PHYS METH IN INORG CHM
PHYS METH IN INORG CHM CHEM 634
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CHEMISTRY 673 IIIDEI Symmetry and Group Theory in Chemistry Tim Hughbanks IIIDEHH 0 BS in Chemistry U of Washington 1977 0 PhD Cornell 1983 0 Faculty member at TAMU since 1987 0 Of ce Chemistry Building Room 330 0 Of ce phone 8450215 0 Of ce Hrs Tues 200 400 PM Other times are OK too 0 email trhmailchemtamuedu CHEMISTRY 673 IIIDEHH 9 This course is for 3 credits 9 Lecture 2 X 75 minweek TTh 1110 1225 Room 2122 9 Web site httpwwwchemtamuedurgrouD hughbanks CHEMISTRY 673 IIIEHH 9 Grades will be based on the homework roughly 50 midterm and final exams 9 Class web site httpwwwchemtamuedu rgrouphughbankscourses673 chem673 html Required Books etc IIIDEHH 0 Chemical Applications of Group Theory by Cotton Wiley 1990 7 text is well written right level some students dislike the style 7 we follow the text most closely in the first half of the course and deviate from it considerably in the 2nd ha 0 Symmetry and Spectroscopy An Introduction by Harris amp Bertolucci Dover 7 text is informally written supplies physical background missing in Cotton goes further than I will but used for Chem 634 anyway Dover edition is cheap 0 some handouts and reference materials will be necessary especially for solids and ligand eld theory Good Inexpensive Paperbacks IIIIEHI 390 Molecular Symmetry and Group Theory by Robert Carter 7 An alternative textbook level is only slightly less demanding than Cotton 390 Group Theory and Quantum Mechanics by Tinkham 7 More difficult than Cotton but probably the most accessible of books written for physicists 7 Has a chapter on solids band theory k space Prerequisites IIll mllll 9 Undergraduate chemistry courses especially inorganic and physical chemistry 9 Usual math courses for scientists especially linear algebra 9 If you have not had linear algebra then familiarity with vectors and matrices acquired elsewhere may suffice don t wait to review these topics do so this week Minimum background Appendix in Cotton s text Reading IIIEHH 9 Please try to keep ahead in reading 9 This will allow me to avoid much mathematical detail in class This is desirable because theorems and proofs become tedious and sometimes get screwed up even when obvious 9 Start with the first 3 chapters now 9 Download and read the material on Determinants and Matrices from the web What is Group Theory IIIIDEHH 9 A fairly recent branch of mathematics Early principles were developed by Evariste Galois killed in a duel in 1832 at age 21 and Niels Abel died in 1829 at age 26 of TB 9 First formal definition of a group was given by Cayley in 1854 Fedorov pioneered the application of group theory to crystallography 9 Group Theory is the closest many chemists get to truly modern mathematics Properties of Groups IIllm 9 Closure product of any two group elements operations is a group element operation including squares 9 One element the identity commutes with all others 9 Associative property holds commutative property does not necessarily hold 9 Every element operation has an inverse which is also a group element operation Simple Examples IIIMHH 390 The integers under the operation of addition 390 The integers under the operation of multiplication 390 Relevant Example A simple symmetry group CZV What are the elements operations how do we define a product Another Relevant Example Rotation Matrices IIIIEHI gt Claim The set of all 2gtlt2 matrices of the form cos 0 sin 0 sin0 cosO forms an continuous infiniteorder group Where the product is assumed to be defined by the usual definition of matrix multiplication gt Proof Symmetry Elements vs Operations IIIDEHH 390 Mathematically the members of a group are called elements 390 In symmetry groups these elements are called operations the term element is reserved for something else 390 The term symmetry element refers to a geometrical entity a point a line or axis or a plane about which the operation is defined The Symmetry Operations IIIIEHI 390 Re ection in a plane 6 390 Inversion through a point i a Rotation about a proper axis Cn through an angle 27tn 390 Improper Rotation Sn about an improper axis 9 Identity do nothing E De ning Properties IIIDEHH 9 The product of any two elements in the group and the square of each element must be an element in the group 9 One element in the group must commute With all others and leave them unchanged 9 The associative law of multiplication must hold De ning Properties cont IIIIEHI 9 Every element must have a reciprocal which is also an element of the group 9 The reciprocal of a product of two or more elements is equal to the product of the reciprocals in reverse order 014 1 6 1114 1 04669 1 c IQ l l Multiplication Tables Rearrangement Theorem IIIIMHH 390 Each row and each column in the group multiplication table lists each of the group elements once and only once Why must this be true From this it follows that no two rows may be identical Thus each row and each column is a rearranged list of the group elements Subgroups IIIIEHI 390 A subgroup is a group within another group a subset of group elements A supergroup is a group obtained by adding new elements to a group to give a larger group 390 The order of any subgroup g of a group of order it must be a divisor of h k where k is an integer Similarity Transforms Classes IIIEEHH 390 is said to be conjugate with 6 if there exists any element of the group X such that flex i Every element is conjugate with itself ii If is conjugate With 6 then 6 is conjugate with iii If is conjugate with 6 and C then 6 and C are conjugate with each other Classes cont IIIIEHI 390 A complete set of elements that are conjugate to one another Within a group is called a class of the group The number of elements in a class is called its order 390 The orders of all classes must be integral factors of the order of the group 20 Electron Paramagnetic Resonance gvalues and gtensors Chem 634 T Hughbanks Angular Momentum The effect of human angular momentum can alter the strength of the Earths spin angular momentum ngmm MBgWWH S l WHRT ARE Vov poms Vquot spmume m RCLDLKWSE EAL Tum was THE WET or ANGva MMENTUM swam rs mu mmmsr BI1 LEWIW 1r NIGHT my quot WISHING Mom DAWN 6mm ME A era r MORE 11m HERE WITH Von httpxkcdcomc162html Magnetic Moments in Atoms Quantum mechanical operator for Classical orbital moment for charge electronic orbital moment 7e in circular orbit of radius r e E Morb m 3L eh p8 7 Bohr magneton 2mc p8 92740092 x 1021 ergG 92740092 gtlt10 24 JT Operator for electronic spin moment spin 2 ge39uB g2 2002319304362215 The Hamiltonian that accounts for interaction between both the orbital and spin magnetic moments and the applied field is given by H H Mm MW 2 ttBH L geS most accurately known physical constant physicsnistgovlconstants Atomic Structure Most of what you know about atomic configurations and states derive from usual kinetic and potential energy terms of the Hamiltonian 2 2 e h 2 02hl27 Where hl Vl 7 1 llt U 1 and second group of terms are the e repulsions However the spin and orbital magnetic moments interact De ning L and S so that they are dimensionless L S gt hL hS H 70 7 SpinOrbit ezhz 1 1 2 no Z2m202ltFgtLs2ltFgt2pBLs note in these de nitions ofL and s in m9p 11 1Ym9qa s2 ls ss1ls factors 0th are not in the eigenvalues How Big is 80 Coupling wsolzltgtllt22gtLs For hydrogenic orbitals 73 NZ3 1 H50 Zlt gt2u L S N Z4 replace With Z21 for manyelectron atoms 7 r I For a 217 3d or 4f hydrogenic orbital lt 1 gt 2 339 7 r 2n na0 36 n principal quantum number 3 l 2 Z more generally for any 711 7 i r3 n3ll1211 a0 4 71 These are too ifso Z 02438 cm L S for 2P ClCClIOHS small for 3d Z4 00145 cm 1L S for 3d electrons by about a factor of 2 Z4 000218 cm 1L S for 4f electrons Details H30 Zltgt2p L S For a 217 3d or 4f hydrogenic orbital n principal quantum number quot1 2n7le7ZZrrmn i 1 Z er rzdr FE r3 2n na0 o ya man 3 H mm L 1 Z X 0 Wilma i 2n 3 3 r3 2n na0 3 3 3 1 Z 1 Z 1 Z 7 7 for 217 7 7 for 3d 7 7 for4f 24 do 405 do 2688 do J xzniSeixdx 0 lt lZ lled gt 12 lled Z increases from 120 cm 1 for Ti2 to 830 cm 1 for Cu2 ltrgt ltragt ltrgt A calc A exp p Ion au au au cm 1 cm l cm1 3d1 39D Sc2 86 79 Ti 2552 1893 7071 159 154 V 3684 1377 3593 255 248 3d2 1quot Sc 35 Ti M 2133 2447 1317 61 60 016 V 3217 1643 5447 106 104 026 Cr 4484 1227 2906 163 164 3d3 F Ti 1706 3508 3162 29 V 2748 2070 9605 57 55 011 Cr3 3959 1447 4297 91 91 017 Mn 5361 1104 2389 135 134 3d 5D V 2289 2819 2071 34 Cr 3451 1781 7211 59 58 012 Mn3 4790 1286 3446 87 88 018 Fe 6332 1000 1986 125 129 025 3ds 6S Cr 2968 2319 1414 Mn2 4250 1548 5513 Fe8 5724 1150 2789 00 7421 09080 1659 3d l D Mn 3683 2026 1087 64 64 Fe2 5081 1393 4496 114 103 018 003 6699 1049 2342 145 Ni4 8552 08371 1423 197 3dquot F Fe 1774 8385 115 119 003 6035 1251 3655 189 178 024 Ni3 7790 09582 1971 272 Cu 9814 07719 1221 320 3d9 F Co 5388 1576 6637 228 228 