In this work, using the so-called self-consistent basis method, solutions to three two-dimensional Schrödinger equations are found, which are invariant under transformations of discrete groups Cnv, n=2,3,4. In the classical limit, classical systems corresponding to these Schrödinger equations allow the existence of both regular and chaotic modes of motion. Using the example of solving the C3v-symmetric Schrödinger equation, all stages of the solution by the self-consistent basis method are presented in sufficient detail and clearly. In this method, solutions to the Schrödinger equations are sought in the form of a trigonometric series in which the coefficient functions are found by exact numerical integration of the original equation and therefore these coefficient functions are consistent with the form of the original differential equation, in particular, with the potential energy surface (PES), which can be quite complex. For example, an equation was solved in which the PES has five local minima and four saddle points. An important positive feature of the self-consistent basis method is the ability to perform all its stages, both analytical and numerical, using well-known computer systems for symbolic calculations such as Maple, Mathematica, Reduce, etc. In this work, the Maple system was used, in which the corresponding programs were compiled, with the help of which all presented results were obtained. For these Schrödinger equations there were various energy levels and wave functions, for some of them three-dimensional images and isolines were constructed. We compared the energy values we calculated with the results of other authors available in the literature and obtained fairly good agreement. In one case, for the Schrödinger equation with C3v symmetry, it was found that high accuracy calculations of energy levels using the self-consistent basis method were achieved with much less computation compared to the diagonalization method. It was also discovered that to calculate energy levels from the energy region where classical motion is chaotic, a more careful adjustment of the two available parameters is required.
In this work, using the so-called self-consistent basis method, solutions to three two-dimensional Schrödinger equations are found, which are invariant under transformations of discrete groups Cnv, n=2,3,4. In the classical limit, classical systems corresponding to these Schrödinger equations allow the existence of both regular and chaotic modes of motion. Using the example of solving the C3v-symmetric Schrödinger equation, all stages of the solution by the self-consistent basis method are presented in sufficient detail and clearly. In this method, solutions to the Schrödinger equations are sought in the form of a trigonometric series in which the coefficient functions are found by exact numerical integration of the original equation and therefore these coefficient functions are consistent with the form of the original differential equation, in particular, with the potential energy surface (PES), which can be quite complex. For example, an equation was solved in which the PES has five local minima and four saddle points. An important positive feature of the self-consistent basis method is the ability to perform all its stages, both analytical and numerical, using well-known computer systems for symbolic calculations such as Maple, Mathematica, Reduce, etc. In this work, the Maple system was used, in which the corresponding programs were compiled, with the help of which all presented results were obtained. For these Schrödinger equations there were various energy levels and wave functions, for some of them three-dimensional images and isolines were constructed. We compared the energy values we calculated with the results of other authors available in the literature and obtained fairly good agreement. In one case, for the Schrödinger equation with C3v symmetry, it was found that high accuracy calculations of energy levels using the self-consistent basis method were achieved with much less computation compared to the diagonalization method. It was also discovered that to calculate energy levels from the energy region where classical motion is chaotic, a more careful adjustment of the two available parameters is required.