# Office of Academic AffairsIndian Institute of Science Education and Research Berhampur

Chemistry

CHM 325: Mathematical Methods for Chemists (4)

Prerequisites: MTH courses taught in first and second years.

Learning Objectives:

To provide students of chemistry with the necessary skills and confidence to apply simple ideas and methods in mathematics for solving problems in physical chemistry and quantum chemistry.

Course Contents:

• Infinite series and power series: Infinite series and their convergence and divergence. Test of convergence – comparison test, integral test, ratio test. Absolute and conditional convergence. Power series and their region of convergence, Maclaurin series, Taylor series. Series expansion of some simple functions.
• Linear algebra: Matrices, determinants, addition, multiplication and inverse of matrices, vectors, linear combination, linear functions, linear operators, linear dependence and independence, special matrices – real, symmetric, anti-symmetric, Hermitian, orthogonal and unitary. Orthogonal transformations – rotations in 2 and 3 dimensions, Euler angles. Linear vector space, orthonormal basis, eigenvalues and eigenvectors of a square matrix, diagonalisation of a square matrix. Matrices and linear operators, Hermitian operators, projection operators.
• Differential equations: Types of differential equations with examples: ordinary differential equations (separable, first, and second order equations) & partial differential equations, Methods for solving differential equations: separation of variables, series solution of ordinary differential equations (Legendre & Bessel equations), Expansion about a regular singular point. Laplace transform and power series (Frobenius) methods of solving linear differential equations. Applications to problems in chemistry.
• Orthogonal functions: Even and odd functions, complete set of functions, orthogonal and orthonormal functions, expansion in terms of orthonornal functions, Fourier series, construction of orthonormal functions by Gram-Schmidt procedure, Schwarz inequality. Hermite, Legendre and Laguerre polynomials and their properties, generating functions and differential equations associated with these polynomials, recursion relations. Highlight their applications in quantum mechanics/chemistry.
• Vector analysis: Scalar and vector multiplications, triple products, differentiation of vectors, directional derivative, gradient, curl, successive application of the differential operator, line integrals, conservative fields, potentials, exact differentials, Green’s theorem in the plane, divergence and divergence theorem, curl and Stokes’ theorem.
• Tensor analysis: Cartesian tensors, tensor notation and operations, Kronecker delta and Levi-Civita symbol. Moment of inertia and other examples of 2nd rank tensors in molecules and materials.
• Curvilinear coordinates: General curvilinear coordinates, vector operators in orthogonal curvilinear coordinates, Laplacian operator in cylindrical, plane polar and spherical polar coordinates. Highlight their applications in quantum mechanics/chemistry.
• Complex numbers: Complex numbers as an ordered pair of numbers, as a point in a 2- dimensional space. Complex plane and Argand diagram, complex algebra, complex power series and disk of convergence, Euler’s formula, powers and roots of complex numbers, exponential and trigonometric and hyperbolic functions, logarithms, complex roots and powers, inverse trigonometric and hyperbolic functions.
• Functions of a complex variable: Analytic functions, Cauchy-Riemann conditions, regular points and singular points, contour integrals, Cauchy’s integral theorem, Cauchy’s integral formula, Laurent series, residue theorem, evaluation of definite integrals by the use of residue theorem.
• Integral transforms: Fourier transform, cosine-sine transforms, Fourier transform of derivatives, convolution theorem, general integral transforms, Laplace transform, convolution, Inversion formula for Laplace transform.