4.3 Accurate Computational Models
121
type are often used as multideterminantal-multiconfigurational reference—mostly
multireference (MR), for short—wave function.
MR-CISD is one of the most accurate ab initio computational schemes that can
be used to describe the electronic structure of systems with a markedly multireference character. Although the ever increasing computing power constantly pushes the
frontiers forward, the applicability of MR-CISD remains limited to small (model)
systems. Moreover, the method suffers from the size-extensivety problem inherent
to any truncated CI method. For these reasons, MR-CISD is hardly ever used in computational studies of molecules with unpaired electrons. There are, however, several
alternative wave function based schemes that can provide very useful information
about the magnetic interactions. In the following sections we will first discuss the insand-outs of a good reference wave function and introduce the difference dedicated
CI (DDCI) method. Thereafter a short account will be given of two implementations of MR perturbation theory, and the chapter will be closed with a discussion of
the consequences of lifting the restrictions of the spin symmetry as done in density
functional theory (DFT).
4.3.1 The Reference Wave Function and Excited Determinants
An important factor in the accurate prediction of magnetic coupling (and other electronic structure) parameters is the proper choice of the reference wave function. There
are many possible ways to construct the reference, but the complete active space
(CAS) approach has emerged as one of the most versatile methods. The molecular
orbitals are divided in three groups: the inactive, the active and the virtual orbitals.
The orbitals in the first group are doubly occupied in all the Slater determinants of
the multireference wave function, while the orbitals in the last group remain always
unoccupied. The orbitals in the second class span the active space. The multiconfigurational wave function is generated by distributing N act electrons—where N act is the
total number of electrons minus two times the number of inactive orbitals—over the
M act active orbitals. This is schematically outlined in Fig. 4.10. The doubly occupied
or empty Hartree Fock orbitals shown on the left are divided in inactive, active and
virtual orbitals. The multiconfigurational wave function is constructed by making a
linear combination of Slater determinants Φ 1 , Φ 2 , etc. that differ by the occupation
of the active orbitals. The CAS procedure generates a MR wave function in which
all possible distributions of the active electrons over the active orbitals are considered. Although this approach often generates many determinants that are very high
in energy and are not specially important in the final wave function, it has several
important practical and conceptual advantages like the good convergence properties, size extensivity, orbital invariance, etc. [11]. Moreover, it has the advantage that
selecting the active orbital space (although far from being trivial) is in most cases
easier than making an unbiased selection of the most important configurations.
121
type are often used as multideterminantal-multiconfigurational reference—mostly
multireference (MR), for short—wave function.
MR-CISD is one of the most accurate ab initio computational schemes that can
be used to describe the electronic structure of systems with a markedly multireference character. Although the ever increasing computing power constantly pushes the
frontiers forward, the applicability of MR-CISD remains limited to small (model)
systems. Moreover, the method suffers from the size-extensivety problem inherent
to any truncated CI method. For these reasons, MR-CISD is hardly ever used in computational studies of molecules with unpaired electrons. There are, however, several
alternative wave function based schemes that can provide very useful information
about the magnetic interactions. In the following sections we will first discuss the insand-outs of a good reference wave function and introduce the difference dedicated
CI (DDCI) method. Thereafter a short account will be given of two implementations of MR perturbation theory, and the chapter will be closed with a discussion of
the consequences of lifting the restrictions of the spin symmetry as done in density
functional theory (DFT).
4.3.1 The Reference Wave Function and Excited Determinants
An important factor in the accurate prediction of magnetic coupling (and other electronic structure) parameters is the proper choice of the reference wave function. There
are many possible ways to construct the reference, but the complete active space
(CAS) approach has emerged as one of the most versatile methods. The molecular
orbitals are divided in three groups: the inactive, the active and the virtual orbitals.
The orbitals in the first group are doubly occupied in all the Slater determinants of
the multireference wave function, while the orbitals in the last group remain always
unoccupied. The orbitals in the second class span the active space. The multiconfigurational wave function is generated by distributing N act electrons—where N act is the
total number of electrons minus two times the number of inactive orbitals—over the
M act active orbitals. This is schematically outlined in Fig. 4.10. The doubly occupied
or empty Hartree Fock orbitals shown on the left are divided in inactive, active and
virtual orbitals. The multiconfigurational wave function is constructed by making a
linear combination of Slater determinants Φ 1 , Φ 2 , etc. that differ by the occupation
of the active orbitals. The CAS procedure generates a MR wave function in which
all possible distributions of the active electrons over the active orbitals are considered. Although this approach often generates many determinants that are very high
in energy and are not specially important in the final wave function, it has several
important practical and conceptual advantages like the good convergence properties, size extensivity, orbital invariance, etc. [11]. Moreover, it has the advantage that
selecting the active orbital space (although far from being trivial) is in most cases
easier than making an unbiased selection of the most important configurations.