Free molecules

Here, superscript F denotes relatively "free" molecules (weak physical forces), and D denotes relatively "bound" or "dimerized" molecules ("chemical" forces). 

Fig. XVIII-16. A four-electron two-orbital interaction that a) has no net bonding in the free molecule but can be bonding to a metal surface if (b) the Fermi level is below the antibonding level. In the lower part of the figure, a zero-electron two-orbital situation (c) has no bonding but there can be bonding to a metal surface as in (d) if the Fermi level is above the bonding level. (From Ref. 160.)...

These simple examples serve to show that instinctive ideas about symmetry are not going to get us very far. We must put symmetry classification on a much firmer footing if it is to be useful. In order to do this we need to define only five types of elements of symmetry - and one of these is almost trivial. In discussing these we refer only to the free molecule, realized in the gas phase at low pressure, and not, for example, to crystals which have additional elements of symmetry relating the positions of different molecules within the unit cell. We shall use, therefore, the Schdnflies notation rather than the Hermann-Mauguin notation favoured in crystallography. 

It should also be remembered that the selection mles derived here are relevant to the free molecule and may break down in the liquid or solid state. This is the case, for example, with the electric dipole forbidden 4q transition in ethylene, where V4 is the torsional vibration shown in Figure 6.23. It is not observed in the infrared specttum of the gas but is observed weakly in the liquid and solid phases. 

Stabilization of Unstable Intermediates. Transition metals can stabilize normally unstable or transient organic intermediates. Cyclobutadiene has never been isolated as a free molecule, but it has been isolated and fully characterized as an iron tricarbonyl complex (138) ... 

Electronic structure studies of free molecules (gas phase), well-defined solid surfaces, and adsorbates on solid surfaces... 

State or vibrational energies. If the current across the metal-insulator-metal sandwich is recorded as a function of applied voltage, the current increases as the threshold for each state or vibrational mode is crossed. The increases in current are in fact very small, and for improved detectability the current is double-differentiated with respect to voltage, thereby providing, in effect, a vibrational spectrum that can be compared directly with free-molecule IR and Raman spectra. 

The most extensive calculations of the electronic structure of fullerenes so far have been done for Ceo- Representative results for the energy levels of the free Ceo molecule are shown in Fig. 5(a) [60]. Because of the molecular nature of solid C o, the electronic structure for the solid phase is expected to be closely related to that of the free molecule [61]. An LDA calculation for the crystalline phase is shown in Fig. 5(b) for the energy bands derived from the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) for Cgo, and the band gap between the LUMO and HOMO-derived energy bands is shown on the figure. The LDA calculations are one-electron treatments which tend to underestimate the actual bandgap. Nevertheless, such calculations are widely used in the fullerene literature to provide physical insights about many of the physical properties. 

The Raman spectrum in Fig. 10 for solid Ceo shows 10 strong Raman lines, the number of Raman-allowed modes expected for the intramolecular modes of the free molecule [6, 94, 92, 93, 95, 96, 97]. As first calculated by Stanton and Newton [98], the normal modes in molecular Ceo above about 1000 cm involve carbon atom displacements that are predominantly tangential... 

K. Siegbahn, C. Nordling, G. Johansson, J. Hedman, P.F. Heden, V. Hamrin, U. Geiius, T. Berg-mark, L.O. Werme, R. Mamie, Y. Baer, ESCA Applied to Free Molecules, Norlh-Holland, Amsterdam 1971. 

Siegbahn, K. Nordling, C. Johansson, G. Hedman, J. Heden, P.F. Hamrin, K. Gelius, U. Bergmark, T. Werme, L.O. Manne, R. Baer, Y. ESCA Applied to Free Molecules . North Holland/American Elsevier New York, 1969, pp. 104fF. 

In ordinary crystalline hydrogen there are three molecules with j = 1 for every one with j = 0. The eigenfunctions for these molecules approximate those for free molecules, namely... 

Additional experimental verification that molecules of hydrogen in condensed phases are in states approximating those for free molecules is provided by the Raman effect measurements of McLennan and McLeod.13 A comparison of the Raman frequencies found by them and the frequencies corresponding to the rotational transitions / = 0—>/ = 2 and/= 1— / = 3 (Table II) shows that the intermolecular interaction in liquid hydrogen produces only a very small change in these rotational energy levels. 

Vibrations of the symmetry class Ai are totally symmetrical, that means all symmetry elements are conserved during the vibrational motion of the atoms. Vibrations of type B are anti-symmetrical with respect to the principal axis. The species of symmetry E are symmetrical with respect to the two in-plane molecular C2 axes and, therefore, two-fold degenerate. In consequence, the free molecule should have 11 observable vibrations. From the character table of the point group 04a the activity of the vibrations is as follows modes of Ai, E2, and 3 symmetry are Raman active, modes of B2 and El are infrared active, and Bi modes are inactive in the free molecule therefore, the number of observable vibrations is reduced to 10. 

The vibrations of the free molecule can be correlated with the vibrations of the crystal by group theoretical methods. Starting with the point group of the molecule Did)> the irreducible representations (the symmetry classes) have to be correlated with those of the site symmetry (C2) in the crystal and, as a second step, the representations of the site have to be correlated with those of the crystal factor group (D2h) [89, 90]. Since the C2 point group is not a direct subgroup of of the molecule and of D211 of the crystal, the correlation has to be carried out in successive steps, for example ... 

Figures 8 and 9 shows a part of the bending region at low temperature containing the components of Vg (150-160 cm ) and Vs (190-200 cm ). The Vg vibration, IR active in the free molecule, has weak components in the Raman spectrum. According to theoretically calculated Raman intensities, which almost perfectly fit the experimental spectrum, the big component has a very low scattering cross-section [87] and is accidentally degenerate with the b2g component at ca. 188 cm. The IR active components of Vg cause strong absorptions in the IR spectrum even if the crystalline sample used for transmission studies is as thin as 400 pm [107, 109].

The V3 mode observed at about 415 cm is inactive in the free molecule and, therefore, the factor group components are very weak in the Raman... 

Fig. 15 Energy level scheme of the isotopic line splitting of V9 in terms of the free molecules ( Sg and 81 87 as the main contributions) and of the orthorhombic crystal with natural abundance of isotopomers, after [109], Numerical values are observed wavenumbers (in cm ), values in brackets came from MD simulations on free rings [131] and in the case of au from LD calculations [116, 117]...

In order to distinguish between symmetry species of the free molecule and of the crystal we have chosen capital letters for the species in case of the molecule and small letters for those of the crystal... 

---