Angles between bonds

With molecules containing three or more atoms, the geometry is not so obvious. Here, the angles between bonds, called bond angles, must be considered. For example, a molecule of the type YX J where Y represents the central atom and X an atom bonded to it, can be—... 

The Angles between Bonds.—The above calculation of tetrahedral angles between bonds when the quantization is changed sets an upper limit on bond angles in doubtful cases, when the criterion is only approximately satisfied. For we can now state that the bond angles in H20 and NH3... 

Angles between Bonds, from Crystal Structure Data ... 

Compound Atom Number of bond9 Angles between bonds... 

Reif, B., Hennig, M., Griesinger, C. Direct measurement of angles between bond vectors in high-resolution NMR. Science im, 276,1230-1233. 

Because lone pair domains are larger and more spread out around the central core than bonding pair domains, the angles between bonding pair domains are smaller than the angles between lone pair domains. Consequently ... 

Since the discovery of the nuclear Overhauser effect (NOE, see previous section) [4, 5] and scalar coupling constants [36, 37] decades ago, NMR-derived structure calculations of biomolecules largely depended on the measurement of these two parameters [38]. Recently it became possible to use cross-correlated relaxation (CCR) to directly measure angles between bond vectors [39] (see also Chapt 7). In addition, residual dipolar couplings of weakly aligned molecules were discovered to measure the orientation of bond vectors relative to the alignment tensor (see Sect 16.5). Measurement of cross-correlated relaxation was described experimentally earlier for homonuclear cases [40, 41] and is widely used in solid-state NMR [42 14]. 

As is common with empirical force fields, MUBFF calculations are carried out using internal molecular coordinates rather than Cartesian coordinates. Internal coordinates describe the structure of a molecule in terms of bond lengths and angles between bonds. As an example, for a bent tri-atomic molecule ABC the three internal coordinates include the lengths of bonds AB (r s) and BC (rec), as well as the angle between them (aABc)- Larger molecules may also... 

All other factors are constants. The angle 9 is the projection angle between bond vectors and is the sole unknown in Eq. 1, and can be readily and precisely determined by measurement of the cross-correlated dipolar relaxation rate. It should be emphasized that 9 is measured directly, without the need of experimental calibration as in the Karplus curve for J-coupling constants for example. 

Modeling programs allow display of distances, bond angles, and dihedral angles between bonded and nonbonded atoms. These measurements float on dotted lines within the model (just like labels) and are often active that is, they are continually updated as the model is changed, as described in the next section. 

As an example, consider a chain of atoms— 1,/ ,/ + 1... with a fixed valence angle so that the angle between bonds / — 1, / and j>. [I + 1 is fixed at 6. Let dj. i = 1,2,3, be the unit base vectors of the fixed laboratory reference frame. In addition, we introduce a moving local Cartesian system with base vectors dj = 1,2,3, defined in Fig. 11. Then any vector r can be written in terms of components with respect to either frame as r = r, c, = fj dj, where we are adopting throughout the convention that superimposed bars denote components with respect to the local frame a,. The vector components with respect to the two frames are related in the usual way as... 

A double bond (4 electrons two pairs) is represented as =. The angle between bonds to the carbon atoms involved in a double bond is 120° and all the bonds lie in the same plane. The ethene molecule is flat in contrast to the tetrahedral arrangement of single bonds. 

If more than two carbon atoms join together to form a chain, they necessarily form a zigzag structure, on account of the tetrahedral angle between bonds. Thus in Fig. XXV-2 we show propane CH3CH2CHJ... 

Bond length, A Angles between bonds (0 ), deg Type of rotation... 

Fig. 3.—Projected Angles Between Bonds Proceeding from Adjacent Carbon Atoms...

The bond angles in the CH4, NH3, and H20 molecules. Note that the bond angle between bonding pairs decreases as the number of lone pairs increases. 

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