Department of Chemical Sciences

FALL SEMESTER

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Graduate Course on

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Mathematical Methods

**1. Linear
algebra**

Matrices, determinants, vectors, linear combination, linear functions, linear operators, linear dependence and independence, special matrices, linear vector space, orthonormal basis, eigenvalues and eigenvectors of a square matrix, diagonalisation of a square matrix, orthogonal transformations, rotations as orthogonal transformations, Euler angles, matrices and linear operators, projection operators.

(from Boas Ch. 3, and parts of Anderson Ch. 3)

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2. Orthogonal functions**

Even and odd functions, complete set of functions, orthogonal and orthonormal functions, expansion in terms of orthonornal functions, the 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. (from Anderson Ch 2)

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3. Complex numbers**

Complex numbers as an ordered pair of numbers, as a point in a 2-dimensional space. Complex plane and Argond 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 (from Boas Ch 2)

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4. 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 (from Boas Ch 14)

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5. Integral transforms**

Fourier transforms, cosine-sine transforms, Fourier transform of derivatives, convolution theorem, transfer functions, discrete Fourier

transform, integral transforms - general, Laplace transform, convolution, Hilbert transform and Kramers-Kronig relations

(from Boas Ch 7, 14)

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6. Vector analysis**

Scalar and vector multiplications, triplet 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 (from Boas Ch 6)

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7. Tensor analysis**

Cartesian tensors, tensor notation and operations, moment of inertia tensor, Kronecker delta and Levi-Civita symbol, pseudovectors and pseudotensors, curvilinear coordinates, vector operators in orthogonal curvilinear coordinates (from Boas Ch 10)

8. 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 transforms, power series (Frobenius) method of solving differential equations, Integral transform solutions, Numerical methods of solving ordinary differential equations: Euler`s method, Runge-Kutta, Predictor-corrector methods (from Boas Ch 8, 12 & 13)

Books: M. L. Boas, *Mathematical methods in
physical sciences*, 3rd editions, Wiley-India (2006)

J. M. Anderson, *Mathematics for
quantum chemistry*, Dover Publications (2005)

**Instructors: Ranjan Das
Ravindra Venkatramani**

Room no.: B-123 NMR Building

Tel. no.: 2258 2792

E-mail: ranjan@tifr.res.in ravi.venkatramani@tifr.res.in

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**Venue: Lecture room AG-80**

**Days: Mondays, Wednesdays and Fridays**

**Time: 9:45 hr to 11:00 hr **

*The first lecture starts on August 17, 2012.*

Graduate Course on

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**Quantum Chemistry**

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· Review of classical mechanics, origins of quantum theory: Black-body radiation, photoelectric effect, Compton effect, Frank-Hertz experiment, wave-particle duality, line spectra of atoms

· Time-dependent and time-independent Schrödinger equations, wavefunctions, observables, operators, expectation values, properties of operators, the uncertainty principle

· Quantum mechanics of free particle, barrier penetration, quantum mechanical tunnelling, potential well, the particle in a box, the harmonic oscillator, hydrogen atom, selection rules

· Angular momentum, spherical harmonics

· Approximate methods: Variation theory, time-independent perturbation theory for non-degenerate and degenerate states. Time-dependent perturbation theory, transition probability, multiphoton transitions

· Many electron systems. The anti-symmetry principle, spin orbitals, Slater determinants. Construction of spin-correct wavefunctions. Addition of angular momenta and atomic term symbols. Hartree-Fock self-consistent field method for atoms

· Molecular structure, Born-Oppenheimer approximation, linear molecules, non-crossing rules

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Semi-empirical molecular orbital
methods, Hückel MO theory, *ab-initio* quantum chemical calculations

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**Instructor: P. K. Madhu**

Room no. NMR Facility

Tel. no. 2874

E-mail: madhu@tifr.res.in

**Venue: NMR conference room**

**Days: Tuesdays and Thursdays **

**Time: 9:30 hr to 11:30 hr**

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*The first lecture starts on August 7, 2012*

Graduate Course on

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· Cell structure and functions of cell organelles

· Bio-organic Chemistry: Carbohydrates, proteins, nucleic acids

· Thermodynamics of macromolecular interactions

· Protein folding

· Enzyme function and kinetics

· Intermediary metabolism

· Bioenergetics and photosynthesis

· Organization of metabolic pathways, channeling of intermediates, processivity

· Metabolism in intact cells and tissues

· Metabolomics – Multiparametric metabolic response of living systems to environmental and physiological stimuli.

· Chemical tools for biology: Molecular imaging and chemo-selective reactions for chemical biology

· Emerging areas in biophysics: Ultrafast reactions in biology and their measurements

__Text Books__

1. ‘Lehninger Principles of Biochemistry’, David L. Nelson and Michael M. Cox, W.H. Freeman and Co., 2009

2. ‘Molecular and Cellular Biophysics’, Meyer Jackson, Cambridge University Press, 2006

**Instructors: Haripal Sonawat
Ankona Datta**

Room no. D-220 B-127

Tel. no. 2394 2078

E-mail: hms@tifr.res.in ankona@tifr.res.in

**Venue: Lecture room AG-80**

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Days: Mondays, Wednesdays & Fridays**

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Time: 11:30 hr to 12:30 hr**

*The first lecture starts on August 27, 2012*