Pure precession

Hinkley et a/.125 improved these calculations by using Morse and RKR curves to produce vibrational wavefunctions. Using the concept of pure precession they calculated. -1-doubling constants for 2n states of OH, BeH,... 

It will be gathered from this summary that, although the formulae for determining spin-splitting and /1-doubling are available, very little quantitative work has been done. Results obtained to date are encouraging. If more refined treatments are available it should be possible to resolve many of the outstanding problems. In particular, molecules such as NO, where simple concepts of pure precession do not apply, are worthy of study. Likewise, little work has been done on A,(J>, etc. states where effects are smaller and quantitative accuracy more difficult to obtain. 

Although it is not the purpose of this Report to consider magnetic properties in detail, the effect of magnetic fields on the -doubling will be briefly considered because further insight is given into the phenomenon of pure precession. Detailed formulae have been given by Radford.135 138 It is found that gas-phase e.p.r. spectra will yield information about differences in lg values, quantities which depend essentially upon p and q, as well as on two similar constants,... 

In this book, we adopt a form in which the function is expressed as a linear combination of spherical harmonics. This form is particularly appropriate for systems with near-spherical symmetry (such as Rydberg states or molecules which conform to Van Vleck s pure precession hypothesis [68, 69]) and is also consistent with the spirit of spherical tensors, which have the same transformation properties under rotations as spherical harmonics. The functional form of the ket rj, A) is written... 

In writing this equation, we have made use of Van Vleck s pure precession hypothesis [12], in which the molecular orbital /.) is approximated by an atomic orbital with well-defined values for the quantum numbers n, l and /.. Such an orbital implies a spherically symmetric potential and its use is most appropriate when the electronic distribution is nearly spherical. Examples of this situation occur quite often in the description of Rydberg states. It is also appropriate for hydrides like OH where the molecule is essentially an oxygen atom with a small pimple, the hydrogen atom, on its side. Accepting the pure precession hypothesis allows the matrix elements of the orbital operators to be evaluated since... 

Later on in this book, we discuss the properties of the CH radical in its X211 state in some detail. The electronic structure of this radical is rather more complicated than that of OH. Despite this, a simple pure precession calculation of the A -doubling parameters reproduces the experimental values (particularly q) reasonably well (see section 10.6.3). 

A particularly simple approach to handling the matrix elements in (8.388) involves what is known as van Vleck s model of pure precession [119], see section 7.8. Applied to LiO the model assumes L to be a good quantum number, with L = 1 since the ground state of O- is 2P. As a result of this assumption,... 

Furthermore, consistent with the pure precession model, one can factorise the M and M2 matrix elements, with the results,... 

We can then make an even more drastic approximation and represent the molecular orbitals in these configurations by pure 2p atomic orbitals on the O atom. This approximation was called pure precession by Van Vleck [41] in this approximation the electrons in these outermost orbitals are in a spherically symmetric environment and they have a well defined value of the orbital angular momentum quantum number / (unity for a p orbital). In the pure precession approximation, we can derive very simple expressions for the g-factors [66], The values for OH predicted on the basis of this very simple model are given in table 9.4. The fact that they agree reasonably well with the experimental numbers suggests that the theoretical model is essentially correct. 

Once again using the pure precession approximation and the parameter values given above we obtain... 

These matrix elements are of two different types. The matrix elements between 2 = +1/2 and -1/2 connect states which are otherwise degenerate, making first-order contributions of opposite sign to the energy for the even (e) and odd (/) combinations of 2 = 1 /2. These contributions to e and / states are represented in the effective Hamiltonian by diagonal terms which are +P (J + 1/2) for 2 = 1/2 states, and if Ja is a good quantum number, Pv has the value 2Bv. This is known as the pure precession limit. 

In the expression for the basis fiinctions, the electronic states which appear in eq. (2) have a well-defined value of the projection of the electronic orbital angular momentum X, even though I itself does not have a definite value, except when the X atom is infinitely lar from the H2.[48] In the pure-precession limit, I would everywhere fixed at its value for the separated halogen atom reactant, namely /= 1. 

