Principal planar moments

In mechanics, the inertial properties of a rotating rigid body are fully described by its inertial moment tensor I. We can simplify the subsequent equations if we employ in place of I the closely related planar moment tensor P, apparently first used by Kraitchman [4], At any stage of the calculations, however, an equivalent equation could be given which involves I instead of P. The principal planar moments P (g = x, y, z) are the three eigenvalues of the planar moment tensor P and the principal inertial moments Ig the eigenvalues of the inertial moment tensor I. Pg, Ig, and the rotational constants Bg = f/Ig are equivalent inertial parameters of the problem investigated (/conversion factor). 

The basic equations of the -method will be presented later within the framework of the more general r -fit problem. A rigid mass point model, which is strictly true only for the equilibrium configuration, is assumed. The application of Kraitch-man s equations (see below) to localize an atomic position requires (1) the principal planar moments (or equivalent inertial parameters) of the parent or reference molecule with known total mass, and (2) the principal planar moments of the isotopomer in which this one atom has been isotopically substituted (with known mass difference). The equations give the squared Cartesian coordinates of the substituted atom in the PAS of the parent. After extracting the root, the correct relative sign of a coordinate usually follows from inspection or from other considerations. The number, identity, and positions of nonsubstituted atoms do not enter the problem at all. To determine a complete molecular structure, each (non-equivalent) atomic position must have been substituted separately at least once, the MRR spectra of the respective isotopomers must all have been evaluated, and as many separate applications of Kraitchman s equations must be carried out. 

Kraitchman s basic idea was to introduce into this first (diagonal) tensor term of Eq. 35 the three experimental principal planar moments of the parent PgWexp(l) as obtained from the MRR spectrum and treat them as independent experimental information. This is the essential distinguishing feature between any retype and any rQ-type method and all differences between the two types of treatment may be traced back to this fact. The first term of Eq. 35 is now written as P p(l). It >s then convenient to replace the notation for the planar tensor P[11(s) (Eq. 35) by I V) to distinguish this new function (Eq. 36) from Eq. 35. Note that Eq. 36, in contrast to Eq. 35, depends explicitly on the positions of only those atoms that have actually been substituted in the isotopomer 5 ... 

Let T(s) be the orthogonal transformation that diagonalizes nI1](s) with eigenvalues II w(s)- Kraitchman equates these eigenvalues to the experimental principal planar moments Pgu 1 exp(s) of the isotopomer s ... 

The other two principal planar moments and expressions for b and c are given by cyclic permutations of a, b, c. Equations of the form of Eq. (11) are known as the Kraitchman equations. Special cases arise for molecules with various symmetry elements. For substitution of an atom in a linear molecule, the magnitude of the a coordinate of the substituted atom is... 

The microwave spectrum of isothiazole shows that the molecule is planar, and enables rotational constants and NQR hyperfine coupling constants to be determined (67MI41700>. The total dipole moment was estimated to be 2.4 0.2D, which agrees with dielectric measurements. Asymmetry parameters and NQR coupling constants show small differences between the solid and gaseous states (79ZN(A)220>, and the principal dipole moment axis approximately bisects the S—N and C(4)—C(5) bonds. 

The first and second moment conditions can be very easily introduced into the r5-fit method as least-squares constraints [7,54] if the number of isotopomers is sufficient for a complete restructure. The effect on the coordinates is not expected to be particularly unbalanced unless the moment conditions are required for the sole purpose of locating atoms that could not be substituted (e.g., fluorine or phosphorus) or that have a near-zero coordinate. While all coordinates may change, the small coordinates will, of course, change more. In the cases tested, the coordinate values of the rs-fit with constraints and those of the corresponding r/e-fit (not of the r0-fit), including errors and correlations, differed by only a small fraction of the respective errors, i.e., much less than reported above. This was true under the provision that all atoms could be substituted and that the planar moments that were excluded from the r -fit because of substitution on a principal plane or axis, were also omitted from the r/E-fit. With these modifications, the basic physical considerations and the input data are the same in both cases, and the results should be identical in the limit where the number of observations equals that of the variables. 

In a recent paper [55], the multitude of possible r0-fitting schemes have been ordered under systematic aspects. Any of the three major types of rotational parameters, principal inertial and planar moments, and rotational constants, or isotopic differences of these quantities between differently chosen members of the available substitution set, could be r0-fitted. The basic experimental information evaluated from the MRR-spectrum of any molecular species will usually consist of... 

Equations (24)-(26) can be used to transform any vector between the two principal axes systems. Except for die signs of the components of rjt and rj, R and t are completely defined by the planar moments P and P of the parent and the daughter isotopomers, respectively. 

The errors of substitution coordinates can be derived ftom die errors of the principal moments of inertia or of the planar moments... 

In these last equations, P, a and r are now the expectation values (solutions) of the quantities defined above, and 0p is the covariance matrix of the parameters. is the number of observables and m the number of parameters. Because the functional dependence of the observables (rotational constants, principal moments of inertia or planar moments) on the structural parameters is strongly non-linear in most cases, an iterative process is essential. Typically, one begins with an assumed structure and expands the moment of inertia functions in terms of the parameters of this structure in a Taylor senes up to the linear term. 

