Role of the action function in defining electronic wavefunctions for large systems

Abstract

The dimension of the Hilbert space needed for the description of an interacting electron system increases exponentially with electron number $N$. As pointed out by W. Kohn this exponential wall problem (EWP) limits the concept of many-electron wavefunctions based on solutions of Schr\"odinger's equation to $N \leq N_0$, where $N_0 \approx 10^3$ \cite{Kohn1999}. This limitation does not hold when the electronic interactions are neglected or treated in a mean-field approximation. The EWP has directed electronic structure calculations for solids to methods like density-functional theory which avoid dealing with many-electron wavefunctions. We show that the highly unsatisfactory limitation of many-electron wavefunctions to $N \leq N_0$ can be overcome by generalizing their definition. We show for the ground state of a large system that it is the logarithm of the solution of Schr\"odinger's equation which should be used to characterize this system. In this case the wavefunctions of independent subsystems add up rather than multiply. This feature is realized by the action function. It provides a simple physical picture for the resolution of the EWP.