Symbolic and numeric Scientific computing for atomic physics
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1 University of Delaware Computational Science Initiative meeting November 9, 2007 Symbolic and numeric Scientific computing for atomic physics Marianna Safronova
2 outline Motivation Symbolic computing Numeric computing Computational challenges & future projects
3 research field Research field: development of high-precision methodologies for the calculation of the atomic properties and applications of such calculations. This research involves: the study of the fundamental physics problems applications of atomic physics to future technological developments
4 motivation: motivation: fundamental fundamental physics physics Cs experiment, University of Colorado
5 motivation: applications to various fields of physics Search for variation of the fundamental finestructure constant with time Development of the high-precision methodologies, benchmark problems for understanding of atomic properties, tests of ab initio theory and experimental methods, and cross-checks of experiment Determination of nuclear properties such as nuclear magnetic and anapole moments Providing data for astrophysics studies
6 motivation: new technologies Quantum communication, cryptography and Quantum information processing
7 Quantum Computer (Innsbruck) P 1/2 D 5/2 Quantum bit S 1/2
8 motiation: next generation atomic clocks Next - generation ultra precise atomic clock Atoms trapped by laser light The ability to develop more precise optical frequency standards will open ways to improve global positioning system (GPS) measurements and tracking of deep-space probes, perform more accurate measurements of the physical constants and tests of fundamental physics such as searches for gravitational waves, etc.
9 motivation Parity Violation Atomic Clocks P 1/2 D 5/2 NEED ATOMIC PROPERTIES S 1/2 quantum bit Quantum information
10 calculation of atomic properites Step 1 Derivation of the formulas Step 2 Conducting possible analytical calculations to simplify expressions for numeric evaluation Step 3 Writing and debugging the programs for numeric calculations Step 4 Numeric calculations
11 calculation of atomic properites Step 1 Derivation of the formulas Step 2 Conducting possible analytical calculations to simplify expression for numerical evaluation Step 3 Writing and debugging the programs for numerical calculations Steps 1 3: Symbolic computing Step 4 Numeric computation
12 coupled-cluster cluster method The coupled-cluster cluster method sums infinite sets of many-body perturbation theory terms. The wave function of the valence electron v is represented as an expansion that includes all possible single, double, and partial triple excitations. Cs: atom with single (valence) electron outside of a closed core. 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 2 4p 6 4d 10 5s 2 5p 6 6s core 1s 2 5p 6 6s valence electron
13 coupled-cluster cluster method Ψ v = Lowest order Core (0) Ψ v core valence electron any excited orbital Single-particle excitations + ρ a a Ψ ma (0) ma m a v + ρ a a Ψ m v (0) mv m v v 1 2 mnab... Double-particle excitations + ρ a a a a Ψ + ρ a a a a Ψ + (0) (0) mnab m n b a v mnva m n a v v mna
14 Step 1: 1: derivation of the formulas H (0) v Need for symbolic computing i j l k d b S2 Ψ > : a a a a : aman ar as a aca aaav 0 : c > 800 terms! The code was developed to implement Wick s theorem and simplify the expression.
15 Symbolic program for coupled-cluster cluster method & perturbation theory Input: expression of the type in ASCII format ijkl ρmnwa i j l k m n a w g a a a : a a :: a a a : Output: simplified resulting formula in the LaTex format, ASCII output is also generated.
16 program features 1) The code is set to work with two or three normal products (all possible cases) with large number of operators. 2) The code differentiates between different types of indices, i.e. core (a,b,c, ), valence (v,w,x,y,...), excited orbitals (m,n,r,s, ), and general case (i, j, k, l, ). 3) The operators are ordered as required in the same order for all terms gmnba ρrswc aman ar as aaabac aw 4) The expression is simplified to account for the identical g = terms and symmetry rules,. ijkl g jilk 5) The direct and exchange terms are joined together, % g = g g ijkl ijkl ijlk.
17 Step 2: 2: analytical calculations Certain summations can be carried out analytically Two symbolic programs are currently available: 1) KENTARO currently interfaced with the previous program. 2) RACAH Maple code very convenient to use, but has compatibility problems.
