chemistry cousre lecture slides from iitb

these slides include all what you need to know about schringer equation

1. CH-111 2023:Physical Chemistry Electronic Structure of Atoms Chemical Bonding Instructor for D3 Batch Dr. Achintya Kumar Dutta Room 380, 2 nd floor, Chemistry Phone: x-7156 Whatsapp No. 9064136238 Email: [email protected] ;

2. Why should you study Physical Chemistry? Use the laws of physics and apply to chemistry Deeper understanding of physical processes/mechanisms and get insights on how things work the way they do! Bridge between novel molecules (materials) prepared by synthetic chemists and their potential applications, (often dealt with by engineers) Biotechnology and healthcare Energy Science – “Solar Energy” conversion Atmospheric Science - “Save the World” Nanoelectronics /Nanotechnology: Molecular Electronics

3. Physics Dynamics of electrons and protons governed by laws of quantum physics O and I Chem Collective behavior of atoms and molecules P Chem Apply QM principles to understand the behavior of atoms and molecules Deeper/better understanding of physical processes Relevance of CH111? You probably “know” what the end-results is… But not necessarily understand the reason “why?” The focus of CH111:P is to understand: Why and How…

4. Schrodinger Equation Fundamentals of Quantum Physics Particle In a Box Tunneling Processes Harmonic oscillator Quantum numbers Yields three Multi-atomic Bonding, Molecular Structure Intermolecular Forces And Interactions Multi-electron Atoms (Periodic Table) Atomic/Molecular Spectroscopy PH-107 Hydrogen Atom CH- 107 Barrier Penetration Band theory Chemical Reactions Molecular Dynamics Biology, (Nano) Materials Science Condensed Matter Physics The bigger picture

5. Outline of this course (18x1hr) Origin of Quantization ~3 lectures Postulates of Quantum Mechanics Particle in a Potential Well Electronic Structure in Atoms ~6 lectures Hydrogen atom Multi-electronic atoms Chemical Bonding ~6 lectures Molecular Orbital Theory Energetics/Electronic structure of homonuclear diatomics Molecular-Electronic Structure ~3 lectures Heteronuclear diatomics and polyatomics , hybridization

6. Textbooks, Materials, and Resources Textbook: Physical Chemistry – I.N. Levine Textbook: Physical Chemistry – P.W. Atkins Reference Text: Physical Chemistry: A Molecular Approach - McQuarrie and Simon CH111 Course Material/lecture slides will be uploaded in IITB Moodle every week All important announcements in moodle and WhatsApp group for D3 with all students & AKD Tutorial problems: Solve ahead of time (MQ!)

7. Grading stuff Weekly mini-quiz (2 marks) – best 4 out of 5 (2x4=8) During tutorials (similar to tut Qs) No make-up or re-mini-quiz Quiz – 12 marks (~50 mins) Mid-Sem - 30 marks (~2 hrs ) Passing marks is typically 30% (<15 out of 50) A+ grade is typically for >96% (>48 out of 50) ATTENDANCE??? ASSISTANCE? UG TAs in hostel!

8. Tutorial Information about TAs: T1: Achintya T2: Sudipta Chakraborty T3: Ankit Kumar T4: Priya Bhandari They will be taking the Tutorial sessions Point of contact: For e.g. Language, any other issues etc.

9. What have you already knew in a nutshell?

10. 11 Davisson and Germer George Thompson

11. Light can have “particle-like” and “wave-like” properties; Matter (electrons & atoms) have “particle-like” and “wave-like” properties (we do not know what they really are!!) Electrons (matter) or light (“ WAVICLE? ”) are both very weird, but their behaviors are identical!! When we refer to a (QM) particle, What we really mean is a “wave-particle” Wave-Particle Duality

12. Exact position and momentum iss “ Indeterministic ” or uncertain for submicroscopic fast moving particles

13. Richard Feynman…

14. Nature of submicroscopic atoms is not understandable and beyond our perception We deduce about their nature based on indirect evidence, and then proceed to describe it… …in terms of quantities which we know from our experience of the macroscopic world ( using analogy and contrast ) However, analogies may not always provide the correct picture (if there is one!)

15. Richard Feynman on Quantum Behavior They behave in their own inimitable yet fascinating manner, which could be called the “ Quantum Mechanical way ” Best to be described mathematically! Inherently, “nature” at the atomic scale is not deterministic, and their exact behaviors can not be predicted – Probabilistic!

16. Need a new theory to understand (dynamics of) electrons and atoms Probabilistic , not deterministic (non- newtonian ) Wavelike equation for describing sub-atomic systems Schrodinger 1925!!! 17 “Where did we get that (equation) from? Nowhere . It is not possible to derive it from anything you know. It came out of the mind of Schrödinger .” - Richard Feynman

17. Why believe in Schrodinger’s Eq.? 18 Why did you believe in N ewton’s LAWs of motion? Did you ever question their validity? Laws  Based on Postulates or Axioms Not been proven wrong experimentally

18. Erase your memory 19

19. Postulates of Quantum Mechanics The state of a system is completely specified by a wave- function Ψ ( r,t ). Square of “ wavefunction ”  probability density Every observable in the classical world is related to a linear Hermitian operator in quantum mechanics Every measurement is associated with operator ( A) ,  Values that will ever be observed are eigenvalues of A The average value of the observable corresponding to A is Any QM system evolves as (TDSE):

20. What can you get from solving the Schrodinger Equation? Y is a mathematical function; real or complex What is the meaning of ψ ( x,t )? A “wave” associated with matter?

