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Group Theoretical Methods and Applications to Molecules and Crystals : And Applications to Molecules and Crystals.

By: Publisher: Cambridge : Cambridge University Press, 1999Copyright date: ©1999Description: 1 online resource (512 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9780511151897
Subject(s): Genre/Form: Additional physical formats: Print version:: Group Theoretical Methods and Applications to Molecules and Crystals : And Applications to Molecules and CrystalsDDC classification:
  • 541.3015122
LOC classification:
  • QD455.3.G75 K56 1999
Online resources:
Contents:
Cover -- Half-title -- Dedication -- Title -- Copyright -- Contents -- Preface -- Acknowledgment -- List of symbols -- 1 Linear transformations -- 1.1 Vectors -- 1.2 Linear transformations and matrices -- 1.2.1 Functions of a matrix -- 1.2.2 Special matrices -- 1.2.3 Direct products of matrices -- 1.2.4 Direct sums of matrices -- 1.3 Similarity transformations -- 1.3.1 Functions of a matrix (revisited) -- 1.4 The characteristic equation of a matrix -- 1.4.1 Diagonalizability and projection operators -- 1.5 Unitary transformations and normal matrices -- 1.5.1 Examples of normal matrices -- 1.6 Exercises -- 2 The theory of matrix transformations -- 2.1 Involutional transformations -- 2.2 Application to the Dirac theory of the electron -- 2.2.1 The Dirac Gamma-matrices -- 2.2.1.1 The Clifford algebra of the order four -- 2.2.1.2 The Clifford algebra for d = 5 -- 2.2.2 The Dirac plane waves -- 2.2.3 The symmetric Dirac plane waves -- 2.3 Intertwining matrices -- 2.3.1 Idempotent matrices -- 2.4 Matrix diagonalizations -- 2.5 Basic properties of the characteristic transformation matrices -- 2.6 Construction of a transformation matrix -- 2.7 Illustrative examples -- 3 Elements of abstract group theory -- 3.1 Group axioms -- 3.1.1 The criterion for a finite group -- 3.1.2 Examples of groups -- 3.2 Group generators for a finite group -- 3.2.1 Examples -- 3.3 Subgroups and coset decompositions -- 3.3.1 The criterion for subgroups -- 3.3.1.1 Cosets -- 3.3.2 Langrange's theorem -- 3.3.2.1 Examples -- 3.4 Conjugation and classes -- 3.4.1 Normalizers -- 3.4.1.1 Normal subgroups -- 3.4.1.2 Examples -- 3.4.2 The centralizer -- 3.4.2.1 Examples (continued) -- 3.4.3 The center -- 3.4.4 Classes -- 3.4.4.1 The class order theorem -- 3.4.4.2 Ambivalent classes -- 3.4.4.3 Examples (continued) -- 3.5 Isomorphism and homomorphism -- 3.5.1 Examples.
3.5.2 Factor groups -- 3.5.2.1 The isomorphism theorem -- 3.5.2.2 Examples -- 3.6 Direct products and semidirect products -- 4 Unitary and orthogonal groups -- 4.1 The unitary group U(n) -- 4.1.1 Basic properties -- 4.1.2 The exponential form -- 4.2 The orthogonal group O(n, c) -- 4.2.1 Basic properties -- 4.2.2 Improper rotation -- 4.2.3 The real orthogonal group O(n, r) -- 4.2.4 Real exponential form -- 4.3 The rotation group in three dimensions O(3, r) -- 4.3.1 Basic properties of rotation -- 4.3.1.1 The characteristic equation of Omega -- 4.3.1.2 The matrix expression of R(Theta) -- 4.3.2 The conjugate rotations -- 4.3.3 The Euler angles -- 5 The point groups of finite order -- 5.1 Introduction -- 5.1.1 The uniaxial group C -- 5.1.2 Multiaxial groups. The equivalence set of axes and axis-vectors -- 5.1.3 Notations and the multiplication law for point operations -- 5.1.3.1 Basis-vector representations -- 5.1.3.2 Examples -- 5.1.3.3 Jones representations -- 5.2 The dihedral group D -- 5.3 Proper polyhedral groups P -- 5.3.1 Proper cubic groups, T and O -- 5.3.1.1 The tetrahedral group T -- 5.3.1.2 The octahedral group O (revisited) -- 5.3.2 Presentations of polyhedral groups -- 5.3.2.1 The Wyle relation -- 5.3.2.2 One- or two-sidedness of a rotation axis -- 5.3.3 Subgroups of proper point groups -- 5.3.4 Theorems on the axis-vectors of proper point groups -- 5.3.4.1 Examples -- 5.4 The Wyle theorem on proper point groups -- 5.5 Improper point groups -- 5.5.1 General discussion -- 5.5.2 Presentations of improper point groups -- 5.5.3 Subgroups of point groups of finite order -- 5.6 The angular distribution of the axis-vectors of rotation for regular polyhedral groups -- 5.6.1 General discussion -- 5.6.2 The icosahedral group Y -- 5.6.3 Buckminsterfullerene C (buckyball) -- 5.7 Coset enumeration -- 6 Theory of group representations.
6.1 Hilbert spaces and linear operators -- 6.1.1 Hilbert spaces -- 6.1.1.1 Orthogonalization -- 6.1.2 Linear operators -- 6.1.2.1 Special operators -- 6.1.3 The matrix representative of an operator -- 6.2 Matrix representations of a group -- 6.2.1 Homomorphism conditions -- 6.2.2 The regular representation -- 6.2.3 Irreducible representations -- 6.3 The basis of a group representation -- 6.3.1 The carrier space of a representation -- 6.3.2 The natural basis of a matrix group -- 6.3.2.1 Examples -- 6.4 Transformation of functions and operators -- 6.4.1 General discussion -- 6.4.2 The group of transformation operators -- 6.4.3 Transformation of operators under G = {R} -- 6.5 Schur's lemma and the orthogonality theorem on irreducible representations -- 6.6 The theory of characters -- 6.6.1 Orthogonality relations -- 6.6.2 Frequencies and irreducibility criteria -- 6.6.2.1 The completeness condition for unirreps -- 6.6.2.2 Exercises -- 6.6.3 Group functions -- 6.7 Irreducible representations of point groups -- 6.7.1 The group C -- 6.7.2 The group D -- 6.7.2.1 Exercises -- 6.7.3 The group T -- 6.7.4 The group O -- 6.7.5 The improper point groups -- 6.8 Properties of irreducible bases -- 6.8.1 The orthogonality of basis functions -- 6.8.2 Application to perturbation theory -- 6.9 Symmetry-adapted functions -- 6.9.1 Generating operators -- 6.9.2 The projection operators -- 6.9.2.1 Concluding remarks -- 6.9.2.2 The projection operators based on the characters -- 6.10 Selection rules -- 7 Construction of symmetry-adapted linear combinations based on the correspondence theorem -- 7.1 Introduction -- 7.2 The basic development -- 7.2.1 Equivalent point space S -- 7.2.2 The correspondence theorem on basis functions -- 7.2.2.1 The SALC of equivalent scalar orbitals -- 7.2.3 Mathematical properties of bases on S.
