Computer Algebraic Tools for Studying the Symmetry Properties of Molecules and Clusters. Katya Rykhlinskaya, University of Kassel

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Computer Algebraic Tools for Studying the Symmetry Properties of Molecules and Clusters Katya Rykhlinskaya, University of Kassel 02. 06. 2005

Computational techniques in the theoretical investigations + Analytic concept Numerical studies

Computational techniques in the theoretical investigations Analytic concept + = Numerical studies Advantages of using CAS in the theoretical research Computer algebraic systems (CAS) (Mathematica, Maple, Reduce,...) Knowledge of the mathematical basis Fast and reliable symbolic manipulation Interactive work (treatment step by step )

Computational techniques in the theoretical investigations Analytic concept + = Numerical studies Advantages of using CAS in the theoretical research Computer algebraic systems (CAS) (Mathematica, Maple, Reduce,...) Knowledge of the mathematical basis Fast and reliable symbolic manipulation Interactive work (treatment step by step ) Application of CAS (in particular!) in the theoretical threatment of many particle systems.

Molecular systems and the symmetry consideration Systems of identical particles Reduction of the standart quantities number

Molecular systems and the symmetry consideration Systems of identical particles Use of the symmetry concept! Reduction of the standart quantities number Possible applications of symmetry in the molecular physics: classification of molecular states derivation of the molecular vibrational modes determination of normal coordinates spectral activities for the IR and Raman spectra level splitting of atoms in the exteral crystal field construction of symmetry orbitals

Molecular systems and the symmetry consideration Systems of identical particles Use of the symmetry concept! Reduction of the standart quantities number Possible applications of symmetry in the molecular physics: classification of molecular states derivation of the molecular vibrational modes determination of normal coordinates spectral activities for the IR and Raman spectra level splitting of atoms in the exteral crystal field construction of symmetry orbitals Computer algebraic tools for dealing with symmetry are required! Group theory mathematical tool for dealing with symmetry!

Symmetry operations. Molecular symmetry and the group theory Proper rotation Cn Reflection Improper rotation Sn Inversion I Groups of operators Families of groups Symmetry groups Cyclic Cn, Cnh, Cnv,Sn Dihedral Dn, Dnh, Dnd Cubic T, Th, Td, O, Oh Identity E Icosahedral I, Ih

Symmetry operations. Molecular symmetry and the group theory Proper rotation Cn Reflection Improper rotation Sn Inversion I Identity E Groups of operators Group representation concept Families of groups Symmetry groups Cyclic Cn, Cnh, Cnv,Sn Dihedral Dn, Dnh, Dnd Cubic T, Th, Td, O, Oh Icosahedral I, Ih

Symmetry operations Group representation concept D3h y z x Induced transformations Representation matrices Depend on the basis! -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0-1 0 0 0-1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0-1 0 0 0 Character table. Similarity transformation Can be reduced to the block diagonal form. 1 0 0 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0-1 0 No further reduction is available! IrRep matrices. Characters!

Symmetry operations Group representation concept D3h y z x Induced transformations Representation matrices Depend on the basis! IrRep matrices. Characters! -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0-1 0 0 0-1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0-1 0 0 0 Can be reduced to the block diagonal form. Character table. Similarity transformation 1 0 0 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0-1 0 No further reduction is available! A1'+ A2'+ E'+ E'+ A2 '+ E ' Wave functions of the molecules form a basis for the irreducible representations. Therefore, the molecular states can be described in terms of irreps!

Molecular systems and the symmetry consideration Systems of identical particles Use of the symmetry concept! Reduction of the standart quantities number Possible applications of symmetry in the molecular physics: classification of molecular states derivation of the molecular vibrational modes determination of normal coordinates spectral activities for the IR and Raman spectra level splitting of atoms in the exteral crystal field construction of symmetry orbitals Computer algebraic tools for dealing with symmetry are required! BETHE! Group theory mathematical tool for dealing with symmetry!

