基于虚时间演化与谱方法的一类基态Wigner函数计算方法

doi:10.19596/j.cnki.1001-246x.8505

摘要/Abstract

摘要：

基于虚时间演化法, 实现基态Wigner函数的数值计算。考虑到Wigner函数是一个高维相空间函数, 通过引入合适的谱方法以降低问题的维度, 并处理所得控制方程中的全局算子。首先, 从Schrödinger波函数的虚时间演化方程出发, 推导得到适用于Wigner函数的控制方程。然后, 根据方程中卷积项的不同表达, 基于简化Grad矩方法和傅里叶伪谱方法, 分别设计两类p空间上的离散方法, 并阐述其中的求解细节。此外, 基于所提出的数值框架, 探讨基于密度泛函理论的基态计算。其中, 针对简化Grad矩方法下的密度泛函计算, 提出一类密度高阶导数的重构方式。最后, 数值算例展示所提方法的有效性, 以及方法在多体问题中的应用前景。

关键词: Wigner函数, 基态, 虚时间演化法, 谱方法, 密度泛函理论

Abstract:

Based on imaginary time propagation method, we realize the Wigner ground state calculation. Since Wigner function is a high dimensional phase space function, appropriate spectral methods are introduced to reduce the dimensionality and to handle the global operators. Firstly, with the aid of the governing equation in Schrödinger imaginary time propagation method, governing equation of Wigner version is derived. Then, according to the expression of the convolution term in the equation, discretization methods of momentum space are designed with simplified Grad moment method and Fourier pseudo-spectral method, respectively. Corresponding numerical details are discussed. In particular, we consider density functional theory calculation in the framework of our methods. A reconstruction method of high order derivative of density is proposed for simplified Grad moment method. Several numerical experiments demonstrate validity of our methods and the potential in many-body problems.

Key words: Wigner function, ground state, imaginary time propagation method, spectral method, density functional theory

詹泓飞, 蔡振宁, 胡光辉. 基于虚时间演化与谱方法的一类基态Wigner函数计算方法[J]. 计算物理, 2022, 39(6): 651-665.

Hongfei ZHAN, Zhenning CAI, Guanghui HU. Wigner Ground State Calculation Based on Imaginary Time Propagation Method and Spectral Method[J]. Chinese Journal of Computational Physics, 2022, 39(6): 651-665.

图/表 7

图1 M=18时，二维谐振子下简化Grad矩方法的误差$ e_{\mathrm{G}}=\max _{|\alpha| \leqslant M}\left\|f_\alpha-f_\alpha^{\text {exc }}\right\|_{\infty}$

Fig.1 Errors $ e_{\mathrm{G}}=\max _{|\alpha| \leqslant M}\left\|f_\alpha-f_\alpha^{\text {exc }}\right\|_{\infty}$ for 2-D harmonic oscillator at M=18

图2 二维谐振子下简化Grad矩方法的误差(a) 整体误差$e_{\mathrm{G}}=\max _{|\alpha| \leqslant M}\left\|f_\alpha-f_\alpha^{\text {exc }}\right\|_\infty $; (b) 密度误差$ e_\rho=\left\|\rho-\rho^{\mathrm{exc}}\right\|_{\infty}$

Fig.2 Errors with simplified Grad moment method for 2-D harmonic oscillator (a) total errors $e_{\mathrm{G}}=\max _{|\alpha| \leqslant M}\left\|f_\alpha-f_\alpha^{\text {exc }}\right\|_\infty $; (b) density errors $ e_\rho=\left\|\rho-\rho^{\mathrm{exc}}\right\|_{\infty}$

图3 二维谐振子下傅里叶伪谱方法的误差(a) 整体误差$ e_{\mathrm{F}}=\max _{\alpha \in I_N}\left\|a_\alpha-a_\alpha^{\text {exc }}\right\|_{\infty}$; (b) 密度误差$e_\rho=\left\|\rho-\rho^\text {exc }\right\|_{\infty} $

Fig.3 Errors with Fourier pseudo-spectral method for 2-D harmonic oscillator (a) total errors $ e_{\mathrm{F}}=\max _{\alpha \in I_N}\left\|a_\alpha-a_\alpha^{\text {exc }}\right\|_{\infty}$; (b) density errors $e_\rho=\left\|\rho-\rho^\text {exc }\right\|_{\infty} $

表1 简化Grad矩方法下, Contact-interacting Hooke atom密度的数值误差‖ρwGrad - ρsfinest‖∞与能量

Table 1 Density errors ‖ρwGrad - ρsfinest‖∞ and energies of contact-interacting Hooke atom with simplified Grad moment method

h	Wigner: 简化Grad矩方法						Schrödinger
	M = 0		M = 2		M = 4		Schrödinger
	error	energy	error	energy	error	energy	energy
2.00 × 10^-1	2.106 8 × 10^-3	1.317 18	1.971 0 × 10^-3	1.317 27	2.287 9 × 10^-3	1.315 68	1.314 76
1.00 × 10^-1	5.821 7 × 10^-4	1.313 98	5.147 9 × 10^-4	1.314 03	6.033 8 × 10^-4	1.313 60	1.313 47
5.00 × 10^-2	1.706 3 × 10^-4	1.313 23	1.368 0 × 10^-4	1.313 26	1.597 9 × 10^-4	1.313 15	1.313 15
2.50 × 10^-2	5.380 8 × 10^-5	1.313 06	3.682 7 × 10^-5	1.313 08	4.265 6 × 10^-5	1.313 05	1.313 07
1.25 × 10^-2	1.772 3 × 10^-5	1.313 04	9.211 2 × 10^-6	1.313 04	1.067 7 × 10^-5	1.313 04	1.313 05
6.25 × 10^-3	5.268 8 × 10^-6	1.313 03	3.002 1 × 10^-6	1.313 04	3.446 0 × 10^-6	1.313 04	1.313 04

