Wigner Ground State Calculation Based on Imaginary Time Propagation Method and Spectral Method

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

Abstract

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

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.

Figures/Tables 7

References 54

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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

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

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

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

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