(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.
S_{1} | S_{2a} | S_{2b} | S_{3x} | S_{3y} | S_{3z} | S_{4x} | S_{4y} | S_{4z} | |
---|---|---|---|---|---|---|---|---|---|
T_{d} | A_{1} | E_{a} | E_{b} | F_{2x} | F_{2y} | F_{2z} | F_{2x} | F_{2y} | F_{2z} |
D_{2d} | A_{1} | A_{1} | B_{1} | E_{x} | E_{y} | B_{2} | E_{x} | E_{y} | B_{2} |
The symmetry species of the lowest vibrational state of the ground electronic
state of methane is, of course, A_{1}. 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
[Γ^{υ}]
[27]. They can be obtained from Table X−l3 of
Ref. [4] (after correcting a small error in the
results for even υ), and are
reproduced here in Table 9. The symmetry species
Γ(υ_{1}, υ_{2}, υ_{3}, υ_{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_{1}^{υ1}] ≡ A_{1}.
Species corresponding to the symmetrized products [F_{1}^{υ}] can be obtained from the second half of this table by exchanging the subscripts 1 and 2 when υ is odd.
υ | [E^{υ}] | |
---|---|---|
6p | p(A_{1} + A_{2} + 2E) + A_{1} | |
6p + 1 | p(A_{1} + A_{2} + 2E) + E | |
6p + 2 | p(A_{1} + A_{2} + 2E) + A_{1} + E | |
6p + 3 | p(A_{1} + A_{2} + 2E) + A_{1} + A_{2} + E | |
6p + 4 | p(A_{1} + A_{2} + 2E) + A_{1} + 2E | |
6p + 5 | p(A_{1} + A_{2} + 2E) + A_{1} + A_{2} + 2E |
υ | [F_{2}^{υ}] |
---|---|
12p | (3p^{2} + p)_{ } Γ + pΓ_{1} + A_{1}^{ } |
12p + 1 | (3p^{2} + p)_{ } Γ − pΓ_{2} + F_{2}^{ } |
12p + 2 | (3p^{2} + 2p)_{ } Γ + pΓ_{1} + A_{1} + E + F_{2}^{ } |
12p + 3 | (3p^{2} + 2p)_{ } Γ − pΓ_{2} + A_{1} + F_{1} + 2F_{2} |
12p + 4 | (3p^{2} + 3p)_{ } Γ + pΓ_{1} + 2A_{1} + 2E + F_{1} + 2F_{2}^{ } |
12p + 5 | (3p^{2} + 3p)_{ } Γ − pΓ_{2} + A_{1} + E + 2F_{1} + 4F_{2}^{ } |
12p + 6 | (3p^{2} + 4p)_{ } Γ + pΓ_{1} + 3A_{1} + A_{2} + 3E + 2F_{1} + 4F_{2}^{ } |
12p + 7 | (3p^{2} + 4p)_{ } Γ − pΓ_{2} + 2A_{1} + 2E + 4F_{1} + 6F_{2}^{ } |
12p + 8 | (3p^{2} + 5p)_{ } Γ + pΓ_{1} + 4A_{1} + A_{2} + 5E + 4F_{1} + 6F_{2}^{ } |
12p + 9 | (3p^{2} + 5p)_{ } Γ − pΓ_{2} + 3A_{1} + A_{2} + 3E + 6F_{1} + 9F_{2}^{ } |
12p + 10 | (3p^{2} + 6p)_{ } Γ + pΓ_{1} + 5A_{1} + 2A_{2} + 7E + 6F_{1} + 9F_{2}^{ } |
12p + 11^{ } | (3p^{2} + 6p)_{ } Γ −
pΓ_{2} + 4A_{1} +
A_{2} + 5E + 9F_{1} +
12F_{2}^{ } Γ = A_{1} + A_{2} + 2E + 3F_{1} + 3F_{2}^{ } Γ_{1} = 2A_{1} − A_{2} + E − 3F_{1}^{ } Γ_{2} = 2A_{2} − A_{1} + E − 3F_{2}^{ } |
A_{1} | A_{2} | E | F_{1} | F_{2} | |
---|---|---|---|---|---|
A_{1} | A_{1} | A_{2} | E | F_{1} | F_{2} |
A_{2} | A_{2} | A_{1} | E | F_{2} | F_{1} |
E | E | E | A_{1} + {A_{2}} + E | F_{1} + F_{2} | F_{1} + F_{2} |
F_{1} | F_{1} | F_{2} | F_{1} + F_{2} | A_{1} + E + {F_{1}} + F_{2} | A_{2} + E + F_{1} + F_{2} |
F_{2} | F_{2} | F_{1} | F_{1} + F_{2} | A_{2} + E + F_{1} + F_{2} | A_{1} + E + {F_{1}} + F_{2} |