We present in this paper two new versions of Rayleigh-Schrödinger (RS) and the Brillouin-Wigner (BW) state-specific multi-reference perturbative theories (SS-MRPT) which stem from our state-specific multi-reference coupled-cluster formalism (SS-MRCC), developed with a complete active space (CAS). They are manifestly sizeextensive and are designed to avoid intruders. The combining coefficients c µ for the model functions φ µ are completely relaxed and are obtained by diagonalizing an effective operator in the model space, one root of which is the target eigenvalue of interest. By invoking suitable partitioning of the hamiltonian, very convenient perturbative versions of the formalism in both the RS and the BW forms are developed for the second order energy. The unperturbed hamiltonians for these theories can be chosen to be of both Mφller-Plesset (MP) and Epstein-Nesbet (EN) type. However, we choose the corresponding Fock operator f µ for each model function φ µ , whose diagonal elements are used to define the unperturbed hamiltonian in the MP partition. In the EN partition, we additionally include all the diagonal direct and exchange ladders. Our SS-MRPT thus utilizes a multi-partitioning strategy. Illustrative numerical applications are presented for potential energy surfaces (PES) of the ground ( 1 Σ + ) and the first delta ( 1 ∆) states of CH + which possess pronounced multi-reference character. Comparison of the results with the corresponding full CI values indicates the efficacy of our formalisms.

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We present in this paper a multi-reference coupled cluster (MRCC) formulation for energy dierences which treats orbital relaxation and correlation eects on the same footing, by invoking a novel cluster ansatz of the valence portion of the wave operator X v . Unlike in the traditional normal-ordered exponential representation of X v , our new relaxation-inducing ansatz, represented symbolically as i r , allows contractions between the spectator lines and also certain other special contractions. By an extensive theoretical analysis, taking as an example the case of one-hole model space (the IP problem), we demonstrate that our ansatz incorporates in a manifestly spin-free form the orbital relaxation to all orders. The traditional Thoulesstype of exponential transformation via one-body excitations can induce the same eect, as is done in the valence-speci®c or the quasi-valence-speci®c MRCC formalisms, but they have to be done in the spin-orbital basis ± making the spin adaptation of the problem a complicated exercise. In contrast, we use a spin-free representation of the cluster operators right from start, but expand the rank of the cluster operators by involving spectator orbitals to distinguish the various spin possibilities. The combinatorial factors entering the contracted power series in i r are chosen in such a way that they correspond to what we would have obtained if we had used a Thouless-like transformation to induce the orbital relaxation. Our working equations generally have only ®nite powers of the cluster operators , resulting in a very compact formulation of the relaxation problem. Pilot numerical applications for the IP computations of HF and H 2 O in the core, the inner valence and the outer valence regions show very good performance of the method vis-a-vis those obtained using the traditional normal ordered ansatz for X v . The improvement in the core IP value is particularly impressive, although even for the valence regions there is an overall improvement of the IP values.

We present in this paper a comprehensive account of an explicitly spin-free coupled cluster theory for treating energy differences of open-shell states relative to a closed-shell ground state, where the open-shell states of interest are dominated by a few simple configuration state functions. We develop a valence-universal coupled cluster formalism to achieve this via a novel cluster expansion ansatz for the valence part of the wave operator, where the orbital relaxation and the correlation relaxation accompanying ionization/excitation from the ground state are taken care of to all orders in compact, efficient, and explicitly spin-free manner. The essential difference of our proposed ansatz from the ordinary and the normal-ordered cluster ansatz in vogue is that (a) we allow the valence cluster operators to be connected among themselves with spectator valence lines only and (b) we use suitable combinatoric factors accompanying powers of cluster operators thus connected, which are equal to the number of ways the operators can be joined, leading to the same excitation (the automorphic factor). We emphasize that such an ansatz does not generate terms (diagrams) with chains of cluster operators joined among themselves via spectator lines only. Barring only a few, almost all the terms in the working equations determining the cluster amplitudes involve contraction of the Hamiltonian with the cluster operators via at least one nonspectator line, leading to what we call a "strongly connected" series. The structure of the working equation is remarkably similar to the single-reference closed-shell equation, with a few additional terms. The presence of contractions among cluster operators via spectator lines introduces the additional physical effects of orbital and correlation relaxation using low-body cluster operators. As an illustrative application of the new multireference coupled cluster (CC) theory, we consider in this paper computation of ionization potentials (IPs) of one-valence problem with only one active orbital. The numerical applications are made for both the core- and the inner- and outer-valence IPs for several molecular systems. The numerical values demonstrate the superiority of the relaxation-inducing CC theory, as compared to the normal-ordered ansatz.

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