Alkali metals show splitting of spectral lines in absence of magnetic field. s lines not split p, d lines split

Transcription

1 Electron Spin Electron spin hypothesis Solution to H atom problem gave three quantum numbers, n,, m. These apply to all atoms. Experiments show not complete description. Something missing. Alkali metals show splitting of spectral lines in absence of magnetic field. s lines not split p, d lines split Na D-line (orange light emitted by excited Na) split by 7 cm - Many experiments not explained without electron "spin." Stern-Gerlach experiment early, dramatic example. Copyright Michael D. Fayer, 07

2 Stern-Gerlach experiment Magnet glass plate oven Ag baffles Ag beam N S m s +/ -/ Beam of silver atoms deflected by magnetic field gradient. Single unpaired electron - should not give two lines on glass plate. s-orbital. No orbital angular momentum. No magnetic moment. Observed deflection corresponds to one Bohr magneton. Copyright Michael D. Fayer, 07

3 Assume: Electron has intrinsic angular momentum called "spin." s square of angular momentum operator S sms sm S sm sm z s s s projection of angular momentum on z-axis s e/mc s s e mc e mc Electron has magnetic moment. One Bohr magneton. Charged particle with angular mom. Ratio twice the ratio for orbital angular momentum. Copyright Michael D. Fayer, 07

4 Electronic wavefunction with spin ( x, yz, ) ( x, y, zm, s ) with spin ms sm s spin angular momentum kets ( m s ), Copyright Michael D. Fayer, 07

5 Electronic States in a Central Field Electron in state of orbital angular momentum Y m (, ) m label as orbital angular momentum quantum number m s spin angular momentum quantum number (/), ( /) m mm Y m s s product space of angular momentum (orbital and spin) eigenvectors Copyright Michael D. Fayer, 07

6 mm s Simultaneous eigenvectors of z L L S S z L Lz m S 4 constant - eigenvalues of S are s(s + ), s = / S z m s linearly independent functions of form mm s. Copyright Michael D. Fayer, 07

7 Example - p states of an electron Orbital functions = orbital angular momentum part 0 Y Y 0 m ( r,, ) R( r) Y (, ) radial spherical harmonics Y j m j m j m spin functions j m j m jm m jm sm s Copyright Michael D. Fayer, 07

8 In the m m representation Y s m m s m m s Y Y Y Y Y Each of these is multiplied by R(r). jj s Copyright Michael D. Fayer, 07

9 Total angular momentum - jm representation Two states of total angular momentum j j j s j j j j j s The jm kets are Copyright Michael D. Fayer, 07

10 0 0 m m j m j = j = / Table of Clebsch-Gordon Coefficients The jm kets can be obtained from the m m kets using the table of Clebsch-Gordon coefficients. For example 0 jm m m m m Copyright Michael D. Fayer, 07

11 Spin-orbit Coupling An electron, with magnetic moment moves in the electric field of rest of atom (or molecule). W E V energy es mc Then e W E V S mc Using eegrad classical energy of magnetic dipole moving in electric field E with velocity V spin magnetic moment of electron Coulomb potential (usually called V, but V is velocity) and p mv Copyright Michael D. Fayer, 07

12 W ( grad p) S mc Hydrogen like atoms ze Coulomb potential 4 0r unit vector ze r ze grad r 4 r r vector derivative of potential 4 r Substituting so 0 0 ze 80mcr H r p S ( r p) L orbital angular momentum H operates on radial part of wavefunction so ze 8 m 0 c r LS operates on orbital ang. mom. part of wavefunction operates on spin ang. mom. part of wavefunction Copyright Michael D. Fayer, 07

13 One unpaired electron central field (Sodium outer electron) V( r) V( r) gradv ( r) r result of operating on r r radial part of wavefunction V( r) H so LS a( r) LS mcr r Many electron system V( ri ) H so ( ) Li Si ai r Li S mc i ri r i i Ignore terms involving, L i S j, the orbital angular momentum of one electron with the spin of another electron. Extremely small. i Copyright Michael D. Fayer, 07

14 Spin-orbit coupling piece of Hamiltonian H so a( r) L S H-like atom spatial part of operator H so ( r) ze 8 m 0 c r E ( rh ) so( rd ) Normalization constant (eq.7.6) contains / / E zz z / z H little S.O. coupling Br sizable S.O. coupling ar ( ) z 4 heavy atom effect similar for all atoms and molecules Heavy atom effect and external heavy atom effect important in many processes. Copyright Michael D. Fayer, 07

15 H so a( r) L S The a(r) part is independent of the angular momentum states. Consider L S LS L S L S L S S x S S S y S S i Lx L L Ly L L i x x y y z z Rearranging the expressions for the angular momentum raising and lowering operators Substituting LS Lz Sz LS LS diagonal off-diagonal Copyright Michael D. Fayer, 07

