Application of point‐group symmetries in chemistry and physics: A

Application of Point-Group Symmetriesin Chemistry and Physics: A Computer-Algebraic Approach

S. FRITZSCHEInstitut fur Physik, Universitat Kassel, Heinrich-Plett-Strasse 40, D-34132 Kassel, Germany

Received 14 June 2005; accepted 14 June 2005Published online 12 September 2005 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.20773

ABSTRACT: The present application of computer algebra is reviewed for its use inphysical and quantum chemistry. After the rapid development of a great deal ofnumerical algorithms and codes during the past five decades, it is time that symbolictechniques become available in the chemical sciences. A careful implementation of suchsymbolic methods will make many tasks more efficient and will facilitate theinvestigation of many quantum systems. Following a brief account of the advantagesand specific requests of applying computer algebra in chemistry and physics, emphasisis placed in this contribution on the Bethe program, a recently developed tool for usingpoint-group symmetries in various fields of chemistry, including molecularspectroscopy, ligand-field theory, and material science, or even for the constructionof molecular wave functions. Besides the great promise of symbolic techniques formodern research, their merits for education are pointed out, as computer algebrasystems are likely to replace the traditional program languages in theundergraduate programs of chemistry and physics. © 2005 Wiley Periodicals, Inc. Int JQuantum Chem 106: 98 –129, 2006

Key words: Clebsch–Gordan coefficients; computer algebra; double group; molecularspectroscopy; molecular vibrations; normal coordinate analysis; point-group symmetry;quantum chemistry; symbolic computations; symmetry orbitals

1. Introduction

D uring the past five decades, computers haveradically changed the image of science. Dur-

ing the 1960s, pure numerical investigations be-

came an accepted path in various branches of mod-ern research and, since then, have often been foundthe only route to obtain a quantitative understand-ing of many systems in chemistry and physics.Today, therefore, a great deal of simulations andextensive computations are carried out in almost allscientific areas, including quantum chemistry,physics, meteorology, robotics, and even biochem-istry.

Contract grant sponsor: Deutsche Forschungsgemeinschaft(DFG; Schwerpunkt 1145).

Contract grant number: FR 1251/8-2.

International Journal of Quantum Chemistry, Vol 106, 98–129 (2006)© 2005 Wiley Periodicals, Inc.

Apart from this numerical revolution in science,however, there has also been a dramatic increase insymbolic techniques, known as computer algebra,in engineering, science, and education. New gener-ations of general-purpose computer algebra sys-tems (CAS), such as Mathematica, Maple, andseveral others, have been developed and enabletheir users today to solve complex tasks within auniform framework, and usually at one and thesame platform. The term “technical computing” hasbeen coined to denote the ability of modern-dayCAS for combining symbolic algorithms, fast nu-merical computations, programming, visualization,and up to the presentation of scientific documents.The ongoing development of these CAS, includingtheir adaptation to parallel environments, gridcomputing, and technical publishing, puts them atthe forefront of scientific computing and enablesone to tackle problems of so far impenetrable com-plexity.

Owing to these merits of computer algebra, infact, advanced scientific computing and simula-tions are about to take a turn from mainly numer-ical toward hybrid solutions, in which symbolictechniques are to be combined with the previouslyestablished numerical algorithms. Although manyof these techniques still need to be worked out, theapplication of computer algebra for physical andquantum chemistry can hardly be overrated. Up tothe present, however, we are still at the beginningof such symbolic developments, and a good deal offurther progress is required before the configura-tion of symbolic and numerical methods will be-come efficient for dealing with chemical com-pounds and reactions beyond very simple ones. Butcomputer algebra is certainly the obvious placewhere new ideas and concepts have to be workedand tested out, if equipped with proper numericalfeatures.

In this contribution, the present and future ap-plication of computer algebra is reviewed for its usein physical and quantum chemistry. To this end, letus begin in Section 2 with a brief account of thecurrently available CAS and their application in thephysical and chemical sciences, as well as in edu-cation. Besides the advantages of symbolic compu-tations, we shall also discuss the particular requestsfor using such techniques in quantum chemistryand physics. The main focus in this review isplaced, however, on the Bethe program, a toolboxfor using the point-group symmetries, which is in-troduced in Section 3. The applications of this pro-gram form the central part of this work, with em-

phasis on molecular spectroscopy (Section 4) andapplications in quantum chemistry (Section 5). Fi-nally, a short summary and outlook are given inSection 6.

2. Use of Computer Algebra Systems

Computer algebra is known today for providingan efficient route in many fields of advanced scien-tific computing. Despite its merits, however, sym-bolic techniques are not yet so widely used in chem-istry, either in research or in education. Let ussummarize the possible applications of computeralgebra, from which a few are later explained inmore detail in Sections 4 and 5.

2.1. THE EARLY DAYS AND THE PRESENT

Of course, this is not the place to outline thedevelopment of computer algebra. CAS have beenavailable for many years now—and almost as longas the conventional languages, such as Fortran orAlgol. For many years, however, symbolic tech-niques have only been secondary when computerswere being used for scientific purposes. Perhaps,the main reason for this shadowy existence was thatmost of the early CAS were restricted to a fewspecific tasks and, hence, to a small group of users.The first general-purpose CAS, called Schoonship[1], was designed by Veltman during the 1960s inorder to perform complex algebraic computationsin particle physics. For his work on gauge theory,including the development of Schoonship, Marti-nus Veltman received the Nobel Prize in Physics in1999 together with one of his students, Gerardus ’tHooft. Reduce, the first CAS with reasonably wide-spread usage, was developed by Anthony Hearn in1968 and has since been promoted by many groupsworldwide [2].

Today, there are more than 30 CAS known withquite different capabilities and applications inmathematics, physics, chemistry, engineering, com-puter science, computational biology, and manyother fields [3]. They are often divided into twomain categories due to their abilities as special-purpose or general-purpose systems. Special-pur-pose systems often deal with just a particularbranch of mathematics, such as differential equa-tions, algebraic geometry, or group theory. Whilethe varied features of special systems are needed bysome users, most scientists and students now prefergeneral-purpose systems, which are characterized

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by three features: (i) a large set of built-in operators,functions, and algorithms; (ii) a high-level syntaxclose to the mathematics in textbooks and scientificpublishing; and (iii) a user-friendly interface. Often,moreover, general-purpose systems provide (iv)programming facilities as well as (v) a varied set ofinterfaces to other applications, databases, or pub-lishing.

Among the general-purpose CAS, Math-ematica and Maple are the most widespread sys-tems in use today. A number of other systems, suchas derive and MuPAD, were developed first forschool applications, but have now become ratherpowerful. All support a large variety of standardtasks from elementary and higher calculus, such aslinear algebra, combinatorics, or systems of polyno-mial and differential equations, to name just a few.Apart from minor faults, and a few occasional bugs,the “mathematics” implemented in these systemsfar exceeds the knowledge that is usually impartedat school and university courses. For many gradu-ates and scientists, therefore, these CAS have be-come almost indispensible during the last decade.

2.2. USE OF COMPUTER ALGEBRA INCHEMISTRY AND PHYSICS

In spite of the vast potential of using computeralgebra for research and education in the naturalsciences, there are a number of practical requeststhat should be taken into account if one wishes todevelop powerful symbolic tools. In this section,therefore, let us first consider the pros and cons ofsymbolic computations before we list a number ofapplications in physical chemistry and in the treat-ment of quantum many-particle systems.

2.2.1. Pros and Cons in Using SymbolicComputations

Several features make general-purpose CAS veryuseful for research in applied mathematics. Apartfrom (i) the fast and reliable manipulation of stan-dard mathematical expressions, these features con-cern in particular the (ii) sound mathematical foun-dation of most CAS. Often, moreover, (iii) thetreatment of large expression is an important pre-condition for dealing with complex chemical orphysical systems and are of utmost practical signif-icance for the description of quantum many-parti-cle systems, an issue we will raise again in Section2.2.3. For these few reasons alone, the importance ofsymbolic methods can hardly be overrated. By de-

veloping a proper hierarchy of algorithms and sym-bolic tools, a great potential is seen especially for allthose formalisms that are well established mathe-matically but that become cumbersome, or evenprohibitive in practice, if applied to complex sys-tems [4].

In practice, however, the advantages of CASmeet very specific demands, something that hasoften restricted the use of symbolic techniques.These demands result from the complexity of chem-ical systems and the diversity of the individualscientific subfields. Some of the important requestson the design and the implementation of symbolictools refer to the particular experiences and habitsof the respective community and can be summa-rized as follows:

1. Since many scientific communities have de-veloped their own jargon and notations, thesoftware (i.e., implementations) first need tobe adapted to the particular user group beforeit becomes widely accepted. This dependenceon previous experiences is found, for instance,in the use of group theory whose language isquite different in atomic physics, physicalchemistry, or crystallography, although theunderlying principles are the same.

2. Because most symbolic tools only provide in-termediate results, a simple and user-intuitivedesign is typically required for all tools.

3. For describing (quantum) many-particle sys-tems, the available data structure must be in-dependent of the number and the particularproperties of the particles and subsystems.Moreover, many tasks require scalable algo-rithms that can be followed up also interac-tively step by step.

4. Simple and instructive test scenarios areneeded in order to offer the potential user notonly a few demonstrations of the tools, butalso examples that can be carried out andextended interactively. Their proper selectionoften decides to which extent symbolic toolsand methods are accepted within a givencommunity.

5. To increase the reliability of the implementa-tions, additional tools for error tracing and fortests on the plausibility of the results arehighly appreciated.

During the past decades, these particular re-quests have led computer algebra to find its way in

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only a few particular areas of physics and chemis-try. In physics, for example, a wider use of CAS isknown only from the fields of general relativity andthe manipulation of Feynman graphs in particlephysics. For the description of many-particle sys-tems, in contrast, only a few developments havetaken place in the past and have led to a rathersmall number of symbolic algorithms, at least whentaking respective publications as a measure.

2.2.2. Symmetry and Spectroscopy

Symmetry considerations are known to play acrucial role in almost all branches of modern phys-ics and chemistry, including the physics of atomsand elementary particles in a similar manner as thebonding of molecules or solids. These consider-ations, formalized in group theory, provide essen-tial insight into the structure and dynamics of mo-lecular behavior. In molecular physics andcrystallography, of course, it is the calculus of thepoint-group symmetries that provides the mathe-matical basis and that is needed to explain theobserved spectra. This ubiquitous need for point-group symmetries has also motivated the design ofMagma [5], Gap [6], and others, that support thedefinition and manipulation of groups. Typically,these systems are designed to provide a mathemat-ically rigorous environment, especially for thoseproblems from algebra and number theory, whichcannot be accomplished by other means. Therefore,these tools are appropriate for mathematicians andfor basic research on group theory but are often lesssuitable for applications as required in chemistryand physics. An alternative view is taken by theBethe program, which has been built on Maple asa standard CAS, and which is further explained inSection 3.

Besides the formal group theory, many otherapplications are possible, where computer algebramight support the study of chemical structures andreactions. To name but a few, these include:

▪ Hybridization of chemical compounds andclassification of molecular states

▪ Derivation of the normal coordinates andmodes of molecules and clusters

▪ Derivation of selection rules and spectral ac-tivities for the fundamental and nonfunda-mental vibrations of molecules

▪ Level spitting of atoms in external crystalfields (ligand field theory)

▪ Magnetic properties of material, if the pointgroup data are properly combined with thetime reversion in order to generate the mag-netic point, or color, groups

▪ Determination of the symmetry of tensorproperties

▪ Analysis of rotational–vibrational spectra andthe Jahn–Teller effect

▪ Electric and magnetic properties of fullerenesand nanotubes

▪ Time series analysis of biochemical models

We return to some of them in Sections 4 and 5.

