# Hamiltonian Computational Chemistry: Geometrical Structures in Chemical Dynamics and Kinetics

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## Abstract

**:**

## 1. Introduction

## 2. Geometrical Structures in Chemical Dynamics

**Potential Energy Surfaces**, are employed to solve the nuclear equations of motion either in quantum mechanics or in classical mechanics. It is, thus, important to investigate common geometrical structures in the foundations of the two basic theories of physics, which in turn may assist in the numerical solutions of the corresponding equations of motion. In the following two subsections, we examine the topological and geometrical properties of classical and quantum mechanics, whereas in Section 3, the relatively new Hamiltonian formulation of thermodynamics is reviewed, all of them in extended phase spaces and at the non-relativistic approximation.

#### 2.1. Canonical Classical Mechanics

#### 2.1.1. Manifolds and Maps

**transpose**of a matrix. Thus, a row vector becomes a column vector, and vice versa, a column vector is converted into a row vector.), $q={({q}^{1},\dots ,{q}^{n})}^{T}$, and the parameter (time) with ${q}^{0}$. Capital letters designate the set of $n+1$ coordinates as a column vector, $Q={({q}^{0},q)}^{T}$. These coordinates parametrize the

**Extended Configuration Manifold**, ${Q}^{n+1}$. Generally, this is a smooth (differentiable)

**nonlinear manifold**.

**canonical projections**taken to be differentiable functions.

**Transition maps**provide the transformation from one coordinate system to another for points that belong to the intersection of two different open subsets.

**tangent space**of ${Q}^{n+1}$ at a point $s\in {Q}^{n+1}\left({T}_{s}{Q}^{n+1}\right)$ is a vector space, and the union of all tangent spaces for all points s of ${Q}^{n+1}$ form the

**tangent bundle**$\left(T{Q}^{n+1}\right)$ with the extended configuration space ${Q}^{n+1}$ to be the

**base space**

**fibers**, and it is a smooth manifold of dimension $2(n+1)$. Since $T{Q}^{n+1}$ is also a smooth manifold, a chart is defined by the diffeomorphism

**potential function**, $\mathcal{V}\left(Q\right)$, is a function on the configuration manifold to real numbers. The

**Lagrangian**, ${\mathcal{L}}_{e}(Q,\dot{Q})$, is a function on the tangent bundle to real numbers.

**dual space**of $T{Q}^{n+1}$ (the set of all linear maps on tangent bundle to real numbers) is the

**cotangent bundle**($M={T}^{*}{Q}^{n+1}$), also named

**phase space**. The phase space is a differentiable manifold of $2(n+1)\u2013$dimension for which the tangent bundle can also be defined with charts described by the generalized coordinates (${q}^{i}$), the conjugate momenta (Notice that we use superscripts for coordinates and subscripts for momenta.) $\left({p}_{j}\right)$, and their velocities, where

**Extended Hamiltonian**, ${\mathcal{H}}_{e}(Q,P)$, is a function on the phase space to real numbers, ${\mathcal{H}}_{e}:{T}^{*}{Q}^{n+1}\mapsto \mathbb{R}$, obtained by a

**Legendre transformation**(${F}_{{\mathcal{L}}_{e}}$) of the Lagrangian. We may consider that the Legendre transformation generates a differentiable map between the tangent and cotangent bundles of ${Q}^{n+1}$, ${F}_{{\mathcal{L}}_{e}}:T{Q}^{n+1}\to {T}^{*}{Q}^{n+1}$. ${\pi}_{{Q}^{n+1}}$ and ${\pi}_{{Q}^{n+1}}^{*}$ are

**canonical projections**to extended configuration manifold of tangent and cotangent bundles, respectively. The tangent bundle of phase space is denoted by $T{T}^{*}{Q}^{n+1}$ and ${\pi}_{{T}^{*}{Q}^{n+1}}$ is the canonical projection to phase space.

