Coordinates, atomic molecule centered

Polyatomic molecules vibrate in a very complicated way, but, expressed in temis of their normal coordinates, atoms or groups of atoms vibrate sinusoidally in phase, with the same frequency. Each mode of motion functions as an independent hamionic oscillator and, provided certain selection rules are satisfied, contributes a band to the vibrational spectr um. There will be at least as many bands as there are degrees of freedom, but the frequencies of the normal coordinates will dominate the vibrational spectrum for simple molecules. An example is water, which has a pair of infrared absorption maxima centered at about 3780 cm and a single peak at about 1580 cm (nist webbook). 

Applications of the theory described in Section III.A.2 to malonaldehyde with use of the high level ab initio quantum chemical methods are reported below [94,95]. The first necessary step is to define 21 internal coordinates of this nine-atom molecule. The nine atoms are numerated as shown in Fig. 12 and the Cartesian coordinates x, in the body-fixed frame of reference (BF) i where n= 1,2,... 9 numerates the atoms are introduced. This BF frame is defined by the two conditions. First, the origin is put at the center of mass of the molecule. 

Here Slater functions (a3 2/7r 1/2) exp( —a r — ra ) with the atom being centered at position vectors ra. The overlap between these functions is given by S. After an FT and integrating over momentum coordinates of one particle, the EMD of H2 molecule within VB and MO theory are derived as... 

Of the 3n coordinates needed to describe an n-atom molecule, three are used for center of mass motion, three describe angular displacement (rotation, hindered rotation, or libration) (two if the molecule is linear, 0 if monatomic), the remaining 3n—6 (3n—5, if linear, 3n—3 = 0, if monatomic) describe atom-atom displacements (vibrations). In some cases it may not be possible to separate translation cleanly from rotation and vibration, but when the separation can be made it is a convenience. Elementary treatments assume... 

Fig. 5.7 A four-atom molecule in a coordinate system. Only one of possibly many electrons is shown. The basis functions are one-electron functions, usually centered on atomic nuclei. Ri, R2, etc., are vectors representing the x, y, z coordinates (conveniently as 3 x 1 column matrices Section 4.3.3) of the nuclei ( of the atoms ), and r is a vector representing thex, y, z coordinates of an electron. The distances of the electron from the centers of the various basis functions are the absolute values of the various vector differences Ir — R l, Ir — R2I, etc. For a particular molecular geometry, Rlt R2, etc. are fixed and enter the functions (x, y, z). Several basis functions may be centered on each nucleus...

The expression in Eq. (10.56) only becomes simple in the special case when the reactants A and B are atoms with no internal degrees of freedom. Then, in the second factor which is related to the gas-phase reaction, Va = Vb = 0, and with no interactions between the reactant molecules, Vint, = 0. The integral over reactant states is therefore simply equal to the square of the volume, V2. The reaction coordinate Q is the distance, r, between the atoms, so the five coordinates involve the center-of-mass coordinates and the rotational coordinates for the linear activated complex. The integrals over them cancel, since both p( ) and V/ t ra are independent of them. There are no other integration variables, so the integrands just take the value they have for <3 = 0. We find... 

Diastereoisomerism is observed when two or more chiral elements appear in the same system. Isolation and characterization of enantiomers or diastereoisomers depend on the structural stability of the chiral element. Therefore the conformation of a chelate ring must generally be considered as labile. The structural stability of the configuration around the metal center and the stability of an asymmetric coordination atom depends on the nature of the central metal atom. In most cases an optically active ligand molecule can be considered as configurationally stable. The presence of a stable chiral element can induce definite chirality in another element which, taken separately, would be labile. 

If we analyze a crystal such as sodium chloride in accordance with the theory here proposed, we do not make any assumptions to the effect that the crystal contains atoms or molecules. We simply assume that the fundamental principles of the theory represent the facts and that the distribution of the diffracting power conforms to the crystal symmetry. In addition, we assume that if we take the origin of coordinates at the center of the heaviest atom, all of the coefficients in the Fourier series have positive values. It will be shown by Dr. Havighurst in subsequent notes that the... 

While small carbon molecules have been found in locations as diverse as interstellar molecular clouds, carbon stars, hydrocarbon flames and laser-ablated carbon, bulk quantities are much less easily produced and have therefore been little studied. However, as with many organic intermediates, stabilization of these reactive molecules by coordination to metal centers can be achieved and is the subject of this review. The preparation of systems containing metal centers linked by carbon-atom chains is both a synthetic challenge (in spite of the first such being obtained over 45 years ago) and of considerable current relevance. 

The theory (7, 8, 9,10,11,12) will be outlined for molecules having n atoms with a total of P valence shell electrons. We seek a set of molecular orbitals (LCAO-MO s), that are linear combinations of atomic orbitals centered on the atoms in the molecule. Since we shall not ignore overlap, the geometry of the molecule must be known, or one must guess it. The molecule is placed in an arbitrary Cartesian coordinate system, and the coordinates of each atom are determined. Orbitals of the s and p Slater-type (STO) make up the basis orbitals, and as indicated above we restrict ourselves to the valence-shell electrons for each of the atoms in the molecule. The STOs have the following form for the radial part of the function (13,18) ... 

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