21. I. Born Interpretation of Wavefunction The state of a QM system is completely specified by Y ( x,y,z,t ) and all possible information can be derived from Y Y is a mathematical function; Probability amplitude Y *. Y dv is the probability that the particle lies in vol. dv, so Y * Y is the probability density at that point 1882-1970 Normalization of Wave function: Probability of finding a particle somewhere in the entire space has to be unity. Dirac’s Bra- Ket notation

22. Restrictions on wavefunction  Well-behaved wavefunctions Physically acceptable wavefunctions 1. Y must be normalizable Y is finite and Y 0 at boundaries/   2. Y must be single-valued because y * y can not have two values! 3. Y is solution of 2 º DE ( Schrod . Eqn ): Quadratically integrable y must be a continuous function in space slope of y i.e. d y /d(space) must also be continuous Single-point discontinuity sometimes acceptable “removable”

23. Postulates of Quantum Mechanics The state of a system is completely specified by a wave- function Ψ ( r,t ). Square of “ wavefunction ”  probability density Every observable in the classical world is related to a linear Hermitian operator in quantum mechanics Every measurement is associated with operator ( A) ,  Values that will ever be observed are eigenvalues of A The average value of the observable corresponding to A is Any QM system evolves as (TDSE):

24. Mathematical Tools-1: Operators I. Operators : Does “ something ” to whatever follows it Takes an Action : Simple operations - addition, multiplication Can be real or complex; Can be represented as matrices. Symmetry operators – rotation, reflection, inversion! Operation  Changes/Perturbs function/system in some way!  act of measurement or perform an experiment 25

25. II. Operator Formalism To every observable in classical mechanics, there corresponds a l inear Hermitian operator (real or complex) in quantum mechanics

26. TDSE: 2 nd order partial differential equation

27. Postulates of Quantum Mechanics The state of a system is completely specified by a wave- function Ψ ( r,t ). Square of “ wavefunction ”  probability density Every observable in the classical world is related to a linear Hermitian operator in quantum mechanics Every measurement is associated with operator ( A) ,  Values that will ever be observed are eigenvalues of A The average value of the observable corresponding to A is Any QM system evolves as (TDSE):

28. Math-2. Eigenfunction /Eigenvalue equations Operator operating on a function re-generates the same function (which is called eigenfunction ) multiplied by a number ( eigenvalue )

29. III. Real (observable) eigenvalues In measurement of a classical variable (operator  satisfying ) only real ( not imaginary ) eigenvalues (a n ) will ever be observed Only rea l eigenvalues will ever be observed , which will specify a number corresponding to a classical variable for an eigenfunction OPERATOR HAS TO BE HERIMITAN OERATOR  REAL EIGENVALUES Y n  eigenfunctions or eigenstates or states a n  Eigenvalues ( outcomes of measurements ) A measurement on system cannot yield an imaginary eigenvalue There are many Eigen-f(n)s (or eigenstates) for the same QM operator  like Sin and Cos are both eigenfunctions of op. D 2

30. Math-3. Probability Distributions: Expectation (Average/Mean value) and Most-Probable Value If P(x) is the probability distribution

31. Postulates of Quantum Mechanics The state of a system is completely specified by a wave- function Ψ ( r,t ). Square of “ wavefunction ”  probability density Every observable in the classical world is related to a linear Hermitian operator in quantum mechanics Every measurement is associated with operator ( A) ,  Values that will ever be observed are eigenvalues of A The average value of the observable corresponding to A is Any QM system evolves as (TDSE):

32. IV. Average or Expectation value So, prescription needed for average value of a classical observable <a> corresponds to the mean value of a classical physical quantity (observable), and A represents the corresponding QM operator

33. Before and after measurement Before measurement: Realist: Value = P (Einstein) Quantum theory is incomplete Orthodox: Entanglement ( Bohr, Copenhagen interpretation ) Measurement produces the value Agnostic: Don’t know, don’t care   Measurement: Property has a value of P

34. Before and after measurement Before measurement: Realist: Value = P (Einstein) Quantum theory is incomplete Orthodox: Entanglement ( Bohr, Copenhagen interpretation ) Measurement produces the value Agnostic: Don’t know, don’t care   Measurement: Property has a value of P Immediately after measurement: Same Value = P Wavefunction collapse

35. Postulates of Quantum Mechanics The state of a system is completely specified by a wave- function Ψ ( r,t ). Square of “ wavefunction ”  probability density Every observable in the classical world is related to a linear Hermitian operator in quantum mechanics Every measurement is associated with operator ( A) ,  Values that will ever be observed are eigenvalues of A The average value of the observable corresponding to A is Any QM system evolves as (TDSE):

36. Math-4: 2 nd order Partial Differential Equations To Solve any PDE  Separation of Variables (or else, can not solve) 2 nd order Normal Differential Equation (DE) With one variable (Simple Harmonic Motion) Partial Differential Equation (PDE) contains two or more variables Two normal DEs with single variable

37. Time-Dependent Schr ö dinger Equation Very often, V( x,t ) = V(x)  then special solutions to TDSE Product of space function and time function: separated out!