**kinetic metric tensor**, written as a function of coordinates q and the particle masses m. The metric is the

**non-degenerate, symmetric, covariant tensor rank-2**that defines the kinetic energy. The momentum ${p}_{i}$ is the

**covector**of the velocity ${\dot{q}}^{i}$,

**interior product (contraction)**of $1\u2013$forms with vector fields. Employing Dirac’s notation, we write

**physical states**are obtained by imposing the two constraints

#### 2.1.2. Equations of Motion

**Hamilton’s principle of stationary action**leads to Hamilton’s equations of motion. Then, Hamilton’s equations with a Hamiltonian ${\mathcal{H}}_{e}\left(x\right)$ are written in the form

**symplectic matrix**. J is the map on the tangent bundle of phase space M

**Hamiltonian vector field**is

**pull-back**${\theta}_{e}$ to ${Q}^{k}$ to produce the $1\u2013$form ${\alpha}^{*}{\theta}_{e}$, which lives on the base manifold ${Q}^{k}$. Then, the

**canonical Poincaré $1\u2013$form**satisfies the relation (

**tautological $1\u2013$form**)

**canonical Symplectic $2\u2013$form**is extracted by taking the

**exterior derivative**(Notice the negative sign in our formulation) of ${\theta}_{e}$

**non-degenerate, skew-symmetric, closed $2\u2013$form**($d{\omega}_{e}=-d\circ d{\theta}_{e}=0$). In local coordinates $(q,p)$, ${\omega}_{e}$ is expressed by the

**wedge products**

**symplectic manifold**. Those local coordinates which satisfy, ${\omega}_{e}={\sum}_{i=1}^{k}d{x}^{i}\wedge d{x}^{k+i}$, are said to be

**canonical**and

**symplectic**. In the following, we shall see that Hamiltonian mechanics and its geometrical properties can be formulated by ${\omega}_{e}$.

**interior product (contraction)**and the triple $(M,{\omega}_{e},{X}_{{\mathcal{H}}_{e}})$ is a

**Hamiltonian system**. In particular, for the variable ${q}_{0}$ we extract the equation

**Poisson bracket**is defined as

**Lie derivative**of a dynamical quantity $\mathcal{O}\left(x\right(t\left)\right)$ with respect to the Hamiltonian vector field ${X}_{{\mathcal{H}}_{e}}$ is defined as the directional derivative of $\mathcal{O}$ along the vector ${X}_{{\mathcal{H}}_{e}}$

- $\{f,g\}$ is bilinear,
- $\{f,g\}=-\{g,f\}$ antisymmetric,
- $\{f,f\}=0$, and
- $\{f,\{g,h\left\}\right\}+\{h,\{f,g\left\}\right\}+\{g,\{h,f\left\}\right\}=0$ (Jacobi identity).

#### 2.1.3. Integrable Hamiltonian Systems

**completely integrable**if it admits n independent constants of motion whose Poisson brackets are in involution, i.e., pairwise commute.

**Lagrangian submanifold**.

**compact phase spaces**, the Lagrangian submanifolds have the structure of a $n\u2013$torus, ${\mathbb{T}}^{n}$. Moreover, in a neighborhood of every such invariant torus, one can find

**angle-action coordinates**$(\varphi ,I)$,

#### 2.1.4. Complexification of Classical Hamilton’s Equations

**complexification**of phase space, i.e., by introducing the complex transformation

**realification**of the quantum Hilbert space, we bring the Schrödinger equation into the form of Hamilton’s equations.

**complex variables**

#### 2.1.5. Jacobi Fields and Variational Equations

**length functional**, i.e., by requiring

**action**or the

**energy**of a physical system. Indeed, Hamilton’s principle of stationary action results in Hamilton’s equations, the solutions of which are geodesics on the phase space manifold.

