Correlated multireference wavefunctions

A Modern First-Principles View on Ligand Field Theory Through the Eyes of Correlated Multireference Wavefunctions... 

We have so far examined the performance of the canonical transformation theory when paired with a suitable multireference wavefunction, such as the CASSCF wavefunction. As we have argued, because the exponential operator describes dynamic correlation, this hybrid approach is the way in which the theory is intended to be used in general bonding situations. However, we can also examine the behavior of the single-reference version of the theory (i.e., using a Hartree-Fock reference). In this way, we can compare in detail with the related... 

CASPT2 is most useful for calculations on excited states and diradicals, where multireference wavefunctions are required. However, there are methods available for including electron correlation for radicals and radical ions for which single-determinantal wavefunctions represent good zero-order approximations, without resorting to multideterminantal (i.e., CASSCF) reference wavefunctions. Two of these methods are discussed in the following sections, and we recommend them over CASPT2 for most calculations on molecules with just one unpaired electron. 

Electron Correlation. Given the open-shell nature of the ground and excited states of lanthanide impurity ions in crystals, electron correlation is exU cmely important. Currently, electronic structure methods based on the use of multireference wavefunctions appear to be the fittest to respond to the requirements. Furthermore, their current evolution towards allowing more and more flexible definitions of the active space makes them even more adequate. Even though the methods used in the applications contained in this chapter are well known and their performance has been proven and documented in many highly correlated systems, we summarize here how to adapt them to the impurity lanthanide ion electronic structure demands. 

Figure 1. The shape of the potential curve for nitrogen in a correlation-consistent polarized double-zeta basis set is presented for the variational 2-RDM method as well as (a) single-reference coupled cluster, (b) multireference second-order perturbation theory (MRPT) and single-double configuration interaction (MRCl), and full configuration interaction (FCl) wavefunction methods. The symbol 2-RDM indicates that the potential curve was shifted by the difference between the 2-RDM and CCSD(T) energies at equilibrium.

Figure 1. Multireference problems involve both dynamical and nondynamical correlation. The nondynamical correlation is accounted for by the CASCI/CASSCF/DMRG wavefunction, which is made of multiple configurations generated in the active space with a fixed number of active electrons. The dynamical correlation is recovered on top of the multiconfigurational reference by correlating the active orbitals with orbitals in the external space (i.e., core and virtual orbitals.)...

The second step of the calculation involves the treatment of dynamic correlation effects, which can be approached by many-body perturbation theory (62) or configuration interaction (63). Multireference coupled-cluster techniques have been developed (64—66) but they are computationally far more demanding and still not established as standard methods. At this point, we will only focus on configuration interaction approaches. What is done in these approaches is to regard the entire zeroth-order wavefunc-tion Tj) or its constituent parts double excitations relative to these reference functions. This produces a set of excited CSFs ( Q) that are used as expansion space for the configuration interaction (Cl) procedure. The resulting wavefunction may be written as... 

For anything but the most trivial systems, it is not possible to solve the electronic Schrodinger equation exactly, and approximate techniques must instead be used. There exist a variety of approximate methods, including Hartree-Fock (HF) theory, single- and multireference correlated ab initio methods, semiempirical methods, and density functional theory. We discuss each of these in turn. In Hartree-Fock theory, the many-electron wavefunction vF(r1, r2,..., r ) is approximated as an antisymmetrized product of one-electron wavefunctions, ifijfi) x Pauli principle. This antisymmetrized product is known as a Slater determinant. 

Constraints were then applied, such that the number of electrons in a orbitals was fixed at six and the number of electrons in n orbitals at four. The results of the two calculations are presented in Table I, where the effects on some of the properties of the nitrogen molecule are given. For comparison the corresponding SCF values are also presented. As can be seen from these results, the effects of the constraints on the CASSCF wavefunction are not negligible. They are, however, considerably smaller than the difference between the CASSCF and the SCF values. Better agreement with experiment can only be obtained by including dynamical correlation effects, for example, by means of a large multireference Cl calculation or a many-body perturbation theory (MBPT) calculation. ... 

Experience in a variety of applications of the C ASSCF method has shown it to be a valuable tool for obtaining good zeroth-order approximations to the wavefunctions. Attempts have been made to extend the treatment to include also the most important dynamical correlation effects. While this can be quite successful in some specific cases (see below for some examples), it is in general an impossible route. Dynamical correlation effects should preferably be included via multireference Cl calculations. It is then rarely necessary to perform very large CASSCF calculations. Degeneracy effects are most often described by a rather small set of active orbitals. On the other hand experience has also shown that it is important to use large basis sets including polarization functions in order to obtain reliable results. The CASSCF calculations will in such studies be dominated by the transformation step rather than by the Cl calculation. A mixture of first- and second-order procedures, as advocated above, is then probably the most economic alternative. 

When MCSCF wavefunctions are used as the reference, the most commonly used methods for recovering the electron correlation are multireference configuration interaction (MRCI) (15) and multireference perturbation theory (MRPT)... 

The incremental scheme based on the wavefunction HF method was extended to the calculation of valence-band energies when the electron-correlation is taken into account. In [176,177] an effective Hamiltonian for the N — l)-electron system was set up in terms of local matrix elements derived from multireference configuration-interaction (MRCI) calcnlations for finite clnsters. This allowed correlation corrections to a HF band strnctnre to be expressed and rehable results obtained for the valence-band structure of covalent semicondnctors. A related method based on an efiective Hamiltonian in locahzed Wannier-type orbitals has also been proposed and applied to polymers [178,179]. Later, the incremental scheme was used to estimate the relative energies of valence-band states and also yield absolnte positions of snch states [180]. 

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