Correction of diffusion calculations when using two types of non-rectangular simulation boxes in molecular simulations

Abstract

Although simulation boxes used in molecular dynamics are normally chosen to be cubic or rectangular, two other cell shapes that are very familiar to crystallographers—the truncated octahedron and the rhombic dodecahedron—could also be used because they are also space-filling cells. Due to their spherical nature, these boxes have been intentionally applied in simulations of biomolecular solutions and liquid structures. Indeed, due to the advantages of running many molecular dynamic codes in parallel, simulations based on these non-rectangular boxes have been growing in popularity in recent years. In this work, the effects of using these two types of boxes on diffusion are explored for the first time, and an appropriate correction formula is derived theoretically within the framework of hydrodynamics. In addition, the range of validity for the correction formula is evaluated by performing molecular dynamic simulations on argon at three different densities.

Appendices

Appendix 1: Proof that a diagonal second-rank tensor with the same diagonal elements is invariant under coordinate rotations. i.e., ${T}_{ij}^{\prime }={T}_{ij}$

Rotation of coordinate axes and orthogonal matrices

We consider two coordinate systems K and K′ with a common origin. Any vector X can be expressed as (x₁, x₂, x₃) in the unprimed system or $\left({x}_1^{\prime },{x}_2^{\prime },{x}_3^{\prime}\right)$ in the primed system (see Fig. 6). $\mathbf{X}={\sum}_{\mathrm{j}=1}^3{x}_j{\mathbf{e}}_j$ or, in the Einstein convention (i.e., if an index occurs twice or more in a term, it implies that the term is to be summed over all possible values of the index),

$$ \mathbf{X}={x}_j{\mathbf{e}}_j, $$

(11)

and in the K′ coordinate system,

$$ \mathbf{X}={x}_k^{\prime }{\mathbf{e}}_k^{\prime }, $$

(12)

where ${\mathbf{e}}_j\ \mathrm{and}\ {\mathbf{e}}_j^{\prime }$ are the unit vectors in the jth direction in the K and K′ coordinate systems, respectively. Combining Eq. 11 and Eq. 12, we obtain

$$ {x}_k^{\prime }{\mathbf{e}}_k^{\prime }={x}_j{\mathbf{e}}_j. $$

(13)

Fig. 6

Rotation of the coordinate axes equivalent to a three-dimensional orthogonal transformation

Left-dotting ${\mathbf{e}}_i^{\prime }$ into Eq. 13, and using ${\mathbf{e}}_i^{\prime}\bullet {\mathbf{e}}_k^{\prime }={\delta}_{ik}$,

$$ {x}_k^{\prime}\left({\mathbf{e}}_i^{\prime}\bullet {\mathbf{e}}_k^{\prime}\right)={x}_k^{\prime }{\delta}_{ik}={x}_j\left({\mathbf{e}}_i^{\prime}\bullet {\mathbf{e}}_j\right). $$

We now define the coordinate rotation operation matrix A, whose elements are

$$ {a}_{ij}={\mathbf{e}}_i^{\prime}\bullet {\mathbf{e}}_j. $$

(14)

Using this definition, we have

$$ {x}_i^{\prime }={a}_{ij}{x}_j, $$

(15)

or, in Dirac bra-ket notation,

$$ \left|\left.{x}^{\prime}\right\rangle =\mathbf{A}\right|\left.x\right\rangle . $$

(16)

The vector in Eq. 16 with one column and three rows is called a column vector. Similarly, if a matrix has one row and three columns, it is called a row vector, 〈x|, with components x_i, i = 1, 2, 3. Clearly, 〈x| is derived from ∣x〉 by interchanging rows and columns, a matrix operation called “transposition.” The transposition of any matrix A yields A^T, called the transpose of A, with matrix elements (A^T)_ij = A_ji. Writing Eq. 16 in row vector form,

$$ \left\langle {x}^{\prime}\mid =\right\langle x\mid {\mathbf{A}}^{\mathrm{T}}. $$

(17)

Matrix A represents an operation that transforms the components of X in the unprimed system to the components in the primed system. By the same argument, if there is an operation matrix A⁻¹ that inversely transforms the components of X in the primed system to those in the unprimed system,

$$ {x}_i={a}_{ij}^{-1}{x}_j^{\prime } $$

(18)

or

$$ \left|\left.x\right\rangle ={\mathbf{A}}^{-\mathbf{1}}\right|\left.{x}^{\prime}\right\rangle . $$

(19)

That is, A⁻¹ produces the opposite rotation to that given by A, meaning that it returns the coordinate system to its original position. Equations 16 and 19 combine to give

$$ \left|\left.x\right\rangle ={\mathbf{A}}^{-\mathbf{1}}\mathbf{A}\right|\left.x\right\rangle . $$

(20)

Since ∣x〉 is chosen arbitrarily,

$$ {\mathbf{A}}^{-1}\mathbf{A}=\mathbf{1}, $$

(21)

the unit matrix. Similarly, using Eqs. 16 and 19 and eliminating ∣x〉 instead of ∣x′〉,

$$ \mathbf{A}{\mathbf{A}}^{-1}=\mathbf{1}. $$

(22)

From Eqs. 21 and 22, A commutes with A⁻¹.

