Molecules

Claimed by Nick Miravete

Molecules

Atoms are essential in the creation of large-scale objects, but along the process of creating something fully realized, molecules are the essential link. Were it not for the chemical bonding between atoms, molecules would not exist, and therefore the infinite amount of forms matter can take on (i.e. a human, an apple, or a mountain) would be impossible. The universe would be a bowl filled with single-atom soup. Molecules allow for diversification in matter and allow the world we live in today to be so unique. While it may be apparent to see the need for molecules, oftentimes the "how" they are created is oversimplified and explained as chemical bonding. The true nature of why these atoms bond to create molecules lies in quantum mechanics.

Molecular Bonding Through Quantum Mechanics

The key principle required to being able to understand why two atoms would choose to chemically bond is the idea that all natural systems (i.e. a collection of atoms) tend to adopt the state of lowest energy. As is the case with many other quantum mechanics problems, the easiest way to visualize molecular bonding is through the use of two hydrogen atoms.

The Hyrdogen Atom

The hydrogen atom is comprised of a single proton and an electron that orbits the proton with said electron creating an electron cloud that surrounds the proton. This electron cloud represents the probabilistic area in which an electron is most likely to be found. As two single hydrogen atoms are introduced to each other, the attractive electrostatic potential force between the opposing electrons and protons increases and therefore starts to pull the two atoms closer together. This will continue until the repulsive electrostatic force between the two protons begins to push the two back apart until an equilibrium state is reached. Once this equilibrium state is reached, the electrons of the two atoms will become "shared" and the gap between the two protons will become part of a combined electron cloud as now the two electrons move freely between the two single protons. However, there is a question that arises from this: What about the repulsion electrostatic force between the two initially separate electron clouds? In this question, all forms of energy must be considered to fully grasp why the two electron clouds do not continue to repel the two atoms.

Energy and the Hamiltonian

Reconsider the initial idea proposed that all natural systems prefer to be in the state of lowest energy. If such were to be the case, then all forms of energy must be considered for this system. The forms of energy found in this two hydrogen atom system are: the kinetic energy of each atom, the electric potential energy between the two protons, the electric potential energy between the two electrons, and the electric potential energy between each set of opposing electrons and protons. The sum of these four energies is what is called in quantum mechanics as the Hamiltonian. The Hamiltonian acts as the operator that corresponds to the energy of system when paired with the Schrodinger equation

[math]\displaystyle{ {\hat {H}} |\Psi \rangle =E|\Psi \rangle }[/math]

The equation for the Hamiltonian itself for the two hydrogen atom system is represented by

[math]\displaystyle{ H = \frac{-ℏ}{2m_{e}}(\nabla _{1}^{2} + \nabla _{2}^{2})+\frac{-ℏ}{2m_{p}}(\nabla _{A}^{2} + \nabla _{B}^{2}) - \frac{e^2}{4\Pi\varepsilon_{0}}(\frac{1}{r_{AB}}+\frac{1}{r_{12}}-\frac{1}{r_{1A}}-\frac{1}{r_{1B}}-\frac{1}{r_{2A}}-\frac{1}{r_{2B}}) }[/math]

In this equation, 1A, 1B, 2A, and 2B denotations represent the relationship between an electron and proton while the AB denotation represents the relationship between the two protons, and the 12 denotation represents the relationship between the two electrons. Using this equation allows us to create this graph that shows the total energy of the system of the two hydrogen atoms as a function of the distance between the two

A Computational Model

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