2.2.3. Symbolic Treatment of Quantum Many-Particle Systems

The study of quantum many-particle systems isanother field in which symbolic computations havebeen found useful in chemistry and physics. In-deed, the evaluation of Feynman graphs is still themost advanced application of computer algebra inphysics. As in spectroscopy, however, there arevarious other tasks in many-particle physics andchemistry for which computer algebra may providesignificant help in the future, including for instance:

▪ Automatic search for symmetries and appro-priate coordinates

▪ Transformation of coordinates and quantummechanical representations

▪ Classification of many-particle quantumstates, including the construction of symme-try orbitals and molecular wave functions forquantum chemical computations

▪ Evaluation of operator products in secondquantization

▪ Derivation of Feynman–Goldstone perturba-tion expansions for open-shell systems

▪ Simplifications of expressions, operators,and/or matrix elements

▪ Use of hyperspherical coordinates in thestudy of few-particle systems

▪ Spin-angular integration of spherical-sym-metric N-electron systems (i.e., the use of Ra-cah’s algebra)

Of course, this list could be continued rathereasily and demonstrates the enourmous potential ofsymbolic methods. For a few selected topics fromthis list, we have begun to develop algorithms and



tools as we briefly describe in the next section.Owing to the large number of different tasks, how-ever, a lot of independent communities now exist,which have more or less lost contact with another,although they often use very similar theoreticalconcepts in their work. Here the development ofsuitable symbolic techniques may help counteractthe tendency for an ever-increasing specializationof the various communities.

2.2.4. “Symmetry Tools”

In the previous sections, we saw the vast poten-tial of symbolic manipulations to facilitate the treat-ment of physical and chemical systems. This in-cludes, in particular, the application of the point-group symmetries and the Bethe program, whichwe shall introduce in the next section. Apart fromthis toolbox of point and double groups, however,we also developed a number of other programs, asdisplayed in Figure 1, which we refer to as the“symmetry tools.” Although these tools serve forquite different tasks in many-particle physics and,hence, were started independent of each other, they

now grow together and are about to become apowerful instrument for describing many-particlesystems.

With the Racah program, for instance, we facil-itate the evaluation of Racah expressions as dis-played in Figure 2, and which occur frequently inthe spin-angular integration of quantum systems.Most generally, these expressions may contain anynumber of Wigner 3n–j symbols, spherical harmon-ics, rotation matrices, and so forth. Since its firstversion in about 1997, the Racah program hasgrown considerably and has been published in anumber of steps since then [8–10]. Today, this pro-gram provides an interactive and user-friendly toolthat is organized in some hierarchical order. Inaddition to the algebraic manipulations, we nowsupport numerical computations for a wide class ofsymbols and functions, as discussed in Ref. [11].

For other systems, it is often sufficient to returnto the one-particle description and to the behaviorof the subsystems, without that it is necessary toconsider the interaction among the particles in de-tail. Mathematically, these systems are often relatedto the hydrogen-like ions [12]. In order to support

FIGURE 2. Structure of Racah expressions as simplified by the RACAH program [8, 9].

FIGURE 1. Symbolic tools for studying quantum many-particle systems. These programs are all based on the gen-eral-purpose CAS MAPLE and have been developed by our group in Kassel during the last decade. Parts of thesetools are now also available from the CPC library [7].

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the work with effective hydrogen-like ions, the Di-rac tools have been developed, i.e., a set of Mapleprocedures for studying the properties and the be-havior of the (hydrogen-like) ions. In a first versionfor the CPC library [13], these tools currently pro-vide the wave functions of these ions in differentrepresentations and both within the nonrelativisticand relativistic framework. In addition, it supportsa fast and accurate computation of a number ofradial integrals, including the one-particle opera-tors rm, e�kr, dm/drm, and jL(kr), as well as for thetwo-particle Slater integrals [14] as needed for theCoulomb repulsion among the electrons.

Two other components in Figure 1 are still underconstruction. The Goldstone module aims for anautomatic derivation of perturbation expansions foropen-shell atoms and ions. Besides the use of anuncoupled Fock basis, the composition of a coupledmany-body perturbation theory appears to be verypowerful for describing many atomic and molecu-lar processes beyond the computation of their totalenergies alone. Here, the efficient use of computeralgebra tools may help to realize this idea withoutgetting stuck in the formal derivations [15]. Finally,the Feynman program [16] has been developed inorder to support the simulation of N-qubit quan-tum systems (quantum registers). Although a firstversion of the program is restricted to unitary trans-formations, it equally supports, whenever possible,the representation of the quantum registers in termsof both their state vectors and density matrices. Inaddition to the composition of two or more quan-tum registers, moreover, the program also performstheir decomposition into various parts by applyingthe partial trace operation and the concept of thereduced density matrix.

3. Symbolic Tools for Dealing WithPoint-Group Symmetries

In chemistry, the point groups, and the closelyrelated double groups, are known to play a centralrole in the analysis of the molecular bonding andspectra. Their basic definition and data can there-fore be found in a large number of textbooks andmonographs on molecular structures and spectros-copy [17–20]. In practice, however, these compila-tions are often incomplete and by far not efficient touse. As an alternative, we briefly introduce theBethe program whose applications in molecularspectroscopy and quantum chemistry will later bediscussed in Sections 4 and 5.

3.1. POINT-GROUP SYMMETRIES ANDGROUP THEORY

Group theory is perhaps one of the most powerfulmathematical tools used in quantum mechanics andspectroscopy. When applied to a quantum mechani-cal system, this theory often allows great simplifica-tion in the mathematical treatment of the system. Inatomic and molecular structure, for instance, the di-agonalization of the Hamiltonian matrix is a fre-quently occurring task that soon becomes intractableas the number of electrons increases. Instead of thebrute force diagonalization of large matrices, it is thenoften more convenient to seek operators that com-mute with the Hamiltonian, and to construct a sym-metry-adapted basis. That is, a proper use of the sym-metry of the system may help divide the Hilbert spaceinto subspaces of much lower dimensions, in whichthe Hamiltonian matrix can be treated independently.This is just one example where group theory helps toachieve simplifications of great elegance and power.Since, for most applications in chemistry and physics,group theory was worked out a long time ago, weshall not go into the mathematical details, but assumethe reader to be familiar with basic concepts of at leastpoint-group theory. From the large number of avail-able texts on this theory, we refer the reader to just theclassical books by Wigner [21], Heine [22], or Elliotand Dawber [23].

For molecules and clusters, the group elementsare the symmetry operations that transform thesystem into a geometrically indistinguishable con-figuration. If we omit the possible permutations ofthe atomic nuclei in a molecule, these transforma-tions are the proper and improper rotations, i.e.,pure rotations around some axis or rotations withan additional inversion at the origin. They formfinite subgroups of the SO3 rotation group in three-dimensional space. For the point and doublegroups, which are of major interest in the chemicalsciences, perhaps the most complete tabulation ofthe group data has been compiled by Altmann andHerzig [24] and has been used as one of the mainreferences in the design of the Bethe program, atleast as far as the notations are concerned.

Despite the power of group theory, however, itsapplication becomes cumbersome if molecular com-pounds with high symmetry are considered, i.e., forsystems with a large number of symmetry operations.For these molecules, the group data are often not sosimple to use, as they are typically compressed into asmall number of tables, using rather arbitrary rules.Moreover, in most compilations neither the exact def-



inition of the symmetry operations nor the data forthe double groups are given explicitly, not to speak ofall the notations and conventions as applied in theliterature. Using a computer–algebraic approach, wemay avoid these difficulties and can easily adapt theoutput of the program to the particular requirements.

3.2. THE BETHE PROGRAM

The Bethe program has been developed withthe intention to provide a simple and reliable access tothe point-group data as required by many applica-tions. Following a brief overview of the program, weshall explain how these data can be manipulated tosolve some particular task. Owing to the interactivedesign of the Bethe program, we expect this tool to beof common interest, both in teaching group theory aswell as for advanced research studies.

3.2.1. Overview. Program Organization, andNotations

Using the framework of Maple, the Bethe pro-gram has been worked out as an interactive tool tofacilitate the use of point-group techniques inchemistry and physics. From the beginning, em-phasis was placed on developing a user-friendlytool that would neither require detailed knowledgeof the theoretical background or the definition ofthe symmetry operations, nor knowledge of the

special names and abbreviations as used in theliterature. Instead, we designed a “language” asclose as possible to the applications. However, noattempt was made to support numerical studies onmolecular structures or to generate the data of ad-ditional groups automatically.

In the present version of the program, Betheprovides the group data for all ordinary groupfamilies, including the cyclic and their relatedgroups Cn, Cnv, and Cnh; the symmetry groups S2n;the dihedral groups Dn, Dnd, and Dnh (n � 2, . . . ,10); the cubic groups O, T, Oh, Th, Td; as well as theicosahedral groups I, Ih. For each of these pointgroups, we provide the definition of the symmetryoperators, the multiplication law, character tables,the irreducible representations, as well as a numberof other properties following Schoenflies’ notation[24]. Moreover, all these data are supported forboth the vector and the double groups.

Following Maple’s philosophy, the Bethe pro-gram has been organized in a hierarchical order. Itpresently includes about 100 procedures that can beinvoked either interactively or simply as languageelements in order to build up commands at somehigher level of the hierarchy. In practice, however,less than 15 main procedures need to be known bythe user; they are briefly explained in Table I toprovide the reader with a first impression of theBethe program. To distinguish these commands

TABLE I ______________________________________________________________________________________________Main commands of the BETHE program to access the data and related information about the point and doublegroups.

BETHE_decompose_representation( ) Determines the irreducible components of a given (reducible) grouprepresentation

BETHE_group_direct_product( ) Returns the direct product of two or more irreducible representationsBETHE_group( ) Provides the basic point group data and notationsBETHE_group_chain( ) Displays the chain structure of the point groupsBETHE_group_character( ) Returns the character of a given irreducible representation and

symmetry operationBETHE_group_class( ) Returns all symmetry operations of the same classBETHE_group_Euler( ) Returns the three Euler angles (�, �, �) for a given symmetry operationBETHE_group_irrep( ) Returns the matrix representation of a given irreducible representation

and symmetry operationBETHE_group_multiplication( ) Returns the product operation of two symmetry operations (i.e., the

group multiplication table)BETHE_group_parameter( ) Specifies the symmetry operations in different parametrizationsBETHE_group_projector( ) Evaluates the projection operator which projects a 3-dim vector into

the subspace L� of the irreducible representation T(�)

BETHE_group_representation( ) Evaluates a few particular representations as displayed in Table IIIBETHE_group_subduction_O3( ) Subduction of a group with respect to the SO3 rotation groupBETHE_group_symmetry( ) Determines the symmetry of a given set of points

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from Maple’s internal functions, they all start withthe prefix BETHE_. More detailed information ofthese procedures can be obtained from the usermanual distributed with the code [25, 26].

Here, we shall not explain the commands of theBethe program in detail. One of the key proceduresof this toolbox is the command BETHE_group(),which provides all the basic information about a par-ticular group, such as the number and names of thesymmetry operations, the classes, irreducible repre-sentations, and many others. This, and many otherprocedures of the program, make use of a group label(Glabel) as the first argument to distinguish betweenthe various point groups. A list of all currently sup-ported group labels is returned by calling BETHE_group() without arguments. In Table II, moreover,

we display the presently supported keywords in al-phabetic order, which can be used for BETHE_group-(Glabel,keyword) in order to obtain the individ-ual information. Figure 3, for example, shows thesymmetry elements and operations of the tetrahedralgroup Td as returned by the program; of course, theoutput depends on the given parameters and can beeither a number, boolean value (a list of) strings, orsimply a null expression if, as in the example above,the procedure just prints some information.

During the past few years, the Bethe program hasbeen published in the CPC library in several steps [25,26]. To make the use of Bethe as simple as possible,we use Maple’s module() facilities. Making use of aproper installation, the module BETHE should be avail-able like any other module of Maple:

� with(BETHE);Welcome to BETHE !

