# 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.

${\displaystyle <\Psi _{0}|{\hat {H}}|\Psi _{0}>}$

${\displaystyle <\Psi _{0}|K.E.+V_{nuc}+\sum _{i}$

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

This can be done for numerical ${\displaystyle \rho }$ 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.

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

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

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

${\displaystyle ......}$

${\displaystyle ......}$

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

When ${\displaystyle \Psi }$ is expanded in terms of the atomic orbitals ${\displaystyle \chi }$ the troublesome ${\displaystyle {\frac {1}{r_{ij}}}}$ term picks out producted pairs of atomic orbitals either side of the operator. This leads to a number of four-centre integralsof order ${\displaystyle n^{4}}$. 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 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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