Direction cosine matrix elements

From the close relationship between the direction cosine matrix elements diagonal in J, matrix elements of the components of the angular momentum, J,K,M Jv J,K My, we have ... 

This justifies the general statement of Eq. (2.3.9) that, for all operators A obeying anomalous molecule-fixed commutation rules, Eq. (2.3.6) (any operator that includes R), the matrix elements of A are real and positive but that A+ and A- act exactly as lowering and raising operators, respectively. This reversal of the lowering/raising roles of A 1 may be viewed as arising from the direction cosine matrix elements [Eq. (2.3.17)] rather than any unusual property of the operator A. 

For a linear molecule, the position of the symmetry axis (the molecule-fixed. z-axis) in space is specified by only two Euler angles, / and 7, which are respectively identical to the spherical polar coordinates 6 and (see Fig. 2.4). The third Euler angle, a, which specifies the orientation of the molecule-fixed x- and y-axes, is unaffected by molecular rotation but appears explicitly as an O- dependent phase factor in the rotational basis functions [Eq. (2.3.41)]. Cartesian coordinates in space- and molecule-fixed systems are related by the geometrical transformation represented by the 3x3 direction cosine matrix (Wilson et al., 1980, p. 286). The direction cosine matrix a given by Hougen (1970, p. 18) is obtained by setting a = 7t/2 (notation of Wilson et al, 1980 6 fi,4)=, x = oi 7t/2). The direction cosine matrix is expressed in terms of sines and cosines of 9 and 4>. Matrix elements (J M O la JMQ), evaluated in the JMQ) basis, of the direction cosines, are expressed in terms of the J, M, and quantum numbers. The direction cosine matrix elements of Hougen (1970, p. 31), Townes and Schawlow (1955, p. 96), and Table 2.1 assume the basis set definition derived from Eq. (2.3.40) and the phase choice a = 7t/2 ... 

Equations (6.3.22a - 6.3.22b) can be simplified, for the case of isotropically oriented molecules, unpolarized radiation, and zero external magnetic or electric fields, by summing over M (see Hougen, 1970, p. 39).f The resultant M-independent (f2 J a f]J) direction cosine matrix elements are listed in Table 6.1. Note that the a Ail = Tl matrix elements have opposite signs for P versus R transitions, whereas the az Afl = 0 matrix elements have the same signs for P and R transitions. 

Table 6.1 M-Independent (fl J) Direction Cosine Matrix Elements...

This expression separates the vibrational dependence, incorporated in p from the rotational dependence in the direction cosine matrix elements /a. Our ability to apply the transformation of Eq. (34) lies in our ability to rewrite both the vibrational contributions ptt and the rotational contributions /a to the right side of Eq. (36) in terms of raising and lowering operators (50). 

If now the rotational quantization were neglected, rnne-, ivhich assumes both positive and negative values, might be omitted from (12) and (15) and the direction cosine matrix elements replaced b their classical average over all directions. 

The perturbation connects adjacent J states. Evaluation of the direction cosine matrix elements in the basis J, M) and the energy level differences gives... 

The matrix elements < n I p I n" ) are primarily associated with vibrational transitions. Since, however, the rotational quantum numbers R and R" also change, in a consistent treatment the transition probabilities of all rotational components of a vibrational absorption band must be evaluated. Usually, an approximate approach is adopted [4] and the direction cosine matrix elements replaced by classical averages over the cosines. These are equal to [3-5]. 

For orientation measurements, this tensor also needs to be expressed in the coordinate system OXYZ, axrz, using the matrix transformation u.xyz = Oaxyz / where O is a matrix whose elements are the direction cosines of the coordinate axes and is its transposed matrix [44]. 

A sinusoidal plot of grf>2 vs.

crystal plane gives another set of Ks that depend on other combinations of the gy, eventually enough data are obtained to determine the six independent values of gy (g is a symmetric matrix so that gy = gy,). The g-matrix is then diagonalized to obtain the principal values and the transformation matrix, elements of which are the direction cosines of the g-matrix principal axes relative to the crystal axes. An analogous treatment of the effective hyperfine coupling constants leads to the principal values of the A2-matrix and the orientation of its principal axes in the crystal coordinate system. 

Note 3 Molecules which constitute nematogens are not strictly cylindrically symmetric and have their orientational order given by the Saupe ordering matrix which has elements 5 aP = (3< a P> - 5ap)/2, where la and ip are the direction cosines between the director and the molecular axes a and P, 5ap is the Kronecker delta and a, p denote the molecular axes X, Y, Z. 

T is the Christoffel matrix. The elements of for arbitrary (i.e. triclinic) symmetry are given in Table 11.1(a). The direction cosines /, are defined by... 

The matrix elements Hkk follow directly from (1) and correspond to directional cosines in a vector space. Transfers between adjacent sites are proportional to tpq. The local structure of VB diagrams limits the outcome to possibilities for Hkki that can readily be enumerated [13]. Spin problems in the covalent basis have even simpler[28] Hkki. The matrix H is not symmetric when the basis is not orthogonal, but it is extremely sparse. This follows because N sites yield about N bonds and each transfer integral gives at most two diagrams. There are consequently 2N off-diagonal Hkk> in matrices of order Ps N, Ne)/4 for systems with inversion and... 

Wc need finally the atomic matrix elements <0, a H i, /i>, the form of which has been given by Slater and Koster (1954). The vector r,. is written (/x + my -t- i z) f/, with x, y, and z unit vectors along the cube axes that is, /, m, and II are the direction cosines of the vector from the left state to the right state. Then the matrix elements arc written as an E, and the states represented by their angular form are written as subscripts, the first for the left state (a), and the second for the right state (/i) for example, the symbol represents <0, a // i, (i),... 

The Slater and Koster (1954) tables of interatomic matrix elements as functions of the direction cosines, /, m, and /i, of the vector from the left state to the right state. Other matrix elements are found by permuting indices. General formulae for these expressions and explicit expressions involving/ and r/ orbitals have been given recently by Sharma (1979). 

For the eight nearest neiglibors in the body-centered cubic structure (Fig. 20-2), the direction cosines /, m, and n entering Table 20-1 are all 3 all combinations of plus and minus being u.sed 3 (11 1 ) 3 (111), 3- / (lll), 3- (lll), 3 (iTl), 3 (111), 3" (fl 1), 3 (111). Those with positive direction cosines n (in the positive z direction) have phase factors those with negative n have pliase factors e Let us then make the evaluation explicitly for states of symmetry zx. Tlie interatomic matrix element docs not appear in Table... 

Table 3. The symmetrical matrix Kajea (only elements on and above the diagonal are given). The elements of this symmetric matrix are the coefficients to (ca — co) =e a of Eq. (16a). They correspond to position k of the (not necessarily linearly ligating) ligand given by the direction cosines (oc,fi,y) referred to the basic space-fixed coordinate system XYZ, relative to which the real (unprimed) d orbitals are defined. <5 has been written as an abbreviation for +]/a2 +...

All. Matrix Elements of the Direction Cosines and Angular Momentum... 

In the following the commutation relations and the nonvanishing matrix elements of the direction cosines and angular momentum operators are summarized. For an excellent discussion of the theory of angular momentum operators the reader is referred to Ref. Derivations of the matrix elements may also be found in many textbooks on quantum mechanics. 

A schematic diagram of the hybrids making up a bond, and the calculation of the bondingantibonding matrix element for a field in a [1001 direction. For this choice of field-direction, all bonds make the same angle (with cosine = 3" ) with the field. 

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