Correction terms Reduced-mass corrections

The occurrence of the term 1 ja% in yj2(0) indicates that a reduced-mass correction factor (14-m/ilf ) 3 is necessary. A small relativistic correction also arises when Dirac functions are used to evaluate y2 0) [21]. Its value in the level is... 

The first term is simply the spin-Zeeman term, the seeond term is the reduced mass correction, and the fliird term is a gauge correction to the SOC, which has been written in somewhat simplified form here (explained in detail elsewhere [60]). 

H[ is the operator of the relative kinetic energy of the nuclei, H is a correction to the kinetic energy of the electrons, is a mass polarisation correction and // denotes the reduced mass of the nuclei. The explicit expression for H R) in terms of elliptic coordinates is given in Ref(Kolos and Wolniewicz, 1964). In the BO approximation the term is neglected. 

The aim of this section is to extract from the measurements the values of the Rydberg constant and Lamb shifts. This analysis is detailed in the references [50,61], More details on the theory of atomic hydrogen can be found in several review articles [62,63,34], It is convenient to express the energy levels in hydrogen as the sum of three terms the first is the well known hyperfine interaction. The second, given by the Dirac equation for a particle with the reduced mass and by the first relativistic correction due to the recoil of the proton, is known exactly, apart from the uncertainties in the physical constants involved (mainly the Rydberg constant R0c). The third term is the Lamb shift, which contains all the other corrections, i.e. the QED corrections, the other relativistic corrections due to the proton recoil and the effect of the proton charge distribution. Consequently, to extract i oo from the accurate measurements one needs to know the Lamb shifts. For this analysis, the theoretical values of the Lamb shifts are sufficiently precise, except for those of the 15 and 2S levels. 

All the symbols have their usual meanings. In the non-recoil limit, the motion of the nucleus is neglected and its finite mass enters only as a reduced mass of the electron. The additional terms arising from the dynamical effects of the nucleus, namely the recoil corrections and radiative-recoil corrections, have been omitted from equation 1 and will not be considered here. For more detailed discussions of the theory, see the review by Sapirstein and Yennie [3] and more recently [4,5,6], The expansion in (Za) is now carried out by expressing F and H as power series in (Za) and ln(Za) 2, as shown below in equations 2 and 3, where a is the ratio of the electron mass to its reduced mass. 

The first term in (1.12) represents the kinetic energy due to translation of the whole molecule through space this motion can be separated off rigorously in the absence of external fields. In the second term, /i is the reduced nuclear mass, M M2/(Mi + M2), and this term represents the kinetic energy of the nuclei. The third term describes the kinetic energy of the electrons and the last term is a correction term, known as the mass polarisation term. The transformation is described in detail in chapter 2 and appendix 2.1. An alternative expression equivalent to (1.12) is obtained by writing the momentum operators in terms of the Laplace operators,... 

The third, fourth and fifth terms in braces in equation (2.141) represent small adiabatic corrections to the potential energy function. They all have a /r 1 reduced mass dependence, unlike Knuci(7 ), and so are the origin of the isotopic shifts in the electronic energy [5],... 

Use of equation (7.205) shows that A4 is proportional to /i 1. We thus see that there is a contribution 6hA4 to the Dunham coefficient Uo (or coexe) since it is the coefficient of (v + 1/2)2 in the expansion (7.198) this term has the expected reduced mass dependence. However, there is also a small contribution 3hA4/2 to the coefficient Too with the wrong reduced mass dependence (/i ). This is an example of the higher-order corrections revealed by Dunham s calculation. 

It is easy to account for what has been called Costain s errors [7,5] by adding to each variance 02(/i ]) on the diagonal of Arm an additional variance term (A/j 11)2. Costain has estimated the uncertainty of a / -coordinate with absolute magnitude h (in pm) as A/i = 15/17/1 pm (when the reduced mass of substitution is assumed as 1). Errors so introduced would be correctly propagated to any further processing of Arm. There is still another, perhaps more fundamental way, to... 

Burdett etal " showed that, to fit isotopic spectra accurately, the EFFF needed two corrections a factor that takes into account the particular isotope of CO used and a further correction to ameliorate the effects of using anharmonic frequencies. The effect of the neglect of M-C/C-O conphng was originally explained by Miller in terms of an x-factor, which depended on the particular CO isotope being nsed. Anharmonicity is accounted for by another correction that could be determined from the all- C 0 molecnle and the reduced mass ratio. 

The Fermi formula (equation 11.2) together with the reduced-mass and relativistic correction terms, evaluated with Dumond and Cohen s values of the atomic constants [39], leads to the hyperfme structure interval... 

A further correction, a type of reduced mass effect, arises from the recoil of the nucleus in the emission of virtual photons which, in the theory, describe the attraction between the nucleus and the electron. The effect is not independent of the structure of the nucleus. It is, however, possible to separate the contributions due to mass and structure, so that the correction appears as the product of two terms, Gr and C8. The magnitude of Cr9 the contribution from a point proton (equation 11.6), has been estimated as parts in 105 [97]. Cs, the structure correction [146], is found to be (1 —2r/a0), where r is an average electromagnetic radius, and a0 is the Bohr radius. G9 is independent of n([146], p. 1773). 

Table 4.1 summarizes the various contributions to the energy, expressed as a double expansion in powers of a 1/137.036 and the electron reduced mass ratio ji./M 10 . Since all the lower-order terms can now be calculated to very high accuracy, including the QED terms of order Ry, the dominant source of uncertainty comes from the QED corrections of order Ry or higher. The comparison between theory and experiment is therefore sensitive to these terms. For the isotope shift, the QED terms independent of /x/M cancel out, and so it is only the radiative recoil terms of order a fx/M 10 Ry ( 10 kHz) that contribute to the uncertainty. Since this is much less than the finite nuclear size correction of about 1 MHz, the comparison between theory and experiment clearly provides a means to determine the nuclear size. 

Therefore, if we know the harmonic vibrational frequencies for our model, we simply add up the 2 0) contribution from each vibrational mode, i, and multiply by Avogadro s number to report energies in units of kJ moP Thus we can see that those modes that correspond to a higher wavenumber (the bond stretches, particularly those that involve hydrogen, as a consequence of the reduced mass term given in the equation in Section 8.2.1) make the greatest contribution to the ZPE correction. 

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