BETHE Maple package for dealing with symmetry. Computer algebraic package BETHE was created to support applications of the molecular symmetry groups in physics, chemistry and biology. Large set of group theoretical parameters Data structure, flexible enough to cover most applications Helpful for occasional use and advanced research work More than 100 procedures within the hierarchical structure Only about 10 commands need to be known by the user Part of the CPC Library (Computer Physics Communications)

BETHE Maple package for dealing with symmetry. Computer algebraic package BETHE was created to support applications of the molecular symmetry groups in physics, chemistry and biology. Large set of group theoretical parameters Data structure, flexible enough to cover most applications Helpful for occasional use and advanced research work More than 100 procedures within the hierarchical structure Only about 10 commands need to be known by the user Part of the CPC Library (Computer Physics Communications) Applications of BETHE package Point group theoretical data Cn, Cnh, Cnv, Sn, Dn, Dnh, O, Oh, T, Th, Td, I, Ih n<11 Generation of the C-G coefficients for the groups Symmetry-adapted basis functions (rel. and nonrel.) Vibrational analysis (generation of the normal coordinates) Level splitting in the external crystal field Selection rules for the IR and Raman spectroscopy (vibrational transitions)

BETHE Maple package for dealing with symmetry. Computer algebraic package BETHE was created to support applications of the molecular symmetry groups in physics, chemistry and biology. Large set of group theoretical parameters Data structure, flexible enough to cover most applications Helpful for occasional use and advanced research work More than 100 procedures within the hierarchical structure Only about 10 commands need to be known by the user Part of the CPC Library (Computer Physics Communications) Applications of BETHE package Point group theoretical data Cn, Cnh, Cnv, Sn, Dn, Dnh, O, Oh, T, Th, Td, I, Ih n<11 Generation of the C-G coefficients for the groups Symmetry-adapted basis functions (rel. and nonrel.) Vibrational analysis (generation of the normal coordinates) Level splitting in the external crystal field Selection rules for the IR and Raman spectroscopy (vibrational transitions)

Generation of point group theoretical data Bethe_group(D3h, operators); [ E, C3+, C3-, C21, C22, C23, sigma_h, S3+, S3-, sigma_v1, sigma_v2, sigma_v3 ] Bethe_group(D3h, irreps); [ A1`, A2`, E`, A1``, A2``, E`` ] Bethe_group_character(D3h, E`, C3+ ); -1 Bethe_group_character(D3h, E` ); [ 2, -1, 0, 2, -1, 0] Bethe_group_irrep(D3h, E`, C3+ ); [ 1/2 ] [-1/2-1/2 I 3 0 ] [ 1/2] [ 0-1/2+1/2 I 3 ]

BETHE Maple package for dealing with symmetry. Computer algebraic package BETHE was created to support applications of the molecular symmetry groups in physics, chemistry and biology. Large set of group theoretical parameters Data structure, flexible enough to cover most applications Helpful for occasional use and advanced research work More than 100 procedures within the hierarchical structure Only about 10 commands need to be known by the user Part of the CPC Library (Computer Physics Communications) Applications of BETHE package Point group theoretical data Cn, Cnh, Cnv, Sn, Dn, Dnh, O, Oh, T, Th, Td, I, Ih n<11 Generation of the C-G coefficients for the groups Symmetry-adapted basis functions (rel. and nonrel.) Vibrational analysis (generation of the normal coordinates) Level splitting in the external crystal field Selection rules for the IR and Raman spectroscopy (vibrational transitions)

Molecular vibrational analysis. Types of molecular motion. Example of the H 20 molecule. Rotational motion Translational motion Vibrational motion Vibrational motion of molecule: interatomic distances and internal angles change periodically without moving of the center of mass.

Molecular vibrational analysis. Types of molecular motion. Example of the H 20 molecule. Rotational motion Translational motion Vibrational motion Vibrational motion of molecule: interatomic distances and internal angles change periodically without moving of the center of mass. = + + Symmetric stretching Assymetric stretching Bending Overall vibrations of a molecule result from the superposition of a number of (relatively small) vibrational motions, known as normal modes of vibrations.

Vibrational motion. Normal modes of vibration. = + + Symmetric stretching Assymetric stretching Bending Normal modes of vibration can be determined as functions of the internal displacement coordinates (interatomic distances or bond angles).

Vibrational motion. Normal modes of vibration. = + + Symmetric stretching Assymetric stretching Bending Normal modes of vibration can be determined as functions of the internal displacement coordinates (interatomic distances or bond angles). ar +r r r α 1 2 1 2 Vibrational modes possess a certain symmetry. Normal coordinates are the basis functions of irreducible representations. Vibrational modes can be classified and determined according to the molecular symmetry!

