Methane Symmetry Operations

6.   Symmetry Properties of Vibrational Functions

(eq. 20)

where δr_i indicates a change in the C−H_i bond length, and δα_ij indicates a change in the H_i−C−H_j angle. Unfortunately, the positions of signs and subscripts in (eq. 20) depend on how numbered hydrogens are associated with the corners of the cube in Fig. 1. Equations (20) are thus written differently in a discussion of the methane force field [26].

While an intuitive application of the permutation-inversion operations on the right side of (eq. 20) will lead to correct results, it is perhaps safer to recall that transformation properties of the displacement vectors d_i have already been fixed, so that transformation properties of δr_i and δα_ij should be determined by noting that they are expressed in terms of the displacement vectors d_i , i = 1, 2, 3, 4, and d_C by the equations

(eq. 21)

where the a_i represent vectors from the equilibrium C position to the equilibrium H_i positions and a = | a_i | . The a_i do not change under the symmetry operations. Application of the substitutions indicated in (eq. 12) and (eq. 16) and use of the relationships in (eq. 13) and (eq. 17) between the equilibrium positions leads immediately to the vibrational symmetry species indicated in Table 8.

The symmetry species of the lowest vibrational state of the ground electronic state of methane is, of course, A₁. The symmetry species of vibrational levels obtained by exciting one quantum of a given vibration are the same as the species of the corresponding vibrational coordinate. The symmetry species of levels obtained by multiple excitation of a single vibration are given by the symmetrized products [] [27]. They can be obtained from Table X−l3 of Ref. [4] (after correcting a small error in the results for even ), and are reproduced here in Table 9. The symmetry species (₁, ₂, ₃, ₄) of levels obtained by multiply exciting several vibrations can be obtained by first determining the species for multiple excitation of each individual vibration, and then determining all possible direct products, as given in Table 10, between those sets of species [4]. Symbolically,

(eq. 22)

where a slight simplification is achieved by noting that [A_1¹] ≡ A₁.

Table 9. Symmetry species [] for multiple excitation of a single degenerate vibration in CH₄

Species corresponding to the symmetrized products [F₁] can be obtained from the second half of this table by exchanging the subscripts 1 and 2 when is odd.

		[E]
	6p	p(A1 + A2 + 2E) + A1
	6p + 1	p(A1 + A2 + 2E) + E
	6p + 2	p(A1 + A2 + 2E) + A1 + E
	6p + 3	p(A1 + A2 + 2E) + A1 + A2 + E
	6p + 4	p(A1 + A2 + 2E) + A1 + 2E
	6p + 5	p(A1 + A2 + 2E) + A1 + A2 + 2E

	[F2]
12p	(3p2 + p)  Γ + pΓ1 + A1
12p + 1	(3p2 + p)  Γ − pΓ2 + F2
12p + 2	(3p2 + 2p)  Γ + pΓ1 + A1 + E + F2
12p + 3	(3p2 + 2p)  Γ − pΓ2 + A1 + F1 + 2F2
12p + 4	(3p2 + 3p)  Γ + pΓ1 + 2A1 + 2E + F1 + 2F2
12p + 5	(3p2 + 3p)  Γ − pΓ2 + A1 + E + 2F1 + 4F2
12p + 6	(3p2 + 4p)  Γ + pΓ1 + 3A1 + A2 + 3E + 2F1 + 4F2
12p + 7	(3p2 + 4p)  Γ − pΓ2 + 2A1 + 2E + 4F1 + 6F2
12p + 8	(3p2 + 5p)  Γ + pΓ1 + 4A1 + A2 + 5E + 4F1 + 6F2
12p + 9	(3p2 + 5p)  Γ − pΓ2 + 3A1 + A2 + 3E + 6F1 + 9F2
12p + 10	(3p2 + 6p)  Γ + pΓ1 + 5A1 + 2A2 + 7E + 6F1 + 9F2
12p + 11	(3p2 + 6p)  Γ − pΓ2 + 4A1 + A2 + 5E + 9F1 + 12F2  Γ = A1 + A2 + 2E + 3F1 + 3F2  Γ1 = 2A1 − A2 + E − 3F1  Γ2 = 2A2 − A1 + E − 3F2

Table 10. Direct product table for the point group Td

Antisymmetrized products are enclosed in curly brackets { }

	A1	A2	E	F1	F2
A1	A1	A2	E	F1	F2
A2	A2	A1	E	F2	F1
E	E	E	A1 + {A2} + E	F1 + F2	F1 + F2
F1	F1	F2	F1 + F2	A1 + E + {F1} + F2	A2 + E + F1 + F2
F2	F2	F1	F1 + F2	A2 + E + F1 + F2	A1 + E + {F1} + F2