q-Vector Generation๏ƒ

Reciprocal Lattice q-Vectors๏ƒ

Let \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) be the basis vectors which span the MD cell. Any position vector in the MD cell can be written as

(46)๏ƒ\[{{\mathbf{r} = x'}{\mathbf{a} + y'}{\mathbf{b} + z'}\mathbf{c}}\]

so that it defines the position vector in the MD cell. With \(x'\), \(y'\), \(z'\) having values between \(0\) and \(1\) if \(\mathbf{r}\) is in the unit cell. The primes indicate that the coordinates are fractional coordinates. A jump due to periodic boundary conditions can cause \(x'\), \(y'\), \(z'\) to jump by \(\pm1\). The set of dual basis vectors \(\mathbf{a}^*\), \(\mathbf{b}^*\), \(\mathbf{c}^*\) where

(47)๏ƒ\[\mathbf{a}^* = \frac{2 \pi}{V} \mathbf{b} \times \mathbf{c}, \qquad \mathbf{b}^* = \frac{2 \pi}{V} \mathbf{c} \times \mathbf{a}, \qquad \mathbf{c}^* = \frac{2 \pi}{V} \mathbf{a} \times \mathbf{b}.\]

If the \(\mathbf{q}\)-vectors are now chosen as

(48)๏ƒ\[\mathbf{q} = h\mathbf{a}^* + k\mathbf{b}^* + l\mathbf{c}^*\]

so that this selection of \(\mathbf{q}\)-vectors produces phase changes for handling jumps in particle trajectories. Here \(h\), \(k\), and \(l\) are integers, jumps in the particle trajectories produce phase changes of multiples of \(2\pi\) in the Fourier transformed particle density, i.e. leave it unchanged.