High-precision%20atomic%20physics%20calculations%20using%20a%20computer%20algebra%20system%20and%20many-body%20perturbation%20theory

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High-precision atomic physics calculations using
a computer algebra system and many-body
perturbation theory

• Warren F. Perger, Michigan Tech University

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Outline

• The many-body problem in physics (a cartoon)
• Quantum field theory (another cartoon, or two)
• Need for high-precision atomic theory
• Many-body perturbation theory (MBPT), atomic
  physics, and computer algebra systems
• Feynman diagrammatic reduction (symbolic)
• Angular algebraic reduction (symbolic)
• Numerical evaluation of Slater (Rk) integrals
• Results for a simple case
• Ideas for future development

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The Many-body Problem...

• Given the initial conditions of n-bodies
  (objects) at a given time, find the positions,
  velocities, and other properties at some later
  time.
• An example celestial objects interacting under
  the force of their mutual gravitational
  attraction

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• This can be solved analytically for n2.
• Perturbation theory can be used for ngt2 objects.

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Why is high-precision important?

• Quantum Electrodynamic (QED) effects calculated
  for hydrogen Lamb shift of 1057.70 MHz Nobel
  prize, 1965, Tomonaga, Schwinger, Feynman for
  Development of QED
• Parity nonconservation requires theory lt 1
  precision to impact development of unified field
  theories, e.g. SU2X U1

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The Atomic System

• Requires quantum-mechanical description solution
  of the Schroedinger (or Dirac) equation

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Quantum mechanics and perturbation theory

• Formalism is systematic and well-known.
• Ability to carry out detailed calculations for an
  arbitrary system is impractical except for a few
  simple systems.
• The symbolic determination of relevant Feynman
  diagrams offers the possibility of an
  error-proof, robust, method by using Wicks
  theorem 1,2, which is an alternative to a
  diagrammatic reduction.

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• Let
• where H0 unperturbed Hamiltonian
• with

and
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• The wavefunctions are given by
• and the energy expressions by

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• Representation of the operators, that is the
  combination of a creation and an annihilation
  operator, is formally equivalent to the
  combination of two free lines in a corresponding
  Feynman (or Goldstone) graph.
• A few examples of symbolically reducing the terms
  using Reduce 3 and Mathematica 4,5,6.
  However, this is but the first, relatively
  simple, step towards a numeric result which can
  be compared with experiment.

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Feyman diagrams and quantum field theory

• R.P. Feynman, Theory of Positrons, Phys. Rev.,
  749-59, 1949.

b
d
time
Propagator
a
c
space
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A
...

A

A
P(f,i) Po(f,i) Po (f,A) P(A) Po
(A,i) Po (f,A) P(A) Po (A,A) P(A)Po
(A,i).
Where P(f,i) probability of propagation from
initial to final state Po(r,s) probability of
free propagation from s to r (intermediate,
virtual, states) P(A) probability of
interaction
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Motivation for using Wicks theorem
multiplication of operator strings (probabilities)
0gt is a string of operators acting on the
vacuum, e.g., 0gt aa 0cgt for the
one-particle, zero-hole, case and likewise V is
a string of second-quantized operators
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Feynman diagrams and Wicks theorem

• Example
• Combine two diagrammatic
• fragments
• Aaa aa and B ab ab ac ad given
• Wicks theorem.
• Then,
• ABABAB
• aa aa ab ab ac ad
• dacaa ab ab ad
• dab ab ac aa ad
• - dab dac ab ad
• dacaa ab ac ad
• dab dab ac ad

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The role of the computer algebra system

• Create a data structure capable of handling all
  attributes of a given list of second-quantized
  operators particle or hole, creation or
  annihilation, core or valence
• Create routines capable of symbolically
  performing Wicks theorem
• Create routines to reduce to numerically
  tractable form, typically Brandow form
• Produces results for both energies and transition
  matrix elements, for open-shell systems and
  multi-configuration atomic states, using a fully
  relativistic approach.
• Example of 1 particle, 0 hole (alkalais)

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Angular reduction

• These g-functions, which carry the many-body
  effects, along with the bra and ket, must next
  undergo a reduction into radial and angular
  parts, typically (again) done with Feynman
  diagrams specialized for this purpose 7.
• But the bra and ket can be each be represented
  with a Wigner 3-J symbol, and each g-function
  with a pair of 3-J symbols.
• The angular reduction is performed by using an
  existing package, Kentaro 8, and a Mathematica
  routine which finds the minimal angular reduction.

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.
.
and
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Symbolic angular reduction

• Once in the form of the product of an arbitrary
  number of 3-J (or Clebsch-Gordon) coefficients,
  Kentaro can be used.
• A Mathematica smart wrapper was developed, in
  order to assure the minimum angular reduction.

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Method of Calculation
WicksThm Mathematica Program to calculate MBPT
terms using Wicks Theorem
LaTeX Output
WTtoTeX Formats MBPT terms in LaTeX. Prepares
data for Kentaro formats results in LaTeX
Kentaro Angular Reduction
LaTeX Output
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Evaluate Calculation of atomic properties, e.g.
energies, transition matrix elements.
Basis set program to calculate relativistic
atomic orbitals
Parallel C/FORTRAN program to calculate
RL(a,b,c,d)
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How do we know its correct?

