Multi-reference MBPT

The main disadvantage of the MBPT/CCM approach, the restriction to a single reference function, is not essential to the theory but only to the current implementation (see references 30-32 for the multi-reference MBPT/CCM theory). To solve open-shell problems we normally use an unrestricted-Hartree-Fock (UHF) reference function. 

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. 

We should mention here that the historically oldest MR many-body theories using 7/gff have been the MR perturbation theories. Successful implementation of all the traditional multi-reference many-body perturbation theories (MR-MBPT), which were developed within the H ff framework [6], was mainly confined to calculation of energy differences of spectroscopic interest, and not to the smdy of PES. 

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]. 

The multi-reference capabilities of ci contrast with the situation in many-body perturbation theory (mbpt) and coupled cluster (cc) theory, for which multi-reference generalizations are substantially more difficult than their... 

MR-MBPT multi-reference many-body perturbation theory MR-MPPT multi-reference Mpller-Plesset perturbation theory CCSD single-reference coupled cluster with single and double replacements... 

Just as single reference Cl can be extended to MRCI, it is also possible to use perturbation methods with a multi-detenninant reference wave function. Formulating MR-MBPT methods, however, is not straightforward. The main problem here is similar to that of ROMP methods, the choice of the unperturbed Hamilton operator. Several different choices are possible, which will give different answers when the tlieory is carried out only to low order. Nevertheless, there are now several different implementations of MP2 type expansions based on a CASSCF reference, denoted CASMP2 or CASPT2. Experience of their performance is still somewhat limited. 

A further restriction on the use of many-body perturbation techniques arises from the (quasi-) degenerate energy structure, which occurs for most open-shell atoms and molecules. In these systems, a single reference state fails to provide a good approximation for the physical states of interest. A better choice, instead, is the use of a multi-configurational reference state or model space, respectively. Such a choice, when combined with configuration interactions calculations, enables one to incorporate important correlation effects (within the model space) to all orders. The extension and application of perturbation expansions towards open-shell systems is of interest for both, the traditional order-by-order MBPT [1] as well as in the case of the CCA [17]. 

PHF methods can, in turn, be classified as the variational and nonvariational ones. In the former gronp of methods the coefficients in linear combination of Slater determinants and in some cases LCAO coefficients in HF MOs are optimized in the PHF calculations, in the latter such an optimization is absent. To the former group of PHF methods one refers different versions of the configuration interaction (Cl) method, the multi-configuration self-consistent field (MCSCF) method, the variational coupled cluster (CC) approach and the rarely used valence bond (VB) and generaUzed VB methods. The nonvariational PHF methods inclnde the majority of CC reaUza-tions and many-body perturbation theory (MBPT), called in its molecular realization the MoUer-Plessett (MP) method. In MP calculations not only RHF but UHF MOs are also used [107]. 

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