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A Primary Unified Geometric Framework of Molecular Reaction Dynamics Based on the Variational Principle

Xingyu Zhang, Jinke Yu, Qingyong Meng

Abstract

This work describes a geometric framework on molecular reaction dynamics based on the variational principle, where the Schr{ö}dinger equation must be solved to ``see'' how a reaction occurs. First, the mathematical preliminaries are given by discussing the principle of least action and the mountain pass theorem. Second, we discuss the physical preliminaries, including the principle of equivalence for deriving the kinetic energy operator (KEO) and artificial intelligence (AI) techniques to build the potential energy surface (PES) in general spacetime. Moreover, we simplified electromagnetic interactions in curved spacetime within the molecular system and consequently, we are able to construct the nuclear Hamiltonian in nonzero curvature spacetime. This indicates possibility to introduce gauge fields through the curvature, such as additional term in the nuclear KEO near a conical intersection. Third, the single-particle approximation provides a powful ansatz to solve the Schr{ö}dinger equation by variational principle. Thus, one can formulate the variational approaches for either electronic structure or quantum dynamics. In this work, based on previous discussions ({\it Phys. Chem. Chem. Phys.} {\bf 27} (2025), 20397) we unified them by a geometric description, where the geometric phase is naturally introduced. Finally, due to optimization characteristic of the present theory, further discussions on the present theory from optimization insight are also given, including two postulates, generative AI techniques, role of perturbation, and Markov process in optimization.

A Primary Unified Geometric Framework of Molecular Reaction Dynamics Based on the Variational Principle

Abstract

This work describes a geometric framework on molecular reaction dynamics based on the variational principle, where the Schr{ö}dinger equation must be solved to ``see'' how a reaction occurs. First, the mathematical preliminaries are given by discussing the principle of least action and the mountain pass theorem. Second, we discuss the physical preliminaries, including the principle of equivalence for deriving the kinetic energy operator (KEO) and artificial intelligence (AI) techniques to build the potential energy surface (PES) in general spacetime. Moreover, we simplified electromagnetic interactions in curved spacetime within the molecular system and consequently, we are able to construct the nuclear Hamiltonian in nonzero curvature spacetime. This indicates possibility to introduce gauge fields through the curvature, such as additional term in the nuclear KEO near a conical intersection. Third, the single-particle approximation provides a powful ansatz to solve the Schr{ö}dinger equation by variational principle. Thus, one can formulate the variational approaches for either electronic structure or quantum dynamics. In this work, based on previous discussions ({\it Phys. Chem. Chem. Phys.} {\bf 27} (2025), 20397) we unified them by a geometric description, where the geometric phase is naturally introduced. Finally, due to optimization characteristic of the present theory, further discussions on the present theory from optimization insight are also given, including two postulates, generative AI techniques, role of perturbation, and Markov process in optimization.
Paper Structure (24 sections, 3 theorems, 79 equations, 4 figures, 2 tables)

This paper contains 24 sections, 3 theorems, 79 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Principle of Least Action
  4. Mountain Pass Theorem
  5. Physical Preliminaries
  6. Principle of Equivalence
  7. Kinetics Energy Operator
  8. Potential Energy Surface
  9. Geometric Theory
  10. Electromagnetic Interaction
  11. Single-Particle Approximation
  12. Variational Framework
  13. Motion and Phase
  14. Optimization Insight
  15. Principle of Least Action in Geometry
  16. ...and 9 more sections

Key Result

Theorem 1

Let function $I$ satisfies (1) there exist $\exists a,r,R\in\mathbb{R}$ such that $0<r<R$ and $I(x)\geq a$ for $\forall x\in A=\{x\in X\vert r<\Vert x\Vert<R\}$, (2) there are $I(0)\leq a$ and $I(e)\leq a$ for some $e\in X$ with $\Vert e\Vert\geq R$, and (3) any sequence $\{x_n\in X\}_{n=1}^{\infty}

Figures (4)

  • Figure 1: Quan-Mol-Geom
  • Figure 2: Quan-Mol-Geom
  • Figure 3: Quan-Mol-Geom
  • Figure 4: Quan-Mol-Geom

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Theorem 1: mountain pass theorem
  • Corollary 1.1
  • Definition 3
  • Proposition 1: Weak principle of equivalence
  • Definition 4
  • Definition 5
  • Definition 6