Principles of kinetic energy ill-conditioning

The phenomenon of kinetic energy or length-scale ill-conditioning is a familiar one within the plane wave approach to electronic structure calculations [1]. It is not, however, restricted to this approach and its effects are seen in many methods which use a large basis set [21,33,34].

The efficiency with which a function can be minimized using iterative techniques such as steepest descents or conjugate gradients is related to the condition number $\kappa=\omega_{\mathrm{max}}/\omega_{\mathrm{min}}$, where $\omega_{\mathrm{max}}$ and $\omega_{\mathrm{min}}$ are the extremal curvatures of the function about the minimum [35]. Minimization is most efficient when the condition number is small and the curvatures have a narrow range of values. On the other hand, when the curvatures take a wide range of values, the number of iterations required for convergence can become unacceptably large and the minimization is said to be ill-conditioned.

The curvatures of the total energy functional are determined by the eigenvalues of the Hamiltonian. Hence, the condition number $\kappa$ depends upon the ratio of the largest and smallest eigenvalues in the basis representation that is being used. With a large systematic basis, these eigenvalues span a broad range. As a result, the condition number is large, rendering the problem ill-conditioned. A significant source of this ill-conditioning is associated with the contribution to the total energy due to the kinetic energy $E_{\mathrm{kin}}$, which is given by

$$E_{\mathrm{kin}} = -\frac{\hbar^{2}}{2m} \sum_{n} f^{\ }_{n}... ...}) \nabla^{2} \psi^{\ }_{n}(\mathbf{r}) \mathrm{d} \mathbf{r}.$$ (13)

It is clear that high energy eigenstates are dominated by their large kinetic energy. These states contribute little to the total ground state energy, as they are unoccupied, yet they contribute greatly to the broadening of the eigenspectrum. The same argument does not hold, however, for the low-lying states for which the potential and kinetic contributions are more closely matched. This ill-conditioning may be alleviated, or preconditioned, by removing the effect of the kinetic energy operator for the high energy states, making them more degenerate, and hence reducing the width of the eigenspectrum, whilst leaving the low energy states unchanged.

In the plane wave approach, the effect of kinetic energy ill-conditioning is reduced by multiplying the steepest descents directions in reciprocal space by a diagonal preconditioning matrix which behaves as the inverse of the kinetic energy operator at high wave vectors and is a constant at low wave vectors [1]. Such a preconditioner, as pointed out in Ref. [21], is qualitatively equivalent to the exact preconditioner for a model Hamiltonian $\hat{X}$ given by

$$\hat{X} = 1 - k^{-2}_{0}\nabla^{2}.$$ (14)

The preconditioner for this model problem may be derived analytically in any basis, as shown in Section 4.