**length of a trajectory**starting from the point $t=0$ and finishing at the point $t={t}_{max}$ is calculated as the line integral

**variation vector**${Y}_{x}\left(t\right)=\delta x\left(t\right)$, (Figure 2). The time derivative of the vector field ${Y}_{x}\left(t\right)$ with respect to the vector field ${X}_{x}$ (Lie derivative) is equal to

**Jacobi field**and the equations

**variational equations.**

**fundamental matrix**$Z(t,{t}_{0})$ satisfies the variational equations with initial condition $Z({t}_{0},{t}_{0})=I$. It is a symplectic matrix, and therefore, the following equations are valid [3]

#### 2.2. Geometrical Quantum Mechanics

#### 2.2.1. Manifolds and Maps

**Hamiltonian operator**of the system that corresponds to its energy, and ${X}_{\widehat{\mathcal{H}}}=-\frac{\U0001d6a4}{\hslash}\widehat{\mathcal{H}}$ is the

**Hamiltonian–Schrödinger vector field**. To any observable $\mathcal{O}$, we assign the operator $\widehat{\mathcal{O}}$ and the

**Schrödinger vector field**${X}_{\widehat{\mathcal{O}}}=-\frac{\U0001d6a4}{\hslash}\widehat{\mathcal{O}}$. Hence, we may consider the vectors $|\psi >$ as

**vector fields**, and thus, the Hilbert space ${\mathcal{H}}^{\mathrm{n}}$ to be also the tangent space ${T}_{|\psi >}{\mathcal{H}}^{\mathrm{n}}$ at the state $|\psi >\in {\mathcal{H}}^{\mathrm{n}}$.

**Hermitian inner product**, ($<\varphi |\psi >=<\psi |\varphi {>}^{*}$).

**self-adjoint linear operators**, $\widehat{\mathcal{O}}={\widehat{\mathcal{O}}}^{\u2020}$, on ${\mathcal{H}}^{n}$, and are thus vector fields. The

**expectation value**of an observable with operator $\widehat{\mathcal{O}}$ at the state $|\psi >$ is the

**real-valued function**

#### 2.2.2. Projective Hilbert Space

**extended Hilbert space**of $(n+1)\u2013$dimension, ${\mathcal{H}}^{n+1}$, which is isomorphic to ${\mathbb{C}}^{n+1}$ with a Hermitian inner product.

**ray**, which we symbolize as $\left\{\right|\psi >\}\equiv \{\lambda |\psi >\}$. A ray is an

**equivalence class of vectors**in ${\mathcal{H}}^{n+1}$: two vectors are equivalent if and only if one is a nonzero complex scalar multiple of the other. Also, adopting normalized vectors (Equation (70)), the

**physical quantum states**are elements of the complex

**Projective Hilbert space**, ${\mathbb{P}}^{n}\left({\mathcal{H}}^{n+1}\right)$ of $n\u2013$dimension obtained by the canonical projection map ${\pi}_{|{\psi}_{\mathbb{P}}>}$ of the extended Hilbert space ${\mathcal{H}}^{n+1}$.

**principal bundle**

**tangent space of the projective space**at point $|{\psi}_{\mathbb{P}}>$, ${T}_{|{\psi}_{\mathbb{P}}>}\left({\mathbb{P}}^{n}\left({\mathcal{H}}^{n+1}\right)\right)$, is isomorphic to the kernel of the ray, ${\left\{\right|\psi >\}}^{\perp}$, i.e.,

**complex conjugate**linear isomorphism onto ${T}_{|{\psi}_{\mathbb{P}}>}\left({\mathbb{P}}^{n}\left({\mathcal{H}}^{n+1}\right)\right)$

**Fubini–Study metric**. ℜ and ℑ are the real and imaginary parts of the Hermitian inner product, respectively, in the extended complex Hilbert space ${\mathcal{H}}^{n+1}$. Both ${\omega}_{|{\psi}_{\mathbb{P}}>}$ and ${g}_{|{\psi}_{\mathbb{P}}>}$ are invariant under all transformations $U\left(1\right)$, for all unitary operators $\widehat{\mathcal{U}}$ on ${\mathcal{H}}^{n+1}$.

