Examples of two-level potentials

One elevated crossing point PDF file P12(E)
Error(E)
Only one crossing point. Even if the energy is small at crossing point, the semiclassical result for P12 is accurate to 4 figures.
Vector(2) potential from the paper M. P.
        Moody et al. Phase corrected ..., 2003 (second example) PDF file P12(E)
Error(E)
Two interfering crossing points. Example from the paper M. P. Moody et al. The mass equals one, but the semiclassical result for P12 is quite accurate (2 figures).
Vector(2) potential from the paper M. P.
        Moody et al. Phase corrected ..., 2003 PDF file P12(E)
Error(E)
Two interfering crossing points. Here, mass and energy are large, and the semiclassical result is extremely accurate.
Two gaussians PDF file P12(E)
Error(E)
Two Gaussians with a crossing point near the summit. Here, one could expect effect of reflection waves, at least for small energies
Gaussian and a constant PDF file P12(E)
Error(E)
Gaussian crosses constant potential. Here, there is around 10% reflection in first channel (occuring out of interaction region). It gives around 30% modulation of the incoming wavefunction.
Two constants PDF file P12(E)
Error(E)
Constant potentials, with Gaussian interaction. There is considerable reflection that occurs in strong-interaction region.
Oscillating Gaussian and a constant PDF file P12(E)
Error(E)
Oscillating Gaussian crosses constant potential six times. Because of complex effects of interference, the wavefunctions behave erratically. Also, there is considerable reflection.
From the paper of Marc Boiron PDF file P12(E)
Error(E)
From the paper of Marc Boiron, Maurice Lombardi and Laurent Wiesenfeld 'Coupled modes semiclassical treatment of nonadiabatic transitions', J. Phys. A: Math. Gen. 30 (1997) 3907–3926. E = 15, V12 = 1.5*(broad Gaussian)
From the paper of Marc Boiron PDF file P12(E)
Error(E)
From the paper of Marc Boiron, Maurice Lombardi and Laurent Wiesenfeld 'Coupled modes semiclassical treatment of nonadiabatic transitions', J. Phys. A: Math. Gen. 30 (1997) 3907–3926. E = 60, V12 = 3*(broad Gaussian)
From the paper of Michael F.Herman CGR PDF file P12(E)
Error(E)
From the paper of Michael F.Herman 'Choosing a good representation of the quantum state wave functions for semiclassical surface hopping calculations', J. Chem. Phys. 11 (1999) 10427, Table I, E = 1.3
From the paper of Michael F.Herman CGR PDF file P12(E)
Error(E)
From the paper of Michael F.Herman 'Choosing a good representation of the quantum state wave functions for semiclassical surface hopping calculations', J. Chem. Phys. 11 (1999) 10427, Table II, E = 2
From the paper of Michael F.Herman CGR PDF file P12(E)
Error(E)
From the paper of Michael F.Herman 'Choosing a good representation of the quantum state wave functions for semiclassical surface hopping calculations', J. Chem. Phys. 11 (1999) 10427, Table III, E = 2
From the paper of Michael F.Herman CGR PDF file P12(E)
Error(E)
From the paper of Michael F.Herman 'Choosing a good representation of the quantum state wave functions for semiclassical surface hopping calculations', J. Chem. Phys. 11 (1999) 10427, Table IV, E = 2
From the paper of Michael F.Herman NSA PDF file P12(E)
Error(E)
From the paper of Michael F.Herman and Michael P.Moody 'Numerical study of the accuracy and efficiency of various approaches for Monte Carlo surface hopping calculations', J. Chem. Phys. 122 (2005) 094104, No. 1 (single crossing)
From the paper of Michael F.Herman NSA PDF file P12(E)
Error(E)
From the paper of Michael F.Herman and Michael P.Moody 'Numerical study of the accuracy and efficiency of various approaches for Monte Carlo surface hopping calculations', J. Chem. Phys. 122 (2005) 094104, No. 2 (no crossing)
From the paper of Michael F.Herman NSA PDF file P12(E)
Error(E)
From the paper of Michael F.Herman and Michael P.Moody 'Numerical study of the accuracy and efficiency of various approaches for Monte Carlo surface hopping calculations', J. Chem. Phys. 122 (2005) 094104, No. 3 (dual crossing)

Explanations of computations

Mathematica programs



Tulane Results of work at Tulane University with M. Herman
Waste icons More Unpublished reports

Designed by A. Sergeev.