Ni2 7094 1130 3003 39 343 324 053 Cu3 9018 08763 1662 438 3d 3D Ni 1401 5264 605 Cu2 8252 1028 2498 830 830 Electron Paramagnetic Resonance of Transition Ions Abragam amp Bleaney lt lZ lled gt 12 lled Z increases from 300 cm 1 for Y2 to 1600 cm 1 for Pd2 ltr3gt ltr2gt ltr4gt exp calc au au au cm 1 cm l 4d1 Y2 2034 5588 5900 300 312 Zr 3160 3857 2533 500 507 Nb 750 Mo5 1030 4d2 Zr2 2706 4526 3786 425 432 Nb 3913 3308 1860 670 644 M0 950 4d3 Nb2 3414 3829 2698 555 560 Mo3 4707 2905 1439 800 812 T04 1150 4d4 Mo2 4175 3319 2022 695 717 T03 990 Ru 1350 4d5 Mo 3662 3954 3298 630 To2 5015 2903 1541 850 Ru3 6496 2313 917 1180 1197 11114 1570 4d6 Ru2 5858 2628 1287 1000 1077 Rh 7447 2117 779 1400 1416 4d7 Rh2 6804 2374 1060 1220 1291 Pd3 8487 1939 659 1640 1664 4d8 Pd2 7814 2158 883 1600 1529 Ag3 9611 1 782 561 1930 1940 4d Ag2 8905 1972 741 1840 1794 Electron Paramagnetic Resonance of Transition Ions Abragam amp Bleaney SpinOrbit Energies Atoms For atoms and most of the lanthanides the magnetic properties are fundamentally derived from the lowest J state WhereJL S andJL SL S l L S The spinorbit energies of J states that are derived from a single RussellSaunders term ZSHL are given by the Lande interval formula LL S JLS gym 1 LL 1 55 1 PJLS Where it igZS for lt half lled shell for gt half lled shell z Energy Z Ion exp calc Term 9 cm 1 80 21L o s 58 4f1 083 640 740 2163 3 0 4 221 2200 59 4f2 Pr3 750 878 3H4 4 0 25 3H5 2100 60 4f3 Nd3 900 1024 1 2 0 lt 12 111166 woo 61 4f Pm 5I4 2 0 gt 12 111166 6 1600 62 4quot sma 1180 1342 Hg 3 0 H 1000 63 4f Eu3 1360 7F 0 0 7F g 400 C Increases from 4 4 Gd 1 8 0 1 P 30000 cm for 65 4f Tb3 1620 1915 7F g 0 7F 3 2000 Ce3 to 66 4f Dys 1820 2182 6H1 3 0 1 3 39 cm for 67 4f10 Ho 2080 2360 51 3 0 SI 68 4f Er3 2470 2610 41 g 0 14 6500 69 4f12 Tm3 2750 2866 8H 3 o 1 Ha T 70 413 Yb 2950 3161 2F g 0 216 11 10000 T This level lies above 3H4 Electron Paramagnetic Resonance of Transition Ions Abragam amp Bleaney Land Interval Rule Derivation 12 JJLsLsL2SZ2Ls So we can rearrange to get a completely general operator identity LosiJ2 L2 sl 2 If we can considerJ states derived from a single RussellSaunders m then we can assume that all the J states have welldefined values ofL and S ie L and S are still good approximate quantum numbers In that case we can operate on any one of the J states I JLS using the above identity to obtain the basis for the Land interval rule AL S I JLS JJ 1 LL1 SS1 I JLS Where l gzs for lt halffilled shell 7 for gt halffilled shell Summary SO Coupling in Atoms Neither S norL are strictly good quantum numbers in the presence of SO coupling butJ is Nevertheless spinorbit splittings are usually small compared to term splittings so J states are usually overwhelmineg derived from a single 23quot1L term Q increases from 120 cm 1 for Ti2 to 830 cm 1 for Cu2 3d from 300 cm 1 for Y2 to 1600 cm 1 for Pd2 4d and from 640 cm 1 for Ce3 to 2950 cm 1 for Tb3 4f Magnetic Moments of Atoms As we39ve seen the Zeeman Hamiltonian has the form WZeeman BH M Mspm MBH L 325 It is customary to express the Zeem an interaction in terms of the total angular momentum J and to define an effective quotgvaluequot g J for this purpose Of course the effective Hamiltonian has to give the same results as if we used the true magnetic moment operator so they are set equal mb WZeeman gJIuBH J 2 EH ges gJJ L geS which is a de nition ong If we assume that g2 2 and again assume that L and S are good quantum numbers we can derive a formula for g J 3 L SS1 LL11L JJ1SS1 LL1 J 2 39 2JJ1 39 2JJ1 Derivation gJJ2LZSJS ifwetakegz 2 dot both sides with J gJJ J J S J gJJZ J2 JsJ2 LssJ2 52 Ls Now plug in the identity L S J2 L2 S2 and solve gj J2 sl L2 which is an operator identity if g2 2 The operator identity is applied to I J39LS to obtain the formula Ss1 LL1 53 2 2JJ1 Why use a gJ value at all The Zeeman splitting splits each Jstate into 2J1 levels each associated with MJ which takes on the values J J1 J l J We therefore want to think about the Zeeman splitting of these 2J1 levels Although magnetic moment operator L geS is not changed because of spinorbit coupling the Zeeman interaction is changed and J is not equal to L geS Carbon Atomic Energy Levels Con gura on Tenn J Levw Experimental atomic lam energy levels are 252222 3 o o 1 1640 shown With 2 434 energies in cm 1 2522p2 1D 2 10 19253 2522p2 1s 0 21 64801 252pa 53 2 33 73520 term 33 73520 2522p2P 35 3P 0 so 33343 1 50 35253 2 50 39314 httpphysicsnistgovPhys m I RefDataASDindexhtml 2522p2P 3s 1P 1 61 98182 term 6 93192 Carbon Atom energies sp3 7777 7 53 18 28906 39 33735 quotI 1 3 3p xquot D spinorbit 0 zeeman 32p2quot 3939 10163 coupllng splitting 3P e e repulsion Energy Spllttlngs differences dlsplayed graphlcally energies in cm 1 uBH 56711x10 5 eV 04574 cm391 atH 10T Copper Atomic Energy Levels Configuration Term J Level cm i 3d104s 2s 1 2 0 term a For TM atoms energy differences between configurations with 9452 2D 5 2 11 quot2555 varying numbers of nd t 3 2 102 5 4quot and n1s electrons 1 can be comparable to 104quot 2Pquot 1 2 3quot 5353 state energy differences 32 30 783656 within a configuration m 3d9204s4p3po 4pc 52 39 018652 40 11399 1 httpphysicsnistgovPhys 2 40 943 73 RefDataASDindexhtml cam 39 7n451 Cu2 ion For TM cations energies of configurations with n1 s electrons are usually large Note the large spin orbit coupling for Cu due to an increased Zeff at the end of the TM series httpphysicsnistgovPhys RefDataASDindexhtml Energy L ziii iiigui a39zfi iwi 3p63d9 3p63d83F4s 3p63d83F4s 3p 53c8i1 D4s Term 1 Lin24 icgr i 2r 52 0 32 2 07169 E 1 52855 4p 92 60 80522 72 62 06509 52 63 14377 32 63 88651 C n 62 13 29 2 72 67 01671 52 68 96378 term 57 23117 2D 52 77 96825 3 78 78000 79 29295 Au 5d electrons CsAu The width of the 5d band in elemental gold is mostly determined by 5d5d overlap In CsAu it is spinorbit coupling The splitting is 52 52 13 eV G K Wertheim in Solid State Chemistry Techniques Cheetham amp Day Eds Au Counting rate Binding energy leVl 1O Sn 3d core electrons The 3d core electrons in tin experience a large effective nuclear charge Ze Consequently the spinorbit coupling is large The splitting is 522 525 8 eV Counting rate Tin Sn 3d52 32 Binding energy eV XPS for Sn See G K Wertheim in Solid State Chemistry Techniques Cheetham amp Day Eds Zeeman interaction for Hatom If a hydrogen atom is placed in a magnetic field H H22 the electron s energy will depend on its mS value The Zeeman interaction between the applied field and the magnetic moment of the electron is illustrated as Nuclear spin neglected H m5 12 ge uBH mS V2 gt g6 200232 11 Zeeman splitting for molecules The Zeeman splitting one observes for a molecule can depend on the direction that the applied magnetic field H makes with the molecular axes 11II I gt 39lt 3 Energy s l5 1 3 H H rS gr 0o x 1 Ad rpt gt I igt g 1 V Energy gt l e la m l ll 3 ll 5 N YJ O Q 13 CE Ad rpt gt Figure 3 EPR experiment for a tetragonal CuII complex A with H Hz and B with H Lz Left orientation of complex center associated energy splitting in a magnetic eld right EPR absorption spectrum Why is the g value directional The spin coordinate of an electron doesn t depend spatial coordinates so how can the direction of the applied field be make any difference in the Zeeman splitting Ans there is an internal orbital contribution to the magnetic field felt by the electron For molecules with nondegenerate HOMOs the orbital contribution arises from orbital mixing due to spinorbit coupling 12 The 9 Tensor To account for the anisotropy of the Zeeman response to an applied magnetic field an effective Hamiltonian using a socalled g tensor is used e yBHgSJBHX Hy HZngx gyy gyz S gm gzy gz 5 The 9 Tensor The gtensor is symmetric 9 9 In the laboratory coordinate system the g tensor can in principle be measured for a single crystal by measuring the Zeeman splitting as a function of the angle that the applied field makes with the crystallographic directions gez g2 M 0052 911x g2 yy 0052 OHy g2 L 0032 QHZ 5 I g2yy I g2zz I The 9 Tensor The gtensor is symmetric 9 9 In the right coordinate system ie using the principal axes the gtensor is diagonalized gm 0 O Sx zuBHgSMBHx Hy Hz 0 g o Sy O O gzz S Z The 9 Tensor While the form just shown is common it isn t really correct We know that when the electron possesses spin angular momentum only the g tensor is isotropic Therefore S in the previous cannot represent the true spin so might add a cap for this fictitious spin gm gw gxz 3 e zyBHgS uBHx Hy HZ gyx gW gyz be g2 g2 gzz S The fictitious spin operators will be defined later 14 The True Zeeman Hamiltonian When an electron has angular momentum L as well as spin angular momentum S the true Hamiltonian for the interaction with an applied field is given by it LBHLgeHS Zeeman The 9 tensor must be constructed such that the energies obtained with the effective Hamiltonian containing 9 and the fictitious spin