3D Schrödinger Eq. Today: Continue with hydrogen. Multi-electron atoms

Transcription

1 3D Schrödinger Eq. Toda: Continue with hdrogen. Multi-electron atoms HWK 13 available online. Please fill out the online participation surve. Worth 10points on HWK 13. Final Eam is Monda Dec :30A-1P HERE Duane G1B0

2 What is Schrodinger Model of Hdrogen Atom? Electron is cloud of probabilit whose wave function t is the solution to the Schrodinger equation: t t i t V t m 1/ Zke r Zke V where: V r

3 Can get rid of time dependence and simplif: Equation in 3D looking for t: Since V not function of time: / iet e t / iet e E t t i t V t m E V m Time independent Schrödinger Equation:

4 E r V mr r r r m r sin 1 sin sin 1 1 Since potential sphericall smmetric easier to solve w/ spherical crds: r = rsincos rsinsin rcos g f r r R Schrödinger s Equation in Spherical Coordinates & no time: Technique for solving = Separation of Variables / e iet g f r R t r Have ou seen this technique for solving different equations? A. es B. no r Zke r V /

5 In 1D electron in a wire: we got quantiation from appling boundar conditions in terms of. r In 3D now have 3 degrees of freedom: Boundar conditions in terms of r What are the boundar conditions on the wavefunction in r? a. must go to 0 at r=0 b. must go to 0 at r=infinit c. at infinit must equal at 0 d. A and B e. A B and C must be normaliable so needs to go to ero Also phsicall makes sense not probable to find electron there

6 In 1D electron in a wire: Have 1 quantum number n. Need to specif value of n to know what state electron is in. n t t n n In 3D now have 3 degrees of freedom: Boundar conditions in terms of r How man quantum numbers are there in 3D? In other words how man numbers do ou need to specif unique wave function? And wh? We ll ask ou to eplain our reasoning! a. 1 b. c. 3 d. 4 e. 5 L sinn L eie nt / Answer: 3 Need one quantum number for each dimension: r: n : l : m If ou said 4 because ou were thinking about spin that s OK too. We ll get to that later. r

7 In 1D electron in a wire: Have 1 quantum number n In 3D now have 3 degrees of freedom: Boundar conditions in terms of r Have 3 quantum numbers n l m r nlm r R r f g nl Shape of depends on n l m. Each nlm gives unique lm m p n= l=1 m=-101 n=1 3 = Principle Quantum Number l=0 1 3 = Angular Momentum Quantum Number =s p d f restricted to 0 1 n-1 m = = -component of Angular Momentum restricted to l to l

8 Comparing H atom & Infinite Square Well: Infinite Square Well: 1D V = 0 if 0<<L otherwise H Atom: 3D Vr = -Zke /r r Energ eigenstates: E n n ml Wave functions: n n t 0 L L sin e n n L ie n t / nlm nlm Energ eigenstates: E n Wave functions: r R r f g r t mz nl k n nlm e 4 lm r e m ie n t /

9 What do the wave functions look like? nlm r R r f g n = 1 3 l restricted to 0 1 n-1 m restricted to l to l 1s s 3s Increasing n nl Increases distance from nucleus Increases # of radial nodes 4s l=0 4p l=1 4d l= Increasing l Increases angular nodes Decreases radial nodes lm m=-3 4f l=3 m=0 m=3 m Much harder to draw in 3D than 1D. Indicate amplitude of with brightness. See pictures: Changes angular distribution

10 Shapes of hdrogen wave functions: nlm r R r f g nl lm Look at s-orbitals l=0: no angular dependence n=1 n= m

11 Higher n average r bigger more spherical shells stacked within each other more nodes as function of r n=1 l=0 n= l=0 Probabilit finding electron as function of r n=3 l=0 0.05nm Radius units of Bohr radius a 0

12 Shapes of hdrogen wave functions: nlm r R r f g nl lm l=1 called p-orbitals: angular dependence n= l=1 m=0: p = dumbbell shaped. l=1 m=-1: bagel shaped around -ais traveling wave l=1 m=+1 m n l 1 m a 3 0 r a 0 e r / a 0 3 cos 4 n l 1 m a 3 0 r a 0 e r / a sin e i w/time dependence e im+it/h Superposition applies: p =superposition addition of m=-1 and m=+1 p =superposition subtraction of m=-1 and m=+1 Dumbbells chemistr

13 Phsics vs Chemistr view of orbits: p wave functions Phsics view n= l=1 Dumbbell Orbits chemistr m=1 m=-1 m=0 p p p p =superposition addition of m=-1 and m=+1 p =superposition subtraction of m=-1 and m=+1