18 Step 3: 3: Symbolic computing for program generation The resulting expressions that need to be evaluated numerically contain very large number of terms, resulting in tedious coding and debugging. The symbolic program generator was developed for this purpose to automatically generate efficient numerical codes for coupled-cluster or perturbation theory terms J j j J j j J j j J k1 k2 k3 jc jd jw jw' c d a b c d ( 1) abcd k1k 2k3 k2 jw' jv ' k3 jw jv k1 jb ja X ( cdab) X ( v ' w' cd) Z ( vwab) k1 k2 k3 ( ε + ε ε ε )( ε + ε ε ε ) c d v' w' a b v' w'
19 Input: Output: Step 3: 3: Symbolic computing for program generation Sample input: list of formulas to be programmed corresponding FORTRAN code for numerical evaluation J j j J j j J j j J k1 k2 k3 jc jd jw jw' c d a b c d ( 1) abcd k1k 2k3 k2 jw' jv ' k3 jw jv k1 jb ja X ( cdab) X ( v ' w' cd) Z ( vwab) k1 k2 k3 ( ε + ε ε ε )( ε + ε ε ε ) c d v' w' a b 1 subroutine term1d(nvp,kvp,nwp,kwp,nv,kv,nw,kw,jtot,res) 2 X[k1](c,d,a,b) X[k2](vp,wp,c,d) Z[k3](v,w,a,b) 3 none 4 4 xx z1=(-1)**(1+jtot+k1+k2+k3+kapc+kapd+kapw+kapwp) xx z2=d6j(2*jtot,jc,jd,2*k2,jwp,jvp) xx z3=d6j(ja,jb,2*jtot,jw,jv,2*k3) xx z4=d6j(jc,jd,2*jtot,jb,ja,2*k1) 5 (e[c,d]-e[vp,wp]) (e[a,b]-e[vp,wp]) 6 No v' w'
20 Step 3: 3: Symbolic computing for program generation Features: Simple input, essentially just type in a formula. Efficient codes are produced Other constructs can be easily added Safety features are build in Works for most formulas Future plans: interface with formula-writing codes
21 Step 4: 4: numerical computation 1. Coupled-cluster method: iterative solution of very large number of equations 2. Relativistic many-body perturbation theory: evaluation of terms such as example considered before 3. Configuration-interaction (CI) method: finding a few eigenvalues of large matrices 4. Combination of coupled-cluster and CI: general super CI code* (in progress)
22 coupled-cluster cluster method
23 coupled-cluster cluster method The excitation coefficients ρ are obtained by the iterative solution of the corresponding equations. Owing to the very large number of the equations the code needs to be very efficient. The inclusion of the triple excitations require new approaches owing to substantially larger memory and computer time requirements. ρ mnab Cs: a,b = 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 2 4p 6 4d 10 5s 2 5p 6 m,n : finite basis set = (35 13) (35 13) equations ρ mnrvab Extra index r gives at least a factor of (35 13).
24 coupled-cluster cluster method ρ mnab These are really complicated equations!!! Quadruple term: gmnrsρrsab rs a,b core (17 shells) Indices mnrs can be ANY orbitals Basis set: n max =35, l max =6 17x17x(35 35x13) 4 =5 x ! Program has to be exceptionally efficient!
25 computational challenges I. Efficiency of the most complicated terms II. III. IV. Efficient parallelization Numerical methods Keeping up with new hardware/software and corresponding compatibility problems
26 computational challenges 1) Efficiency of the most complicated terms: gmnrsρrsab Can it be improved further? rs Is any time being wasted owing to imperfect loop ordering? Automated code optimization? 2) Efficient parallelization, especially for triple excitations How to do efficient parallelization of iterative procedures with large local files?
27 computational challenges 3) Efficient parallel codes for configuration interaction method (finding few eigenvalues of large matrices) 4) New approaches for the numerical basis sets (currently using B splines) 5) Resolving convergence issues for the iterative procedure in the cases of large excitations from nearby core shells. 6) Computing on multicore machines
28 conclusion Fundamental Physics Atomic Clocks Symbolic and Numeric Computing for Atomic Calculations P 1/2 D 5/2 S 1/2 Quantum bit Quantum information
29 Group web site: other theory collaborations: graduate StudentS: bindiya arora rupsi pal Jenny tchoukova dansha Jiang Charles Clark and Carl Williams (NIST) Walter Johnson (University of Notre Dame) Ulyana Safronova (University of Nevada-Reno) Michael Kozlov (Petersburg Nuclear Physics Institute) Victor Flambaum and Vladimir Dzuba (UNSW, Australia)
30 nsf workshop on cyber-enabled enabled discovery and innovation ( cdi ) NSF solicitation Workshop page
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