This work presents a mathematically rigorous, unified statistical-thermodynamic framework that connects the microscopic structural features of the hydrogen-bond network to the macroscopic anomalies of liquid water. By representing water as a two-state statistical mixture of Low-Density (LDS) and High-Density (HDS) structures, we incorporate the physical principle of shared-bond coordination (accounting for the fact that each hydrogen bond is shared by two neighboring molecules) to establish a direct link between the macroscopic transition enthalpy and the single-bond dissociation energy (ΔHₜrans = 0. 5 · Eₑff). Through a first-order perturbative expansion of the transcendental equation of state, we derive an explicit, closed-form analytical expression for the Temperature of Maximum Density (TMD), Tₘax (P), as a function of the characteristic coexistence temperature Tc (P): Tₘax (P) ≈ (3/2) ·Tc (P) - B·Tc (P) ² We rigorously demonstrate from classical thermodynamic stability that the pressure derivative dTₘax/dP is strictly negative in the liquid phase, a consequence of the lower volume (ΔVₜrans 0) of the close-packed state. The model qualitatively predicts the celebrated "retracing TMD line" (the "nose" of the TMD curve) under negative pressures. Furthermore, using Maxwell relations, we mathematically prove that the negative slope of the TMD line is inextricably linked to the anomalous temperature dependence of the isothermal compressibility ( (∂κT / ∂T) P < 0 at the TMD), showing that both anomalies are manifestations of the same underlying structural transition. Key Physical & Mathematical Highlights of the Model: Microscopic-Macroscopic Bridge: We resolve the parameter-space redundancy of two-state models by physically locking the transition enthalpy to the single hydrogen-bond breaking energy: ΔHₜrans (P) = 0. 5 · Eₑff (P). First Closed-Form TMD Equation: Instead of relying on multi-parameter transcendental computer fits, this model analytically collapses the equation of state specifically around the structural coexistence line (Tc), where the structural fluctuation susceptibility reaches its maximum. Parameter Dimensionality Reduction: The volume coefficient B = 2·R·β₀ / (ΔVₛ·ΔSₜrans) is explicitly constrained by water's experimental molar volume (V ≈ 18 cm³/mol) and background thermal expansivity (α ≈ 2×10⁻⁴ K⁻¹), yielding a baseline molar expansivity of β₀ ≈ V·α ≈ 3. 6×10⁻⁹ m³/ (mol·K). This eliminates arbitrary parameter-fitting. The Density-Compressibility Link: Using exact Maxwell relations along the TMD line (where thermal expansion α = 0), we mathematically prove that the negative slope of the TMD line (dTₘax/dP < 0) strictly dictates the anomalous temperature dependence of the isothermal compressibility: (∂κT / ∂T) P < 0.
Vakhtang Mchedlishvili (Mon,) studied this question.