BETHE_save_framework � nonrelativistic[AO, Abasis, BETHE_decompose_representation, BETHE_direct_product, . . . ,

. . . , BETHE_spectral_activity, BETHE_transform_vector, SO, atom, molecule]

TABLE II ______________________________________________________________________________________________Optional arguments of the command BETHE_group(Glabel,. . . ).*

Keyword(s) Output of the procedure

crystallographic Boolean value true for the crystallographic groups or falsecrystall_system Name of the crystallographic systemcubic Boolean value true for the cubic groups or falsecyclic Boolean value true for the cyclic groups or falsedihedral Boolean value true for the dihedral groups or falseexamples Prints a few examplesgroup_table Prints a summary about all the presently supported point groupsirreps List of irreducible representation identifiersirreps, double List of irreducible representations identifiers in the double groupNo_class Number of classesNo_class, double Number of classes in the double groupNo_irregular Number of irregular classesNo_irreps Number of irreducible representationsNo_irreps, double Number of irreducible representations in the double groupNo_operators Number of symmetry operationsNo_operators, double Number of symmetry operations in the double groupNo_regular Number of regular classesoperator_details Prints a description of all symmetry operationsoperators List of symmetry operation identifiersoperators, double List of symmetry operation identifiers in the double groupproper Boolean value true for proper groups or false (improper groups)subgroup List of subgroup labelssymmetry_elements Prints a description of all symmetry elements

* An empty parameter set can be used to return a list of all point groups (Glabel’s) supported by the BETHE program.



FIGURE 3. Symmetry elements and operations of the tetrahedral group Td as defined internally within the BETHE

program. In the lower part, moreover, the irreducible representations are shown for the point and double group.

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3.2.2. Interactive Work With Group Data

In the previous section, we saw how the commandBETHE_group() can be invoked in order to obtain thebasic group data as displayed in many textbooks andmonographs. Of course, the advantage of a symbolicapproach is that these data can be manipulated tosolve some particular problem. To demonstrate fur-ther the access and use of the group data, let usconsider the tetrahedral group Td, which describes thesymmetries of the tetrahedron, including the inver-sion of the coordinates. This non-Abelian group hasthe order 24 and is isomorphic to the group A4 � C2,where A4 is an alternating group and C2 the lowestcyclic group with a two-fold rotation about some axis.

A tetrahedral symmetry is fulfilled approximately bythe methane molecule CH4 (to which we will returnlater in several examples), but also by a few organicmolecules, such as neo-pentane and others. In theBethe program, we can determine the type of thesymmetry operations for the group Td by typingBETHE_group(Td,operators) as shown in thelower part of Figure 3. As seen from this output, allsymmetry operations, and irreducible representa-tions, are handled by means of appropriate stringidentifiers as derived from the Schoenflies’ notation.For each of these strings, moreover, we can also de-termine the parametrization of this symmetry opera-tion in terms of the Euler angles:

� BETHE_group_Euler(Td, “C2x”), BETHE_group_Euler(Td, “S4x-”);

[0, Pi, Pi], [�Pi—-2,Pi—-2,Pi—-2]

or the parameters (�, n), which specifies the angle andthe axis of rotation. In addition, we can easily confirm

that the “multiplication” of any two operator stringsgives rise to an operator (string) of the same group

� BETHE_group_multiplication(Td, “C2x”, “S4y-”),BETHE_group_multiplication(Td, “C2x”, “RS4y-”);

“sigma_d3”, “Rsigma_d3”

and which shows explicitly that the successive trans-formation of an object under the symmetry operationsof a group always results in one of the indistinguish-able configurations. The whole “multiplication table”is then obtained simply by cycling through all pairs ofsymmetry operations. There are other important pro-cedures, such as BETHE_group_character() andBETHE_group_irrep(), which provide the charac-ters and irreducible representations of a group.These are summarized in Table I and will be usedbelow to determine, for instance, the spectral activ-ities of molecules or their symmetry orbitals.

In our discussion of the Bethe program, we of-ten refer to the point groups independent ofwhether the vector or the double point groups aremeant. While the vector groups just allows theproper and improper rotations of some object, it ispossible to “add” the concept of the electron spin tothese groups with the consequence, that the num-

ber of symmetry operations is doubled when com-pared with the number of the corresponding vectorgroup, i.e. without the spin. These extended groupsare usually called the double groups; they basicallyarise from the fact that a s � 1/2 spin function isinvariant only under a rotation of 4� (around anyaxis in space). In the Bethe program, we alwayssupport both, the vector point groups and the cor-responding double groups together. The doublegroups are important in various chemical applica-tions including, for example, the theory of the tran-sition metal ions, as well as relativistic quantumchemistry [27].

3.2.3. Reducible and IrreducibleRepresentations

The symmetry operations of an molecule or crys-tal, or its symmetry group itself, would be of minor



interest, if they would not lead to induced transfor-mations and to a great simplification in describingsuch systems by using group theory. In fact, therelationship between the symmetry operations andtheir induced transformations is the topic of therepresentation theory of the groups and one of themain reason for studying symmetries in Nature,maybe apart from their beauty. Typically, such in-duced transformations can be expressed by matricesand are called the representations of the group, byassigning one matrix to each of the symmetry opera-tors. Since, in general, we may choose the basis for arepresentation in some given vector space L ratherfreely, the matrix representations of a group are notunique and usually depend on the choice of the co-ordinates, as well as on further parameters.

Here, we shall not say much about the represen-tations of the point groups and their manipulationby means of the Bethe program. For the sake ofbrevity, instead, let us recall only the role of theirreducible representations of a group. A set oflinear operators, which are defined in the vector

space L, are said to form a representation of a groupG, if there is assigned an operator T(Sa) to eachsymmetry operator Sa of the group and if theseoperators fulfill the same “multiplication rule” likethe symmetry operations: T(Sa)T(Sb) � T(SaSb) andT(E) � 1. The vector space L is then called therepresentation space of T and its dimension thedimension of the representation [23].

Of course, the great advantage in using grouptheory is that any reducible representation T(Sa) canbe decomposed into a rather small number of irre-ducible representations T(�)(Sa), which are uniqueand independent of the basis up to a unitary trans-formation. Again, several notations are knownfrom the literature to denote these irreducible rep-resentations owing to the dimension and the “phys-ical origin” of some given transformation. In theBethe program, we follow the chemical (Mullican)notation and identify the irreducible representa-tions by some appropriate string identifiers, similaras for the symmetry operations. For the tetrahedralgroup, these string identifiers are

� BETHE_group(Td, irreps); BETHE_group(Td, irreps, double);

[“A1”, “A2”, “E”, “T1”, “T2”][“A1”, “A2”, “E”, “T1”, “T2”, “E1/2”, “E5/2”, “F3/2”]

for the vector and double group, respectively. Inthis notation (and similar to those in Ref. [24]), theone-dimensional representations are labeled by Aor B in dependence of whether the character of thesmallest rotation about the principal axis is �1 or�1. In addition, the two-, three-, and four-dimen-sional representations are labeled E, T, and F, whilefive- and six-dimensional representations are de-noted by H and I, respectively.

For each irreducible representation T(�), the Be-the program can provide either the explicit matrixrepresentation or simply the characters of the rep-resentation, that is, the traces of these matrices,

which are sufficient for most practical applications.The characters of a group representation are oftendenoted by � and are used, for instance, to deter-mine the number of (inequivalent) irreducible rep-resentations involved in some reducible represen-tation. In the output above, as usual, the first stringdenotes the (one-dimensional) total symmetric rep-resentation with characters all equal to 1, while, forexample, T1 and T2 denote three-dimensional irre-ducible representations of this group. For the irre-ducible representation T1, we may obtain the char-acter and the explicit matrix, either for a singlesymmetry operation

� BETHE_group_character(Td, “T1”, “C2x”), BETHE_group_irrep(Td, “T1”, “C2x”);

[�1 0 0][ ]

�1, [ 0 �1 0][ ][ 0 0 1]

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108 VOL. 106, NO. 1

or for all the operations simultaneously as defined above

� BETHE_group_character(Td, “T1”); BETHE_group_irrep(Td, “T1”):

[3, �1, �1, �1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, �1, �1, �1, �1, �1, �1]

and where the full printout of the matrices is omit-ted by using a double point at the end of the line.From these few examples, however, it becomesclear how the corresponding data are obtained forother representations and/or symmetry operators.Table III displays a few important representationswhose characters or explicit matrices can be gener-ated automatically by using the command BETHE_representation(). For these typically reduciblerepresentations, the irreducible components arethen obtained by using standard techniques [17,20].

3.2.4. Symmetry-Adapted LinearCombinations

A common task of group theory refers to thegeneration of symmetry-adapted functions, that is,the generation of basis functions for the irreduciblerepresentations. For a given set of orthonormalfunctions {��,i, i � 1, . . . , n}, such as the eigenspaceof a Hamiltonian with eigenenergy E� for example,these basis functions have to be constructed as lin-ear combinations of the functions {��,i} and, hence,are called a symmetry-adapted linear combination

(SALC). The efficient construction of these SALCs isusually the first step in any molecular orbital cal-culation [19], but it has also been found importantfor the analysis of Walsh diagrams [28] or the de-scription of d-metal complexes [29, 30].

Independent of the particular context in whichthe SALC are used, there is a universal techniquefor generating these functions based on projectionoperators. These operators are built on the irreduc-ible representations T(�) of the group and are givenby

Pi��

s�

g �a

g

Tii��*�Sa�Sa, (1)

where g denotes the order of the group and s� thedimension of the irreducible representation T(�),and where the summation runs over all the sym-metry operations {Sa} of the group. By construction,the projection operator (1) leaves all those functionsunchanged, which constitute to the basis of T(�),while all other “parts” of these functions (vectors)are projected out. Since, in Eq. (1), the irreduciblematrix representations are used explicitly, these

TABLE III _____________________________________________________________________________________________Reducible representations as supported by the BETHE program.*

Representation Output of the procedure

“polar_vector” Transformation of the polar vector r � (x, y, z) as induced by the group(elements)

“axial_vector” Transformation of the axial vector R � (Rx, Ry, Rz) as induced by the group“Ylm” Transformation of the spherical harmonics of (spherical tensor) rank l, i.e., of

Ylm(, �), m � l, l 1, . . . , l as induced by the group“jm” Transformation of the spinor functions �jm� of half-integer (spherical tensor) rank

j, i.e., of �jm� m � j, j 1, . . . , j“cartesian_tensor” Transformation of cartesian tensor functions of given rank“Euler” Euler representation of the group“regular” Regular representation of the group“total” Total matrix representation for a given set of atomic displacements“vibrational” Representation of the vibrational motion of a given set of atomic coordinates

* For these representations, either the characters or the explicit matrix representations can be obtained by means of the commandBETHE_group_representation(Glabel, “keyword”, . . .).



projection operators are called complete, in contrastto the case

P�� i

Pi��

s�

g �i

�a

Tii��*�Sa�Sa

�s�

g �a� �

i

Tii��*�Sa��Sa

�s�

g �a

��*�Sa�Sa, (2)

in which only the characters �(�) � ¥i Tii(�) are used.

These projectors are called incomplete, as they still“project” a function onto the space of the represen-tation T(�), but without that an orthogonal set offunctions is obtained. Therefore, SALC, which areconstructed by means of the incomplete projectors(2) still need to be orthonormalized to represent auseful basis for further computations. In the Betheprogram, the projection operators for a given pointgroup can be obtained from the command BETHE_group_projector().

We shall return to the generation of SALCs atvarious points in the discussion, including the con-struction of symmetry orbitals for multi-center mol-ecules. However, before we continue with ap-plications of the Bethe program in molecularspectroscopy, let us refer to an extension of thepoint group symmetries, which, although nothinghas yet been implemented along these lines, mightbe useful in the future for studying the properties ofmagnetic materials.