7.2.4 Illustrative examples of the SALCs of equivalent scalars -- 7.3 SALCs of equivalent orbitals in general -- 7.3.1 The general expression of SALCs -- 7.3.2 Two-point bases and operator bases -- 7.3.3 Notations for equivalent orbitals -- 7.3.4 Alternative elementary bases -- 7.3.5 Illustrative examples -- 7.4 The general classification of SALCs -- 7.4.1 D SALCs from the equivalent orbitals… -- 7.5 Hybrid atomic orbitals -- 7.5.1 The Sigma-bonding hybrid AOs -- 7.5.2 General hybrid AOs -- 7.6 Symmetry coordinates of molecular vibration based on the correspondence theorem -- 7.6.1 External symmetry coordinates of vibration -- 7.6.2 Internal vibrational coordinates -- 7.6.3 Illustrative examples -- 8 Subduced and induced representations -- 8.1 Subduced representations -- 8.2 Induced representations -- 8.2.1 Transitivity of induction -- 8.2.2 Characters of induced representations -- 8.2.3 The irreducibility condition for induced representations -- 8.3 Induced representations from the irreps of a normal subgroup -- 8.3.1 Conjugate representations -- 8.3.2 Little groups and orbits -- 8.3.3 Examples -- 8.4 Irreps of a solvable group by induction -- 8.4.1 Solvable groups -- 8.4.2 Induced representations for a solvable group -- 8.4.3 Case I (reducible) -- 8.4.3.1 The induced representation from the identity representation -- 8.4.4 Case II (irreducible) -- 8.4.5 Examples -- 8.5 General theorems on induced and subduced representations and construction of unirreps via small representations -- 8.5.1 Induction and subduction -- 8.5.2 Small representations of a little group -- 8.5.3 Induced representations from small representations -- 9 Elements of continuous groups -- 9.1 Introduction -- 9.1.1 Mixed continuous groups -- 9.2 The Hurwitz integral -- 9.2.1 Orthogonality relations -- 9.3 Group generators and Lie algebra.
9.4 The connectedness of a continuous group and the multivalued representations -- 10 The representations of the rotation group -- 10.1 The structure of SU(2) -- 10.1.1 The generators of SU(2) -- 10.1.2 The parameter space Omega' of SU(2) -- 10.1.2.1 The connectedness of SU(2) -- 10.1.3 Spinors -- 10.1.4 Quaternions -- 10.2 The homomorphism between SU(2) and SO(3, r) -- 10.3 The unirreps…of the rotation group -- 10.3.1 The homogeneity of… -- 10.3.2 The unitarity of… -- 10.3.3 The irreducibility of… -- 10.3.4 The completeness of the unirreps… -- 10.3.5 Orthogonality relations of… -- 10.3.6 The Hurwitz density function for SU(2) -- 10.4 The generalized spinors and the angular momentum eigenfunctions -- 10.4.1 The generalized spinors -- 10.4.2 The transformation of the total angular momentum eigenfunctions under the general rotation U -- 10.4.3 The vector addition model -- 10.4.4 The Clebsch-Gordan coefficients -- 10.4.4.1 Covariance -- 10.4.4.2 Contravariance -- 10.4.5 The angular momentum eigenfunctions for one electron -- 11 Single- and double-valued representations of point groups -- 11.1 The double-valued representations of point groups expressed by the projective representations -- 11.1.1 The projective set of a point group -- 11.1.2 The orthogonality relations for projective unirreps -- 11.2 The structures of double point groups -- 11.2.1 Defining relations of double point groups -- 11.2.2 The structure of the double dihedral group D -- 11.2.3 The structure of the double octahedral group O -- 11.2.3.1 The class structure of the double group O -- 11.3 The unirreps of double point groups expressed by the projective unirreps of point groups -- 11.3.1 The uniaxial group C -- 11.3.2 The group C -- 11.3.3 The group D -- 11.3.4 The group D -- 11.3.5 The group O -- 11.3.6 The tetrahedral group T -- 12 Projective representations -- 12.1 Basic concepts.
12.2 Projective equivalence.
Summary: From the pioneering author in the field, this book is ideal for condensed matter physicists and physical chemists.
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Cover -- Half-title -- Dedication -- Title -- Copyright -- Contents -- Preface -- Acknowledgment -- List of symbols -- 1 Linear transformations -- 1.1 Vectors -- 1.2 Linear transformations and matrices -- 1.2.1 Functions of a matrix -- 1.2.2 Special matrices -- 1.2.3 Direct products of matrices -- 1.2.4 Direct sums of matrices -- 1.3 Similarity transformations -- 1.3.1 Functions of a matrix (revisited) -- 1.4 The characteristic equation of a matrix -- 1.4.1 Diagonalizability and projection operators -- 1.5 Unitary transformations and normal matrices -- 1.5.1 Examples of normal matrices -- 1.6 Exercises -- 2 The theory of matrix transformations -- 2.1 Involutional transformations -- 2.2 Application to the Dirac theory of the electron -- 2.2.1 The Dirac Gamma-matrices -- 2.2.1.1 The Clifford algebra of the order four -- 2.2.1.2 The Clifford algebra for d = 5 -- 2.2.2 The Dirac plane waves -- 2.2.3 The symmetric Dirac plane waves -- 2.3 Intertwining matrices -- 2.3.1 Idempotent matrices -- 2.4 Matrix diagonalizations -- 2.5 Basic properties of the characteristic transformation matrices -- 2.6 Construction of a transformation matrix -- 2.7 Illustrative examples -- 3 Elements of abstract group theory -- 3.1 Group axioms -- 3.1.1 The criterion for a finite group -- 3.1.2 Examples of groups -- 3.2 Group generators for a finite group -- 3.2.1 Examples -- 3.3 Subgroups and coset decompositions -- 3.3.1 The criterion for subgroups -- 3.3.1.1 Cosets -- 3.3.2 Langrange's theorem -- 3.3.2.1 Examples -- 3.4 Conjugation and classes -- 3.4.1 Normalizers -- 3.4.1.1 Normal subgroups -- 3.4.1.2 Examples -- 3.4.2 The centralizer -- 3.4.2.1 Examples (continued) -- 3.4.3 The center -- 3.4.4 Classes -- 3.4.4.1 The class order theorem -- 3.4.4.2 Ambivalent classes -- 3.4.4.3 Examples (continued) -- 3.5 Isomorphism and homomorphism -- 3.5.1 Examples.

