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- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Symmetry_(Vallance)Group Theory is a branch of the mathematical field of algebra. One important application, the theory of symmetry groups, is a powerful tool for the prediction of physical properties of molecules and c...Group Theory is a branch of the mathematical field of algebra. One important application, the theory of symmetry groups, is a powerful tool for the prediction of physical properties of molecules and crystals. It is for example possible to determine whether a molecule can have a dipole moment. Many important predictions of spectroscopic experiments (optical, IR or Raman) can be made purely by group theoretical considerations.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Symmetry_(Vallance)/01%3A_Chapters/1.20%3A_Calculating_Orbital_Energies_and_Expansion_CoefficientsCalculation of the orbital energies and expansion coefficients is based on the variation principle, which states that any approximate wavefunction must have a higher energy than the true wavefunction....Calculation of the orbital energies and expansion coefficients is based on the variation principle, which states that any approximate wavefunction must have a higher energy than the true wavefunction. This follows directly from the fairly common-sense idea that in general any system tries to minimize its energy. If an ‘approximate’ wavefunction had a lower energy than the ‘true’ wavefunction, we would expect the system to try and adopt this ‘approximate’ lower energy state.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Symmetry_(Vallance)/01%3A_Chapters/1.17%3A_Determining_whether_an_Integral_can_be_Non-zeroAs we continue with this course, we will discover that there are many times when we would like to know whether a particular integral is necessarily zero, or whether there is a chance that it may be no...As we continue with this course, we will discover that there are many times when we would like to know whether a particular integral is necessarily zero, or whether there is a chance that it may be non-zero. We can often use group theory to differentiate these two cases.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Symmetry_(Vallance)/01%3A_Chapters/1.12%3A_Reduction_of_Representations_IA block diagonal matrix can be written as the direct sum of the matrices that lie along the diagonal. Note that a direct sum is very different from ordinary matrix addition since it produces a matrix ...A block diagonal matrix can be written as the direct sum of the matrices that lie along the diagonal. Note that a direct sum is very different from ordinary matrix addition since it produces a matrix of higher dimensionality.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Symmetry_(Vallance)/01%3A_Chapters/1.28%3A_SummaryHopefully this text has given you a reasonable introduction to the qualitative description of molecular symmetry, and also to the way in which it can be used quantitatively within the context of group...Hopefully this text has given you a reasonable introduction to the qualitative description of molecular symmetry, and also to the way in which it can be used quantitatively within the context of group theory to predict important molecular properties.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Symmetry_(Vallance)/01%3A_Chapters/1.21%3A_Solving_the_Secular_EquationsAny set of linear equations may be rewritten as a matrix equation Ax = b, which can be classified as simultaneous linear equations or homogeneous linear equations, depending on whether b is non-zero o...Any set of linear equations may be rewritten as a matrix equation Ax = b, which can be classified as simultaneous linear equations or homogeneous linear equations, depending on whether b is non-zero or zero. the set of equations only has a solution if the determinant of A is equal to zero. The secular equations we want to solve are homogeneous equations, and we will use this property of the determinant to determine the molecular orbital energies.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Symmetry_(Vallance)/01%3A_Chapters/1.14%3A_Character_TablesA character table summarizes the behavior of all of the possible irreducible representations of a group under each of the symmetry operations of the group. In many applications of group theory, we onl...A character table summarizes the behavior of all of the possible irreducible representations of a group under each of the symmetry operations of the group. In many applications of group theory, we only need to know the characters of the representative matrices, rather than the matrices themselves. Luckily, when each basis function transforms as a 1D irreducible representation there is a simple shortcut to determining the characters without having to construct the entire matrix representation.
- https://chem.libretexts.org/Courses/BethuneCookman_University/BCU%3A_CH_332_Physical_Chemistry_II/Text/7%3A_Approximation_Methods/7.02%3A_Linear_Variational_Method_and_the_Secular_DeterminantA special type of variation widely used in the study of molecules is the so-called linear variation function, a linear combination of N linearly independent functions (often atomic orbitals). Quite o...A special type of variation widely used in the study of molecules is the so-called linear variation function, a linear combination of N linearly independent functions (often atomic orbitals). Quite often a trial wavefunction is expanded as a linear combination of other functions (not the eigenvalues of the Hamiltonian, since they are not known) .
- https://chem.libretexts.org/Courses/BethuneCookman_University/B-CU%3ACH-331_Physical_Chemistry_I/CH-331_Text/CH-331_Text/07._Approximation_Methods/7.2%3A_Linear_Variational_Method_and_the_Secular_DeterminantA special type of variation widely used in the study of molecules is the so-called linear variation function, a linear combination of N linearly independent functions (often atomic orbitals). Quite o...A special type of variation widely used in the study of molecules is the so-called linear variation function, a linear combination of N linearly independent functions (often atomic orbitals). Quite often a trial wavefunction is expanded as a linear combination of other functions (not the eigenvalues of the Hamiltonian, since they are not known) .
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Symmetry_(Vallance)/01%3A_Chapters/1.13%3A_Irreducible_representations_and_symmetry_speciesWhen two one-dimensional irreducible representations are seen to be identical, they have the ‘same symmetry’, transforming in the same way under all of the symmetry operations of the point group and f...When two one-dimensional irreducible representations are seen to be identical, they have the ‘same symmetry’, transforming in the same way under all of the symmetry operations of the point group and forming bases for the same matrix representation. As such, they are said to belong to the same symmetry species. There are a limited number of ways in which an arbitrary function can transform under the symmetry operations of a group, giving rise to a limited number of symmetry species.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Symmetry_(Vallance)/01%3A_Chapters/1.19%3A_Bonding_in_Polyatomics-_Constructing_Molecular_Orbitals_from_SALCsIn the previous section we showed how to use symmetry to determine whether two atomic orbitals can form a chemical bond. How do we carry out the same procedure for a polyatomic molecule, in which many...In the previous section we showed how to use symmetry to determine whether two atomic orbitals can form a chemical bond. How do we carry out the same procedure for a polyatomic molecule, in which many atomic orbitals may combine to form a bond? Any SALCs of the same symmetry could potentially form a bond, so all we need to do to construct a molecular orbital is take a linear combination of all the SALCs of the same symmetry species.
