Time-dependent wave packet propagation using quantum hydrodynamics

Abstract

A new approach for propagating time-dependent quantum wave packets is presented based on the direct numerical solution of the quantum hydrodynamic equations of motion associated with the de Broglie–Bohm formulation of quantum mechanics. A generalized iterative finite difference method (IFDM) is used to solve the resulting set of non-linear coupled equations. The IFDM is 2nd-order accurate in both space and time and exhibits exponential convergence with respect to the iteration count. The stability and computational efficiency of the IFDM is significantly improved by using a “smart” Eulerian grid which has the same computational advantages as a Lagrangian or Arbitrary Lagrangian Eulerian (ALE) grid. The IFDM is generalized to treat higher-dimensional problems and anharmonic potentials. The method is applied to a one-dimensional Gaussian wave packet scattering from an Eckart barrier, a one-dimensional Morse oscillator, and a two-dimensional (2D) model collinear reaction using an anharmonic potential energy surface. The 2D scattering results represent the first successful application of an accurate direct numerical solution of the quantum hydrodynamic equations to an anharmonic potential energy surface.

Appendices

Appendix 1: Analytic derivatives of F ₁ and $\tilde F_{2}$

The functions F ₁(C,v) and $\tilde F_2(C,v)$ are defined by Eqs. 9 and 26, respectively. The C and v are short-hand notation for C = C ⁿ⁺¹ _i and v = v ⁿ⁺¹ _i′ . Thus, the analytic derivatives of F ₁(C, v) are obtained by taking the derivative of Eq. 9 with respect to C ⁿ⁺¹ _i and v ⁿ⁺¹ _i′

$$ {\partial F_1\over\partial C} = 1, $$

(52)

$$ {\partial F_1\over\partial v} = {\Updelta t\over \Updelta x}\left [1/16 + (C^{n+1}_{i+1} - C^{n+1}_{i-1})/8\right]. $$

(53)

Similarly, the analytic derivatives of $\tilde F_2(C,v)$ are obtained by taking the derivative of Eq. 26 with respect to C ⁿ⁺¹ _i and v ⁿ⁺¹ _i′

$$ {\partial{\tilde F}_2\over\partial C} = -{\Updelta t\over 2 m} {\partial f^{n+1}_{q i'}\over\partial C}, $$

(54)

$$ {\partial{\tilde F}_2\over\partial v} = 1 + {\Updelta t\over \Updelta x}\left[(v^{n+1}_{i'+1} - v^{n+1}_{i'-1})/4 + \Updelta t^2 \tilde\gamma^{n+1}_{i'}/(m \Updelta x)\right]. $$

(55)

The derivative of f ⁿ⁺¹ _{q i′} with respect to C ⁿ⁺¹ _i is given by

$$ {\partial f^{n+1}_{q i'}\over\partial C} ={\hbar^2\over 2m}\left[ 1/(2\Updelta x^3) - \langle \partial^2_x C \rangle_{i'}/(2\Updelta x) - \langle \partial_x C \rangle_{i'}/\Updelta x^2\right]. $$

(56)

Substituting the 2nd-order finite difference expressions for the derivatives of C into Eq. 56 gives

$$ {\partial f^{n+1}_{q i'}\over\partial C} ={\hbar^2\over 2m}\left [ 1 + C^{n+1}_{i} - C^{n+1}_{i+2}\right]/(2\Updelta x^3). $$

(57)

In the above derivations, the derivatives of the artificial viscosity coefficient $\tilde\gamma$ with respect to C and v are ignored.

