Impressive advances have been made throughout the years in the study of atomic
structure, at both the experimental and theoretical levels. For atomic hydrogen and
other equivalent two-body systems, exact analytical solutions to the
nonrelativistic Schroedinger equation are known. It is now possible to calculate
essentially exact nonrelativistic energies for helium (He) and other three-body
(two-electron) systems as well. Even for properties other than the nonrelativistic
energy, the precision of the calculation has been referred to as essentially exact
for all practical purposes, i.e., the precision goes well beyond what can be
achieved experimentally. These high-precision results for two-electron systems
have been produced using wave functions which include interelectronic coordinates,
a trademark of the classic Hylleraas (Hy) calculations in the 1920s. The Hy-CI method
is a method which includes interelectronic coordinates in the wave function to mimic
the high precision of Hy methods, but also includes configurational terms that are
the trademark of the conventional Configuration-Interaction (CI) methods employed in
calculating energies for many-electron atomic (and molecular) systems. The Hy-CI method
has been called a hybrid method, since it includes both configurations and
interelectronic coordinates in the terms of a variational expansion (wave function).
In any attempt to get very precise energies, large basis sets have to be employed,
which means that linear dependence in the basis set is never very far away. To
proceed to several thousand terms in a wave function, extended precision
arithmetic is needed to obviate the linear dependence problem, which in turn leads to
higher CPU costs. The use of several thousand terms in a wave function leads to
memory problems arising from storage of the matrix elements prior to the matrix
diagonalization step. And this is true already for the two-electron case
of helium (He) and its isoelectronic ions (He-like systems). Where three electron
atomic systems (lithium (Li) and other members of its isoelectronic series) have been treated
essentially as accurately as He-like systems, demand on computer resources has
increased by 6000 fold. Because of these computational difficulties, already in the
four electron case (beryllium (Be) and other members of its isoelectronic series) there
are no calculations of the ground or excited states with an error of less than 10
microhartrees (0.00001 a.u.). The challenge for computational scientists is to
extend the phenomenal He accomplishments to three, four, and more electron
atomic systems. This is where the Hy-CI method becomes important, because the use
of configurations whereever possible leads to less difficult integrals than in
a purely Hy method, and if one restricts the wave function to at most a single
interelectronic coordinate to the first power, then the most difficult integrals
are already dealt with at the four electron level and the calculation retains the
precision of Hy techniques, but is greatly simplified.
The Hy-CI Method has been parallelized.
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Why Parallelize the Hy-CI Method?
Even using the Hy-CI Method for two electrons, extended precision arithmetic, large
basis sets, and extended precision places high CPU and memory costs on a high
precision calculation. The solution to these problems, for both CPU speed
and memory needs, is to parallelize the calculation. This enables high precision
Hy-CI method calculations and opens up the possibility of Hy-CI method calculations
for three, four (and hopefully more) electron atomic systems.
The benefits from this work should be obvious from the fact that
the technique is still being used today and that the
original work of SAVG Computational Scientist James Sims, in collaboration with
Stanley Hagstrom of Indiana University, from 1971 to
1976 is still being referenced in the (peer-evaluated) literature.
In 1996, in a review article in Computational Chemistry, it was declared that this method
is nearly impossible to use for more that 3 or 4 electrons. We believe that
while that may have been true in 1996, it is no longer true today due to the availability
of cheap CPUs which can be connected in parallel to enhance both the CPU power and the
memory that can be brought to bear on the computational task.
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How is the Parallelization Realized?
The variational method solution to Schroedinger's equation involves computing matrix
elements for a well-known matrix eigenvalue (secular) equation and then solving the
equation by solving the N-dimensional generalized eigenvalue problem
HC = lambda . SC
This generalized eigenvalue problem is solved using
a technique called inverse iteration. Since the
inverse iteration solver matrix representation is a blocked one, we modified the
secular equation step to generate matrix elements in the appropriate block order.
Then Message Passing Interface (MPI) standard was used to run the same
program on multiple processors (on the same or
different hosts) and give each host a block of the matrix, with no need to
redistribute the matrices for the inverse iteration step. The MPI code uses blocking
sends and receives to add up the pieces of the matrix scattered across the
processors (by doing the equivalent of an MPI_reduce and then a MPI_gather).
Hence the calculation of the blocks of the matrix runs in parallel, and the blocks
of the matrix are spread across processors, hence solving both the memory problem
(by spreading the arrays across the entire memory of the cluster) and the CPU
speed problem (by running the calculation in parallel on different processors in
the cluster).
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What is the Performance of the Parallel Code?
NIST's Scientific Computer Facility cluster of 16 PCs running Windows NT was utilized
for parallel computation of the ground state of He and He-like ions. Typical run times
for a calculation of this size about are 8 hours on a single CPU, but only 30 - 40 minutes
on the parallel processing cluster.