#### 2.2.3. Realification of Hilbert Space and Kähler Manifolds

**almost complex structure**is introduced in ${\mathcal{H}}_{r}^{2n}$ by replacing the imaginary number $\U0001d6a4=\sqrt{-1}$ with the symplectic matrix $-J$, Equation (15). We also derive

**Kähler structure**, and thus, ${\mathcal{H}}_{r}^{2n}$ is a

**Kähler manifold**[2].

**Expectation value**of $\widehat{\mathcal{H}}$ at the state $|\psi >$

**dispersion**) of a quantum observable $\widehat{O}$ in a normalized state $|\psi >\in {\mathcal{H}}^{n}$ is defined as [8,31]

**covariance or correlation function**is expressed as

**Schrödinger vector fields**are expressed as usually

**Schwartz inequality**implies

**Robertson-Schrödinger uncertainty relation**is written as (Notice that to be consistent with the literature in discussing the covariance of two observables, we introduce the factor $1/2$ in the commutator and anticommutator).

#### 2.2.4. Equations of Motion

**wavefunction**and it is a complex function, $\psi \left(s\right)\phantom{\rule{0.277778em}{0ex}}\in {\mathbb{C}}^{n}$, which we assume normalized to one

**completeness relation**is written as

**Schrödinger picture**, $\psi (s,t)$. We expand $\left|\psi \right(s,t)>$ of a dynamical system in an arbitrary orthonormal basis set, $|{\chi}_{k}\left(s\right)>,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}k=1,\dots ,n$

**complex variables**(${Q}_{k},{P}_{k}$) by introducing the real functions ${q}^{k}\left(t\right)$ and ${p}_{k}\left(t\right)$ to correspond to real and imaginary parts of the complex variables, respectively,

**transition frequencies**. It is also worth noting that the quantum Hamiltonian is similar to that of a harmonic oscillator after complexification (Equation (56)).

**canonical Poincaré $1\u2013$form**

#### 2.2.5. Quantum Systems as Totally Integrable Hamiltonian Systems

## 3. Hamiltonian Chemical Thermodynamics

#### 3.1. Manifolds and Maps

**extensive**physical properties of entropy (S), internal energy (U), volume (V), and the number of molecules (or moles) (${N}^{1},\dots ,{N}^{M}$) of M chemical species to consist of the coordinates of the extended configuration manifold ${Q}^{n+1}$, where $n=2+M$. If we attribute entropy to be a homogeneous first-degree function of $q={({q}^{1},{q}^{2},\dots ,{q}^{n})}^{T}\equiv {(U,V,{N}^{1},\dots ,{N}^{M})}^{T}\in {Q}^{n+1}$, then according to Euler’s theorem for homogeneous functions we can write

**conjugate intensive**variables $\left(\right)$ as the partial derivatives of entropy

**contact manifold**$\left({C}^{2n+1}\right)$, where the physical states of the system live. The extended configuration manifold and the contact manifold are generally nonlinear, and the maps $({\varphi}_{Q},{U}_{Q})$ and $({\varphi}_{C},{U}_{C})$ determine local coordinate systems in Euclidean space.

**physical thermodynamic submanifold**(PTS) of a thermodynamic system

**Legendrian submanifold**, and it is what we represent with the equation of states [32].

**kernel**of ${\theta}_{c}$ provides the maximal dimension hyperplanes tangent to ${L}_{c}^{n}$.

**Thermodynamic Extended Physical State Submanifold (TEPSS)**in the

**extended phase space**, ${T}^{*}{Q}^{n+1}\equiv {P}^{2n+2}$, named

**Lagrangian submanifold**.

**projective space**of thermodynamic extended phase space ${P}^{2n+2}$ (see Appendix A.4.1)

**homogeneous in momenta Lagrangian submanifold**(see Table A1). This means that $(d{\theta}_{e}=0)$ as well as

**ray**$\{Q,P\}$. Then, it is valid ${\theta}_{e}=0$ for every vector field tangent to ${L}_{p}^{n+1}$. Also, every

**homogeneous Lagrangian submanifold**originates from a Legendrian submanifold of the form ${\pi}^{-1}\left({L}_{c}^{n}\right)$. We can also state that a submanifold is homogeneous if its generating function is homogeneous.

