A generalised methodology using conformal geometric algebra for mathematical chemistry. (English) Zbl 1448.92395
Summary: This paper presents a novel and generalised mathematical formulation for mathematical and theoretical chemistry. The formulation presented employs network theoretic methods and standard conformal geometric algebra (conformal Clifford algebra) with a unique algebraic extension that introduces special non-integer basis vectors, known as ‘shades’. The methodology developed lays preliminary foundations for both generic and more specialised characterisations and modelling of chemical systems by computing a multi-vector function, termed the ‘hyperfield’ (an algebra-geometric characterisation of a many-agent chemical system). In this paper, the formulation has been applied to inorganic compounds, in the context of their emergent properties (such as boiling and melting points) and intra-molecular characteristics. Case studies have been presented for the calculation of melting point, boiling point, maximum solubility in water and equilibrium bond length of 21 diatomic molecules. This work finds that a network theoretic model of many-agent systems modelled in \(\mathbf{R}^{4,1}\) space, using an algebraic extension of non-integer basis vectors, presents a promising and unifying method for theoretical mathematical chemistry.
MSC:
| 92E10 | Molecular structure (graph-theoretic methods, methods of differential topology, etc.) |
| 15A67 | Applications of Clifford algebras to physics, etc. |
Keywords:
mathematical chemistry; conformal geometric algebra; Clifford algebra; theoretical chemistry; many agent systemsReferences:
| [1] | Paldus, J.; Sarma, CR, J, Chem. Phys., 83, 5135-5152 (1985) |
| [2] | Paldus, J.; Jeziorski, B., Theor. Chim. Acta., 73, 81-103 (1988) · doi:10.1007/BF00528196 |
| [3] | Paldus, J.; Rettrup, S.; Sarma, CR, J. Mol. Struct. Theochem, 199, 85-101 (1989) · doi:10.1016/0166-1280(89)80044-2 |
| [4] | Li, X.; Paldus, J., Int. J. Quantum Chem., 41, 117-146 (1992) · doi:10.1002/qua.560410112 |
| [5] | Havel, TF, Encycl, Comput. Chem., 120, 723-742 (1998) |
| [6] | Chys, P., J. Chem. Phys., 128, 104-107 (2008) · doi:10.1063/1.2831776 |
| [7] | Lavor, C.; Alves, R.; Figueiredo, W.; Petraglia, A.; Maculan, N., Adv. Appl. Clifford Algebras, 25, 925-942 (2015) · Zbl 1327.15052 · doi:10.1007/s00006-015-0532-2 |
| [8] | Alves, R.; Lavor, C., Adv. Appl. Clifford Algebras, 27, 439-452 (2017) · Zbl 1373.92148 · doi:10.1007/s00006-016-0653-2 |
| [9] | Thiruvengadam, S.; Murphy, ME; Tan, JS; Watling, RJ; Stewart, JI; Miller, K., Math. Geosci. (2020) · Zbl 1451.86014 · doi:10.1007/s11004-020-09856-3 |
| [10] | Thiruvengadam, S.; Tan, JS; Miller, K., Neurocomputing (2020) · doi:10.1016/j.neucom.2019.12.116 |
| [11] | Thiruvengadam, S.; Tan, JS; Miller, K., Adv. Appl. Clifford Algebras, 30, 37 (2020) · Zbl 1444.37064 · doi:10.1007/s00006-020-01061-z |
| [12] | Thiruvengadam, S.; Murphy, ME; Tan, JS; Miller, K., J. Surfactants Deterg. (2019) · doi:10.1002/jsde.12360 |
