1. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 1 Ch121a Atomic Level Simulations of Materials and Molecules William A. Goddard III, wag@wag.caltech.edu Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics, California Institute of Technology BI 115 Hours: 2:30-3:30 Monday and Wednesday Lecture or Lab: Friday 2-3pm (+3-4pm) Teaching Assistants Wei-Guang Liu, Fan Lu, Jose Mendoza, Andrea Kirkpatrick Lecture 7, April 15, 2011 Molecular Dynamics – 3: vibrations

2. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 2 Outline of today’s lecture Vibration of molecules –Classical and quantum harmonic oscillators –Internal vibrations and normal modes –Rotations and selection rules Experimentally probing the vibrations –Dipoles and polarizabilities –IR and Raman spectra –Selection rules Thermodynamics of molecules –Definition of functions –Relationship to normal modes –Deviations from ideal classical behavior

3. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 3 Simple vibrations Starting with an atom inside a molecule at equilibrium, we can expand its potential energy as a power series. The second order term gives the local spring constant We conceptualize molecular vibrations as coupled quantum mechanical harmonic oscillators (which have constant differences between energy levels) Including Anharmonicity in the interactions, the energy levels become closer with higher energy Some (but not all) of the vibrational modes of molecules interact with or emit photons This provides a spectroscopic fingerprint to characterize the molecule

4. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 4 Vibration in one dimension – Harmonic Oscillator Consider a one dimensional spring with equilibrium length x e which is fixed at one end with a mass M at the other. If we extend the spring to some new distance x and let go, it will oscillate with some frequency,  which is related to the M and spring constant k. To determine the relation we solve Newton’s equation  M (d 2 x/dt 2 ) = F = -k (x-x e ) Assume x-x 0 =  = A cos(  t) then –M    cos(  t) = -k A cos(  t) Hence –M   = -k or  Sqrt(k/M). Stiffer force constant k  higher  and higher M  lower  No friction E= ½ k  2

5. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 5 Reduced Mass M1M1 M2M2 Put M 1 at R 1 and M 2 at R 2 CM = Center of mass Fix R cm = (M 1 R 1 + M 2 R 2 )/(M 1 + M 2 ) = 0 Relative coordinate R=(R 2 -R 1 ) Then P cm = (M 1 + M 2 )*V cm = 0 And P 2 = - P 1 Thus KE = ½ P 1 2 /M 1 + ½ P 2 2 /M 2 = ½ P 1 2 /  Where 1/  = (1/M 1 + 1/M 2 ) or  = M 1 M 2 /(M 1 + M 2 ) Is the reduced mass. Thus we can treat the diatomic molecule as a simple mass on a spring but with a reduced mass, 

6. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 6 (n + ½) 2 For molecules the energy is harmonic near equilibrium but for large distortions the bond can break. The simplest case is the Morse Potential : Exact solution Real potentials are more complex; in general: (n + ½) 2 (n + ½) 3 Successive vibrational levels are closer by (Philip Morse a professor at MIT, do not manufacture cigarettes)

7. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 7 Now on to multiple atoms N atoms => 3N degrees of freedom However, 3 degrees for translation, get = 0 3 degrees for rotation is non-linear molecule, get = 0 2 degrees if linear (but really a restriction only for diatomic The remaining (3N-6) are vibrational modes (just 1 for diatomic) Derive a basis set for describing the vibrational modes by solving the eigensystem of the Hessian matrix Eigenvalue problem or

8. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 8 Vibration for a molecule with N particles F k = -(∂E( R new )/∂ R k ) = -(∂E/∂ R k ) 0 -  m (∂ 2 E/∂ R k ∂ R m ) (  R ) m Where we have neglected terms of order  2. Writing the Hessian as H km = (∂ 2 E/∂ R k ∂ R m ) with (∂E/∂ R k ) 0 = 0, we get F k = -  m H km (  R ) m = M k (∂ 2 R k /∂t 2 ) To find the normal modes we write (  R ) m = A m cos  t leading to M k (∂ 2 R k /∂t 2 ) = M k  2 (A k cos  t) =  m H km (A m cos  t) Here the coefficient of cos  t must be {M k  2 A k -  m H km A m }=0 There are 3N degrees of freedom (dof) which we collect together into the 3N vector, R k where k=1,2..3N The interactions then lead to 3N net forces, F k = -(∂E( R new )/∂ R k ) all of which are zero at equilibrium, R 0 Now consider that every particle is moved a small amount leading to a 3N distortion vector, (  R ) m = R new – R 0 Expanding the force in a Taylor’s series leads to

9. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 9 Solving for the Vibrational modes The normal modes satisfy {M k  2 A k -  m H km A m }=0 To solve this we mass weight the coordinates as B k = sqrt(M k  A k leading to Sqrt(M k )  2 B k -  m H km [1/sqrt(M m )]B m }=0 leading to  m G km B m =  2 B k where G km = H km /sqrt(M k M m ) G is referred to as the reduced Hessian For M degrees of freedom this has M eigenstates  m G km B mp =  kp B k (  2 ) p where the eigenvalues are the squares of the vibrational energies. If the Hessian includes the 6 translation and rotation modes then there will be 6 zero frequency modes

10. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 10 Saddle points If the point of interest were a saddle point rather than a minimum, G would have one negative eigenvalue. This leads to an imaginary frequency

11. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 11 For practical simulations We can obtain reasonably accurate vibrational modes from just the classical harmonic oscillators N atoms => 3N degrees of freedom However, there are 3 degrees for translation, n = 0 3 degrees for rotation for non-linear molecules, n = 0 2 degrees if linear The rest are vibrational modes

12. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 12 Normal Modes of Vibration H 2 O 1595 cm -1 3657 cm -1 3756 cm -1 H2OH2OD2OD2O 1178 cm -1 2671 cm -1 2788 cm -1 Sym. stretch Antisym. stretch Bend Isotope effect:  ~ sqrt(k/M): Simple D / H ~ 1/sqrt(2) = 0.707: Ratio: 0.730 Ratio: 0.735 Ratio: 0.742 More accurately, reduced masses   H = M H M O /(M H +M O )   D = M D M O /(M D +M O ) Ratio = sqrt[M D (M H +M O )/M H (M D +M O )] ~ sqrt(2*17/1*18) = 0.728 Most accurately M H =1.007825 M D =2.0141 M O =15.99492 Ratio = 0.728

13. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 13 EM energy absorbed by interatomic bonds in organic compounds frequencies between 4000 and 400 cm -1 (wavenumbers) Useful for resolving molecular vibrations http://webbook.nist.gov/chemistry/ http://wwwchem.csustan.edu/Tutorials/INFRARED.HTM The Infrared (IR) Spectrum Characteristic vibrational modes

14. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 14 Normal Modes of Vibration CH 4 2917 cm -1 3019 cm -1 1534 cm -1 CH 4 CD 4 1178 cm -1 2259 cm -1 1092 cm -1 Sym. stretch 1 Anti. stretch 3 Sym. bend 2 3 A1A1 T2T2 E T2T2 1306 cm -1 996 cm -1

15. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 15 Fitting force fields to Vibrational frequencies and force constants Hessian-Biased Force Fields from Combining Theory and Experiment; S. Dasgupta and W. A. Goddard III; J. Chem. Phys. 90, 7207 (1989) H 2 CO MC: Morse bond stretch and cosine angle bend MCX: include 1 center cross terms

16. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 16 The Schrödinger equation H  =   for harmonic oscillator The QM Harmonic Oscillator energy wavefunctions reference http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html#c1

17. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 17 Raman and IR spectroscopy IR –Vibrations at same frequency as radiation –To be observable, there must be a finite dipole derivative –Thus homonuclear diatomic molecule (O2, N2,etc.) does not lead to IR absorption or emission. Raman spectroscopy is complimentary to IR spectroscopy. –radiation at some frequency, n, is scattered by the molecule to frequency, n ’, shifted observed frequency shifts are related to vibrational modes in the molecule IR and Raman have symmetry based selection rules that specify active or inactive modes

18. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 18 IR and Raman selection rules for vibrations The electrical dipole moment is responsible for IR The polarizability is responsible for Raman  For both, we consider transition matrix elements of the form The intensity is proportional to d  /dR averaged over the vibrational state where  is the external electric field at frequency

19. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 19 IR selection rules, continued For IR, we expand dipole moment  We see that the transition elements are  The dipole changes during the vibration  Can show that n can only change 1 level at a time

20. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 20 Raman selection rules For Raman, we expand polarizability substitute the dipole expression for the induced dipole Same rules except now it’s the polarizability that has to change For both Raman and IR, our expansion of the dipole and alpha shows higher order effects possible =

21. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 21 center of mass translation  x  =  x  y  =0  z  =0  x  =0  y  =  y  z  =0  x  =0  y  =0  z  =  z center of mass rotation (nonlinear molecules)  x  =0  y  =-c   x  z  =b   x  x  = c   y  y  =0  z  =-a   y  x  = -b   z  y  =a   x  z  =0 linear molecules have only 2 rotational degrees of freedom The translational and rotational degrees of freedom can be removed beforehand by using internal coordinates or by transforming to a new coordinate system in which these 6 modes are separated out Both  and V are constant  =0 Translation and Rotation Modes Both K and V are constant  =0

22. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 22 Classical Rotations The moment of inertia about an axis q is defined as x k (q) is the perpendicular distance to the axis q Can also define a moment of inertia tensor where (just replace the mass density with point masses and the integral with a summation. Diagonalization of this matrix gives the principle moments of inertia! the rotational energy has the form

23. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 23 Quantum Rotations The rotational Hamiltonian has no associated potential energy For symmetric rotors, two of the moments of inertia are equivalent, combine: Eigenfunctions are spherical harmonic functions Y J,K or Z lm with eigenvalues

24. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 24 Transition rules for rotations For rotations –Wavefunctions are spherical harmonics –Project the dipole and polarizability due to rotation It can be shown that for IR –Delta J changes by +/- 1 –Delta M J changes by 0 or +/-1 –Delta K does not change For Raman –Delta J could be 1 or 2 –Delta K = 0 –But for K=0, delta J cannot be +/- 1

25. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 25 Raman scattering Phonons are the normal modes of lattice vibrations (thermal + zero point energy) When a photon absorbs/emits a single phonon, momentum and energy conservation  the photon gains/loses the energy and the crystal momentum of the phonon. –q ~ q` => K = 0 –The process is called anti-Stokes for absorption and Stokes for emission. –Alternatively, one could look at the process as a Doppler shift in the incident photon caused by a first order Bragg reflection off the phonon with group velocity v = ( ω / k)*k

26. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 26 Raman selection rules For Raman, we expand polarizability substitute the dipole expression for the induced dipole Same rules except now it’s the polarizability that has to change For both Raman and IR, our expansion of the dipole and alpha shows higher order effects possible =

27. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 27 Another simple way of looking at Raman Take our earlier expression for the time dependent dipole and expose it to an ideal monochromatic light (electric field) We get the Stokes lines when we add the frequency and the anti- Stokes when we substract The peak of the incident light is called the Rayleigh line

28. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 28 The external EM field is monochromatic Dipole moment of the system Interaction between the field and the molecules Probability for a transition from the state i to the state f (the Golden Rule) Rate of energy loss from the radiation to the system The flux of the incident radiation c: speed of light n: index of refraction of the medium The Sorption lineshape - 1

29. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 29 Absorption cross section  (  ) Define absorption linshape I(  ) as It is more convenient to express I(  ) in the time domain I(  ) is just the Fourier transform of the autocorrelation function of the dipole moment ensemble average Beer-Lambert law Log(P/P 0 )=  bc The Sorption lineshape - II

30. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 30 Non idealities and surprising behavior Anharmonicity – bonds do eventually dissociate Coriolis forces –Interaction between vibration and rotation Inversion doubling Identical atoms on rotation – need to obey the Pauli Principle –Total wavefunction symmetric for Boson and antisymmetric for Fermion

31. © copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L07 31 Figure taken from Streitwiser & Heathcock, Introduction toOrganic Chemistry, Chapter 14, 1976 Electromagnetic Spectrum How does a Molecule response to an oscillating external electric field (of frequency  )? Absorption of radiation via exciting to a higher energy state ħ  ~ (E f - E i )