Is Large-Scale Ab initio Hartree-Fock Calculation Chemically Accurate? Toward Improved Calculation of Biological Molecule Properties

Numerical errors in total energy values in large-scale Hartree-Fock calculations are discussed. To obtain total energy values within chemical accuracy, 0.01 kcal/mol, stricter numerical accuracy is required as basis size increases. In molecules with 10,000 basis sizes, such as proteins, numerical accuracy for total energy values must be retained to at least 11 digits (i.e., to the order of 1.0D-10) to keep accumulation of numerical errors less than the chemical accuracy (0.01 kcal/mol). With this criterion, we examined the sensitivity analysis of numerical accuracy in Hartree-Fock calculation by uniformly replacing the last bit of the mantissa part of a double-precision real number by zero in the Fock matrix construction step, the total energy calculation step, and the Fock matrix diagonalization step. Using a partial summation technique in the Fock matrix generation step, the numerical error for total energy value of molecules with basis size greater than 10,000 was within chemical accuracy (0.01 kcal/mol), whereas with the conventional method the numerical error with several thousand basis sets was larger than chemical accuracy. Computation of one Fock matrix element with parallel machines can include the partial summation technique automatically, so that parallel calculation yields not only high-performance computing but also more precise numerical solutions than the conventional sequential algorithm. We also found that the numerical error of the Householder-QR diagonalization routine is equal to or less than chemical accuracy, even with a matrix size of 10,000.