Mathematical Base


Symmetry

Invariance

Symmetry is invariance after an operation. This fundamental idea is a continual driving force in my research; if collision is the operation, then total energy is the invariant. Also, a governing rule like this allows a reduction of complexity around certain collision geometric configurations. The symmetry of naturally occuring atomic and molecular processes observed in experiment provide solid ground for a theoretical framework.

Transformations

Perhaps the most general and abstract way of thinking about symmetry is the use of operator algebra. Consider a general transformation operator with the ability to reflect, rotate, translate, dilate and invert. Is there an operator that can perform any of these operations with one standard form? What is the space it acts on?  I will consider these questions in detail with abstract algebra.

Algebra

Definition

together with three kinds of operations

Geometric Algebra

Geometric algebra comes in different flavors, here are a few examples.

Algebra

g3

g31

g41

Example

{1,x,y,z,xy,yz,zx,xyz}

16 elements!

{o,x,y,z,i}

Group

SO(3) or SU(2)

SO(3,1) or Affine

CO

A discussion of vectors as instances of an algebra class leads to how easy it is for rotors to transform, dilate, rotate or reflect these vector in their spaces with object oriented programming. Consider a new kind of product for two vectors, a combination of the dot and the wedge. Using this goemetric product cleans up many derivations and representations in physics. Suppose we wanted to rotate a vector t into a vector g. Given two vectors r and o which represent normals of two reflection planes we could simply reflect twice with the operation.

g = -rotor

Or said another way, a rotation of t about an axis perpindicular to the plane formed by r and o is the above.  The result is a rotation around an axis orthogonal to both vectors o and r. Here the geometric product of the o and r produces a bivector called a rotor instance.  Therefore, the operator and what it acts on are in the same algebra!

Lie Algebra

A Lie algebra is a set of elements closed under commutation. For example, the set of operators {x,p,1} over a complex field are closed under commutation. Take any two and see that they either commute to zero [x,1]=0 or to an element of the algebra, i.e. [x,p]=i.

Lie algebra comes in different flavors, here are a few examples.

Algebra

u(2)

su(2)

so(3)

Example

{1,a,a*,N}

{1,a,a*}

{Lx,Ly,Lz}

Group

U(2)

SU(2)

SO(3)

Lie algebras make quantum mechanics easier to calculate in that you work with no wave equations or differential operators. The Hamiltonian is simply formed out of Lie algebra elements, such as the ladder operator version of the simple harmonic oscillator. Abstract algebras have nontrivial applications in many areas. One area that seems to be a research frontier is molecular dynamics. The vibron model assigns a U(2) algebra for each normal or local energy mode of a molecule's internal dynamics.

Intramolecular Dynamics

Two Formalisms

Analytic motion of an object is best generalized in one of two ways.

Lagrange Hamilton
L = T - V.\quad H
\frac{d~}{dt} \ \left( \, \frac{\partial L}{\partial \dot{x}_i} \, \right) \ - \ \frac{\partial L}{\partial x_i} \ = \ 0 \dot p = -\frac{\partial \mathcal{H}}{\partial q} \dot q =~~\frac{\partial \mathcal{H}}{\partial p}.

Given explicit forms of the kinetic and potential energies in a coordinate system these result in differential equations. The solutions to these equations reveal the analytic motion between initial and final conditions. Notice that the Lagrangian formalism gives position and velocity while the Hamiltonian formalism gives position and momentum. One possible familiar result is Newton's equation F=ma. For many bodies, the math bulks up beyond control. To this end molecular dynamics has become a field where approximation methods flourish.

Algebraic methods are used to construct the Hamiltonian such that one may avoid differential operators and/or wave functions. the familiar ladder operator form of the harmonic oscillator Hamiltonian is an example of this using a Lie algebra. When approaching molecular dynamics with this method the calculations becomes cleaner. A more common procedure, perturbation theory, results in an infinite amount of differential equations while the algebraic approach sustains analyticity with a finite amount equations. They are however are coupled and nonlinear.

To calculate dynamics of many atom systems, approximations are needed. A particular way is the Born-Oppenheimer Approximation. For processes with no electron transfer this method is analogous to the adiabatic theorem in that the nuclei move much slower than the electrons.

Born-Oppenheimer Approximation

A 5 step procedure

  1. Assume that the nuclei are clamped in fixed positions, therefore eliminating the nuclear kinetic energy term
  2. Solve the electronic structure problem and obtain the electronic wave functions for the given nuclear configurations
  3. Assume that the total wave function for a given electronic state can be written as a product of an electronic and nuclear wave function
  4. Substitute the product wave function into the time independant Schrodinger equation
  5. Neglect the effect of the nuclear kinetic energy operator on the electronic wave function

The mathematics follow these steps. Consider a Hamiltonian of 5 general terms. The first is the sum of all the nuclear kinetic energies. The second is the sum of all electronic kinetic energies. The third is the potential energy between the elect ,rons and the nuclei. The fourth term is potential energy between nuclei. The last one is potential energy between electrons.

hboc

Applying the first procedure from above we eliminate the nuclear kinetic energy term assuming it is negligible because of the relatively large masses and the dynamics being adiabatic, meaning the nuclei move slow compared to the electronic motion.

he

We then use procedure two and solve the electronic structure problem

defa

and obtain the electronic wave functions for the given nuclear configurations. Then assume that the total wave function for a given electronic state can be written as a product of an electronic and nuclear wave function

hse

Substitute the product wave function into the time independant Schrodinger equation. Neglect the effect of the nuclear kinetic energy operator on the electronic wave function. Multiply from the left with the electronic solution conjugated, then integrate over the electronic coordinates.

htis

The energy eigenvalue of this equation is a function of the "parametrized" nuclear coordinates. This is considered the potential energy surface for the nuclei.. This surface is available from electronic structure calculations or obtained by some semiempirical or emperical method. With this surface, reactions between different molecular species can be visuallised intuitively. An important detail of these reactions is seen when different types of energy are transferred/exchanged during the scattering process. To see more go to Collisions.


Tim Wendler timoth500@yahoo.com

Manuel Berrondo   Jean-Francois Van Huele   J. Ward Moody   Scott Bergesen  Gus Hart