Term value

Valuable results have been attained with these programs, but much more is possible in the future. For one there are the term values of highly ionized heavy atoms, which are difficult to access experimentally, but also the corresponding transition probabilities important for the explicit simulation of a high-temperature plasma. Another aspect, which has attracted attention, is the hyperfine structure of atomic spectra and with it the determination of nuclear moments in the combination of computation and high-resolution experiments. 

Desclaux s list of Dirac-Fock expectation values for almost all atoms of the periodic table (Desclaux 1973), which has served as a very valuable reference in the past, has been updated and extended (Visscher and Dyall 1997). 

---

Flartree D R 1928 The wave meohanios of an atom with a non-Coulomb oentral field. Part III. Term values and intensities in series in optioal speotra Proc. Camb. Phil. See. 24 426-37... 

In practice, what is measured experimentally is not energy but frequency, in the millimetre wave and microwave regions, or wavenumber, in the far infrared. Therefore we convert the energy levels of Equation (5.10) to what are known as term values F J) having dimensions of either frequency, by dividing by h, or wavenumber, by dividing by he, giving... 

Figure 5.2 shows a set of rotational energy levels, or, strictly, term values, for the CO molecule. 

Figure 5.2 Rotational term values F J) (horizontal lines) relative populations Nj/Nq (calculated from Equation 5.15) and transition wavenumbers v (for the transitions indicated by the vertical arrows) for CO...

There is a stack of rotational levels, with term values such as those given by Equation (5.19), associated with not only the zero-point vibrational level but also all the other vibrational levels shown, for example, in Figure 1.13. However, the Boltzmann equation (Equation 2.11), together with the vibrational energy level expression (Equation 1.69), gives the ratio of the population of the wth vibrational level to Nq, that of the zero-point level, as... 

The rotational constants B and D are both slightly vibrationally dependent so that the term values of Equation (5.19) should be written... 

For an oblate symmefric rotor, such as NFI3, fhe rofafional term values are given by... 

When the effects of centrifugal distortion are included the term values of a prolate symmetric rotor are given by... 

Although these molecules form much the largest group we shall take up the smallest space in considering their rotational spectra. The reason for this is that there are no closed formulae for their rotational term values. Instead, these term values can be determined accurately only by a matrix diagonalization for each value of J, which remains a good quantum number. The selection mle A/ = 0, 1 applies and the molecule must have a permanent dipole moment. 

At a simple level, the rotational transitions of near-symmetric rotors (see Equations 5.8 and 5.9) are easier to understand. For a prolate or oblate near-symmetric rotor the rotational term values are given, approximately, by... 

Examples of prolate near-symmetric rotors are the s-trans and s-cis isomers of crotonic acid, shown in Figure 5.8, the a axis straddling a chain of the heavier atoms in both species. The rotational term values for both isomers are given approximately by Equation (5.37) but, because A and B are different for each of them, their rotational transitions are not quite coincident. Figure 5.9 shows a part of a low-resolution microwave spectmm of crotonic acid in which the weaker series of lines is due to the less abundant s-cis isomer and the stronger series is due to the more abundant s-trans isomer. 

Starting from an expression for rotational term values show that... 

Figure 1.13 shows the potential function, vibrational wave functions and energy levels for a harmonic oscillator. Just as for rotation it is convenient to use term values instead of energy levels. Vibrational term values G(v) invariably have dimensions of wavenumber, so we have, from Equation (1.69),... 

However, unlike electrical anharmonicity, mechanical anharmonicity modifies the vibrational term values and wave functions. The harmonic oscillator term values of Equation (6.3) are modified to a power series in (u + ) ... 

One effect of the modification to the harmonic oscillator term values is that, unlike the case of the harmonic oscillator, cOg cannot be measured directly. Wavenumbers AO y2 for v+ ) — V transitions are given by... 

When a molecule has both vibrational and rotational energy the total term values S are given by the sum of the rotational term values F J), given in Equation (5.23), and the vibrational term values G v), given in Equation (6.16) ... 

The reason for the subscript 2 in the A FiJ) symbol is that these are the differences between rotational term values, in a particular vibrational state, with J differing by 2. 

In an approximation which is analogous to that which we have used for a diatomic molecule, each of the vibrations of a polyatomic molecule can be regarded as harmonic. Quantum mechanical treatment in the harmonic oscillator approximation shows that the vibrational term values G(v ) associated with each normal vibration i, all taken to be nondegenerate, are given by... 

Neglecting centrifugal distortion, the rotational term values for a spherical rotor in an A j vibrational state are... 

Although, as in linear and symmetric rotor molecules, the term values are slightly modified by Coriolis forces in a degenerate (T2) state, the rotational selection rules... 

The vibrational term values for a polyatomic anharmonic oscillator with only nondegenerate vibrations are modified from the harmonic oscillator values of Equation (6.41) to... 

For an anharmonic oscillator with degenerate vibrations the term values are modified from those of Equation (6.88) to... 

The only types of anharmonic potential function we have encountered so far are the two illustrated in Figure 6.38, both of which show only a single minimum. There are, however, some vibrations whose potential functions do not resemble either of those but show more than one minimum and whose term values are neither harmonic, nor are they given by Equation (6.88) or Equation (6.89). Such vibrations can be separated into various types, which will now be discussed individually. 

Just as in the ground electronic state a molecule may vibrate and rotate in excited electronic states. The total term value S for a molecule with an electronic term value T,... 

The vibrational term values for any electronic state, ground or excited, can be expressed, as in Equation (6.16), by... 

If a sufficient number of vibrational term values are known in any electronic state the dissociation energy Dq can be obtained from a Birge-Sponer extrapolation, as discussed in... 

Section 6.13.2 and illustrated in Figure 6.5. The possible inaccuracies of the method were made clear and it was stressed that these are reduced by obtaining term values near to the dissociation limit. Whether this can be done depends very much on the relative dispositions of the various potential curves in a particular molecule and whether electronic transitions between them are allowed. How many ground state vibrational term values can be obtained from an emission spectrum is determined by the Franck-Condon principle. If r c r" then progressions in emission are very short and few term values result but if r is very different from r", as in the A U — system of carbon monoxide discussed in Section 7.2.5.4, long progressions are observed in emission and a more accurate value of Dq can be obtained. 

In Figure 7.25 are shown stacks of rotational levels associated with two electronic states between which a transition is allowed by the -F -F and, if it is a homonuclear diatomic, g u selection rules of Equations (7.70) and (7.71). The sets of levels would be similar if both were states or if the upper state were g and the lower state u The rotational term values for any X state are given by the expression encountered first in Equation (5.23), namely... 

Saturation kinetics are also called zero-order kinetics or Michaelis-Menten kinetics. The Michaelis-Menten equation is mainly used to characterize the interactions of enzymes and substrates, but it is also widely applied to characterize the elimination of chemical compounds from the body. The substrate concentration that produces half-maximal velocity of an enzymatic reaction, termed value or Michaelis constant, can be determined experimentally by graphing r/, as a function of substrate concentration, [S]. 

---