16 Spin-Orbit Coupling in the m m representation LS Lz Sz LS LS not diagonal in the m m representation States in the m m representation 0 0 The spin-orbit coupling Hamiltonian matrix in the m m representation is a 6 6. Calculating the matrix elements Kets are diagonal for the LSpiece of the Hamiltonian. For example z z LS z z L z brings out m and S z brings out m s. m m s m m s Copyright Michael D. Fayer, 07

17 L S L S not diagonal Apply L S L S to the six kets. Examples L S LS 0 can t raise above largest values L 0 S LS Left multiply by 0 0 L S LS matrix element Copyright Michael D. Fayer, 07

18 H SO a( r) Lz Sz LS LS put kets along top 0 0 H so = a(r) put bras down left side example LS LS Matrix is block diagonal. Copyright Michael D. Fayer, 07

19 Matrix Block Diagonal Each block is independent. Diagonalize separately. Form determinant from block with (eigenvalue) subtracted from diagonal elements. 0 The two blocks are the same. Only need to solve one of them. The two blocks are diagonal eigenvalues. Expand the determinant. Solve for. Each block multiplied by a(r). / / 0 /4/4 [/ or ] ar ( ) ar ( ) The two blocks each give these results. The two blocks each give /a(r). Therefore, 4 eigenvalues of /a(r). eigenvalues of -a(r). Copyright Michael D. Fayer, 07

20 Energy level diagram 0.5a(r) 0 four-fold degenerate E no spin-orbit coupling E = 6,980 cm - fluorescence p -a(r) two-fold degenerate Na D-line (p to s transition) split by 7 cm -. 7 cm - = / a(r). a(r) =. cm -. s Ratio ar ( ) E 70 4 Ratio small, but need spin-orbit coupling to explain line splitting. Copyright Michael D. Fayer, 07

21 Spin-orbit Coupling in the jm representation The jm kets are However ar ( ) L S is in the m m representation. Can t operate H SO directly on the jm kets. Use table of Clebsch-Gordan coefficient to take jm kets into m m rep., operate, then convert back to jm rep. Alternatively, rewrite operator so it can operate on the jm kets. Copyright Michael D. Fayer, 07

22 We need the operator, ar ( ) L S, which is an m m, in the jm J L S ( ) J L S L S LS So LS J L S /( ) Recall that a jm ket, Jm, is an abbreviation for jjjm. So for the spin-orbit coupling problem, the jm ket is lsjm l is an eigenket of the L operator. s is an eigenket of the S operator. and J is an eigenket of the J operator. Therefore, the jm kets are eigenkets of the L S operator. Copyright Michael D. Fayer, 07

23 ar ( ) L S Operate on L S J L S /( ) ar ( ) LS ar ( )/( J L S) ar J L S ( )/( ) 5 ar ( )/ 4 4 / ar ( ) Therefore, ar ( ) LS ar ( ) is an eigenket of ar ( ) L S with eigenvalue /a(r) Copyright Michael D. Fayer, 07

24 The jm kets are Since /( J L S ) does not operate on the m of the jm kets, lsjm, all of the kets with J = / give the same eigenvalue, /a(r). Then operating on ( )/( ) ar J L S ( )/( ) ar J L S ar ( )/ 4 4 ar ( ) Copyright Michael D. Fayer, 07

25 Applying H SO to all jm kets Eigenkets The jm representation kets are the eigenkets because H SO couples the orbital and spin angular momenta. Coupled representation eigenkets. The jm kets Na - p unpaired electron +0.5a(r) 0 eigenvalues a(r) eigenvalues - -.0a(r) -.0a(r) Copyright Michael D. Fayer, 07

26 The jm representation kets will be the eigenkets whenever a term in Hamiltonian couples two types of angular momenta. Examples: Hyperfine Interaction - I S Coupling of electron spin and nuclear spin. Small splitting in Na optical spectrum. Structure in ESR spectra. Electronic triplet states - coupling of two unpaired electrons. Copyright Michael D. Fayer, 07

27 Have diagonalized H SO in m m representation. Have shown that jm kets are eigenkets. A unitary transformation U mm H so will take the non-diagonal matrix in the m m representation jm into the diagonal matrix in the jm representation. H so H jm m m U H U so so U is the matrix of Clebsch-Gordan Coefficients. Copyright Michael D. Fayer, 07

28 0 0 H so = a(r) m m matrix Both are block diagonal. Only need to work with corresponding blocks. j = j = / j m 0 0 Table of Clebsch-Gordan coefficients m m Copyright Michael D. Fayer, 07

29 Multiplying the blocks U H U 0 SO / 0 0 diagonal matrix with eigenvalues on the diagonal Copyright Michael D. Fayer, 07

