 (Parte 1 de 5)

References and Acknowledgements

The material discussed here is following closely the textbook below

MAE 715 –Atomistic Modeling of Materials N. Zabaras (2/18/2009)

¾Methods of Electronic structure calculations: From molecules to solids, M. Springborg (Chapter 13)

Correlation effects and the Møller-Plesset method

Correlation effects

MAE 715 –Atomistic Modeling of Materials N. Zabaras (2/18/2009)

 ¾We derived the H-F equations by approximating the N-electron

eigenfunction Ψ, to the electronic Schrodinger equation usinga single Slater determinant.

¾Although this might provide an accurate estimate for the total electronic energy

Ee , other properties might be less well described.

¾In particular, within the UHF approximation Φ was in the general case not an

 eigenfunction of the total operator With the projection technique it was

possible to account for this deficiency, but the resulting wavefunction was now a sum of more Slater determinants.

Correlation effects

MAE 715 –Atomistic Modeling of Materials N. Zabaras (2/18/2009)

¾Assuming that the Hartree-Fock equations can be solved exactly (e.g. when we have a complete set of basis functions within the Hartree-Fock-Roothaan approach) –this is the Hartree-Fock limit --the differences between ' the single-determinant wavefunction and the many-determinant wavefunction are called (static) correlation effects. In some cases the differences can be significant.

¾How to include the correlation effects and how to estimate when they might be important?

Revisiting the Hydrogen molecule

MAE 715 –Atomistic Modeling of Materials N. Zabaras (2/18/2009)

 non-interacting. For we have a hydrogen molecule.

¾We consider two Hydrogen atoms at a distance D. For large D, the two atoms are ¾For anyD, the two electrons occupy two orbitals that only differ in their spin.

¾The orbital φ1 is the bonding combination of two atomic statescentered at each atom where

Revisiting the Hydrogen molecule

MAE 715 –Atomistic Modeling of Materials N. Zabaras (2/18/2009)

Revisiting the Hydrogen molecule

MAE 715 –Atomistic Modeling of Materials N. Zabaras (2/18/2009)

Both electrons are on the same atom. This corresponds to the arrangements (ionic term)

The two electrons are at different atoms.

This corresponds to the

H--H arrangements (covalent term)

¾For the molecule (small D), this description might be acceptable, but for large D one would intuitively expect that the system consists of two neutral atoms. The fact that the ionic and covalent terms are given the same weight does not seem to be appropriate.

Revisiting the Hydrogen molecule

MAE 715 –Atomistic Modeling of Materials N. Zabaras (2/18/2009)

¾One may improve on this by considering a more general wavefunction

¾The parameter c depends on D and approaches 0 for D

¾Similarly to the bonding orbital, one may also construct the corresponding antibonding orbital:

The antibonding state

MAE 715 –Atomistic Modeling of Materials N. Zabaras (2/18/2009)

¾The Slater determinant constructed from the antibonding orbital is the following:

¾We can now modify the generalized wavefunction discussed earlier that had the form:

Two Slater determinants: Correlation effects

MAE 715 –Atomistic Modeling of Materials N. Zabaras (2/18/2009)

⇓ This wavefunction consists of more than one

Slater determinant, i.e. is one containing (static) correlation effects

Two determinants: Correlation effects

MAE 715 –Atomistic Modeling of Materials N. Zabaras (2/18/2009)

The two Slater determinants correspond to the two configurations shown. The vertical axis

Two determinants: Correlation effects

MAE 715 –Atomistic Modeling of Materials N. Zabaras (2/18/2009)

The single-particle energies of φ1 and φ2 as a function of D.

Excited configurations become important when their energy difference from the ground state is small.

¾For very large D, the single-particle energies of φ 1 and φ2 approach each other

(Parte 1 de 5)