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Methane Symmetry Operations - Nuclear spin functions
9. Symmetry Properties of Laboratory-Fixed Nuclear Spin Functions, Nuclear Spin Statistics, and Parities
9.1 Symmetry properties of the nuclear spin functions
Spin functions, whether electronic or nuclear, must of necessity be handled somewhat formally, since the introduction of spin "variables" does not facilitate the mathematical representation of observed physical phenomena. We therefore often restrict our consideration to the spin functions themselves, characterizing them by a total spin angular momentum and a projection along some axis. Two particularly convenient types of spin functions can be defined: one in which spin projections are taken along the laboratory-fixed Z axis, the other in which spin projections are taken along the molecule-fixed z axis. Most frequently, nuclear spin functions are considered in which projections are taken along the laboratory-fixed Z axis, and for simplicity we shall consider only that case here. A discussion of molecule-fixed proton spin functions would require the introduction of a double group of Td [3], since the proton spin is half-integral.
Because laboratory-fixed nuclear spin functions are analogues of positional functions of laboratory-fixed coordinates of individual particles, it is possible to apply permutation-inversion operations to them directly [see equation (eq. 8)]. Thus, for example
$$\begin{eqnarray*} (234) \, | m_a m_b m_c m_d\rangle &\equiv & (234) \, | m_a(1), m_b(2), m_c(3), m_d(4)\,\rangle\\ &=& \phantom{(234)} \, | m_a(1), m_b(3), m_c(4), m_d(2)\,\rangle \equiv | m_a m_d m_b m_c\rangle ~ . \end{eqnarray*}$$
(eq. 34)
Furthermore, laboratory-fixed spin functions are invariant with respect to the laboratory-fixed inversion operation, so that we need consider only the permutation effects of permutation-inversion operations.
Let us represent the four-proton spin functions by the symbols |αααα⟩, |αααβ⟩, etc., where α and β indicate individual proton spin projections mi along the Z axis of +½ and −½, respectively, and where the spin projections of protons i = 1, 2, 3, 4 are given in that order. It then follows from the permutation-inversion operations given in Sec. 3, that the species of the proton spin functions under the operations of the full molecular symmetry group are [17] as given in Table 15. Values for the total spin I, where I = I1 + I2 + I3 + I4, and for its projection mI along the laboratory-fixed Z axis are also given.
| I | mI | Species | Function | |||
|---|---|---|---|---|---|---|
| 2 | +2 | A1 | + |αααα⟩ | |||
| 2 | +1 | A1 | 4−½ [ + |βααα⟩ | + |αβαα⟩ + |ααβα⟩ + |αααβ⟩ ] | ||
| 2 | 0 | A1 | 6−½ [ + |αβαβ⟩ | + |βαβα⟩ + |βααβ⟩ + |αββα⟩ + |ααββ⟩ + |ββαα⟩ ] | ||
| 2 | −1 | A1 | 4−½ [ + |αβββ⟩ | + |βαββ⟩ + |ββαβ⟩ + |βββα⟩ ] | ||
| 2 | −2 | A1 | + |ββββ⟩ | |||
| 1 | +1 | F2x | 4−½ [ − |βααα⟩ | + |αβαα⟩ − |ααβα⟩ + |αααβ⟩ ] | ||
| 1 | 0 | F2x | 2−½ [ + |αβαβ⟩ | − |βαβα⟩ ] | ||
| 1 | −1 | F2x | 4−½ [ + |αβββ⟩ | − |βαββ⟩ + |ββαβ⟩ − |βββα⟩ ] | ||
| 1 | +1 | F2y | 4−½ [ + |βααα⟩ | − |αβαα⟩ − |ααβα⟩ + |αααβ⟩ ] | ||
| 1 | 0 | F2y | 2−½ [ + |βααβ⟩ | − |αββα⟩ ] | ||
| 1 | −1 | F2y | 4−½ [ − |αβββ⟩ | + |βαββ⟩ + |ββαβ⟩ − |βββα⟩ ] | ||
| 1 | +1 | F2z | 4−½ [ − |βααα⟩ | − |αβαα⟩ + |ααβα⟩ + |αααβ⟩ ] | ||
| 1 | 0 | F2z | 2−½ [ + |ααββ⟩ | − |ββαα⟩ ] | ||
| 1 | −1 | F2z | 4−½ [ + |αβββ⟩ | + |βαββ⟩ − |ββαβ⟩ − |βββα⟩ ] | ||
| 0 | 0 | Ea | 12−½ [ + 2|ααββ⟩ | + 2|ββαα⟩ − |αβαβ⟩ − |βαβα⟩ − |βααβ⟩ − |αββα⟩ ] | ||
| 0 | 0 | Eb | 4−½ [ + |αβαβ⟩ | + |βαβα⟩ − |βααβ⟩ − |αββα⟩ ] | ||
Table 16 presents similar information for the 19 four-deuteron spin functions with mI = +I. The remaining 62 functions can be constructed from these entries by ladder operator techniques [29], which generate a function |I,mI −1,Γ⟩ from the function |I,mI,Γ⟩. In the compact notation of Table 16, |++0−⟩ represents a four-deuteron spin function |m1 = +1, m2 = +1, m3 = 0, m4 = −1⟩; |I,mI,Γ; α = m⟩15 represents a four-deuteron function obtained from the four-proton function |I,mI,Γ⟩ of Table 15 by giving α the indicated value m and β the value required to obtain a total mI value equal to that in column 2 of Table 16; |I,mI,Γ; α = m,n⟩15 is similar but requires a symmetric substitution |αiαj⟩ → 2−1/2[|mn⟩ + |nm⟩]; |I,mI, F1x; n = 2⟩16 indicates an altered entry from Table 16.
