Internuclear axis

Rotational diffusion coefficient, Dg, internal motion rate parameter, angle between the internal rotation axis and the internuclear axis... 

FIGURE 2 17 The carbon-carbon double bond in ethylene has a cr component and a tt compo nent The cr component arises from overlap of sp hybridized orbitals along the internuclear axis The tt component results from a side by side overlap of 2p orbitals... 

FIGURE 3 10 Bent bonds in cyclopropane (a) The orbitals involved in carbon-carbon bond formation overlap in a region that is displaced from the internuclear axis (b) The three areas of greatest negative electrostatic potential (red) correspond to those predicted by the bent bond description... 

A bond in which the orbitals overlap along a line connecting the atoms (the inter-ntidear axis) is called a sigma (a) bond. The electron distribution in a a bond is cylin-drically symmetric were we to slice through a a bond perpendicular to the internuclear-axis, its cross-section would appear- as a circle. Another way to see the shape of the electron distribution is to view the molecule end-on. 

Each carbon of ethylene uses two of its sp hybrid orbitals to form a bonds to two hydrogen atoms, as illustrated in the first par-f of Figure 2.17. The remaining sp orbitals, one on each carbon, overlap along the internuclear- axis to give a a bond connecting the two carbons. 

Strong sp -sp a bonds are not possible for cyclopropane, because the 60° bond angles of the ring do not permit the orbitals to be properly aligned for effective overlap (Figure 3.10). The less effective overlap that does occur leads to what chemists refer to as bent bonds. The electron density in the carbon-carbon bonds of cyclopropane does not lie along the internuclear- axis but is distr-ibuted along an arc between the two carbon atoms. The r-ing bonds of cyclopropane are weaker than other carbon-carbon a bonds. 

Now imagine that we rotate the molecule about the internuclear axis. The curved contour will trace out a surface. If we draw a unit outward normal vector to this surface, it will be everywhere perpendicular to the gradient vector (because the gradient vector points along the trajectory). 

The antisymmetric stretching vibration. The molecule loses its original symmetry during the vibration. At the two extrema of the vibration the shapes of the molecule will be identical. Because the molecular polarizability is essentially the summation of all bond polarizabilities and is independent of direction along the internuclear axis, it will have identical values at the extrema. Consequently, the vibration is Raman inactive. 

FIGURE 3.8 When electrons with opposite spins (depicted as t and 1) in two hydrogen 1s-orbitals pair and thes-orbitals overlap, they lorm a boundary surface of the electron cloud. The cloud has cylindrical symmetry around the internuclear axis and spreads over both nuclei. In the illustrations in this book, cr-bonds are usually colored blue... 

The Greek letter sigma, electron distribution resembles that of an s-orbital. 

We encounter a different type of bond in a nitrogen molecule, N2. There is a single electron in each of the three 2p-orbitals on each atom (33). However, when we try to pair them and form three bonds, only one of the three orbitals on each atom can overlap end to end to form a (T-bond (Fig. 3.10). Two of the 2/7-orbitals on each atom (2px and 2py) are perpendicular to the internuclear axis, and each one contains an unpaired electron (Fig. 3.11, top). When the electrons in one of these p-orbitals on each N atom pair, the orbitals can overlap only in a side-by-side arrangement. This overlap results in a TT-bond, a bond in which the two electrons lie in two lobes, one on each side of the internuclear axis (Fig. 3.11, bottom). More formally, a 7T-bond has a single nodal plane containing the internuclear axis. Although a TT-bond has electron density on each side of the internuclear axis, it is only one bond, with the electron cloud in the form of two lobes, just as a p-orbital is one orbital with two lobes. In a molecule with two Tr-bonds, such as N2, the... 

FIGURE 3.9 A o-bond can also be formed when electrons in 1a- and 2p,-orbitals pair (where z is the direction along the internuclear axis). The two electrons in the bond are spread over the entire region of space enclosed by the boundary surface. 

FIGURE 3.30 Two p-orbitals can overlap side by side to give bonding (below) and antibonding (above) u-orbitals. Note that the latter has a nodal plane between the two nuclei. Both orbitals have a nodal plane through the two nuclei and look like p-orbitals when viewed along the internuclear axis. 

The radial wave functions used are thus the hydrogen-like 2p and 3d functions, J ai(r) and J 32-(r), for all orbitals of the L and M shells, respectively the symbols pts, and i 3j, piP, p3d represent these multiplied by the angular parts 1 (for s), /3 cos 8 (for p), and /5/4 (3 cos2 0-1) (for d), rather than the usual hydrogen-like orbitals. The 2-axis for each atom points along the internuclear axis toward the other atom. 

Ethyne has two jt bonding orbitals at right angles to each other and a resultant jt electron density that is cylindrically symmetric with respect to the internuclear axis. Complexes of ethyne with HF [133], HC1 [134], HBr [135], C1F [66], CI2 [47], BrCl [50], Br2 [92] and IC1 [95] have been characterised by... 

There are two general conclusions of importance. First, the distance r(Z- X), where Z is the electron donor atom/centre in the complex B- XY, is smaller than the sum of the van der Waals radii ax and ax of these atoms. This result has been shown [179] to be consistent with the conclusion that the van der Waals radius of the atom X in the dihalogen molecule X is shorter along the XY internuclear axis than it is perpendicular to it, i.e. there is a polar flattening of the atom X in the molecule XY of the type suggested by Stone et al. [180]. This result has been shown to hold for the cases XY = CI2 [174], BrCl [175], C1F [176] and IC1 [178], but not for F2, in which the F atom in the molecule appears (admittedly on the basis of only a few examples) to be more nearly spherical [177]. 