are the same as one would get with the true Hamiltonian Huh Can you clarify The effect ofthe internal molecular orbital angular momentum can alter the strength of an external field necessary to produce the Zeeman splitting so that an electron s spin flips at a given excitation frequency w LBHLgeHS Zeeman The 9 tensor must be constructed so that the effect of the internal orbital angular momentum is accounted for correctly As a consequence the g tensor contains information about the wavefunction ofthe molecule in its electronic ground state Nondegenerate HOMO case Suppose we have a molecule like a d9 square planar complex with a single unpaired electron with wavefunction 1100 with either spin aor 6 We calculate the Zeeman energy as follows states III006 or IIIOB we takeH H22 Ea ltw0au8H oL gEH Sw0a 2 lat WO XIHZLZ ngzSz lV006 ZI LBHzOIo L2 W0g239uBHza Sz 0 1 L W0EgzluBHz LZ Ea BHZ W0 2 l W07 g2BHZ Eu 7 Eli gELLBHz but this isjust same likewise E uBHZ10 as the usual Zeeman splitting Nondegenerate HOMO case Where did we go wrong In the presence of SO coupling lyloor and lylo are not correct wavefunctionsl Instead we should use perturbed wavefunctions where SO coupling is taken into account Excited states H iwoa are mixed into A H i W 13 ghround stateliby 0 e so cou in for lt halffilled shell p g 7 for gt half lled shell L s Lzsz Ls L7S 11 L2 W n LX iL W llwoa7 wa Enigj gt 1m 11 L2 W n LX iiL W 7 W0 IltEK7E00gt Wn iznt iEnigoi o What See Carrington amp McLachlan Introduction to Magnetic Resonance Fictitious Spin Definition The ctitious spin operators are defined so that they operate on and in exactly the same way the true spin operators Sx Sy Sz operate on la and B A 1 A l S 5lgt Sz l 52 g Hug g ll H like Si la Saloon3 A A 7 S 0 S 0 s i0 s 70 39li gt fl and so on When the field is aligned along 2 HI Hy 0 the effective Hamiltonian containing the g tensor simpli es A gm 8W 8x2 SAX H2 MBH g SHglo 0 Hzlgyx 8W gyz S gm gzy g2 SZ if BHZ gan gzySy 8232 The Hamiltonians Matrices Must be Equivalent Ifthe effective Hamiltonian means anything in the basis of the true wavefunctions and its matrix must be equal to the Zeeman Hamiltonian in the same basis HRH gz y 82251 Wzm 301sz ggHzSz lvrzemml l emf l or imyimmi HZ l 2m l il zem if Now we evaluate all the matIiX elements WW 0 WW i e i Haitih i e i 1H Hg lHH g ig lg 1g fig 11 J 2 5 z z z 5 z zx zy H z z 2 DC zy e stgzxigzy JBHzgz 5 Z gzxigzy is i LzgeSz LzgeSz 7 Wham JBHZ if L2 geszltl L2 gas We can now set the mattix elements equal to obtain g2 2 Lz geSz gzx igzy 27 L2 geSz G o ry Detai Is We can now set the matrix elements equal to obtain gZ 2 LzgeSz gzxigzy 2 7 LzgeSz gzz 2 I geSz gt 7 evaluate to Ist order in C g2 204004 T gESz ulna n L2 14 n LX iLy u realmgszlwia lwoal ggSzllwi l ZWWKQeszllthWWmllegSleal 1 g2 g2 72wwlaluiuMM 2wwlelu wlLlun g g ZCXWJ Wol lwnllwnlLllro E 7E n efgxz i Interpreting g values 1Ol39llln Wu jyO lt Ill X Ll En E0 gy geZCX In the nondegenerate case the departure of the gvalues from ge z 200232 is due to the mixing of some orbital angular momentum into the ground state by spinorbit coupling The factors determining this mixing are The change in orbital angular momentum the numerator is 1 when LX and Ly are involved and zero when L2 is involved more explanation below The larger C is the larger the mixing and the larger the 9 shift from 200232 The smaller the energy gaps are the larger the mixing and the largerthe 9 shift from 200232 d orbital conversions In coordination complexes the ligand eld splits the d orbitals into familiar ligand field diagrams g tensor expressions involve orbital angular momentum operators that are evaluated with complex orbitals To interconvert we lwi l ll el 7 39 0 0 quota 1 F 1 iii lwkfllmlel gxfrx E cos 7 il 1 refr iizwrzn isinzeeizw lxy7Tl2il72 signs have to be consistent sin000509 w To help crank things out Effect of orbital angular momentum operators on the real d orbitals Lxdxz 4de Lyd 56122 idxziyz dez idyz Lxdyz ixgdzz idxziyz Lydyz 139de dey2 ag Ld id Ld id Ld 2id2 2 x xy 2 y xy yz 2 xy X y LXdXLy2 idyZ Lydxty2 4d dexziyz 2de L312 iJ dyZ Lydz 143d de2 0 Reference Ballhausen C J Introduction to Ligand Field Theory Examples L x newLgtiogt ii1gti4gti4 ax i1gti4gtigti iyzgt xziyz1Lz 2L 7 z72 lzlzrzlezilzfllzrlezilZixfllzlelez ilw L z Magic Pentagon for S 12 systems m 11 6 22 6 0 g g L 4 2 e E E xz gt yz i1 11 0 L lt gt 2 L u 2 2 x2 y2ltgtXy i2 The pentagon gives values of 11 but remember the horizontal connections apply to L2 and the vertical connections apply to LX and L The LX and Ly operators that couple the orbitals are shown Again 2 is the principal axis Example Cuphthalocyanine All dx2395 2 N N N CU N N A2 A1 l N v Observed gvalues are 9 2165 and Qi 2045 What are A1 and A2 What information is needed C 830 cm1 for Cu2 20 Table 3 Calculated g Values for dquot Ions in PseudoOctahedral Coordination Con guration S Ground State obsa gx gy gz d1 12 2ng E ge 2mmquot ge 2AA2b ge 8kA3b d2 1 3T1g VD g 92t2AC ge 9A2Ac ge d3 32 4Azg E ge taxA1 ge six12quot ge 8mg d4 HS 2 5Eg VD 6AA1 6AAtge ge 2AA1f 2AAzf 8iA3f d5 HS 52 6mg E ge ge ge d5 LS 12 ZTZg D 2 2ay s2 2 2ay p32 2a2 2y2 k az y2g 143ch2 yzng sz yzng d6 HS 2 5T2g VD ge ZAAlh g ZAAlh g3 8AA2h d7 HS 32 4T2g D 25 y3i 25 W3i 25 y3i 05 0 23 we 4m 4m 2m d3 1 3A2g VD ge 8AA1quot ge SitA2quot ge 8AA3quot d9 12 213g E ge ZAAlo ge 2iA20 86 8AA30 ge 62tA1p ge 6AA2p ge aE easy to observe generally at room temperature Ddif cult to observe usually liquid helium temperature is re quired VDv ry dif cult to observe zero eld splitting can be large to prevent observation except in very high magnetic elds The x and y axes of the reference system bisect the equatorial L M L angle The symmetries of the d orbitals are lxy lxz lyz e T2g and lxz yz Izz 6 Eg A should be taken as positive in any case bGround state Ix y A and A2 are the energies of lyz and lxz degenerate in tetragonal symmetry A3 is the energy of lxz yz c A is the splitting of the ground term in tetragonal symmetry dA are the states arising from the splitting of the 4T2g state The g tensor is generally nearly isotropic even in distorted chromophores eGround state con guration tzg3 eg1 5 xyl lcz1 lyz1 x2 y2 tz20 A1 and A2 are the energies of the excitations lyz gt lz2 and lxz gt Izz respectively They are degenerate in tetragonal symmetry f Ground state con guration tzg3eg39 E Pml Iral ly2l Jr2 322O l22 A and A2 are the energies of the excitations lyz gt lxy and lxz gt lxy respectively They are degenerate in tetragonal symmetry 8The ground Kramers doublet arising form the 2T2 state is written as 2T2 i ialil 3 b HZ IF l Z 42 i yFl iz After Grif th 18 M01 Phys 1971 21 135 hElongated tetragonal Ground state 582 A1 and A2 are the energies of the excitations to 5 E and 581 i0 symmetry y l in strong ligand elds and 32 in weak elds lTetragonal elongated complexes mTetragonal compressed complexes Ai are the states arising from the splitting of the 3T2g excited state A1 E3B3g A2 E3B2g A3 23mg A1 and A2 are degenerate in tetragonal symmetries Complexes having lxz yz have the ground state tetragonally elongated A1 and A2 are the energies of lxz and lyz degenerate in tetragonal symmetry A3 is the energy of lxy p Complexes having zz have the ground state tetragonaily compressed A and A2 are the energies of lxz and Jyz degenerate in tetragonal symmetry 21 Effect of ligand delocalization Delocalization reduces the effective spinorbit coupling by M 1 K a significant fraction K39Kquot xy 2 g value shifts are therefore decreased in comparison to those expected for pure 1 orbitals The appearance of spectra FIGURE 13 3 idealized absorption A and derivative B spectra for an unoriented system with S 12 axial symmetry and no hyperfine interaction g gt g 9L v9 H inc gt H inc gt A B At left is an absorption spectrum for an axial system eg D4h or D3h EPR spectra are usually plotted as derivatives to changes in slope and extrema more evident 22 Orthorhombic case gX gy and gZ 39 9X lt 9y lt 92 Most probable orientations have 9 values near the middle gvalue FIGURE 13 4 Powder epr H spectra of S 12 systems A An orthorhombic system with I O More reality Absorption Simulated EPR spectrum of a normal copper complex 91 A JJEZOB tetragonal CuH2062 dH g 24o B at Xband n 950 GHz A EPR absorption B first A 150x104cm1 derivative spectrum without and C with AIS copper hyperfine splitting IA ll lt 35x10394 cm1 IlllllllllllllllllJll 2600 2800 3000 3200 3400 3600 Field Gauss More than one unpaired electron When there are more than one unpaired electrons the spin Hamiltonian ignoring hyper ne interactions is written as arm uBHg D z2 E f j Notes The new terms are due to zero field splitting D is the axial splitting parameter the rhombic E term disappears when the system is axially symmetric The main origin of ZFS is different for organic triplets than for TM complexes In the former it arises from dipoledipole interactions between the