4.3 Accurate Computational Models
127
hhba| ˆ
V |aabp==hhba| ˆ
H|aabp=−−hhba| ˆ
H|apba=
−−hh|
1 − ˆ
P 12
r 12
|ap=
h(1)h(2)a(1)p(2)
r 12
dτ 1 dτ 2
(4.39b)
The same reasoning holds for Φ S and Φ ′
S showing that taking into account the 2h-1p
and 1h-2p excitations only causes a uniform shift of the diagonal matrix elements,
and hence, they can be left out of the calculation of the energy difference between
the states of the model space.
This variant of the difference dedicated CI is commonly known as DDCI2 and
gives reasonable energy differences for systems with a moderate importance of the
ionic determinants. This is specially interesting for the treatment of organic biradicals
or TM complexes with weakly coupled spin moments. However, the DDCI2 energy
difference becomes increasingly more approximate when the CAS reference wave
function contains non-negligible contributions from non-degenerate determinants.
In these cases, one necessarily has to rely on the more expensive DDCI procedure.
Finally, the external space is sometimes even further reduced by eliminating also
the 2h and 2p determinants from the CI. The resulting CAS+S or DDCI1 method
can be used to obtain a first impression of the relative size of the parameters, but
does normally not provide accurate answers. Moreover, one should be aware of
the practical problem that the implementations of this variant in different computer
programs do not consider exactly the same list of determinants.
4.3.3 Multireference Perturbation Theory
As an alternative for the variational methods, one can also apply multireference
perturbation theory (MRPT) to calculate magnetic interactions. In principle, this
type of calculations makes it possible to treat larger systems with more unpaired
electrons. Among the many different implementations, two schemes are especially
popular in the field of magnetic interactions: CASPT2 [13] and NEVPT2 [14, 15],
which will be shortly overviewed here.
The standard Møller–Plesset perturbation theory uses a single determinant reference wave function and defines the zeroth-order Hamiltonian as the sum of the Fock
operators
ˆ
H
(0)
MP =
ˆ
F i
(4.40)
By Koopmans’ theorem, the eigenvalues of the Fock operator applied on occupied
orbitals are proportional to ionization potentials and the eigenvalues corresponding
to unoccupied orbitals are related to the electron affinities. CASPT2 extends the
applicability to multireference cases by defining an effective one-electron Fock-type
operator as zeroth order Hamiltonian in the following way
127
hhba| ˆ
V |aabp==hhba| ˆ
H|aabp=−−hhba| ˆ
H|apba=
−−hh|
1 − ˆ
P 12
r 12
|ap=
h(1)h(2)a(1)p(2)
r 12
dτ 1 dτ 2
(4.39b)
The same reasoning holds for Φ S and Φ ′
S showing that taking into account the 2h-1p
and 1h-2p excitations only causes a uniform shift of the diagonal matrix elements,
and hence, they can be left out of the calculation of the energy difference between
the states of the model space.
This variant of the difference dedicated CI is commonly known as DDCI2 and
gives reasonable energy differences for systems with a moderate importance of the
ionic determinants. This is specially interesting for the treatment of organic biradicals
or TM complexes with weakly coupled spin moments. However, the DDCI2 energy
difference becomes increasingly more approximate when the CAS reference wave
function contains non-negligible contributions from non-degenerate determinants.
In these cases, one necessarily has to rely on the more expensive DDCI procedure.
Finally, the external space is sometimes even further reduced by eliminating also
the 2h and 2p determinants from the CI. The resulting CAS+S or DDCI1 method
can be used to obtain a first impression of the relative size of the parameters, but
does normally not provide accurate answers. Moreover, one should be aware of
the practical problem that the implementations of this variant in different computer
programs do not consider exactly the same list of determinants.
4.3.3 Multireference Perturbation Theory
As an alternative for the variational methods, one can also apply multireference
perturbation theory (MRPT) to calculate magnetic interactions. In principle, this
type of calculations makes it possible to treat larger systems with more unpaired
electrons. Among the many different implementations, two schemes are especially
popular in the field of magnetic interactions: CASPT2 [13] and NEVPT2 [14, 15],
which will be shortly overviewed here.
The standard Møller–Plesset perturbation theory uses a single determinant reference wave function and defines the zeroth-order Hamiltonian as the sum of the Fock
operators
ˆ
H
(0)
MP =
ˆ
F i
(4.40)
By Koopmans’ theorem, the eigenvalues of the Fock operator applied on occupied
orbitals are proportional to ionization potentials and the eigenvalues corresponding
to unoccupied orbitals are related to the electron affinities. CASPT2 extends the
applicability to multireference cases by defining an effective one-electron Fock-type
operator as zeroth order Hamiltonian in the following way