^{1,a)}, Sverker Holmgren

^{1}and Hans O. Karlsson

^{2}

### Abstract

We solve the time-dependent Schrödinger equation for molecular dynamics using a pseudospectral method with global, exponentially decaying, Hagedorn basis functions. The approximation properties of the Hagedorn basis depend strongly on the scaling of the spatial coordinates. Using results from control theory we develop a time-dependent scaling which adaptively matches the basis to the wave packet. The method requires no knowledge of the Hessian of the potential. The viability of the method is demonstrated on a model for the photodissociation of IBr, using a Fourier basis in the bound state and Hagedorn bases in the dissociative states. Using the new approach to adapting the basis we are able to solve the problem with less than half the number of basis functions otherwise necessary. We also present calculations on a two-dimensional model of CO_{2} where the new method considerably reduces the required number of basis functions compared to the Fourier pseudospectral method.

Discussions with Bengt Carlsson, Vasile Gradinaru, Magnus Gustafsson, Katharina Kormann, Hans Norlander, and Elias Rudberg are gratefully acknowledged.

I. INTRODUCTION

II. HAGEDORN WAVE PACKETS

III. TIME STEPPING

IV. ADAPTIVE SCALING OF THE BASIS SET

V. THE CONTROLLER

VI. PHOTODISSOCIATION OF IBR

A. Coupling of the electronic states

B. Determination of parameters for the Hagedorn wave packets

C. Computational results

VII. PROPAGATION IN TWO DIMENSIONS

VIII. CONCLUSIONS

### Key Topics

- Wave functions
- 38.0
- Excited states
- 10.0
- Control theory
- 6.0
- Ground states
- 6.0
- Collocation methods
- 5.0

## Figures

Error as function of the time step for Galerkin (crosses) and collocation (plain lines). Errors compared to a reference solution within the Hagedorn space are shown with dashed lines, and errors compared to a spatially well-resolved reference are shown with solid lines. The TDSE is integrated up to *t* = 1 with 64 basis functions in (a), and to *t* = 5 with 8 basis functions in (b).

Error as function of the time step for Galerkin (crosses) and collocation (plain lines). Errors compared to a reference solution within the Hagedorn space are shown with dashed lines, and errors compared to a spatially well-resolved reference are shown with solid lines. The TDSE is integrated up to *t* = 1 with 64 basis functions in (a), and to *t* = 5 with 8 basis functions in (b).

Projection of to two different Hagedorn spaces, both with *K* = 32 basis functions. The figures show *u*(*x*) with solid lines and its projection with dashed lines. In both cases, ɛ = 1, *q* = *p* = 0, *P* = i/*Q*. In (a), *Q* = 1 and in (b), *Q* = 0.5. The ℓ_{∞}-errors are 0.93 and 1.8 × 10^{−5}, respectively. In (b), the two lines are indistinguishable.

Projection of to two different Hagedorn spaces, both with *K* = 32 basis functions. The figures show *u*(*x*) with solid lines and its projection with dashed lines. In both cases, ɛ = 1, *q* = *p* = 0, *P* = i/*Q*. In (a), *Q* = 1 and in (b), *Q* = 0.5. The ℓ_{∞}-errors are 0.93 and 1.8 × 10^{−5}, respectively. In (b), the two lines are indistinguishable.

(a) The time evolution of the uncertainty parameter *QQ**. The solid line indicates control-based propagation and the dotted semiclassical propagation. The dashed line indicates the control signal. (b) A lin-log plot of the squared modulus of the wave function at time *t* = 5. The crosses show the location of the collocation points in the control scheme.

(a) The time evolution of the uncertainty parameter *QQ**. The solid line indicates control-based propagation and the dotted semiclassical propagation. The dashed line indicates the control signal. (b) A lin-log plot of the squared modulus of the wave function at time *t* = 5. The crosses show the location of the collocation points in the control scheme.

The PES of the IBr molecule with sketched wave packets and grids. The equidistant Fourier grid of the state stays fixed, while the Gauss-Hermite grids in the two excited states follow the wave packets.

