MIT OpenCourseWare http ocw mit edu 5 80 Small Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 5 80 Lecture 26 Fall 2008 Page 1 of 7 pages Lecture 26 Polyatomic Vibrations II s Vectors G matrix and Eckart Condition Last time q S Q Q L 1 D M1 2 q S 1 2T S G S 2V S F S G 1 D 1 D 1 because q D 1 S and D q S 2 V Fij Si S j 0 L S S T S V S d L L 1 secular equation 0 F G 0 dT S j S j 1 2 assuming that S t A cos t all j 1 2 3N 6 j j i e that all internal coordinates oscillate at same frequency and relative phase but with 1 2 different amplitudes 2 3N 6 possible different values of obtained from 3N 6 3N 6 secular equation TODAY Finish discussion of secular equation forms of various transformations descriptions of each normal coordinate s Vectors definition and properties imposition of translational and rotational constraints derivation of G from s Vectors NEXT TIME SOME EXAMPLES OF s VECTOR CALCULATIONS Last time we derived 0 F G 1 left multiply by G 0 GF 1 must diagonalize GF to get 3N 6 eigenvalues k Similarity not unitary transformation 5 80 Lecture 26 Fall 2008 Page 2 of 7 pages 1 0 0 1 L GFL 0 0 0 0 3N 6 Although G and F are both real and symmetric GF is not symmetric so the diagonalizing transformation is not unitary L 1 L For example the product of two real and symmetric matrices a b c d ac bd ad bc b a d c bc ad bd ac is a matrix that is not symmetric What do we already know about L from prior requirements that T and V must be put into separable forms Q Q 2k Want 2T Q L 1 S where Q k We had previously 2T q q S D 1 D 1 S S G 1 S G L G 1L Q Q So L G 1L 1 is required thus L 1 L G 1 is needed to keep T in separable form Want 2V Q Q k Q 2k k previously 2V S F S Q L FL Q So L FL is required to keep V in separable form L is not unitary we already know that L 1 L G 1 L so the s are not eigenvalues of F L FL is not a similarity transformation putting it all together the secular equation requires L 1GFL The i are eigenvalues of GF not of F multiply on left by L 1 L G 1 L G 1GFL L FL is self consistent 1 D 1 D 1 5 80 Lecture 26 Fall 2008 Now we are ready to ask F matrix Page 3 of 7 pages where do we get G can we determine F from spectrum how do we define 3N 6 independent internal coordinates subject to the two constraints of NO center of mass translation NO Rotation about center of mass 3N 6 3N 6 symmetric 3N 6 3N 7 linearly independent off diagonal elements these are 2 symmetric 3N 6 diagonal elements 3N 6 3N 5 2 N 3 4 5 6 independent elements elements 6 21 45 78 Too many even for isotopes modes 3 6 9 12 Must use tricks group theory insight set non adjacent F s 0 ab initio calculations borrowing from similar groups in other molecules and isotopic substitution to determine a complete set of Fij s even at only the harmonic quadratic level Imagine how many cubic and quartic force constants might be needed Internal Coordinates Stretch Bend rAB ABC A C B Torsion dihedral angle A A D D B C B C angle between ABC and BCD planes 5 80 Lecture 26 Fall 2008 Page 4 of 7 pages how to select 3N 6 independent internal coordinates B e g can t have rAB ABC rBC BCA rCA CAB C A how to limit to only 3 S s group theory projection operators will be helpful Actually no problem if we define F and G matrices that have extra rows and columns Get normal modes with zero frequency How to impose constraints algebraic and trigonometric nightmare no rotation about center of mass turns out that all cases have been worked out and tabulated imposing these constraints is an essential part of the process of generating the G matrix no center of mass translation G matrix elements are tabulated in Wilson Decius and Cross pages 303 306 in concise but obscure notation and diagrams Derived by s vector method St atoms G tt st 1 st st m specifies an arbitrary translation of atom not mass weighted to be derived later in this lecture 5 80 Lecture 26 Fall 2008 Page 5 of 7 pages s vectors recall St Bti i s B i power series expansion of S for small displacements from equilibrium retain only leading term St i St i i 0 plus neglected higher terms Bti now rewrite St as sum over atoms rather than over individual i N St 1 St St St x y z z 0 y 0 x 0 but we should recognize that this is expressed compactly in terms of the gradient of S N St 1 St Arbitrary displacement of atom St s t St St S S x t y t z x y z x x y y z z Note that choice of parallel to St gives largest change in St Thus s t is vector pointing in the direction in which a displacement of atom gives the LARGEST increase in St s t s t s vectors are derivable by vector analysis for all conceivable topological situations and are These The actual increase in St that results from UNIT DISPLACEMENT of atom in direction that increases St most rapidly tabulated by WILSON DECIUS AND CROSS pages 55 61 This will tell us how to build in translation and rotation constraints St s t 5 80 Lecture 26 1 Fall 2008 Page 6 of 7 pages Rigid translation of molecule prohibited by ensuring that rigid translation all 1 2 N has zero projection on each St want 0 St s t s t satisfied if s t 0 3 constraints one each for x y and z 2 Rotation prohibited A much more difficult problem How do we define a body fixed coordinate system if the atoms are not a rigid framework This question is at the heart of the distinction between vibration and rotation We must specify how the body fixed coordinate system is defined at all times based on the observable instantaneous positions of all atoms in a laboratory fixed coordinate system with its origin at the center of mass We have several choices of how to do this A key criterion is avoiding large high frequency angular accelerations in the …