SOCI methods

Spin-orbit Cl methods versus the full two-component treatments 

whether or not the correlation treatment and SO coupling are carried out separately, 

A large number of spectroscopic applications using spin-orbit Cl methods are nowadays performed using effective core potentials, especially when heavy atoms are involved. However, by nature all SOCI methods can in principle be used either in an all-electron or in an ECP scheme, provided the code can compute the appropriate integrals (see section 2.1.1). Although we are dealing in 

If spin-orbit coupling is considered as a perturbation, the total Hamiltonian decouples into a spin-free (referred as // for a scalar relativistic Hamiltonian) and a spin-orbit part H = The electrostatic and spin-orbit inter- 

A straightforward generalisation of the above perturbation Cl/SO treatment is to use the correlated scalar relativistic eigenfimctions 4 ) of the scalar Hamiltonian /f as a truncated set of contracted many-electron basis functions for the total Hamiltonian. Introducing the subscript im for a given wave functions to mark out the spatial and spin degenerate components of this multi-plet, the matrix representation of the Hamiltonian writes 

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The reliability of SOCI, coupled with its essentially a priori selection of the Cl space, makes it an attractive alternative to full Cl. Unfortunately, for reasonable active spaces the dimension of the SOCI grows very rapidly with system size and thus the method is applicable only to small molecules. Nevertheless, for quite a few molecules it is possible to use the SOCI method in conjunction with large one-particle basis sets and hence to obtain wavefunctions very close to the exact nonrelativistic Born-Oppenheimer limit. Below, we will attempt to give the reader a sense of the types of problems to which the SOCI method has been applied. Bauschlicher, Langhoff, and Taylor have already given15 an excellent review of the related CASSCF MR-CISD method, and we refer the reader to their article for a discussion of additional important studies. 

CISD[TQ] and SOCI methods employed CISD natural orbitals. 

It is remarkable that the PW91 DFT method can reproduce the results of the much more expensive SOCI method so well for the actinide excited states. Given the huge difference in the amount of computer time demanded by these two methods, the application of the DFT method to excited states of other actinide compounds with /" ( > 1) configurations promises to be a challenging venture (because of the problem of state multiplets) but potentially a very fruitful one. 

Finally, some spectroscopic applications for pseudopotentials within SOCI methods are presented in section 3. We focus our attention on applications related to relativistic averaged and spin-orbit pseudopotentials (other effective core potentials applications are presented in chapters 6 and 7 in this book). Due to the large number of theoretical studies carried out so far, we have chosen to illustrate the different SOCI methods and discuss a few results, rather than to present an extensive review of the whole set of pseudopotential spectroscopic applications which would be less informative. Concerning the works not reported here, we refer to the exhaustive and up-to-date bibliography on relativistic molecular studies by Pyykko [21-24]. The choice of an application is made on the basis of its ability to illustrate the performances on both the pseudopotential and the SOCI methods. One has to keep in mind that it is not easy to compare objectively different pseudopotentials in use since this would require the same conditions in calculations (core definition, atomic basis set, SOCI method). The applications are separated into gas phase (section 3.1) and embedded (section 3.2) molecular applications. Even if the main purpose of this chapter is to deal with applications to molecular spectroscopy, it is of great interest to underline the importance of the spin-orbit coupling on the ground state reactivity of open-shell systems. A case study is presented in section 3.1.4. 

The atoms are the simplest examples which are chosen to illustrate the ability of the SOCI methods to obtain accurate results. In this case the notation W stands for the usual notation where L and 5 are good atomic quantum numbers. 

Effective Hamiltonian-based contracted SOCI methods (CI /SO)... 

Effective Hamiltonian-based uncontracted SOCI methods (DGCF )... 

As one of the main difficulties of the SOCI methods lies in the electronic correlation treatment, we choose an example with a large niunber of valence electrons to correlate, namely a di-halogen molecule, in order to illustrate how main-group pseudopotentials perform for spectroscopic constants. Moreover to test the ability of the various SOCI methods to handle accurately very large spin-orbit splittings, we deal with the heaviest experimentally known, the iodine molecule. 

Gathering the experience of the previously discussed examples, we can say that the theoretical study of spectroscopic properties of transition metal complexes is far from being simple. Relativistic pseudopotentials (AREP and SOREP) were shown to be efficient and accurate tools to tackle this problem. From the methodological point of view, recently developed effective Hamiltonian SOCI methods that can treat correlation and spin-orbit coupling on the same footing exist (see section 2.2.5), and efforts have to be invested in applying... 

As in the case of NpOj, no experiment was available to compare this calculations with. As can be seen from Table 8, the 23 lowest fine-structure states arise from a limited number of A-S states. This calculation shows that it is in principle possible for some simple actinide molecules to use a contracted CI /SO method. But it is worthwhile to note that it would have been much easier to use an adapted SOCI method, namely a DGCP method which includes automatically all determinants coupled with the states of interest (see section 2.2.5). 

To conclude this part, it should be noticed that lanthanide and actinide electronic spectroscopy can now be handled by ab initio methods with a sufficient accuracy. In particular, the very recent progresses on the SOCI methods are promising for the treatment of excited states of lanthanide and actinide compounds. 

CI /SO effective hamiltonian-based contracted SOCI methods CI/SO contracted SOCI... 

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