The PES of the IBr molecule with sketched wave packets and grids. The equidistant Fourier grid of the state stays fixed, while the Gauss-Hermite grids in the two excited states follow the wave packets.

The real part of ψ_{2} at time *t* _{0}, when the laser pulse peaks. The cross marks the coordinate of strongest coupling with the ground electronic state. The circle marks where the classical coordinate *q* would have been located if no control were used.

The real part of ψ_{2} at time *t* _{0}, when the laser pulse peaks. The cross marks the coordinate of strongest coupling with the ground electronic state. The circle marks where the classical coordinate *q* would have been located if no control were used.

The solution, real parts and moduli, at the three potential energy surfaces at time .

The solution, real parts and moduli, at the three potential energy surfaces at time .

(a) The front of ψ_{3} in modulus at time . The dashed line indicates , and the cross *x* _{δ}. As grows, the dashed line will rise above the top of the hump, and *x* _{δ} will encounter a discontinuity. The response to this is shown in (b). The dashed line shows the control signal for |*Q*|, and the solid line its actual value.

(a) The front of ψ_{3} in modulus at time . The dashed line indicates , and the cross *x* _{δ}. As grows, the dashed line will rise above the top of the hump, and *x* _{δ} will encounter a discontinuity. The response to this is shown in (b). The dashed line shows the control signal for |*Q*|, and the solid line its actual value.

Level curves for the excited state PES of CO_{2}. The location of the saddle point is indicated by a cross, and the centrepoint of the initial Gaussian wave packet by a ring. The level curves are separated by 0.015 hartree.

Level curves for the excited state PES of CO_{2}. The location of the saddle point is indicated by a cross, and the centrepoint of the initial Gaussian wave packet by a ring. The level curves are separated by 0.015 hartree.

Time evolution of the wave function. The modulus of the wave function, |Ψ(*s*, *a*)|, is plotted with a separation of 1 between the level curves. The cross indicates the location of the saddle point in the potential. The wave function is plotted at times *t* = (a) , (b) , (c) , and (d) .

Time evolution of the wave function. The modulus of the wave function, |Ψ(*s*, *a*)|, is plotted with a separation of 1 between the level curves. The cross indicates the location of the saddle point in the potential. The wave function is plotted at times *t* = (a) , (b) , (c) , and (d) .

(a) The modulus of *Q* _{2}, which determines the spread of the collocation points along the asymmetric stretch coordinate axis, plotted against time. Semiclassical propagation of *Q* and *P* is shown with solid lines, and control-based propagation with dashed lines. (b) The time evolution of the maximum pointwise error for the two propagation schemes.

(a) The modulus of *Q* _{2}, which determines the spread of the collocation points along the asymmetric stretch coordinate axis, plotted against time. Semiclassical propagation of *Q* and *P* is shown with solid lines, and control-based propagation with dashed lines. (b) The time evolution of the maximum pointwise error for the two propagation schemes.

The performance of the controller. |*Q* _{1}| (below) and |*Q* _{2}| (above) are plotted against time. The square root of the control signal is shown with solid lines, and the actual parameter values with dashed lines.

The performance of the controller. |*Q* _{1}| (below) and |*Q* _{2}| (above) are plotted against time. The square root of the control signal is shown with solid lines, and the actual parameter values with dashed lines.

Level curves for the pointwise error at time for solutions calculated with (a) a control-propagated Hagedorn basis with 32^{2} basis functions, and (b) a Fourier basis with 52^{2} basis functions. The curve separation is 1.0 × 10^{−5}, and the maximum pointwise error is 8.7 × 10^{−5} and 7.2 × 10^{−5} for the Hagedorn and Fourier solutions, respectively.

Level curves for the pointwise error at time for solutions calculated with (a) a control-propagated Hagedorn basis with 32^{2} basis functions, and (b) a Fourier basis with 52^{2} basis functions. The curve separation is 1.0 × 10^{−5}, and the maximum pointwise error is 8.7 × 10^{−5} and 7.2 × 10^{−5} for the Hagedorn and Fourier solutions, respectively.

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