#### Modeling of high energy quantum states and global computations of molecular spectra

Modeling of high energy quantum states of molecules and global calculations of their absorption/emission spectra has recently become a challenge for applied mathematics. This required processing of huge amount of data and accurate intensive parallel computations that are of major importance for various domains: dynamics of molecular formation and dissociation, radiation transfer and greenhouse effects in the terrestrial and planetary atmospheres, interpretation of spectral signatures in astrophysical observations of exosolar planets and modeling of stars evolution.

A breakthrough in this domain was possible due to a progress in global theoretical predictions of spectral transitions and Einstein probability coefficients among all high-energy quantum states. In order to converge the opacity of hydrocarbons, which are considered as possible organic-matter- markers for various astrophysical objects, we need to compute and to include in the models hundreds of billions of lines.

A necessary step is a construction of multi-dimensional potential energy hyper-surfaces for the nuclear motion using ab initio electronic structure methods. Accurate variational prediction of ro-vibrational spectra in a “brut-force” direct product approach implies a diagonalization of the Hamiltonian matrix of the dimension where is the number of basis functions for each of N degrees of freedom. To make calculations efficient, this required a development of appropriate truncation-compression techniques, careful checking of the basis set convergence and a full account of symmetry using irreducible tensors.

On the mathematical side this is linked to stability of inverse problems (regularization methods), non-linear dynamics involving Kolmogorov-Arnold-Moser and bifurcation theory in the phase spaces of the nuclear motion, Hamiltonian Contact transformations / Poincaré-Birkhoff normal forms, optimization of eigenvalue-eigenvector solvers , convergence of variational calculations for very high dimensionality.

A modeling and development of efficient computational method in this field of science would benefit from complementary expertises and collaborations in quantum chemistry, astrophysics, mathematics and “big-data” simulations.

*This work was done in collaboration with D. Lapierre, V. Kokoouline, A. Alijah, M. Rey, A. Nikitin, S. Tashkun, R. Kochanov and E. Starikova.*