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

Warren F. Perger, Michigan Tech University

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

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:

12

21

r

mGmV

This can be solved analytically for n=2. “Perturbation” theory can be used for

n>2 objects.

Zeroth-ordertrajectory

Sun Planet 1

Perturbed trajectory

Planet 2

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 < 1% precision to impact development of unified field theories, e.g. SU2X U1

The Atomic System Requires quantum-mechanical

description; solution of the Schroedinger (or Dirac) equation:

)()()](2

[2

2

rErrm V ij

EH

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 Wick’s theorem [1,2], which is an alternative to a diagrammatic reduction.

Let:

where H0 = unperturbed Hamiltonian

with

i j k cl dijklcdijkl

V g a a a a a a

HH Vo

ijijji aaaa

( ')( ) ( ') ( )* *

| ' |i j k l

ijkl

rr r rg

r r

and

0|1|)(2|)(

0|)(1|)(

00|)(

)2()1()0(

)1()0(

)0(

EVEEH

VEEH

EH

o

o

o

2||0

1||0

0||0

)3(

)2(

)1(

V

V

V

EEE

The wavefunctions are given by:

and the energy expressions by:

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.

Feyman diagrams and quantum field theory

a

b d

c

Propagator

time

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

space

+ 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”

Motivation for using Wick’s theorem: multiplication of operator strings (probabilities)

0|1|)(2|)(

0|)(1|)(

00|)(

)2()1()0(

)1()0(

)0(

EVEEH

VEEH

EH

o

o

o

|0> is a string of operators acting on the “vacuum”, e.g., |0> = aa

+ |0c> for the one-particle, zero-hole, case

and likewise V is a string of second-quantized operators:

cd

dckljiijkl

ijkl aaaaaagV

Feynman diagrams and Wick’s theorem

Example:

Combine two diagrammatic

fragments:

A=a+ aa and B= ab

+ a ac

+ ad given

Wick’s theorem.

Then,

AB={AB}+{AB}

= {a+ aa ab

+ a ac

+ ad }

+ ac{a+ ab

+ a ad }

+ ab+ ac

+ aa ad

- ac ab+ ad

+ ac{a+ aac

+ ad }

+ ab ac+

ad

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 Wick’s 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 (alkalai’s)

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.

112

1/ 2

11 2 1 2 1 2

( 1)

( 1)

| | ( , ) ( , )

( , ) || || || ||

|| ||0 0 0

( , ) ( ) ( ) ( ) ( )

[(2 1)(2 1)]

ka b c d

k

k k ka b c d a c b d

k a cka c

kk

k

ca

abcd

a b c d

l l l l

l l l l l l l l

kab cd X ab cdR

X k

kl l

ab cdR d d

r

l lll

C C

C

P r P r P r P r r rrr

g

|f

….

( ,....)X k ( ,....)X k | i….

and…

1 2 3

1 2 3

j j j

m m m

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.

Method of Calculation

WicksThmMathematica Program to calculate

MBPT terms usingWick’s Theorem

WTtoTeXFormats MBPTterms in LaTeX.Prepares data for Kentaro; formats results in LaTeX

LaTeX Output

KentaroAngular Reduction

LaTeX Output

EvaluateCalculation of atomic

properties, e.g. energies,transition matrix

elements.

Basis set programto calculate

relativistic atomicorbitals

Parallel C/FORTRAN program to

calculate RL(a,b,c,d)

How do we know it’s 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).

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.

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)

Total execution time

The total execution time can be estimated as:

LRn tLNt ~

Where tRL is the time to evaluate a given Slater integral, N=number 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 takesroughly 402 x 9 x 10-3 ~ 14 seconds. A quadruples term, however, takes roughly 194 days.

Solutions to the numerical problems

1) Improve on speed of Rk integrals

2) Employ parallel processing.

Improving speed of Rk integrals

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

1

1

11 2 1 2 1 2

11 1 2 2 20 0 1

112 2 2 1

2

1 1 10

2

( , ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( , )

[

]

kk

k

k

k

k

k

k

a b c d

a c b d

b d

a c

ab cdR d d

rd

d dr

b d d

P r P r P r P r r r

P r P r P r P r r

P r P r r r

P r P r r

rr

rr

rr

Y

5 B-splines of 3rd order (2nd degree polynomial

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

( ) ( ), ii

a r iNow BcP

Distributed computation model

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

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.

Sodium, E(2), S=Single, D=Double, etc., 40 B-splines, Lmax=9

E(2) =1.3578E-3 (S) -7.2266E-3 (D)

= -5.8689E-3

Sodium, E(3), S=Single, D=Double, etc., 40 B-splines, Lmax=9

-2.4754E-4 (S) 1.4249E-2 (T) 3.7681E-4 (D) 1.3498E-3 (T)

-1.4543E-2 (D) 5.4543E-4 (T)

2.5139E-4 (D) -1.6760E-3 (T)

-1.5717E-3 (D) 1.0875E-3 (Q)*

1.1834E-4 (D) E(3) = -4.1143E-4

-3.5146E-4 (T) * used Lmax=6

Sodium, Etotal, S=Single, D=Double, etc., 40 B-splines, Lmax=9

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

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

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.