electrons in the latter the dominant mechanism is spinorbit coupling EPR usually easily observed in S 12 systems often difficult for S integer this arises from two aspects of ZFS Relaxation times rT1 can be very short due to mixing ofground state with excited states that are close in energy Fluctuations in the environment modulate the mixing and yield rapidly relaxing states It is often dif cult to observe EPR in solution usually need lowT in crystals As we ll see below ZFS may move the allowed AMS 1 transitions too high in energy out of the range accessible magnetic fields 24 Zsz2n Some missed questions Problem Set 2 The ligand eld splitting parameters for octahedral complexes are normally considerably larger than for tetrahedral complexes A0 gt A Nevertheless the energy of the rst 6161 transition 5A1 a 4T2 of FeIHO6 chromophores is around 11000 cm l but is around 22000 cm 1 for FeHIO4 chromophores Give a concise but complete explanation Midterm Suppose you were studying an orthorhombic crystal with cell parameters a 30A b 50 A c 90 A At what diffraction angle 26 would you expect to observe the 246 re ection if you are using CuKM radiation 1 154056 A Group theory for halfintegral J sinJ 1301 2 sinJ 12a 21 D71 sinoc 2702 sinoc2 IfJ is halfintegral then 2 1 is an even integer if J is an integer then 2 1 is an odd integer 95Cw2 95CD ifJ is an integer and 95Cw2 ZCa ifJ is halfintegral When 1 or S is halfintegral rotations by 21 are treated as if they are a new distinct symmetry operation since othenNise the the characters ofthe representations for which the Jstates form a basis wouldn t be uniquely defined This leadsto the use of double groups Cotton Sec 97 to handle these cases 25 Double Group Example Cuphthaocyanine D E R 2c 2c3 2CRC2 2 42c2 42cz 4 RC4 RC4 C4RC4 2RC2 2RC2 A1 1 1 1 1 1 1 1 x2y2zz A2 1 1 1 1 1 1 1 zRz 131 1 1 1 1 1 1 1 x y2 32 1 1 1 1 1 1 1 xy E 2 2 o o 2 o o xy RXRy xzyz F6 2 2 J2 2 0 0 0 F12 F7 2 2 2 J2 0 0 A172 F6F6 13172 r6r7 E F6F6 F7 FmF6 F7 4121 22 2A11quot6 f i L 4 2 2 x xz u yz E1quot61quot7 A2 A L 4 2 2 y L gt 2 2 Z X2y2 8 yxy 28233107 dxz a dy V39 in a distorted octahedral complex symmetry aspects of 80 Coupling 3E 3T I E H ir Q How are the statee split inthe J J presence of SpInOrblt Coupling J x T E J r JT Any angular momentum states w1th a value J 5 distortion ie 21 1 states withMJ JJ 1 J that l l 13A are eubj ect to the in uence by the symmetry of the I 2quot enV1ronment the character of a rotatlon IS glven by quot C sinJ12or Oh D4h 9 a sin 062 D4 E 2C4 C2Cj 2C2 2C2 A1 1 1 1 1 1 x2 yz 22 A2 1 1 1 1 1 2 R B1 1 1 1 1 1 x2 y2 132 1 1 1 1 1 xy E 2 0 2 0 0 x y RyRy xzyz PH 3 1 1 1 1 1quotS1gtEA2 F3A2A2 E A2E Al F3EE E A2A1 A2 Bl Bz E 26 3 E quot 39 i Zero Field Tl i1 J 9 H T JT NJ jsAdistortion 2 1i rmquot ls L1SILZISZ Oh in lg ffl LaSwLi 1 L1S1 E ZFS L2XS2X 25 L21S21 3E 1L1S17 L17S1L12S12 u4 xy xz 3Awa MLSZJMHSZZ W5 iii Wlle Wu or 9 W6 M1 12 wille yz 9 9 a 7 xyrwr W2 lij H m M wt lial inHim5a l W3Hxz W9 Mam2H Gory Detail Typical Matrix Element If We expand out the determinants We can evaluate the a typical matrix element 14 gamma 7 ulxy2al z 31062 w xzmmz a z1xz2tlltlz Vs Wmlwo 5 Wow 7 ult1xylt2gtal z ng 2 2 was as Lust was with Lush tamer wlt1gtnlt2gtlaiaz All the terms involving LIZSlz and LZZSZZ drop out because of spin orthogonality Also the terms involving SH and S drop out because they kill the a spin states they operate on So We have so far mxyamzy xz1xy2zzl z laz LlSli L2S27 xzlyz2r yz1xz2zzlzzz C in out the in lowerin o erations and doin the trivial in inte rals ields my 8 5P g P g 5P g Y immune a ult1gtxylt2gtll L1 llxzamz a J Z1xz2 immune a xzlt1gtxylt2gtl L2 l ammo a J Z1xz2 In the top integral We can trivially carry out the integrals over electron 2 coordinates and in the bottom integral We can do the same for electron 1 coordinates and things simplify mul L l yz1 7 imam L l yz2 identical integrals V ltwlalyzltwllxHIM wsiwmiwi 2J5 Where We used the table for operations on d orbitals given earlier only the Ly term survived 27 The spinorbit energy matrices involving the mixing of 145 and 148 into 141 and wgare iwl iws W8 ilk iws ills vi 0 rel 0 ii5 i 5 0 ii5 i 5 0 Wsi i 0 5 Wsi i 0 5 22 22 The spinorbit energy matrix involving the 3 mixing of 144 146 147 149 into 14Z is E W2 W4 W6 W7 W9 1 L i i II2 0 NE 2J5 2 2 NE 6 i 5 0 l5 0 II4i 22 2 g II6 E 0 5 0 7 II7 i 5 0 5 0 ltw l Z 9 m 0 7 0 5 More Detail if 2 Shi of wlandlu32XE Shi of uZ4gtlt 5 E M341 w Mg0 Spin Hamiltonian and EPR spectrum with ZFS Now that We39ve shown the physical oIigin of zero eld splitting E let39s see how it ts together With the spin Hamiltonian E 0 s W217uBHg DS 77SS31u8Hg D f7 ltEH sinceS1 The states on which this opemtes have and are D MS 71 i39 7 Msi1 MS0 l0 MS0 H at u gHlD I 7 gyuBgHjD When D gt 0 the ground state Is 7 0 75 3 diamagnetic If D is too large no EPR 7 3 signal is observed When D lt 0 the ground state is paramagnetic but transitions with AMS 2 are forbidden allowed in second order only 28 For an axially symmetric quanet state E 0Ll16 spin Hamiltonian is Hg u8Hg D 337 uBHg D 337 since S The states on Whicthis opemies are Hgyl BgHD He lig7u8gHD ZFS quartet state axial symmetry MS32 When D gt O and too large the 12 9 32 transition will not be observable in EPR The 12 a 32 transition would only be observable at high fields The 12 9 12 transition is 32 always observable since it is independent of ZFS WeilJBEH D WMl F JBgH D MS 12 MS 7 12 Ms PM 1 6 913 in D3 6A1 states HS Mn and Fe39 Because ground state has no orbital angular momentum only higherorder spinorbit andor direct dipole dipole spinspin couplings give ZFS ZFS parameters are small though measurable 5 iZ 4D i2 2 i 2 small D 10 2 cm 1 29 Electron Paramagnetic Resonance Hyper ne Interactions Chem 634 T Hughbanks Reading 9 Unfortunately Drago s Physical Methods for Chemists is probably still the best text for this section but it is virtually unavailable and very expensive lfyou can find a copy to read from recommend it For EPR Chapters 9 and 13 a See also Inorganic Electronic Structure and Spectroscopy Volume 1 Chapter 2 by Bencini and Gatteschi and Volume 2 Chapter 1 by Solomon and Hanson a Background references Orton Electron Paramagnetic Resonance Abragam amp Bleaney Electron Paramagnetic Resonance of Transition Ions Carri ngton amp McLachlan Introduction to Magnetic Resonance Wertz amp Bolton Electron Spin Resonance What is EPR ESR Electron Paramagnetic Spin Resonance Applies to atoms and molecules with one or more unpaired electrons An applied magnetic eld induces Zeeman splittings in spin states and energy is absorbed from radiation when the frequency meets the resonance condition hv AE oc IJB X H 1l 1 cm l v cl 1010 s 1 microwave GHz What information do we get from EPR Chemists are mostly interested in two main pieces of information gvalues and hyperfine couplings though spinrelaxation information can also be important to more advanced practitioners gvalues these are in general the structuredependent ie directiondependent proportionality constants that relate the electron spin resonance energy to the direction ofthe applied magnetic eld hv gXJyJZpBH The directiondependence is determined by electronic structure Hyperfine coupling arise from interactions between magnetic nuclei and the electron spin and give information about the delocalization of the unpaired spins on to those nuclei What background do we need We need an understanding of the magnetic properties of atoms andor openshell ions In particular we will need to understand the role of spinorbit coupling which is essential for understanding gvalue anisotropy Quantum mechanical tools manipulation of angular momentum operators and understanding of how perturbation theory works Basic ideas from MO theory and ligand field theory must be familiar Spectroscopy The Big Picture SPECTRAL RANGE I l Hard Soft Vacuum Near Visible Near Mid Far Sub mm MiCro Radio I yrayl Xray I Xray UV I UV 39blue red IR i IR 4 IR lmmw I wave Pwave Jwave lt01A 5A 100A 2000A 07 pm 25 pm 25 um 1mm39 10 Cm gt 10 nm 200nm 400 nm 700nm 2500 nm gt109 2x107 10 5x10 25x10 14x10 4000 400 10 01 lCmquotl 12x109240x10 12gtlt10 600x103300x103170x10J 48x103 5x103 120 12 E J moiquot 120000 2400 39120 6 3 17 05 005 0001 000001 eVl 3x10 9 6x10 3x10 51o5x10 575X10 agtlt10 12gtlt10 12gtlt10 J 3x10 3x109 uin XRF 1 Electronic 4 r Rotational 4 NMR Mossbauer XPS ups i Vibrational l v ESR NOR SPECTROSCOPIC TECHNIQUES Q GED XRD HNuclear energies Chemical energies Molecular energies Spin energies Experimental Aspects EPR performed at fixed frequency ie field is varied Some common frequencies Xband v 95 GHz 95 x109 s 1 9 3 cm typical required eld is 3400 G 034 T most common hvgeuB 338994 G for v 9500 GHz Kband v 24 GHz 24 x1010 s 1 9 1 cm typical required eld is 8600 G 086 T less common hvgeuB 856406 G for v 2400 GHz Qband v 35 GHz 35 x1010 s 1 9 08 cm typical required eld is 12500 G 