30 Electron Spin, Antisymmetrization of Wavefunctions, and the Pauli Principle In treating the He atom, electron spin was not included. Electron - particle with intrinsic angular momentum, spin S / m s / / / / / Copyright Michael D. Fayer, 07

31 Excited States of He neglecting Spin Ground State of He - perturbation theory H ' e 4 r 0 perturbation Zeroth order wavefunctions (any level) Product of H atom orbitals () () () () n m n m n m n m () electron ; coordinates () electron ; coordinates ( r,, ) ( r,, ) Copyright Michael D. Fayer, 07

32 Zeroth order energy 0 Enn 4Rhc n n R 4 e 8 hc 0 - reduced mass of H atom First excited state energy zeroth order E 0 5Rhc since n n and n and n same energy Copyright Michael D. Fayer, 07

33 First Excited Configuration States corresponding to zeroth order energy - 8 fold degenerate s s s p p s s s p s p s x s p p s x y z y z All have same zeroth order energy. All have or n n and n and n Copyright Michael D. Fayer, 07

34 Degenerate perturbation theory problem System of equations (matrix) Form determinant s() s( ) s() s( ) s() ( ) x p x 4 p () s( ) 5 s() ( ) p y 6 py () s ( ) 7 s() ( ) p z 8 pz( ) s( ) J s E' K s K J s E' s J p x E' K px K p J p E' x x J p E' K y p y E' K p = 0 y J p y J p E' K p z K p z z J p z E' E' - eigenvalues J's - diagonal matrix elements K's - off diagonal matrix elements Block diagonal Four x blocks Copyright Michael D. Fayer, 07

35 e Js ()() s s ()() s s dd 4 r e Ks ()() s s ()() s s dd 4 r e J () s p () () s p () d d px x x 4or px x x 4or o o e K s() p () p () s() d d J, K, J, K replace x with y and z py py pz pz Copyright Michael D. Fayer, 07

36 J's - Coulomb Integrals Classically represent average Coulomb interaction energy of two electrons with probability distribution functions () s and () s for J s K's - Exchange Integrals No classical counter part. Product basis set not correct zeroth order set of functions. Exchange - Integrals differ by an Exchange of electrons. e Ks s( ) s( ) s( ) s( ) dd 4 r o Copyright Michael D. Fayer, 07

37 Other matrix elements are zero. Example e s() s() s() pz () dd 0 4 r o odd function - changes sign on inversion through origin Other functions are even. Integral of even function times odd function over all space = 0. Copyright Michael D. Fayer, 07

38 Eigenvalues (E') ) Set determinant = 0 ) Expand Four blocks blocks for p orbitals identical E J K J J J s s pi pi K K K s s pi pi triple root, p x, p y, p z triple root, p x, p y, p z Energy level diagram first excited states n,n =, SP SS P P S triply degenerate triply degenerate S no off-diagonal elements ground state n,n =, S S S configuration more stable. S orbital puts more electron density close to nucleus - greater Coulombic attraction. Copyright Michael D. Fayer, 07

39 Eigenfunctions E J K s s ()() s s ()() s s E J K s s ()() s s ()() s s pi pi s p i pi s E J K () () () () i x, y, z s p p s E J K pi pi () () () () i i Copyright Michael D. Fayer, 07

40 Symmetric and Antisymmetric Combinations + combination symmetric. Symmetric with respect to the interchange of electron coordinates (labels). P P is the permutation operator. P interchanges the labels of two electrons (labels - coordinates) Applying P to the wavefunction gives back identical function times. Copyright Michael D. Fayer, 07

41 combination antisymmetric. Antisymmetric with respect to the interchange of electron coordinates (labels). Switching coordinates of two electrons gives function back multiplied by. P P s () s () s () s () s () s () s () s () s() s() s() s() Copyright Michael D. Fayer, 07

42 Both + and functions are eigenfunctions of permutation operator. Symmetric function eigenvalue + Antisymmetric function eigenvalue All wavefunctions for a system containing two or more identical particles are either symmetric or antisymmetric with respect to exchange of a pair of electron labels (positions). Many electron wavefunctions must be eigenfunctions of the permutation operator. Copyright Michael D. Fayer, 07

43 Inclusion of electron spin in He atom problem Two spin / particles 4 possible states. In m m representation () () () () () () () () The two functions, ( ) ( ) and ( ) ( ) are neither symmetric nor antisymmetric. m m representation not proper representation for two (or more) electron system. Copyright Michael D. Fayer, 07

44 This image cannot currently be displayed. Transform into jm representation In jm representation () ( ) () ( ) () ( ) () ( ) 0 symmetric spin = () ( ) () ( ) 00 antisymmetric spin = 0 Copyright Michael D. Fayer, 07

