92
3 Ab initio Electronic Structure for Few-Body Systems
minimum basis set to adopt would include, accordingly, the functions associated
with valence orbitals.
8
As we have already seen, the reformulation of STO atomic orbitals in terms of
Cartesian Gaussian ones STO-N G is used to increase the computational efficiency
of the calculations. However, in order to achieve reasonable accuracy one would
have to take into account of two times their number. This type of basis is called DZ
(Double Zeta).
9 Further improvement can be obtained, for example, by adopting the
basis DZP (Double Zeta Polarization) that includes orbitals having higher values of
l and thus takes into account the orbital distorsion due to polarization. However, in
order to approach the HF limit it is necessary to use an even larger basis set. For
example, in order to deal with highly excited states, diffuse Rydberg functions need
to be added.
3.2.2 The CI and MC-SCF Methods
The method of election for recovering the residual (correlation) energy missing in
the HF treatment is, however, to be found outside the blind extension of the number of HF orbitals and rely on descriptors of the instantaneous interactions between
electron pairs. In other words, the main limitation of the HF theory is the neglecting
of the correlation among electronic motion even if the SCF method accounts for
interelectronic repulsion through the already defined Coulomb and exchange terms.
Obviously, a single determinantal function (that is a single HF configuration) cannot
represent the true eigenfunction even for closed-shell molecules. This type of correlation (coulomb interaction) is known as dynamic correlation and the difference
between the HF energy and the unknown, E exact , true one
E exact = E H F + E correlation .
(3.20)
is, indeed, the correlation energy. Two methods have been proposed in order to
cope with that problem configuration interaction (CI) and multiconfiguration self
consistent field (MCSCF).
In the CI method, the molecular wavefunction is a multiconfigurational (multireference) function that is expanded as a sum of Slater determinants
=
I
C I I
(3.21)
in which the C I coefficients are determined using a variational procedure (i.e., by
minimizing the total energy).
8 As an example for CH 4 the minimal set is given by the functions 1s, 2s e 2p of C and 1 s of H.
9 e.g., For the CH 4 molecule one would take, accordingly, two atomic orbital of each type.
3 Ab initio Electronic Structure for Few-Body Systems
minimum basis set to adopt would include, accordingly, the functions associated
with valence orbitals.
8
As we have already seen, the reformulation of STO atomic orbitals in terms of
Cartesian Gaussian ones STO-N G is used to increase the computational efficiency
of the calculations. However, in order to achieve reasonable accuracy one would
have to take into account of two times their number. This type of basis is called DZ
(Double Zeta).
9 Further improvement can be obtained, for example, by adopting the
basis DZP (Double Zeta Polarization) that includes orbitals having higher values of
l and thus takes into account the orbital distorsion due to polarization. However, in
order to approach the HF limit it is necessary to use an even larger basis set. For
example, in order to deal with highly excited states, diffuse Rydberg functions need
to be added.
3.2.2 The CI and MC-SCF Methods
The method of election for recovering the residual (correlation) energy missing in
the HF treatment is, however, to be found outside the blind extension of the number of HF orbitals and rely on descriptors of the instantaneous interactions between
electron pairs. In other words, the main limitation of the HF theory is the neglecting
of the correlation among electronic motion even if the SCF method accounts for
interelectronic repulsion through the already defined Coulomb and exchange terms.
Obviously, a single determinantal function (that is a single HF configuration) cannot
represent the true eigenfunction even for closed-shell molecules. This type of correlation (coulomb interaction) is known as dynamic correlation and the difference
between the HF energy and the unknown, E exact , true one
E exact = E H F + E correlation .
(3.20)
is, indeed, the correlation energy. Two methods have been proposed in order to
cope with that problem configuration interaction (CI) and multiconfiguration self
consistent field (MCSCF).
In the CI method, the molecular wavefunction is a multiconfigurational (multireference) function that is expanded as a sum of Slater determinants
=
I
C I I
(3.21)
in which the C I coefficients are determined using a variational procedure (i.e., by
minimizing the total energy).
8 As an example for CH 4 the minimal set is given by the functions 1s, 2s e 2p of C and 1 s of H.
9 e.g., For the CH 4 molecule one would take, accordingly, two atomic orbital of each type.