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Multi-Electron Atoms

Multi-Electron Atoms. Complete Description of a Ground State Wavefunction ψ. A total of three quantum numbers appear from the solution of. n = principal quantum number. l = angular momentum quantum number. m l = magnetic quantum number.  n l m (r, θ , Φ ).

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Multi-Electron Atoms

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  1. Multi-ElectronAtoms

  2. Complete Description of a Ground State Wavefunction ψ • A total of three quantum numbers appear from the solution of n = principal quantum number l = angular momentum quantum number ml= magnetic quantum number nlm(r,θ,Φ)

  3. What will a ground state wave-function be called??

  4. Correlation of Wavefunctions to Orbitals • Using the terminology of chemists: • 100(r,,) is instead called the 1s orbital. • n designates the shell (1, 2, 3, 4,…..) • l designates the sub-shell (s, p, d, f….) • ml completes the description of the orbital l = 1 (p orbital) l = 0 (s orbital) l = 2 (d orbital) l = 3 (f orbital) When l =1 and ml = 0 then orbital is pz When l=1 and ml= ±1 then orbital is pxor py

  5. Hydrogen Atom Wavefunctions 100 n = 1 l = 0 m = 0 1s 100 -RH / 12 -2.18 x 10-18J n = 2 l = 0 m = 0 200 200 2s -RH / 22 -5.45 x 10-19J n = 2 l = 1 m = +1 2px (or2py) -RH / 22 211 - 5.45 x 10-19J 211 n = 2 l = 1 m = 0 210 -RH / 22 - 5.45 x 10-19J 210 2pz n = 2 l = 1 m = -1 21-1 21-1 2py (or2px) - 5.45 x 10-19J -RH / 22

  6. What is the corresponding orbital for a 4,1,0 state? • 1. 1s • 2. 2s • 3. 4s • 4. 5s • 5. 4px • 6. 4py • 7. 4pz • 8. 4dz

  7. En = -RH n2 Arrangement of Shells/Subshells/Orbitals and Corresponding Quantum Numbers For a H-atom, orbitals with same value of n have equal energy. For any shell n there are n2 degenerate orbitals.

  8. Energy Levels For a Hydrogen Atom: 3s 3px 3pz 3py 3dxy 3dyz 3dz2 3dxz 3dx2-y2 2s 2px 2pz 2py 1s

  9. Concept Check! • How many orbitals in a single atom can have the following two quantum numbers: n = 4, ml = -2 • one • two • three • four • five • six • seven • eight • zero

  10. Degeneracy of states • States having the same energy are called degenerate. • For every value of n there are n2 degenerate states.

  11. Figure from MIT Open CourseWare Physical Interpretation of Ψ • Quantum world is very different from the macroscopic world that we are used to seeing. • Therefore unfortunately a physical interpretation of Ψdoes not exist. • However a physical interpretation for Ψ2 does exist! | nlm(r,,)|2 = Probability Density Probability / Volume Max Born

  12. Electron clouds • Although we cannot know how the electron travels around the nucleus we can know where it spends the majority of its time (thus, we can know position but not trajectory). • The “probability” of finding an electron around a nucleus can be calculated. • Relative probability is indicated by a series of dots, indicating the “electron cloud”. • 90% electron probability or cloud for 1s orbital (notice higher probability toward the centre)

  13. Solution to the Wave function for a H-atom Any wave function Ψcan be divided into two components Ylm(,) Rnl(r) radial Ψ angular Ψ Ylm(,) Rnl(r) For all s orbitals (1s, 2s, 3s, etc,) Y is a constant. Where a0 = Bohr radius (constant) = 52.9 pm

  14. Figure 1.23 Chem Principles Shape of an s-orbital • The shape of an s-orbital is spherically symmetrical, independent of  and .

  15. 2a0 Probability Density Plots of s-orbitals Figures from MIT OCW NODE a value of r,  and  for which both  and 2 = 0 RADIAL NODE: a value of r for which both  and 2 = 0 Radial Nodes = n - 1 - l For 1s radial nodes = 1 – 1 – 0 = 0 For 2s radial nodes = 2 – 1 – 0 = 1 For 3s radial nodes = 3 – 1 – 0 = 2

  16. How many radial nodes does a hydrogen atom 3d orbital have? • One • Two • Three • Four • Five • Six • Seven • Eight • Zero

  17. Radial Probability Distribution • The probability of finding an electron in a shell of thickness dr at a distance r from the nucleus. • For s-orbitals RPD = 4πr22 dr rmp = Bohr radius = 0.529Å http://www.emu.edu.tr/mugp101/PHYSLETS/physletprob/ch10_modern/radial.html

  18. rmp = 6a0 rmp = 11.5a0 node node node RPD for a 2s and a 3s orbital for H-atom: volume as n increases rmpalso increases.

  19. Concept Check • Identify the correct RPD plot (and radial node number) for a 4s orbital 3. 1. 2. 4.

  20. Radial Probability Distributions for other orbitals: < 3d 3s 3p

  21. Quantum Tunneling Scanning Tunneling Microscope

  22. p-orbitals (l = 1): • For any sub-shell l = 1 there are three p orbitals; • m = +1 or -1 (pxorpy) and m = 0 (pz) • Difference from the s-orbitals lies in the fact that p-orbitals wave-functions depend on  and . p-orbitals are notspherically symmetrical!

