Multi-reference functions

Today, there remain a number of problems in molecular electronic structure theory. The most outstanding of these is undoubtedly the development of a robust theoretical apparatus for the accurate description of dissociative processes which usually demand the use of multi-reference functions. This requirement has recently kindled a renewal of interest in the Brillouin-Wigner perturbation theory and its application to such problems. This contribution describes the application of... 

We turn now to the Brillouin-Wigner perturbation theory for a system described in zero order by a multi-reference function. The multi-reference formalism closely parallels that given in the previous section for the case of a single-reference function. Let us begin by defining a reference space V. Let... 

As in the single-reference formulation of the theory, the exact eigenstate is constructed by exploiting the completeness of the unperturbed basis. For a single state (but multi-reference function), Eq. 2.19 is now written as... 

The formalism presented in the preceding section for the case of a single-reference function can be readily generalized to the multi-reference case. The structure of the present section mirrors that of the previous one, in that in the first subsection we consider the choice of model function in the multi-reference case and define various projection operators. Remember that for the case of a single-reference function, the term model function is synonymous with zero-order function . For the multireference case, we shall see that the term model function is employed in a somewhat different manner. This is followed by two subsections, the first defining the wave operator and the second defining the reaction operator in the case of a multi-reference function. 

We are now in a position to obtain perturbation expansions by expanding the inverse operator in the effective Hamiltonian, the wave operator and the reaction operator. We begin, as we did in our discussion of the partitioning technique, by considering the case of a single-reference function and then turn our attention to the multi-reference function case. 

The appropriate choice of multi-reference function for quasi-degenerate problems is a significant problem and one which we do not address here. The use of a multireference formalism is required for problems as simple as the dissociation of the ground state of the hydrogen molecule. The choice of multi-reference function is dictated by the physics and chemistry of the systems under study. For more complicated problems the choice of reference requires considerable care. This choice certainly represents a significant barrier to the development of black box quantum chemical software packages for problems demanding the use of a multi-reference formalism. 

It is well-known that such functions can suffer from large amounts of spin contamination and are not suited to obtaining any surfaces except those that are the lowest of a given S3niraietry. However the UHF function, unlike an RHF function, will usually allow a molecule to separate correctly into its fragments for all decomposition channels. In contrast multi-reference-function techniques that include all configurations required to achieve correct separation would be intractable for even most three- and four-atom molecules. To limit the uncertainty introduced in using a UHF function for open shells, we monitor the multiplicity in the calculations. For some cases, such as the A A" state of HNO in the present paper, it offers a caution on the interpretation of the results, while for other cases, such as the A HCO surface, no multiplicity problems are encountered. 

Eq. (15b) for OH + H2 using multi reference configuration interaction wave functions. 

It is possible to construct a Cl wave function starting with an MCSCF calculation rather than starting with a HF wave function. This starting wave function is called the reference state. These calculations are called multi-reference conhguration interaction (MRCI) calculations. There are more Cl determinants in this type of calculation than in a conventional Cl. This type of calculation can be very costly in terms of computing resources, but can give an optimal amount of correlation for some problems. 

The relative importance of tlie different excitations may qualitatively be understood by noting tliat the doubles provide electron correlation for electron pairs, Quadruply excited determinants are important as they primarily correspond to products of double excitations. The singly excited determinants allow inclusion of multi-reference charactei in the wave function, i.e. they allow the orbitals to relax . Although the HF orbitals are optimum for the single determinant wave function, that is no longer the case when man) determinants are included. The triply excited determinants are doubly excited relative tc the singles, and can then be viewed as providing correlation for the multi-reference part of the Cl wave function. 

Specifically, if T] < 0.02, the CCSD(T) metliod is expected to give results close the full Cl limit for the given basis set. If is larger than 0.02, it indicates that the reference wave function has significant multi-determinant character, and multi-reference coupled cluster should preferentially be employed. Such methods are being developedbut have not yet seen any extensive use. 

Grimme, S., Waletzke, M., 1999, A Combination of Kohn-Sham Density Functional Theory and Multi-Reference Configuration Interaction Methods , J. Chem. Phys., Ill, 5645. 

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>. 

The reference (zeroth-order) function in the CASPT2 method is a predetermined CASSCF wave function. The coefficients in the CAS function are thus fixed and are not affected by the perturbation operator. This choice of the reference function often works well when the other solutions to the CAS Hamiltonian are well separated in energy, but there may be a problem when two or more electronic states of the same symmetry are close in energy. Such situations are common for excited states. One can then expect the dynamic correlation to also affect the reference function. This problem can be handled by extending the perturbation treatment to include electronic states that are close in energy. This extension, called the Multi-State CASPT2 method, has been implemented by Finley and coworkers.24 We will briefly summarize the main aspects of the Multi-State CASPT2 method. 

Density Functional Theory, DFT (B3LYP), CASSCF (Complete Active-State Self-Consistent Field) and MRSD-CI (Multi-Reference Single-Double Correlation Interaction) calculations on the diatomic units AuO, AuO", AuO " and AuO " clearly show that stability of Au-0 bond reduces in this order. This trend is consistent with the molecular orbital diagram of AuO molecule presented in Fig. 10. 

Lastly, the SF approach implemented within the time-dependent. (TD) density functional theory (DFT) extends DFT to multi-reference situations with no cost increase relative to the non-SF TD-DFT. Similarly to DFT and TD-DFT, the SF-DFT model (27) is formally exact and therefore will yield exact answers with the exact density functional. With the available inexact ftmctionals, the SF-DFT represents an improvement over its non-SF counterparts. It has been shown to yield accurate equilibrium properties and singlet-triplet energy gaps in diradicals (27). 

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