3.5.2 Factor groups -- 3.5.2.1 The isomorphism theorem -- 3.5.2.2 Examples -- 3.6 Direct products and semidirect products -- 4 Unitary and orthogonal groups -- 4.1 The unitary group U(n) -- 4.1.1 Basic properties -- 4.1.2 The exponential form -- 4.2 The orthogonal group O(n, c) -- 4.2.1 Basic properties -- 4.2.2 Improper rotation -- 4.2.3 The real orthogonal group O(n, r) -- 4.2.4 Real exponential form -- 4.3 The rotation group in three dimensions O(3, r) -- 4.3.1 Basic properties of rotation -- 4.3.1.1 The characteristic equation of Omega -- 4.3.1.2 The matrix expression of R(Theta) -- 4.3.2 The conjugate rotations -- 4.3.3 The Euler angles -- 5 The point groups of finite order -- 5.1 Introduction -- 5.1.1 The uniaxial group C -- 5.1.2 Multiaxial groups. The equivalence set of axes and axis-vectors -- 5.1.3 Notations and the multiplication law for point operations -- 5.1.3.1 Basis-vector representations -- 5.1.3.2 Examples -- 5.1.3.3 Jones representations -- 5.2 The dihedral group D -- 5.3 Proper polyhedral groups P -- 5.3.1 Proper cubic groups, T and O -- 5.3.1.1 The tetrahedral group T -- 5.3.1.2 The octahedral group O (revisited) -- 5.3.2 Presentations of polyhedral groups -- 5.3.2.1 The Wyle relation -- 5.3.2.2 One- or two-sidedness of a rotation axis -- 5.3.3 Subgroups of proper point groups -- 5.3.4 Theorems on the axis-vectors of proper point groups -- 5.3.4.1 Examples -- 5.4 The Wyle theorem on proper point groups -- 5.5 Improper point groups -- 5.5.1 General discussion -- 5.5.2 Presentations of improper point groups -- 5.5.3 Subgroups of point groups of finite order -- 5.6 The angular distribution of the axis-vectors of rotation for regular polyhedral groups -- 5.6.1 General discussion -- 5.6.2 The icosahedral group Y -- 5.6.3 Buckminsterfullerene C (buckyball) -- 5.7 Coset enumeration -- 6 Theory of group representations.

