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.
Suggested Readings :
- Boas, M. L, Mathematical methods for the physical sciences, Kaye Pace, Ed.
3rd, 2006.
- McQuarrie, D. A., Mathematical methods for scientists and engineers,
University Science Books, 2003.
- Anderson, J. M., Mathematics for quantum chemistry, Dover Publications
2005.
- Riley,K. F., Hobson, M. P., Bence, S. J., Mathematical methods for physics
and engineering, Cambridge University Press, 3rd Ed., 2012.
- Bell, W. W., Special functions for scientists and engineers, Dover Publications
2004.
- Arfken, G., Weber, H., and Harris, F., Mathematical methods for physicists,
Academic Press, Ed. 7th, 2012.
- Kreyszig, E. Advanced engineering mathematics, 10th Ed. Wiley, 2015.
Previous