3.3. MAGNETIC POINT GROUPS

Closely related to the point-group and double-group symmetries discussed above are the so-calledmagnetic or color group symmetries. These symme-tries arise, for instance, if an additional antisymme-try operation T is present in some material owing tosome nonvanishing current density J(r) and/or spindensity S(r). Such an antisymmetry operator is notpresent in any of the ordinary point groups. Theconcept of the color groups was worked out byShubnikov and other during the 1960s [31] andplays an important role today in investigations ofmagnetic materials or disordered phase transitions.For details on these topic, we refer the reader to thereviews by Schwarzenberger [32] and Lifshitz [33],which contain extensive bibliographies.

Usually, three types of magnetic groups are dis-tinguished, which are called (i) the uncolored orordinary point groups, (ii) the gray point groups, aswell as (iii) the black-and-white groups. If we con-sider just the 32 ordinary crystallographic groups, atotal of 122 color groups can be constructed for-mally. Suppose that G denotes one of the ordinary,type (i) groups and T the antisymmetry operator,then the gray point groups (type ii) are formed by

M � G � TG,

while the type (iii) magnetic groups

M � H � T�G � H�

are built up by means of some halving unitarysubgroup H of the group G. By construction, there-fore, the color groups always contain an equal num-ber of unitary and antiunitary elements. Here, anoperator U is said to be unitary if U��U�� and is antiunitary for U��U�� * � ��,respectively.

In practice, the magnetic groups can be derivedquite easily by analyzing the subgroups and char-acter tables for the ordinary point groups. Since theoperator T is antiunitary, these groups are nonuni-tary and contain both unitary and antiunitary op-erators. For this reason, the conventional theory ofrepresentations cannot be applied but has to bereplaced by the formalisms of corepresentations(coreps) as developed by Wigner [21]. The distinc-tion between the irreducible representations andthe corepresentations needs to be made because thecorep matrices do not multiply in the same way asthe operators, but include a slightly more compli-cated “multiplication law.” At present, neither thegroup classification of the magnetic groups northeir (co-)representations are supported by the Be-the program. These groups could be implementedin the future without that much additional codework required. However, in order to be useful forthe field of crystallography, one would have tocombine the 90 colored (crystallographic) pointgroups, of types (i) and (ii) from above, with thecorresponding colored Bravais lattices, which thengives rise to a total of 1,651 magnetic space groups[34].

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110 VOL. 106, NO. 1

4. Applications in MolecularSpectroscopy

Over the years, the point group symmetries havebecome an integral part of molecular spectroscopyand the quantum theory of molecules and solids.A few selected examples of these fields, therefore, willserve us to demonstrate how the Bethe programhelps in solving typical subtasks from daily (research)work. This includes the derivation of the symmetrycoordinates as well as the interaction of moleculeswith the radiation field. Although not many detailsare given below, we wish to give an impression ofwhat the Bethe program is able to do.

4.1. MOLECULAR SYMMETRY AND GEOMETRY

A simple but frequently occurring task in phys-ical chemistry refers to the specification of molecu-lar symmetry and geometry. If the symmetry of a

molecule or cluster is known, for instance, we mayask about the coordinates of all atoms, if they aregiven for just one or few atoms from each set ofequivalent atoms. Vice versa, we may wish to de-termine the highest symmetry of a molecule if theatomic coordinates are given.

To demonstrate the use of the Bethe program inthis context, let us consider methane (CH4) as one ofthe simplest but most abundant molecules in Na-ture. Methane is a colorless, odorless gas and is(with about 75%) the principal component of natu-ral gas; moreover, it is known to obey a Td symme-try with the carbon atom in the center and a COHbond length of about a 1.2 Å. Hence, there aretwo set of equivalent atoms; apart from the carbonatom at the origin (0, 0, 0), we can try to place oneof the hydrogen atoms, H1, at the coordinates (0, 0,a) along the z-axis. By using the Bethe program,this gives rise to

� w_carbon :� BETHE_generate_sites(Td,[0,0,0]);w_hydrogen :� BETHE_generate_sites(Td,[0,0,a]);

w_carbon :� [[0, 0, 0]]

w_hydrogen :� [[0, 0, a], [0, 0, �a], [0, a, 0], [0, �a, 0], [a, 0, 0], [�a, 0, 0]],

which tells us explicitly that the coordinates of theatoms cannot be chosen arbitrarily but have to beconsistent with the underlying group symmetry. A

proper set of equivalent sites is obtained for thehydrogen atoms, if we choose H1 at the position (a,a, a),

� w_hydrogen :� BETHE_generate_sites(Td, [a,a,a]);

w_hydrogen :� [[a, a, a], [a, �a, �a], [�a, a, �a], [�a, �a, a]]

Conversely, we can determine the symmetry of themolecule if the coordinates of all the atoms aregiven explicitly. For example, we may test whether

the two sets of the carbon and hydrogen atomsaltogether obey a D2 symmetry

� BETHE_group_symmetry(D2, w_carbon, w_hydrogen);

true

or even ask for the highest symmetry

� BETHE_group_symmetry(highest, w_carbon, w_hydrogen);

Td,



which confirms our assumptions above about thesymmetry of methane. Although this example istrivial from the viewpoint of molecular geometry, itdemonstrates how one can generate the atomic co-ordinates and symmetries and how one can usethem later in other applications, such as the input ofquantum chemical computations. In addition, aquick specification of the atomic coordinates isneeded for determining the normal coordinates of amolecule, as we consider in the next section.

Several topics in theoretical chemistry make useof the close relation between the symmetry of asystem and the atomic coordinates. In order to de-termine, for instance, the equilibrium structure of achemical compound, one has to consider the totalenergy as a function of the positions of the nucleiand to search for minima and saddle points on thepotential energy surface (PES). This is achieved bycalculating the first and second derivatives of theenergy functional that permit us to characterize thestationary points on the PES. In several programs,such as the Crystal code [35], the geometry opti-mization is performed in symmetrized Cartesiancoordinates to exploit the symmetry of the lattice.

A similar interrelation also occurs in the study ofcarbon nanotubes whose symmetry is described bythe so-called “line groups” [36, 37], i.e., the one-dimensional analogue of the crystallographic pointgroups. Besides the rotations, these groups mustalso incorporate the possible translations of the sys-tem along the tube axis. For carbon tubes, the sym-metry of a given fragment is determined by thenumber of hexagons (N) along the axis and alongthe tube circumference (n), which can be either evenor odd. Using the four possible combinations ofthese numbers, i.e., the Dnh and Dnd point symmetryof the carbon tubes, one is then able to explain theirdifferences in the molecular binding [38].

4.2. NORMAL MODES AND SYMMETRYCOORDINATES

Knowing the symmetry of a molecule, grouptheory can be applied to determine the normal co-ordinates and their activity in various fields of spec-troscopy. Usually, such an analysis is based on theBorn–Oppenheimer approximation, i.e., the separa-tion of the electronic and nuclear coordinates. Forlow excitations, moreover, the harmonic approxi-mation is often used to describe the motion of theatomic nuclei as independent harmonic “oscilla-tions” along the normal coordinates. In the follow-ing, we explain how Bethe can be used to expressthe normal coordinates in terms of either the Car-tesian or internal displacements from the equilib-rium geometry and how to learn more about themolecular vibrations. See Table IV for a short list ofcommands of the Bethe program which help in theanalysis of the molecular vibrations and spectra.

4.2.1. Harmonic Approximation

In the semiclassical picture of molecules, thepure vibrational spectra is thought of as the conse-quence of the harmonic oscillations of the nucleiaround their equilibrium positions. This is equiva-lent, of course, to assuming that the molecularbonds are replaced and attached by springs in orderto represent the vibrational stretches and bendsaround the equilibrium. While, in detail, the vibra-tional motion of molecules is much more compli-cated because of the anharmonic coupling betweendifferent vibrational modes or with the rotationalmotion of the molecules, this simple picture oftenhelps obtain insight into the geometrical structureof the molecules. Mathematically, the harmonic ap-proximation starts with a Taylor expansion of thepotential energy close to the equilibrium position.If, for instance, the N nuclei of the molecule are

TABLE IV _____________________________________________________________________________________________Important commands of the BETHE program for the determination of symmetry coordinates and the analysis ofmolecular spectra.

atom( ) Represents an atom in terms of its individual coordinatesmolecule( ) Represents a molecule in terms of its individual atomsBETHE_generate_sites( ) Generates all equivalent sites of a given point under the operations of a groupBETHE_normal_coordinates( ) Calculates the normal coordinates of a molecule in terms of its Cartesian

displacement vectors or internal displacement vectorsBETHE_normal_display( ) Displays the vibrational motion of a (planar) molecule graphicallyBETHE_spectral_activity( ) Determines the (infrared or Raman) activity of an vibrational mode

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112 VOL. 106, NO. 1

described by the coordinates q1, q2, . . . , q3N, theclassical Hamiltonian of the nuclear motion can bewritten as

H �12 �

i�1

3N

miqi2 �

12 �

ij

kijqiqj, (3)

if we neglect all terms beyond the quadratic one in theexpansion. In Eq. (3), mi denotes the masses and the kij

the force constants of the nuclei. Instead of the 3NCartesian coordinates of the nuclei, of course, it issufficient to use only 3N � 6 vibrational coordinates,Q1, Q2, . . . , Q3N�6, which are associated with the vi-brational degrees of freedom. Then, by taking aproper linear combination of the nuclear coordinates

Qi � �k

aikqk, (4)

it is always possible to bring the Hamiltonian of themolecule into the form

H �12 �

k

Qk2 �

12 �

k

k2Qk

2, (5)

in which the Qk are the normal coordinates and k thenormal frequencies of the system. As seen in Eqs. (4)and (5), the normal modes just describe simple har-monic oscillations that do not interact with each otherand that refer to a concerted motion of several atoms,passing through their equilibrium positions at thesame time. Since the Hamiltonian is a sum of nonin-teracting oscillations, the total vibrational energy isthe sum of the 3N � 6 individual energies, while thetotal wave function � is formed by the product of thewell-known oscillator wave functions �nk

(Qk), one foreach of the normal coordinates.

The great advantage of the normal coordinates isthat they must reflect the symmetry of the system;i.e., they have to form a basis of the irreduciblerepresentations {T(�)} of the corresponding symme-try group. Symmetry considerations therefore helpin the classification of the normal modes and their

spectral activities concerning the interaction of themolecules with the radiation field. In practice, oneoften starts from the reducible representation of the3N Cartesian displacements, as induced by thesymmetry group. In a second step, later, this reduc-ible representation is then decomposed into its ir-reducible components in order to obtain the sym-metry properties of all the normal modes involvedin the motion of the nuclei.

4.2.2. Construction of Normal Coordinates

In the Bethe program, we can construct the normalcoordinates either in terms of the Cartesian or internaldisplacements by following the two steps from above.Most conveniently, this is performed by applying pro-jection operators (1) as introduced in Section 3.2.4. Inthese projectors, the matrix operations T(Sa) now referto the transformation of the displacement vectors asinduced by the symmetry operations {Sa} of thegroup. For example, if we start from the atomic (Car-tesian) displacements {xj, yj, zj, j � 1, . . . , N} andapply the complete projector (1)

Pi��xj � Q�,i � cx1x1 � cy1y1 � · · · � czNzN, (6)

we immediately obtain the normal coordinates (dis-placements) in terms of these Cartesian displace-ments. Of course, the application of the projectorPi

(�) onto the coordinate xj may also give a zerocontribution, which simply means that this givencomponent is not contained in the normal coordi-nate Q�i. To obtain all the normal coordinates, onehas to cycle, as appropriate, through all the Carte-sian or internal displacements [20], respectively.

