On the Use of Lie Group Homomorphisms for Treating Similarity Transformations in Nonadiabatic Photochemistry

Table 9

Expressions of 𝓡_𝑘,𝑘′(𝜙,𝜃,𝜓) for Euler angles (𝑘,𝑘^′∈{1,2,3,4,5}≡𝑚,𝑚^′∈{1,3,4,6,8}).

	𝑘=1	cos2𝜙cos𝜃cos2𝜓−sin2𝜙3+cos2𝜃4sin2𝜓	𝑘=2	−sin2𝜙cos𝜃cos2𝜓−cos2𝜙3+cos2𝜃4sin2𝜓
𝑘^′=1	𝑘=3	cos𝜙sin𝜃cos2𝜓−sin𝜙sin2𝜃2sin2𝜓	𝑘=4	sin𝜙sin𝜃cos2𝜓+cos𝜙sin2𝜃2sin2𝜓
	𝑘=5	−√3(1−cos2𝜃)4sin2𝜓
	𝑘=1	cos2𝜙cos𝜃sin2𝜓+sin2𝜙3+cos2𝜃4cos2𝜓	𝑘=2	−sin2𝜙cos𝜃sin2𝜓+cos2𝜙3+cos2𝜃4cos2𝜓
𝑘^′=2	𝑘=3	cos𝜙sin𝜃sin2𝜓+sin𝜙sin2𝜃2cos2𝜓	𝑘=4	sin𝜙sin𝜃sin2𝜓−cos𝜙sin2𝜃2cos2𝜓
	𝑘=5	√3(1−cos2𝜃)4cos2𝜓
	𝑘=1	−cos2𝜙sin𝜃cos𝜓+sin2𝜙sin2𝜃2sin𝜓	𝑘=2	sin2𝜙sin𝜃cos𝜓+cos2𝜙sin2𝜃2sin𝜓
𝑘^′=3	𝑘=3	cos𝜙cos𝜃cos𝜓−sin𝜙cos2𝜃sin𝜓	𝑘=4	sin𝜙cos𝜃cos𝜓+cos𝜙cos2𝜃sin𝜓
	𝑘=5	−√3sin2𝜃2sin𝜓
	𝑘=1	cos2𝜙sin𝜃sin𝜓+sin2𝜙sin2𝜃2cos𝜓	𝑘=2	−sin2𝜙sin𝜃sin𝜓+cos2𝜙sin2𝜃2cos𝜓
𝑘^′=4	𝑘=3	−cos𝜙cos𝜃sin𝜓−sin𝜙cos2𝜃cos𝜓	𝑘=4	−sin𝜙cos𝜃sin𝜓+cos𝜙cos2𝜃cos𝜓
	𝑘=5	−√3sin2𝜃2cos𝜓
	𝑘=1	√sin2𝜙3(1−cos2𝜃)4	𝑘=2	√cos2𝜙3(1−cos2𝜃)4
𝑘^′=5	𝑘=3	√−sin𝜙3sin2𝜃2	𝑘=4	√cos𝜙3sin2𝜃2
	𝑘=5	1+3cos2𝜃4