125 T hvgeuB 1248925 Gfor v 3500 GHz Solvents generally avoid water alcohols high dielectric constants why Experimental Aspects Form of samples Gases liquids solids crystals frozen solutions Glasses are best for solution samples measured below solvent freezing points because they give homogeneity whereas solutions have cracks and frozen crystallites and scatter radiation Note glasses can form with pure solvents or mixtures Unsymmetrical molecules have a greater tendency to form glasses eg cyclohexane crystallizes methylcyclohexane tends to form glasses Hbonding promotes crystallization Experimental Aspects Containers Pure silica quartz Lot Pyrex borosilicate glass the latter absorbs microwaves to too great an extent Constant homogeneous magnetic elds Microwave generator klystron sets up a standing wave for which the frequency is xed by the geometry ofthe cavity To improve the SN ratio reduce the effect of 1fnoise the eld is generally modulated 100200 kHz and detected with a lockin detector amplifier Phase sensitive detection gives a true differential of line shapes For this reason EPR spectra are normally displayed as derivatives this is actually convenient since it accentuates features of the absorption in which we are interested anyway The Experiment EPR Fixed microwave frequency sweep the field ln NMR frequencies are varied at xed fields The power absorbed is measured as a function of field MODULATED POWER SUPPLV KLYSTRON MICROWAVE GENERATOR MAGNET 41 STAN DIN G ICROWAVE absorbed Modulated Field Ht ll I 00501 H at constant v wim dd i sommmmmmma m l CiOW ave cosmt modulation frequency The eld is modulated to improve the SN ratio Hatom Spin Hamiltonian H geuBH S gNuNH IaS I Spin 87v 2 a geuBgNHN 1st term electronic Zeeman 2nd term nuclear Zeeman 3rd term Fermi Contact hyperfine isotropic magnitude depends on the electron density of the unpaired electron on the nucleus 140 Zeeman interaction for Hatom If a hydrogen atom is placed in a magnetic field H H22 the electron s energy will depend on its mS value The Zeeman interaction between the applied field and the magnetic moment of the electron is illustrated as ms 1 2 Nuclear spin ge39uBH neglected H ms 2 12 gt ge 200232 Hatom Spin Energies 4 1 laoc g 1 H f e N lace I quotg igf39B N 1 39 39 39 I H 1g u elgt 3 I 2 N N h We N a z 4 IO e N I la 0 quot e N I geauBH39S gN39uN I aSI lie320 WJXNgt 1 a 4 v a 1 I e Ngt g M H l39239 N N I39l e Ngt 1 l eO Ngt 2 geMBH gNJN I Electronic Nuclear ia Zeeman Zeeman 18L0rder hyperfine hvl hvz a Cl H I I geluBHl gNluNHl EgeHBH2gNHNH2E 1MHI IMHZZ a geluB ge uB geIUB Spectrum 0r H1 H2 MH1 112 geHB geHB a 1 This is grouped conveniently since geuB gtgt g NM N ge uB I 111 sz a l ge uB 1420 MHz X V From Drago Physical A Note on units of energy Three different units of energy none of them really are energy are used in EPR cm 1 1m GHz or Hz or MHz v and gauss G The conversions are done as follows cm 1 x c 299793 x 1010 cms e Hz or cm 1 x 299793 e GHz Hz gtlt hge B 35683 x 10397 Gs e G or MHz gtlt 035683 e G The conversion to gauss yields the field necessary to induce an equivalent Zeeman splitting for a free electron Example a zero field splitting parameter of 01012 cm 1 could aso be reported as 3034 GHz or 1082 G Methyl radical splitting diagram e Nico Mid Nla mic ll The four transitions 1 A for the methyl radical 3 m states are lowest for m8 12 and the m state lowest for mS 12 from the lS term l I mica rota MI I le Methods Fig 97 corrected Methyl radical splitting diagram The four transitions for the methyl radical ml states are lowest for m8 12 and the m state lowest for m8 12 from the lS term Plotted to reflect the constant frequency experimental conditions 3 M1 2 A E M1 1 12 M1 2 M12 v39 v 39 3 M1 12 I I I M 2 u 39 39 M1 l I l I 3 2 i EM1 2 i i i l l H l I I I I I I I Organic 1 Radicals 375 Gauss 39l W 321 237 7 Gauss WW aHzQpH Q225G ESR Spectrum Napthalide ESR Spectrum Napthalide Anion am 1379 MlIz N 492 G a 494 M111 N 175 G 1111 1111 Napthalide Anion SOMO 0429 0425 a 0258 0263 Numbers are coefficients from ESR Hijckel 7N TABLE 11 Nuclear Moments and Spins y radians yN radians Q Nucleus I 4 sec 1 gauss 1 Nucleus I 57 sec gauss l 10 2 cm H1 12 5585 26753 D2 1 0857 4107 000274 C13 12 1405 6728 L17 32 2 71 10398 002 N15 12 v0567 2712 B 32 1791 8853 000355 F19 12 5257 25179 N 1 0403 1934 002 Si29 12 1111 5319 O17 52 0757 3628 00265 P31 12 2263 10840 Na23 32 1478 7081 100 or 0836 Pt 12 1 120 5747 3 3 32 0429 2054 0064 Cl35 32 0548 2624 0079 Cl37 32 0456 2184 0062 K39 32 0261 1250 0113 V 72 1468 7033 03 NUCICi With 110 Spin C O16 015 Si S32 C3 The quadrupole moment of Na23 is uncertain 11 Nuclei with 1 gt 12 A nucleus with spin l splits the electron resonance into 2 1 peaks Again the electronic Zeeman interaction is much larger than the hyperfine interaction H 11 1 0 1 VOH2052 v4 0H s 12 51V1OO I 72 g I P9658 A ll S mT l Characteristic vanadyl EPR gtensor is nearly isotropic g 19658 12 Nitronylnitrcxide Two 14N For n identical nuclei hyperfine splitting yields 2111 1 lines aS I1 12 19 1 0 0 10 1 1 10 01 0 1 hyper ne 19 2 11 1 1 SUMO Delocalized Electrons in Clusters 5900 100 abundance COco I 72 ocCOquot c 23gtlt721 22 expected lines SOMO 32 a 728 G Single crystal 77K H H z 13 Anisotropic hyperfine coupHng Simulated EPR spectrum of a normal copper complex tetragonal CuH2062 at Xband n 950 GHz A EPR absorption B first derivative spectrum without and C with copper hyperfine splitting From Drago corrected 5 2 4D 2 ltEEEE 68 3quot 2 1 1 2D i3 x 2 3 2 Z i 2 Zero field Applied Nuclear levels field splitting A I S Excited 4ng state mixes into 6A1g B lllilies 4 Absorption dx n gi208 AI39S lAil lt 35x1o4 cm1 63Cu 691 I 32 A0 495 GHz 65Cu 309 I 32 A0 530 GHz Fine structure IAHI 150x10394 cm1 g I I I I I I I I I I I I I I I I I I I I 2600 2800 3000 3200 3400 3500 Field Gauss l 55Mn 100 Mn2 doped into MgVZO6 gX 20042 1 00005 gy 20092 1 0001 gz 20005 1 00005 DX 21815 G Dy 87i5 G D2 306i20 G from H N Ng and C Calvo Can J Chem 50 3619 1972 500 G ll Hill J Assign all the transitions with initial and final values of MS and m Multiple hyperfine splittings EPRderivative spectrum of bissalicylaldimine copperl H2 H1 with isotopically pure 63Cu N 0 Askerisk indicates calibration CU peak from DPPH From A H O N Maki and B R McGarvey J Chem Phys 29 35 1958 63Cu132 100 enriched 14NI1 1HI12 A gorgeous case of accidental degeneracy r W 4 multiplets each multiplet has 11 lines with relative i intensity ratio 12345654321 EPR external standard DPPH diphenyl lt picrylhydrazide g 20037 i 00002 Decreasing H gt Organic Radicals in Solids 7r geuBHoS gNuNHISoTI S39T39IS39TquotIaS I Spin 1st term electronic Zeeman gtensor anisotropy neglected 2nd term STI nuclear Zeeman 3rd term Hyperfine includes the isotropic Fermi contact term and the anisotropic dipoar part of the hyperfine ST I S T IS T IaS I 15 Hyperfine Anisotropy The classical energy of interaction between lmPolar geHBgNHNS 39 T 391 two magnetic dipoles ue and uN is T T a1 E 2 lie 39 N 3019 39I XPN 39 1 N i r2 3x2 3xy 3xz dinolar r3 r5 r3 r5 r5 r5 Where r 1s the vector separat1ng the d1poles I 3xy r2 3y2 3yz The corresponding QM Hamiltonian is T r5 r5 r5 SI 3SrIr Wdipolar geF BgNF Ni 3 5 3 3 7 7 r5 r5 r5 Matrix elements of T should be interpreted as expectation values of the operators shown over the electronic wavefunction Typical Spectra Pitfalls Even reasonably straightfonvard systems can fail to give typical spectra o39ll l ModernEPR interpretation is almost always accompanied by computer simulation if Spin gquBHzS39z gquBthS gyuBHy y D FIGURE 13 13 A Small zerolield and magnetic field splitting of the A ground 2 5 e2 n2 state eld along lpr a a 3 case and the resulting spectrum D Sz Sy B Trans CrC5H5Nl2 in DMF H20 CHZO CHSOH glassmy at 93 GHZ Dgt 04 4 cmquotElt001 AA AA A A C TransCrCsH5NaCzl in DMF H20 CH30H glass at 9211 GHz I A I 5 I D Computer slmulation 01 C with g g 199 D 0164 cmquot E O l 2 z J X X y y Reprinted with permission lrom E Pedersen and H Toitlund Inorg Chem 13 1603 1974 Copyright by the American Chemical Societyl 16 Exchange Coupling Exchangecoupled N NW dimers usually HQN i FC quotNHQN antiferromagnetic are quotM N usually EPRsilent at N2N om N NH low temperatures Ns If the coupling is very 6H2 weak transitions from the S 0 ground state can be observed next slide but it is unusual Exchange coupled lll 5 dimers Ht i39 n N 39 i CUN N NHN JV IJL J 1 HQN N7 N 39 2N i NH2 NVN N N gi N gll 5H2 l AMS 2 Field G 17 Nuclear Magnetic Resonance Chem 634 T Hughbanks Books Web sites etc 0 J Iggo NMR Spectroscopy in Inorganic Chemistry 0 J K M Sanders and B K Hunter Modern NMR Spectroscopy A Guide for Chemists 0 H Friebolin Basic One and TwoDimensional NMR Spectroscopy 0 D Canet Nuclear Magnetic Resonance Concepts and Methods 0 E A V Ebsworth D W H Rankin S Cradock Structural Methods in Inorganic Chemistry Chapter 2 0 R S Drago Physical Methods Chapters 7 8 and 12 