#### 3.2. Equations of Motion

**Poisson brackets Equation (34)**

#### 3.2.1. Contact Equations of Motion

#### 3.2.2. Riemannian Metric on Lagrangian Submanifold

#### 3.3. Embedding Systems in Homogeneous Media

#### 3.4. Chemical Kinetics

#### 3.4.1. Thermodynamics of Chemical Reactions

**stoichiometric matrix**for the K elementary reactions.

**molar reaction Gibbs free energy**.

**Affinity**

#### 3.4.2. Thermodynamic Hamiltonian in Massieu-Gibbs Representation

## 4. Numerical Implementations

#### 4.1. High-Order Finite-Difference and Pseudospectral Methods

**Lagrange interpolating polynomials**[46,47]. Given that variable order finite-difference algorithms are among the most suitable for modern high-performance computing, the computer technology to which computational chemistry is mainly addressed, as well as their programming simplicity, finite-difference methods emerge as one of the best choices for studying chemical dynamics.

**cardinal set of basis functions**, $\left\{{u}_{j}\left(x\right)\right\}$, also called discrete variable representation, by choosing N grid points, $\left\{{x}_{i}\right\}$, at which the function is calculated. The cardinal functions obey the $\delta \u2013$Kronecker property

**differentiation matrix**${D}^{m}$ contains the coefficients necessary for calculating the mth- derivative at the collocation points, and T is the column vector of dimension N containing the basis functions.

#### 4.2. The Hénon–Heiles Model

#### 4.2.1. A Classical Time-Dependent Hénon–Heiles System

#### 4.2.2. The Quantum Hénon–Heiles System

#### 4.2.3. Energy Dissipation of Hénon–Heiles System in Homogeneous Media

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

FD | Finite Difference |

PS | Pseudospectral |

DOF | degrees of freedom |

TEPS | Thermodynamic Extended Phase Space |

TEPSS | Thermodynamic Extended Physical State Submanifold |

TCS | Thermodynamic Contact Space |

PTS | Physical Thermodynamic Submanifold |

ODEs | Ordinary Differential Equations |

HNN | Hamiltonian neural networks |

PNN | Physics neural networks |

nD | $n\u2013$dimensional |

## Appendix A

#### Appendix A.1. Proof of Equation (89)

#### Appendix A.2. Proof of Equation (90)

#### Appendix A.3. Proof of Equation (152)

**thermodynamic contact vector field**is

#### Appendix A.4. Tables

#### Appendix A.4.1. Projection Maps between Thermodynamic Extended Phase Space and Thermodynamic Contact Space

**Table A1.**Projection maps $\left(\pi \right)$ between thermodynamic extended phase space (TEPS) and thermodynamic contact space (TCS). ${\pi}^{-1}$ denotes the inverse map, ${\pi}_{*}$ the push-forward of vector fields operation and ${\pi}^{*}$ the pull-back of functions ($0\u2013$forms). ${P}^{2n+2}$ is the extended phase space, ${C}^{2n+1}$ the contact state space, ${L}_{p}^{n+1}$ the Lagrangian submanifold in TEPS, which describes the thermodynamic extended physical state submanifold (TEPSS), ${L}_{c}^{n}$ the Legendrian submanifold in TCS, which describes the physical thermodynamic submanifold (PTS), ${H}_{e}$ the extended Hamiltonian, ${X}_{{H}_{e}}$ the extended Hamiltonian vector field, ${H}_{c}$ the contact Hamiltonian and ${X}_{{H}_{c}}$ the contact Hamiltonian vector field.

