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 groud 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.

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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