We found that the processing speed could be
predicted, as a function of cluster size, by the simple scaling
law T = constant (s + (1 - s) /N), where T is the
runtime in seconds, N is the number of processors, the constant is 6419 for this case,
and s is the inherently sequential part of the calculation.
As far as we know, this is the first high
precision calculation for few electron atomic systems to employ parallel computing.
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Hy-CI method has been used to compute not only energy levels, but also other atomic properties such as ionization potentials, electron affinities, electric polarizabilities, and transition probabilities of two, three, and four electon atoms and other members of their isoelectronic sequences. In a recent work, Hy-CI method variational calculations with up to 4648 expansion terms have been carried out for the ground singlet S state of neutral helium and 4 of its isoelectronic ions, H-, Li+, Be++, and B+++. This has resulted in high-precision results for the nonrelativistic energies that are believed to be accurate to 20 decimal digits. This work employs a very novel wave function, namely, one consisting of at most a single r12 raised to the first power combined with a conventional non-orthogonal configuration interaction (CI) basis. We believe that this technique can be extended to multielectron systems (more than 3 or 4 electrons). The combination of computational simplicity of this form of the wave function, compared to other wave functions of comparable accuracy, as well as the use of parallel processing and extended precision arithmetic, make it possible (we believe) to achieve levels of accuracy comparable to what has been achieved for He (a 2 electron atom), for atoms with more than 2 electrons. Work is in progress, for example, to see what precision can be obtained for atomic beryllium, the key to multielectron (more than 4 electrons) systems, since the integrals that arise for more than 4 electrons are of the same type as the ones that arise in the 4 electon systems. A recent publication of ours discusses the most difficult integral arising in Hy-CI calculations, the three-electron triangle integral. We find that a direct plus tail Levin-u transformation convergence acceleration is the best method for overcoming the slow convergence of this integral, and removes the real bottleneck to highly accurate Hy-CI calculations. Now that this bottleneck has been removed, doing really accurate calculations on atoms with N greater than or equal to 5 becomes a real possibility.
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Papers/Presentations
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James S. Sims and Stanley A. Hagstrom, Hy-CI Study of the 22S Ground State of Neutral Lithium and the First Five Excited 2S States,
Physical Review A, 80,
2009.
ID: 052507.
Note: DOI: 10.1103/PhysRevA.80.052507 |
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James S. Sims and Stanley A. Hagstrom, Math and computational science issues in high-precision Hy-CI calculations II. Kinetic Energy and electron-nucleus int
eraction integrals,
Journal of Physics B: Atomic, Molecular, and Optical Physics, 40,
2007,
pp. 1575-1587.
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James S. Sims and Stanley A. Hagstrom, James S. Sims and Stanley A. Hagstrom, High Precision Variational
Born-Oppenheimer Energies of the Ground State of the Hydrogen Molecule ,
Journal of Chemical Physics, 124
(9)
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3/7/2006,
p. 7.
Note: 09410
Links:
postscript, pdf and html.
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James S. Sims and Stanley A. Hagstrom, Math and computational science issues in high-precision Hy-CI calculations I. Three-electron integrals,
Journal of Physics B: Atomic, Molecular, and Optical Physics, 37
(7)
,
2004,
pp. 1519-1540.
Links:
postscript and pdf.
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James S. Sims and Stanley A. Hagstrom, Analytic Value of the Atomic Three-electron Integral with Slater Wave Functions,
Physical Review A, 68,
2003,
p. 016501.
Note: Comment: Phys. Rev. A 44,5492(1991)
Links:
postscript and pdf.
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James S. Sims and Stanley A. Hagstrom, High Precision Hy-CI Variational
Calculations for the Ground State of Neutral Helium and Heliumlike Ions,
International Journal of Quantum Chemistry, 90
(6)
,
2002,
pp. 1600-1609.
Note: DOI 10.1002/qua.10344
Links:
postscript, pdf and html.
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James S. Sims, William L. George, Steven G. Satterfield, Howard K. Hung, John G. Hagedorn, Peter M. Ketcham, Terence J. Griffin, Stanley A. Hagstrom, Julien C. Franiatte, Garnett W. Bryant, W. Jaskolski, Nicos S. Martys, Charles E. Bouldin, Vernon Simmons, Olivier P. Nicolas, James A. Warren, Barbara A. am Ende, John E. Koontz, B. James Filla, Vital G. Pourprix, Stefanie R. Copley, Robert B. Bohn, Adele P. Peskin, Yolanda M. Parker and Judith E. Devaney, Accelerating Scientific Discovery Through Computation
and Visualization II,
NIST Journal of Research, 107
(3)
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May-June, 2002,
pp. 223-245.
Links:
postscript and pdf.
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