Rotational energy levels, ortho

Rotational energy levels of para 1 — 0) and ortho (/=1) hydrogen molecule [2],... 

Furthermore, the NH3 molecules consist of two nuclear spin modifications of total spin 3/2 (parallel) and 1/2 (antiparallel). The Fermi statistics of the three hydrogen nuclei divide the NH3 rotational energy levels into an ortho-and para-species, respectively, depending on whether AT is a multiple of 3 or not. Transitions between the two modifications are strongly forbidden. As a further consequence, for K = 0 only alternating levels occur. 

Fig. 19. Rotational energy levels of HjO divided into para- and ortho-species and sorted according to Kc. The strongest dipole transitions are indicated by thin lines. Heavy arrows indicate some microwave and millimeter wave transitions, together with the transition frequency in MHz. Double arrows indicate transitions of potential astrophysical interest which may appear in maser emission. It may be noted that these four transitions are the only way out of the series of lines with KC = J...

Ah diatomic molecnles for which the constituent nnclei have spin exhibit the phenomenon of spin isomerism. The nnclear spins can be parallel (the ortho isomer) or opposed (the para isomer). For most diatomic molecules, which might be expected to exhibit, spin isomerism the energy separation of the rotational states is small compared to kT, even at low temperatures. However, in the case of hydrogen molecules, which have the smallest moment of inertia of any diatomic molecnles, the energy difference between the rotational energy levels is relatively large and only the lowest states are popnlated at room temperature. 

Figure 7 PHOFEX spectrum of the lowest rotational state of ortho singlet methylene near the threshold for CH2CO CH2 + CO. The smoother line is the phase-space theory rate constant. The step positions match the rotational energy levels for free CO.

These represent the nuclear spin Zeeman interaction, the rotational Zeeman interaction, the nuclear spin-rotation interaction, the nuclear spin-nuclear spin dipolar interaction, and the diamagnetic interactions. Using irreducible tensor methods we examine the matrix elements of each of these five terms in turn, working first in the decoupled basis set rj J, Mj /, Mi), where rj specifies all other electronic and vibrational quantum numbers this is the basis which is most appropriate for high magnetic field studies. In due course we will also calculate the matrix elements and energy levels in a ry, J, I, F, Mf) coupled basis which is appropriate for low field investigations. Most of the experimental studies involved ortho-H2 in its lowest rotational level, J = 1. If the proton nuclear spins are denoted I and /2, each with value 1 /2, ortho-H2 has total nuclear spin / equal to 1. Para-H2 has a total nuclear spin / equal to 0. 

Note, for completeness, that the vibrational and electronic spin parts of the total wave function are both unaffected by I i, that is, they are symmetric. We now have all the information required to derive expressions for the zero-field spin spin and spin rotation energies of the N, J levels for both para- and ortho-H2, excluding nuclear magnetic hyperfine interaction for ortho-H2 which we will come to in due course. These are given in table 8.6. 

In molecular hydrogen, the existence of nuclear-spin energy levels is responsible for the distinction between ortho and para hydrogen, which correspond to the triplet and singlet (i.e., parallel and antiparallel) orientations, respectively, of the two nuclei in H2. Because of the coupling of the rotational and spin levels, ortho and para hydrogen differ in specific heat and certain other properties. The correlated orientation of the nuclear spins in para H2 has re-... 

Vibration-rotation-tuimehng energy levels for the 4- 4 (ortho-ortho) states. The dashed arrows indicate perpendicular bands, whereas solid arrows correspond to parallel bands. The transitions are tentatively assigned to an out of plane vibration with a = 0 state at 89.141305(47), and aK= state at 86.77785 (9) cm [06Lin]. 

Same as hydrogen, under normal circumstances, deuterium (normal-D2) consists of ortho-D2 and para-D2. However, because the D atom nuclear spin quantum number / = 1 is boson, nuclear exchange symmetry of the total wave function is symmetrical. Para-D2 corresponds to the energy levels at even-number rotational quantum states (7 = 0, 2,4,...) ortho-D2 corresponds to odd-number states 0 =1 3, 5,...). The ratio of ortho-D2 to para-D2 is 2 1. We prepared ortho-D2 using the same method as para-H2. 

A free molecule of hydrogen sulphide has energy levels shown in Fig. 2. Since the two hydrogen atoms in the molecule are equivalent, there are ortho and para species of hydrogen sulphide. If the molecule rotates freely in the clathrate cavity, the distinction between the two species should be relevant to the low temperature property of the compound. 

Figure 4 The potential function for hindered internal rotation of the transition state for triplet ketene, V(0) = (l/2)Vq(l-cos20) + (l/2)Vi(l cos40), where Vq = 240 cm" and Vj = 20 cm The ab initio transition-state geometry is drawn. At the maximum, the O is out of the plane with the CCO plane bisecting the HCH angle. The calculated energy levels are shown for K = 0 ( —, ortho —, para). The zero-point level splitting is less than 1 cm [11].

We are now almost in a position to calculate the zero-field spin-rotation and spin spin dipolar energies of the N, J levels, but we have first to discuss the parity restrictions on the levels for para- and ortho-H2(see also chapter 6). 

Lichten [3 5] studied the magnetic resonance spectrum of the para-H2, N = 2 level, and was able to determine the zero-field spin-spin and spin-orbit parameters we will describe how this was done below. Before we come to that we note, from table 8.6, that in TV = 2 it is not possible to separate Xo and X2. Measurements of the relative energies of the J spin components in TV = 2 give values of Xo + fo(iX2, and the spin-orbit constant A the spin rotation constant y is too small to be determined. In figure 8.18 we show a diagram of the lower rotational levels for both para- and ortho-H2 in its c3 nu state, which illustrates the difference between the two forms of H2. This diagram does not show any details of the nuclear hyperfine splitting, which we will come to in due course. 

J=0), then H (298.15 K)-H (0 K) would be 0.254 kcal mol less for "normal" than for "equilibrium" Hg. This would change the difference between AjH (0 K) and a H (298.15 K) for all species involving hydrogen (8, 10). No such change would occur if we chose the lowest level (v=0, J=0) as the energy zero for ortho-H. "Equilibrium" Hg is the form which parallels most substances, i.e., those maintaining equilibrium among all rotational levels (9). 

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