Asymmetric molecules rotations

Equation XVI-21 provides for the general case of a molecule having n independent ways of rotation and a moment of inertia 7 that, for an asymmetric molecule, is the (geometric) mean of the principal moments. The quantity a is the symmetry number, or the number of indistinguishable positions into which the molecule can be turned by rotations. The rotational energy and entropy are [66,67]... 

As is the case for diatomic molecules, rotational fine structure of electronic spectra of polyatomic molecules is very similar, in principle, to that of their infrared vibrational spectra. For linear, symmetric rotor, spherical rotor and asymmetric rotor molecules the selection mles are the same as those discussed in Sections 6.2.4.1 to 6.2.4.4. The major difference, in practice, is that, as for diatomics, there is likely to be a much larger change of geometry, and therefore of rotational constants, from one electronic state to another than from one vibrational state to another. 

Of the visible spectroscopic techniques, CD spectroscopy has seen the most rapid and dramatic growth. The far-UV circular dichroism spectrum of a protein is a direct reflection of its secondary structure [71]. An asymmetrical molecule, such as a protein macromolecule, exhibits circular dichroism because it absorbs circularly polarized light of one rotation differently from circularly polarized light of the other rotation. Therefore, the technique is useful in determining changes in secondary structure as a function of stability, thermal treatment, or freeze-thaw. 

As mentioned in Section 1.2, the presence of an asymmetric carbon is neither a necessary nor a sufficient condition for optical activity. Each enantiomer of a chiral molecule rotates the plane of polarized light to an equal degree but in opposite directions. A chiral compound is optically active only if the amount of one enantiomer is in excess of the other. Measuring the enantiomer composition is very important in asymmetric synthesis, as chemists working in this area need the information to evaluate the asymmetric induction efficiency of asymmetric reactions. 

Equations relating and the spin-rotation constants have been given by Ramsey 9, 77, 78) for linear molecules and by Flygare 24) for symmetric top, spherical top, and asymmetric molecules. A simple expression for the general case, and neglecting vibrational effects, is... 

The phenomenon of melting point also involves the entropy of melting, which has a much smaller value for a symmetrical molecule (such as methane, with a symmetry number of 12) than with an asymmetrical molecule (such as CH2CIF, with a symmetry number of 1). When we introduce the fourth parameter of the entropy of rotational symmetry given by R In(cr), the regression results is... 

The eigenfunctions of J2, Ja (or Jc) and Jz clearly play important roles in polyatomic molecule rotational motion they are the eigenstates for spherical-top and symmetric-top species, and they can be used as a basis in terms of which to expand the eigenstates of asymmetric-top molecules whose energy levels do not admit an analytical solution. These eigenfunctions IJ,M,K> are given in terms of the set of so-called "rotation matrices" which are denoted Dj m,K ... 

Absorption of radiation giving rise to transitions between opposite spin orientations of unpaired electrons in a magnetic field Rotation of the plane polarized light by asymmetric molecules in solution without and with variation in wavelength... 

The assignment of an asymmetric centre as (R) or (S) has nothing to do with whichever direction the molecule rotates plane-polarised light. Optical rotation can only be determined experimentally. By convention, molecules which rotate plane-polarised light clockwise are written as (+) or d. Molecules which rotate plane-polarised light counterclockwise are written as (-) or 1. The (R) enantiomer of lactic acid is found to rotate plane-polarised light counterclockwise and so this molecule is defined as (R)-(-)-lactic acid. 

O. Methods of various degrees of sophistication, for the V electron density at the nitrogen atom. Taking the ft population as constant, there is a good correlation between calculated and observed a values 70>. To extend 71> 72> the calculation to asymmetric molecules, like pyrimidine or pyridazine, the new orientation of the electric field gradient tensor must be considered. In pyridazine the two nitrogen Oz axes are rotated in a way that brings them nearer parallel and, from microwave measurements, this rotation was estimated at 9° 73 ... 

The two compounds differ physically only in the orientation of their crystals and the direction in which their solutions rotate polarized light. These may seem like trivial differences, but when such optical isomers interact with other asymmetric molecules (especially enzymes) they may acquire very different chemical reactivities. This principle of optical selectivity is central to the biochemistry of carbohydrates (and other classes of naturally occurring compounds as well). The manner of drawing the two isomers as shown in the top part of Figure III-I is called the Fischer projection. In this projection, the most oxidized (aldehyde) end of the molecule is drawn up and is considered to be behind the plane of the paper. The -CH2OH end is drawn down and is also behind the plane. Because the groups are... 

Two independent molecules are found in the monoclinic Pa crystal of 41, both in the bisected form. The distal bond in the cyclopropane ring is shorter by 0.018 A than the vicinal bonds. The phenyl groups of 42 are rotated (as calculated from atomic coordinates) by 19" and -85° from the bisecting position, away from each other. Such conformation-ally asymmetric molecules make up the chiral orthorhombic crystals. The phenyl... 

When a molecule rotates in a liquid, the motion of the electrons (seen as an asymmetric charge distribution attached to the molecule) generates a magnetic field at the nucleus, which couples to the nuclear spin I. The Hamiltonian Msr can be given as [4] ... 

Rotations of Asymmetric Molecules and the Hydrogen Atom in Free and Confined Configurations... 

Rotations of asymmetric molecules revisited Matrix evaluation and generating function of LamS functions... 

Rotations of asymmetric molecules in semi-infinite spaces with elliptical cone boundaries... 

The exact solution of the Schrbdinger equation for the rotations of asymmetric molecules is possible because of the existence of complete sets of commuting operators, namely the respective Hamiltonians H and H, and the square of angular momentum H. The practical advantage of using the Hamiltonian H over H is that the latter involves three independent parameters a,b,c), whereas H involves only one independent parameter. 

Both works [2] and [3] show the separations of the eigenvalue equations for H and H, and H and H, in their respective spheroconal coordinates, into Lame differential equations in the individual elliptical cone angular coordinates. The corresponding solutions are Lam6 spheroconal polynomials included in the classic book of Whittaker and Watson [12]. In practice, the numerical evaluation of such Lame functions was not developed in an efficient manner so that the exact formulation of Ref. [2] did not prosper. Consequently, the analysis of rotations of asymmetric molecules took the route of perturbation theory using the familiar basis of spherical harmonics. 

REVIEW OF EXACT FORMULATIONS AND EVALUATIONS OF ROTATIONS OF ASYMMETRIC MOLECULES AND THE HYDROGEN ATOM... 

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