Appendix 2: Derivation of γ(C, v)

2.1 One-dimensional case

The second-order truncation error with respect to $\Updelta t$ in the finite-difference representation of the velocity equation (Eq. 6) is given by $\varepsilon$ defined in Eq. 24. If the classical potential does not depend explicitly on time, then the time derivatives of the total force become equal to the time derivatives of just the quantum force, and Eq. 24 becomes

$$ \varepsilon =\frac{1}{12}\Updelta t^2 \left [ \partial^2_t f^{n'}_{q i'} - m \partial^2_t(v^{n'}_{i'}\partial_x v^{n'}_{i'})\right]. $$

(58)

By expanding the time derivatives of the two terms in brackets in Eq. 58 and using the definition of the quantum force (Eq. 8), Eq. 58 becomes a complicated expression involving many terms of mixed time and spacial derivatives of C and v. By taking the appropriate time and spacial derivatives of the equations of motion for both C (Eq. 4) and v (Eq. 3), all of the various time derivatives of C and v can be eventually expressed in terms of derivatives of x only. The coefficient of the second derivative of v can then be identified as the effective numerical diffusion coefficient (i.e., $\Upgamma(C,v) \partial^2_x v$). The algebra is straightforward but tedious, and for one dimension, it is feasible to pursue. However, it is much faster and straightforward to implement the analysis on a symbolic algebra solver such as Mathematica [42]. The resulting expression for $\Upgamma(C,v) = \Updelta t^2 \gamma$ is given by

$$ \begin{aligned} \Upgamma(C,v) =& \frac{1}{12}\Updelta t^2 [ 3 f v - 9 m v^2 \partial_x v +18 {\hbar^2\over 2m} (\partial_x C)^2 \partial_x v \\ +& 30 {\hbar^2\over 2m} v \partial_x C \partial^2_x C +36 {\hbar^2\over 2m} \partial_x v \partial^2_x C \\ +& 21 {\hbar^2\over 2m} v \partial^3_x C + 8 {\hbar^2\over 2m} \partial^3_x v ], \end{aligned} $$

(59)

where for clarity, the superscripts (n′) and subscripts (i′) of the C and v have been dropped. The force f in the first term in brackets in Eq. 59 is the total force (f = f _q  + f _c ). Each term in brackets will be identified as γ _i where i = 1–7 so that γ = ∑ ⁷ _i=1 γ _i /12. Equation 59 shows that γ contains both classical and quantum contributions. The classical contributions correspond to 3 f _c v + γ₂. The remaining terms are all proportional to $\hbar^2$ and are therefore quantum in nature. For a Gaussian wave packet, C is quadratic and v is linear with respect to x − x ₀ where x ₀ is the center of the packet. Thus, the first four γ_1–4 are quadratic (i.e., they are proportional to (x − x ₀)²). The fifth term γ₅ is a constant term proportional to the product of the width of the wave packet and slope of the velocity field. At the edges of the grid, this term is much smaller than the quadratic terms. The last two terms γ₆ and γ₇ involve third derivatives of C and v and are zero for a Gaussian wave packet. For the above reasons, the original IFDM included only the γ_1–3 [14]. Note that for a Gaussian wave packet, the fourth term γ₄ has the same functional form as the quantum force term in γ₁. Due to the accumulation of numerical noise, the γ₆ and especially γ₇ can become significant at the edges of the grid even for a Gaussian wave packet. As shown in Sect. 3.2, the long-term stability of the method is improved by including these terms (especially γ₇). Furthermore, for anharmonic potentials (such as the Morse potential), the third derivative of C in γ₆ can be non-zero even without numerical noise.

In practice, $\tilde\gamma$ in Eq. 26 is computed from evaluating the following expression at each grid point

$$ \tilde\gamma = \frac{1}{12} \{\gamma_0 {\rm MAX}[\vert\gamma_1\vert, \vert\gamma_2\vert,\vert\gamma_3\vert,\vert\gamma_4\vert,\vert\gamma_5\vert] + \gamma'_0 {\rm MAX}[\vert\gamma_6\vert,\vert\gamma_7\vert ]\}, $$

(60)

where the MAX function returns the maximum value of its arguments. The first five γ _i are grouped separately from the γ₆ and γ₇ in order to allow for separate scaling via the two independent constants γ₀ and γ′₀. The artificial viscosity constants γ₀ and γ′₀ are determined from convergence and stability tests. The goal is to choose them as small as possible but still large enough to maintain the stability of the calculation.

