Introduction to Molecular Spectroscopy

Transcription

1 Chem 5.6, Fall 004 Leture #36 Page Introduton to Moleular Spetrosopy QM s essental for understandng moleular spetra and spetrosopy. In ths leture we delneate some features of NMR as an ntrodutory example to spetrosopy sne the Hamltonans and wavefuntons are partularly smple. Thus, NMR permts us to demonstrate several fundamental prnples rasng and lowerng operators, transton probabltes, et -- wth a mnmal amount of algebra. These are onepts you have lkely enountered elsewhere (5., 5.3, 5.33, et.) or other ourses. The prnples and prodeures are applable to other areas of spetrosopy -- eletron, vbratonal, rotatonal, et. but the algebra s more extensve nergy Levels of Nulear Spn Systems Zeeman Hamltonan For an unoupled nulear spn n a magnet feld the energy s µ Ths leads to a Zeeman Hamltonan, H Z, of the form H Z I Where magnet feld (Gauss) µ I Plank s onstant µ s the magnet moment I spn operator gyromagnet rato (rad/s Gauss) Nule ome dressed wth eletron and that rulate about the feld dreton, 0, generatng a feld,, that s opposte n dreton to the appled feld as shown on the left. Thus the Zeeman energy levels of the bare nuleus are shfted as shown n the fgure on the rght by the sheldng effet of the eletrons.

2 Chem 5.6, Fall 004 Leture #36 Page More spefally, the eletron rulaton produes a feld, opposed to 0 and of magntude equal to 0. Thus, the effetve feld,, at the nuleus s ( ) Note that s dfferent for eah hemally dfferent nulear spn - - ths s the famous hemal shft - - and permts resoluton of lnes n NMR spetra orrespondng to hemally dfferent stes. The Hamltonan s modfed aordngly H Z ( ) Z Wave funtons that desrbe spn are (MQuarre Seton 8.4 p300) d * d * orthonormal * * 0 Note that we have ntrodued the braket notaton to smplfy the formulas. In partular, the bra and the ket are used as shorthand notaton for the ntegral * d qenvalues of and ˆ * ˆ + + ˆ ˆ The energy levels assoated wth the two states are therefore. ( ) * ˆ + ( ) ( ) ( ) * ˆ ( ) ( ) Note the sgn of the Hamltonan s hosen so that the state ( spn parallel to 0 ) s lower n energy than the state ( spn antparallel to 0 ). In addton, 0 0 s the nulear Larmor frequeny (rad/se). The energy dfferene between the states s ( )

3 Chem 5.6, Fall 004 Leture #36 Page 3 Two Spns the Spn System For two unoupled spn s, denoted and, wth dfferent hemal shfts ( ) we obtan four energy levels as llustrated n the dagram below. Formulas for the energy levels are also provded below and are alulated n a straghtforward manner. ( + ( + [( ) + ( )] [( ) ( )] ( + ( ( ) ( + ( ( ) ( + ( Rasng and Lowerng Operators Spetrosop Transtons nergy levels are deteted va spetrosop transtons. To ndue these transtons we rradate wth a perturbng feld wth that has energy To treat these transtons quantum mehanally, we need some addtonal relatons ˆ ˆ ˆ Y ˆ Y It takes some effort to justfy these relatons so we smply state the results. refly they an be derved from the Paul spn matres I Z 0 I I Y 0 0 and note that they satsfy the ommutaton relatons

4 Chem 5.6, Fall 004 Leture #36 Page 4 [, Y ] Z We an now defne two new operators + + Y - What s the effet of I + and I on the wave funtons? Y So ( + Y ) + Y + ( Y ) Y + and These operators transform the wave funton and onnet dfferent energy levels. For example * * + * * In an NMR experment the extng rf (rado frequeny) feld s appled perpendular to laboratory feld and s represented by the Hamltonan H rf I ( + + ) x Reall the laboratory feld s taken as along Z as s onventonal. For the spn system above we obtan for the transton probablty W W x x x x x x x x + x x x x - Here I + and I operate on both states and. Note that the allowed transtons orrespond to M ± transtons. Ths SLCTION RUL arses from the propertes of the rasng and lowerng operators! The ntensty of the spetrosop transton, that s the ntensty of the lne n the spetrum, s proportonal to Intensty I +

5 Chem 5.6, Fall 004 Leture #36 Page 5 xpetaton value of the Magnet Moment Larmor Preesson Reall the tme dependent Shrödnger quaton (MQuarre Seton 4.3, p. 0) d ψ Ĥ ψ Usng the general ntal state ψ α α + β β We obtan d ψ Multple from the left by α d α α + d β β α Ĥ α β Ĥ β Sne α β 0 d α α α + d β α β α α Ĥ α β α Ĥ β d α α H αα β H ββ Repeat the proedure wth a left multply by β to fnd an expresson for d β β d β α H βα β H ββ Ths yelds the expresson for the tme dependene of α and β α β H αα H αβ H βα H ββ α β or n another form Usng ψ Hψ H ω ο I Z and I Z 0 We obtan Substtutng n the above Integraton yelds H ω ο I Z α ω ο α ω ο ο ο ω ο β ω ο β 0

6 Chem 5.6, Fall 004 Leture #36 Page 6 Suppose (t ) ()e t () () I + I x ( + * * * ) Re( ) * * * I y ( ) Im ( ) * Re (0)e t I (0) x os t * (0)e t I Im (0) y sn t (t) ()e To obtan the expresson above let a+b and g+f. Form the omplex onjugates and perform the ndated algebra to obtan the results. Substtutng for (t) and (t) we obtan Thus I * I (0) (0) (0) * (0) 0 and I osllate and I does not hange wth tme the spn exeutes Larmor x y preesson about the Z-axs, the dreton of 0! and t