Calculation of lattice sums of general type
- PDF / 1,770,863 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 65 Downloads / 187 Views
Calculation of lattice sums of general type A. Popov1 · V. Popov2 Received: 1 April 2020 / Accepted: 13 September 2020 © Springer Nature Switzerland AG 2020
Abstract An efficient calculation of lattice sums is given. The proposed algorithm is based on Ewald decomposition and gives a new insight into the calculation of lattice sums with an arbitrary degree of |𝐑 − 𝐫| , that arise in a solution to many problems of the crystalline state. Its implementation to the calculation of the Madelung constant is presented for PC, BCC and FCC lattices. Keywords Lattice sums · Numerical methods · Periodic structures · Convergence · Ewald method · Madelung constant Mathematics Subject Classification 33F05 · 40A25 · 40A30 · 40B05 · 40C15 · 40F05 · 40H05 · 65D20 · 65Z05
1 Introduction A theoretical description of the physical properties of the crystalline state of matter [1–4] requires often an efficient and accurate computation of lattice sums [5]. An elegant method of summation of divergent series proposed by Ewald almost a hundred years ago [6] is still the most popular [7]. According to this method the sum of the vectors of a crystal lattice R in three-dimensional space at an arbitrary r:
S−1 (𝐫, 𝟎) =
∑ 𝐑
1 |𝐑 − 𝐫|
(1)
is replaced by two sums with the exponential rate of convergence: * A. Popov [email protected] V. Popov [email protected] 1
Department of Modern Special Materials, Polzunov Altai State Technical University, Barnaul, Russia
2
Department of Physics, Polzunov Altai State Technical University, Barnaul, Russia
13
Vol.:(0123456789)
Journal of Mathematical Chemistry
S−1 (𝐫, 𝟎) =
� erfc 𝐑
�
�√
� 𝛼�𝐑 − 𝐫�
�𝐑 − 𝐫�
� � � 1 K2 4𝜋 � 1 . − exp i𝐊 ⋅ 𝐫 − + Ω 𝐊≠𝟎 K 2 4𝛼 4𝛼
(2)
Here erfc(x) is the complementary error function, Ω is the volume of a three-dimensional space cell, K is the reciprocal lattice vector, 𝛼 is the convergence parameter
Kmax , 2Rmax
𝛼=
(3)
chosen from the condition of equally fast convergence of the both lattice sums up to the radii Rmax and Kmax of the spheres covering the number of nodes close to each other, both in R and in K.The criterion of correct representation of the expression for S−1 (𝐫, 𝟎) is the independence of the right-hand side of (2) from the convergence parameter 𝛼 , the optimal choice of which is determined by the relation (3). For an arbitrary vector 𝐤 ≠ 0 the sum of a more general form
S−1 (𝐫, 𝐤) ≡
∑ ei𝐤⋅𝐑 4𝜋 ∑ ei(𝐊+𝐤)⋅𝐫 = |𝐑 − 𝐫| Ω 𝐊 |𝐊 + 𝐤|2 𝐑
is replaced by two exponentially fast converging sums: �√ � i𝐤⋅𝐑 𝛼�𝐑 − 𝐫� � e erfc S−1 (𝐫, 𝐤) = �𝐑 − 𝐫� 𝐑 � � �𝐊 + 𝐤�2 1 4𝜋 � . + exp i(𝐊 + 𝐤) ⋅ 𝐫 − Ω 𝐊 �𝐊 + 𝐤�2 4𝛼 In particular, for 𝐫 = 𝟎 and 𝐤 ≠ 𝟎 :
S−1 (𝟎, 𝐤) =
i𝐤⋅𝐑 � e erfc 𝐑≠𝟎
�√
R
�
�
𝛼R −
4𝛼 𝜋
� � �𝐊 + 𝐤�2 1 4𝜋 � exp − + , Ω 𝐊 �𝐊 + 𝐤�2 4𝛼 for 𝐫 = 𝟎 and 𝐤 = 𝟎 :
13
(4)
(5)
(6)
Journal of Mathematical Chemistry
S−1 (𝟎, 𝟎) =
�√ � 𝛼R � erfc 𝐑≠𝟎
�
R
� −
4𝛼 𝜋
(7)
� � � 4𝜋 � 1 1 K2 . + − exp − Ω 𝐊≠𝟎 K 2 4𝛼 4𝛼
Note that the method for calculation of divergent series by removal of the constants, which are infinitely large in magnitude,
Data Loading...
