  In this work we propose a novel approach to solve the Schr\"{o}dinger
equation which combines projection onto the ground state with a path-filtering
truncation scheme. The resulting projector configuration interaction (PCI)
approach realizes a deterministic version of the full configuration interaction
quantum Monte Carlo (FCIQMC) method [Booth, G. H.; Thom, A. J. W.; Alavi, A. J.
Chem. Phys. 2009, 131, 054106]. To improve upon the linearized imaginary-time
propagator, we develop an optimal projector scheme based on an exponential
Chebyshev expansion in the limit of an infinite imaginary time step. After
writing the exact projector as a path integral in determinant space, we
introduce a path filtering procedure that truncates the size of the
determinantal basis and approximates the Hamiltonian. The path filtering
procedure is controlled by one real threshold that determines the accuracy of
the PCI energy and is not biased towards any determinant. Therefore, the PCI
approach can equally well describe static and dynamic electron correlation.
This point is illustrated in benchmark computation on N$_2$ at both equilibrium
and stretched geometries. In both cases, the PCI achieves chemical accuracy
with wave functions that contain less than 0.5% of the full CI space. We also
report computations on the ground state of C$_2$ with up to quaduple-$\zeta$
basis sets and wave functions as large as 200 million determinants, which allow
a direct comparison of the PCI, FCIQMC, and density matrix renormalization
group (DMRG) methods. The size of the PCI wave function grows modestly with the
number of unoccupied orbitals and its accuracy may be tuned to match that of
FCIQMC and DMRG.
