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Correlation effects and the Møller-Plesset method
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MAE 715 – Atomistic Modeling of Materials N. Zabaras (2/18/2009)
Methods of Electronic structure calculations: From molecules to solids
MAE 715 – Atomistic Modeling of Materials N. Zabaras (2/18/2009)
We derived the H-F equations by approximating the N-electron^ eigenfunction
, to the electronic Schrodinger equation
using
a single Slater
determinant. ¾
Although this might provide an accurate estimate for the total electronic energy
, other properties might be less well described. e
In particular, within the UHF approximation
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.
MAE 715 – Atomistic Modeling of Materials N. Zabaras (2/18/2009)
We consider two Hydrogen atoms at a distance
. For large
, the two atoms are
non-interacting. For
we have a hydrogen molecule.
For any
, the two electrons occupy two orbitals that only differ in their spin.
The orbital
φ
1
is the bonding combination of two atomic states
centered
at each atom
where
MAE 715 – Atomistic Modeling of Materials N. Zabaras (2/18/2009)
MAE 715 – Atomistic Modeling of Materials N. Zabaras (2/18/2009)
One may improve on this by considering a more general wavefunction ¾
The parameter
c
depends on
and approaches
for
Similarly to the bonding orbital, one may also construct the corresponding^ antibonding orbital:
→ ∞
MAE 715 – Atomistic Modeling of Materials N. Zabaras (2/18/2009)
The Slater determinant constructed from the antibonding orbital is the following:^ ¾
We can now modify the generalized wavefunction discussed earlier that had the^ form:
MAE 715 – Atomistic Modeling of Materials N. Zabaras (2/18/2009)
The two Slater determinants correspond to the two configurations shown. The vertical axis gives the single particle energies of
φ
1
and
φ
MAE 715 – Atomistic Modeling of Materials N. Zabaras (2/18/2009)
The single-particle energies of
φ
1
and
φ
2
as a function of
Excited configurations become important when their energy difference
from the ground state is small.
For very large
the single-particle energies of
φ
1
and
φ
2
approach each other
(the energies of the bonding and the antibonding orbitals do not differ), whereas for small
they have markedly different energies.
MAE 715 – Atomistic Modeling of Materials N. Zabaras (2/18/2009)
¾^ The figure shows a set of orbitals obtained from the^ Hartree-Fock-Roothan calculation. On the left^ the orbitals without their occupancies; on the^ middle the occupancies for the ground state;^ and on the right the excited states.
We may, however, also use energetically higher orbitals in constructing Slater^ determinants for excited states. ¾
A more general wavefunction contains all the possible configurations that can be^ constructed from the orbitals of the Hartree-Fock-Roothaan equations.
MAE 715 – Atomistic Modeling of Materials N. Zabaras (2/18/2009)
The excited configurations are obtained from the single electron orbitals that are^ calculated using the Fock operator for the ground state. ¾
All single-electron orbitals (both those that for the ground state are occupied and^ those that are empty) become orthonormal -- they are all eigenfunctions of^ the Fock operator. ¾
Instead of writing ¾
We introduce the notation: ¾
Here we denote with
the different configurations and
is the number of configurations that can be constructed from the orbitals obtained from the H-F-R equations.
Taking
, for
i=1,2…
gives the Hartree-Fock method
MAE 715 – Atomistic Modeling of Materials N. Zabaras (2/18/2009)
The following eigenvalue problem results: ¾
And the eigenvalue is the needed energy: ¾
We need to study in some detail the matrix elements in the above eigenvalue^ problem using:
MAE 715 – Atomistic Modeling of Materials N. Zabaras (2/18/2009)
We first consider the general case of an operator that is a sum of identical^ single-electron operators:
MAE 715 – Atomistic Modeling of Materials N. Zabaras (2/18/2009)
Note from this equation (by taking
), that different configurations
are orthonormal: ¾
For any single-electron operator,
there will be non-vanishing matrix
elements only between configurations that differ at most in one orbital. ¾
Thus there will be non-vanishing matrix elements only between^ configurations that can be obtained from each other by applying at^ most one pair of creation and annihilation operators
MAE 715 – Atomistic Modeling of Materials N. Zabaras (2/18/2009)
We can apply the same process for: ¾
Thus there will be non-vanishing matrix elements only for configurations^ differing by at most two orbitals.
