Multireference Second Order Perturbation Theory

There are two popular approaches to recovering dynamical correlation through second order perturbation theory. Whilst both take a CASSCF wavefunction as their starting point, their definition of the zeroth order Hamiltonian, Ho, required for the perturbational treatment, differ. 

It can be shown that within the CASPT2 approach, the reference state directly interacts only with those states that differ from it through either single or double excitation. It is not uncommon to find that the zeroth order energy of one or more of these excited states can be similar to, or even below, the energy of the reference. Such intmder states cause the perturbational approach to fail, sometimes in dramatic fashion, and must be eliminated through either a redefinition of the active space or via the application of level-shifting techniques [30], 

We finish this chapter with some example applications from the literature. These serve to illustrate the fundamental questions in f-element chemistry that can be probed with the CASSCF method. 

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Figure 1. The shape of the potential curve for nitrogen in a correlation-consistent polarized double-zeta basis set is presented for the variational 2-RDM method as well as (a) single-reference coupled cluster, (b) multireference second-order perturbation theory (MRPT) and single-double configuration interaction (MRCl), and full configuration interaction (FCl) wavefunction methods. The symbol 2-RDM indicates that the potential curve was shifted by the difference between the 2-RDM and CCSD(T) energies at equilibrium.

Equilibrium Bond Distance and the Harmonic Frequency for N2 from the 2-RDM Method with 2-Positivity (DQG) Conditions Compared with Their Values from Coupled-Cluster Singles-Doubles with Perturbative Triples (CCD(T)), Multireference Second-Order Perturbation Theory (MRPT), Multireference Configuration Interaction with Single-Double Excitations (MRCI), and Full Configuration Interaction (FCI)". 

Levine BG, Coe JD, Martinez TJ (2008) Optimizing conical intersections without derivative coupling vectors application to multistate multireference second-order perturbation theory (MS-CASPT2). J Phys Chem B 112 405... 

The methods of choice must be adequate for manifolds of electronic states that are localized around a lanthanide ion in a solid host. The combination of a solid environment, a heavy element, and 4/, 5d, and other open-shells, demands the consideration of the effects of the solid host, the use of relativistic Hamiltonians up to spin-orbit coupling, the correct treatment of static and dynamic correlation, and handling large manifolds of quasi-degenerate excited states. We decided to use embedded-cluster wavefunction theory-based (EC-WFT) methods, with a two-component relativistic Hamiltonian to be used in two-steps, a multi-configurational variational treatment of static correlation, and a multireference second-order perturbation theory treatment of dynamic correlation. 

Several different versions of second-order perturbation theory for multireference wave functions have been implemented but the one currently in widest use is probably the CASPT2 method. This method was developed by Roos and co-workers in Lund, Sweden, and it is available in their MOLCAS package of computer programs. 

RELATIVISTIC MULTIREFERENCE PERTURBATION THEORY COMPLETE ACTIVE-SPACE SECOND-ORDER PERTURBATION THEORY (CASPT2) WITH THE FOUR-COMPONENT DIRAC HAMILTONIAN... 

CASPT2 Second order perturbation theory on top of a CASSCF wave function yp n5 Cheapest for multireference states... 

Many-body Brillouin-Wigner second-order perturbation theory using a multireference formulation an application to bond breaking in the diatomic hydrides BH and FH Molecular Physics 104, 2367 (2006)... 

K. Hirao. Relativistic Multireference Perturbation Theory Complete Active-Space Second-Order Perturbation Theory (CASPT2) With The Four-Component Dirac Hamiltonian. In Radiation Induced Molecular Phenomena in Nucleic Adds, Volume 4 of Challenges and Advances in Computational Chemistry and Physics, p. 157-177. Springer, 2008. 

MC approaches [30] involve the optimization of molecular orbitals within a restricted subspace of electronic occupations provided such active space is appropriately chosen, they allow for an accurate description of static electron correlation effects. Dynamical correlation effects can also be introduced either at the perturbation theory level [complete active space with second-order perturbation theory (CASPT2), and multireference Mpller-Plesset (MR-MP2) methods] [31] or via configuration interaction (MR-CI). 

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