Multi-reference levels

The tautomerism of furoxan (l,2,5-oxadiazole-2-oxide) has been investigated by different computational methods comprising modern density functions as well as single-reference and multi-reference ab initio methods. The ring-opening process to 1,2-dinitrosoethylene is the most critical step of the reaction and cannot be treated reliably by low-level computations (Scheme 2). The existence of cis-cis-trans- 1,2-dinitrosoethylene as a stable intermediate is advocated by perturbational methods, but high-level coupled-cluster calculations identify this as an artifact <2001JA7326>. 

Structures were optimized at the theoretical level described in Table 1, footnote a. Final energies were obtained by using multi-reference configuration interaction (MRCI) wave functions including single and double excitations from die reference wave functions. cFrom Reference 8. 

There are many computational investigations of transition metal oxides, see, e.g., the recent review by Harrison [7]. Some studies have included the whole sequence of 3d metal oxides. In one of these studies, Bauschlicher and Maitre [17], employed different high-level ab initio methods. It was found that ScO - MnO and CuO were well described by single-reference based techniques and that the CCSD(T) method gave spectroscopic constants (re, uje and D0) in good agreement with experiments. For FeO - NiO multi-reference based techniques (CASSCF/ICACPF) were necessary to get good results. 

Current and future volumes in the Series include both single-authored and multi-authored research monographs and reference level works as well as edited collections of themed reviews and articles. They all contain comprehensive bibliographies. 

Because of its size-extensivity and faster convergence with respect to excitation level Coupled cluster theory has replaced Cl theory as the dominant approach in ab initio correlation calculations. Like MBPT the theory is still mainly applied in cases where the exact wave function is dominated by a single determinant, but multireference methods have been formulated and begin to enter mainstream quantum chemistry. Generalization of the algorithms to the relativistic no-pair level can again be achieved via the spinorbital formulation of the methods. I will first discuss the single reference method and then consider the Fock space method [40] that uses multi-reference wavefiinctions for ionized or excited states. 

In order to go beyond the (6/6)CASSCF level, while maintaining the ability of a (6/6)CASSCF wave function to describe the two possible diradical extremes for the Cope TS, CASPT2 calculations proved ideal. This method, which had been developed by Roos and co-workers [19], applies multi-reference, second-order perturbation theory to CASSCF wave functions and in 1993 CASPT2 had become available in the MOLCAS suite of at initio calculations from the Roos group [20]. 

CC theory is inherently better than an equivalent level of Cl because it eliminates unlinked diagrams and as a consequence, is size-extensive [13]. It is also inherently better than an equivalent level of MBPT because it is not hmited to finite-orders, or potential difficulties encountered in the convergence of perturbation theory. It is well known, e.g. that ordinary MP perturbation theory does not converge for the electron gas, and this has also been emphasized recently for molecules [47], though resummations (including CC theory) work fine [48]. But, the operable word is equivalent level . For Cl, that meant at least single and double excitations, and frequently some more, perhaps even from a multi-reference space. MBPT had been done with single excitations in fourth-order SDQ-MBPT(4) in the above two papers [13,46]. 

Due to the multi-configurational nature of TI2, a multi-reference treatment appears to be necessary to obtain quantitative spectroscopic constants, and spin-orbit interactions are essential even for a qualitative description of the bonding situation. For a system like TI2, the KRCI approach is a very useful tool for molecular calculations. From the larger spin-orbit splitting in element 113, the (113)2 bond is expected to be even weaker than that of TI2. A preliminary study on (113)2 indicates that the spectroscopic constants calculated at the REP-KRCCSD(T) level are 3.82 A, 17 cm and 0.05 eV with the corresponding spin-orbit effects of 0.56 A, -46 cm and -0.81 eV, confirming this expectation. 

The effects of intruder states are generally more severe for molecules than for atoms, due to more dense energy levels. Therefore, even if there are ways of avoiding - or at least reducing - the effect of intruder states in the multi-reference approach, it is when the interest lies entirely in one or a few particular states, more advantageous to study one state at a time in a state-specific approach. 

Since beryllium only has four electrons, CISDTQ is a full Cl treatment and completely equivalent to a CCSDTQ calculation. The multi-reference character displays itself as a relatively slow convergence of the perturbation series, with millihartree accuracy being attained at the MP6 level and inclusion of terms up to MP20 is required in order to converge the energy to within 10 au of the exact answer. Note also that the correlation energy is overestimated at order seven, i.e. the perturbation series oscillates at higher orders. The contribution from triply excited states is minute, as expected... 

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