o httpwwwcemmsuedureuschVirtuaITeXtSpectrp ynmrnmr1htm NMR is Broadly Applicable Nuclear Magnetic Resonance Applies to atoms and molecules with nuclear spin quantum numbers greater than zero Applied magnetic field induces Zeeman splittings in spin states and energy is absorbed from radiation when the frequency meets the resonance condition hv AE oc uN gtlt H0 11 10 3 cm i v 01 25 500 MHz Familiar Matters Nuclei with l gt12 have quadrupole moments even solution spectra are broadened when such nuclei experience electric field gradients For solution spectra the immediate chemicalstructural information is conveyed by the chemical shifts and the spinspin couplings More Familiar Matters Resonance frequencies for two interacting nuclei are given by E hszmAthABmAmB A AltB where mA 1s the zcomponent the sp1n on A and J A B is the coupling constant forA interacting with B This expression applies when chemical shift differences are much greater than couplings vA vB gtgt JAB Zeeman interaction for proton If a proton is placed in a magnetic field H sz the proton s energy will depend on its ml value The Zeeman interaction between the applied field and the magnetic moment of the proton is illustrated as F t m1 2 12 or pro on gN 5585 E gNHNHo rNhHo YN 226 753 magnetogyric ratio m1 12 radians s 1 0 G71 Resonance Larmor Frequency AE yNhHO M 7N VL nH0 gt wozyNHo including chemical shift v y N H 0 1 0 275 NMR transitions are stimulated absorption and emission by an additional oscillating electromagnetic field H is applied at the Larmor frequency Basic Spectra 1H V 900 750 600 liSO 300 150 0 T f 90 MHZ 1H NMR spectra TMS Basic Spectra 1H 3 900 750 600 450 300 150 D lt v Hz QCHZOCOCH3 12 90 MHZ 1HNMR Basic Spectra 1H CH 3CH 2 Br Ethyl hrumitle has two types of hydrogerL The down elll peak at about 35 ppm is the CH2 leshieltled by lhe Br and appears as quanel due m the adjacem methyl group n3 The up elll peak at about 17 ppm is lhe CH3 and appears as a triplet due lhe adjacent methylene n2 Mogul I I I I I I I I I I I 39 7 T E 2 1 m D ppm Basic Spectra 1H CH3CH2CH2 Br 1 triplet sn quot2 1r lei sn deshielded shielded s Iinas sn quot5 3 yres an middle methylene appeals 256 Iinas due m 5 neighhnuls 3 o 2 mang 2 2 1 1 1 1 1 1 1 1 1 1 1 1 11 1o 9 8 7 6 5 4 3 z 1 0 PP Aczmphennne hns mpes nYczrhnn sn 5 paaks Wm me I types n annualquot ammus helwezn 1Z 1rl mlmlhu kulunl39 unlluuy n mayquot uml ma mamyx qrnllp mum 230 18r 1m 40 12 phenolsOH alcohols OH thioalcohols SH amines NH2 carboxylic acidsOH Ho cfo H H aldehydes DEC heteroaromatics Ch arenes alkenes alcohols alkynes 1 12 I cyclopropyl 39 MCH3 l l 12 11 10 9 8 7 6 5 4 3 2 1 0 Chemical shifts of 1H nuclei in organic compounds 6 ketones aldehydes acids esters amides thbkefones azomethMes heteroaromatics mkenes arenes nilriles NEC O alkynes IFCC I Cquaternary L 0 C l I l l lC C3 Inorganlc and organometalhc chemISts I I must look well outside this range HalC I C lerl39ary e g aromatic compounds usually lie JJ H between 8 110170 ppm but nbonded 351 arenes maV be shifted bV up to 100 ppm to S H 39 a E secondar lower HaluCH C Y 13C C14 5 293 ppm DCH 39 CHz quot39quotquotquotquot cc 39 13C MCOx 5 170 290 ppm 3NCH 13C LXMCR2 5 250 370 ppm 2 H2CS C Interstitial carbides are highly deshielded Hal CH2 mmary 13c Zr6CX12L6 5 450 520 ppm 0 CH CH3 3 N ICH3 schems CH312D0 882 CF3 DOH C5h5 CE3C00H CElL CHEla CHg H Hal CH3 I 1ADioxan I39DMSU 230 210 190 170 150 130 110 90 70 50 3O 10 e Chemical shifts of I3C nuclei in organic compounds 3 TABLE 11 Nuclear Moments and Spins yN radians yN radians Q Nucleus I gN sec 1 gauss 1 Nucleus I 9 sec 1 gauss 1 10 24 cm2 H1 12 5585 26753 D2 1 0857 4107 000274 C13 12 1405 6728 Li7 32 2171 10398 002 N15 12 0567 2712 B11 32 1791 8853 000355 F19 12 5257 25179 N 1 0403 1934 002 47 Si29 12 1111 5319 O17 52 0757 3628 00265 100 P31 12 2263 10840 Na23 32 1478 7081 100 or 195 0836quot 337 Pt 12 1120 5747 S33 32 0429 2054 0064 C135 32 0548 2624 0079 Cl37 32 0456 2184 0062 K39 32 0261 1250 0113 Nuclei with no spin C12 O16 O18 Si28 S32 Ca4 Sensitivity Populations Even at 300 MHz H 705 T the population difference of the two spin states for a proton is 10 5 Since other nuclei have smaller magnetogyric ratios they exhibit even smaller differences at the same field N N06 3 e AEkBT 1 A E 099999135 k T B H 705T T 2 298K The quadrupole moment of Na23 is uncertain Time Scale NMR is generally considered to be a slow technique with a characteristic time scale 1 10 7 s Recall 11 v 100 MHz Molecular events occurring in times much faster than 10 7 s are time averaged in NMR a more careful distinction to come later Relaxation Refers to processes by which spins nonradiatively lose energy The phenomenon of relaxation is crucial to NMR In the oldfashioned continuous wave method signals would rapidly saturate because spin populations can become equal In FTNMR relaxation is necessary to bring system to equilibrium between pulses T1 spinlattice relaxation dissipation of energy to surroundings nonspin degrees of freedom T2 spinspin relaxation transfer of energy to other spins T1 2 T2 Quadrupole moments Q cause T2 to shorten greatly leading to linebroadening lggo Sec 110 Relaxation Time sec Correlation times and Relaxation 103 nT poisesdeg 10 3 102 10 1 1 10 10 1 I T I A o o 0 10quot v 102 o 15 E s 2 10 3 4 1 0 5 3 0 6 O l Solid quot 3 4 oe m 10 r o wOTc 1 L T2 o I 10 9 i l l I Proton relaxation times in glycerine between 10 15 103911 107 103 60 C and 35 C Measured at 29 Mcs Correlation Time 1c sec Relaxation effects depend on the frequency spectrum of the local fields experienced by a nuclear spin The correlation time TC is a characteristic time scale of fluctuations due to molecular tumbling rotational Brownian motion in solution Relaxation is most efficient when 1TC V T hl 11 NMR properties of 301113 quadrupolar nucleia Relative Natural Magnetogyric NMR Quadrup ole abundanc e ratio frequency Relative moment 2 isotope Spin in IIIT rad T quot3 u MHZ receptivity III mm1 a 2H 1 111115 41933 134 15 3 11176 23 x 19 Li 1 24 39321 142 63 3 19quot 3 4 111 7111 312 9213 1113925 339 22 x 10quotquot 4 x 191 2 Be 322 10119 3596 141 14 5 19 5 x 10 N13 3 1913 23243 1112 39 gt1 113 3 35 x 1939 3 322 3134 35343 321 13 x 11139I 41 3 1351 quotN 1 993 19333 22 111 x 111 31 1 x 11139 1 391 512 111132 33229 136 11 x 111395 23 3 111 1 2313131 322 10911 2123111 265 93 3 111 2 1 x 19quot 1quotMg 522 1111 1339 31 22 x 111 22 x 111 I 41 532 111912 1392511 261 21 x 19 15 5 111quot1 3 312 1123 21135 22 12 4 111395 55 x 111 is 322 255 23243 93 33 5 19quot 1 x 111 quot C1 322 245 21342 32 32 9 111 29 3 111quot 33 342 931 12493 42 43 51 113 49 4 19 45131 242 1115 13025 32 32 x 1124 2 5 111 3931 232 11399 331131 243 311 51 111 22 x 111 3 1 532 23 15193 56 15 x 13 1 29 gt1 111 1 39 222 55 4151119 53 21 1 113 1 24 5 113 quot 9d 222 993 29453 233 33 x 111 l 45 4 19 1 53Cr 322 913 1512 52 33 13 111quot5 3 3 111 2 Mn 532 11100 33113 242 13 54 112 I 4 gt1 19 Co 222 111011 15312 23151 23 2439 111quot 33 51 111quot Ni 342 12 2394 39 41 3 111 5 16124 19quot cu 342 391 20924 265 35 x 111 2 21 3 19quot Cu 342 3119 23331 234 33 3 111 2 20 x 19 5 11 522 41 13233 33 12 3 111 16 x 19 596a 322 304 34323 2411 42 9 113 2 19 3 19 Ga 332 393 31231 3915 52 3 111 2 12 gt1 10 G 922 23 99352 35 11 x 19 13 gt1 111quot 43 3x2 113113 4595 122 25 x 111 2 29 x 111quotI 79Br 322 595 32223 251 411 x 10quotquot 32 14 111quot1 Br 322 495 22433 221 49 x 111 1 31 3 111 1 Table 22 NMR properties or some quadrupolzr nuclei cont d Relative Natural Magnelogyric NMR uadrupnlc abundancc ratiobl frequency Relative momcnlb Isotope Spin 0 107rad T45quot MHz receptivity lO Z m2 quotRbquot 32 27 9 37307 323 4 9 x 10quot 13 x 10 quot51 92 70 1153 43 19 x 10 3 x 10quot Zr 52 112 724959 93 11 gtlt 10quot 21 x 10 10 92 1000 5554 24 5 49 x 10 22 x 10quot quotMo 52 15 7 1750 5 5 51x10 12 x10quot quotMo 52 9 5 41787 57 33 x 10quot 211 9quot 52 127 125439 45 15 x 10quot 75 x 10 1 Ru 52 171 1333 5 2 23 gtlt 10quot 44 x 10 I 1gtd 52 22 2 7123 45 2 5 X10quot 3 xiiquot 111 92 95 7 53903 22 0 3 4 10 l 33 x 10quot quot3951 2 s7 3 54355 24 0 9 3 x 10quot 723 x 10quot 39 53 72 427 34343 150 0 10quot 35 x 10quot 39271 52 100 0 5 3317 201 9 5 x 10quot 79 x 1039 x 32 212 22 32 5 9 x 10quot 12 X 10 I quot 65 72 1000 3 5277 132 4 3 x 10 1 3 x 10quot Baquot 32 113 2933 111 7 9 x 10quot 23 x 10 I 9121 2 999 3301 142 50 x 10 2 22 x 10quot quot Hr 72 13 5 10111 4 o 2 5 x 10 4 5 quot m 2 13 3 0579 25 7 4 x 10quot 51 39 Ta 72 3 22 120 3 7 x 10 3 quot c 52 371 5077 227 5 1 x 10 2 23 11 2 5 9 5133 22 33 x 10quot 22 Os 32 151 095 7 3 39 x 10quot 3 x 10 quot 11 2 373 04543 1 7 93 x 10quot 1 1 11 32 527 05054 19 2 1 x 10 10 7A 32 1000 04525 17 25 x 10 59 x 10quot m39Hg 32 132 17775 55 19 x 10quot 44 x 10quot quot131 92 1000 42342 152 14 x 10quot 733 x 10quot 392 h1an 39 h r r a 39 39 39 map cerium have potentially useful NMR isotopes 011 Fuller I Phyx c1122 Ref