An Investigation of The Kinetic Energy Operator of A Polyatomic Molecule with Geometric Algebra

Abstract

Physics concepts require mathematical frameworks to be understood and supported as an algebraic expression. Mathematicians and physicists have introduced and explored a variety of algebras throughout history. One of these is Clifford Algebra, often known as geometric algebra.

This work developed a general and useful method for deriving the operators of the kinetic energy of polyatomic molecules using Geometric Algebra. The kinetic energy operator of a polyatomic molecule contains the vibrational and rotational kinetic energy operators.

The gradients of vibrational coordinates form the exact vibrational kinetic energy operator of a polyatomic molecule. The conventional methods utilized for obtaining these gradients can often be extremely laborious. However, the gradients for any vibrational coordinate can be readily computed using geometric algebraic techniques. These gradients are the measuring vectors. so, the components of the reciprocal metric tensor  readily form that emerges in the exact internal kinetic energy operators of polyatomic molecules. On the other hand, Finding the measuring vectors for the rotational degrees of freedom is more difficult because the components of the total angular momentum operator are not conjugated to any rotational coordinates. Nonetheless, using geometric algebraic methods without any restrictions on the number of particles in the system, rotational measuring vectors for any geometrically defined body frame may be easily computed and this is what we show in this paper.