Abstract: In this paper, by constructing hard examples of CSP (with large domains) and SAT (with long clauses), we prove that such examples cannot be solved without exhaustive search, which implies a weaker conclusion P NP. This constructive approach for proving impossibility results is very different (and missing) from those currently used in computational complexity theory, but is similar to that used by Kurt Gödel in proving his famous logical impossibility results. Just as shown by Gödel's results that formal unprovability is provable in mathematics, the results of this paper show that proving computational hardness is not hard in mathematics. Abstract: We propose a nonvariational scheme for geometry optimization of molecules for the first-quantized eigensolver, a recently proposed framework for quantum chemistry using the probabilistic imaginary-time evolution (PITE) on a quantum computer. While the electrons in a molecule are treated in the scheme as quantum mechanical particles, the nuclei are treated as classical point charges.