Other Dam 5 335 1975 except whereotherwiscstated Aspin 12 isotope also exists s1 39 R Neumann F magquot J Kowalski and Gzu Pullitz z PhysikA279ZA91976 I s ButlgenbachRDicke HGebauer R1ltu1men and F maker 2 PI1ysikA285 125 1973 c Br vard and 1 Granger J Chem Phys 75 417511931 NMR Spectrometers Typical now are 51OT magnets must be very stable and highly homogeneous RF transmitter and pickup coils Decoupler Recording device computer for Fast Fourier Transforms FlD storage More Experimental Aspects Must lock on a resonance frequency of a nucleus that is not the one being measured 1H NMR uses D to maintain stability Probe is at the center of the eld and consists of a sample holder with coils The radiofrequency of the nucleus under investigation is applied to the sample via the transmitter coil and at resonance a voltage is induced in the receiver coil which detects sample magnetization The sample is spinning during the experiment to ensure optimum eld homogeneity The sample holder acts as a turbine driven by compressed air Most instruments have another coil used as decoupler and a variable temperature device Macroscopic Magnetization The molar bulk nuclear magnetization H 0 applied eld 7 3kg M NA 10 1H0 2 2 gN N 11 1H0 NA 3kg This applies to a mole of nuclei with spin I and is derived in exactly the same way as for Curie s law for electronic magnetic moments It is very small since yN MB1836 Resultant Magnetization At equilibrium there is no transverse component to the magnetization In many instances the behavior of MO is envisioned as it evolves in NMR experiments m12 Lower energy Higher energy m 12 lggo Sec 112 Basic Pulse Experiment ltagt Assume one type of nucleus a Begin at equilibrium magnetization for the applied field b l b Perturb the system C with a transverse RF 00 pulse and detect emitted radiation as system returns to equilibrium 13 a A b Basic Pulse Experiment Assume one type of nucleus a Apply a perturbing 19 field H1 oscillating at angular frequency 00 along the xaxis H1 can be described as a superposition of two fields rotating away from the xaxis We can restrict attention to the circular component moving in the same direction as the precession of M Basic Pulse Experiment My Fig 15 The relative orientations of the magnetization M its xy component Mxy and the rotating radio frequency eld H1 H1 oscillating at angular frequency 00 in the laboratory frame follows M s motion around the z axis and applies a torque to M that drives M into the xy plane lf H1 is applied as a pulse for time t a few us the tip angle 6 through which M is tilted is 6 yHltp Usually the time of the pulse is referred to by the tip angle ie as a 7t2 pulse for example 14 Data Acquisition h C 2 m 19 One second or the FID and 5 mm containing ones is 30 Hz Hz from the central frequency to Fourier transformed spectrum from b Afterthe pulse is applied data is collected as Free Induction decays Fle until the signal begins to descend into the noise Fle are Fourier transformed to pull our their characteristic frequencies ortheir differences with respect to a reference frequency Basic 1Pulse FT NMR Experiment Acqnmnan Tum Pulse Width Recycle Delay 2v Pulse Width l mwgl Time 4gt Figural Schm ckcmma mofalh nnmlixpuimm Summary After the pulse the timedomain FID data is collected over the acquisition time Then there is a delay time to allow the spin system to return to equilibriumthen another pulse httpwwwscs uiuc ed umainzvBasicsbasics htm Peak intensities and Relaxation times The intensity observed for a given resonance IA is be proportional to the extent to which the pulse induced magnetization decays to its equilibrium value A oc MO MZO1 eTT1A A I 1 6 7T13 I A X B B 1 eTT1A MO is the equilibrium longitudinal magnetization MZO is the longitudinal magnetization after the perturbing pulse 1 is the acquisition time and TIA is the spinlattice relaxation time for nucleus A See Canet Chapter 4 1 fiQiall Inversion f Recovery Method TPF r l b c d e JL it if T A Pulse sequence for determining the 13C spinlattice relaxation time T1 by the inversion recovery method with continuous 1H BB decoupling 1803 T 903 FID The pulse durations along the time axis are not to scale for a 180 pulse IP is several us whereas 77 is of the order of seconds B The vector diagrams a to e and the signals below them Show for five different values of T the effects of spin lattice relaxation on M and on the amplitudes of the signals obtained after applying a 90 pulse and performing the Fourier transformation of the FID 16 323 CHZCH3 7 ili A J 1 15 r L 10 1443 lrzsg 297 138 1286 1281 Figure 73 2263 MHZ 13C NMR spectra of ethyibenzene 1 recorded by the inversion recovery method Fig 72 with r 1 5 10 1530 50 and 100 s 1 More on Spinspin Coupling The nuclear spin Hamiltonian for two interaction nuclei A and X is if hvA1f4 vX1 hJAZIA IX where I 1 and I g are operators for the zcornponent the spin on A and X and J AX is the coupling constant for A interacting with X o This Hamiltonian will yield the energy expression given earlier when chemical shift differences are much greater than couplings vA vBl gtgt JAB 1397 istOrder Spectral Case 7f7hvA1vX1j hJAZIAIX W lliligt 42 WV ilJAcAf saw wa 2cxr aAgt1vn it 1 H ewxrcpianAJAX ifsagtiafsgt lt c 7c vn lap 2cArcXvnr AJAX 1 Fi gap cxgtlvn 71J2aAcxvn VAJAX W limo wn sh ldd r mg 615 i pif n In the firstorder case 11m where vA vX gtgt I I H H JAX mixing ofthe H H spin functions is neglected d 0 d S t l C 2 r er peC ra ase 1 2lt6Acxgt1vU AIAX w m hJAXIA 1X 1 r r z z 2 hJAXE1A1X IAIX IAIX 4 1 sin9l0 gt cosQl a 2ltaxrcAgt1vw AJAX ifw ux uAWEI Viiquot wow Vr J law 2 A x u AAX coseiamismei g mixes 04 and ma 11 5an 71 6 a V J i gt i gt 2 A x n A Ax w own All 5 5323553 33 In the 2ndorder lslrorder 23nd70rd26r caSeY IVA VXI quotjJAXY Ii 2I 1I 4 I1 miXing ofthe spin functions leads to H H intensity changes I N 1 7 stQ See Drago Chapter 7 1 1 HM the rooftop effect and line shifting Range of cases lt AV gt 2 3 lt Ap gt I 2 3 4 JAX JAX 1 JAB JAB 4 AX System AllJ Large AB System AvJ 2 4 Ay gt 2 3 1 4 pl 4 H JAB JAB JAB JAB l L 11 14 AB System AVJ 4 AB System AuJ Very Small FIGURE 1211 Calculated spectra oftwoSpin system transRhCOCPh2PCH2POPh223 31P1H First and Secondorder Coupling PO A 103Rh 12 abundance 100 19 Coupling to Quadrupole A Nuclei Calculated band shapes for a spin 12 nucleus 1H coupled to a spin 1 nucleus MN The shape depends on the ratio of 14N 39 relaxation rate to the NH coupling For very fast relaxation just a single line is observed AAA while for slow relaxation there are three lines of equal intensity Because spinspin coupling Correlations like the wellknown Karplus relation shown here ultimately derive from the manner in which s electron density is involved in wavefunctions that spread over two coupled nuclei 12 l I l l l l T l l l l l l I l l l JHH HZ l 1 L 1 l l 1 100 120 140 160 180 Fig 46 The vicinal Karplus correlation Relationship between dihedral angle and coupling constant for vicinal protons induces spin wavefunction mixing the relaxation of a quadrupole nucleus is partially transferred to spin12 nuclei J L to which it is coupled Coupling Structural Correlations For 3bond coupling constants the empirical relation here is useful 3JXY AcosZ Bcos C 20 Cou pl i ng Correlations ultimately derive from the manner in which s electron density is involved in wavefunctions that spread over two coupled nuclei I b F9th CSPPhZH flanslngH m m JPaPc Jan JPaPc co P JPaH 47 prpc prpc Mquot 21 J c I I 14 P H H 1th l l 39 T Fig 219 The size of the coupling constant is a good indication of the interbond angle being smaller between similar groups in a cis orientation than when the coupling nuclei are mutually trans Here the P NMR spectrum of a palladium hydride complex is shown The larger couplings are between trans groups From data In Braunstein et a Inorg Chem 1992 31 411 Not so basic Spectra 13C 2 2 2 2 l 150 140 130 Cis isomer 513Clppm transisomer 3i3Cppm i O N 393 N II I C N NRr N O NquotminquotJumN O C N ON i No CI 21 Misc Coupling Examples a b Coupling of spin12 nuclei works in pretty U much the onz same way in r 1 a 31P spectrum of POMe and b 2S Si spectrum of 3 all cases SiMe4 In the second case the outermost lines are too weak to be seen Weak lines between the strong ones arise from small amounts of 13 C SiZH6 1H spectrum 298i 47 abundance l JSIH l V 2J SOHz SH 22 Coupling to nonspin12 Nuclei Wt 100 Hz l t Fig 214 H NMR spectrum of GeH4 The ten evenly spaced lines are due to the 8 of the molecules which contain 73Ge 1 92 The intense central line arises from all other isotopic species 1st and 2nd order spectra 10 Hz 73Ge I 92 8 abundance couples to equivalent protons in 73GeH4 1H spectrum l L M l 14 81Hppm b a 360 MHz 1H spectrum for I I l l l L 32 30 28 26 39 15 14 81Hppm CHsCHZSPFZ 1St order J J b 80 MHz 1H spectrum for CHsCHZSPFZ 2nol order 23 Zr6BCI12MeCN6XCIX139X 11B NlVlR spectra for RbSZr