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通过将平面波与高斯函数或者样条函数结合到一起, 本文构建了一种新的复合基组. 利用格拉姆-施密特正交化方法或者Löwdin正交化方法, 对复合基组进行正交归一化. 通过选择平面波函数中波矢的绝对值, 选择性地求解某个能量区间内的本征态, 将大型哈密顿矩阵的计算转变为多个小型矩阵的计算, 以及通过减少电子势能平缓部分展开基矢数目, 极大地加快了计算速度. 以一维有限深势阱为例, 通过与严格计算方法的对比, 验证了本文复合基组能够在加速计算的情况下保证求解精度. 同时, 本文还研究了不同的参数设置对计算精度的影响, 包括复合基矢的疏密度、高斯函数的宽值, 以及样条函数不同区域占函数总宽度的比值等参数. 最后该复合基组可以直接应用到对大尺寸纳米金属结构的等离激元数值计算当中.By combining plane waves with Gaussian or spline functions, a new composite basis set is constructed in this work. As a non local basis vector, the plane wave basis group needs a large number of plane waves to expand all parts of the physical space, including the intermediate regions that are not important to our problems. Our basis set uses the local characteristics of Gaussian function or spline function at the same time, and controls the energy interval by selecting different plane wave vectors, in order to realize the partition solution of Hamiltonian matrix. Orthogonal normalization of composite basis sets is performed by using Gram-Schmidt’s orthogonalization method or Löwdin’s orthogonalization method. Considering the completeness of plane wave vector, a certain value of positive and negative should be selected at the same time. Here, by changing the absolute value of wave vector, we can select the eigenvalue interval to be solved. The plane wave with a specific wave vector value is equivalent to a trial solution in the region with gentle potential energy. The algorithm automatically combines local Gaussian or spline functions to match the difference in wave vector value between the trial solution and the strict solution. By selecting the absolute value of the wave vector in the plane wave function, the calculation of large Hamiltonian matrices turns into the calculation of multiple small matrices, together with reducing the basis numbers in the region where the electron potential changes smoothly, therefore, we can significantly reduce the computational time. As an example, we apply this basis set to a one-dimensional finite depth potential well. It can be found that our method significantly reduce the number of basis vectors used to expand the wave function while maintaining a suitable degree of computational accuracy. We also study the influence of different parameters on calculation accuracy. Finally, the above calculation method can be directly applied to the density functional theory (DFT) calculation of plasmons in silver nanoplates or other metal nanostructures. Given a reasonable tentative initial state, the ground state electron density distribution of the system can be solved by self-consistent solution through using DFT theory, and then the electromagnetic field distribution and optical properties of the system can be solved by using time-dependent density functional theory.
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Keywords:
- composite basis set /
- plane wave /
- spline function /
- numerical calculation of plasmonic properties
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图 1 用来复合的高斯函数与样条函数 (a)高斯函数; (b)样条函数
Fig. 1. Gaussian wave function and spline function for composition: (a) Gaussian wave function; (b) spline function.
图 2 仅使用少量基矢时, 计算值与理论值的相对误差(x轴为不同的本征态, y轴为计算值和理论值之间的相对误差, 后文相同)
Fig. 2. Relative error between calculated and theoretical values when only a small number of base vectors are used (x-axis, different eigenstates; y-axis, relative error between calculated and theoretical values; the following image is the same as the Fig. 2).
图 6 耦合部分与函数宽度的比值对精度的影响
Fig. 6. Effect of the ratio of coupling part to function width on precision.