#### Appendix A.4.2. Thermodynamic Manifolds in Entropy Representation

Manifold | ${P}^{2n}\equiv {T}^{*}{Q}^{n}$ | ${C}^{2n+1}\equiv P\left({T}^{*}{Q}^{n+1}\right)$ | ${P}^{2n+2}\equiv {T}^{*}{Q}^{n+1}$ |

Coordinates | |||

$n=2+r$ | ${q}^{1}=U,{q}^{2}=V,$ | ${q}^{0}=S,{q}^{1}=U,$ | ${q}^{0}=S,{q}^{1}=U,$ |

${q}^{k}={N}^{k-2}$ | ${q}^{2}=V,{q}^{k}={N}^{k-2}$ | ${q}^{2}=V,{q}^{k}={N}^{k-2}$ | |

Momenta | |||

${p}_{0}=-1$ | ${\gamma}_{1}=\frac{1}{T}$ | ${\gamma}_{1}=\frac{1}{T}$ | ${p}_{1}=-{p}_{0}{\gamma}_{1}$ |

${\gamma}_{2}=\frac{P}{T}$ | ${\gamma}_{2}=\frac{P}{T}$ | ${p}_{2}=-{p}_{0}{\gamma}_{2}$ | |

${\gamma}_{k}=-\frac{{\mu}_{k-2}}{T}$ | ${\gamma}_{k}=-\frac{{\mu}_{k-2}}{T}$ | ${p}_{k}=-{p}_{0}{\gamma}_{k}$ | |

$1\u2013$form | |||

$\theta ={\sum}_{i=1}^{n}{\gamma}_{i}d{q}^{i}$ | ${\theta}_{c}=d{q}^{0}-{\sum}_{i=1}^{n}{\gamma}_{i}d{q}^{i}$ | ${\theta}_{e}={\sum}_{i=0}^{n}{p}_{i}d{q}^{i}$ | |

$2\u2013$form | |||

$\omega =-d\theta $ | $\omega ={\sum}_{i=1}^{n}d{q}^{i}\wedge d{\gamma}_{i}$ | ${\omega}_{e}={\sum}_{i=0}^{n}d{q}^{i}\wedge d{p}^{i}$ | |

PTS | ${L}_{p}^{n}$ | ${L}_{c}^{n}$ | ${L}_{p}^{n+1}$ (TEPSS) |

$\omega =0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathcal{H}=S$ | ${\theta}_{c}=0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}F=S$ | ${\theta}_{e}=0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\omega}_{e}=0$ | |

${\mathcal{H}}_{e}=0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\mathcal{L}}_{{X}_{{\mathcal{H}}_{e}}}{\theta}_{e}=0$ | |||

Metric | $d{l}^{2}=-{\sum}_{i=1}^{n}d{q}^{i}\otimes d{\gamma}_{i}$ | $d{l}^{2}=-{\sum}_{i=1}^{n}d{q}^{i}\otimes d{\gamma}_{i}$ | $d{l}^{2}=-{\sum}_{i=1}^{n}d{q}^{i}\otimes d{\gamma}_{i}$ |

#### Appendix A.4.3. Thermodynamic Manifolds in Energy Representation

Manifold | ${P}^{2n}\equiv {T}^{*}{Q}^{n}$ | ${C}^{2n+1}\equiv P\left({T}^{*}{Q}^{n+1}\right)$ | ${P}^{2n+2}\equiv {T}^{*}{Q}^{n+1}$ |

Coordinates | |||

$n=2+r$ | ${q}^{0}=S,{q}^{2}=V$ | ${q}^{0}=S,{q}^{1}=U$ | ${q}^{0}=S,{q}^{1}=U$ |

${q}^{k}={N}^{k-2}$ | ${q}^{2}=V,{q}^{k}={N}^{k-2}$ | ${q}^{2}=V,{q}^{k}={N}^{k-2}$ | |

Momenta | |||

${p}_{1}=-1$ | ${\beta}_{0}=T$ | ${\beta}_{0}=T$ | ${p}_{0}=-{p}_{1}{\beta}_{0}$ |