6.1 Hilbert spaces and linear operators -- 6.1.1 Hilbert spaces -- 6.1.1.1 Orthogonalization -- 6.1.2 Linear operators -- 6.1.2.1 Special operators -- 6.1.3 The matrix representative of an operator -- 6.2 Matrix representations of a group -- 6.2.1 Homomorphism conditions -- 6.2.2 The regular representation -- 6.2.3 Irreducible representations -- 6.3 The basis of a group representation -- 6.3.1 The carrier space of a representation -- 6.3.2 The natural basis of a matrix group -- 6.3.2.1 Examples -- 6.4 Transformation of functions and operators -- 6.4.1 General discussion -- 6.4.2 The group of transformation operators -- 6.4.3 Transformation of operators under G = {R} -- 6.5 Schur's lemma and the orthogonality theorem on irreducible representations -- 6.6 The theory of characters -- 6.6.1 Orthogonality relations -- 6.6.2 Frequencies and irreducibility criteria -- 6.6.2.1 The completeness condition for unirreps -- 6.6.2.2 Exercises -- 6.6.3 Group functions -- 6.7 Irreducible representations of point groups -- 6.7.1 The group C -- 6.7.2 The group D -- 6.7.2.1 Exercises -- 6.7.3 The group T -- 6.7.4 The group O -- 6.7.5 The improper point groups -- 6.8 Properties of irreducible bases -- 6.8.1 The orthogonality of basis functions -- 6.8.2 Application to perturbation theory -- 6.9 Symmetry-adapted functions -- 6.9.1 Generating operators -- 6.9.2 The projection operators -- 6.9.2.1 Concluding remarks -- 6.9.2.2 The projection operators based on the characters -- 6.10 Selection rules -- 7 Construction of symmetry-adapted linear combinations based on the correspondence theorem -- 7.1 Introduction -- 7.2 The basic development -- 7.2.1 Equivalent point space S -- 7.2.2 The correspondence theorem on basis functions -- 7.2.2.1 The SALC of equivalent scalar orbitals -- 7.2.3 Mathematical properties of bases on S.