• Use software, where possible, to test itself,
  e.g. calculate E3 in two ways.
• Prohibit use of Brandow simplification, then
  numerically re-calculate and compare.
• Compare with experiment (last resort).

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Parallel processing and Many-body Perturbation
Theory (MBPT)

• MBPT terms can be summed in any order.
• Terms require a numerical basis set for
  evaluation.
• B-splines used to represent orbitals.
• CPU time on the order of minutes to hours per
  term.
• No terms can be left out.

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Numerical evaluation

• Evaluate written in C to exploit pointers
• ONE subroutine used for all terms (singles,
  doubles, etc.)
• Organization of terms into singles, doubles, etc.
  facilitates course-grained parallel processing
• Example of 1p0h illustrates need for parallel
  processing quadruple-excitation term requires
  more time than all other terms combined
• Client-proxy scheme permits utilization of
  processors from wide range of locations
• Heterogeneous collection of computing resources
  used (SGI Origin 2000, Linux cluster, Sun
  Solaris)

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Total execution time

• The total execution time can be estimated as

Where tRL is the time to evaluate a given Slater
integral, Nnumber of B-splines (40), n is the
order of perturbation theory, and L is the
maximum multipole moment (10). For a 750MHz
Pentium III running RedHat Linux 7.2, this is
about 1ms. Therefore, a second-order energy term
takes roughly 402 x 9 x 10-3 14 seconds. A
quadruples term, however, takes roughly 194 days.
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Solutions to the numerical problems

1. Improve on speed of Rk integrals
2. Employ parallel processing.

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Improving speed of Rk integrals

• B-spline representation is an efficient basis set
  (Bottcher Strayer Notre Dame group)

5 B-splines of 3rd order (2nd degree polynomial
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• Therefore, Yk(b,d) can be integrated analytically
  exactly with Mathematica
• Mathematica then used to create very efficient
  Fortran (or C) code
• 600,000 lines of Fortran created and compiled
• Pre-computed Yk(b,d) requires 3GBytes of memory
• Re-compiled Linux kernel
• Running on 64-bit SGI Origin2000
• With Yk(b,d) pre-computed, results in factor of
  400 improvement in speed of Rk integral
• Must next demonstrate that this rate can be
  sustained

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Distributed computation model
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• Parallel strategy utilizes heterogenous group of
  machines, across the Internet (cf. Globus)
• Uses stunnel to prevent malicious intrusion
• Robust tested by deliberately stopping either
  client or host, system recovers automatically,
  resuming where it left off

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Discussion and Results

• We have achieved a fully integrated approach for
  symbolically calculating the expressions in MBPT
  using Mathematica.
• We have incorporated an automated angular
  reduction package to further reduce the
  expressions to a form which can be numerically
  evaluated.
• We have written a general-purpose program for
  numerically evaluating the terms in MBPT for an
  arbitrary problem up to third-order.
• We have made numerous consistently checks, such
  as calculating E(3) two ways and deliberately not
  combining terms before angular reduction.
• Rk integral speed shows great improvement.
• Development of grid-based parallel machine now
  operational.

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Sodium, E(2), SSingle, DDouble, etc., 40
B-splines, Lmax9

• E(2) 1.3578E-3 (S) -7.2266E-3 (D)
• -5.8689E-3

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Sodium, E(3), SSingle, DDouble, etc., 40
B-splines, Lmax9
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Sodium, Etotal, SSingle, DDouble, etc., 40
B-splines, Lmax9

• Etotal E(0) E(1) E(2)
  E(3)
• -0.18203269 0 -5.8689E-3 -
  4.11E-4
• -0.18831
• Eexpt -0.18886 (C. E. Moore, Atomic Energy
  Levels, NBS Circ. No. 35, 1971)
• Roughly 0.3 disagreement
• Breit/QED corrections not yet included

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Future work

• Refine parallelization strategy
• Extend to 4th-order
• Apply to problems requiring high precision, e.g.
  Thallium parity violation
• Testing on many architectures 32 bit, 64 bit,
  different compilers testing and more testing
• Publish codes currently alpha version

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References

• 1 W. F. Perger, et al, Computers in Science and
  Eng., 3, No1, 38, Jan/Feb (2001).
• 2 G. C. Wick, Phys. Rev. 80, 268 (1950).
• 3 S. A. Blundell, D. S. Guo, W. R. Johnson, and
  J. Sapirstein, At. Data Nucl. Tables 37, 103
  (1987).
• 4 W. F. Perger, J. Dantuluru, M. Idrees, and K.
  Flurchick, Proceedings of the 6th Joint EPS-APS
  International Conference on Physics Computing,
  edited by R. Grueber and M. Tomassini (European
  Physical Society, Geneva, Switzerland, Lugano,
  Switzerland, 1994), pp. 507-510.
• 5 W. Perger, J. Dantuluru, Ken Flurchick, and
  M. I. Bhatti, in Bulletin of the American
  Physical Society, APS, Washington, DC, 1995, No.
  2, p. 999.
• 6 W. F. Perger, J. Dantuluru, M. I. Bhatti, and
  K. Flurchick, in Bulletin of the American
  Physical Society, Toronto, Canada, 1995, No. 4,
  p. 1291.
• 7 I. Lindgren and J. Morrison, Atomic Many-body
  Theory, Springer-Verlag, 1986.
• 8 K. Takada, Comput. Phys. Commun., 69, 142,
  1992.