Rotational quantum number allowed values

As was shown in the preceding discussion (see also Sections Vin and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. 

Any molecule with a permanent electric dipole moment can interact with an electromagnetic field and increase its rotational energy by absorbing photons. Measuring the separation between rotational levels (for example, by applying a microwave field which can cause transitions between states with different values of /) let us measure the bond length. The selection rule is A/ = +1—the rotational quantum number can only increase by one. So the allowed transition energies are... 

This expression is roughly consistent with the band structure of rotation-vibration spectra. Since the rotational quantum number J assumes integral values, the lines comprising a rotation-vibration band of a rigid rotator are equally spaced. The separation of such lines allows calculation of the moment of inertia of the molecule without the necessity for exploring the far infrared. 

The principal use of the nuclear symmetry character is in determining the allowed values of the rotational quantum number K of the molecule. The complete wave functions for a molecule (including the nuclear-spin function) must be either symmetric or antisymmetric in the nuclei, depending on the nature of the nuclei involved. If the nuclei have no spins, then the existent functions are of one or the other of the types listed below. 

An optical-optical double resonance (OODR) scheme exists, utilizing two continuous-wave (cw), monochromatic, tunable lasers, whereby the rotational quantum numbers of all observed lines may be established without ambiguity, prior knowledge of B-values, trial-and-error searches for consistent combination differences, or redundant confirmation lines that are weak because of level population may be distinguished from those that are weak because of intrinsic linestrength and forbidden transitions may be made to appear with comparable peak intensities but considerably narrower widths than allowed transitions. 

Figure 9.7 The constraint of angular momentum conservation and the allowed values of the orbital angular momentum ( ) and the product rotational quantum number, /, imposed by PST. The total, and conserved, angular momentum is I. The case of low I O,) and high / (/ ) are illustrated.

For a homonuclear diatomic molecule composed of even(odd) mass-ntmber nuclei, the total wave function, which we assume to be a product of electronic, vibrational, rotational, and nuclear-spin functions, must be symmetric (antisymmetric). If the electronic wave function is symmetric, and if the nuclear spin is zero, as in the ground state of 0a, only even values of J, the rotational quantum number, are allowed. If the nuclear spin is not zero, both even and odd values of J (l.e., symmetric and antisymmetric rotational wave functions) are allowed, but with different statistical weights. These may be determined from the nuclear-spin part of the wave function. 

The following are sets of rotational quantum numbers (7, Mj, K). Label each indicated transition as either allowed or forbidden. Hint Remember the rules for allowed values of the various quantum numbers. 

This equation gives the frequency in cm of the photon whose energy has the right value to increase the rotational quantum number from J to J + 1. This is the only allowed transition for absorption. Therefore, B can be calculated from the observed frequencies in cm For example, the spacing between two adjacent absorption lines is equal to 2B from Eq. (1.66). From B we can calculate the moment of inertia [Eq. (1.56)] and from that the internuclear distance [Eq. (1.51)]. Figure 1.18 shows the energy levels, transitions, and origin of the rotational absorption spectrum of a diatomic molecule. 

The fact that CO2 is linear and symmetric leads, just as with homonuclear diatomic molecules, to certain simplifications in its spectra. CO2 has no permanent dipole moment and therefore no pure rotational spectrum. Furthermore, transitions involving only a change in v are infrared inactive, since this mode of vibration maintains the symmetry (see Fig. 2.1). In the case of the laser bands mentioned above, transitions between vibrational levels are accompanied by a change in rotational quantum number J = 1, but those terminating on odd values of J are not allowed, also for reasons of symmetry. None of these remarks would apply to N2O, for example, because this molecule has the structure N-N-0 and is therefore not symmetric. 

