Hydrogen molecular ion in a strong magnetic field by the Lagrange-mesh

method

Cocoyoc, February 2007

Daniel BayeUniversité Libre de Bruxelles, Belgium

with M. Vincke, J.-M. Sparenberg

Hydrogen molecular ion in a strong magnetic field by the Lagrange-mesh method

• Introduction

• Lagrange-mesh method

• H2+ in a strong magnetic field (aligned)

• Other systems

• H2+ in a strong magnetic field (general)

• Three-body systems

• Conclusion

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

H2+ 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

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

Schrödinger equation (1D)

Variational wave function

System of variational equations

Principle: potential matrix at Gauss approximation

Mesh equations

- simplicity of mesh equations but approximately variational- Tij : simple functions of xi and xj

- Lagrange basis hidden: only appears through • mesh points xi

• kinetic matrix elements Tij

- wave function known everywhere

D. B., P.-H. Heenen, J. Phys. A 19 (1986) 2041M. Vincke, L. Malegat, D. B., J. Phys. B 26 (1993) 811D. B., M. Hesse, M. Vincke, Phys. Rev. E 65 (2002) 026701D. B., Phys. Stat. Sol. (b) 243 (2006) 1095

● 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

H2+ in an aligned magnetic field

Prolate spheroidal coordinates

Potential

Coulomb singularity regularized by volume element

Laplacian

Singularities for m > 0

→ Regularized basis functions

Lagrange mesh

h : scaling parameter

Lagrange basis

ν: regularization index

-1

0

1

2

3

-1 -0,5 0 0,5 1

-1

-0,5

0

0,5

1

1,5

2

-1 -0,5 0 0,5 1

Lagrange-Legendre basis

N = 4

-1

-0,5

0

0,5

1

1,5

2

-1 -0,5 0 0,5 1

Lagrange-Laguerre basis

-0,6

-0,4

-0,2

0

0,2

0,4

0,6

0,8

0 1 2 3 4

-0,6

-0,4

-0,2

0

0,2

0,4

0,6

0 1 2 3 4

-0,6

-0,4

-0,2

0

0,2

0,4

0,6

0 1 2 3 4

N = 4

h = 0.2

Parity-projected basis

Wave function

Potential matrix diagonal and simple!

Choice of regularization: ν depends on m

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) 3569TL: A.V. Turbiner, J.C. López Vieyra, Phys. Rev. A 69 (2004) 053413

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

A test on the hydrogen atom

KLJ: Y.P. Kravchenko, M.A. Liberman, B. Johansson, Phys. Rev. A 54 (1996) 287

Other systems

He23+

TL: A.V. Turbiner, J.C. López Vieyra, Phys. Rep. 424 (2006) 309

H2+ in an arbitrary magnetic field

Gauge choice?

Molecule axis fixed, rotated field

Symmetries: parity

Properties of basis

Real matrix if

General gauge

Simplest calculation with

Hamiltonian

Wave function

Potential matrix still diagonal and simple!Real band matrix with couplings of m values

Convergence

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

Energy surface

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

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

Examples (B = 0)M. Hesse, D. B., J. Phys. B 32 (1999) 5605

Helium atom (infinite mass)Eg.s. = - 2.903 724 377 034 14 a.u. (N = 50, Nz = 40)

Positronium ionEg.s. = - 0.262 005 070 232 97 a.u. (N = 50, Nz = 40)

Hydrogen molecular ion (finite masses, no Born-Oppenheimer approximation!)Eg.s. = - 0.597 139 063 122 8 a.u. (N = 50, Nz = 40)

Basis size

-0.6

-0.55

-0.5

0 5 10 15 20 25 30 35

J

E (

u.a

.)

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

Helium atom in a strong magnetic field

• 5-dimensional problem • 6-digits accuracy • 104 to 105 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

Conclusions

Lagrange-mesh method: ● highly accurate approximate variational method ● simple but singularities may destroy accuracy

H2+ 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

Kinetic-energy matrix element

- exact calculation possible as a function of xi

- Gauss approximation for Tij : identical to collocation (pseudospectral) method

- if not symmetrical :

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