Spreadsheet Methods for Group Theoretical Calculations



Spreadsheet Methods for Point Group Theoretical Calculations

Ed Vitz

Kutztown University

Kutztown, PA 19530

When Crawling Arnold, the title character in a Jules Feiffer play (1), was challenged to grow up and take his part in our “complex adult world”, he said, “Children are complex; adults are just complicated”. Crawling Arnold would likely point out that while there is good reason for calling group theoretical treatments of molecules and ions “complex”, the linear algebraic and matrix methods we use in the application of group theoretical principles are, in most cases, merely complicated. Accordingly, a spreadsheet can reduce hours of linear algebra needed to describe bonding or vibrational and rotational spectroscopy to the essence of simplicity, allowing students to spend relatively more time on truly complex group-theoretical issues. It is not suggested that spreadsheets replace more conventional approaches to any of the group theoretical calculations; but as a supplement, spreadsheets can remove some of the tedium once the basic ideas are mastered. As will be shown, they can provide some guidance in constructing reducible representations.

Character tables for all the common point groups have been collected in one Excel( workbook (CHARTABL2000.XLS, ~300 kB) available through JCE. Appropriate Mullikan Symbols have been entered for each representation, and standard labels used for the group operations. The entries have been checked for accuracy by use in two offerings of Advanced Inorganic Chemistry. Instructors can use the templates for projection in an appropriately equipped lecture hall to demonstrate the use of Excel for calculations, and the templates can be used to create printouts with spaces left for manual calculations by students during lectures. A user might begin by using the Edit>Move or Copy Sheet Excel command to make a clean copy of one of the character table sheets. The new worksheet can then be dedicated to the application at hand. Formulas appropriate to the task may be entered below the character table in a manner designed to closely mimic a logical, manual, (non-spreadsheet) tabular approach to solving problems. This encourages students to practice manual methods, and aids in student comprehension and retention by emphasizing the tabular relationships among terms in a calculation.

The value of the spreadsheet approach can be illustrated by considering how it might be used to prove orthogonality of representations (or their bases). This is one of a number of exercises typically presented in early chapter of texts on group theory to introduce students to properties of groups, to the terminology of character tables, and to mathematical operations involving the tables. One popular text (2), gives the formula for proving orthogonality of irreducible representations, in this case A2 and E in Td:

(1) [pic]

While this formula may be abstruse in the linear form presented here, it may become much clearer if is entered as a formula in tabular form on a spreadsheet as described below. Figure 1 shows the template

[pic]

Figure 1: A spreadsheet template used to prove orthogonality of representations.

with the Td character table. An instructor can enter an Excel formula in cell C11 by clicking on cell C3 while telling the audience that the contents of this cell (the number 1) represents gc, the order of the class of operations Rc. The process is continued by entering “*” and clicking on C5 while defining its contents (the number 1) as (i, the character of A2 under the operation E. The formula is completed by entering “*C7” while defining the contents (the number 2) as (j, the character of the two dimensional representation E under the identity operation E (making the distinction between the two Es clear). The formula can then be copied over the range D11 to G11, giving the array shown, and it becomes clear that the sum of the entries is zero, demonstrating orthogonality. We have reinforced conceptual relationships by combining them with spatial relationships in the table to make the formula more meaningful, and at the same time calculated a result accurately, avoiding the many trivial mistakes that a formula like (1) invites. It is to be emphasized that the operations just illustrated are rather mechanical and in themselves “just complicated” as Crawling Arnold would say. Use of the spreadsheet to do them should allow more time to discuss the complex, conceptual importance of orthogonality.

Example worksheets for the calculation of hybrid orbitals, for rotational/vibrational spectral analysis, for demonstrating that transformation matrices form a set closed to all group operations, and others (a total of over 30) have been included in the workbook accompanying this article. The examples given below will make it easy to see how the spreadsheet could also be used to demonstrate other important group theoretical properties. Although examples for this article have been drawn from three of the most popular texts, the techniques demonstrated here are consistent with the approaches of virtually all contemporary textbooks on the subject.

Group Theory spreadsheets have several advantages in addition to facilitating calculations. In classrooms equipped with projection systems, computer spreadsheets make it convenient to display character tables for discussion, and to display accurate calculations without extremely tedious use of the blackboard. We make the workbooks available to students in Advanced Inorganic Chemistry over the web so that they can do problems of interesting complexity while avoiding the frustration with arithmetic minutae that has plagued previous practitioners. Students are required to use non-spreadsheet methods on tests administered in class, so they are urged to practice manual methods. They do so grudgingly after experiencing spreadsheet efficiency. Students who demonstrate proficiency with the Excel templates on homework problems did similar problems on tests successfully without Excel.

