Multi-Configurational Self Consistent Field methods – MCSCF.

As mentioned before the Hartree-Fock method neglects the instantaneous correlation between all the electrons that comes from the Coulombic repulsion between electrons. Instead, each electron is treated as if it was moving in a mean field of other electrons and the repulsion between them is averaged. The energy of the system will then be too high because the electrons are able to move to close than they should to each other. However, to obtain a lower energy of the system dynamic electron correlation needs to be included. It is especially important for the transition metal systems, which usually require correlation for qualitative results due to the large number of close lying excited states and for the systems that exhibit Jahn-Teller distortions. Jahn-Teller distortions will be described later in this chapter. The Hartree-Fock wavefunction is constructed as a single SD containing a set of one electron MOs and it can describe only a single configuration within a basis set. One way to describe the energy of the system better is for all possible configurations to be included thus the wavefunction needs to be represented as a linear combination of multiple SD as in equation below [5]:

One of the common extensions of Hartree-Fock theory that does this is configuration interaction (CI). The wavefunction is built as a linear combination of many Slater determinants, and is created by the mixing of many-electron wave functions obtained from different electronic configurations, generated by exciting electrons from occupied orbitals to virtual orbitals. A viable CI model can be obtained by limiting the CI expansion to only specific excitation levels (single excitations – CIS, single and double excitations – CISD…) as a full CI (FCI) is too expensive for all but the smallest of systems. CI method includes so called non-dynamic (static) correlation, which is a common feature of MCSCF methods and accounts for more than one electronic configuration at the same time in a balanced manner.

By including all possible N-electron excitations one obtains the Full-Configuration Interaction (Full-CI) method, which is the best calculation possible in a given basis set. However, this method is very expensive and the total number of possible SD is given by the binomial coefficient [3]:

K N ! "# $ %& ' K

( )

(

)

K – total number of Hartree-Fock spin-orbitals N – number of electrons

Full CI calculations can be performed only for small systems. For larger systems calculations can get extremely expensive and not practical because of the high number of determinants that need to be generated.

In the CI method discussed above the initial HF orbitals are kept fixed in the CI expansion, however a class of methods have been developed where the orbitals are optimized for the CI wavefunction rather than the HF one. The most popular variant of the MCSCF method that does this is the Complete Active Space Self Consistent Field method (CASSCF). Molecular orbitals which build the wavefunction are optimized in addition to the CI expansion coefficients. Hartree-Fock orbitals can be taken as an initial guess, but will invariably be different after convergence of the wavefunction. Here the molecular orbitals which build the wave function are divided into active and inactive ones and are chosen manually, dependent on the studied system. CASSCF

introduces the correlation between the active electrons in active orbitals by performing a Full CI amongst them (Figure 2.2.). It is necessary in CASSCF to optimize AO coefficients because the CI is performed only in the active space. By doing this one can be sure that the obtained energy will be the lowest possible for a given active space, and also ensures that the optimized molecular orbitals are flexible enough to describe the entire potential energy surface.

!

Figure 2.2. Comparison of the molecular orbital partitions in the single-configurational Hartree-Fock method and multi-configurational CASSCF [5].

For systems in which the multi-configurational character of the wavefunction is crucial (systems included in this thesis), HF theory guess orbitals may not be accurate enough. Building the CASSCF wavefunction starting from an incorrect wavefunction may give false or inaccurate results. Thus, the most efficient way to get the multideterminantal nature of the wavefunction is to use so called natural orbitals (NO) that diagonalize the density matrix and its eigenvalues are the occupation numbers of the orbitals. The occupation number is the number of electrons in each natural orbital and it is a real number between 0 and 2. For occupied orbitals occupation numbers are usually close to 1 or 2 and for unoccupied orbitals close to 0. The occupation numbers of orbitals that have to be included in the active space need to be different from exactly 0 or 2. An RHF wavefunction gives occupations of exactly 0 or 2 because of the lack of the electron correlation. UHF natural orbitals can be used as an alternative because they provide fractional occupation numbers [3].

In the case when all valence electrons will be chosen for the active space one can obtain an extremely accurate description of the wave function. However it is possible only for small systems for the same reason as Full CI computations. For large molecules (e.g.,

coordination compounds) where the active space may only contain some of the valence electrons the CASSCF method can overestimate the properties of the system studied; that is why the choice of correct orbitals is crucial for this method. The most important advantage of this method is that it can be used to describe conical intersections, and any multi-configurational states involved. The only limitation of this method is the number of the active space. The maximum active space one can be routinely applied nowadays is around 15 electrons in 15 orbitals (about 1.5 billion different configurations).

There is also a variation of the CASSCF method called Restricted Active Space Self Consistent Field (RASSCF). The orbitals are divided in three spaces RAS1, RAS2 and RAS3 (Figure 2.3) [11]. They have a restricted number of allowed excitations. RAS1 corresponds to doubly occupied orbitals and the RAS3 consists of empty orbitals. There is a full CI performed in the RAS2 space. Additional configurations can be generated by allowing for example a maximum of 2 electrons to be excited from the RAS1 space, and maximum 2 electrons to be excited into the RAS3 space, from the RAS2 space. The main concept of the RASSCF method is to generate configurations by combination of full CI, but in a small space RAS2 (which reduces the cost of calculations), and CISD in a larger space (RAS1, RAS3).

Figure 2.3. Orbital partitions in RASSCF method (picture reproduced from Reference [3]). RAS3 RAS2 RAS1 Inactive Virtual Full CI 0, 1 or 2 excitations