${\beta}_{2}=-P$ | ${\beta}_{2}=-P$ | ${p}_{2}=-{p}_{1}{\beta}_{2}$ | |

${\beta}_{k}={\mu}_{k-2}$ | ${\beta}_{k}={\mu}_{k-2}$ | ${p}_{k}=-{p}_{1}{\beta}_{k}$ | |

$1\u2013$form | |||

$\theta ={\beta}_{0}d{q}^{0}+{\sum}_{i=2}^{n}{\beta}_{i}d{q}^{i}$ | ${\theta}_{c}=d{q}^{1}-{\beta}_{0}d{q}^{0}-$ | ${\theta}_{e}={\sum}_{i=0}^{n}{p}_{i}d{q}^{i}$ | |

${\sum}_{i=2}^{n}{\beta}_{i}d{q}^{i}$ | |||

$2\u2013$form | |||

$\omega =-d\theta $ | $\omega =d{q}^{0}\wedge d{\beta}_{0}+{\sum}_{i=2}^{n}d{q}^{i}\wedge d{\beta}_{i}$ | ${\omega}_{e}={\sum}_{i=0}^{n}d{q}^{i}\wedge d{p}^{i}$ | |

PTS | ${L}_{p}^{n}$ | ${L}_{c}^{n}$ | ${L}_{p}^{n+1}$ (TEPSS) |

$\omega =0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathcal{H}=U$ | ${\theta}_{c}=0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}F=U$ | ${\theta}_{e}=0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\omega}_{e}=0$ | |

${\mathcal{H}}_{e}=0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\mathcal{L}}_{{X}_{{H}_{e}}}{\theta}_{e}=0$ | |||

Metric | $d{l}^{2}=d{q}^{0}\otimes d{\beta}_{0}+$ | $d{l}^{2}=d{q}^{0}\otimes d{\beta}_{0}+$ | $d{l}^{2}=d{q}^{0}\otimes d{\beta}_{0}+$ |

${\sum}_{i=2}^{n}d{q}^{i}\otimes d{\beta}_{i}$ | ${\sum}_{i=2}^{n}d{q}^{i}\otimes d{\beta}_{i}$ | ${\sum}_{i=2}^{n}d{q}^{i}\otimes d{\beta}_{i}$ |

#### Appendix A.5. Hamiltonian Chemical Kinetics: A Simple Chemical Kinetic Example: Consecutive First-Order Elementary Reactions

#### Appendix A.5.1. Discussion

**Figure A1.**(

**a**) Concentrations of the constituent chemical species as functions of time, (

**b**) reaction metric, (

**c**) reaction coordinates $({\xi}^{1},{\xi}^{2})$, (

**d**) conjugate momenta $({p}_{{\xi}^{1}},{p}_{{\xi}^{2}})$ to reaction coordinates. The quantities have been calculated from a trajectory run with initial concentrations, $\left[{\mathrm{Q}}_{1}\right]=1.5,$ $\left[{\mathrm{Q}}_{2}\right]=0.03,$ $\left[\mathrm{P}\right]=0.001$. The rate constants for the forward reactions are taken equal to ${k}_{f1}=1.5,$ ${k}_{f2}=1.0$, and the backward reactions ${k}_{b1}=0.5,{k}_{b2}=0.25$.

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**Figure 1.**Manifolds and functions which determine the geometrical structures of a classical system with $n+1$ coordinates. ${q}^{0}$ denotes a parameter and, specifically, the time for time-dependent systems. $q=({q}^{1},\dots ,{q}^{n})$ are the n coordinates that define the configurations of the system and $p=({p}_{1},\dots ,{p}_{n})$ their canonical conjugate momenta. Details are given in the text.

**Figure 2.**The description of the

**variation vector field**, ${Y}_{x}$, with respect to a reference trajectory with vector field ${X}_{x}$ and initial condition ${x}_{0}$. ${\Phi}_{{x}_{0}}\left(t\right)$ denotes the Hamiltonian flow.

