Multi-reference second-order perturbation

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]. 

Subsequently, Kozlowski et al. [24] also revisited the Cope rearrangement with inclusion of dynamic correlation between the active and inactive electrons. However, they used Davidson s own version of multi-reference, second-order perturbation theory [25], which allows the coefficients of the configurations in the CASSCE wave function to be recalculated after inclusion of dynamic electron correlation. Kozlowski et al. found that the addition of dynamic correlation to the (6/6)CASSCE wave function for the Cope TS causes the weight of the RHE configuration to increase at the expense of the pair conhgurations that are necessary to describe the two diradical extremes in Eig. 30.1. Thus, without the inclusion of dynamic electron correlation in the wave function, (6/6) CASSCF overestimates the diradical character of the C2 wave function [24]. 

In the early 90s, a second improvement in the field of computational photochemistry took place. The Roos group proposed and implemented the multi-reference second-order perturbation theory (CASPT2), which reduced the CASSCF errors to the level of the accepted chemical accuracy (0.2 eV). This promoted the computational photochemistry to a new stage. Since then, the CASSCF/CASPT2 method is the most general and robust method for qualitative and quantitative accurate studies of any type of photochemical or photophysical process. 

Multi-reference second order perturbation theory (CASPT2) provides an optimum approach for determining the rovibrational properties of the inner transition state. Such calculations combine high accuracy with suitable efficiency, particularly now that anal3dic CASPT2 gradients are available in electronic structure programs such as MOLPRO. ... 

Multi-reference second order perturbation method (CASPT2 method) multi-reference MP3 method (MR-MP3) is also available. 

The main importance of Cl is that the FCI calculations provide results that are used as benchmarks for testing other post-Hartree-Fock methods. Less important seems to be the use of Cl as a post-Hartree-Fock method in routine chemical applications, because results of about the same accuracy may be obtained more economically by other methods. The size inconsistency of CI-SD may also be a drawback in some applications. Still, the recent progress in the development of Cl programs indicates that Cl might regain its importance even in this field. The traditional domain of Cl has been in electronic spectroscopy and excited electronic states in general. This is still true for semiempirical calculations. For ab initio calculations, however, it may be preferable to use multi-reference second-order perturbation theory, SAC-CI, or the equation-of-motion CC approach. 

We turn, in this section, to the multi-reference Brillouin-Wigner perturbation theory. We divide our discussion into two parts. In Section 4.4.2.1, we survey the basic theoretical apparatus of multi-reference second-order Brillouin-Wigner perturbation theory. In Section 4.4.3, we describe an a posteriori correction to multi-reference Brillouin-Wigner perturbation theory. 

Multi-reference second-order Brillouin-Wigner perturbation theory... 

U.S. Mahapatra, S. Chattopadhyay, R.K. Chaudhuri, Second-order state-specific multi-reference Mller-Plesset perturbation theory Application to energy surfaces of diimide, ethylene, butadiene and cyclobutadiene, J. Comput. Chem. 32 (2011) 325. 

We continue this section by pointing out that calculations exist that combine (PT) and MC methods. For example, methods such as complete active space, second order perturbation theory (CASPT2) and multi-configuration quasidegenerate perturbation theory MCODPT use a MC wave function as the reference. PT, generally to second order, is used to estimate the contribution from excited states that arise from excitation outside the original active space of the MC calculation. 

Our group has already studied all the fluoro-, chloro-and bromo-carbenes [11-16]. We used the complete active space self-consistent field (CASSCF) method as well as complete active space second-order perturbation theory (CASPT2) and multi-reference configuration interaction (MRCI) approaches to compute the geometries, force constants, and vibrational frequencies of the (singlet) X and A states as well as the (triplet) a states. Our theoretical studies of most of these carbenes were carried out specifically to complement LIF studies that were pursued in our laboratories by Kable et al. [6]. In addition to the determination of spectroscopic constants, the spectroscopic and theoretical studies considered dynamics on the A surfaces, i.e. whether photodissociation or internal conversion to the ground state would occur. 

Abstract The purpose of this paper is to introduce a second-order perturbation theory derived from the mathematical framework of the quasiparticle-based multi-reference coupled-cluster approach (Rolik and Kallay in J Chem Phys 141 134112, 2014). The quasiparticles are introduced via a unitary transformation which allows us to represent a complete active space reference function and other elements of an orthonormal multi-reference basis in a determinant-like form. The quasiparticle creation and annihilation operators satisfy the fermion anti-commutation relations. As the consequence of the many-particle nature of the applied unitary transformation these quasiparticles are also many-particle objects, and the Hamilton operator in the quasiparticle basis contains higher than two-body terms. The definition of the new theory strictly follows the form of the single-reference many-body perturbation theory and retains several of its beneficial properties like the extensivity. The efficient implementation of the method is briefly discussed, and test results are also presented. 

Additionally, one needs to remember that for a powerful tool such as pertnrbation theory, there is no obstacle to applying the multi-determinant reference function as the unperturbed function in perturbation calculus. Thus, similar to the SCF-HF and MP2 approaches, CASPT2 would be the second-order perturbation theory complete active space method - the perturbationaUy corrected CASSCF. 

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