Most Accurate Molecular Quantum Computations to Date Performed
January 2006
Computational scientists at NIST and Indiana University have achieved record
levels of accuracy in the development of computational methods for the virtual
measurement of fundamental properties of molecules. Their recent result for
the ground state of dihydrogen (H_{2}) represents the highest level of
accuracy ever reached (1012 hartree) in molecular quantum computations (except
for trivial oneelectron cases). H_{2} has the distinction of being
the first molecule whose dissociation energy was correctly predicted by quantum
mechanical calculation (1968) before being measured reliably by experiment.
Today we may be witnessing again a situation in which quantum mechanical
calculations yield more accurate determinations of this fundamental property
than can be measured experimentally.
Exact analytical solutions to the nonrelativistic molecular electronic
Schrodinger equation are known (in the BornOppenheimer approximation) only for
the oneelectron H_{2}^{+} ion and other equivalent one
electron systems. However, very high precision approximations are now
available for molecular hydrogen (a twoelectron system). MCSD computational
scientist James Sims, in collaboration with Stanley Hagstrom of Indiana
University, has calculated BornOppenheimer energies of
_{1}Σ_{g}^{+} states of H_{2} using up to
7034 expansion terms in confocal elliptical coordinates with explicit inclusion
of interelectronic distance coordinates up through r127. Sims and Hagstrom
calculate
BornOppenheimer (BO) energies for various internuclear distances
in the range of 0.4 to 6.0 bohr with an error of 1 in the 13th digit. For
example, the nonrelativistic energy is 1.1744 7571 4220(1) hartree at
R = 1.4 bohr, and is 1.1744 7593 1399(1) hartree at the equilibrium
R = 1.4011 bohr distance. In each case Sims and Hagstrom obtained lower
(i.e., more accurate) energies than previously reported results. In an article
recently accepted by the Journal of Chemical Physics, Sims and Hagstrom
discuss how these best calculations to date on the ground state of neutral
hydrogen were accomplished.
Almost all results reported in their paper were obtained using quadruple
precision (approximately 30+ digit) floating point subroutines written in
Fortran 90.
In addition, multiple precision floating point arithmetic was used for the
Rudenberg f function, on which all integrals depend, and which is subject to
differencing problems. To address these differencing problems the authors
systematically increased the number of decimal digits used for the f part of
the calculation up to a maximum of 160 decimal digits. Parallel processing
proved essential for obtaining results over a range of R from 0.4 to 6.0 bohr.
The authors solved the secular equation using their own portable parallel
inverse iteration eigensolver. The construction of the H and S matrices was
also parallelized since that allows the total memory needed to be spread across
multiple processors and eliminates a need to communicate matrix elements
between processors. For a 4190 term wave function they achieved a factor of
30 speedup on 32 processors for the most computationally intensive portion of
the computation while running on a NIST 147 processor cluster of Pentium,
Athlon, and Intel processors running RedHat Linux.