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[1] Wang L W, Kang Y F, Liu X H, Zhang S M, Huang W P, Wang S R 2012 Sensor. Actuat. B-Chem. 162 237
Google Scholar
[2] Huynh W U, Dittmer J J, Alivisatos A P 2002 Science 295 2425
Google Scholar
[3] Jain P K, Huang X, El-Sayed I H, El-Sayed M A 2008 Acc. Chem. Res. 41 1578
Google Scholar
[4] Jiang N N, Zhuo X L, Wang J F 2018 Chem. Rev. 118 3054
Google Scholar
[5] Baumberg J J, Aizpurua J, Mikkelsen M H, Smith D R 2019 Nat. Mater. 18 668
Google Scholar
[6] Benz F, Schmidt M K, Dreismann A, Chikkaraddy R, Zhang Y, Demetriadou A, Carnegie C, Ohadi H, De Nijs B, Esteban R 2016 Science 354 726
Google Scholar
[7] Hao E, Schatz G C 2004 J. Chem. Phys. 120 357
Google Scholar
[8] Ghosh S K, Pal T 2007 Chem. Rev. 107 4797
Google Scholar
[9] Romero I, Aizpurua J, Bryant G W, De Abajo F J G 2006 Opt. Express. 14 9988
Google Scholar
[10] Schuller J A, Barnard E S, Cai W, Jun Y C, White J S, Brongersma M L 2010 Nat. Mater. 9 193
Google Scholar
[11] Chen H J, Shao L, Li Q, Wang J F 2013 Chem. Soc. Rev. 42 2679
Google Scholar
[12] Xu H X, Bjerneld E J, Käll M, Börjesson L 1999 Phys. Rev. Lett. 83 4357
Google Scholar
[13] Michaels A M, Jiang J, Brus L 2000 J. Phys. Chem. B 104 11965
Google Scholar
[14] Pines D 1953 Phys. Rev. 92 626
Google Scholar
[15] Hopfield J 1958 Phys. Rev. 112 1555
Google Scholar
[16] Elson J, Ritchie R 1971 Phys. Rev. B 4 4129
Google Scholar
[17] Waks E, Sridharan D 2010 Phys. Rev. A 82 043845
Google Scholar
[18] Li Z P, Xu H X 2007 J. Quant. Spectrosc. Ra. 103 394
Google Scholar
[19] Flatau P J, Fuller K A, Mackowski D W 1993 Appl. Opt. 32 3302
Google Scholar
[20] Futamata M, Maruyama Y, Ishikawa M 2003 J. Phys. Chem. B 107 7607
Google Scholar
[21] Mao L, Li Z P, Wu B, Xu H X 2009 Appl. Phys. Lett. 94 243102
Google Scholar
[22] Zuloaga J, Prodan E, Nordlander P 2009 Nano. Lett. 9 887
Google Scholar
[23] Parr R G 1983 Annu. Rev. Phys. Chem. 34 631
Google Scholar
[24] Engel E, Dreizler R M 2011 Density Functional Theory (Berlin: Springer) pp11–55
[25] Burke K 2012 J. Chem. Phys. 136 150901
Google Scholar
[26] Argaman N, Makov G 2000 Am. J. Phys. 68 69
Google Scholar
[27] Runge E, Gross E K 1984 Phys. Rev. Lett. 52 997
Google Scholar
[28] Marques M A, Gross E K 2004 Annu. Rev. Phys. Chem. 55 427
Google Scholar
[29] Perdew J P, Schmidt K 2001 AIP Conf. Proc. 577 1
Google Scholar
[30] Perdew J P, Ruzsinszky A, Tao J, Staroverov V N, Scuseria G E, Csonka G I 2005 J. Chem. Phys. 123 062201
Google Scholar
[31] Meng L Y, Yam C Y, Koo S K, Chen Q, Wong N, Chen G H 2012 J. Chem. Theory. Comput. 8 1190
Google Scholar
[32] Yam C Y, Meng L Y, Chen G H, Chen Q, Wong N 2011 Phys. Chem. Chem. Phys. 13 14365
Google Scholar
[33] 何禹, 王一波 2017 物理化学学报 33 1149
Google Scholar
He Y, Wang Y B 2017 Acta Phys. Chim. Sin. 33 1149
Google Scholar
[34] 段宜武, 吴为平, 鲍诚光, 安伟科 1991 高能物理与核物理 15 42
Duan Y W, Wu W P, Bao C G, An W K 1991 Phys. Ener. Fort. Phys. Nucl. 15 42
[35] Hu H P, Wang M, Ding Z Y, Ji G F 2016 Acta Phys. Chim. Sin. 32 2059
Google Scholar
[36] Wachters A J H 1970 J. Chem. Phys. 52 1033
Google Scholar
[37] Rice J R 1966 Math. Comput. 20 325
Google Scholar
[38] Björck Å 1994 Linear. Algebra. Appl. 197 297
[39] Aiken J G, Erdos J A, Goldstein J A 1980 Int. J. Quantum. Chem. 18 1101
Google Scholar
[40] 曾谨言 2007 高等量子力学(第一卷) (北京: 科学出版社) 第72页
Zeng J Y 2007 Advanced Quantum Mechanics (Vol. 1) (Beijing: Science Press) p72
[41] Mao L, Wu B 2011 Surf. Sci. 605 1230
Google Scholar
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