A New Approach to Analyse $H\otimes (2h\oplus g)$ Jahn-Teller System for $C_{60}$

It is now well known that electron (hole)-vibron coupling and hence Jahn-Teller (JT) effect is important understanding the properties of $C_{60}$ and related molecules. In this paper, we study $H\otimes (2h\oplus g)$ coupling case to find the potential energy surfaces for the positively charged $C_{60}$ molecule due to distortion. The $H\otimes (2h\oplus g)$ Jahn-Teller system is of particular importance as this will be the JT effect displayed by $C_{60}$ molecules removed with an electron. $C_{60}^{+}$ is obtained by removing one electron from fivefold degenerate Hu highest occupied molecular orbital (HOMO) and a hole in HOMO interacts with the vibrational modes of $C_{60}$ and symmetry is broken. We apply the method of symmetry breaking mechanism to obtain expressions for the potential energy surface.