Wave functions multireference coupled-clusters

One way of achieving size-consistency for a dissociation process is to use an MCSCF wave function as the reference. Unfortunately, as noted above, there are as yet no general multireference perturbation theory or multireference coupled-cluster treatments that can be applied to such an MCSCF reference function. For rather few electrons, as we shall see, the MRCI approach performs acceptably. 

As long as a satisfactory multireference coupled-cluster theory is missing, there are various options for states that need a zeroth-order multiconfiguration wave function. One possibility is to start from an MC-SCF calculation and to improve this by selected Cl. Since the MC-SCF part is basically extensive, while the Cl part is not, and since one can hardly go beyond external double excitations, one tends to include as many configurations in the MC-SCF part as possible. However, MC-SCF is usually of CAS (complete active space) [154] type, e.g. like full Cl, which restricts the possible size of the active space. Such multireference Cl scheme have been very popular for describing excited states, reaction barriers, dissociation processes etc. 

Multireference coupled cluster (MRCC) models provide a generalization of the single-reference CC approach (3.1) for applications where several reference determinants contribute with similarly large weights to the wave function of the molecular system under consideration. The MRCC models can be divided into state-specific (SS) and multi-state approaches. One of the state-specific MRCC (SSMRCC) approaches is the active space CC method proposed in the works of Adamowicz and co-workers [14—18]. This method established the foundation for the approach developed and implemented by the authors of this article [19-27]. 

This is not to say that the CCSD(T) method provides an accurate description of the ground states of all molecules. The coupled cluster method is based on a Hartree-Fock reference wave function and thus will fail when the HF wave function does not provide an adequate zero-order description of the molecule. The development of multireference coupled cluster methods is being actively pursued by several groups. 

In addition to the encouraging numerical results, the canonical transformation theory has a number of appealing formal features. It is based on a unitary exponential and is therefore a Hermitian theory it is size-consistent and it has a cost comparable to that of single-reference coupled-cluster theory. Cumulants are used in two places in the theory to close the commutator expansion of the unitary exponential, and to decouple the complexity of the multireference wave-function from the treatment of dynamic correlation. 

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. 

Studies of rare earth or transition metal complexes often necessitate use of multireference wave functions. Among the Coupled Cluster type methods one can distinguish two main lines of approach to incorporate multireference character in the reference wave function. In the Hilbert space method one computes a single wave function for a particular state, while in the Fock space method one tries to obtain a manifold of states simultaneously. Since the latter method [40] has recently been implemented and applied in conjunction with the relativistic Hamiltonian [48-50] we will focus on this approach. 

The appropriate choice of the wave function is essential if chemically meaningful results are to be achieved. When only near equilibrium properties on the lowest potential energy surface are required, the single reference based coupled cluster wave function describes that region of the potential energy surfaces to near spectroscopic accuracy. However, nonadiabatic processes require wave functions that are necessarily multireference in character. For this reason, MRCI wave functions have been the wave function of choice in this field. These wave functions are developed from molecular orbitals determined from a self-consistent field (SCF), a multiconfigurational SCF... 

There are two major routes to improve the Cl methods. The first is by including size-consistent corrections for dynamical EC leading to the CIS(D) and CIS-MP2 methods " (which are closely related to the coupled-cluster methods described in a later section). The second is by improving the reference wave functions, which leads to multireference Cl (MRCI) methods, that, however, remain size-inconsistent. 

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