To demonstrate the construction of the normalcoordinates by means of the Bethe program, let usreturn to methane. Using the variables w_carbonand w_hydrogen from above, we have definedalready the atomic coordinates of the carbon andhydrogen atoms. In Bethe, alternatively, we mayconsider also the methane molecule just as a collec-tion of individual atoms and assign the molecule asa whole to the single variable methane by

� methane :� molecule(atom(C, [0,0,0]), atom(H1, [a,a,a]), atom(H2, [a,�a,�a]),atom(H3, [�a,a,�a]), atom(H4, [�a,�a,a]));

methane :� molecule(atom(C, [0, 0, 0]), atom(H1, [a, a, a]),atom(H2, [a, �a, �a]), atom(H3, [�a, a, �a]), atom(H4, [�a, �a, a])),



where we used the two auxiliary proceduresatom() and molecule() from Table IV in orderto keep together the relevant information aboutsome given atom or molecule. To derive the nor-mal coordinates, next we have to determine eitherthe 3N � 3N total representation of the group(as associated with the Cartesian displacementsof the nuclei) or just that “part” of the represen-tation which refers to the 3N � 6 vibrationalcoordinates, leaving the translational and rota-tional motion apart. In the Bethe program, wecan obtain the characters of the vibrational rep-resentation directly from the command [cf. TablesI and III]

� VR :� BETHE_group_representation(Td,“vibrational”, methane);

VR :� [9, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0,0, �1, �1, �1, �1, �1, �1, 3, 3, 3, 3,

3, 3]

and by using the keyword (string) “vibrational.” Inthe above output, the sequence of the characters cor-responds to the sequence of symmetry operations asobtained by a call to BETHE_group(Td, . . . ).From the list VR of the characters, the irreduciblerepresentations are derived by making the de-composition of the (characters of the) vibrationalrepresentation into its irreducible components,

� BETHE_decompose_representation(D3h, VR);

[“A1”, “E”, “T2”, “T2”]

and which shows that there is a total symmetric mode(A1), as well as three degenerate modes, which to-gether represent the nine vibrational modes of themethane molecule. For each irreducible component,of course, the dimension of the representation givesus directly the number of the energetically degeneratevibrational modes. Once having determined these ir-reducible components of the vibrational representa-tion, we can also obtain the normal coordinates interms of the Cartesian displacements

�c1x�1�, c1y

�1�, . . . , cNz�1��, �c1x

�2�, c1y�2�, . . . , cNz

�2��, . . .

�c1x�3N�6�, c1y

�3N�6�, . . . , cNz�3N�6��.

In the Bethe program, this is achieved, for instance,for the total symmetric mode, by calling theprocedure

� Q_A1 :� BETHE_normal_coordinates(Td, methane, “A1”, Cartesian);

Q_A1:�[ [ [0, 0, 0, 1/12, 1/12, 1/12, 1/12,�1––12

,�1––12

,�1––12

,1/12,�1–�12

,�1–�12

,�1–�12

,1/12 ] ] ],

where the second and third parameter, methaneand “A1”, again describe the molecule and one ofthe irreducible components as obtained from thetotal vibrational representation. In addition, the lastparameter Cartesian is used as a keyword in

order to specify that the normal coordinates are tobe returned in terms of the Cartesian displace-ments. Similarly, we may also obtain the two nor-mal coordinates associated with the doubly degen-erate irreducible representation E

� Q_E :� BETHE_normal_coordinates(Td, methane, “E”, Cartesian);

Q_E:�[ [ [0,0,0,1/12,�1––24

,�1––24

, 1/12, 1/24, 1/24,�1––12

,�1––24

, 1/24,�1––12

, 1/24,�1––24

],

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114 VOL. 106, NO. 1

[0, 0, 0, 0,31/2

––24

, �31/2

––24

, 0, �31/2

––24

,31/2

––24

, 0,31/2

––24

,31/2

––24

, 0, �31/2

––24

, 24 24]]]

and where each sublists of Q_E defines one of theallowed normal coordinates.

4.2.3. Normal Coordinates in Terms ofInternal Displacement Vectors

For complex molecules, the Cartesian displace-ments become quickly unconvenient as the dimen-sion of the vibrational representation and, hence,the number of normal coordinates (4) increases�3N with the number of atoms in the system. In-stead of the Cartesian coordinates, one often wishesto apply internal coordinates, which refer to theinteratomic distances and/or the bond angles at theequilibrium configuration. That is, the displace-ments from the equilibrium are now described bymeans of stretching vectors r1, r2, . . . and/or thebond-angle deformation vectors �1, �2, . . . , whilethe other steps in the derivation of the normalcoordinates remain similar if use is made of thecorresponding projection operators (1).

In practice, however, there is little agreement in theliterature on the question of how to choose the internalcoordinates. As a rule of thumb, one often starts outfrom a number of stretching vectors that refer to strongbonds between pairs of neighboring atoms and thenadds as many bond-angle deformation vectors asneeded, in order to obtain a total set of 3N � 6 internaldisplacements (displacement vectors). In the CH4 mole-cule, for example, we need at least nine displacementvectors in order to represent the normal modes of meth-ane. A convenient way is to take the four stretching

vectors r1, . . . , r4 along the COH bonds and the angles�HOCOH as given by the six pairs of hydrogen atoms,with the carbon atom in between. Of course, only five ofthese angles are linear independent of each other. Theadvantage of internal coordinates is that they enable usto classify the stretching and bending modes separately;i.e., we obtain the two different reducible representa-tions T(stretch) and T(bend), for which the stretching vectorsand the bond angles form a basis, respectively. To thisend, in the Bethe program, we first define the internalstretching vectors in terms of the atoms involved (i.e., alist [C, H] here refers to C3 H):

� BETHE_internal_coordinates(Td,methane, stretching);

[[C, H1], [C, H2], [C, H3], [C, H4]].

With this definition of the stretching vectors, wecan interpret the total symmetric vibration

� Q_A1 :� BETHE_normal_coordinates(Td,methane, “A1”, stretching);

Q_A1 :� [[[1/2, 1/2, 1/2, 1/2]]]

as the one in which all the hydrogen atoms oscillatealong their COH bonds equally through the equi-librium position. Moreover, one of the triply degen-erate T2 vibrational modes from above has stretch-ing characters and is given by

� Q_E :� BETHE_normal_coordinates(Td, methane, “T2”, stretching);

Q_E :� [[[1/2, �1/2, 1/2, �1/2]], [[1/2, �1/2, �1/2, 1/2]],

[[1/2, 1/2, �1/2, �1/2]]],

while the E modes do not include the stretchingvectors of the molecule

� Q_E :� BETHE_normal_coordinates(Td,methane, “E”, stretching);

Q_E :� [[], []].

Instead, these modes refer to oscillations of the atomsaround the bond angles at equilibrium and are definedby means of (six) proper angles in the Bethe program.

4.3. MOLECULAR VIBRATIONS

Vibrational spectroscopy is certainly known asthe key experimental tool used to resolve the struc-



ture and bonds of molecules or to understand theiradsorption at surfaces [39–41]. While most of thevibrational transitions of molecules belong to theinfrared (IR) region of the electromagnetic spec-trum, accounting for the frequent application of IRspectroscopy, a number of other techniques havebeen developed as well during the past decades,such as Raman spectroscopy, vibrational circulardichroism, or the measurement of the energy loss ofscattered electrons. For the interpretation of theobserved spectra, then, the theory of point anddouble groups is often needed and forms the bridgeto extract most, if not all, the relevant informationabout the molecules or compounds.

4.3.1. Classification of Molecular Vibrations

Starting from the Born–Oppenheimer approxi-mation, the molecular vibrations are considered in-dependent of the state and motion of the electrons.For a wide range of temperatures and pressures, ofcourse, the molecules are found predominantly inthe electronic ground state, so that the vibrations ofthe nuclei can be simply classified in terms of theirnormal modes. Most easily, perhaps, this is seen bymeans of the total vibrational wave function

��n1, n2, . . . , n3N�6� � �nk

�nk�Qk�, (7)

which is simply the product of the oscillator func-tions �nk

(Qk) as associated with the normal coordi-nates k � 1, . . . , 3N � 6. The vibrational groundstate refers to no quanta of excitations, i.e., nk � 0for all k, and hence to the wave function �(0, 0 . . . ,0), which must be invariant under all the symmetryoperations of the group. For this reason, the vibra-tional ground state always transforms according tothe totally symmetric irreducible representationwith characters � � �1 for all operators {Sa, a �1, . . . , g}. In practice, of course, only those excita-tions can be resolved energetically that belong todifferent irreducible, and nondegenerate, represen-tations of the group. However, since the normalvibrations do not interact with each other, the fol-lowing vibrational transitions can be distinguishedand are labeled by just the set of quantum numbers{nk} in order to count the absorbed quanta in thevarious modes:

Fundamental transitions: Refer to the low excita-tions with just a single quantum in one of thenormal modes, i.e., nk � �k,ko

for some mode 1 �ko � 3N � 6. (The set of this low excited levels are

known also as the fundamental (vibrational) levelsof the molecule and give rise to the Raman lines andthe IR bands in the region of �100–5,000 cm�1; thecorresponding fundamental transitions are oftenmore intense than any other kind of transition by atleast one order of magnitude.)

Overtones: Combine the vibrational ground statewith excited states in which more than one quan-tum is absorbed in a particular normal mode 1 �ko � 3N � 6, i.e., for nko

� 2Combination bands: Refer to all transitions in

which more than a single vibrational mode is ex-cited

Hot bands: Arise if an already excited vibration isfurther excited by absorption, a process whose in-tensity also depends on the population of the lowerexcited states

From a group-theoretical viewpoint, of course,the determination of the normal coordinates andnormal vibrations follows very similar lines. Oncehaving generated the vibrational representationT(vib) of a molecule as described in Section 4.2.2, thenormal vibrations and selection rules for the vari-ous types of vibrational spectroscopy are obtainedfrom the decomposition of this representation intoits irreducible components. Apart from the symme-try type of the normal vibrations, this reduction

T(vib) � ��

m�T�� (8)

also gives rise to the number of modes m�, with aparticular symmetry, and hence to the shape of thespectrum as observed experimentally. In general,the total number of irreducible representations inEq. (8) gives the number of vibrational modes of themolecule, while the dimensions denote the degreeof degeneracy for the corresponding frequencies.

4.3.2. Selection Rules for IR and RamanSpectroscopy

IR and Raman spectroscopy are likely the mostwidely applied techniques today for extracting infor-mation about the geometrical and electronic structureof molecules. Although both techniques use the inter-action of the electrons with the radiation field, theyare based on quite different physical principles. Whilethe absorption of IR light is observed in IR spectros-copy, Raman spectra refer to the scattering of light.Both techniques have been improved considerablyover the past few decades and now cover a widerange of applications as described in various text-

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books [40, 42]. Here, we shall not explain the details ofthese methods but demonstrate how computer alge-bra may help in a fast and reliable vibrational analy-sis, even if complex molecules become involved in theexperiments.

Apart from the proper frequency of light, onecentral question in spectroscopy refers to the spec-tral activity and the strength of transitions, as theline intensities in the spectra may differ by manyorders of magnitude. For a given transition, theobserved strength is usually related to the symme-try of the wave functions (7) in the initial and finalstates and, hence, group theory can be used topredict the number and intensity of the observedbands. Conversely, for a given shape of the spec-trum, the symmetry of a molecule may be inferredby analyzing the number and distribution of lines.If we denote the total wave functions for the initialand final vibrational states by �i and �f, respec-tively, a transition from the ground to the funda-mental levels, which is associated with the jth nor-mal mode, can be written as

�i � �k

�nk 3 �nj �k�j

�nk � �f (9)

and used to obtain the symmetry selection rules forIR and Raman spectroscopy. In fact, these rulesdetermine which of the possible transitions (owingto the Ritz’ combination principle) are likely to beobserved in the spectrum. Or, in other words, theytell us which of the vibrational modes are active inone or the other type of the spectra.

