Diatomic state symmetry

The state symmetries of heteronuclear diatomic molecules are simpler than those of homonuclear diatomics the g and u states of Doo/i simply coalesce in Coot . Table 3.2 summarizes the correlations of the irreps of Doo/i with those of its subgroup Coov and between the latter and those of C2v ... 

In homonuelear diatomics an additional symmetry element is needed to classify state symmetries in the D f, point group, and we can use i (molecule-fixed) for this purpose. A new complication, peculiar to homonuelear molecules... 

Since the 6 orbital carries no orbital angular momentum, the orderii of the levels according to 12 is the same as it would be vor a /7 (D i,) diatomic state (25 a) with the same number of 7r-symmetry electrons. However, the bi symmetry of d in exchanges the D41, symmetry labels A, B of the 12 = 0 and 12 = 2 spin-orbit levels relative to those expected by correlation to the diatomic triplet state [25 a] d > 17> Z, 2 (D )... 

H(I) and H(II). This fact does not provide any information on the nuclear sti ucture of this species at the energy minimum. By symmetry, it is clear that the system has three equivalent minima on the ground-state surface, which were designated as the three diatomic pairs. The nuclear geometry of each of these minima is quite different from that of the other two. 

For atoms, electronic states may be classified and selection rules specified entirely by use of the quantum numbers L, S and J. In diatomic molecules the quantum numbers A, S and Q are not quite sufficient. We must also use one (for heteronuclear) or two (for homonuclear) symmetry properties of the electronic wave function ij/. ... 

The first is the g or m symmetry property which indicates that ij/ is symmetric or antisymmetric respectively to inversion through the centre of the molecule (see Section 4.1.3). Since the molecule must have a centre of inversion for this property to apply, states are labelled g or m for homonuclear diatomics only. The property is indicated by a postsubscript, as in... 

The second symmetry property applies to all diatomics and concerns the symmetry of with respect to reflection across any (n ) plane containing the intemuclear axis. If is symmetric to (i.e. unchanged by) this reflection the state is labelled -I- and if it is antisymmetric to (i.e. changed in sign by) this reflection the state is labelled —as in or Ig. This symbolism is normally used only for I states. Although U, A, doubly degenerate state is... 

In the case of atoms, deriving states from configurations, in the Russell-Saunders approximation (Section 7.1.2.3), simply involved juggling with the available quantum numbers. In diatomic molecules we have seen already that some symmetry properties must be included, in addition to the available quantum numbers, in a discussion of selection rules. 

For the orbital parts of the electronic wave functions of two electronic states the selection rules depend entirely on symmetry properties. [In fact, the electronic selection rules can also be obtained, from symmetry arguments only, for diatomic molecules and atoms, using the (or and Kf point groups, respectively but it is more... 

The first step in the solution of equation (10.28b) is to hold the two nuclei fixed in space, so that the operator drops out. Equation (10.28b) then takes the form of (10.6). Since the diatomic molecule has axial symmetry, the eigenfunctions and eigenvalues of He in equation (10.6) depend only on the fixed value R of the intemuclear distance, so that we may write them as tpKiy, K) and Sk(R). If equation (10.6) is solved repeatedly to obtain the ground-state energy eo(K) for many values of the parameter R, then a curve of the general form... 

As mentioned earlier, heavy polar diatomic molecules, such as BaF, YbF, T1F, and PbO, are the prime experimental probes for the search of the violation of space inversion symmetry (P) and time reversal invariance (T). The experimental detection of these effects has important consequences [37, 38] for the theory of fundamental interactions or for physics beyond the standard model [39, 40]. For instance, a series of experiments on T1F [41] have already been reported, which provide the tightest limit available on the tensor coupling constant Cj, proton electric dipole moment (EDM) dp, and so on. Experiments on the YbF and BaF molecules are also of fundamental significance for the study of symmetry violation in nature, as these experiments have the potential to detect effects due to the electron EDM de. Accurate theoretical calculations are also absolutely necessary to interpret these ongoing (and perhaps forthcoming) experimental outcomes. For example, knowledge of the effective electric field E (characterized by Wd) on the unpaired electron is required to link the experimentally determined P,T-odd frequency shift with the electron s EDM de in the ground (X2X /2) state of YbF and BaF. 

For linear molecules or ions the symbols are usually those derived from the term symbols for the electronic states of diatomic and other linear molecules. A capital Greek letter E, n, A, O,... is used, corresponding to k — 0,1,2,3,..., where A. is the quantum number for rotation about the molecular axis. For E species a superscript + or - is added to indicate the symmetry with respect to a plane that contains the molecular axis. 

These selection rules are affected by molecular vibrations, since vibrations distort the symmetry of a molecule in both electronic states. Therefore, an otherwise forbidden transition may be (weakly) allowed. An example is found in the lowest singlet-singlet absorption in benzene at 260 nm. Finally, the Franck-Condon principle restricts the nature of allowed transitions. A large number of calculated Franck-Condon factors are now available for diatomic molecules. 

From the quantum mechanical standpoint the appearance of the factor 1/2 = 1/s for the diatomic case means the configurations generated by a rotation of 180° are identical, so the number of distinguishable states is only one-half the classical total. Thus the classical value of the partition function must be divided by the symmetry number which is 1 for a heteronuclear diatomic and 2 for a homonuclear diatomic molecule. 

Proceeding in the spirit above it seems reasonable to inquire why s is equal to the number of equivalent rotations, rather than to the total number of symmetry operations for the molecule of interest. Rotational partition functions of the diatomic molecule were discussed immediately above. It was pointed out that symmetry requirements mandate that homonuclear diatomics occupy rotational states with either even or odd values of the rotational quantum number J depending on the nuclear spin quantum number I. Heteronuclear diatomics populate both even and odd J states. Similar behaviors are expected for polyatomic molecules but the analysis of polyatomic rotational wave functions is far more complex than it is for diatomics. Moreover the spacing between polyatomic rotational energy levels is small compared to kT and classical analysis is appropriate. These factors appreciated there is little motivation to study the quantum rules applying to individual rotational states of polyatomic molecules. 

The electronic contributions to the g factors arise in second-order perturbation theory from the perturbation of the electronic motion by the vibrational or rotational motion of the nuclei [19,26]. This non-adiabatic coupling of nuclear and electronic motion, which exemplifies a breakdown of the Born-Oppenheimer approximation, leads to a mixing of the electronic ground state with excited electronic states of appropriate symmetry. The electronic contribution to the vibrational g factor of a diatomic molecule is then given as a sum-over-excited-states expression... 

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