1. Hydrogen molecular ion in a strong magnetic field by the Lagrange-mesh method Cocoyoc, February 2007 Daniel Baye Université Libre de Bruxelles, Belgium with M. Vincke, J.-M. Sparenberg

2. Hydrogen molecular ion in a strong magnetic field by the Lagrange-mesh method Introduction Lagrange-mesh method H 2 + in a strong magnetic field (aligned) Other systems H 2 + in a strong magnetic field (general) Three-body systems Conclusion

3. Introduction Lagrange-mesh method: - approximate variational method - orthonormal basis associated with a mesh - use of Gauss quadrature consistent with the basis - simplicity of mesh calculation D. B., P.-H. Heenen, J. Phys. A 19 (1986) 2041 D. B., Phys. Stat. Sol. (b) 243 (2006) 1095 H 2 + in a strong magnetic field - Born-Oppenheimer approximation - prolate spheroidal coordinates - simple but highly accurate (aligned) - extension to non-aligned case M. Vincke, D. B., J. Phys. B 39 (2006) 2605

4. Lagrange-mesh method N Lagrange functions (infinitely differentiable) N associated mesh points (i)Lagrange condition (ii) Gauss quadrature exact for products Corollary: Lagrange functions are orthonormal

5. Schrödinger equation (1D) Variational wave function System of variational equations Principle: potential matrix at Gauss approximation

6. Mesh equations - simplicity of mesh equations but approximately variational - T ij : simple functions of x i and x j - Lagrange basis hidden: only appears through mesh points x i kinetic matrix elements T ij - wave function known everywhere D. B., P.-H. Heenen, J. Phys. A 19 (1986) 2041 M. Vincke, L. Malegat, D. B., J. Phys. B 26 (1993) 811 D. B., M. Hesse, M. Vincke, Phys. Rev. E 65 (2002) 026701 D. B., Phys. Stat. Sol. (b) 243 (2006) 1095

7. ● When it works, it is - simple - highly accurate ● When does it work? - no singularities (Gauss quadrature!) - if singularities are regularized Principle of regularization for a singularity at x = 0 ● Coulomb remains the big problem (solved for 2 and 3 particles) Main properties of the Lagrange-mesh method

8. H 2 + in an aligned magnetic field Prolate spheroidal coordinates

9. Potential Coulomb singularity regularized by volume element Laplacian Singularities for m > 0 → Regularized basis functions

10. Lagrange mesh h : scaling parameter Lagrange basis ν: regularization index

11. Lagrange-Legendre basis N = 4

12. Lagrange-Laguerre basis N = 4 h = 0.2

13. Parity-projected basis Wave function Potential matrix diagonal and simple! Choice of regularization: ν depends on m

14. Equilibrium distances and energies m = 0 M. Vincke, D. B., J. Phys. B 39 (2006) 2605 GLT: X. Guan, B. Li, K.T. Taylor, J. Phys. B 36 (2003) 3569 TL: A.V. Turbiner, J.C. López Vieyra, Phys. Rev. A 69 (2004) 053413

15. Equilibrium distances and energies M. Vincke, D. B., J. Phys. B 39 (2006) 2605 GLT: X. Guan, B. Li, K.T. Taylor, J. Phys. B 36 (2003) 3569 KS: U. Kappes, P. Schmelcher, Phys. Rev. A 51 (1995) 4542

16. A test on the hydrogen atom KLJ: Y.P. Kravchenko, M.A. Liberman, B. Johansson, Phys. Rev. A 54 (1996) 287 Other systems

17. He 2 3+ TL: A.V. Turbiner, J.C. López Vieyra, Phys. Rep. 424 (2006) 309

18. H 2 + in an arbitrary magnetic field Gauge choice? Molecule axis fixed, rotated field

19. Symmetries: parity Properties of basis Real matrix if General gauge Simplest calculation with

20. Hamiltonian Wave function Potential matrix still diagonal and simple! Real band matrix with couplings of m values

21. Convergence TL: A.V. Turbiner, J.C. López Vieyra, Phys. Rev. A 68 (2003) 012504

22. Energy surface Similar to: U. Kappes, P. Schmelcher, Phys. Rev. A 51 (1995) 4542

23. Three -body systems Lagrange-mesh calculations in perimetric coordinates (+ Euler angles) M. Hesse, D. B., J. Phys. B 32 (1999) 5605 Regularization Applications to three-body atoms and molecules

24. Examples (B = 0) M. Hesse, D. B., J. Phys. B 32 (1999) 5605 Helium atom (infinite mass) E g.s. = - 2.903 724 377 034 14 a.u. (N = 50, N z = 40) Positronium ion E g.s. = - 0.262 005 070 232 97 a.u. (N = 50, N z = 40) Hydrogen molecular ion (finite masses, no Born-Oppenheimer approximation!) E g.s. = - 0.597 139 063 122 8 a.u. (N = 50, N z = 40) Basis size

25. Ground-state rotational band of hydrogen molecular ion J = 0 to 35 - 12-digit accuracy - radii, interparticle distances, quadrupole moments, … M. Hesse, D. B., J. Phys. B 36 (2003) 139

26. Helium atom in a strong magnetic field 5-dimensional problem 6-digits accuracy 10 4 to 10 5 basis functions γ < 5 M. Hesse, D. B., J. Phys. B 37 (2004) 3937 BSD: W. Becken, P. Schmelcher, F.K. Diakonos, J. Phys. B 32 (1999) 1557

27. Conclusions Lagrange-mesh method: ● highly accurate approximate variational method ● simple but singularities may destroy accuracy H 2 + in a strong magnetic field ● accurate results or short computation times ● non-aligned case in progress ● goal: comparison with purely quantum calculations (evaluation of center-of-mass corrections?) Applicable to selected systems only

28. Kinetic-energy matrix element - exact calculation possible as a function of x i - Gauss approximation for T ij : identical to collocation (pseudospectral) method - if not symmetrical :

29. Lagrange functions in perimetric coordinates and regularization (equivalent to an expansion in Laguerre polynomials) Lagrange condition Regularization factor M. Hesse, D. Baye, J. Phys. B 36 (2003) 139