MOLECULES BONDS. ENERGY LEVELS electronic vibrational rotational. P461 - Molecules 1

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1 BONDS MOLECULES Ionic: closed shell (+) o open shell (-) Covalent: both open shells neutal ( shae e) Othe (skip): van de Waals (He- He) Hydogen bonds (in DNA, poteins, etc) ENERGY LEVELS electonic vibational otational P46 - Molecules

2 Ionic Bonds - NaCl Fist appoximation. Both atoms ae ions (no electon shaing) and bond due to EM foce between two chaged bodies Na + bond Cl Atom valence ionization ~effz adius Na 3s 5. ev.8.7 nm Cl 3s 3p 5 3 ev.9.7 nm A 3s 3p 6 6 ev nm E 3.6Z n A moe tightly bound than Cl. But Cl - looks like A and moe tightly bound then neutal Cl Has effective Z ~ 3 E( Cl need ) E( Cl : (5. 3.8) ev eff ) 3.8eV Na+ Cl P46 - Molecules Na + + Cl

3 Atomic Popeties. nm ev P46 - Molecules 3

4 NaCl potential vs R 3 tems Pauli epulsion at small R shielding of nuclei becomes incomplete electons wave functions ovelap and electons foced to highe states (p 3s) P46 - Molecules 4

5 Ionic Bonds - KCl anothe example. What is the enegy equied to dissociate a KCl molecule into K atom and Cl atom given Ionization enegy K 4.34 ev electon affinity Cl 3.8 ev sepaation distance R.79 angstoms attactive EM potential fo.79 A -5.6 ev takes 5.6 ev to sepaate K+ and Cl- ions to infinity takes 3.8 ev to pull electon fom Cl- to fom neutal Cl gain 4.34 ev when K+ captues electon and foms neutal K ev enegy equied P46 - Molecules 5

6 Ionic vs Covalent As R >>.5 (size of p obit), thee is little ovelap in the electon wave function between the Na and Cl ions mostly ionic bond 94% ionic and 6% covalent (DH makes up numbes) look at HFl molecule H ionization enegy 3.6 ev Fl electon affinity 3.4 ev need. ev in electostatic enegy e.4evnm R. 4nm πε U.eV 4 as the size of filled p in Fl is about.5 nm and the nominal s in an H atom is.5 nm, the electons ae attached to both atoms covalent bond % ionic and 9% covalent (DH made up numbes) the nuclei will stat to not be shielded fom each othe some ++ epulsion P46 - Molecules 6

7 Covalent Bonds - Diatomic Molecules assume all valence electons ae shaed often S lowest enegy but not always (Oxygen is S) if both atoms ae the same then ψ same if switch atom() and atom() --- electon densities aound each atom ae the same (even sot of holds if diffeent atoms like CO) H(s) <-- vey fa apat ---> H(s) close togethe H( s )H( s ) electon wavefunctions ovelap - shaed two enegy levels (S,) which have ψ (,) ψ (,) E Rinfinity (atoms) s*s bands Vib and ot P46 - Molecules 7

8 Covalent Bonds - Hydogen even if only electon, bond is covalent look fist at ionized diatomic H H + have epulsive potential between potons depends on R p-p sepaation (about. nm) e 4πε R V pp but then have attactive enegy V e fo the electon. It will depend on R (and one calculates R by looking at the minimum of V e + V pp ) -3.6 V e lowest enegy states P46 - Molecules 8

9 Covalent Bonds - Hydogen guess at a 3D solution fo the wave function. Must not depend on vs fo two atoms. Only one electon and, ae locations of two potons ψ (,) ψ (,) (, spatial) at lage sepaation just two H atoms ψ S ( H ψ (,) ) ± e A( e / a two possibilities: symmetic and antisymmetic when the sepaation becomes small / a ± e / a ) ψ ψ ψ symmetic p p ψ antisymmetic lage sepaation small sepaation P46 - Molecules 9

