Describing Molecules in Motion by Quantum Many-Body Methods
For a complete quantum description of molecular systems, it is necessary to solve Schrödinger equations for both electrons and nuclei. In this chapter, focus is given to approximate methods for solving the nuclear Schrödinger equation. Similarities and di
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Describing Molecules in Motion by Quantum Many-Body Methods Ove Christiansen
Abstract For a complete quantum description of molecular systems, it is necessary to solve Schrödinger equations for both electrons and nuclei. In this chapter, focus is given to approximate methods for solving the nuclear Schrödinger equation. Similarities and dissimilarities compared to the practice employed for the electronic case will be noted. A many-body view on potential energy surfaces will be used to motivate a many-body view on the general problem of solving the nuclear Schrödinger equation. A second quantization multimode formalism will be outlined and used to formulate many-body wave functions for nuclear motion. The vibrational selfconsistent field (VSCF) method is introduced. Full vibrational configuration interaction (FVCI) is introduced as the reference, before primary attention is given to vibrational coupled cluster (VCC) theory. VCC theory is furthermore analysed from a tensor decomposition perspective and with a perspective to scaling with system size. Keywords Molecules in motion ⋅ Anharmonic molecular vibrations ⋅ Vibrational coupled cluster ⋅ Many-body methods ⋅ Potential energy surfaces ⋅ Second quantization ⋅ Tensors
9.1 Introducing Predicting the behaviour of molecules on the basis of quantum mechanics requires a quantum treatment of both the electrons and the atomic nuclei. In the BornOppenheimer approximation, we can can consider the electronic and the nuclear problem as two separate, but coupled problems. Through the years a number of computational tools have been developed for solving the electronic Schrödinger equation. Among the most successful of these methods are many-body methods such as perturbation theory and coupled cluster theory, and they are now text book methods [1, 2]. Today coupled cluster calculations are in many contexts the golden standard. O. Christiansen (✉) Department of Chemistry, Aarhus University, Aarhus, Denmark e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 M.J. Wójcik et al. (eds.), Frontiers of Quantum Chemistry, https://doi.org/10.1007/978-981-10-5651-2_9
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O. Christiansen
In this chapter, we consider many-body methods for the nuclear Schrödinger equation. For solving the nuclear Schrödinger equation the application of many-body methods has been considerable less common, and for nuclear motion variational treatments using a linear expansion in a basis is prevailing. Nevertheless, there is still key advantages of a many-body approach. In the next chapter, I will highlight some general differences between electronic structure theory and vibrational structure theory. Thereafter, I will introduce hierarchies of approximations for the Hamiltonian that can be denoted many-body like and subsequently discuss construction of wave functions in second quantization, including, in particular, vibrational self-consistent field (VSCF), vibrational configuration interaction (VCI) and vibrational coupled cluster (VCC) methods. I will also present a chapter with a tensor per
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