Chris Pardi , University of Surrey

1. Applying absorbing boundary conditions to time-dependent Hartree-Fock calculations of giant resonances Chris Pardi, University of Surrey ECT* Trento, Advances in time-dependent methods for quantum many-body systems

2. Introduction The aims of this talk are to highlight: • The significance of the boundary condition in the computational solution of physical problems. • An absorbing boundary condition (ABC) suitable for nuclear time-dependent Hartree-Fock (TDHF) calculations of giant monopole resonances. • Complications to the ABCs caused by the inclusion of charge. • Extension of the method to more realistic calculations: full skyrme force and 3d calculations.

3. TDHF Calculations P. Ring, P. Schuck, The Nuclear Many-Body Problem

4. Potentials For all results shown here the t0-t3 Skyrme potential is used for the nuclear interaction and the Coulomb potential for the electrostatic interaction. This gives the terms in the two body potential as where and the terms in the three body potential as J.-S. Wu et al. Phy. Rev. C 60(4) 044302 (1999).

5. TDHF Equations

6. TDHF Boundary Conditions At the origin the radial single particle wavefunction satisfy However, wavefunctions of the Schroedinger equation decay to zero as the space variable tends to infinity. This is satisfied by enforcing that the single particle wavefunctions satisfy This poses a problem computationally as the boundary must be evaluated at a finite value, known as the artificial boundary.

7. Reflecting Boundary Conditions (RBCs) An initial way to treat the boundary is to impose where R is the artificial boundary. Physically this corresponds to a boundary that reflects all matter coming into contact with it. This type of boundary condition has the advantage of being easy to implement and fast to evaluate. However, for solutions on a unbounded domain, the following animation shows it only remains accurate so long as matter does not come into contact with it.

8. Using TDHF with RBCs to study Giant Resonances • This is a problem when using TDHF with RBCs to study giant resonances. Giant resonances generally decay by particle emission and so some density will move away from the nucleus. • This is shown by the animation below showing a calculation of the evolution of the density of Helium undergoing the giant monopole resonance. Giant Resonances by M. N. Harakeh, Adriaan Woude

9. Using TDHF with RBCs to study Giant Resonances Although reflection is clearly observed the amount of density interaction with the boundary is tiny. However, the strength functions are sensitive to these reflections. The following plots show the strength functions of Helium plotted with RBCs at varying distances:

10. RBCs: Timings Plot showing how long it takes to calculate solutions to the TDHF equations up to a particular timestep. The particular calculation is of a Calcium nucleus undergoing a resonance. • The red lines a calculation where the RBC’s distance from the origin is chosen to maintain accuracy in the results • The black line shows a calculation with a RBC close to the nucleus, as a rough guide to the optimum calculation time.

11. Absorbing Boundary Conditions (ABCs)

12. ABCs This assumption allows the TDHF equations to be written in the previous form. Given the small amount of density crossing the boundary the assumption appears reasonable. The specific form of the exterior potential can now be written as The TDHF equations in the exterior are now simple enough for some ABCs to be derived.

13. ABCs M. Mangin-Brinet et al. Phys. Rev. A 57, 3245–3255 (1998). M. Heinen et al. Phys. Rev. E, 79(5) 056709 (2009).

14. M. Mangin-Brinet et al. Phys. Rev. A 57, 3245–3255 (1998). M. Heinen et al. Phys. Rev. E, 79(5) 056709 (2009).

15. To test the ABCs accuracy, solutions to the following Schroedinger equation are found: The maximum difference occurring between calculations using ABCs and RBCs is plotted as a function of radius. The ABCs are applied at 10 and the RBCs are applied at 200.

16. ABCs

17. Least Squares [1] Xu, Kuan; Jiang, Shidong; A Bootstrap Method for Sum-of- Poles Approximations. J. Sci. Comput. 55 (2013), no. 1, 16–39. [2] F.W.J. Olver et al. NIST Handbook of Mathematical Functions.

18. Kernel on the Imaginary axis

19. Least Squares Results

20. Least Squares Results The table shows the number of poles require to approximate the kernel for various parameter sets. Each kernel approximation takes around 0.6s to complete.

21. ABC Testing To see any additional error coming from the approximation of the kernel the solution to the following equation is calculated: Analytic Kernel Sum-of-exponentials kernel

22. Strength Functions The expected strength, a calculation using RBCs such that no reflection occurs, is plotted along side the result using ABCs. There seems to be no adverse affect from the assumption

23. Strength Functions

24. ABCs: Timings Plot showing how long it takes to calculate solutions to the TDHF equations up to a particular timestep. • The light blue line shows the calculation time using ABCs. • The addition time required to calculate the kernel approximation is excluded.

25. ABCs: Timings Plot showing how long it takes to calculate solutions to the TDHF equations up to a particular timestep. • Closer inspection shows the effect of the ABC’s time nonlinearity. • The time required by each timestep increases as more steps are completed.

26. Fast ABCs using recursion

27. Fast ABCs: Timings Plot showing how long it takes to calculate solutions to the TDHF equations up to a particular timestep. • The purple line shows the calculation time for the fast ABC. • The inefficiency form the non-linearity in the ABC has been removed.

28. Faster, Fast ABCs • The speed of the boundary calculation is now dependent only on the number of terms in the sum-of-exponentials. • Reducing the number of terms in sum-of-exponentials could by done by: • Keeping track of when each exponent’s contribution becomes insignificant and removing it. • Considering approximations made on a smaller portion of the imaginary axis or with a higher error. Closer examination may yield the same accruaccy with less exponentials. • Using a sum-of-exponential reduction code, such as that described in [1]. [1] Xu, Kuan; Jiang, Shidong; A Bootstrap Method for Sum-of- Poles Approximations. J. Sci. Comput. 55 (2013), no. 1, 16–39.

29. Outlook J. C. Slater Phys. Rev. 81, 385–390 (1951)

30. Conclusion • Seen the importance of the boundary condition applied at the artificial boundary when calculated solutions that are unbounded in space. • Seen that although results can be gained by using a large region, it results in an inefficient calculation. • Shown that ABCs suitable for TDHF calculations of linear response can be derived and that they are efficient and accurate. • Shown that the extension of the method to 3d calculations is feasible.