In general, a vibrational transition of a moleculeis possible only if the dipole moment changes due

to the interaction with the light field. For IR transi-tions, of course, this requires a change in the per-manent dipole moment at a certain frequency as theangular momentum of the photon has to be ab-sorbed completely by the molecule. Since the prob-ability for a photon absorption is directly propor-tional to the (square of the) transition moment

� d��*f��i, (10)

an excitation of the jth vibrational mode is forbid-den in the IR spectrum if the integral (10) vanishesfor symmetry reasons, i.e., if the totally symmetricrepresentation of the group is not contained in atleast one of the direct products

��i� � ��( x)�( y)�( z)

� � ��f�. (11)

This follows from the fact that the components ofthe dipole moment � � (�x, �y, �z) � (x, y, z)behave under the symmetry operations of thegroup like a translation of the molecule [41].

In the Bethe program, we can analyze the spec-tral activity of the fundamental transitions if thesymmetry of the corresponding mode is known.Using the irreducible components of the vibrationalrepresentation of methane from Section 4.2.2, forinstance, we can ask about the IR activity of thebending mode E

� IR_active_E :� BETHE_spectral_activity(Td, “E”, infrared);

IR_active_“E” :� false

or the stretching mode T2

� IR_active_T2 :� BETHE_spectral_activity(Td, “T2”, infrared);IR_active_T2 :� true

and where the third argument infrared is takenas a keyword to specify the kind of the spectro-scopic activity.

In Raman spectroscopy, in contrast, the vibra-tional frequencies are observed as Raman shifts� � �0 with respect to the incident light at fre-quency � within the visible region of the electro-magnetic spectrum. Since these shifts correspond

to an inelastic scattering of the photons, here weneed to study the dipole moment

�ind � E�, (12)

which is induced by the external field E and where� denotes the polarizability, which is a measure for



how easily the electronic configuration of the mol-ecule can be distorted by the field. In general, thepolarizability is a 3 � 3 Cartesian tensor whosenine components �jk � �kj, j, k � { x, y, z} refer tothe various directions in space and thus have totransform under the symmetry operations of the

molecular (point) group like the products x2, xy,xz, . . . .

In the Bethe program, the Raman activity of avibrational mode can be obtained similarly asbefore if the keyword Raman is used as, for ex-ample, for the stretching mode T2

� Raman_active_T2 :� BETHE_spectral_activity(Td, “T2”, Raman);

Raman_active_T2 :� true

Of course, similar lines can be followed to under-stand the vibrational activities of more complexmolecules as explained in various textbooks andmonographs.

4.3.3. Nonfundamental Vibrations: Overtones

Apart from the fundamental transitions of a mol-ecule, there are typically a vast display of otherlines and bands in most vibrational spectra. Theselines (bands) are associated with excitations inwhich one or several vibrational modes are inducedat high frequencies and are known as the overtones,combination, or hot bands of a molecule. In theharmonic approximation of the vibrational motion,these lines can be interpreted as the absorption ofseveral fundamental photons {ni � 1} in the final-state wave function (9). Although such high-fre-quency lines are often much less intense than thefundamental transitions, they can be observed bymeans of modern (tunable) light sources. Moreover,

resonance mechanisms have been observed thatlead to an enhanced absorption at high frequenciesdue to coupling of these lines with nearby funda-mental transitions.

Using group theory, all these lines (bands) can beunderstood quite similarly to the fundamental tran-sitions, i.e., by analyzing the irreducible compo-nents associated with the upper and lower state ofsome given transition. However, because severalphotons are involved, care has to be taken about thedegeneracy of the normal modes. Therefore, for thesake of simplicity, let us restrict our discussions tothe case of the overtones. If a vibrational mode k isnot degenerate, such as A1, in the case of methane,the final-state (overtone) function is totally symmet-ric if the number of photons nk is even and has thesymmetry of this particular mode if nk is odd, aproperty that can be derived from the harmonicoscillator. In Bethe, we can determine the spectralactivity for A1 in IR spectroscopy

� BETHE_spectral_activity(Td, “A1”, infrared, 2);

false

by just providing the number of (fundamental)photons as a fourth argument. As seen from theoutput, the first overtone is not active in the IRspectrum but would be active in the Raman spec-trum (not shown). For degenerate modes, more-over, there are several possibilities for how theenergy can be shared by vibrations along thecorresponding normal coordinates. For thesemodes, a number of general formulas have beenderived in the literature [41] in order to deter-mine the possible symmetries of the excitedstates.

4.4. CRYSTAL FIELD SPLITTINGS

Apart from vibrational spectroscopy, the sym-metry properties of a system are also reflected bythe splitting of the ground-state level of atoms andions if placed into an external crystal field. For thetransition metals with an open d-shell configura-tion, in particular, a field splitting is caused by anelectrostatic field because the orientation of the d-functions will increase the energy in a region ofhigh electron density and will decrease it otherwise.In ligand field theory, moreover, one assumes the

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metal–ligand interaction as a covalent bond basedon the overlap between the d-orbitals of the metalsand the ligand donor orbitals. Today, therefore,crystal and ligand fields are mainly applied in orderto understand the bonding in transition metal com-plexes.

4.4.1. Classification of One-Electron Statesin a Crystal Field

Crystal field theory was worked out originallyby Bethe, Van Vleck, and others during the 1930s.These investigators assumed a purely electrostaticinteraction between a metal cation (embedded in acrystal field) and the negatively charged electronsof the surrounding ligands. If, for simplicity, wejust consider a single electron in the valence shell ofthe transition metal ion, its angular (wave) functionis given by the spherical harmonics

Ylm��, �� 1

2��lm��eim� (13)

with orbital angular momentum l, leaving out theradial and the spin part of the wave function for thepresent. For such a wave function, which is degen-erate in m � �l, �l � 1, . . . , l in the free atom, wehave the matrix representation

T�l ��R�� eil� 0 · · · 0 00 ei�l�1�� 0 0 0� �

· · · � �

0 0 · · · e�i�l�1�� 00 0 · · · 0 eil�

� (14)

for any rotation by the angle �. In the Bethe pro-gram, we can evaluate this representation for thefinite rotations of an octahedral environment (pointgroup O) of the ligands

� wa :� BETHE_group_representation(0, Ylm, 2);

wa :� [5, 1, 1, 1, �1, �1, �1, �1, �1, �1, �1, �1,�1, �1, �1, �1, �1, �1, 1, 1, 1, 1, 1, 1],

where l � 2 refers to a single d-electron and wherewe restrict ourselves to the characters of the repre-sentation. The full matrix representation would be

obtained by adding the keyword matrix as afourth argument. From the decomposition of thesecharacters, we find

� wb :� BETHE_decompose_representation(0, wa);

wb :� [“E”, “T2”]

i.e., that the five-fold degenerate level of the d-electron is split by the octahedral environment intotwo levels, a doublet E and the triplet T2. Again,group theory could be used here to demonstratethat, for an octahedral field, the (real) dx2–y2 and dz2

orbitals belong to the irreducible representation E,while the dxy, dxz, and dyz orbitals belong to T2. Forother symmetries of the external electrostatic field,the representation T(l ) in Eq. (14) might be irreduc-ible leaving the ionic level degenerate as before.

In practice, the level splitting of atoms and mole-cules in a given crystal field often follows a certainhierarchy owing to some additional interactions andperturbations. Suppose the crystal field is distorted byimpurities or external fields, a further level splitting is

expected because the symmetry of the system is thenreduced. If, for example, the octahedral symmetry Oof the crystal field from above is reduced to D4 sym-metry, the further level splitting of the E and T2 levelsis obtained by carrying out a subduction of the group

� BETHE_group_subduction(0, “E”, D4);BETHE_group_subduction(0, “T2”, D4);

[“A1”, “B1”][“B2”, “E”]

and which shows that each of these levels nowsplits into a pair of “fine-structure” levels with onlyone (E) still degenerate.



4.4.2. Many-Electron Atoms in a Crystal Field

The level splitting of a single electron within acrystal field can also be generalized quite easily tothe case of (effective) many-electron atoms andions, taking Pauli’s principle into account. Owing tothe coupling scheme and the inter-electronic inter-action, however, different fine-structure regimesneed to be distinguished. If, for example, the crystalfield is weak compared with the electron–electronrepulsion within the valence shell (and if we neglectagain the spin of the electrons for the present), thesplitting of a given (LS-) term with total orbitalangular momentum L is basically the same as for asingle l-shell electron, simply because the totalwave function has the same behavior under therotation of the coordinates. For atoms or ions withan outer d2 configuration, for instance, we have thefive LS terms 1S, 3P, 1D, 3F, and 1G, which transformunder rotations like the spherical harmonics YLM

with L � 0, . . . , 4, respectively. For each of theseterms, we can determine the irreducible compo-nents for an octahedral crystal environment bymeans of the Bethe program:

�we_S:�BETHE_group_subduction_O3(0, 0);we_P :� BETHE_group_subduction_O3(0, 1);we_D :� BETHE_group_subduction_O3(0, 2);we_F :� BETHE_group_subduction_O3(0, 3);we_G :� BETHE_group_subduction_O3(0, 4);

we_S :� [“A1”]we_P :� [“T1”]we_D :� [“E”, “T2”]we_F :� [“A2”, “T1”, “T2”]we_G :� [“A1”, “E”, “T1”, “T2”]

and, as found in various textbooks [43, 44]. More-over, since the (electrostatic) crystal field does notinteract directly with the spins of the electrons, allterms from above have, in good approximation,the same spin multiplicity as the original LS par-ent term, that is, 1S 31 A1, 3P 33 T1, 1D 3 {1E,1T2}, etc.

A rather different level splitting is found if thecrystal field becomes comparable or even strongerthan the interaction among the electrons in the va-lence shell. For such a strong field, the couplingwith the electrons must be treated before the elec-tron–electron interaction is taken into account. Us-ing the results from above for the splitting of asingle d-electron, we then have the three possibleconfigurations, E2, ET2, and (T2)2, for two d-elec-trons in the presence of a strong field. For theseconfigurations, the symmetry properties of the lev-els involved can be determined by taking the directproduct of the corresponding irreducible represen-tations. For the configuration (T2)2, for instance, weobtain

� BETHE_group_direct_product(0, “T2”, “T2”);[“A1”, “E”, “T1”, “T2”],

which shows that the 9-fold degenerate level (T2)2

now splits into the four sublevels with symmetryA1, E, T1, and T2. A similar procedure can also beapplied to the other two configurations E2 and ET2,respectively.

4.4.3. Ligand-Field Theory and Transition–Metal Chemistry

The electrostatic treatment of the crystal fieldand the valence-bond model [45], an alternativechemical approach, certainly help explain differentfeatures in the chemistry of transition metals. Nei-ther of these models, however, is really appropriateto predict the bonds of transition–metal complexesin different chemical environments. Therefore, athird model had to be developed, which is knownas ligand field theory, and which allows more than

a purely electrostatic treatment of the surroundingligands [43]. In practice, transition elements oftensupport the formation of very stable coordinationcomplexes that are known not only to be very col-orful but also to exhibit paramagnetism.

Here we shall not consider the details of thistheory in which the interaction between differentatomic shells is considered in order to form molec-ular orbitals. For an octahedral transition–metalcomplex, such as the Co(NH3)6

3� ion, for instance,one assumes that the 15 occupied 3d, 4s, and 4porbitals of the cobalt atom overlap with the sixligands to form a total of 15 molecular orbitals.These overlaps hold the complex together and formthe five (central) orbitals of T2 and E symmetry asdiscussed above in Section 4.4.1 for a single 3delectron (or single vacency, which is formally

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equivalent). The advantage of the ligand field the-ory is that the 3d orbitals can be used in two differ-ent ways: to form the covalent bond skeleton and inorder to “bind” the electrons that belonged origi-nally to the transition metal. Therefore, the symme-try relations and point-group theory can be usedagain for studying such metal complexes and theirreactivity [46].

