Symmetry orbits and molecular frameworks.

Here is a simple illustration of evaluating the dihedral orbits of D_2, introduced earlier on here, in analogy to the field orbits introduced here. We want to study the orbits generated by the symmetries of

$$

D_2 = {1,h,v,o}

$$

when applied to each one of the x- and y-  displacements shown in the 5-element framework below. We recall that o indicates the double reflection, that is,  o=hv=vh. The multiplication table of D_2 was introduced earlier here.

For example, applying the {1, h, v, o} symmetries, as planar transformations, to the displacement  x_1 gives the orbit

{ x_1, x_4, -x_2, -x_3 }.

There are two remaining orbits for x-displacements, specifically,

{ x_0, x_0, -x_0, -x_0 },   

{ x_2, x_3, -x_1, -x_4 };

and three orbits for y-displacements, specifically:

{ y_0, -y_0, y_0, -y_0 }, 

{ y_1, -y_4, y_2, -y_3 }, 

{ y_3, -y_2, y_4, -y_1 }.

These resulting displacements are then examples of data indexed by the symmetries of D_2, in analogy with the examples introduced here, or  here. The resulting orbit invariants, in each orbit, are the summaries of the data that were introduced earlier, in the same pages indexed above. In each orbit, therefore, we evaluate the invariants 

I_1 = d_1 + d_h + d_v + d_o,

I_h = d_1 + d_h - d_v - d_o,

I_v = d_1 - d_h + d_v - d_o,

I_o = d_1 - d_h - d_v + d_o,

where d_s indicated the result of applying the symmetry s to the displacement d.

For example, applying I_1 to the orbit

 { x_1, x_4, -x_2, -x_3 }

gives

 x_1 + x_4 - x_2 - x_3. 

When I_1 is applied to the remaining orbits we find that they become matched with (alternating or same-sign) companions of the same displacement type among the outer framework elements, and within the center-of-mass element. The same matching applies to the other invariants I_h, I_v, and I_o.

The resulting joint effect is called a normal mode of (potential) framework vibration, each mode being indexed by an orbit invariant (later on to be recognized as an irreducible representation) of D_2.

This simple construction retains, however, the main elements of several classically important areas in physics, chemistry and biology, and in particular in the foundations of vibrational spectroscopy.

Here is the superposition (d_x + d_y) of the modes associated with I_1:

Here is the mode associated with I_h:

Here is the mode associated with I_v:

And here is the mode associated with the I_o:

The animation of  1-mode is available here, the h-mode here, the v-mode here, and the o-mode  here.

In future postings we will revisit this mechanism to obtain the classification of normal modes of real molecules. The simple adaptation will require the introduction of 3D space symmetries operation on 3D frameworks, with x-, y-, and z-displacements. 

Posted: 10/30/2011

Last revised: 10/30/2011
These  postings are based on "Symmetry Studies" An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

Note: The computer code to generate the graphical displacements and corresponding animations was developed with Mathematica (TM) and is available from the author (viana@uic.edu).

Visual Perception of Symmetry - III

Background:

Visual Perception of Symmetry II

Visual Perception of Symmetry I

The flag preference experiments described above were repeated in the Fall semester of 2011, with the same purpose of ranking the flags according to one's preference. The orbit structure of the flags was precisely the same. However, the flags were square in shape and the colors were the basic RGBY colors. Here are the flags presented for ranking:

The joint distribution of frequency counts for first-choice flag (rows) and second-choice flags (columns) is shown below. A total of 13 students ranked the flags in each of the 6 rows, thus together producing the 78 rankings for the joint distribution below.

In future postings we will be studying these data using the methods of symmetry orbits.

Posted: 10/04/2011

Last revised: 01/28/2015
These  postings are based on "Symmetry Studies" An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)