Computation of Atomic Properties with the Hy-CI Method
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).
Additional Technical Details:
- Why is the Hy-CI Method Important?
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 0.1 microhartrees (0.0000001 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 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 the early parallel work, the 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 to achieve levels of accuracy comparable to what has been achieved for He (a 2 electron atom), for atoms with more than 2 electrons. Indeed, the technique has been recently used to obtain results for the nonrelativistic energies for four excited states of the lithium atom which represent the highest level of accuracy ever reached (less than 10-9 hartree) in atomic quantum computations with more than two electrons. Work is in progress 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. One of our publications 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 the 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.
- J. S. Sims and S. A. Hagstrom,Hylleraas-configuration-interaction study of the 1S ground state of neutral beryllium, Physical Review A, 83, 2011. ID: 032518.
Note: DOI: 10.1103/PhysRevA.83.032518
- 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. ( DOI: 10.1103/PhysRevA.80.052507 )
- 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.
- 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) , 3/7/2006, p. 7. (09410)
- 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.
- 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. ( Comment: Phys. Rev. A 44,5492(1991) )
- 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
- 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) , May-June, 2002, pp. 223-245.
Additional Project Information:
The Hy-CI Method has been parallelized.
- Why Parallelize the Hy-CI Method?
Even using the Hy-CI Method for two electrons, large basis sets and extended precision arithmetic place 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 HPCVG 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.
- 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. The 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 an MPI_gather). 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).
- 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 was the first high precision calculation for few electron atomic systems to employ parallel computing. Our current parallel environment for this work is NIST's Central Computing Facility Linux cluster.
- Parallel Algorithms and Implementation: James S. Sims
- Scientists: James S. Sims and Stanley A. Hagstrom
- Group Leader: Judith E. Terrill