Vibrational theory for monatomic liquids

The construction of the vibration-transit theory of liquid dynamics is being presented in three sequential research reports. The first is on the entire condensed-matter collection of $N$-atom potential energy valleys and identification of the random valleys as the liquid domain. The present (second) report defines the vibrational Hamiltonian and describes its application to statistical mechanics. The following is a brief list of the major topics treated here. The vibrational Hamiltonian is universal, in that its potential energy is a single $3N$-dimensional harmonic valley. The anharmonic contribution is also treated. The Hamiltonian is calibrated from first-principles calculations of the structural potential and the vibrational frequencies and eigenvectors. Exact quantum-statistical-mechanical functions are expressed in universal equations and are evaluated exactly from vibrational data. Exact classical-statistical-mechanical functions are also expressed in universal equations and are evaluated exactly from a few moments of the vibrational frequency distribution. The complete condensed-matter distributions of these moments are graphically displayed, and their use in statistical mechanics is clarified. The third report will present transit theory, which treats the motion of atoms between the $N$-atom potential energy valleys.

Figure 1

The straight line is the vibrational potential energy ${\mathrm{\Phi }}_{\mathrm{vib}}=\frac{3}{2}{k}_{B}T$, and dots fitted by the dashed line are total potential energy from MD ${\mathrm{\Phi }}_{\mathrm{MD}}$. ${\mathrm{\Phi }}_{tr}$ is ${\mathrm{\Phi }}_{\mathrm{MD}}-{\mathrm{\Phi }}_{\mathrm{vib}}$. The volume of the system is constant at $V_{lm}=278a_0^3$, the volume of the liquid at melt. $T_m$ is the temperature of the liquid at melt.

Figure 2

Histogram of $g\left(\omega \right)$ from the set of vibrational frequencies ${\omega }_{\lambda }$, calculated from the dynamical matrix. Calculations are based on the Na pair potential $\varphi (r;V)$.

Figure 3

Histogram of $g\left(\omega \right)$ made as in Fig. 2 from calculations based on first-principles DFT.

Figure 4

Distribution of ${\theta }_2$ from 1000 quenches to structures, graphed against the structure potential ${\mathrm{\Phi }}_0$. Solid dots are the random distribution, and the revealed cross indicates the liquid structure. Open circles are the symmetric distribution. The bcc crystal ${\mathrm{\Phi }}_0$ is set to zero for graphical clarity.

Figure 5

Distribution of ${\theta }_1$ (details are as in Fig. 4).

Figure 6

Distribution of ${\theta }_0$ (details are as in Fig. 4).