14 Chemistr: Shells set of orbitals with similar energ 1s s p 6 p p p 3s 3p 6 3d 10 These are the wave functions orbitals we just found: n=1 3 = Principle Quantum Number E n E 1 / n l=s p d f = Angular Momentum Quantum Number =0 1 3 restricted to 0 1 n-1 L l l 1 m = = -component of Angular Momentum restricted to l to l L m for Hdrogen same as Bohr

15 n=1 3 = Principle Quantum Number E n E 1 / n l=s p d f = Angular Momentum Quantum Number =0 1 3 restricted to 0 1 n-1 m = = -component of Angular Momentum restricted to -l to l L l l 1 L m for Hdrogen same as Bohr An electron in hdrogen is ecited to Energ = -13.6/9 ev. How man different wave functions nlm in H have this energ? [graded indep. but use groups] a. 1 b. 3 c. 6 d. 9 e. 10

16 An electron in hdrogen is ecited to Energ = -13.6/9 ev. How man different wave functions in H have this energ? a. 1 b. 3 c. 6 d. 9 e. 10 n= Principle Quantum Number: l=restricted to 0 1 n-1 m=restricted to -l to l n l m s states 3p states l=1 3d states l= E n E 1 / n Answer is d: l=01 n=3 9 states all with the same energ Isn t this cool Chemists had alread figured out rules for how man electrons could be in each shell. Didn t know wh. Solving Schrödinger equation eplains WHY!

17 Energ Diagram for Hdrogen l=0 s l=1 p l= d n=3 3s 3p 3d n= s p In HYDROGEN energ onl depends on n not l and m. NOT true for multi-electron atoms! n=1 1s l=0m=0

18 n=1 3 = Principle Quantum Number E n E 1 / n l=s p d f = Angular Momentum Quantum Number =0 1 3 restricted to 0 1 n-1 m = = -component of Angular Momentum restricted to -l to l L l l 1 L m for Hdrogen same as Bohr What is the magnitude of the angular momentum of the ground state of Hdrogen? a. 0 b. h c. sqrth d. not enough information Answer is a. n=1 so l=0 and m=0... Angular momentum is 0

19 Schrodinger finds quantiation of energ and angular momentum: n=1 3 l=0 1 3 restricted to 0 1 n-1 E n E 1 / n L l l 1 How does Schrodinger compare to what Bohr thought? I. The energ of the ground state solution is same II. The angular momentum of the ground state solution is different III. The location of the electron is different a. same same same b. same same different c. same different different d. different same different e. different different different Bohr got energ right but he said angular momentum L=nh and thought the electron was a point particle orbiting around nucleus.

20 Solved S s equation for hdrogen: wave functions energies angular momentum In atom with multiple electrons what do ou epect to change in the wa ou set up the problem? and in the solutions? Student Ideas: A. B. C. D. E. F. G.

21 How does Schrodinger model of atom compare with other models? Wh is it better? Bohr model: Gives correct energies. Postulates fied energ levels. Doesn t eplain WHY energ levels fied. Describes electron as point particle moving in circle. debroglie model: Also gives correct energies. Eplains fied energ levels b postulating electron is standing wave not orbiting particle. Onl looks at wave around a ring: basicall 1D not 3D Gets angular momentum wrong. Can t generalie to multi-electron atoms. + +

22 How does Schrodinger model of atom compare with other models? Schrodinger model: Gives correct energies. Wh is it better? Gives correct angular momentum. Describes electron as 3D wave of probabilit. Quantied energ levels result from boundar conditions. Schrodinger equation can generalie to multi-electron atoms. How?

23 Wh is each model useful? Bohr useful for thinking about energ levels predicting spectral lines. debroglie useful for giving simple model of how wave properties lead to quantiation. Schrodinger useful for describing how atoms interact shells chemistr atoms with more than one electron.

24 Total Energ A brief review of chemistr Electron configuration in atoms: How do the electrons fit into the available orbitals? What are energies of orbitals? 3s 3p 3d s p 1s

25 Total Energ A brief review of chemistr Electron configuration in atoms: How do the electrons fit into the available orbitals? What are energies of orbitals? Filling orbitals lowest to highest energ e s per orbital H He Li Be B C N O 3s s 1s Ogen = 1s s p 4 e e 3p p e e e e e e 3d Shell not full reactive Shell full stable

26 Total Energ Will the 1s orbital be at the same energ level for each atom? Wh or wh not? What would change in Schrodinger s equation? No. Change number of protons Change potential energ in Schrodinger s equation 1s held tighter if more protons. H He Li Be B C N O The energ of the orbitals depends on the atom. 3s s 1s e e 3p p e e e e e e 3d Shell not full reactive Shell full stable

27 A brief review of chemistr Electron configuration in atoms: How do the electrons fit into the available orbitals? What are energies of orbitals? 1 3 principle quantum number tells ou some about energ s p d tells ou some about geometric configuration of orbital 3s 3p 3d Shell s p e e e e e e Shell 1 1s e e