38. Separation of variables       Separation constant    

39. Separation of variables                 ,Hamiltonian operator  

40. Time-Independent Schr ö dinger Equation

41. Schrödinger being inducted as an angel…

42. Time-independent Schrodinger equation For a free particle V(x) =0 There are no external forces acting Free Particle

43. Let us assume Trial Solution Second-order linear differential equation Free Particle

44. Second-order linear differential equation Let us assume Trial Solution Free Particle

45. Free Particle

46. There are no restrictions on k E can have any value Energies of free particles are continuous Free Particle No Quantization All energies are allowed de Broglie wave

47. There are no restrictions on k E can have any value Energies of free particles are continuous Free Particle No Quantization All energies are allowed de Broglie wave Is this a good wavefunction?

48. For x < 0 and x > L  V = ∞ Particle in 1-D Box For 0 ≤ x ≤ L  V = 0 Trial Solution: Energy: Wavefunction should be continuous: Boundary condition Zero

49. For x < 0 and x > L  V = ∞ Particle in 1-D Box For 0 ≤ x ≤ L  V = 0 n = 1, 2, 3, 4, ….. n ≠ 0, as wavefunction cannot be zero everywhere Boundary condition Trial Solution: Energy:

50. Particle in 1-D Box: Normalization n = 1, 2, 3, 4, ….. n ≠ 0, as wavefunction cannot be zero everywhere      

51. Particle in 1-D Box: Wavefunctiona n = 1, 2, 3, 4, ….. n ≠ 0, as wavefunction cannot be zero everywhere   Orthogonality .       Is the first derivative continuous? Not at x = 0 and x = L

52. Particle in 1-D Box: Energy n = 1, 2, 3, 4, ….. n ≠ 0, as wavefunction cannot be zero everywhere   Energy is quantized! Boundary conditions are the origin of quantization Energy separation increases with increasing values of n Lowest possible energy is non-zero: Zero point energy Larger the box, smaller the energy of h ν  

53. Wavefunction: n=1,3.. Symmetric wrt inversion (even function) n=2,4.. Anti-Symmetric (odd function) Number of Nodes (zero crossings) = n-1 Particle in 1-D Box: Spectroscopy Transition is allowed when Transition Moment Integral       Non-zero integral: Symmetric integrand Antisymmetric If one wave function is symmetric, then the other should be antisymmetric Selection rule:   Odd to even, even to odd transitions are allowed

54. Particle in 1-D Box: Examples in Chemistry n = 1, 2, 3, 4, …..   Compare with the experimental value of 258 nm Particle in a box is a good first approximation Hexatriene: a linear molecule of length 7.3 Å It absorbs at 258 nm Use particle in a box model to explain the results.    Six  electrons fill lower three levels

55. Particle in 1-D Box: Examples in Chemistry n = 1, 2, 3, 4, …..   Increase in bridge length increase the emission wavelength. Predicts correct trend and gets the wavelength almost right. Particle in a box is a good first approximation Electronic spectra of conjugated molecules Β -carotene is orange because of 11 conjugated double bonds

56. Particle in 1-D Box: Examples in Chemistry n = 1, 2, 3, 4, …..   CB VB CB VB CB VB Quantum Dots – Quasi-particle ( exciton ) in a Box!

57. 58 Applications of Q-Dots/–Rods/-Wells range from biology to solar photovoltaics to LED based displays Quantum-Dot based Solar Cells Quantum-Dot based in-vivo imaging for tumor/cancer

58. Expectation value: Position

59. Expectation value: Position Probability in a thin strip for different n and x values

60. Particle in a 2-D box V=0 L x L y Square Box  L x = L y = L Separation of variables

61. Expectation value: Momentum   Eigenfunctions: Equal magnitude, opposite direction Equal probability for propagation in the two directions

62. Degeneracy is manifestation of symmetry Energy 63 Recap: Crystal Field splitting

63. Particle in a 2-D box: Wavefuntiona Number of nodes = n x +n y -2

64. Rectangular box V=0 L x L y (1, 2) and (2, 1) levels, for example, have same energy in square box, but not in rectangular box Symmetry and degeneracy go hand in hand

65. 3D box

66. Particle in a box: Take home messages Schrodinger equation is exactly solvable Boundary conditions: Quantization More nodes in wavefunction, higher is the associated energy Eigenfunction of linear momentum operator Simple model, finds application in Chemistry Increase in dimensionality: Separation of variable Symmetry and degeneracy go hand in hand Beyond 3D functions Testing ground for more sophisticated treatment What happens if the potential barrier is finite?