Exact ODE Framework for Classical and Quantum Corrections for the Lennard-Jones Second Virial Coefficient.

Area of Science:

• Thermodynamics
• Statistical Mechanics
• Mathematical Physics

Background:

• The second virial coefficient (SVC) is crucial for molecular theory.
• Traditional SVC calculations involve complex integrations of pair potentials.
• A need exists for more efficient and insightful calculation methods.

Purpose of the Study:

• To introduce a novel method for calculating the SVC of the Lennard-Jones fluid.
• To reformulate SVC calculation using ordinary differential equations (ODEs).
• To uncover new mathematical structures underlying thermodynamic properties.

Main Methods:

• Reformulation of SVC calculation in terms of ODEs.
• Generalization of confluent hypergeometric and Weber-Hermite equations for the classical SVC.
• Development of new ODEs and exact-analytical solutions for the first quantum correction.

Main Results:

• ODEs for SVC can be transformed into Schrödinger-like equations.
• The classical SVC term relates to a harmonic oscillator.
• The quantum correction involves inverse-power potential terms.
• SVC can be expressed as a linear combination of special functions (Kummer, Weber, Whittaker).

Conclusions:

• A new mathematical framework for SVC calculation has been established.
• This ODE-based approach reveals a deep connection between classical thermodynamics and quantum mechanics.
• The method offers a versatile way to express and compute the SVC.