10 Covalent Bonds - Hydogen+ symmetic wave function has lowe enegy less cuvatue. node vs nodes (compae to paticle in a box) also geate shielding of potons fom each othe as highe pobability fo the electon to be between the two potons (antisymmetic goes to at midpoint) can extapolate to R --- symmetic becomes a S state of He and antisymmetic (with wavefunction at oigin) becomes a P state total E V pp + E e detemine this as a function of R intenuclea sepaation. Find thee is a minimum fo symmetic but not fo antisymmetic covalent bond P46 - Molecules

11 Covalent Bonds - Hydogen+ E- P46 - Molecules

12 Enegy Levels fo given electonic states (s,3p, etc S, S) detemine effective V(R) and see if a minimum (bound state) exists as NOT V() potential, Sch. Eq. Not sepaable into (THETA,PHI) pats L not eigenfunction, L not good eigenvalue but often phi symmety L z m good will then have H.O. vibations aound minimum Rnuclea sepaation V P46 - Molecules

13 Neutal Hydogen Molecule H H + +.7eV + 4.7eV moe tightly bound with electons. Have: additional shielding of potons (lowe E) e-e epulsion (highe E) end up: R.7 nm (compaed to about.9 nm with single electon) the size of a H atom is about.5 nm and so the s wavefunctions of the atoms ae ovelapping and need to use Femi-Diac statistics Pauli exclusion and a totally antisymmetic wavefunction H H + + H H + ψ ( e, e) ψspace ψspin ψ ( e, e ) if S ψ if S ψ spin spin sym, ψ space antisym, ψ antisym space sym P46 - Molecules 3

14 Neutal Hydogen Molecule the antisymmetic space has ψ when gives: lowe electon pobability between potons less shielding highe enegy in this case (and in most cases) have covalent bond when electons ae paied with antipaallel spin S S E E e E e + V + pp R pp S P46 - Molecules 4

15 Covalent Bonds #bonds #unpaied electons O : H O N : 3 NH 3 C : CH (s p ) o C : 4 CH 4 (s p 3 ) can numeically detemine wave functions. Lots of appoximations; often use elliptical coodinates can cove in physical chemisty. Now thee is a cottage industy poviding calculations fo often complicated molecules. Often need some guiding by hand. Use exp(- ) fo electons instead of exp(-) as faste to calculate pp p equal electon pobability contous P46 - Molecules 5

16 Molecula Bonding compute code can now calculate many molecula bonds. A common one is called GAUSS to save time usually set up by hand configuations. p P46 - Molecules 6

17 Covalent p Bonding bonding in N and O (and sot of in molecules with C and othe atoms) depend on p obital shapes 3 diffeent 3p wave functions. Along x y and z diections. (If have sp 3 then along 4 legs of tetahedon) Fo covalent bonds with identical atoms, symmety equies that both atoms have the same electon distibutions. So ψ will have xx,yy,zz combinations only. Let x be the axis of the molecule and y,z be pependicula. y,z give same enegy eigenvalues and so can have mixing φsymmety p P46 - Molecules 7

18 Covalent p Bonding just based on symmety between the atoms electon distibutions, can sketch out the bond and antibonding wave functions. If electons ae between nuclei lowe enegy BOND YY,ZZ ANTIBOND N N N N BOND XX ANTIBOND p N N N N P46 - Molecules 8

19 Covalent p Bonding give enegy levels which ae then filled enegy odeing changes between O and N as diffeent electon distibutions usually think of covalent bonds as S but have S fo O N O x P anti x y z y z x P bond y z y z x p S antibond S bond S P46 - Molecules 9

20 MOLECULAR ENERGY LEVELS Have Schod. Eq. Fo H (same ideas fo moe complicated). Fo poton and electon, [ ] h m p + p+ e+ eψ + [ V + V + V + V + V + V ] ψ Eψ pp ee ep ep ep ep eal solution: numeic. But solve by sepaating ψ ψ electonic ψ otational ψ vibational these ae faily independent. e.g. electonic sets potential well vs distance between potons vibational modes (which don t eally change sepaation by much). Rotational also set by distance E E e ot visible, UV, E vib micowave, IR P46 - Molecules

