Lie algebra has been proved to be one of the most important mathematical tool to understand many natural phenomena in different length scale of physics. The primary objective of this essay is to give a brief overview of some original works on Lie algebra to understand the chronological development of this subject, its application in the field of molecular spectroscopy and give a brief outline of the work carried out by the author and his group.
Posted by: Nirmal Kumar Sarkar, Department of Physics, Karimganj College, Karimganj 788710, INDIA
Email: nks_sds@rediffmail.com
In recent years, molecular spectroscopy is undergoing through an exciting change. In terms of quantitative change, better initial state preparation, improved light sources and specially designed pumping schemes [1-2] and more sensitive detection techniques [3] are providing ever improved resolution and a wider range of accessible final states. Regarding qualitative change, a closer view suggests that not only better results but also new ideas are forthcoming.
To have a proper description and understanding of modern spectroscopy, one needs a theoretical framework, which can discuss both level structure beyond the harmonic limit and the corresponding dynamics. Hence, a Hamiltonian is unavoidable here to reach the goal since it is the generator of time evolution. Yet, one needs a practical method for the determination of the eigenvalues of this Hamiltonian. In the traditional Dunham-like expansion [4], it will be useful if the spectra can be well approximated by a small number of constants. However, one needs to be able to relate the parameters in this expansion directly to a Hamiltonian. The traditional familiar way of doing this proceed in two steps is: first, the electronic problem is solved in the Born-Oppenheimer approximation, leading to the potential for the motion of the nuclei. Then, the Schrodinger equation for the eigenvalues of this potential is solved. Since for any, but diatomic molecules, the potential is a function of many coordinates, neither the first nor the second step is simple to implement. For a number of test cases, this procedure has been carried out; and for diatomic molecules of lower row atoms, it can challenge experiments in its precision. For larger molecules, it is still not practical to compute the required potential with sufficient accuracy. It is, therefore, often approximated using convenient functional forms. Not too far from a deep equilibrium point, the potential can be expanded in the displacement coordinates relative to the equilibrium configuration. Such a ‘force field’ representation is quite convenient but is of limited validity for higher lying states due to the slow convergence of such a power series expansion. More flexible functional forms that can describe the asymptotic dissociation plateaus [5] require many parameters. The purely numerical solution of the Schrodinger differential equation for the eigenvalues of such a potential makes a large-scale numerical problem complicated.
Marius Sophus Lie (December 17, 1842 - February 18, 1899)
Since the last part of the 20th century, an algebraic approach [6-7] has been used in the study of the molecular spectra. The algebraic approach attracted a wider scientific community in recent years for the analysis and interpretation of experimental rovibrational spectra of small and medium sized molecules. This approach is based on the idea of dynamical symmetry and is expressed through the language of Lie algebras. This approach can account for any specific mechanism relevant for the correct characterization of the molecular dynamics and spectroscopy. Applying algebraic techniques, in this approach, one obtains an effective Hamiltonian operator that conveniently describes the rovibrational degrees of freedom of the physical system. The algebraic approach is especially useful when the potential energy surface is unknown or when it is too difficult to calculate the spectrum by starting with the potential surface using the traditional approaches [8]. For example, in a large molecule (larger than a diatomic), the potential surface is a very complex function, composed of a discouragingly large number of coordinates. Under such circumstances, the traditional approaches face a tremendous difficulty in approximating the molecule as soon as we consider the highly excited levels. Further, a large number of parameters are also required here to obtain meaningful results. On the other hand, the algebraic approach needs a much more economical set of parameters to provide the fits of the spectra [9,10]. This is a very important practical advantage of the algebraic approach over the traditional approaches. In algebraic approach, there are general forms of algebraic Hamiltonian and by using a common Hamiltonian entire classes of molecules can be approximated by changing only the parameters (typically, linear) for different molecules. This is another remarkable advantage of the algebraic approach over the traditional approaches. Finally, we note the comparative ease of the algebraic operations used in the algebraic approach – a technical advantage of the algebraic approach over the traditional approaches.
The matrix formulation of quantum mechanics was introduced as early as 1925 [11-14]. But till today, the algebraic (or matrix) formulation of quantum mechanics is less familiar than the differential (or wave) formulation. As per the view point, this is a disadvantage, and one purpose of this article is to indicate, by explicit examples, the benefits of the algebraic approach. The interested readers may judge whether the benefits are sufficient enough to overcome the barrier to the understanding of a new approach. Through this literature, the author intend to demonstrate that the algebraic formulation is indeed a viable and better alternative in describing various complex molecular systems.
As told earlier, the algebras used in the newly introduced algebraic approach are Lie algebras. These algebras were introduced at the end of the nineteenth century by Marius Sophus Lie but it is only in the first part of the 20th century that its systematic large scale application could be seen in physics. It was in the 1930s that Lie algebras were being used in physics [15-20]. Most of the early applications dealt with the algebra of rotations (Wigner-Racah algebra). An approach that starts from the algebra as the key tool for the construction of spectra originated in elementary particle physics in the 1960s [21-22] and had major applications in nuclear and particle physics [23-25]. Later, since 1979, the applications of Lie algebras have been extended to the study of molecular spectra and some other related fields in physics by Levine [6], Iachello [7] and other researchers [10, 26-42].
In the study of polyatomic molecules, both the U(2) and U(4) algebraic models [28, 43-44] of the new algebraic approach have been attracting a wider scientific community of the globe in recent years. The models already have been applied in the study of vibrational spectra of linear triatomic, linear tetratomic and some other small, medium and large-sized molecules [26-27, 29, 44-47]. In many aspects, the U(4) and U(2) algebraic models are in advantageous positions with respect to their traditional counterparts in the study of vibrational spectra of a molecule. However, it is to be noted that the U(2) algebraic model is one dimensional in nature and it does not take into account rotational motions of a molecule. Simply to provide a global vibrational picture of a certain molecule, the rotational degrees of freedom are completely disregarded in the U(2) algebraic model. Such a drawback is overcome with the introduction of the U(4) algebraic model. The full treatment of molecular rovibrational degrees of freedom can be achieved in the three-dimensional framework of the U(4) algebraic model. The three dimensional framework of the U(4) algebraic model leads us to a more complex and realistic picture of a molecular system as well as producing a more exacting algebraic treatment. The three-dimensional algebraic model is definitely much more difficult to manipulate than the one-dimensional one, purely for algebraic aspects. Due to this reason, the U(4) algebraic model could proceed till today only up to the approximation of linear tetratomic molecules [33, 47-48]. So far the cases of bent polyatomic molecules are concerned, successful applications of the U(4) algebraic model could be reported so far only for a few bent triatomic molecules [49-50]. Thus, for the U(4) algebraic model, more or less, the entire kingdom of large molecules in general and the bent polyatomic molecules in particular have been left unattended till today ! Obviously, here is enormous scope for the interested researchers to make a breakthrough in near future !
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