quantumbiochemistry.org

Quantumbiochemistry.org

The quantum theory applied to biology

The time independent energy and wave-function for a molecule

In classical mechanics we describe a model as:

  1. a trajectory (also called an orbit) for the comprising particles
  2. the energy of the comprising particles.
From the trajectory and the energy we can then calculate various properties that we find useful.

In quantum mechanics the wave-function is the logical analogue for the trajectory.

We consider models that consist of moving electrons and atomic nuclei (a gas of electrons and atomic nuclei). Any molecule can thus be modeled as a gas of electrons and atomic nuclei and assume the total energy is the sum of :

  1. the kinetic energy of the electrons,
  2. the kinetic energy of the nuclei,
  3. the potential energy due the electrostatic forces between the nuclei,
  4. the potential energy due to the electrostatic forces between the electrons,
  5. the potential energy due to the electrostatic forces between the electrons and the nuclei.
We neglect other terms like dipole interactions.

With this in mind we can write down a general expression for the time independent Schrödinger equation for molecules.

Hψ = Eψ
where :
  1. ψ is the unknown molecular wave-function
  2. E the unknown energy and
  3. H the molecular Hamiltonian.

The hamiltonian H for a model with i(1 → N) electrons and A (1 → M ) nulceï is:

Molecular Hamiltonian
  1. ZA is the nuclear charge of the Ath atom
  2. Laplacian is the Laplacian for the ith electron
  3. Laplacianis the Laplacian for the nuclei
  4. RAB is the distance between the nuclei
  5. rij is the distance between the electrons
  6. riA the distance between an electron and a nucleus

Currently it is impossible to find an exact mathematical solution to such a differential equation. Scientists found an approximate solution for the molecular wave-function by using various physical and mathematical principles. In de sequel we explain what you need to understand in order to think up a model and to analyse the calculated results.


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