21 Rotations Go to cente-of-mass fame fo two atoms M M R E ot with R c. m. R M µω R µ wite down Schod. Eq. Fo otational enegy H spheically symmetic in cm fame. Rotationally confined to sphee R (E&R Ch 7-6) ψ L ot ot ψ ot Y m ( L I M + m I R mm m µ µ v L ψot Eotψot ( V I ( θ, φ) + ) h spheical intege hamonics ) L z mh < m < int ege P46 - Molecules

22 Rotations As know angula momentum eigenvalues E ot E E L I E ( + ) h I h I spacing use NaCl as example. R.4 nm 3 35 µ h ( hc) I µ c R 3.9 (97eV fm) MeV (.4 [ ] h ( + ) ( ) fm) kt.5 ev fo T3. Easy to be in an excited state and elative amount is anothe way to measue tempeatue 3.5eV I P46 - Molecules

23 find n n Ex. - and Pob -5a # quantum states # quantum states ( + ) e # in level # in level ( + ) h / IkT e e E E / kt / kt -5a which level is most highly populated? n pob h (+ ) Ikt IkT.5 ( ).5 h # states E.5 /.5.5 e n h I E / kt ( + + ) P46 - Molecules 3 7 NaCl at T3

24 Absoption/Emission-Rotation occus if molecule has an electic dipole moment (if not will have electonic-vibational-otational) can patially calculate using 47 EM. QM mechanics selection ules simila to atoms d+- (can have d+- in highe ode) E h E ( + ) (above v/e) I ( + ) E R ER 3E R E.4eV 3 6E ν R 3 h 4. ev Hz equally spaced absoption enegies / sec but if lage angula momentum (lage ), not symmetic and R sepaation inceases, I inceases, enegy spacing changes P46 - Molecules 4

25 Absoption/Emission-Rotation Diffeent isotopes have diffeent mass, diffeent moments of inetia I enegy shift Cl vscl µ % diffeent excited states (both electonic and vibational) will have diffeent sepaation between the molecules. So diffeent I and diffeent de and photon enegy spectum has boad, ~continuous spectum with absoption peaks supeimposed E γ P46 - Molecules 5

26 Spectum and Molecula Popeties can use measued spectum to detemine molecula popeties CO molecule. Measue enegy spacing of absobed photons h I R hν µ ν h. πνµ Hz measues aveage sepaation between the C + atoms 6.9 I m µ R P46 - Molecules 6

27 Molecula Vibations E tot R sepaation minimum in sepaation distance can be appoximated as a paabola PHYS46 gave QM solutions to Schod. Eq. Fo hamonic oscillato let vvibational quantum numbe,,... E vib ( v+ Etot C R R NaCl : hν ) hν R.4eV ν π cuvatue of HCl C µ : hν paabola.4ev P46 - Molecules 7

28 Molecula Vibations Ex. -3. Knowing the foce constant in HCl, find the photon enegy (in eality use measued photon enegy to undestand the shape of Etotal) foce cons tan t 47nt / m 9eV / fm µ m m H H mcl + m Cl m H 93MeV / c E γ hν h π C µ hc C µ c 97MeVfm 9eV / fm 93MeV.35eV P46 - Molecules 8

29 Rotational-Vibational Enegy Levels The # of otational levels only fixed by the top of the finite enegy well diffeent vibational levels can ovelap ) h C + µ E v ( v+ µ ( + ) a often just called otational-vibational band spacing (between vibational levels and otational levels) will vay as move to the top of the well. The electon distibution changes and so aveage sepaation changes. Well non-symmetic at oom T, most molecules in lowest vib. State n e e n # states # states hν / kt.35/.5 7 well is finite limited numbe of vibational states (~4 fo some befoe dissociation) h P46 - Molecules 9

30 Molecula Specta P46 - Molecules 3

31 Electonic Enegy Levels Electons can be in highe enegy states (equivalent to p 3s, 3d etc) can still have a molecula bond as long as a minimum in the E total vs R distibution the potential well tends to be shallow > fewe vibational modes > diffeent vibational and otational enegy spacing as diffeent moment of inetia (spacing) and sping constant P46 - Molecules 3