45 Eight spatial functions. H independent of spin. Each of 8 orbital functions can be multiplied by the 4 spin functions. total (spin orbital) functions. s() s() () () s() s() () () () s p () () () x s() s() () () () () Copyright Michael D. Fayer, 07

46 Secular determinant looks like 8 8 = 0 Block diagonal same as before except the correct zeroth order functions are obtained by multiplying each of the previous spatial functions by the four spin functions. Example ss orbitals Copyright Michael D. Fayer, 07

47 ()() s s ()() s s ()() s s ()() s s ()() s s ()() s s ()() s s ()() s s Totally Symmetric (- spat. & spin sym. 4 spat. & spin antisym.) ()() ()() ()() ()() s s s s ()() s s ()() s s s s s s ()() s s ()() s s Totally Antisymmetric (5-7 spat. antisym; spin sym. 8 spat. sym.; spin antisym.) Copyright Michael D. Fayer, 07

48 Energy Level Diagram Divided according to totally symmetric or totally antisymmetric. Totally Symmetric Totally Antisymmetric sp sp P sp sp P ss ss ss ss S S s symmetric spin function s antisymmetric spin function S No perturbation can mix symmetric and antisymmetric states (H symmetric). Copyright Michael D. Fayer, 07

49 Which functions occur in nature? Totally symmetric or totally antisymmetric Answer with experiments. All experiments States occuring in nature are always Totally Antisymmetric (Example - ground state of He atoms not paramagnetic, spins paired.) Copyright Michael D. Fayer, 07

50 Assume: The wavefunction representing an actual state of a system containing two or more electrons must be totally antisymmetric in the coordinates of the electrons; i.e., on interchanging the coordinates of any two electrons the wavefunction must change sign. Q.M. statement of the Pauli Exclusion Principle Copyright Michael D. Fayer, 07

51 To see equivalence of Antisymmetrization and Pauli Principle Antisymmetric functions can be written as determinants. A() represents an orbital spin function for one electron, for example, s()(), and B, C, N are others, then A() B() N() A() B() N() A( N) B( N) N( N) Expand the determinant gives the antisymmeterized wavefunction. Totally antisymmetric because interchange of any two rows changes the sign of the determinant. Copyright Michael D. Fayer, 07

52 Example - He ground state () s () () s () s () () s (no bar means α spin) (bar means spin) ()() s s ()() s s () s () s () s () s () s ()() s () () s ()() s () ()() s s () () Correct antisymmetric ground state function. Copyright Michael D. Fayer, 07

53 Another important property of determinants Two Columns of Determinant Equal Determinant Vanishes Pauli Principle For a given orbital there are only two possible orbital spin functions, i. e., those obtained by multiplying the orbital function by or spin functions. Thus, that is, no more than two electrons can occupy the same orbital in an atom, and the two must have their spins opposed, no two electrons can have the same values of all four quantum numbers n,, m, m s otherwise two columns will be equal and the determinant, the wavefunction, vanishes. Copyright Michael D. Fayer, 07

54 Example - He ground state with both spins s() s() ()() s s ()() s s ()() s s () s ()() s () () s ()() s () 0 Requirement of antisymmetric wavefunctions is the equivalent of the Pauli Principle. Copyright Michael D. Fayer, 07

55 Singlet and Triplet States Totally Antisymmetric term symbols sp P sp P ss ss S S s S symmetric spin function antisymmetric spin function singlet states, sym. orbital function (single) antisym. spin function. triplet states, antisym. orbital function (three) sym. spin functions. Copyright Michael D. Fayer, 07

56 For same orbital configuration Triplet States lower in energy than singlet states because of electron correlation. Triplet sym. spin. Therefore, antisym. orbital For example: () () T [ s() s() s() s()] ( () () () ()) () () 0 Singlet antisym. spin. Therefore, sym. orbital S [ s() s() s() s()] ( () () () ()) 00 Copyright Michael D. Fayer, 07

57 Singlet - electrons can be at the same place. Consider a point with coordinates, q. S [ s() s() s() s()] S [ sq ( ) sq ( ) sq ( ) sq ( )] D S [ sq ( ) sq ( )] () Correlation Diagram Fix electron. Plot prob. of finding. Prob. of finding electron () at r given electron () is at q. q r (schematic illustration) Copyright Michael D. Fayer, 07

58 Triplet - electrons cannot be at the same place. electrons have node for being in same place. Consider a point with coordinates, q (any q). T [ s() s() s() s()] T [ sq ( ) sq ( ) sq ( ) sq ( )] T () 0 Correlation Diagram Fix electron. Plot prob. of finding. Prob. of finding electron () at r given electron () is at q. () D D q r r q (schematic illustrations) Copyright Michael D. Fayer, 07

59 In triplet state Electrons are anti-correlated. Reduces electron-electron repulsion. Lowers energy below singlet state of same orbital configuration. Copyright Michael D. Fayer, 07