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Principles of Quantum Chemistry

First Principle - Dual Nature of Electrons

Any matter can act as a wave, based on the equation:

\[ \lambda=\frac{h}{m\cdot v}=\frac{h}{p} \]
  • For a car with a massive mass, this is quite tiny - \(2\e{-28}\:nm\)
  • For an electron it is not - \(0.3\:nm\)

Second Principle - Uncertainty Principle

There is a natural floor to the precision we can get for the properties of a quantum particle, governed by the equations:

\[ \begin{gather} \Delta x\cdot \Delta p \geq\frac{h}{2\pi}\\ \Delta E \cdot \Delta t \geq\frac{h}{2\pi} \end{gather} \]

In quantum chemistry, we describe the wave as a wavefunction, with \(i\) bringing in the imaginary component and \(a_0\) giving the real component:

\[ \Psi(x,t)=a_0\cdot e^{2\pi i \big(\frac{x}{\lambda}-v\cdot t\big)} \]

To get any useful values out of this, we need to square the wavefunction (multiply by it’s complex conjugate, as \(\sqrt{-1}^2=-1\)):

\[ \begin{gather} |\Psi|^2=\text{real component}\\ |\Psi|^2=\psi^*\cdot\psi=\frac{1}{\sqrt{2\pi}}e^{\big(-\frac{x^2}{2}\big)} \end{gather} \]

This gives us a probability density for the wavefunction

Third Principle - Derivation of the Schrödinger Equation

When we substitute in the kinetic energy \(T\) and the potential energy \(V\) to the Schrödinger equation, with our wavefunction, we get the time dependent Schrödinger equation:

\[ \hat{H}\Psi(x,y,z,t)=i\hbar\frac{\partial}{\partial t}\Psi(x,y,z,t) \]

Fourth Principle - Time Dependence

\(\hat{H}\) is an operator (indicated by the hat). An operator is a function that transforms one function into another by a certain set of rules.

An operator acting on an eigenfunction (such as \(\Psi\)) returns an eigenvalue (scalar) multiplied by the original eigenfunction

\[ \begin{align} <operator><eigenfunction>&=<eigenvlue><eigenfunction>\\ \hat{H}\Psi&=E\Psi \end{align} \]

There are operators for every quantum property, such as dipole moment and polarisability, though not all experimental properties can be attributed to an operator and thus cannot be accurately measured with the Schrödinger equation.

When we remove the time dependence of the Schrödinger equation, we can get the time independent equation:

\[ \hat{H}\Psi(x,y,z)=E\Psi(x,y,z) \]

Fifth Principle - Average Values

The properties that are calculate from the Schrödinger equation will not be logical/useful unless we average them:

\[ \langle E\rangle=\frac{\int_{-\infin}^{\infin}\psi^*\hat{H}\psi dr}{\int_{-\infin}^{\infin}\psi^*\psi dr} \]

This separates the eigenvalue from the returned eigenfunction.