表2 傅里叶伪谱方法下, Contact-interacting Hooke atom密度的数值误差‖ρwFourier - ρsfinest‖∞与能量

Table 2 Density errors ‖ρwFourier - ρsfinest‖∞ and energies of contact-interacting Hooke atom with Fourier pseudo-spectral method

h	Wigner: 傅里叶伪谱方法						Schrödinger
	N = 16		N = 32		N = 64		Schrödinger
	error	energy	error	energy	error	energy	energy
2.00 × 10^-1	4.518 0 × 10^-4	1.212 67	2.218 5 × 10^-3	1.317 50	2.218 9 × 10^-3	1.317 52	1.314 76
1.00 × 10^-1	2.237 4 × 10^-3	1.209 54	5.3788 × 10^-4	1.314 20	5.382 8 × 10^-4	1.314 22	1.313 47
5.00 × 10^-2	2.667 7 × 10^-3	1.208 75	1.3675 × 10^-4	1.313 32	1.371 5 × 10^-4	1.313 34	1.313 15
2.50 × 10^-2	2.775 3 × 10^-3	1.208 56	3.6469 × 10^-5	1.313 10	3.646 9 × 10^-5	1.313 12	1.313 07
1.25 × 10^-2	2.802 2 × 10^-3	1.208 51	1.1414 × 10^-5	1.313 04	1.181 5 × 10^-5	1.313 06	1.313 05
6.25 × 10^-3	2.808 9 × 10^-3	1.208 45	5.1510 × 10^-6	1.313 03	5.552 2 × 10^-6	1.313 05	1.313 04

表3 简化Grad矩方法下，Contact-interacting Hooke atom的相对误差$\operatorname{max}_{0 \leqslant k \leqslant K}\left\|f_{2 k}-f_{2 k}^{\text {finest }}\right\|_{\infty} $与收敛阶

Table 3 Relative errors $\operatorname{max}_{0 \leqslant k \leqslant K}\left\|f_{2 k}-f_{2 k}^{\text {finest }}\right\|_{\infty} $ and convergence order with simplified Grad moment method for contact-interacting Hooke atom

h	K = 0		K = 1		K = 2
h	error	rate	error	rate	error	rate
2.00 × 10^-1	2.101 5 × 10^-3		1.970 0 × 10^-3		2.286 6 × 10^-3
1.00 × 10^-1	5.769 0 × 10^-4	1.865 0	5.137 8 × 10^-4	1.939 0	6.020 1 × 10^-4	1.925 3
5.00 × 10^-2	1.653 6 × 10^-4	1.802 7	1.358 0 × 10^-4	1.919 7	1.584 1 × 10^-4	1.926 1
2.50 × 10^-2	4.854 0 × 10^-5	1.768 4	3.582 0 × 10^-5	1.922 6	4.128 2 × 10^-5	1.940 1
6.25 × 10^-3	1.245 4 × 10^-5	1.962 5	8.204 2 × 10^-6	2.126 3	9.303 2 × 10^-6	2.149 7

表4 傅里叶伪谱方法下，Contact-interacting Hooke atom的相对误差$\max _{\alpha \in \mathcal{I}_N}\left\|a_\alpha-a_\alpha^{\text {finest }}\right\|_{\infty} $与收敛阶

Table 4 Relative errors $\max _{\alpha \in \mathcal{I}_N}\left\|a_\alpha-a_\alpha^{\text {finest }}\right\|_{\infty} $ and convergence order with Fourier pseudo-spectral method for contact-interacting Hooke atom

h	N = 16		N = 32		N = 64
h	error	rate	error	rate	error	rate
2.00 × 10^-1	2.357 1 × 10^-3		2.213 4 × 10^-3		2.213 3 × 10^-3
1.00 × 10^-1	5.715 5 × 10^-4	2.044 1	5.327 3 × 10^-4	2.054 8	5.327 2 × 10^-4	2.054 8
5.00 × 10^-2	1.412 7 × 10^-4	2.016 5	1.316 0 × 10^-4	2.017 3	1.316 0 × 10^-4	2.017 3
2.50 × 10^-2	3.362 4 × 10^-5	2.070 8	3.131 8 × 10^-5	2.071 1	3.131 8 × 10^-5	2.071 1
6.25 × 10^-3	6.726 6 × 10^-6	2.321 6	6.263 3 × 10^-6	2.322 0	6.263 2 × 10^-6	2.322 0

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