Proper symmetry operations

Here /, are the three moments of inertia. The symmetry index a is the order of the rotational subgroup in the molecular point group (i.e. the number of proper symmetry operations), for H2O it is 2, for NH3 it is 3, for benzene it is 12 etc. The rotational partition function requires only information about the atomic masses and positions (eq. (12.14)), i.e. the molecular geometry. 

Proper symmetry operation A symmetry operation that maintains the handedness of the object. Such operations include translations, rotation axes, and screw axes. 

Rotations are known as proper symmetry operations whereas the operations involving reflection and inversion are improper. Proper symmetry operations may be performed physically using molecular models, whereas improper opera-... 

Homotopic. The relationship between two regions of a molecule that are related by a proper symmetry operation. 

In summary, proper spin eigenfunetions must be eonstmeted from antisymmetrie (i.e., determinental) wavefunetions as demonstrated above beeause the total and total Sz remain valid symmetry operators for many-eleetron systems. Doing so results in the spin-adapted wavefunetions being expressed as eombinations of determinants with eoeffieients determined via spin angular momentum teehniques as demonstrated above. In... 

One Must be Able to Evaluate the Matrix Elements Among Properly Symmetry Adapted N-Electron Configuration Eunctions for Any Operator, the Electronic Hamiltonian in Particular. The Slater-Condon Rules Provide this Capability... 

The cyclobutene-butadiene interconversion can serve as an example of the reasoning employed in construction of an orbital correlation diagram. For this reaction, the four n orbitals of butadiene are converted smoothly into the two n and two a orbitals of the ground state of cyclobutene. The analysis is done as shown in Fig. 11.3. The n orbitals of butadiene are ip2, 3, and ij/. For cyclobutene, the four orbitals are a, iz, a, and n. Each of the orbitals is classified with respect to the symmetiy elements that are maintained in the course of the transformation. The relevant symmetry features depend on the structure of the reacting system. The most common elements of symmetiy to be considered are planes of symmetiy and rotation axes. An orbital is classified as symmetric (5) if it is unchanged by reflection in a plane of symmetiy or by rotation about an axis of symmetiy. If the orbital changes sign (phase) at each lobe as a result of the symmetry operation, it is called antisymmetric (A). Proper MOs must be either symmetric or antisymmetric. If an orbital is not sufficiently symmetric to be either S or A, it must be adapted by eombination with other orbitals to meet this requirement. 

P is a symmetry operator ensuring the proper spatial symmetry of the function, (1,2) stands for the permutation (exchange) of both electrons coordinates and the sign in Eq. (22) determines the multiplicity for (+) Ft, represents the singlet state and for (-) the triplet state. 

Vector projection operators act by cancelhng out all components of a vector except the one it is designed to select. The decomposition of a function in an analogous way requires expression of the function as a sum of components each of a proper symmetry species. For example, it is possible to write any function of three variables as a sum of components that are symmetric or antisymmetric with respect to inversion ... 

If the molecule is rotated around the z axis by 120° (360°/3), an equivalent configuration of the molecule is produced. The boron atom does not change its position, and the fluorine atoms exchange places depending upon the direction of the rotation. The rotation described is the symmetry operation associated with the C3 axis of symmetry, and the demonstration of its production of an equivalent configuration of the BF3 molecule is what is required to indicate that the C3 proper axis of symmetry is possessed by that molecule. 

There are other proper axes of symmetry possessed by the BF3 molecule. The three lines joining the boron and fluorine nuclei are all contained by C2 axes (from hereon the term proper is dropped, unless it is absolutely necessary to remove possible confusion) as may be seen from Figure 2.2. The associated symmetry operation of rotating the molecule around one of the C2 axes by 360°/2 = 180° produces an equivalent configuration of the molecule. The boron atom and one of the fluorine atoms do not move whilst the other two fluorine atoms exchange places. There are, then, three C2 axes of symmetry possessed by the BF3 molecule. 

Our convention is that a symmetry operation R changes the locations of points in space, while the coordinate axes remain fixed. In contrast in Section 1.2 we considered a change (proper or improper rotation) of coordinate axes, while, the points in space remained fixed Let x y z be a set of axes derived from the xyz axes by a proper or improper rotation. Consider a point fixed in space. We found that its coordinates in the x y z system are related to its coordinates in the xyz system by (1.120) or (2.29) ... 

The characters of Trot are closely related to those of Ttrans. Any symmetry operation is either a Cn or an Sn operation. Consider first the effect on, say, Rx of a Cn rotation about some axis, not necessarily the x axis. This Cn rotation will move the rotation displacement vectors in such a manner as to transform Rr into a vector R where R r is the vector obtained by applying C directly to Rx an example is shown in Fig. 9.7. Thus for proper rotations, the matrices describing how Rx, R, and Rr transform are exactly those matrices that describe how ordinary (polar)... 

We consider four kinds of symmetry elements. For an n fold proper rotation axis of symmetry Cn, rotation by 2n f n radians about the axis is a symmetry operation. For a plane of symmetry a, reflection through the plane is a symmetry operation. For a center of symmetry /, inversion through this center point is a symmetry operation. For an n-fold improper rotation axis Sn, rotation by lir/n radians about the axis followed by reflection in a plane perpendicular to the axis is a symmetry operation. To denote symmetry operations, we add a circumflex to the symbol for the corresponding symmetry element. Thus Cn is a rotation by lit/n radians. Note that since = o, a plane of symmetry is equivalent to an S, axis. It is easy to see that a 180° rotation about an axis followed by reflection in a plane perpendicular to the axis is equivalent to inversion hence S2 = i, and a center of symmetry is equivalent to an S2 axis. 

All symmetry operations that we wish to consider can be regarded as either proper or improper rotations. 

The final requirement, that every element of the group have an inverse, is also satisfied. For a group composed of symmetry operations, we may define the inverse of a given operation as that second operation, which will exactly undo what the given operation does. In more sophisticated terms, the reciprocal S of an operation R must be such that RS = SR = E. Let us consider each type of symmetry operation. For <7, reflection in a plane, the inverse is clearly a itself a x a = cr = E. For proper rotation, C , the inverse is C" m, for C x C "m = C" = E. For improper rotation, S , the reciprocal depends on whether m and n are even or odd, but a reciprocal exists in each of the four possible cases. When n is even, the reciprocal of S% is S m whether m is even or odd. When n is odd and m is even, S% — C , the reciprocal of which is Q m. For S" with both n and m odd we may write 5 = C a. The reciprocal would be the product Q ma, which is equal to and which... 

We have already seen that, if a molecule possesses a proper axis, C , and also a twofold axis perpendicular to it, there must then necessarily be n such twofold axes. The n operations, E, C , CJ,. . . , . C" l, plus the n twofold rotations constitute a complete set of symmetry operations, as may be verified... 

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