the Kohn-Sham equations:

$${\hat {\cal H}}^\mathrm{KS} \psi_i = \left[ -\frac{\hbar^2}{... ...rho]}{\delta \rho({\bf r})} \right] \psi_i = \epsilon_i \psi_i$$ (43)

Thus far, the DFT expressions are exact. However, an exact expression for $E_\mathrm{xc}[\rho]$ is not known, and one is forced to use an approximate functional. Also, there is no straightforward way in which the exchange correlation functional can be systematically improved.

For an homogeneous electron gas or an electron gas with slow varying density, one can show that

$$E^\mathrm{LD}_\mathrm{xc} = \int\! \mathrm{d}{\bf r}\rho({\bf r}) \, \epsilon_\mathrm{xc} (\rho({\bf r}))$$ (44)

The exchange correlation energy density function $\epsilon_\mathrm{xc}(\rho({\bf r}))$ for the uniform electron gas is known to high accuracy from Monte Carlo calculations of Ceperly and Alder[35]. Several parameterizations were used for this type of functional, which are known as local density approximation (LDA) functionals. Especially in solid-state physics, LDA works surprisingly well despite its approximate nature. However, most chemical applications do not satisfy the restriction of slowly varying electron density and the LDA fails. Much better results are obtained by taking into account the density fluctuations, via the gradient of the density. This has led to the development of various so-called generalized gradient approximations (GGAs), of which the most popular is the Becke[36] exchange functional:

$$\epsilon_\mathrm{xc}^\mathrm{B} = -\beta \rho^{1/3} \frac{x^2}{(1+6\beta x \sinh^{-1} x) }$$ (45)

$$x = \frac{\vert\nabla \rho \vert}{\rho^{4/3}}$$ (46)

which has led to the acceptance of DFT as a valuable tool for computational chemistry, after its introduction in 1988. This functional contains one adjustable parameter $\beta$ which was chosen so that the sum of the LDA and Becke exchange terms accurately reproduce the exchange energies of six noble gas atoms, $\beta=0.0042$. A large number of other GGA functionals for both correlation and exchange have been developed, of which the most popular probably the 4 parameter LYP correlation functional[37] , the parameter-free Perdew86 correlation functional[38] and the Perdew91 exchange+correlation functional[39]. Throughout the present work, we will make use of Becke's exchange functional and the Perdew86 correlation functional. However, the search for the holy grail continues. Routes to improvement are for example the mixing in of exact (Hartree-Fock) exchange (so-called hybrid functionals, such as the B3LYP functional), the use of highly parameterized GGA functionals which are fitted to large sets of empirical molecular properties (e.g. the HCTH functional[40]), taking the Laplacian of the density into account (e.g. the BLAP functional[41]), correcting for the approximate kinetic energy term[42], correction for the spurious Coulombic self interaction (so-called SIC functionals[43]) and imposing of the correct asymptotic behavior (e.g. the VanLeeuwen-Baerends functional[44]). Other important progress, which makes DFT particularly popular by chemists, is made in the development of computer programs that calculate chemical and physical properties, such as spectroscopic observables, a posteriori, using the Kohn-Sham density. The reader interested in the technical details of density functional theory and its application in chemistry might wish to read references BeGr97,DFT,DFT2,BiBa00.