| I | mI | Species | Function | |
|---|---|---|---|---|
| 4 | +4 | A1 | | 2, +2, A1; α= +1⟩15 ≡ | + + + +⟩ | |
| 3 | +3 | F2x | | 1, +1, F2x; α= +1⟩15 ≡ 4−½ [ − | 0 + + +⟩ + | + 0 + +⟩ − | + + 0 +⟩ + | + + + 0⟩ ] | |
| 3 | +3 | F2y | | 1, +1, F2y; α= +1⟩15 | |
| 3 | +3 | F2z | | 1, +1, F2z; α= +1⟩15 | |
| 2 | +2 | A1 | 7−½ [ square root of 6 | 2, +1, A1; α = +1⟩15 − | 2, 0, A1; α = +1⟩15 ] | |
| 2 | +2 | Ea | | 0, 0, Ea; α = +1⟩15 | |
| 2 | +2 | Eb | | 0, 0, Eb; α = +1⟩15 ≡ 4−½ [ | + 0 + 0⟩ + | 0 + 0 +⟩ − | 0 + + 0⟩ − | + 0 0 +⟩ ] | |
| 2 | +2 | F2x | 3−½ [ square root of 2 | 1, +1, F2x; α = +1⟩15 − | 1, 0, F2x; α = +1⟩15 ] | |
| 2 | +2 | F2y | 3−½ [ square root of 2 | 1, +1, F2y; α = +1⟩15 − | 1, 0, F2y; α = +1⟩15 ] | |
| 2 | +2 | F2z | 3−½ [ square root of 2 | 1, +1, F2z; α = +1⟩15 − | 1, 0, F2z; α = +1⟩15 ] | |
| 1 | +1 | F1x | 8−½ { + | + + 0 −⟩ − | + + − 0⟩ + | 0 − + +⟩ − | − 0 + +⟩ + (−1)n [ + | 0 + + −⟩ − | − + + 0⟩ − | + 0 − +⟩ + | + − 0 +⟩ ] }, n = 1 |
|
| 1 | +1 | F1y | 8−½ { + | + 0 + −⟩ − | + − + 0⟩ − | 0 + − +⟩ + | − + 0 +⟩ + (−1)n[ + | + + 0 −⟩ − | + + − 0⟩ − | 0 − + +⟩ + | − 0 + +⟩ ] }, n = 1 |
|
| 1 | +1 | F1z | 8−½ { + | 0 + + −⟩ − | − + + 0⟩ + | + 0 − +⟩ − | + − 0 +⟩ + (−1)n [ + | + 0 + −⟩ − | + − + 0⟩ + | 0 + − +⟩ − | − + 0 +⟩ ] }, n = 1 |
|
| 1 | +1 | F2x | 5−½ [ | 1, +1, F1x; n = 2⟩16 + square root of 2 | 1, 0, F2x; α = 0, −1 ⟩15 + square root of 2 | 1, −1, F2x; α = +1⟩15 ] | |
| 1 | +1 | F2y | 5−½ [ | 1, +1, F1y; n = 2⟩16 + square root of 2 | 1, 0, F2y; α = 0, −1 ⟩15 + square root of 2 | 1, −1, F2y; α = +1⟩15 ] | |
| 1 | +1 | F2z | 5−½ [ | 1, +1, F1z; n = 2⟩16 + square root of 2 | 1, 0, F2z; α = 0, −1 ⟩15 + square root of 2 | 1, −1, F2z; α = +1⟩15 ] | |
| 0 | 0 | A1 | 15−½ [ 2 square root of 2 | 2, 0, A1; α = +1⟩15 − 2| 2, 0, A1; α = +1, −1⟩15 + square root of 3 | 0 0 0 0⟩ ] | |
| 0 | 0 | Ea | 3−½ [ | 0, 0, Ea; α = +1⟩15 + square root of 2 | 0, 0, Ea; α = +1, −1⟩15 ] | |
| 0 | 0 | Eb | 3−½ [ | 0, 0, Eb; α = +1⟩15 + square root of 2 | 0, 0, Eb; α = +1, −1⟩15 ] ≡ 3−½ [ | 0, 0, Eb; α = +1⟩15 + 4−½ [ | + 0 − 0⟩ + | − 0 + 0⟩ + | 0 + 0 −⟩ + | 0 − 0 +⟩ − | 0 + − 0⟩ − | 0 − + 0⟩ − | + 0 0 −⟩ − | − 0 0 +⟩ ] } |
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