# Computational Chemistry/Molecular quantum mechanics

** Previous chapter - ** Molecular dynamics

### Introduction[edit]

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.

The summation indices are over all electron pairs. It is the 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 corresponds to the -electron density
. If we know we can solve
So we *guess* and solve independent Schrödinger equations.
Unfortunately each solution then depends on which we guessed.
So we extrapolate a new and solve the temporary Schrödinger equation again. This continues until stops changing. If our initial guessed was
appropriate we will have the SCF approximation to the ground state.

This can be done for numerical 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.

When is expanded in terms of the atomic orbitals the troublesome
term picks out producted pairs of atomic orbitals
either side of the operator. This leads to a number of *four-centre integrals*of order . These fill up the disc space and take a long time to compute.

### Bibliography[edit]

- A. Szabo and N. S. Ostlund,
*Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory*, (Macmillan, New York 1989). *Computational Quantum Chemistry*, Alan Hinchliffe,(Wiley, 1988).- Tim Clark,
*A Handbook of Computational Chemistry*, Wiley (1985). - Cramer C.J.,
*Essentials of Computational Chemstry,Second Edition*,John Wiley, 2004. - Jensen F. 1999,
*Introduction to Computational Chemistry*,Wiley, Chichester. - Web link on Hartree-Fock theory http://vergil.chemistry.gatech.edu/notes/hf-intro/hf-intro.html

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Semiempirical quantum chemistry