Use of the BETHE for the vibrational analysis of the water molecule. >Bethe_group(C2v, operators); [ E, C2, sigma_x, sigma_y ] >Bethe_group(C2v, irreps); [ A1, A2, B1, B2 ] Classification of the molecular vibrations in terms of the irreducible representations. Construction of normal coordinates H2O := molecule(atom(h1, [-a,0,-b]), atom(o, [0,0,0]), atom(h2, [a,0,-b])); >VR := Bethe_group_representation(C2v, vibrational, H2O) VR := [3, 1, 1, 3]; >Bethe_decompose_representation(C2v, VR); [ A1, A1, B1 ] > Bethe_internal_coordinates(C2, H2O, bending); [ [H1, O, H2] ]; >Bethe_normal_coordinates(C2v, H2O, A1, bending); [[1]] >Bethe_normal_coordinates(C2v, H2O, B1, bending); [ ] > Bethe_internal_coordinates(C2, H2O, stretching);[[o, H1],[O,H2]]; >Bethe_normal_coordinates(C2v, H2O, B1, stretching); 1/2 1/2 2 2 [[- ----, ----]] 2 2

BETHE Maple package for dealing with symmetry. Computer algebraic package BETHE was created to support applications of the molecular symmetry groups in physics, chemistry and biology. Large set of group theoretical parameters Data structure, flexible enough to cover most applications Helpful for occasional use and advanced research work More than 100 procedures within the hierarchical structure Only about 10 commands need to be known by the user Part of the CPC Library (Computer Physics Communications) Applications of BETHE package Point group theoretical data Cn, Cnh, Cnv, Sn, Dn, Dnh, O, Oh, T, Th, Td, I, Ih n<11 Generation of the C-G coefficients for the groups Symmetry-adapted basis functions (rel. and nonrel.) Vibrational analysis (generation of the normal coordinates) Level splitting in the external crystal field Selection rules for the IR and Raman spectroscopy (vibrational transitions)

Level sptitting in the external crystal field Consider a free atom, containing one d electron ( l = 2 ) outside a filled shell. This atom has 2l +1=5 orbitals. The d state is (5 fold) degenerate. Therefore, the energies of the five atomic d orbitals are identical. The atomic states can be classified according to the irreducible representations of the corresponding symmetry group. The free atom belongs to the continuous rotation group O3. Its states refer to the atomic functions with the angular factor Ylm. Atom, placed into the crystal environment belongs to the crystal symmetry point group.

Level sptitting in the external crystal field The states of one electron atom, placed into the crystal field, become split. This splitting can be classified in terms of the irreducible representations of the surrounding crystal point group. Free Oh D4h Distortion of the crystal configuration leads to the additional splitting. Example demonstrate the energy levels behaviour in the octahedral Oh and dihedral D4h crystal environments. Bethe_group_subduction_O3(Oh, 2); [ Eg, T2g ] (irreps of Oh group) Bethe_group_subduction(Oh, Eg, D4h); [ A1g, B1g ] (irreps of D4h group) Bethe_group_subduction(Oh, T2g, D4h): [ B2g, Eg ] (irreps of D4h group)

Summary: BETHE package for applications of symmetry Molecular geometry Point group theoretical data Cn, Cnh, Cnv, Sn, Dn, Dnh, O, Oh, T, Th, Td, I, Ih n<11 Double group data Level splitting in the external crystal field MAPLE package BETHE Symmetry-adapted basis functions (rel. and nonrel.) Vibrational analysis (generation of the normal coordinates) Selection rules for the IR and Raman spectroscopy (vibrational transitions) Generation of the C-G coefficients for the groups

Outlook: future development of the BETHE package. Problem of the molecular symmetry distortion (known as the Jahn Teller effect) Further treatment of the atomic energy levels behaviour in the external (crystal) field and studying the magnetic properties of materials Development of the vibrational analysis of the molecule as well as the related problems, such as resonance Raman spectroscopy, polarization of the vibrational modes and many others..... your request

Outlook: future development of the BETHE package. Problem of the molecular symmetry distortion (known as the Jahn Teller effect) Further treatment of the atomic energy levels behaviour in the external (crystal) field and studying the magnetic properties of materials Development of the vibrational analysis of the molecule as well as the related problems, such as resonance Raman spectroscopy, polarization of the vibrational modes and many others..... your request Thank you for attention!

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