2.2 Two-dimensional case

The second-order truncation error with respect to $\Updelta t$ in the finite-difference representation of the velocity equation (Eq. 30) is given by $\varepsilon_{\rm 2D}$ defined in Eq. 43. If the classical potential does not depend explicitly on time, then the time derivatives of the total force become equal to the time derivatives of just the quantum force, and Eq. 43 becomes

$$ \varepsilon_{\rm 2D} =\frac{1}{12}\Updelta t^2 \left [ \partial^2_t f^{n'}_{q i',j} - m \partial^2_t(v^{n'}_{i',j}\partial_x v^{n'}_{i',j}) - m \partial^2_t(u^{n'}_{i',j}\partial_y v^{n'}_{i',j})\right]. $$

(61)

By expanding the time derivatives of the two terms in brackets in Eq. 61 and using the definition of the quantum force (Eq. 32), Eq. 61 becomes a complicated expression involving many terms of mixed time and spacial derivatives of C,  v, and u. By taking the appropriate time and spacial derivatives of the equations of motion for both C (Eq. 29), v (Eq. 27), and u (Eq. 28), all of the various time derivatives of C,  v, and u can be eventually expressed in terms of derivatives of x and y only. The coefficient of the second derivatives of v with respect to x and y can then be identified as the effective numerical diffusion coefficients: $^1\Upgamma(C,v,u)$ and $^2\Upgamma(C,v,u)$ (and similarly for u). The algebra is straightforward but tedious, and for two dimensions, a symbolic algebra solver such as Mathematica [42] is much preferred. The resulting expression for $^1\Upgamma(C,v,u) = {^1}\gamma \Updelta t^2$ is given by

$$ ^1\Upgamma(C,v,u) = \frac{1}{12}\Updelta t^2 \sum_i {^1\gamma_i }$$

(62)

where ¹γ = ∑ _i ¹γ _i /12 and the individual terms are

$$ \begin{aligned} ^1\gamma_1 =& +3 f v , {^1}\gamma_2 = -9 m v^2 \partial_x v , {^1}\gamma_3 = +18 {h} (\partial_x C)^2 \partial_x v ,\\ {^1}\gamma_4 =& +30 {h} v \partial_x C \partial^2_x C , {^1}\gamma_5 = +36 {h} \partial_x v \partial^2_x C , {^1}\gamma_6 = +21 {h} v \partial^3_x C , \\ {^1}\gamma_7 =& +8 {h} \partial^3_x v , {^1}\gamma_8 = -6 m v u \partial_y v , {^1}\gamma_9 = +4 {h} \partial_x C \partial_y C\partial_y v ,\\ {^1}\gamma_{10} =&+8 {h} \partial_x C \partial_y C\partial_x u , {^1}\gamma_{11} = +6 {h} v \partial_y C \partial^2_{xy} C , {^1}\gamma_{12} = +8 {h} \partial^2_{xy} C \partial_y v ,\\ {^1}\gamma_{13} =&+12 {h} u \partial_x C \partial^2_{xy} C , {^1}\gamma_{14} =+16 {h} \partial^2_{xy} C \partial_x u , {^1}\gamma_{15} = +4 {h} \partial^2_{xy} v \partial_y C ,\\ {^1}\gamma_{16} =&+4 {h} \partial^2_{xy} u \partial_x C , {^1}\gamma_{17} = +8 {h} \partial^2_x u \partial_y C , {^1}\gamma_{18} =+12 {h} u \partial^3_{x^2 y} C ,\\ {^1}\gamma_{19} =&+4 {h} \partial^3_{x^2 y} u , {^1}\gamma_{20} =+2 {h} \partial^2_{y} v \partial_x C , {^1}\gamma_{21} =+3 {h} v \partial^3_{x y^2} C ,\\ {^1}\gamma_{22} =&+2 {h} \partial^3_{x y^2} v, \end{aligned} $$

(63)

where for clarity, the superscripts and subscripts on the C,  v, and u have been dropped and $h={\hbar^2/2m}$. The first seven terms are identical to those derived for the one-dimensional case (see Eq. 59). The remaining fifteen new terms involve mixed derivatives, or x derivatives of u and y derivatives of v. For the 2D application considered in this work, most of these new terms for i > 7 are small relative to those for i ≤ 7.

