3.2 Higher Level Ab initio Methods
93
In a full CI approach all of the excited states contributing in the sum (3.21), that is
extended to all the configurations, can be built by taking into account all the possible
electronic excitations to the different molecular orbitals of the system. A full CI
wavefunction will consist of a large number of SCF I terms. For example, for
the H 2 O molecule, using a DZ basis the full CI eigenfunction will consist of about
250,000 configurations.
A full CI calculation is possible only for fairly small systems. For larger systems,
one truncates the list of considered configurations to those contributing significantly
to the wavefunction.
A sufficiently simple CI scheme is the CISD (single and double configuration interaction) one in which only the singly and doubly excited configurations are included.
However, a truncated CI is not “size consistent”.
10
In the alternative MC-SCF (multiconfiguration-SCF) method a simultaneous optimization of both the C I coefficients of (3.21) among the various configurations and
the c iq coefficients of the (3.19) (for example, of the LCAO expansion) is made. The
simultaneous optimization of both sets of coefficients makes the MC-SCF procedure
extremely heavy. For this reason, quite often in the (3.21) expansion, one considers
only the configurations obtainable from a limited optimized number of molecular
orbitals (active or valence orbitals). The CASSCF (Complete Active Space) method
minimizes the number of configurations to be considered by dividing the molecular
orbitals in three sets: the two inactive sets (the extreme cases of either the doubly
occupied orbitals or the nonoccupied ones in all the configurations) and the active
set of the intermediate orbitals occupied only at certain configurations (for small
molecules the valence orbitals or the antibonding ones).
3.2.3 Perturbation Methods
An alternative method is the perturbative [22] one in which an explicit construction
of the Hamiltonian matrix is needed. In this approach, the Hamiltonian operator is
written as a sum of two terms
H = H
(0)
+ λV
(3.22)
where H
(0) is the unperturbed (zeroth order) Hamiltonian while V is a perturbation
term modulated by the λ coefficient. Then for the i-th state the wavefunction i
and the energy E i are expanded in terms of subsequent corrections
(n)
i and E
(n)
i
as
follows:
i =
n=0
λ
(n)
(n)
i
(3.23)
10 A method is said “size consistent” when the energy computed for a molecular system by bringing
two of its subsystems (say A and B) at infinite distance is equal to the sum of that of the two
subsystems A and B computed as separate ones.
93
In a full CI approach all of the excited states contributing in the sum (3.21), that is
extended to all the configurations, can be built by taking into account all the possible
electronic excitations to the different molecular orbitals of the system. A full CI
wavefunction will consist of a large number of SCF I terms. For example, for
the H 2 O molecule, using a DZ basis the full CI eigenfunction will consist of about
250,000 configurations.
A full CI calculation is possible only for fairly small systems. For larger systems,
one truncates the list of considered configurations to those contributing significantly
to the wavefunction.
A sufficiently simple CI scheme is the CISD (single and double configuration interaction) one in which only the singly and doubly excited configurations are included.
However, a truncated CI is not “size consistent”.
10
In the alternative MC-SCF (multiconfiguration-SCF) method a simultaneous optimization of both the C I coefficients of (3.21) among the various configurations and
the c iq coefficients of the (3.19) (for example, of the LCAO expansion) is made. The
simultaneous optimization of both sets of coefficients makes the MC-SCF procedure
extremely heavy. For this reason, quite often in the (3.21) expansion, one considers
only the configurations obtainable from a limited optimized number of molecular
orbitals (active or valence orbitals). The CASSCF (Complete Active Space) method
minimizes the number of configurations to be considered by dividing the molecular
orbitals in three sets: the two inactive sets (the extreme cases of either the doubly
occupied orbitals or the nonoccupied ones in all the configurations) and the active
set of the intermediate orbitals occupied only at certain configurations (for small
molecules the valence orbitals or the antibonding ones).
3.2.3 Perturbation Methods
An alternative method is the perturbative [22] one in which an explicit construction
of the Hamiltonian matrix is needed. In this approach, the Hamiltonian operator is
written as a sum of two terms
H = H
(0)
+ λV
(3.22)
where H
(0) is the unperturbed (zeroth order) Hamiltonian while V is a perturbation
term modulated by the λ coefficient. Then for the i-th state the wavefunction i
and the energy E i are expanded in terms of subsequent corrections
(n)
i and E
(n)
i
as
follows:
i =
n=0
λ
(n)
(n)
i
(3.23)
10 A method is said “size consistent” when the energy computed for a molecular system by bringing
two of its subsystems (say A and B) at infinite distance is equal to the sum of that of the two
subsystems A and B computed as separate ones.