The vibrational transitions in question are centred at 10.4 /im and 9.4 //m and are usually referred to as the 10 //m and 9 //m bands or I and II, in keeping with the above. Within each of these are two allowed possibilities for the change in the rotational quantum number J, namely AJ = — 1 (P branch) or AJ = +1 (R branch). Conventionally then, a transition is designated by (for example) 9P34, indicating the 9 /im band P-branch transition down to the J = 34 level. Owing to the symmetry of the molecule, only transitions terminating on even values of J are allowed in these bands. 

In H2, the two nuclear spins of the H atoms can be combined in two different ways, resulting in the molecular species with the total nuclear spin of the two hydrogens of / = 1 or / = 0. In the ground electronic and vibrational state, each H2 molecule can be characterized by a combination of a certain rotational state with a certain nuclear spin state. As protons are fermions, Pauli s principle requires that the total wave function of a molecule should be antisymmetric with respect to the permutation of the two nuclei. The rotational states with even values of the rotational quantum number / are symmetric with respect to such permutations, including the rotational state with the lowest energy, which has / = 0. Such rotational states can be combined only with the antisymmetric nuclear spin state with 1 = 0, and such combinations correspond to parahydrogen (PH2). All rotational states with odd / values are antisymmetric and are only allowed in combination with the symmetric (/ = 1) nuclear spin states. These combinations correspond to orthohydrogen (oH2). At the same time, H2 molecules with even-even and odd-odd combinations of the two quantum numbers do not exist. As a result. 

We next consider the rotational partition functions for diatomic molecules. According to section 14c, there is no restriction on the allowed values of the rotational quantum number J if the nuclei are different. If the two nuclei have spins si and S2 the nuclear spin statistical weifeht is (2si + l)(2s2 + 1). The rotational energy levels are... 

The rotational quantum numbers J and K can have integral values 0, 1,2,... with both 3iK allowed. The quantum numbers obey the constraint A < 7. 

The spinor that describes the spherical rotation satisfies Schrodinger s equation and specifies two orientations of the spin, colloquially known as up and down (j) and ( [), distinguished by the allowed values of the magnetic spin quantum number, ms = . The two-way splitting of a beam of silver ions in a Stern-Gerlach experiment is explained by the interaction of spin angular momentum with the magnetic field. 

The symmetry of an isolated atom is that of the full rotation group R+ (3), whose irreducible representations (IRs) are D where j is an integer or half an odd integer. An application of the fundamental matrix element theorem [22] tells that the matrix element (5.1) is non-zero only if the IR DW of Wi is included in the direct product x of the IRs of ra and < f. The components of the electric dipole transform like the components of a polar vector, under the IR l)(V) of R+(3). Thus, when the initial and final atomic states are characterized by angular momenta Ji and J2, respectively, the electric dipole matrix element (5.1) is non-zero only if D(Jl) is contained in Dx D(j 2 ) = D(J2+1) + T)(J2) + )(J2-i) for j2 > 1 This condition is met for = J2 + 1, J2, or J2 — 1. However, it can be seen that a transition between two states with the same value of J is allowed only for J 0 as DW x D= D( D(°) is the unit IR of R+(3)). For a hydrogen-like centre, when an atomic state is defined by an orbital quantum number , this can be reduced to the Laporte selection rule A = 1. This is of course formal, as it will be shown that an impurity state is the weighted sum of different atomic-like states with different values of but with the same parity P = ( —1) These states are represented by an atomic spectroscopy notation, with lower case letters for the values of (0, 1, 2, 3, 4, 5, etc. correspond to s, p, d, f, g, h, etc.). The impurity states with P = 1 and -1 are called even- and odd-parity states, respectively. For the one-valley EM donor states, this quasi-atomic selection rule determines that the parity-allowed transitions from Is states are towards np (n > 2), n/ (n > 4), nh (n > 6), or nj (n > 8) states. For the acceptor states in cubic semiconductors, the even- and odd-parity states labelled by the double IRs T of Oh or Td are indexed by + or respectively, and the parity-allowed transition take place between Ti+ and... 

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