4.4.4. Vibronic Coupling in Molecules

In molecular spectroscopy, the (combined) dy-namics of the nuclei and electrons is treated con-ventionally within the Born–Oppenheimer approx-imation, that is, the nuclear motion is considered tobe separable from the electrons. While this approx-imation is usually appropriate for the low-lyingexcitations of a molecule, it is less well fulfilled forthe highly excited states, where the Born–Oppen-heimer approach is more the exception than therule. For highly excited molecules and clusters,many new effects arise from the coupling of theelectronic and nuclear degrees of freedom, includ-ing depolarization of Raman lines, Jahn–Teller dis-tortions, and various others. Then, group theoryoften provides almost the only route to analyze thecorresponding spectra. Although, up to the present,the Bethe program has not been applied in thisfield, it might allow a more efficient treatment ofthese concepts or if quantum mechanical matrixelements need to be evaluated.

4.5. TENSOR PROPERTIES OF CRYSTALS

To conclude this section on the use of computeralgebra for the analysis of spectra, let us also men-tion briefly the tensor properties of crystals. Follow-ing Neumann’s principle [47], of course, the intrin-sic symmetry of a crystal has to be reflected by itsmacroscopic properties, such as the dielectric sus-ceptibility, the electrical conductivity, or the ther-mal expansion, which can all be represented bymeans of second-rank tensors. For instance, the di-electric susceptibility of an anisotropic medium canbe described by a second-rank tensor that expressesthe relation between the polarization of the me-dium and the electric field (vectors).

In general, any set of 3r quantities that transformunder a rotation of the coordinates like the compo-nents of (r) vectors is called a polar (or true) tensorof rank r. For a crystal with a given space group G,this set must also be covariant under the symmetry

operations of the group, a condition that often re-duces the number of the independent tensor com-ponents significantly. We shall not discuss how onecan derive the relationship between various tensorcomponents for a given symmetry, but note thatthis can be done by means of the Bethe program.An efficient access to the symmetry properties oftensors might be helpful, for instance, for the inves-tigation of magnetic materials. In these studies, themagnetic vector quantities such as the magneticfield, magnetic induction, or the magnetization vec-tor must be treated by means of so-called C-tensors[48, 49], perhaps a future application of Bethe.

5. Applications in Quantum Chemistry

During the past five decades, quantum chemis-try has not only become a soaring subject but hasalso influenced the development of computers ow-ing to its very specific demands. Therefore, a fur-ther impact on the design of computer hardwarecan be expected, if the traditional methods of quan-tum chemistry are combined with computer-alge-braic techniques, i.e., if hybrid methods are devel-oped. These methods should enable the user toexploit algebraic techniques and the symmetry ofmolecules and solids at a much deeper level than inpurely numerical applications. From the large num-ber of possible applications, we shall display just afew examples that are intended to demonstrate theusefulness of computer algebra.

5.1. HYBRID ORBITALS AS LINEARCOMBINATION OF ATOMIC ORBITALS

Since the late 1930s, Pauling’s hypothesis of hy-brid orbitals has been perhaps one of the mostfrequently applied concepts in all chemistry. Thisconcept, in which the molecular hybrid orbitals areconsidered as linear combinations of atomic orbitals(AO), has been used since then in order to suggestand explain the geometrical structure and the bondorders of many molecules. The hybridization ofatomic orbitals, in particular, helps distinguish be-tween the �- and �-bonding and is usually suffi-cient to envision the shape of molecular orbitals.Owing to the overlap of the atomic orbitals, more-over, it shows immediately that the electrons arelocalized mainly within the ‘bond region’.

As in many other concepts in chemistry, hybrid-ization has its mathematical basis in group theory



and can be combined with molecular orbital theoryin order to define molecular (sub-) shells. In prac-tice, of course, there are different forms of hybrid-ization such as sp or sp3 (hybrid) orbitals. In meth-ane, our working example from above, the four sp3

hybrid orbitals point toward the vertices of a tetra-hedron and overlap each with the 1s orbital of thecorresponding H atom, similar to those found forother molecular ions. To generate the proper linearcombinations (SALCs) of atomic orbitals, severalsteps need to be carried out, including:

▪ Definition of equivalent atomic orbitals withrespect to the molecular symmetry

▪ Generation of the group representation foreach set of equivalent orbitals

▪ Decomposition of the representations into itsirreducible components

▪ Determine the combinations of the atomic or-bitals which possess the proper symmetry ofthe system

For the first three steps from above, we have, infact, demonstrated already how the Bethe programcan be used to solve these or very similar tasks.Then, in order to find the proper combination of theatomic orbitals as well, different methods are avail-able including the application of the projection op-erators as discussed below in the next subsection.For a quick start, however, perhaps the simplestmethod is to match the symmetry of an atomicorbital onto the central atom of the molecule, if sucha center really exists. Here, we shall not discuss thismatching in detail which is often made simply byhand and for which the advantages of computeralgebra, if any, mainly concerns the teaching of thebasic principles and consequences of symmetries inmolecules and solids.

In spite of the great success of the hybridizationmodel, several difficulties show that this “picture”of the molecular bonds is not fully consistent withthe experiment. In fact, hybridization is the extremecase of orbital mixing. For the sp3 hybrid orbitals ofmethane, for instance, this picture implies that allfour orbitals are identical and that each has 25%s-character and 75% p-character. While this is truefor methane, it is not fullfilled for monochlormeth-ane, where the bond angles are no longer all thesame. Theoretically, such differences can be treated(with more or less success) by means of fractionalhybridization schemes, an idea that was developedearlier but that never gained wide acceptance. By

using Bethe, such sophisticated schemes might betreated perhaps more efficiently.

5.2. CONSTRUCTION OF SYMMETRYORBITALS

Despite the great success of Pauling’s hybridiza-tion model as a first step toward a qualitative un-derstanding of the chemical bonds and the struc-ture of molecules, we need to return to the point-group symmetries (and theory) in order to obtain amore reliable picture. This becomes clear, in partic-ular, if we wish to construct symmetry orbitals asthe basic ingredients to determine molecular wavefunctions. For a full set of molecular orbitals, ofcourse, we need to solve some sort of Schrodingerequation.

To explain the relation between the various typesof (one-electron) wave functions, let us begin withthe atomic orbitals

�anlm� �Pnl�ra�

raYlm�a, �a� (15)

which, as usual, are given in spherical coordinates(ra, a, �a) and which are centered at the position aof some of the atoms. Using the LCAO method, i.e.,the linear combination of atomic orbitals, the mo-lecular orbitals can be constructed from the atomicorbitals (15) either directly

�� anlm

C�,anlm�anlm� (16)

or by first making use of an expansion in terms ofsymmetry orbitals

�� i�

B�,�i��i��. (17)

These latter (symmetry) orbitals are sought to beinvariant under the symmetry operations of thegroup and, hence, are classified most convenientlyby means of the group indices i, �, and � of theirreducible representations of the correspondingpoint group. For a given atomic basis, of course, thesymmetry orbitals are found as linear combinationsof atomic orbitals

��i�� anlm�i�� a�nlm�

A�i�;am,a�m��nl � �a�nlm�� (18)

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with coefficients A�i�;am,a�m�(nl ) , which are determined

geometrically by the symmetry and the position ofthe (equivalent) atoms, and where a� refers to all theequivalent atoms for the atom at position a. Inmolecular computations, either the coefficientsC�,anlm or B�,�i� are obtained variationally by solv-ing a corresponding secular equation. The use ofsymmetry orbitals (18) has the advantage that theyhelp reduce the size of the (position) space, inwhich the molecular orbitals need to be treatedexplicitly. For complex but still symmetric mole-cules this often results in a significant gain in theefficiency of the computations.

The symmetry orbitals are invariant with respectto the symmetry operations of the group and aredifferent for the point and double groups, i.e., innonrelativistic and relativistic computations. Sincetheir generation is a purely geometrical task (i.e.,independent of all details of the electronic structureof some molecule), it can be carried out algebra-ically for any given symmetry group and actuallybefore the quantum chemical computations start[50]. In the Bethe program, we apply the groupprojection operator technique to construct the sym-metry orbitals (18) in terms of a given set of atomicorbitals. By applying the projection operator

P��i� �

si

g �r

T��i�*�Sr�Sr, (19)

all the (involved) irreducible components ofthe atomic basis can be calculated for thegroup [cf. Eq. (2)] and yields the symmetry coef-ficients

A�i�;am,a�m��nl � � �

r

�a�,SraT��i�*�Sr��1�l�Rm�m��, (20)

where the summation runs over all the symmetryoperators of the group.

To show how the Bethe program helps in theconstruction of symmetry orbitals, let us take againmethane as one of the most simple polyatomic mol-ecule. For a very first impression on the chemicalbonds of methane, we can restrict ourselves to the(minimal) atomic basis, including the 1s orbitals ofthe four hydrogen atoms as well as the 2s and 2porbitals (n � 2, l � 0, 1) of carbon. In the Betheprogram, this atomic basis is defined by

� basis_C :� Abasis(“C”, [0,0,0], [2,0], [2,1]);basis_H :� Abasis(“H”, [a,a,a], [1,0]);

basis_C :� Abasis(“C”, [0, 0, 0], [2, 0], [2, 1])basis_H :�Abasis(“H”, [a, a, a], [1, 0])

if the carbon atom is found at the origin (of thecoordinates), and if we assume �3a2 as the COHbond length. Similar, as atom() and molecule(),here the command Abasis() is an auxiliary pro-cedure which serves for keeping all the relevantinformation about the atomic basis together foreach sort of equivalent atoms [cf. Table V]. In par-

ticular, providing the principal and orbital angularquantum momentum quantum numbers for each(atomic) subshell, all the magnetic substates �nlm�are then taken into account automatically. For thischoice of the atomic basis, we can define the corre-sponding symmetry orbital basis for the hydrogenatoms by

� BETHE_generate_SO_basis(Td, basis_H, print);

1) Td � H [a, a, a], n�1, l�0, m�0; A1(1, 1) �2) Td � H [a, a, a], n�1, l�0, m�0; T2(1, 1) �3) Td � H [a, a, a], n�1, l�0, m�0; T2(2, 1) �4) Td � H [a, a, a], n�1, l�0, m�0; T2(3, 1) �,



and where a third argument, print, has beenused to enforce the program to output the sym-metry orbitals in a line mode; a null expressionis returned in this case. As seen from the output,each line represents one of the symmetry orbitalsin a notation similar to Eq. (18). The last columnin this output clearly indicates the irreduciblerepresentation of the group T(i), including the

indices of the individual matrix elements. This isstill a rather formal classification of the symmetryorbitals and, perhaps, of not much help withoutknowing their expansion in terms of the atomicorbitals at the different (but equivalent) sites ofthe molecule. An explicit representation is ob-tained by adding this keyword as one of the lastarguments

� BETHE_generate_SO_basis(Td, basis_H, print, explicit);

1. SO: Td � H [a, a, a], n�1, l�0, m�0; A1(1, 1) �1) .5000000000, � [a, a, a], n�1, l�0, m�0�2) .5000000000, �[a, �a, �a], n�1, l�0, m�0�3) .5000000000, �[�a, a, �a], n�1, l�0, m�0�4) .5000000000, �[�a, �a, a], n�1, l�0, m�0�

2. SO: Td � H [a, a, a], n�1, l�0, m�0; T2(1, 1) �1) .5000000000, � [a, a, a], n�1, l�0, m�0��

and where, again, the line mode (keyword print) isused to list the contributions from the atomic orbitalsat different sites and with different m quantum num-bers. For each symmetry orbital, the expansion coef-

ficients are normalized due to ¥i ci2 � 1. Without the

optional argument print, the expansion of the sym-metry orbitals are returned in a list structure [SO1,SO2, . . .] which could be processed further.

TABLE V ______________________________________________________________________________________________Important commands of the BETHE program for the computation of symmetry orbitals and Clebsch–Gordancoefficients.

