The quantum theory applied to biology

We can calculate an approximate solution to the Schrödinger Equation

Even with only the electronic Hamiltonian it is still not possible to find an exact solution to the Schrödinger equation. Fortunately we can find an approximate solution by making use of functional variation.

The variation principle states : The expectation value of the Hamiltonian is an upper bound to the exact ground state energy for a normalized wave-function that satisfies the correct boundary conditions. We can write this symbolically as: Variation Principle

The electronic energy E0 is the expectation value of the electronic Hamiltonian. The wave-function then becomes a variable on which the energy depends. Mathematicians say the wave-function is a functional of the energy E0.

There is a procedure to obtain a wave-function that corresponds to the minimum of the energy. This wave-function will be an approximation to the real wave-function. Much scientific research has been done on this subject. Various methods exist that give good approximate wave-functions.

The Hartree-Fock approximation uses the variation principle to obtain a wave-function and an energy. The Hartree-Fock approximation consists of two distinct methods:

  • Hartree-Fock Roothaan-Hall restricted closed shell method
  • Hartree-Fock Pople-Nesbet unrestricted open shell method

The Rayleigh-Schrödinger perturbation theory and configuration interaction allow us to further improve the Hartree-Fock approximation.

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