28 Can Schrodinger make sense of the periodic table? Schrodinger s solution for multi-electron atoms Need to account for all the interactions among the electrons Must solve for all electrons at once! use matrices V for q 1 = kq nucleus *q 1 /r n-1 + kq q 1 /r -1 + kq 3 q 1 /r

29 Schrodinger s solution for multi-electron atoms What s different for these cases? Potential energ V changes! Now more protons AND other electrons V for q 1 = kq nucleus q 1 /r n-1 + kq q 1 /r -1 + kq 3 q 1 /r Need to account for all the interactions among the electrons Must solve for all electrons at once! use matrices Gets ver difficult to solve huge computer programs! Solutions change: - wave functions change higher Z more protons electrons in 1s more strongl bound radial distribution quite different general shape p-orbital s-orbital similar but not same - energ of wave functions affected b Z # of protons higher Z more protons electrons in 1s more strongl bound more negative total energ

30 Energ For a given atom Schrodinger predicts allowed wave functions and energies of these wave functions. 4s l=0 l=1 l= 4p 3d m= s 3p Li 3 e s Na 11 e s s n= 1s n=1 p m=-101 Wh would behavior of Li be similar to Na? a. because shape of outer most electron is similar. b. because energ of outer most electron is similar. c. both a and b d. some other reason

31 Wave functions for Li vs Na Li 3 e s p 1s s 3s Na 11 e s In case of Na what will energ of outermost electron be and WHY? a. much more negative than for the outermost electron in Li b. similar to the energ of the outermost electron in Li c. much less negative than for the outermost electron in Li

32 Wave functions for sodium What affects total energ of outermost electron? p 1s 3s s 1. The effective charge force it feels towards center of atom.. It s distance from the nucleus. What effective charge does 3s electron feel pulling it towards the nucleus? Close to 1 proton 10 electrons closer in shield cancel a lot of the nuclear charge. What about distance? In H 3s level is on average 9 further than 1s so 9*Bohr radius. In Na 11 protons pull 1s s p closer to nucleus distance of 3s not as far out. Electron in 3s is a bit further than 1s in H but ~same as s in Li. Proimit of electrons in 1s s p is what makes 3s a bit bigger. In case of Na what will energ of outermost electron be and WHY? b. ver similar to the energ of the outermost electron in Li AND somewhat within a factor of 3 of the ground state of H

33 Energ Schrodinger predicts wave functions and energies of these wave functions. 4s l=0 l=1 l= 4p 3d m= s s 3p p m=-101 Li Na 1s Wh would behavior of Li be similar to Na? a. because shape of outer most electron is similar. b. because energ of outer most electron is similar. c. both a and b d. some other reason

34 Wh does ioniation energ increase and sie decrease as add electrons in p orbitals? Ioniation energ Sie distance of outermost e

35 s 1s p As go from Li to N end up with 3 electrons in p one in each orbital Wh is ioniation energ larger and sie smaller than in Li? Develop reasoning P orbitals each have direction electrons in p do not effectivel shield electrons in p from the nucleus. So electrons in p orbitals: 1. feel larger effective positive charge. are held closer to nucleus.

36 Valence n All atoms in this row have common filling of outer most shell valence electrons common shapes similar energies so similar behavior l=0 s-orbitals l=1 p-orbitals l= d-orbitals l= f-orbitals

37 ENERGY n=3 n= Hdrogen 1p 1e l=0 s s l=1 p p l= d 3s 3p 3d 1s p s Boron 5p 5e s NOT TO SCALE! 4p 3d 4s 3p 3s p m=-101 s n=1 1s l=0m=0 Splitting of s and p energ levels shielding Energ onl depends on n Energ depends on n and l 1s

38 Energ In multi-electron atoms energ of electron level depends on n and l quantum numbers: 4s l=0 l=1 m=-101 4p 3d l= m= s s 1s 3p p What is electron configuration for atom with 0 electrons? Write it out 1s etc! a. 1s s p 6 3s 3p 4 b. 1s s p 6 3s 3p 6 3d c. 1s s p 6 3s 3p 6 4s 3d 6 d. 1s s p 6 3s 3p 6 4s e. none of the above Answer is d! Calcium: Fills lowest energ levels first Which orbitals are occupied effects: chemical behavior bonding reactivit etc.

39 Energ In multi-electron atoms energ of electron level depends on n and l quantum numbers: 4s 3s l=0 l=1 m=-101 l= m= p 3d 3p 3 rd Shell 4 th Shell Calcium has 3 complete shells. Incomplete shell: Chemical behavior & bonding determined b electrons in outer most shell furthest from the nucleus. 4 p s st Shell 313 1s 1 st Shell

40 Electronic structure of atom determines its form metal semi-metal non-metal: - related to electrons in outermost shell - how these atoms bond to each other Semiconductors