In many chemical modeling problems, only a small part of the full system needs t o be treated at a high level of accuracy; the remaining part of the system can be treated in a more approximate manner. Such models are said to be multiscaling, involving varying accuracy requirements in different regions of the chemical system. Our aim is the development of methods where different regions of space are treated at different resolutions and accuracies, thus allowing for a fully quantum-mechani cal partitioning of the molecular system without introducing arbitrary couplings between for instance a QM and a MM system. This can be realized through the use of scaling and detail (also known as wavelet) functions.
The key to the success of the approach is a separation of variables for operator kernels such as the Coulomb potential, allowing these operators to be written as a one-dimensional product of functions. The sparse, banded structure of the wavelet representation of the operator kernels ensures that the approach inherently scales linearly with system size.
Since the basic equation for solving the self-consistent field equations of wavelet theory is the same as the basic equation for continuum models, namely the Poisson and Helmholtz kernels, a seamless integration of the multiscale wavelet model to an infinite continuum can be achieved. The numerical accuracy problems plaguing continuum models due to the discretization of the boundary surface may be resolved since wavelets allow for strict error control.