  23. p and d-orbitals p-orbitals look like a dumbell with 3 orientations: px, py, pz(“p sub z”). p-orbitals consist of two lobes seperated by a nodal plane. There is zero probability of finding a p-electron at the nucleus. Only electrons in the s orbitals have a substantial probability of being very close to the nucleus Electrons in the s orbitals are LEAST shielded.

  24. d-orbitals Four of the d orbitals resemble two dumbells in a clover shape. The last d orbital resembles a p-orbital with a donut wrapped around the middle!

  25. Multi-Electron Atoms and the spin quantum number ms When the Schrodinger wave equation is solved for multielectron atoms, a fourth quantum number ms the spin quantum number also appears. ms = +1/2 (spin up) or ms = -1/2(spin down) ms completes the description of an electron and is NOT dependant on the orbital.

  26. Atoms with Many Electrons and the Periodic Table The underlying physical laws necessary for …the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble! Paul Dirac (1929) Paul Dirac at a Super-Collider workshop in the early 1930s.

  27. Uhlenbeck and Goudsmit Ne 1s2 2s2 2px2 2py2 2pz2 Wolfgang Pauli Discovery of Electron Spin: http://www.ilorentz.org/history/spin/goudsmit.html

  28. Pauli Exclusion Principle • No two electrons in the same atom can have the same four quantum numbers. • The Pauli exclusion principle limits us to two electrons per orbital. Ne 1s2 2s2 2px2 2py2 2pz2

  29. Concept Check! • How many electrons in a single atom can be in a 2p state? • one • two • three • four • five • six • seven • eight • zero

  30. How many electrons in a single atom can have the following two quantum numbers: n = 4, ml = -2 • One • Two • Three • Four • Five • Six • Seven • Eight • Zero

  31. (r111r222) (r111r2 22) (r111r222r333) (r111r2 22r333) Shrodinger Equation for Multielectron Atoms NEED AN APPROXIMATION!

  32. Hartree Orbitals • One electron orbital approximation: e- # 1 e- # 2 1s (1) 1s (2) 100+1/2 100-1/2 200+1/2 1s(2) 2s(1)

  33. Electronic Configurations • Electronic configurations are basically short hand notations for different wavefunctions, using the “1 electron orbital approximation”. 1s22s2 1s22s22p1

  34. Multi-electron vs. Hydrogen Atom Wave Functions e.g. Ar 1s22s22p63s23p6 Similarities to H-atom Wave functions: • Similar in shape • Identical nodal structure Differences to H-atom Wave functions: • Each multi-electron orbital is smaller than the corresponding hydrogen atom orbital. • In multi-electron atoms, orbital energies depend not only on n (shell)they also • depend on l (sub-shell).

  35. Multi-electron vs. Hydrogen Atom Energy Levels more negative

  36. Zeff≠ Z • Zeff differs from Z because of shielding.

  37. Shielding and Zeff Case A # 1 # 2 2 • Electron #2 cancels part of the charge experienced by electron #1. • Electron #1 experiences a force on average of Zeff = ___ , not Zeff = +2e. • The energy of electron #1 is that of an electron in a H (1-electron) atom. (2.18 x 10-18 J) +1 total Shielding

  38. Shielding and Zeff Case B # 2 # 1 2 • Electron #2 does not cancel the charge experienced by electron #1. • Electron #1 experiences a force on average of Zeff = ___ • The energy of electron #1 is that of an electron in a He+1(1-electron) ion. • (8.72 x 10-18 J) +2 No Shielding

  39. Extreme case A: Zeff = 1, IEHe= 2.18 x 10–18 J total shielding • Extreme case B: Zeff = 2, IEHe= 8.72 x 10–18 J no shielding • Experimental IEHe = 3.94 x 10–18 J So the reality is somewhere between total shielding and no shielding.

  40. We can calculate the Zeff from the experimentally determined IE: Our calculated Zeff should be a reasonable value, it should fall between total shielding and no shielding.

  41. Which value(s) below is a possible Zeff for the 2s electron in a Li (Z = 3) atom? • Zeff = 0.39 • Zeff = 0.87 • Zeff = 1.42 • Zeff = 3.19 • Option 1 and 2 • Option 1, 2, and 3 • Option 2 and 4

  42. Why is E2s<E2p and E3s<E3p<E3d?

  43. Energy differences of s and p-orbitals: multi single can get Also for a given n state, electrons in the s-orbitals are less shieldedfrom the nucleus as compared to the p-electrons and hence experience a greater Zeff.

  44. Radial Probability Distributions for other orbitals: < 3d 3s 3p

  45. Consider why the electronic configuration for Li is 1s2 2s1and not 1s2 2p1. The s-orbital is less shielded. Averaging over the RPD yields Zeff2p < Zeff2s E2s< E2p

  46. Aufbau (building up) principle • Fill energy states that depend on (n & l) one electron at a time, starting at the lowest energy state. O (Z = 8) parallel 1s22s22p4

  47. Identify the correct electron configuration for the carbon (Z = 6) atom. • 1s22s23s2 • 1s22s22px2 • 1s22s22py2 • 1s22s22pz2 • 1s22s22px12pz1 • 1s22s22px12pz12py1 E 3s 2px 2pz 2py 2s 1s

  48. Periods in the Periodic Table • Period in a periodic table refers to the value of the principal quantum number “n”. core valence 3d10 3d5

  49. Writing Down the Electronic Configurations:

  50. Electron Configurations for Ions 3d24s2 3d2

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