32 Absoption and Emission Spectum Will depend on the electic dipole moment (edm) If edm (in symmetic molecules) mostly need to have electonic tansition fo non-zeo matix element and theefoe tansition in UV if asymmetic (CO, etc) then have non-zeo edm. Can have pue otational (micowave) and vibational-otational (IR) tansitions obtain selection ules fom petubation theoy finalv pet simila wavefunctions and so same H.O. and angula momentum selection ules as in 46: otational vibational initial v ±, ± if change in electonic state, the vib. and ot. wave functions in the two states ae often vey diffeent and beaks the selection ules P46 - Molecules 3

33 Electonic Tansitions Use Fank-Conden and Heitle-London pinciples compae initial and final state wavefunctions. Want them to be simila and ovelap in space otational and vibational selection ules can be boken matix element. Need to wok out integals v fi ψ * f v pet ψdvolume i E R Geen ovelaps oange doesn t P46 - Molecules 33

34 Photon Scatteing Both light passing though a gas and a technique γ in M b ~M a M a γ out if photon is at a esonant fequency. Then M b is an excited state and the outgoing photon enegy depends on the details of the enegy levels. If photon not at a esonance, M b is vitual. But electic-dipole selection ules hold at each vetex ± ± each vetex v v ( Rayleigh), ± fo o ( Raman) both P46 - Molecules 34

35 Molecula Symmety Effects Identical nuclei O, H etc ψ (,) spin spin N N ψ (,),,..( Boson) ψ, 3 M..( Femion) ψ M symmetic antisymmetic diffeent components of molecula wavefunction ψ ψ ψ ψ M elec vib ot ψ electonic symmetic symmetic ( ) ψ vibational Paity the symmety of the nuclea pat of the wavefunction will depend on the nuclea spin combination (same as in atoms) e R ψ otational x x nucleus e covalentbond ψ P46 - Molecules 35

36 Molecula Symmety Effects Femionic Nuclei H : total wavefunction must be antisymmetic. So otational*nuclea odd combine nuclea spin S S S N N N ψ S ( tiplet ) ψ ψ N otation (sin otation + S N N antisymmet ic glet) ψ both N symmetic S symmetic antisymmet ic even odd otho paa othosymmetic nuclea spin paaantisymmetic A given molecule is eithe otho o paa. It stays that way fo a long time since vey weak intenuclea foces which might flip the spin. Raman scatteing peseves this as d look at spectal lines and count numbe of odd vs even tansitions gives nuclea spin P46 - Molecules 36

37 Molecula Symmety Effects obseved tansitions Femionic nuclei * 5 * Bosonic nuclei * 4 * * 3 * * * * * * * otho paa paa otho othosymmetic nuclea spin paaantisymmetic look at spectal lines and count numbe of odd vs even tansitions gives nuclea spin. Each tansition unique enegy h E ( + ) I h E E+ E (4+ 6) I P46 - Molecules 37

38 Otho vs Paa in Diatomic Molecules Assume identical nuclei each with Si (i+)*(i+) total states i+ - both m ae the same symmetic have (i+)(i+)-(i+) i(i+) emaining states: half symmetic and half antisymmetic total # symmetic (i+)+i(i+) (i+)(i+) total # antisymmetic i(i+) # otho # sym i + # paa # antisym i Example: spin / nuclei S,,, Sz Sz, Sz ( ), +, ( ),, + 3,, P46 - Molecules 38

39 Molecula Symmety Effects Bosonic Nuclei O, N etc. have totally symmetic wavefunction nuclea-otational: sym-sym o antisym-antisym O N o S S Ni onlyψ : : ψ ψ Ni otation S N otation otation S S N N symmetic symmetic antisymmetic antisymmetic symmetic symmetic even even odd O(6)-O(6) o C()-C() molecules can only have even states. C(3) discoveed by seeing fobidden ( 3) tansitions in Raman scatteing N(4)-N(4) even tansitions (, etc) most intense odd i even i+ +, P46 - Molecules 39