Tetratomic molecules

In tetratomic molecules,1 there are three independent vector coordinates, iq, r2, and r3, which we can think of as three bonds. The general algebraic theory tells us that a quantization of these coordinates (and associated momenta) leads to the algebra... 

Recoupling coefficients are important in computing matrix elements of operators. Consider, for example, the C operators defined, for triatomic molecules, in Eq. (4.68). For three bonds (tetratomic molecules) one has... 

For tetratomic molecules there are three possible geometric arrangements (1) linear, (2) bent planar, and (3) aplanar, examples of which are shown in Figure 5.2. We discuss here only linear tetratomic molecules. 

The procedure for studying tetratomic molecules is identical to that followed in the study of diatomic and triatomic molecules. One begins with a local-mode Hamiltonian... 

In tetratomic molecules there is only one such operator, Ci23. The local Hamiltonian (5.16) is diagonal in the basis (5.4) with eigenvalues... 

The quantum numbers va,vh,vc denote the three local stretching modes, while vd and v1 denote the two bending modes. For tetratomic molecules, it becomes... 

Summary of interbond couplings in linear tetratomic molecules... 

The types of couplings needed to describe accurately vibrational spectra of linear tetratomic molecules are summarized in Table 5.4. In most molecules, only the C, M, and C operators are needed. The S and Cf, operators are necessary only in some exceptional cases. 

Beyond the early work on acetylene (van Roosmalen et al., 1983a see also van Roosmalen, Benjamin, and Levine, 1984, and Benjamin, van Roosmalen, and Levine, 1984, for the work on the stretch modes), much of the algebraic approach to tetratomic molecules is yet to be fully published. We specifically draw attention to the thesis work of Lemus (1988), which contains important details on the Clebsh-Gordan coefficients of 0(4), and the theses of Viola (1991) and Manini (1991). The formalism necessary to describe linear and quasilinear molecules can be found in Iachello, Oss, and Lemus (1991b) Iachello, Manini, and Oss (1992) and Iachello, Oss, and Viola (1993a,b). See also Bemardes, Hornos, and Homos (1993). 

It is instructive to analyze the effect of the interaction terms (Majorana operators) in Eq. (6.24). These terms split the degeneracies of the multiplets of Figure 6.1, as shown in Figure 6.3. Thus, the Majorana terms remove the degeneracies of the local modes and bring the behavior of the molecule towards the normal limit, precisely in the same way as in tri- or tetratomic molecules. 

The methods discussed in Sections 7.14—7.20 can also be used for polyatomic molecules. Shao, Walet, and Amado (1993) have analyzed linear tetratomic molecules and found that even for these molecules the error introduced by the 1 /N expansion is of order 1%. Their results are shown in Table 7.3. 

Table 7.3 Comparison between vibrational frequencies of linear tetratomic molecules obtained by exact diagonalization of the Hamiltonian and the 1 IN (mean field) result.

This appendix provides a summary of the functional form of the algebraic Hamiltonian used in the text for tri- and tetratomic molecules. Values of the parameters are reported both for a low-order realistic representation of the spectrum and for accurate fits using terms quadratic in the Casimir operators. 

Linear (and quasilinear) tetratomic molecules. For linear and quasilinear tetratomic molecules we use a notation similar to that of Eqs. (C.2) and (C.4). Written explicitly, the Hamiltonian of linear tetratomic molecules is... 

B. More strongly Bound Molecules HCO Tetratomic Molecules Conclusions... 

Mode Selectivity in Larger Than Tetratomic Molecules... 

In addition to the VMP of water isotopologues described above, VMP of many other triatomic, as well as tetratomic molecules, has been extensively studied and reviewed in detail in Refs. [30, 31] and, on ethyne isotopologues, in Ref. [32]. As explained in these papers, VMP theories and experiments that deal with tetratomic... 

The first tetratomic molecule we shall treat is H202. The structure of H202, from electron-diffraction measurements, is helical, as shown in Figure 7-1. 

Let us illustrate two examples of v-j correlation, the dissociation of a triatomic and of a tetratomic molecule, H2O and H2O2, for example. In the first case, the correlation between recoil velocity and final fragment... 

Ll Cl" " (Figure 12) was the most abundant species, and hence the easiest to measure. A reasonably linear segment is apparent over ca. one decade, although this curve is less steep than in the case of Li. This may reflect the more complex Boltzmann population of states for this tetratomic molecule, where a pre-exponential factor is necessary. The departure from linearity is also less abrupt, and occurs at ca. 10.20 eV. We take this to be the adiabatic I.P. of Li2Cl2. Our earlier photoelectron spectroscopic studies (18) yielded 10.22 and 10.17 eV for this quantity, employing two alternative methods of extrapolation. Hence, the inference seems quite plausible that departure from linearity on the semi-logarithmic plot yields a value very close to the adiabatic ionization potential. 

Vapor pressure data, as reviewed by Brewer and Kane 37), can be represented by assuming the tetratomic molecule to be the gaseous species, with a heat of sublimation at 298 K. of 34,500 cal./mole, an entropy at 298 K. of 75.00 e. u., and a reasonable estimate of the heat capacity. According to this view, there is no appreciable concentration of the diatomic species in the vapor at saturation pressure below 1000 K., which sets a lower limit of about 48,000 cal./mole for the heat of sublimation at 298 K. for the diatomic gas. Comparison with the bond energies of P4 and Sb4 gives support to this value. 

A sample of hydrogen gas consists of these diatomic molecules (H2)—pairs of atoms that are chemically bound and behave as an independent unit—not separate H atoms. Other nonmetals that exist as diatomic molecules at room temperature are nitrogen (N2), oxygen (O2), and the halogens [fluorine (F2), chlorine (CI2), bromine (Br2), and iodine (E)]. Phosphorus exists as tetratomic molecules (P4), and sulfur and selenium as octatomic molecules (Sg and Scg) (Figure 2.14). At room temperature, covalent substances may be gases, liquids, or solids. 

---