The resulting expression for $^2\Upgamma(C,v,u) = {^2}\gamma \Updelta t^2$ is given by

$$ ^2\Upgamma(C,v,u) = \frac{1}{12}\Updelta t^2 \sum_i {^2\gamma_i} , $$

(64)

where ²γ = ∑ _i ²γ _i /12 and the individual terms are

$$ \begin{aligned} {^2}\gamma_1 =& 3 g u , {^2}\gamma_2 = -3 m u^2 \partial_x v , {^2}\gamma_3 = -6 m u^2 \partial_y u ,\\ {^2}\gamma_4 = &+6 {h} u \partial_y C \partial^2_{x} C , {^2}\gamma_5 = +5 {h} \partial_x v \partial^2_{x} C , {^2}\gamma_6 = +3 {h} v \partial^3_{x} C ,\\ {^2}\gamma_7 = &+{h} \partial^3_{x} v , {^2}\gamma_8 = -6 m u v \partial_x u , {^2}\gamma_{9} = +2 {h} \partial_x C \partial_y C \partial_y v ,\\ {^2}\gamma_{10} = &+4 {h} \partial_x C \partial_y C \partial_x u , {^2}\gamma_{11} = +4 {h} \partial_y u \partial^2_{x} C , {^2}\gamma_{12} = +3 {h} \partial^2_{xy} C \partial_y v ,\\ {^2}\gamma_{13} =& +6 {h} u \partial_x C \partial^2_{xy} C , {^2}\gamma_{14} = +10 {h} \partial^2_{xy} C \partial_x u , {^2}\gamma_{15} = +2 {h} \partial^2_{x} v \partial_x C ,\\ {^2}\gamma_{16} = &+4 {h} \partial^2_{xy} u \partial_x C , {^2}\gamma_{17} = +4 {h} \partial^2_{x} u \partial_y C , {^2}\gamma_{18} = +9 {h} u \partial^3_{x^2 y} C ,\\ {^2}\gamma_{19} = &+3 {h} \partial^3_{x^2 y} u. \end{aligned} $$

(65)

The various terms in Eqs. 63 and 65 have been ordered so that they match each other in form except for the terms with i = 3,  11, and 15. For the 2D application considered in this work, most of the terms in Eq. 65 for i > 7 are small relative to those for i ≤ 7.

In practice $^1\tilde\gamma$ in Eq. 45 is computed from evaluating the following expression at each grid point

$$ ^1\tilde\gamma = \frac{1}{12} \left\{ ^1\gamma_0 {\rm MAX}\left[\sum_{i\ne 6,7}\vert ^1\gamma_i\vert\right] + {^1}\gamma'_0 {\rm MAX}\left[\vert ^1\gamma_6\vert,\vert ^1\gamma_7\vert\right]\right\}, $$

(66)

and similarly for $^2\tilde\gamma$

$$ ^2\tilde\gamma = \frac{1}{12} \left\{ ^2\gamma_0 {\rm MAX}\left [\sum_{i\ne 6,7}\vert ^2\gamma_i\vert\right] + {^2}\gamma'_0 {\rm MAX}\left[\vert ^2\gamma_{6}\vert,\vert ^2\gamma_{7}\vert\right]\right\}, $$

(67)

where the MAX function returns the maximum value of its arguments. The artificial viscosity constants ¹γ₀,  ¹γ′₀,  ²γ₀, and ²γ′₀ are determined from convergence and stability tests. The goal is to choose them as small as possible but still large enough to maintain the stability of the calculation. For the 2D application considered in this work, stability was obtained using only the first seven terms in Eqs. 63 and 65 (i.e., i ≤ 7) for the diffusion coefficients $^1\Upgamma$ and $^2\Upgamma$ multiplying $\partial^2_x v$ and $\partial^2_y v,$ respectively. Similar expressions are used for the diffusion coefficients multiplying ∂ ² _x u and ∂ ² _y u. These expressions can be obtained from those in Eqs. 63 and 65 by interchanging $x\leftrightarrow y,\,v\leftrightarrow u,$ and $f\leftrightarrow g$. In evaluating the various ¹γ _i and ²γ _i , it is important to use the appropriate four-point averages so that they are properly evaluated on the staggered v or u grids (see Fig. 1).