Use in Bonding Calculations: Group theoretical methods are often used to determine which central atom atomic orbitals are appropriate for bonding in complexes of any coordination number. Example worksheets have been developed for sigma and pi bonding in molecules of Td, C4v (square pyramidal (3) systems), Oh (for example (4) SF6), D3h (for example BF3), C2v (for example SF4), and D2h symmetry. They can be used to develop either a valence bond (central atom hybridization) or molecular orbital approach.

Figure 2 shows part of the worksheet used to find orbitals of symmetry appropriate for ( bonding in the Td case (5). The analysis begins by drawing four sigma bond vectors from a point representing a central atom towards the vertices of a tetrahedron representing the ligands. Then two vectors are drawn at

[pic]

Figure 2: A spreadsheet is used to reduce a representation having as bases two mutually perpendicular vectors on each ligand which are also perpendicular to the bond. This representation reduces to E + T1 + T2.

each vertex, perpendicular to each other and to the sigma bond vector. These eight vectors represent the symmetry of the ( bonds. The representation ((, spanning the set of 8 vectors, is determined by standard methods (presented in virtually all texts on the subject), and it must now be reduced to irreducible representations of the group in order to discover which orbitals have the appropriate symmetry (are bases for the representations). The reduction is accomplished as follows: The formula in cell C14 is “=C$12*C5*C$3”, and it is simply copied over the range C14..G18 to give the direct products of (( with each of the irreducible representations of the group. Cell J14 contains the formula =SUM(C14:G14)/$J$3, which determines the contribution of A1 to the reducible representation. The order of the group is entered in cell J3, which contains the formula =SUM(C3:G3). When the formula in J14 is copied to cells J15..J18, the results show that representations A1 and A2 do not enter into (( (zeros are calculated) but that E, T1 and T2 are spanned by (( (each entering into (( once) so their bases ([pic], [pic]for E and dxy, dxz, and dyz , or px, py, pz, for T2) are of appropriate symmetry for pi bonding.

As much or as little of the spreadsheet automation can be deleted as judged appropriate for student exercises. If the students are proficient at reducing representations, only the reducible representation (( may be deleted to create a student exercise, leaving in the Excel formulas for the reduction process. Generation of (( will then be guided by the spreadsheet. For example, suppose a student entered all the characters of (( correctly, except he absentmindedly substituted 6 for 8 under E. This would give values of –0.083, -0.083, 0.833, 0.75 and 0.75 in cells J14-J18, the non-integral values indicating immediately that something is wrong. A student using paper and pencil would waste time on meaningless arithmetic only to discover the dead end, and the process might be repeated many times as frustration grew. With the spreadsheet, if the student suspects the error and replaces the 6 with 8, there is immediate feedback that (( is now correct. The odds against students guessing all the characters are prohibitive, so the spreadsheet cannot be used to circumvent understanding of how to generate representations, but educated guesses are encouraged. If an educated guess proves correct, students can then be asked to demonstrate its correctness by writing the corresponding transformation matrices.

It is clear that students use the spreadsheet approach effectively because they often develop unique ways to solve a variety of homework problems, starting with the templates. For example, to solve the pi bonding problem discussed above, one group of students entered the Excel formula =(($C$3*C5*$C$12)+($D$3*D5*$D$12)+($E$3*E5*$E$12)+($F$3*F5*$F$12)+($G$3*G5*$G$12))/$J$3

in cell C14 and copied it into cells C15…C18 to calculate the same results displayed in cells J14…J18 of Figure 2. This illustrates the fact that templates don’t constrain students to a particular style. The formula above is clearly an application of a standard equation analogous to (1) that some students prefer.

Use in Spectroscopic Calculations: In both undergraduate and graduate level inorganic chemistry courses, students may be asked to determine the number of IR- or Raman-active vibrations in species of various symmetries. Spreadsheets have been included for the octahedral case (6) and several species of lower symmetry like cis- and trans-Fe(CO)4Cl2, which are C2v and D4h respectively (7), BCl3 (D3h), and for water (C2v).

