Transcript The Lagrange-mesh method for quantum

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
Hydrogen molecular ion in a strong magnetic
field by the Lagrange-mesh method
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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
N associated mesh points
(infinitely differentiable)
(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) 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
Main properties of the Lagrange-mesh method
● 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)
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
Lagrange-Legendre basis
3
2
N=4
1
0
-1
-0,5
0
0,5
1
-1
2
2
1,5
1,5
1
1
0,5
0,5
0
-1
-0,5
0
0
-0,5
-1
0,5
1
-1
-0,5
0
-0,5
-1
0,5
1
Lagrange-Laguerre basis
0,8
0,6
0,4
N=4
0,2
h = 0.2
0
0
1
2
3
4
-0,2
-0,4
-0,6
0,6
0,6
0,4
0,4
0,2
0,2
0
0
0
1
2
3
0
4
-0,2
-0,2
-0,4
-0,4
-0,6
-0,6
1
2
3
4
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) 3569
TL: 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
Other systems
A test on the hydrogen atom
KLJ: Y.P. Kravchenko, M.A. Liberman, B. Johansson, Phys. Rev. A 54 (1996) 287
He23+
TL: A.V. Turbiner, J.C. López Vieyra, Phys. Rep. 424 (2006) 309
H2+ in an arbitrary magnetic field
Molecule axis fixed, rotated field
Gauge choice?
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 ion
Eg.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
Ground-state rotational band of hydrogen molecular ion
J = 0 to 35
E (u.a.)
-0.5
-0.55
-0.6
0
5
10
15
20
25
30
J
- 12-digit accuracy
- radii, interparticle distances, quadrupole moments, …
M. Hesse, D. B., J. Phys. B 36 (2003) 139
35
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