AO( ) Auxiliary procedure to represent an atomic orbital r�anlm� which is centered atthe position a � (a1, a2, a3)

SO( ) Auxiliary procedure to represent a symmetry orbital r�(Ga)nlm; T(�)��Abasis( ) Auxiliary procedure to represent an atomic basis set {r�anlm�} which is

centered at the position a � (a1, a2, a3)BETHE_generate_AO( ) Generates a list of atomic orbitals (including all m’s) at the site a � (a1, a2, a3)

and for an atom with the identifier stringatom

BETHE_CG_coefficient( ) Calculates the Clebsch–Gordan coefficients for a given point groupBETHE_generate_AO_basis( ) Generates an atomic basis by applying all symmetry operations of the point

group G to the atomic orbitals AO1, AO2, . . . of a given orbital basisBETHE_generate_SO( ) Expands a symmetry orbital r�(Ga)nlm; T(�)�� in terms of the atomic orbitals

of a set of equivalent atomsBETHE_generate_SO_basis( ) Generates a complete, but linear independent basis of symmetry orbitals for

the point group G with label Glabel from the set of atomic orbitals asdescribed by the atomic basis sets Abasis1, Abasis2, . . .

BETHE_set( ) Defines either a relativistic or nonrelativistic framework for the generation of theatomic orbitals and the internal interpretation of the quantum numbers

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Instead of the nonrelativistic atomic orbitals (15),we may start equivalently from a relativistic de-scription of the atoms, based on Dirac’s Hamilto-nian. In this case, the atomic orbitals (for an electronwith spin s � 1/2) are given by the Dirac spinors[51]

�an�m� � Pn�(ra)

ra��m(a, �a, s)

iQn�(ra)

ra��m(a, �a, s) , (21)

where � is the relativistic angular momentum quan-tum number and, again, spherical coordinates areused and centered at the position a of the atom. TheDirac spinors consists out of the two (upper andlower) Pauli spinors with the (s � 1/2) spinorspherical harmonics

��m�a, �a, s�

� �ms

lm ms, sms�jm�Yl,m�ms�a, �a��sms (22)

for the spin-angular part of the atomic orbitals,sometimes known also as the spherical (or Dirac)spin orbitals. When compared with Eq. (15), therelativistic quantum number

� � ��( j � 1/2) � �(l � 1) for j � l � 1/2j � 1/2 � l for j � l 1/2

(23)

replaces the orbital quantum number l, but nowcontains information about the total angular mo-mentum j and the parity (l ) of the atomic orbital.Since, moreover, the relativistic orbitals �an�m� in(21) always refer to half-integer total angular mo-menta j, the double-valued irreducible representa-tions (with the corresponding superscript j) need tobe considered in this case in defining the projectionoperators (19). In practice, however, the summationover the symmetry operators can still be restrictedto those of the corresponding point group becausethe contributions from the two (double group) op-erations S and S� turn out to be equal. To generatethe symmetry orbitals within a relativistic frame-work, therefore, we may follow very similar lines asbefore but by “redefining” first the framework torelativistic

� BETHE_set(framework � relativistic);

Framework is changed to relativistic

5.3. CLEBSCH–GORDAN COEFFICIENTS FORPOINT GROUPS

In physics, the Clebsch–Gordan (CG) coefficientsare best known for the SO3 rotation group from thecoupling of angular momenta. More general, theCG coefficients are defined in group theory as the(geometrical) constants that allow us to perform thedecomposition of tensor products of two (irreduc-ible) representations into its irreducible compo-nents. For symmetric molecules, of course, we needto consider the CG coefficients of the point or dou-ble groups in order to construct many-electronwave functions of well-defined symmetry. Below,therefore, let us briefly summarize the meaning ofthe CG coefficients and how they can be derived bymeans of the Bethe program.

5.3.1. Clebsch–Gordan Expansion

The CG coefficients are closely related to thetensor product of two irreducible representationsand its decomposition into a (so-called) direct sum

T��V��

�

m�T�� t

�

T��,t�, (24)

which is known also as the Clebsch–Gordan expan-sion from the literature [23]. In this expansion,weights m� are obtained from the characters of therepresentation T(�V�) and T(�) by using the greatorthogonality theorem [17, 23]. In Eq. (24), the dotover the summation indicates that not the usualmatrix addition is meant but the decompositionof the representation T(�V�) into a block-diagonalform. For all finite groups, this decomposition canbe achieved by a proper unitary matrix C(�,�) owingto the similarity transformation

�C��,��1T��V��C��,�� T��1,1�

· · ·T��k,t�

· · ·� ,

(25)



and whose matrix elements �i, �k��tl� are just theClebsch–Gordan coefficients (CG) of the symmetrygroup.

In quantum mechanics, the CG coefficients usu-ally appear in the “coupling” of (many-particle)wave functions if a symmetry-adapted basis is re-quired. For molecules with several electrons in anopen molecular subshell, the total wave functioncan be treated either in the uncoupled product basis(of the subsystems) or within a coupled basis whosebasis functions transform like the irreducible com-ponents of the tensor product. Suppose we havetwo electrons with coordinates r1 and r2 whosewave functions, �j

(�)(r1) and �l(�)(r2), form a basis for

the irreducible representations T(�) and T(�) of thegroup G, respectively. Instead of the product func-tions �j

(�)(r1)�l(�)(r2), we can also construct the linear

combinations

�l��,t��r1, r2� � �

ik

�i�k��tl��i��r1��k

��r2� (26)

for the composite system, which now transformslike a basis of the irreducible representation T(�). Inthis expansion, the CG coefficients �i, �k��tl� sim-ply define the “weights” of the individual productfunctions, while the ��i, �k��tl��2 is the probabilityto find each subsystem in the corresponding (one-particle) state �i

(�)(r1) and �k(�)(r2), respectively. For

interacting many-particle systems, moreover, thecoupled basis functions are often physically moreappropriate than the uncoupled product functionsin order to characterize the state of the system, even

if the corresponding Hilbert (sub-)spaces remainsthe same.

5.3.2. Computation of the Clebsch–GordanCoefficients

One can use standard techniques from linearalgebra to derive the unitary matrix C(�,�) and,hence, the Clebsch–Gordan coefficients explicitly.Using the orthogonality property of the irreduciblerepresentations of the point groups, they must ful-fill the relation

�t

�i, �k��tl��j, �l��tn�*

�s�

g �a

Tij��Sa�Tkl

��Sa�Tln��*�Sa�. (27)

Starting from the case i � j, k � l, and m � n, Eq.(27) can be used to determine all the CG coeffi-cients, apart from an overall phase. In the Betheprogram, all the CG coefficients are taken to be realfollowing the phase convention by Altmann andHerzig [24].

Using Bethe, we can calculate either individualCG coefficients or the full unitary matrix C(�,�) fromabove. For the Td symmetry of the methane mole-cule, we have the irreducible representations A1, A2,E, T1, and T2 together with all the direct products A1V A1, A1 V A2, . . . . For the product E V T1, forinstance, we have the direct sum T1 � T2 and candetermine the CG coefficients associated with T2(and some proper indices ikl ):

� BETHE_CG_coefficient(Td, “E”, 1, “T1”, 3, 1, “T2”, 3);

�(1 � 31/2 I) 21/2

———————————————4

or even the full 6 � 6 matrix of CG coefficients [forBETHE_CG_matrix(C3v, “E”, “T1”)], leavingthe output apart for the sake of brevity.

5.4. SYMMETRY REDUCTION OF MATRIXELEMENTS

So far, we have considered the symmetry prop-erties of the (molecular) wave functions at either theone- or many-particle level. However, symmetryconsiderations also serve for another purpose con-

cerning the simplification of matrix elements. InSection 4.3, for instance, we saw how the symmetryof the vibrational modes helps understand the(spectral) activities of chemical compounds in dif-ferent fields of molecular spectroscopy. In fact, thissort of selection rule is one very important class ofsymmetry relations between different matrix ele-ments which always apply, if the wave functionsand the operators involved in some integral obeycertain symmetry properties. To understand theserules, let us consider the wave functions �i, �f and

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126 VOL. 106, NO. 1

the operator T, which transform under some sym-metry operation Sa as

Sa�i � ��1�ni�i

Sa�f � ��1�nf�f

SaT � ��1�nTT. (28)

Then, the amplitude

Sa�f�T��i� � ��1�nf�ni�nT�f�T��i� � �f�T��i� (29)

is nonzero only if nf � ni � nT is an even integer andmust vanish in all other cases.

Equation (28) displays a strict selection rule be-cause the wave functions and operator all trans-form into some “constant” times themselves. An-other situation in which symmetry still helps in theevaluation of the (transition) amplitudes arises if �i,�f and the operator T transform like

Sa�i � ci��i, Sa�f � cf��f, SaT � cTT�, (30)

i.e., into some constant times functions with well-defined symmetry. When this occurs, we will notobtain a selection rule but rather a relation betweentwo different matrix elements

�f�T��i� � �c*fcTci��f�T��i� (31)

if the symmetry operation Sa�f�T��i� � �f�T��i�changes the value of the amplitude in a well-de-fined way. Further simplifications often occur in theevaluation of matrix elements if the symmetryproperties (and the structure) of the operators areconsidered in detail.

The symmetry reduction of matrix elements hasgreat importance, for instance, in studying the elec-tronic–vibrational coupling of molecules. For highexcitations, namely, the Born–Oppenheimer ap-proximation usually breaks down due to the cou-pling of the electronic and vibrational motion. Inthe literature, this coupling is known also as Jahn–Teller coupling and gives rise often to the (Jahn–Teller) effect, that a polyatomic system with a de-generate electronic ground state is unstable andleads to a spontaneous symmetry break and termsplitting [52, 53]. Here, an efficient treatment of thesymmetry may help in analyzing these couplingmore consistently also for complex species, such asthe ionic states of C60 [54].

6. Summary and Outlook

The recent work on developing computer alge-braic techniques has been reviewed for its use in(quantum) chemistry and physics. Following ashort account on the advantages of CAS in thesefields, emphasis was placed on the Bethe programand how this code can be applied in molecularspectroscopy. Apart from the analysis of the normalmodes and the spectral activity of molecules withpoint-group symmetry, in particular, it was dem-onstrated how Bethe can be used to understand thefield splitting in crystals or to construct molecularwave functions of proper symmetry. In the currentversion, the data of 72 frequently applied pointgroups can be used, together with the data for theall the corresponding double groups.

Several examples are worked out in Sections 4and 5 to display (some of) the present features ofthe Bethe program. While we cannot show all thedetails explicitly, these examples certainly demon-strate the potential in applying computer algebraictechniques in research and education. Besides theBethe program, the other components of the Sym-metry Tools [cf. Section 2.2.4] have been found use-ful in different applications, ranging from the anal-ysis of the coherence transfer in Auger cascades [55]over the two-photon ionization of highly chargedions [56] and up to the control of entanglement inatomic photoionization [57]. Although we are cer-tainly still near to the beginning in developing suchalgebraic tools, they have shown the capacity ofalgebraic techniques, also compared with the(much more advanced) numerical tools and librar-ies in scientific computing. Of course, the fullpower of these techniques will become apparentonly if further algebraic software packages and li-braries are developed. In the long run, moreover, acloser combination of algebraic and numerical tech-niques is desirable.

There are various directions in which the Betheprogram could be developed in the future. Apartfrom (i) improved tools for the vibrational analysisof molecules and for the level splitting of atoms invarious regimes of the crystal field, interest hasbeen pointed out for (ii) the spontaneous Jahn–Teller distortion of molecules due to the electronic–vibrational coupling. For studying the magneticproperties of material, moreover, one wishes (iii) toaccess the data from the magnetic point or the colorgroups as well. For education, finally, a number ofexamples and demonstrations are needed for which



suggestions from the site of the users are very wel-come. Owing to the large number of possible (andsurely desirable) extensions of these and additionalpackages, collaboration with other groups is alwaysappreciated.

ACKNOWLEDGMENTS

The development of large symbolic packagessuch as Bethe, Racah, and others is certainly im-possible without the help of coworkers. Therefore,the long collaboration with Gediminas Gaigalas,Peter Koval, Thomas Radtke, Katja Ryklinskaya,Oliver Scharf, Andrey Surzhykov, and DanielUrsescu on one or the other of these projects isgratefully acknowledged.

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