When using configuration interaction, sometimes it is clear that one needs to use more than one determinant to describe the system accu- rately. More contributions to the wavefunction are necessary, so the CI expansion needs to be balanced adequately. One example where this kind of approach plays a role is for example, on open-shell systems when different states are important for the ground state. Other systems where these kind of methods are important are: when a bond is being broken, or the opposite, a bond is being formed, several determinants have to be considered for constructing the wavefunction. The determinants for the construction of the wavefunction also have similar weights and should all be included. This part is tricky since it is up to the researcher to use his/her chemical intuition to decide what determinants to include in order not to make the calculation too demanding. Each of the determi- nants is built from molecular spin-orbitals which cannot be fixed. The classical MC SCF approach is:
Ψ =X
I
dIΦI (2.48)
Equation 2.48 differs from the previous CI equation where the d rep- resents variational coefficients and ΦI individual Slater determinants or
CSFs. In the classical MC SCF approach:
orbitals are calculated using an initial guess. 2. Optimisation of coefficients in CI expansion.
3. Then, LCAO coefficients are varied in the orbitals and this way obtain the optimum molecular orbitals for that CI expansion. 4. If convergence is not reached you have to return to step 1.
2.9.1 Complete active space self-consistent field (CASSCF) A widely used case of the MC SCF method is CASSCF [45] [46] [47]. For CASSCF the orbitals are divided in virtual, inactive and active. The active space always depends on the system studied. CASSCF is the main MC SCF wavefunction method used for geometry optimisations, because it allows the gradient and second derivatives to be computed analytically. The procedure to do a CAS calculation can be very tricky so one has to use chemical intuition to build an active space successfully. For the active space all the possible excitations and occupancies (0 to 2 electrons) are considered. This way all the determinants from the MC SCF expansion are considered with all excitations within the active space and size consistency is achieved. Since the results do not depend on any linear transformation of the molecular spin orbitals, it makes the result invariant regarding the localization of the molecular orbitals. The only limitation is the basis set, that depends on the size of the system. Figure 2.9 summarizes what has been being discussed.
CASSCF has numerous advantages over other methods:
• All the regions on the potential energy surface are treated in a balanced way.
• Any number of electronic states can be treated.
• It is a multiconfigurational method, meaning that it can treat all the possible electronic configurations of the orbitals of the system. • It is one of the best methods to describe conical intersections.
Figure 2.9: CASSCF - Scheme how orbitals are divided for a calculation.
Although this multiconfigurational method has numerous advantages, it also has a few disadvantages:
• It is not a black-box.
• It cannot include all the orbitals in the active space, except for very small molecules.
• It is very expensive for large active spaces.
CASSCF has been widely used due to its robustness to describe a lot of system properties even though it can be very difficult to set up and the region of full CI is limited due to computational resources.
CASSCF can be used to calculate conical intersections. As it was men- tioned in the beginning of this chapter, rich photochemical reactions are very important to research, because they can help the scientists to understand phenomena that happen when molecules are irradiated with light.
2.9.2 Restricted active space self-consistent field (RASSCF) An extension of the complete active space self-consistent field is the re- stricted active space self-consistent field (RASSCF). In this method the active orbitals are partitioned into three different sub-spaces: RAS1, RAS2 and RAS3. RAS2 is equivalent to CASSCF where a full CI cal- culation is performed. RAS1 and RAS3 have restrictions, RAS1 has a maximum number of holes allowed and in RAS3 a maximum number of electrons. As it happens for CASSCF there is a core of inactive or- bitals that make up the wavefunction. The large number of coupling coefficients present in these calculations create problems in terms of computational time. This method can be very helpful as it is described for example is Chapter 4. The limitation of a CASSCF calculation is around 16 orbitals and 16 electrons, which already produces a very large number of configurations. RASSCF performs calculations that include more orbitals and electrons and allows a large space to be chosen with specific restrictions.
2.9.3 Perturbation approaches to multireference methods CASSCF is able to recover the static correlation successfully, although is usually lacking dynamic correlation. One can use perturbation the- ory to solve this problem. One of the most popular methods nowadays is CASPT2. This method was proposed by Andersson et al initially [48]. In CASPT2 a multiconfigurational wavefunction generated from a CAS- SCF calculation is used as the zeroth-order wavefunction, which has a perturbation approach on top of that. The configuration space can be divided into four subspaces: V0, VK, VSD and VT Q.... V0 is the one-
dimensional space composed by the CAS reference function. VK is the
space spanned by the orthogonal complement to the CAS reference func- tion. VSD is the space spanned by the single and double replacement
states usually in reference to V0 and VT Q... gathers the higher excita-
of the necessary dynamic correlation that many systems need. Nowa- days, most of the multireference methods have been limited to second order perturbation. Analytical gradients are not available yet and geom- etry optimisation steps are too demanding so, numerical gradients have to be used instead. Another variant of perturbation theory used on this thesis (Chapter 5 and 6) is NEVPT2. This variant of multireference per- turbation theory is very powerful and uses different approximations to the system. Depending on the computational cost different approaches to this method can be used (totally uncontracted, strongly contracted and partially contracted). Another advantage is the fact that it avoids intruder states in perturbation series i.e avoids large denominators in the perturbation expansion.
2.10
Density functional theory
Nowadays, methods based on density functional theory that use the Kohn-Sham formalism are considered a valuable alternative to the tra- ditional ab initio methods of quantum mechanics. Kohn and Sham tried to find solutions from first principles of SCF and treat electronic corre- lation differently from wavefunction techniques. DFT methods can be applied to large systems, like coordination compounds, inorganic or bi- ological systems. DFT has been established as a valuable research tool because it can serve either to validate the conclusions that have been reached from experiments or to distinguish between possibilities that are left open. Density functional theory using the Kohn-Sham approach is considered an improvement over Hartree-Fock theory. The system can be modelled as a function of the electronic density. The biggest disad- vantage of DFT is the non-systematic approach to improve the results towards an exact solution.