7.2.4 Illustrative examples of the SALCs of equivalent scalars -- 7.3 SALCs of equivalent orbitals in general -- 7.3.1 The general expression of SALCs -- 7.3.2 Two-point bases and operator bases -- 7.3.3 Notations for equivalent orbitals -- 7.3.4 Alternative elementary bases -- 7.3.5 Illustrative examples -- 7.4 The general classification of SALCs -- 7.4.1 D SALCs from the equivalent orbitals… -- 7.5 Hybrid atomic orbitals -- 7.5.1 The Sigma-bonding hybrid AOs -- 7.5.2 General hybrid AOs -- 7.6 Symmetry coordinates of molecular vibration based on the correspondence theorem -- 7.6.1 External symmetry coordinates of vibration -- 7.6.2 Internal vibrational coordinates -- 7.6.3 Illustrative examples -- 8 Subduced and induced representations -- 8.1 Subduced representations -- 8.2 Induced representations -- 8.2.1 Transitivity of induction -- 8.2.2 Characters of induced representations -- 8.2.3 The irreducibility condition for induced representations -- 8.3 Induced representations from the irreps of a normal subgroup -- 8.3.1 Conjugate representations -- 8.3.2 Little groups and orbits -- 8.3.3 Examples -- 8.4 Irreps of a solvable group by induction -- 8.4.1 Solvable groups -- 8.4.2 Induced representations for a solvable group -- 8.4.3 Case I (reducible) -- 8.4.3.1 The induced representation from the identity representation -- 8.4.4 Case II (irreducible) -- 8.4.5 Examples -- 8.5 General theorems on induced and subduced representations and construction of unirreps via small representations -- 8.5.1 Induction and subduction -- 8.5.2 Small representations of a little group -- 8.5.3 Induced representations from small representations -- 9 Elements of continuous groups -- 9.1 Introduction -- 9.1.1 Mixed continuous groups -- 9.2 The Hurwitz integral -- 9.2.1 Orthogonality relations -- 9.3 Group generators and Lie algebra.

9.4 The connectedness of a continuous group and the multivalued representations -- 10 The representations of the rotation group -- 10.1 The structure of SU(2) -- 10.1.1 The generators of SU(2) -- 10.1.2 The parameter space Omega' of SU(2) -- 10.1.2.1 The connectedness of SU(2) -- 10.1.3 Spinors -- 10.1.4 Quaternions -- 10.2 The homomorphism between SU(2) and SO(3, r) -- 10.3 The unirreps…of the rotation group -- 10.3.1 The homogeneity of… -- 10.3.2 The unitarity of… -- 10.3.3 The irreducibility of… -- 10.3.4 The completeness of the unirreps… -- 10.3.5 Orthogonality relations of… -- 10.3.6 The Hurwitz density function for SU(2) -- 10.4 The generalized spinors and the angular momentum eigenfunctions -- 10.4.1 The generalized spinors -- 10.4.2 The transformation of the total angular momentum eigenfunctions under the general rotation U -- 10.4.3 The vector addition model -- 10.4.4 The Clebsch-Gordan coefficients -- 10.4.4.1 Covariance -- 10.4.4.2 Contravariance -- 10.4.5 The angular momentum eigenfunctions for one electron -- 11 Single- and double-valued representations of point groups -- 11.1 The double-valued representations of point groups expressed by the projective representations -- 11.1.1 The projective set of a point group -- 11.1.2 The orthogonality relations for projective unirreps -- 11.2 The structures of double point groups -- 11.2.1 Defining relations of double point groups -- 11.2.2 The structure of the double dihedral group D -- 11.2.3 The structure of the double octahedral group O -- 11.2.3.1 The class structure of the double group O -- 11.3 The unirreps of double point groups expressed by the projective unirreps of point groups -- 11.3.1 The uniaxial group C -- 11.3.2 The group C -- 11.3.3 The group D -- 11.3.4 The group D -- 11.3.5 The group O -- 11.3.6 The tetrahedral group T -- 12 Projective representations -- 12.1 Basic concepts.

12.2 Projective equivalence.

From the pioneering author in the field, this book is ideal for condensed matter physicists and physical chemists.

Description based on publisher supplied metadata and other sources.

Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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