Performing shell model calculations for heavy, deformed nuclei has been a challenging problem in nuclear physics. The projected shell model (PSM) idea makes it possible to bridge two traditional nuclear physics methods: the deformed mean-field and the traditional shell model. By using the angular-momentum-projection technique, in which one starts with (in principle, any) deformed single-particle states to construct a shell model basis, one can demonstrate the simplicity and efficiency of this method for the description of heavy, deformed nuclei.

The original version of the PSM assumes restricted types of quasi-particle configurations in its model space, which severely limits the application for high-spin states and/or high-excitations. The problem lies in the fact that in applying the generalized Wick's theorem to calculate rotational matrix elements, the number of the terms becomes so large that it is very difficult to write down expressions explicitly for more than 4-qp states. Recently, the Pfaffian formulae have been proposed to calculate the rotated matrix element. The proposal was largely inspired by the initial introduction of Pfaffian by Robledo. The configuration space of the PSM has been expanded to include up to 10-qp states for both positive and negative parities. This development enables us to study some interesting high-spin phenomena. As the first applications of the Pfaffian algorithm in spectroscopy calculation, we take 166Hf as an example and show that 6-qp states become the main configuration in the yrast band beyond spin I~34, which explains the observed third back-bending in moment of inertia. Multi-qp high-K isomers in 176Hf with different configurations are investigated as another example. We also discuss other potential applications for the structure calculation with the chaotic motion and in nuclear astrophysics when states are highly excited.

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