Computational Chemistry/Molecular quantum mechanics

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Introduction
Our applications of quantum theory here involve solving the wave equation for a given molecular geometry. This can be done at a variety of levels of approximation each with a variety of computing resource requirements.

Our applications of quantum theory here involve solving the wave equation for a given molecular geometry. This can be done at a variety of levels of approximation each with a variety of computing resource requirements. We are assuming here a vague familiarity with the Self Consistent Field wavefunction and its component molecular orbitals.

$$ < \Psi_{0} | \hat{H} | \Psi_{0} > $$

$$ < \Psi_{0} | K.E. + V_{nuc} + \sum_{i < j} {\frac {1} {r_{ij}}} | \Psi_{0} > $$

The $$ij$$ summation indices are over all electron pairs. It is the $$\frac {1} {r_{ij}}$$ which prevents easy solution of the equation, either by separation of variables for a single atom, or by simple matrix equations for a non spherical molecule.

The electron density $$\rho (x,y,z)$$ corresponds to the $$N$$-electron density $$\rho (N)$$. If we know $$\rho (N-1)$$ we can solve $$< \Psi_{0} | \hat{H} | \Psi_{0} >$$ So we guess $$\rho (N)$$ and solve $$N$$ independent Schrödinger equations. Unfortunately each solution then depends on $$\rho (N)$$ which we guessed. So we extrapolate a new $$\rho^{'} (N)$$ and solve the  temporary Schrödinger  equation again. This continues until $$\rho$$ stops changing. If our initial guessed $$\rho$$ was appropriate we will have the SCF approximation to the ground state.

This can be done for numerical $$\rho$$ or we can use LCAO (Linear Combination of Atomic Orbitals) in an algebraic form and integrate into a linear algebraic matrix problem. This use of a basis set is our normal way of doing calculations.

Our wavefunction is a product of molecular orbitals, technically in the form of a Slater determinant in order to ensure the antisymmetry of the electronic wavefunction. This has some technical consequences which you need not be concerned with unless doing a theoretical project. Theoreticians should make Szabo and Ostlund their bedtime reading.

$$ \Phi_{k} (x_{1}, x_{2}, x_{3}, .....x_{N}) ~=~ \frac {1} {\sqrt {N!} } ~determinant $$

$$ \psi _{A} (x_{1}) \psi _{B} (x_{1}) ..........\psi _{N} (x_{1}) $$

$$ \psi _{A} (x_{2}) \psi _{B} (x_{2}) ..........\psi _{N} (x_{2})$$

$$

......$$

$$

......$$

$$

\psi _{A} (x_{N}) \psi _{B} (x_{N}) ..........\psi _{N} (x_{N}) $$

When $$\Psi$$ is expanded in terms of the atomic orbitals $$\chi$$ the troublesome $$\frac {1} {r_{ij}}$$ term picks out producted pairs of atomic orbitals either side of the operator. This leads to a number of four-centre integralsof order $$n^4$$. These fill up the disc space and take a long time to compute.