The rotation given by A keeps the length from the origin to the point the same in both systems. Squaring for convenience gives

$$ \left\langle x|\left.x\right\rangle \right.=\left\langle {x}^{\prime}\right.\mid \left.{x}^{\prime}\right\rangle =\left\langle x|{\mathbf{A}}^{\mathrm{T}}\mathbf{A}|\left.x\right\rangle \right. $$

(23)

Since vector ∣x〉 is arbitrary,

$$ {\mathbf{A}}^{\mathrm{T}}\mathbf{A}=\mathbf{1}. $$

(24)

Multiplying Eq. 24 by A⁻¹ from the right and using Eq. 22, we have

$$ {\mathbf{A}}^{\mathrm{T}}={\mathbf{A}}^{-1}, $$

(25)

another definition of an orthogonal matrix. Multiplying Eq. 25 by A, we obtain

$$ \mathbf{A}{\mathbf{A}}^{\mathrm{T}}=\mathbf{1} $$

(26)

or

$$ {a}_{ji}{a}_{ki}={\delta}_{jk}. $$

(27)

Diagonal tensor and its transformation under coordinate rotations

In physics, the mobility tensor T_ij is defined as the ith component of the drift velocity of an object under the influence of unit external force in the jth direction,

$$ {V}_i={T}_{ij}{F}_j, $$

(28)

if the system is symmetric about the three principal axes in the K coordinate system (j  = 1, 2, 3). T_ij must be a diagonal tensor with the same values as T_jj. Its components in the K′ coordinate system, ${T}_{ij}^{\prime },$ conform to the transformation

$$ {T}_{ij}^{\prime }={a}_{ik}{a}_{jl}{T}_{kl}. $$

(29)

Because T_kl = 0 for k ≠ l and T_kl = T₁₁ for k = l, any diagonal elements must be equal to T₁₁. Taking i = j = 2 for example, we expand Eq. 29 as

$$ {\displaystyle \begin{array}{l}{T}_{22}^{\prime }={a}_{21}{a}_{21}{T}_{11}+{a}_{21}{a}_{22}{T}_{12}+{a}_{21}{a}_{23}{T}_{13}\\ {}+{a}_{22}{a}_{21}{T}_{21}+{a}_{22}{a}_{22}{T}_{22}+{a}_{22}{a}_{23}{T}_{23}\\ {}+{a}_{23}{a}_{21}{T}_{31}+{a}_{23}{a}_{22}{T}_{32}+{a}_{23}{a}_{33}{T}_{33}\\ {}=\left({a}_{21}{a}_{21}+{a}_{22}{a}_{22}+{a}_{23}{a}_{23}\right){T}_{11}\\ {}={a}_{2i}{a}_{2i}{T}_{11}\end{array}}. $$

From Eq. 27, j = k = 2 and a_2ia_2i = δ₂₂ = 1, so we have

$$ {T}_{22}^{\prime }={T}_{11}. $$

Similarly, for the off-diagonal elements ${T}_{ij}^{\prime },$

$$ {T}_{ij}^{\prime }=0\mathrm{for}\ i\ne j. $$

Thus,${T}_{ij}^{\prime }$ has the same form as T_ij.

Appendix 2: Equivalence of the descriptions of motion in three different primitive cells in molecular dynamics

Let us consider the three different primitive cells for the lattice shown in Fig. 7. The first type of cell, a parallelogram formed by two translational vectors a₁ and a₂, is primitive because there is only one lattice point in each cell (Fig. 7a). The rectangles in Fig. 7b are another form of primitive cell. One edge of the rectangular cell overlaps with the edge of the parallelogram, while another edge is just the projection of the parallelogram on the y-axis. If we suppose that all the lattice points lie at the geometrical centers of the parallelograms, then the area enclosed by the perpendicular bisectors of the lines joining the nearest lattice points is the Wigner–Seitz cell (Fig. 7c).

Fig. 7a–c

Graphic illustrating the equivalence of the three cell types in two dimensions: parallelogram (a), rectangular box (b), and Wigner–Seitz cell (c)

It is easy to prove that the areas of the three cells are the same. If N molecules are placed in the parallelogram and periodic boundary conditions are applied (only the ith and the jth molecules are shown in Fig. 7a), each molecule will appear exactly once in both the rectangular cell (Fig. 7b) and the Wigner–Seitz cell (Fig. 7c). Of course, some molecules that appear in the rectangular cell (or the Wigner–Seitz cell) are just image molecules in the parallelogram. Whether the molecule or its image is included makes no difference because the two possess exactly the same physical quantities (mass, velocity, spatial orientation, etc.). For a given molecule, e.g., the jth one, the other molecules around it are exactly the same in all three cases, so the forces exerted on the jth molecule are identical, which in turn gives rise to the same trajectory of motion in each case. In fact, if the boundaries of the cells are removed, the spatial locations of all the molecules are exactly the same. This best illustrates the equivalence of the descriptions of motion in the three cell types.