The vibrational/rotational modes for water are derived in the worksheet printed as Figure 3. In this case, the method of Huheey (8) has been used to advantage, simplifying the determination of the representation spanning all motions of all atoms. Since all motions are described in terms of three Cartesian vectors on each atom, the transformation matrix for water with 3 atoms would be a 9 x 9 matrix and the six ligands and central atom in an octahedral complex would require a 21 x 21 matrix. Generation of the characters of the 9 x 9 matrix for water can be simplified by imagining it as the product of two simpler matrices as follows: Since vectors on atoms which are interchanged by a group operation cannot have elements on the character of the transformation matrix (8), one representation is developed whose elements are the numbers of ligands unshifted by the operation, and it is multiplied by another representation whose elements are the characters of the transformation matrices for one of the

[pic]

Figure 3: A spreadsheet for the analysis of the translational, vibrational, and rotational motions of a water molecule. Three vibrations have the dipolar symmetry of the coordinate axes and will be IR active.

Cartesian vector sets which is unshifted. (, the product of the “unshifted” and “contribution” representations, calculated in row 23, will then contain the characters of the 9 x 9 overall matrix. For example, the (v operation on the plane bisecting the H-O-H angle, and the C2 operation on the axis through the O of H2O leave only the O atom unshifted (the H atoms are interchanged), while E and (’v leave all 3 atoms unshifted. This is the source of the values 3,1,1,3 in the representation labeled “unshifted.” A second representation has the characters of the transformation matrices for one of the Cartesian coordinate systems centered on an unshifted atom. That is, C2 will transform x to –x, y to –y, and z to itself on a coordinate system centered on the O with the z axis splitting the H-O-H angle, so it has a character of -1 in the representation labeled “contribution.” As before, mistakes in ( can be caught before a lot of time is invested in calculations by making sure that integral values are returned in cells H25..H28. To do so, the formula =C$23*C16*C$14 is entered in cell C25 and copied to the range C25..F28. If non-integral values are returned, the offending character may often be found as before by an educated guess, in effect teaching the user something about how the characters are determined, without losing the point of group theoretical calculations in a frustrating session of repeated trivial arithmetic. Once more, it is not the purpose of this article to discuss the implications of the calculated results (which can be found in any group theory text), only to describe how the results can be obtained efficiently by spreadsheet methods so that more time can be spent studying the interesting and complex issues.

Matrix Manipulations: one worksheet has been included to show how a spreadsheet can be used to do the matrix mathematics that is necessary to demonstrate that the matrix representations of group operations are closed to the operations of the group. In other words, if the matrix representing C4 is a legitimate group operation, so must be C42, C43, C44, etc. Once the transformation matrices for any representation are defined, Excel can multiply or take the inverse of matrices with the functions MMULT and MINVERSE. These functions return a 3x3 matrix, for example, when two 3x3 matrices are multiplied, as long as the control+shift+enter keys are pushed simultaneously to complete the operation (these functions otherwise return only a 1x1 matrix). Instructions for using these functions and a few leading questions are provided for students in one of the worksheets.

Conclusion: The workbook described here is offered as a tool, rather than a complete solution to every problem of interest. Spreadsheets simplify the solution of many types of group theoretical problems, and only a few illustrative examples are provided here, along with the character tables and methods needed for further work. Spreadsheet character tables can be adapted to a variety of teaching preferences and needs, and have been remarkably useful in our Advanced Inorganic course. While we certainly wouldn’t advocate complete abandonment of hand calculations, spreadsheets can save an enormous amount of time and frustration by automating trivial work.

Literature Citations

1. Feiffer, J. Crawling Arnold, Dramatists Play Service, Incorporated: New York, 1963; p. 20.

2. Carter, R. L. Molecular Symmetry and Group Theory, John Wiley & Sons, Inc: New York, 1998; p. 64.

3. Miessler, G.L.; Tarr, D.A. Inorganic Chemistry, 2nd ed.; Prentice Hall: Upper Saddle River NJ, 1998; p. 107, problems 5-11 and 5-12.

4. Huheey, J.E.; Keiter, E.A; Keiter, R.L. Inorganic Chemistry: Principles of Structure and Reactivity,” 4th ed.; HarperCollins College Publishers: New York, 1993; p. 88, problem 3.29 b,c.

5. Cotton, F.A., Chemical Applications of Group Theory, 2nd ed.; Wiley Interscience: New York, 1971; p. 212.

6. Miessler, ibid., p. 108 problem 4-19.

7. Miessler, ibid, p. 107, problem 4-18.

8. Huheey, ibid, p. 67-8.

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