Linear response methods

Conventional density functional theory calculations are always iterative, irrespective of the method of solving the Hamiltonian matrix, since the effective potential $V({\bf r})$ which includes electron-electron interaction depends on the electronic charge density $n({\bf r})$, but $n({\bf r})$ is calculated from the wavefunctions which are the ground states of $V({\bf r})$. A self-consistent solution is thus found from an initial guess for the wavefunctions (or, equivalently, the charge density or potential). The electronic energy is then given by $E = \int \! n({\bf r}) V({\bf r}) d {\bf r}$,

However, the electronic structure and energy will change in response to a perturbation on the potential. If the perturbed potential is written as

$$V_\lambda({\bf r}) = V_0({\bf r}) + \lambda \Delta V({\bf r})$$ (4.7)

then

$$\frac{\partial E_\lambda}{\partial \lambda} = \int n_\lambda({\bf r}) \Delta V({\bf r}) d {\bf r}$$ (4.8)

and

$$\frac{\partial^2 E_\lambda}{\partial \lambda^2} = \int \frac{... ...lambda({\bf r})}{\partial \lambda} \Delta V({\bf r}) d {\bf r}$$ (4.9)

Thus the second order change in energy requires only the first order change in the charge density and potential, hence the term linear response. If the displacement of one atom is taken as $\lambda$ then (4.9) becomes exactly what is needed for the force constants in (4.2).

If the equations of density functional theory are linearised, as in the form of (4.7), then a similar methodology to a conventional DFT calculation may be followed, but working instead with the derivatives of the potential, the wavefunctions and the charge density with respect to atomic displacements (18). This is by no means trivial, but has the great advantage that the wavevector ${\bf q}$ of a periodic perturbation may take any value (although in practice it is calculated over a grid) since the perturbation may be made periodic and dealt with analytically, whilst only requiring a calculation with a single primitive cell. The anharmonicity inherent in the finite-displacement method is also removed. This method naturally gives elements of the Fourier-transformed dynamical matrix rather than the real-space force constants.

The dielectric tensor $\epsilon$$^\infty$ may be found using a similar procedure, since it is related to the first order change in the wavefunctions under a perturbing electric field (17). The Born effective charges may be found either within the linear response formalism (17) or from finite difference displacements and direct calculation of the resulting change in polarisation (19). Results for stishovite are shown with the finite-displacement results in Table 4.1.

This method has been successfully applied in several systems, particularly those with a small primitive cell so that effort may be focussed on obtaining results at many wavevectors (17; 18; 20; 21). With a sufficiently fine grid of wavevectors, real-space force constants may be deduced from the Fourier-transformed dynamical matrix to not only produce smooth curves at all wavevectors, but study the real-space interactions (22; 23).

The academic code has been independently developed to include linear-response methods, and results have been used to parametrise model Hamiltonians (24). It is currently being implemented in the commercial MSI version, with a release scheduled for late 1999.