If the value of l is 1, o+ = 2l/2A. This is known as part of the pure precession hypothesis, which is approximately true only for hydrides, Rydberg states of an l > 0 complex, and certain highly ionic molecules (see Section 5.5). Although the pure precession hypothesis is frequently applied to homonuclear molecules, there is no justification for this. 

Value obtained using the pure precession approximation. (Sections 3.5.4 and 5.5). Calculated value is —0.08 cm-1 (Lefebvre-Brion unpublished calculation). 

If the 2n and 2E+ states (with identical potential curves) in question interact appreciably only with each other and not with other 2E or 2II states, then p = 7. When the pair of interacting 2II and 2E states are found to have p = 7, it is said that they are in a unique perturber relationship, which has nothing to do with the pure precession hypothesis, discussed below and in Section 5.5. 

In Van Vleck s pure precession hypothesis (Van Vleck, 1929 Mulliken and Christy, 1931 see Section 5.5),... 

For example, the spin-rotation splitting (called p-doubling by Van Vleck) of the OH A2E+ state is given mainly by interaction with the inverted X2n state. Consequently, for A< 0 and En — E% < 0, one predicts 7 > 0. The e-levels of the A2 X+-state are well above the /-levels, in qualitative agreement with the pure precession prediction. Consequently p and q have the same sign for A > 0 and p and q have opposite signs for A < 0 this is the case for the X2n... 

The idea of an effective Hamiltonian for diatomic molecules was first articulated by Tinkham and Strandberg (1955) and later developed by Miller (1969) and Brown, et al., (1979). The crucial idea is that a spectrum-fitting model (for example Eq. 18 of Brown, et al., 1979) be defined in terms of the minimum number of linearly independent fit parameters. These fit parameters have no physical significance. However, if they are defined in terms of sums of matrix elements of the exact Hamiltonian (see Tables I and II of Brown, et al., 1979) or sums of parameters appropriate to a special limiting case (such as the unique perturber approximation, see Table III of Brown, et al., 1979, or pure precession, Section 5.5), then physically significant parameters suitable for comparison with the results of ab initio calculations are usually derivable from fit parameters. 

The hypothesis of pure precession (Van Vleck, 1929) is often used in the estimation of A-doubling constants. These constants, in the o,p, q notation suggested by Mulliken and Christy (1931), are introduced into the effective Hamiltonian... 

This discussion is intended to distinguish the levels of approximation often hidden behind the name pure precession. For clarity, only 2E 2II interactions will be discussed. The considerably more complicated 3E 3II case is treated by Brown and Merer (1979). 

An important and frequently encountered result, often mistakenly taken as evidence for pure precession, is that, in the unique perturber, identical potential curve limit, the effective spin-rotation constant of the 2E+ state, jv, is equal to Py. This result is a direct consequence of the second-order perturbation theoretical definition of the contribution of a 2II state to the spin-rotation splitting in a 2E state (see Section 3.5.4) ... 

Finally, the pure precession approximation requires, in addition to the unique perturber, identical potential assumptions, that the interacting 2II and 2 states axe each well described by a single configuration, that these configurations are identical except for a single spin-orbital, and that this spin-orbital is a pure... 

Similar equations for o,p, and q may be derived for generalized pure precession situations involving two or more open shells. The validity of these pure precession equations depends on many assumptions, yet experimentalists are very quick to use them to explain the sign or even the magnitude of observed A-doubling and spin-rotation constants or to use these constants to infer /eff values for valence-shell a and 7r orbitals. 

For the OH radical, the values of p and q for the X2n (ground state can be attributed to a unique perturber interaction with the A2E+ (ow4) state. The pure precession approximation simply ignores the contribution of the atomic orbital to the per molecular orbital. For all hydrides, the oTsh orbital makes a negligible contribution to Hso and BL+ matrix elements. For OH, the H-atom contributions to a+ and b are 4 x 10 4% and 1%, respectively (Hinkley, et al., 1972). 

This generalization to cases where the interacting n and E states are derived from more complex configurations than 7T1 and cr1 is the basis for generalized pure precession. The p and q parameters for CH X2n are... 

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