We assume that, on formation of B- XY, a fraction 5j (i = intermolecular) of an electronic charge is transferred from the electron donor atom of Z of the Lewis base B to the npz orbital of X and that similarly a fraction 5p (p = polarisation) of an electronic charge is transferred from npz of X to n pz of Y, where z is the XY internuclear axis and n and n are the valence-shell principal quantum numbers of X and Y. Within the approximations of the Townes-Dailey model [187], the nuclear quadrupole coupling constants at X and Y in the hypothetical equilibrium state of B- -XY can be shown [178] to be given by ... 

The equilibrium angular geometry of a halogen-bonded complex B- XY can be predicted by assuming that the internuclear axis of a XY or X2 molecule bes ... 

Figure 6.5 Contour plot of the electron density in a plane containing the nuclei of (a) CO and (b) CI2. both drawn at the same scale. Along the internuclear axis, the electron density reaches its minimum value at a point marked by a square. For CI2 this is the midpoint.

Figure 6.6 Relief map of the electron density for CO in a plane containing the molecular axis. The electron density falls off more rapidly for displacements perpendicular to the internuclear axis than along the internuclear axis.

Bond paths are observed between bonded atoms in a molecule and only between these atoms. They are usually consistent with the bonds as defined by the Lewis structure and by experiment. There are, however, differences. There is only a single bond path between atoms that are multiply bonded in a Lewis structure because the electron density is always a maximum along the internuclear axis even in a Lewis multiple bond. The value of pb does, however, increase with increasing Lewis bond order, as is shown by the values for ethane (0.249 au), ethene (0.356 au), and ethyne (0.427 au), which indicate, as expected, an increasing amount of electron density in the bonding region. 

The state of the superoxide ion has been summarized by Naceache et al. 22). It appears probable that an ionic model is most suitable for the adsorbed species since the hyperfine interaction with the adjacent cation is relatively small. Furthermore, the equivalent 170 hyperfine interaction suggests that the ion is adsorbed with its internuclear axis parallel to the plane of the surface and perpendicular to the axis of symmetry of the adsorption site. Hence, the covalent structures suggested by several investigators have not been verified by ESR data. 

After the cr bond has formed, further interaction of the p orbitals on the two atoms is restricted to the px and py orbitals, which are perpendicular to the pz orbital. When these orbitals interact, the region of orbital overlap is not symmetrical around the internuclear axis but rather on either side of the internu-clear axis, and a 7r bond results. Orbital overlap of this type is also shown in Figures 3.5 and 3.6. The combinations of wave functions for the bonding -n orbitals can be written as... 

For diatomic molecules, there is coupling of spin and orbital angular momenta by a coupling scheme that is similar to the Russell-Saunders procedure described for atoms. When the electrons are in a specific molecular orbital, they have the same orbital angular momentum as designated by the m value. As in the case of atoms, the m value depends on the type of orbital. When the internuclear axis is the z-axis, the orbitals that form a bonds (which are symmetric around the internuclear axis) are the s, pz, and dzi orbitals. Those which form 7r bonds are the px, p, dlz, and dyi orbitals. The cip-y2 an(i dxy can overlap in a "sideways" fashion with one stacked above the other, and the bond would be a 8 bond. For these types of molecular orbitals, the corresponding m values are... 

The molecular orbitals are labelled a and ir depending on whether they are symmetrical about the internuclear axis or have a nodal plane passing through the nuclei. The m.o. s are numbered in sequence of increasing energy, independent of the numbering of the atomic orbitals. This numbering serves to avoid any confusion in cases where atomic orbitals from different levels are combined, as in... 

A fascinating feature of the G 2 species occurs in the excited singlet state. The 1G 2 species is bound by 29.0 kcalmol-1 at Re = 2.7444 A and has ordinarylooking bond order b = 1. However, the bonding character is remarkably different in the ot and (3 spin sets, corresponding to bent banana bonds of opposite curvature with respect to the internuclear axis. Figure 3.32 displays the form of one of these spin-NBOs, showing the off-axis curvature with respect to the Ga—Ga line of... 

Fig. 7.1 The electron density p(t) is displayed in the and

Fig. 11 (see p. 86/87) contains all trigonal orbitals which were encountered in the molecules considered. The bonding orbitals, in the left column, exhibit the increasing polarization from N2 to LiF. Moreover, the inclination of the contributing (sp) hybrid of the right atom into the bond region diminishes as the polarization increases, i.e., the axis of this hybrid is much closer to being perpendicular to the internuclear axis in LiF than in N2. Clearly, an increase in (p) character accompanies the diminshed inclination. 

The effective hamiltonian in formula 29 incorporates approximations that we here consider. Apart from a term V"(R) that originates in nonadiabatic effects [67] beyond those taken into account through the rotational and vibrational g factors, other contributions arise that become amalgamated into that term. Replacement of nuclear masses by atomic masses within factors in terms for kinetic energy for motion both along and perpendicular to the internuclear axis yields a term of this form for the atomic reduced mass. 

The fluctuations in the orientation of anisotropically polarizable molecules in liquids also cause frequency broadening of the scattered light, as investigated for CSj in CCI4 239). CS is a highly polarizable molecule with very different polarizabilities along and perpendicular to the internuclear axis. CCU on the other hand, is a poor scatterer because it is an isotropic molecule. Thus, if CSj is mixed with CCI4, the CSj molecules can be studied in a new environment. 

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