**Figure 3.**Manifolds and functions which determine the geometrical structures of a quantum system. The states of the quantum system are the elements of a $n\u2013$dimensional complex vector space (Hilbert space ${\mathcal{H}}^{n}\cong {\mathbb{C}}^{\mathrm{n}})$ that includes the vectors $|\psi >$, their complex conjugate $<\psi |$, and linear transformations $|\dot{\psi}>$. Hermitian inner products, $<\varphi |\psi >$, are employed for Hilbert spaces. The tangent bundle $({\mathcal{H}}^{n}\times {\mathcal{H}}^{n})$ is mapped to the Extended Hilbert Space $({\mathcal{H}}^{n+1}\times {\mathcal{H}}^{n+1})$ by the inclusion of a complex phase $\lambda $, the elements of the unitary group $U\left(1\right)$, that produces the rays $\left\{\right|\psi >\}:=\{\lambda |\psi >\}$. The canonical projection, ${\pi}_{|{\psi}_{\mathbb{P}}>}$, projects the rays in ${\mathcal{H}}^{n+1}$ onto the Projective Hilbert Space ${\mathbb{P}}^{n}\left({\mathcal{H}}^{n+1}\right)$, the space where the physical states of the system live. Details are given in the text.

**Figure 4.**Manifolds and functions which determine the geometrical structures of a thermodynamical system with $n+1$ coordinates. S denotes the entropy and $q={({q}^{1},\dots ,{q}^{n})}^{T}$ are the coordinates of $n\u2013$extensive properties. $\gamma $ are the partial derivatives of entropy that correspond to the intensive properties of the system. With the inclusion of gauge ${P}_{S}$, the conjugate momenta p are defined with respect to which homogeneous Hamiltonians, ${\mathcal{H}}_{e}$, of first-degree are determined. Details are given in the text.

**Figure 5.**Potential energy surface of the Hénon–Heiles model and isocontours in the configuration plane.

**Figure 6.**(

**a**) A dissociating trajectory of a time-dependent Hénon–Heiles potential. (

**b**) The trajectory is initialized from a periodic orbit (red thick line) and with energy ${\mathcal{H}}_{d}=0.1575$.

**Figure 7.**(

**a**) The initial wavepacket centered at the minimum of the potential well. (

**b**) The evolved wavepacket after 2000 time units.

**Figure 8.**(

**a**) The autocorrelation function for the initial Gaussian wavepacket. (

**b**) The power spectrum was obtained by taking the Fourier transformation of the autocorrelation function.

**Figure 9.**Projections of two representative trajectories in the Hénon–Heiles phase space are shown with friction parameters ${b}_{11}=0.3\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}{b}_{22}=0.001$; (

**a**) in the $({\sigma}^{1},{\sigma}^{2},{\pi}_{1})$ space and (

**b**) in the $({\sigma}^{1},{\sigma}^{2},{\pi}_{2})$ space.

**Figure 10.**Panel (

**a**) is the time evolution of the entropy production and (

**b**) the trajectory length calculated with the Ruppeiner metric for the two trajectories shown in Figure 9.

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Farantos, S.C.
Hamiltonian Computational Chemistry: Geometrical Structures in Chemical Dynamics and Kinetics. *Entropy* **2024**, *26*, 399.
https://doi.org/10.3390/e26050399

**AMA Style**

Farantos SC.
Hamiltonian Computational Chemistry: Geometrical Structures in Chemical Dynamics and Kinetics. *Entropy*. 2024; 26(5):399.
https://doi.org/10.3390/e26050399

**Chicago/Turabian Style**

Farantos, Stavros C.
2024. "Hamiltonian Computational Chemistry: Geometrical Structures in Chemical Dynamics and Kinetics" *Entropy* 26, no. 5: 399.
https://doi.org/10.3390/e26050399