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Chemistry 3451 (Physical Chemistry III) Information Sheet Dimensions, Physical Constants, Conversion Factors and other useful information Dimensions: Page 2 of 31 Physical Constants: Useful Conversion Factors: Page 3 of 31 Useful Prefixes for Powers of Ten: The Greek Alphabet: Page 4 of 31 Integrals: Useful Mathematical Topics (Integrals, Derivative Properties, Imaginary numbers and Relationships) Page 5 of 31 Page 6 of 31 Page 7 of 31 Page 8 of 31 Some Calculus Rules: Differential Equations: Second order differential equations have two solutions Page 9 of 31 Hermite’s Differential Equation (associated) Legendre’s Differential Equation for m < 0 Separation of Variables Assumed, without initial justification, that the dependent variable can be written as a product now can be decomposed into two ordinary differential equations and solve Page 10 of 31 Trigonometry: Page 11 of 31 Page 12 of 31 Complex Numbers: ; ; ; ; Page 13 of 31 Vectors and Matrices: Vectors are ordered N-­‐tuples of numbers dot-­‐ or inner-­‐product (scalar) norm (or generalized length) matrix is a square N x N array Unit Matrix Multiplication of a vector by a matrix on the left gives a new vector the product of two matrices is a matrix The transpose of a matrix is given by Page 14 of 31 the adjoint (or Hermitian conjugate) is given by The determinant of a matrix the expectation value in matrix form The eigenvalue problem in matrix formulation the cross-­‐product (a new matrix of size N x M) Page 15 of 31 Some useful identities involving scalar and cross-­‐products Angular Momentum: Bra-­‐Ket notation: Page 16 of 31 More Useful Relationships: The Taylor Series Expansion of a (well-­‐behaved) function f(x) about the point x=a: where The gradient-­‐squared (or Laplacian) operator in Cartesian, Polar and Spherical-­‐Polar coordinates: Page 17 of 31 Group Theory Flow Scheme for Group Symbols: Character Tables: Page 18 of 31 Special, High Symmetry Groups Page 19 of 31 Page 20 of 31 Page 21 of 31 Groups with no axis of symmetry The S2n Groups Page 22 of 31 The Cn Groups Page 23 of 31 The Cnv Groups Page 24 of 31 The Cnh Groups Page 25 of 31 Page 26 of 31 The Dn Groups Page 27 of 31 The Dnd Groups Page 28 of 31 The Dnh Groups Page 29 of 31 Page 30 of 31 Page 31 of 31