CllgB dissolved in acetonitrile with a no added ligand b 6 eq added TlPF and c 10 eq added PPNCl Labeling on the peaks x 0 6 indicates the number of tenninal chlorides in Zr BCll2NCCH36XC1X 1 Zr6BCI12PEt36 b 31p 3 0 36 42 48 5 4 ppm a B 200 199 198 197 196 PPm a 11B spectrum of ZraBCI12PEt36 plus an impurity b 31P spectrum of ZraBCI12PEt36 plus an impurity What is the impurity 24 ZchCI12M90H6xpyx 11B spectrum of 3 Zr6BCI12MeOH6 1 in methanol solvent 4 to which pyridine was added until the 5 0 pyridine mole MW imam fraction was 5 Tricky Spectrum ll rr rIrrr 800 700 600 600 700 Hz l 800 19F NMR spectrum of SPF22 The spectrum is centrosymmetric and half of the total intensity falls in the two most intense lines which are truncated in the figure The two very weak lines are spinning sidebands 25 A Typical Simple Structural problem 24 The 13C NMR spectrum of the carbonyl groups in one isomer of WCO4POMe3SPh is shown in Fig 246 Identify the isomer and account for the form of the spectrum What would you expect to see in the 13C spectrum of the other isomer The gure is adapted with permission from D Darensbourg KM Sanchez and J Reibenspies Inorg Chem 27 3636 1988 Copyright 1988 American Chemical Society lillllillllilt 210 205 200 alVSClppm Fig 246 Chemical Shift Equivalence isochronous nuclei Hc 1R3Rd d Two Isomers of13 dibromo13 diphenylpropane How many chemicalshift equivalent sets of aliphatic protons in each isomer 26 Magnetic lnequivalence p uoronitrobenzene No2 Hc Hd Ha Hb F difluoroethylenes Fb Hb la FD la Hb Magnetically equivalent nuclei are coupled equally to every third nucleus in the spin system In each molecule the protons are chemical shift equivalent Which sets are magnetically equivalen Notation for labeling nuclei label eg A eg AA for two AA A forthree etc Chemically and magnetically equivalent nuclei given the same Chemically but not magnetically equivalent nuclei given primes Nuclei with small chemical shift differences smaller or comparable to coupling between them given alphabetically close labels eg A B c D etc lnequivalent nuclei with large chemical shift differences given alphabetically distant labels eg AMX for three One can describe a mix of types of nuclei AZBMXY is a sixspin system with two equivalent nuclei A strongly coupled compared to AVAB to one nucleus B weakly coupled to one nucleus M vs AVAM and very weakly coupled to X amp Y M is weakly coupled to all nuclei Bis only strongly coupled to A X amp Y are strongly coupled to each other and weakly to all other nuclei 27 Examples ignoring phenyl groups 1R3Rdl Ho Hd Ph AA39MM39 ABM2 Examples P uoronitrobenzene No2 c Ha AA BB in a sufficiently high eld AA MM Ha Hb F difluoroethylenes Fb Hi Fa Fl Fa Hi all AA XX AA MM equally good 28 More Examples 31P PMe3 Cl PA Cl C PEt3 eCI PB M M 82 ivi M all Baal all PBI PMe3 Cl PA Cl Both 31PMe3 couple to each of the 31PEt3 in an equivalent fashion and vice versa More Examples PMea PMea Cl CI I PA 039 C 33 Cl M Cl ls CI M E Me P MLPEI M M 4 3 I 3 Cl lMeaP I PBS oi PA PEta Cl CI PB Cl AA BB The 31PMe3 couple to each of the 31PEt3 differently and vice versa magnetically inequivalent Sample data JAB J 104 Hz JAA J 20 Hz JBB JBB 192 Hz JAB JAE JEN J 23 Hz 29 Expected 31P H Spectrum PA or PAv PB 01 PB39 PA CI s CI s PBI CI l PAl l JAB or PB Cl JA39B39 AA BB JAB39 JBA39 Sample data J J JABJAB104Hz AA39 3339 JAA JAA 20 Hz FE F JBB JBB 192 Hz I I II I JA BZJAB ZJBA ZJB A223HZ two sets of multiplets 2391d order 13 1H AMX can be complicated X 73 HL 47le gt JMx 4 JMX x 35 t 30 25 5 ppm vs TMS FIGURE 1212 lOO MHz spectrum ofstyrene oxide 25397 in CCI The par oflhe spectrum due to aromatic proton is not 3 ow 30 Time Scale AE yNhHO th 739N 27 L H w7NH0 0 O LN including chemical shift v H01 o NMR is generally considered to be a slow technique with a characteristic time scale I 10 7 s Recall 11 v 100 MHz Molecular events occurring in times much faster than 10 7 s are time averaged in NMR a slightly more careful distinction to comes now i Hindered H Wm H H TfCH3 Rotation CH3 CH3 A 2 site example illustrating different lineshape regimes in a dynamic process amenable to NMR study Fig 240 a Calculated band shapes for various values of exchange rate relative to frequency difference for two equally populated sites 11 Gas phase 393C NMR spectra of NNdimethyl 3CZfurmarnide 2XXXI The spectra recorded at 427 415 410 Kilt Near Fast exchange Coalescence Intermediate exchange 397 and 373 K show two distinct resonances at low temperatures when rotation about the central C N bond is rapid Spectra in b are taken with permission from 3D Ross and NS True 1 Am Chem Soc 106 2451 1984 Copyright 1984 American Chemical Society Slow exchange Stopped exchange ewe 31 n L m Cyclohexarte quotquotm Interconverslon Ne L Sim Anumerzrsme exampte W I mustratrrrg mrrererrt j Hner hape regrm r my 1 magma ad rrarm p cess M amenaptetu NMR L mm 5 WW r r JWW M M I m r A mm m smeaumm Complete Line Shape Analysis Return spectra at mrrererrt temperature r 3 shmrexchanu Enema MM 1 51 e 4Uspecvamcua escenceVEEWE m quot2314351 and1a51 VEmme ectrum m slapped Exchange regrme J s nrrewrutns Avm s s e a r Ppe exchanga 3 sp me Me sp u ate spectra m the cuarescerrce regtme vanmg ks unm upseweu and car mated e tra match RE eata u r s W necessary carcmate spectra m fastrexchange nrmt TC and kc If a the exchange process is firstorder b there is no coupling between exchanging nuclei and c the two singlets have equal intensities For the coalescence temperature T C the rate constant kc is 7rAv k 7 222Av J2 C Av is the chemical shift difference in Hz in the absence of exchange determined at lowest practical temperature Even when a b and c are not exactly fulfilled this is usually a decent estimate for kc TC and kc more When expressed in ppm chemical shifts are field independent But Av is in Hz which means that Av oc H Therefore TC and kC increase with the field used in the NMR experiment and the highest available field may not be the most appropriate Coupling between exchanging nuclei can be handled If nuclei A and B are exchanging and are coupled with constant J AB kc is k sz 222 Av2 6wa C 33 L2diphenyldiazctidinone An example HA with Coupling HSCBJQ O 6H5 HA and H3 are clearly inequivalent at 55 C m in cm W 14 2 but in the fast exchange regime at 35 OC39 53929 Al60 35 C in acetone d have changed if carried out on a 300 MHz instrument Ass 6 60 MHZ 39H NMR signals of the ring protons HA and HB Activation Parameters Use of the Arrhenius plots Ink vs lT are a lot of work since they require data for a range of temperatures many measurements and full lineshape analyses Estimate for AGi using the Eyring equation is easy with TC in hand k T k T k N57eAGiRT 857eAsiReAHiRT N is usually taken to be 1 An Eyring plot lnkT vs lT would be better but again much more work 34 3 How does the ring whiz l l 3 EX FeCpCO2 E 4 3 19 A I H 725 N I H H r 40quot A g A Fe CEO 52 5 H H 5 C 56 A T Cp A H 0 60 l A 4 H2 l H A s4 M l39 A 3 00 A B I H1 B l I I l I I I 4L l 65 60 55 44 40 35 30 Peaks at 6w 60 B amp 63 A are assigned to H314 amp H215 respectively A resonances collapse faster than B resonances Competing Mechanisms At 3 C the COD ethylenic protons and the diastereotopic CH3P are not exchanging At 67 C the COD ethylenic protons are exchanging but the diastereotopic CH3P groups are not At 117 C the COD ethylenic protons and the diastereotopic CH3P are both exchanging 35 Competing Mechanisms At 3 C the COD ethylenic protons and the diastereotopic CH3P are not exchanging At 67 C the COD ethylenic protons are exchanging but the diastereotopic CH3P groups are not At 117 C the COD ethylenic protons and the diastereotopic CH3P are both exchanging Problem Reaction of lF7 and SbF5 gives a 11 product The 19F NMR spectrum contains two sets of resonances one with 6 lines of equal intensity the other with overlapping patterns of 6 lines of equal intensity and 8 lines of equal intensity the former being somewhat stronger Explain 1213b 52 573 1213b 72 427 12752 100 36 Problem What coupling pattern would you expect to observe for a single proton coupled to a one and b two equivalent 11B nuclei c Same as b but with a real sample 11B 32 804 1 B 3 196 Problem The 1H NMR spectrum of GeFH3 consists of two lines separated by 42 Hz What are the relative positions and intensities of all the lines in the 19F spectrum of a GeDFH2 b GeDZFH c GeD3F 37 Problem WCO6 reacts with NaBH4 to give an anionic product Its tetraethylammonium salt has the empirical composition C18H21NO10W2 Its 1H NMR spectrum includes a triplet intensity ratio 161 at 6 225 ppm with a separation between the outer peaks of 42 Hz What can you deduce about the structure of the anion 183w s 12 144 Characterize indicated protons as to chemical shift andor magnetic equivalence H H HaC H quot CCC CHCH ch H lt D 3 5 H H30 H H CHCH32 CI CI H H o CI H30 EH3 H T l 38
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