Abstract
The hyperspherical method is a widely used and successful approach for the quantum treatment of elementary chemical processes. It has been mostly applied to three-atomic systems, and current progress is here outlined concerning the basic theoretical framework for the extension to four-body bound state and reactive scattering problems. Although most applications only exploit the advantages of the hyperspherical coordinate systems for the formulation of the few-body problem, the full power of the technique implies representations explicitly involving quantum hyperangular momentum operators as dynamical quantities and hyperspherical harmonics as basis functions. In terms of discrete analogues of these harmonics one has a universal representation for the kinetic energy and a diagonal representation for the potential (hyperquantization algorithm). Very recently, advances have been made on the use of the approach in classical dynamics, provided that a hyperspherical formulation is given based on “classical” definitions of the hyperangular momenta and related quantities. The aim of the present paper is to offer a retrospective and prospective view of the hyperspherical methods both in quantum and classical dynamics. Specifically, regarding the general quantum hyperspherical approaches for three- and four-body systems, we first focus on the basis set issue, and then we present developments on the classical formulation that has led to applications involving the implementations of hyperspherical techniques for classical molecular dynamics simulations of simple nanoaggregates.
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References
Kuppermann A (1996). J Phys Chem 100:2621–2636; 100:11202 (Erratum)
Aquilanti V, Cavalli S, De Fazio D (1998). J Chem Phys 109:3792–3804
Aquilanti V, Cavalli S, De Fazio D, Volpi A (2001). Int J Quantum Chem 85:368–381
Wang DS, Kuppermann A (2003). J Phys Chem A 107:7290–7310
Aquilanti V, Cavalli S (1997). J Chem Soc Faraday Trans 93:801–809
Kuppermann A (1997). J Phys Chem A 101:6368–6383
Littlejohn RG, Mitchell KA, Aquilanti V (1999). Phys Chem Chem Phys 1:1259–1264
Wang DS, Kuppermann A (2001). J Chem Phys 115:9184–9208
Aquilanti V, Beddoni A, Lombardi A, Littlejohn RG (2002). Int J Quantum Chem 89:277–291
Aquilanti V, Lombardi A, Littlejohn RG (2004). Theor Chem Acc 111:400–406
Kuppermann A (2004). J Phys Chem A 108:8894–8904
Smith FT (1960). Phys Rev 120:1058–1069
Smith FT (1962). J Math Phys 3:735–748
Whitten RC, Smith FT (1968). J Math Phys 9:1103–1113
Aquilanti V, Cavalli S (1986). J Chem Phys 85:1355–1361
Aquilanti V, Cavalli S, Grossi G (1986). J Chem Phys 85:1362–1375
Aquilanti V, Tonzani S (2004). J Chem Phys 120:4066–4073
Ragni M, Bitencourt ACP, Aquilanti V (2007). Prog Theor Chem Phys 16:133–158
Pogrebnya SK, Echave J, Clary DC (1997). J Chem Phys 107:8975–8984
Skouteris D, Castillo JF, Manolopoulos DE (2000). Comp Phys Commun 133:128–135
Littlejohn RG, Mitchell KA, Aquilanti V, Cavalli S (1998). Phys Rev A 58:3705–3717
Littlejohn RG, Mitchell KA, Reinsch M, Aquilanti V, Cavalli S (1998). Phys Rev A 58:3718–3738
Aquilanti V, Beddoni A, Cavalli S, Lombardi A, Littlejohn RG (2000). Mol Phys 98:1763–1770
Aquilanti V, Cavalli S, De Fazio D, Volpi A, Aguilar A, Giménez X, Lucas JM (2003). Chem Phys Lett 371:504–509
Aquilanti V, Cavalli S, Simoni A, Aguilar A, Lucas JM, De Fazio D (2004). J Chem Phys 121:11675–11690
Aquilanti V, Cavalli S, De Fazio D, Volpi A, Aguilar A, Lucas JM (2005). Chem Phys 308:237–253
Aquilanti V, Cavalli S, De Fazio D, Simoni A, Tscherbul TV (2005). J Chem Phys 123:054314-15
Zickendraht W (1969). J Math Phys 10:30–37
Zickendraht W (1971). J Math Phys 12:1663–1674
Kuppermann A (2006). J Phys Chem A 110:809–816
Aquilanti V, Lombardi A, Yurtsever E (2002). Phys Chem Chem Phys 4:5040–5051
Horn RA, Johnson CR (1990). Matrix analysis, 2nd edn. Cambridge University Press, Cambridge
Aquilanti V, Lombardi A, Sevryuk MB (2004). J Chem Phys 121:5579–5589
Sevryuk MB, Lombardi A, Aquilanti V (2005). Phys Rev A 72:033201-28
Aquilanti V, Lombardi A, Sevryuk MB, Yurtsever E (2004). Phys Rev Lett 93:113402-4
Aquilanti V, Carmona Novillo E, Garcia E, Lombardi A, Sevryuk MB, Yurtsever E (2006). Comput Mater Sci 35: 187–191
Lombardi A, Aquilanti V, Yurtsever E, Sevryuk MB (2006). Chem Phys Lett 430:424–428
Calvo F, Gadéa FX, Lombardi A, Aquilanti V (2006). J Chem Phys 125:114307–114313
Grossi G, Peroncelli L, Rahman N (1999). Chem Phys Lett 313:639–646
Capecchi G, De Fazio D, Grossi G, Peroncelli L, Rahman N (2001). Mol Phys 99:443–453
Peroncelli L, Grossi G, Aquilanti V (2004). Mol Phys 102:2345–2359
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Lombardi, A., Palazzetti, F., Peroncelli, L. et al. Few-body quantum and many-body classical hyperspherical approaches to reactions and to cluster dynamics. Theor Chem Account 117, 709–721 (2007). https://doi.org/10.1007/s00214-006-0195-0
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DOI: https://doi.org/10.1007/s00214-006-0195-0