Computational Chemistry/Molecular quantum mechanics

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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.


< \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})<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[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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