| [13] | Lounesto, P.; Wene, GP, Acta Appl. Math., 9, 165-173 (1987) · Zbl 0658.15030 · doi:10.1007/BF00047537 |
| [14] | Newman, M., Networks (2010), Oxford: Oxford University Press, Oxford · Zbl 1195.94003 |
| [15] | Doran, C.; Lasenby, A., Geometric Algebra for Physicists (2003), Cambridge: Cambridge University Press, Cambridge · Zbl 1078.53001 |
| [16] | Hestenes, D.; Sobczyk, G., Clifford Algebra to Geometric Calculus (1984), Dordrecht: D. Reidel, Dordrecht · Zbl 0541.53059 |
| [17] | C. Doran, A. Lasenby, J. Lasenby, Conformal geometry, Euclidean space and Geometric Algebra, in Uncertainty in Geometric Computations (2002), pp. 41-58. · Zbl 1291.70011 |
| [18] | D. Hildenbrand, Conformal Geometric Algebra, in Foundations of Geometric Algebra Computing (2013), pp. 27-44. · Zbl 1268.65038 |
| [19] | E. Bayro-Corrochano, Geometric Algebra Tensor Voting, Hough Transform, Voting and Perception Using Conformal Geometric Algebra, vol. 1, in Geometric Algebra Applications (2018), pp. 559-611. |
| [20] | Anton, H.; Rorres, C., Elementary Linear Algebra: Applications Version (2013), Hoboken, NJ: Wiley, Hoboken, NJ |
| [21] | Hitzer, E., J. Control Meas. Syst. Integr., 4, 1-11 (2013) |
| [22] | Allen, LC, J. Am. Chem. Soc., 111, 9003-9014 (1989) · doi:10.1021/ja00207a003 |
| [23] | Rahm, M.; Hoffmann, R.; Ashcroft, NW, Chem. A Eur. J., 22, 14625-14632 (2016) · doi:10.1002/chem.201602949 |
| [24] | J.R. Rumble, D.R. Lide, T.J. Bruno, CRC Handbook of Chemistry and Physics (2017). |
| [25] | K.P. Huber, G. Herzberg, Molecular Structure and Molecular Spectra. IV. Constants of Diatomic Molecules. (New York: Van Rostrand-Reinhold, 1979). |
| [26] | Anwar, A.; Frenkel, D.; Noro, M., J. Chem. Phys., 118, 728-735 (2003) · doi:10.1063/1.1522375 |
| [27] | Aragones, J.; Sanz, E.; Valeriani, C.; Vega, C., J. Chem. Phys., 137, 104507 (2012) · doi:10.1063/1.4745205 |
| [28] | Krzyzaniak, J.; Myrdal, P.; Simamora, P.; Yakowsky, S., Ind. Eng. Chem. Res., 34, 2530-2535 (1995) · doi:10.1021/ie00046a039 |
| [29] | Jain, A.; Yang, G.; Yalkowsky, S., Ind. Eng. Chem. Res., 43, 7618-7621 (2004) · doi:10.1021/ie049378m |
| [30] | Zhang, Y.; Maginna, E., J. Chem. Phys., 136, 144116 (2012) · doi:10.1063/1.3702587 |
| [31] | Mester, Z.; Panagiotopoulos, AZ, J. Chem. Phys., 143, 044505 (2015) · doi:10.1063/1.4926840 |
| [32] | Sanz, E.; Vega, C., J. Chem. Phys., 126, 014507 (2007) · doi:10.1063/1.2397683 |
| [33] | Paluch, AS; Jayaraman, S.; Shah, JK; Maginn, EJ, J. Chem. Phys., 133, 124504 (2010) · doi:10.1063/1.3478539 |
| [34] | Moucka, F.; Lísal, M.; Škvor, J.; Jirsák, J.; Nezbeda, I.; Smith, WR, J. Phys. Chem. B, 115, 7849-7861 (2011) · doi:10.1021/jp202054d |
| [35] | Pyykkö, P.; Atsumi, M., Chem. A Eur. J., 15, 186-197 (2009) · doi:10.1002/chem.200800987 |
| [36] | Segal, GA, J. Chem. Phys., 47, 1876-1877 (1967) · doi:10.1063/1.1712197 |
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