# 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.
$\Phi _{k}(x_{1},x_{2},x_{3},.....x_{N})~=~{\frac {1}{\sqrt {N!}}}~determinant$
$\psi _{A}(x_{1})\psi _{B}(x_{1})..........\psi _{N}(x_{1})$
$\psi _{A}(x_{2})\psi _{B}(x_{2})..........\psi _{N}(x_{2})$
$\psi _{A}(x_{N})<th>\psi _{B}(x_{N})..........\psi _{N}(x_{N})$
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.

### Bibliography

* 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 Chemistry,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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