Solving the Schrödinger equation using method using a multidimensional grid such as DVR, FBR-DFR or pseudo-spectral approaches [1] is limited by the number of degrees of freedom, d, of the system. With these methods using direct-product, the numerical complexity grows exponentially with d and it is relatively easily to study four (d=6) or five (d=9) atomic systems. This limit can be pushed away using contraction techniques [2-4], pruned basis sets [5] or schemes with a basis function selection. [6]

Somehow, the Smolyak scheme [7-9] can be viewed as an approach with a selection of basis functions and a sparse grid adapted to that basis set. The parameter, L, controls the selection and thus the size of the basis set and the grid. With this scheme, the numerical complexity grows as a polynomial of degree L with d. Therefore, systems up to 12 degrees of freedom can be studied, [10-12] although larger systems (d=21) are possible if few states are needed.[13] 

This limit can be pushed away using a system-bath separation, and in our Smolyak scheme, three Lparameters are required: [14] (i) Ls for the degrees of freedom associated with the system part. (ii) Lb associated with the bath mode. (iii) Lm a parameter controlling the coupling between the system and the bath. 

This system-bath separation has no limitation with respect to the form of the Hamiltonian and it works well when the coupling between the system and the bath is weak and typically: Ls >> Lb and Lm > Ls (Lm=Ls+1 or Ls+2 can be used). 

To illustrate the advantages and the limitations of this approach, time-independent and time-dependent applications are presented:

 - The effect of the rotation-translation motions (bath modes, up to 120) of the water shell on the rotation-translation H2 motions (system modes, 5) in clathrate hydrate. [14]

- The photoisomerization of a retinal chromophore model with 2 active modes (system) and 23 modes (bath). [15]

References:

[1] Light, J. C., Carrington Jr, T. Advances in Chemical Physics, 2000, 114, 263–310.

[2] Wang, X.-G., Carrington, T. The Journal of Chemical Physics, 2008, 129, 234102.

[3] Felker, P. M., Bacic, Z. J Chem Phys, 2022, 2021, 244111

[4] O. Vendrell, H.-D. Meyer, J Chem Phys 2011, 134, 044135.

[5] Cooper, J., Carrington, T. The Journal of Chemical Physics, 2009, 130, 214110.

[6] J. M. Bowman, S. Carter, X. Huang, Int Rev Phys Chem 2003, 22, 533–549.

[7] S. A. Smolyak, Soviet Mathematics Doklady 1963, 4, 240.

[8] G. Avila, T. Carrington, J Chem Phys 2009, 131, 174103.

[9] D. Lauvergnat, A. Nauts, Physical chemistry chemical physics 2010, 12, 8405–12.

[10] D. Lauvergnat, A. Nauts, Spectrochimica Acta Part A: 2014, 119, 18–25.

[11] G. Avila, T. Carrington, J Chem Phys 2017, 147, 064103.

[12] G. Avila, E. Mátyus, J Chem Phys 2019, 150, 174107.

[13] Lauvergnat, D., Nauts, A., ChemPhysChem, 2023, 24, 202300501.

[14] A. Chen, D. M. Benoit, Y. Scribano, A. Nauts, D. Lauvergnat, J Chem Theory Comput 2022, 18, 4366–4372.

[15] Pereira, A., Knapik, J., Chen, A. et al. Eur. Phys. J. Spec. Top. 2023, 232, 1917–1933