1. Defining the Vector System
Before calculating correlations, we must define what is moving. The authors use a vector connecting second-nearest neighbors ($C_n$ to $C_{n+2}$) to capture local backbone dynamics.
Vector rmn: Connecting C2 to C4
Normalized to unit vector umn
The Setup
We track the position $\mathbf{R}$ of atoms $n$ and $m$. The raw vector is:
$$\mathbf{r}_{mn}(t) = \mathbf{R}_n(t) - \mathbf{R}_m(t)$$
Since we care about orientation, not bond stretching, we normalize it:
$$\mathbf{u}_{mn}(t) = \frac{\mathbf{r}_{mn}(t)}{|\mathbf{r}_{mn}(t)|}$$
Why Cn to Cn+2?
This specific vector spans two bonds. It is short enough to sense local Crankshaft motions and dihedral flips, but long enough to represent the backbone direction. An "End-to-End" vector would result in extremely slow decay (global relaxation).
2. The Legendre Polynomials
The correlation function is based on the angle $\theta$ the vector rotates. We don't just use the angle; we use Legendre Polynomials ($P_L$) to capture specific symmetries.
P1(x) = x
Sensitive to direction. If vector flips 180°, P1 becomes -1. Used in Dielectric relaxation.
P2(x) = ½(3x² - 1)
The Standard. Symmetric. If vector flips 180°, P2 stays 1. Decays to 0 for random isotropy (avg x² = 1/3).
P3(x) = ½(5x³ - 3x)
More sensitive to large angle jumps. Used to detect non-diffusional motion.
3. Time Evolution C2(t)
In MD simulations, we average $P_2$ over many time origins. The result is a decay curve. The speed of decay tells us about the material's state (Melt vs Glassy).
System State
Computational Method
- Sliding Window: Start calculation at $t_0 = 0, 10, 20...$ ps.
- Ensemble Avg: Average over all polymer chains.
- Result: $C_2(t)$ starts at 1.0 and decays towards 0.0.
4. Interpretation: Ratios & Motion Types
By calculating relaxation times ($\tau$) for different orders ($P_1, P_2, P_3$), researchers can identify how the molecule moves: small steps or big jumps.