In the columns identifying the experimental method, MW stands for any method studying the pure rotational spectrum of a molecule except for rotational Raman spectroscopy marked by the rot. Raman entry. FUR stands for Fourier transform infhired spectroscopy, IR laser for any infiured laser system (diode laser, difference frequency laser or other). LIF indicates laser induced fluorescence usually in the visible or ultraviolet region of the spectrum, joint marks a few selected cases where spectroscopic and diffraction data were used to determine the molecular structure. A method enclosed in parentheses means that the structure has been derived from data that were collected by this method in earlier publications. The type of structure determined is shown by the symbols identifying the various methods discussed in section II. V/ refers to determinations using the Kraitchman/Chutjian expressions or least squares methods fitting only isotopic differences of principal or planar moments (with or without first... 

One effect of the use of a diagonal weight matrix is to make the solution to Eq. (18) depend on the particular linear combination of moments of inertia used. Thus, use of principal moments of inertia, or principal planar second moments, or differences in either of these all lead to different structural parameters even if the same set of rotational constants is used to derive the moments. By comparing structural parameters derived from each set of moments, some insight into the effect of the correlations between the moments and the magnitudes of the model errors may be obtained. 

Note that the planar moment is obtained from the principal moments of inertia not always by the above combination, the latter is determined by the orientation of the principal axes. 

As in diatomic molecules the structure of greatest importance is the equilibrium structure, but one rotational constant can give, at most, only one structural parameter. In a non-linear but planar molecule the out-of-plane principal moment of inertia 4 is related to the other two by... 

Determination of the moments of inertia of a molecule generally allows one to decide whether the molecule is planar. If aa, ba, and ca are the equilibrium principal-axis coordinates of nucleus a, then the equilibrium principal moments of inertia are... 

For a planar molecule, one principal axis must be perpendicular to the molecular plane, and clearly this must be the axis of greatest moment of inertia, the c axis. Thus ca = 0 for all nuclei in a planar molecule. It follows that... 

Dielectric studies have been applied principally to the problem of deciding between several possible structures. A typical H bonded example may be taken from one of Curran s papers (467) o-methoxy-phenol (guaiacol) has two possible planar configurations whose dipole moments, as calculated by the vector addition method, are widely different (Fig. 5-3). The measured value of 2.41 d indicates that the intra-molecularly H bonded form exists in the benzene and dioxane solutions used. A large number of phenols, aniline derivatives, and other disub-stituted aromatic compounds have been studied in a similar fashion, and the wide occurrence of the ortho effect has been demonstrated. If two adjacent positions of the ring have proper substituents, a H bond will form. 

A planar structure with D2jj symmetry was suggested by Leroi et al. (4), Thompson and Carlson (5). Two iron and two chlorine atoms form a square. The outer Fe-Cl distance is estimated to be the same as in FeCl2(g) and the square Pe-Cl distance is assumed to be slightly larger. The Cl-Pe-Cl(bridge) angle is estimated to be 135. The principal moments of... 

The molecular structure is assumed to be planar with as in CrO(g) (2). The principal moments of inertia are I. 

The radius of gyration can also be calculated from the - principal moments of inertia I for planar molecules (Ic = 0) it is defined as ... 

We define the order of the singular values as a > a2 > 31. The planar and collinear configurations give a3 0 and a2 a3 = 0, respectively. Furthermore, we let the sign of a3 specify the permutational isomers of the cluster [14]. That is, if (det Ws) = psl (ps2 x ps3) > 0, which is the case for isomer (A) in Fig. 12, fl3 >0. Otherwise, a3 < 0. Eigenvectors ea(a = 1,2,3) coincide with the principal axes of instantaneous moment of inertia tensor of the four-body system. We thereby refer to the principal-axis frame as a body frame. On the other hand, the triplet of axes (u1,u2,u3) or an SO(3) matrix U constitutes an internal frame. Rotation of the internal frame in a three-dimensional space, which is the democratic rotation in the four-body system, is parameterized by three... 

As an example consider a planar, near oblate symmetric top molecule. If small amplitude vibrations are assumed, we may expect that the ju-tensor elements are of the order of the reciprocal principal moments of inertia. However, for a planar configuration, withIc-Ia+Ib, it is easily seen that the generally smallest element, ncc, may reach extreme values when evaluated in a PAS using Eq. (2.63). A numerical example illustrates this ... 

In most molecules described in this secton, lx 4- Iy + Iz approximates very closely to N, and so when the moments of inertia are displayed in 3-dimensions, the points are approximately co-planar, lying near the plane lx + Iy + Iz = N. To simplify the 3-dimensional plots, the projection of the points onto the lx + Iy + Iz = N plane is illustrated. The projection onto this plane is calculated by performing a principal axis transformation. 

Inertial defect - In molecular spectroscopy, the quantityfor a molecule whose equilibrium configuration is planar, where and / are the effective principal moments of inertia. The inertial defect for a rigid planar molecule would be zero, but vibration-rotation interactions in a real molecule lead to a positive inertial defect. 

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