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CH1. Atomic Structure orbitals p eriodicity

CH1. Atomic Structure orbitals p eriodicity. Schrodinger equation. - (h 2 /2 p 2 m e 2 ) [d 2 Y /dx 2 +d 2 Y /dy 2 +d 2 Y /dz 2 ] + V Y = E Y. h = constant m e = electron mass V = potential E E = total energy. gives quantized energies.

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CH1. Atomic Structure orbitals p eriodicity

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  1. CH1. Atomic Structure • orbitals • periodicity

  2. Schrodinger equation - (h2/2p2me2) [d2Y/dx2+d2Y/dy2+d2Y/dz2] + V Y = E Y h = constant me = electron mass V = potential E E = total energy gives quantized energies

  3. Yn,l,ml (r,q,f) = Rn,l (r) Yl,ml (q,f) Rn,l(r) is the radial component of Y • n = 1, 2, 3, ...; l = 0 to n – 1 • integral of Y over all space must be finite, so R → 0 at large r Spherical coordinates

  4. Rn,l (r)  = 2 Zr / na0

  5. Radial Distribution Function (RDF) RDF max is the Bohr radius • R(r)2 is a probability function (always positive) • The volume increases exponentially with r, and is 0 at nucleus (where r = 0) • 4pr2R2 is a radial distribution function (RDF) that takes into account the spherical volume element

  6. Yn,l,ml (r,q,f) = Rn,l(r) Yl,ml(q,f) Yl,ml (q,f) is the angular component of Y • ml = - l to + l • When l = 0 (s orbital), Y is a constant, and Y is spherically symmetric

  7. Some Y2 functions When l = 1 (p orbitals) ml = 0 (pz orbital) Y = 1.54 cosq, Y2 cos2q, (q = angle between z axis and xy plane) xy is a nodal plane Y positive Y negative

  8. Orbitals an atomic orbital is a specific solution for Y, parameters are Z, n, l, and ml • Examples: 1s is n = 1, l = 0, ml= 0 2px is n = 2, l = 1, ml= -1

  9. From SA Table 1.2 for hydrogenic orbitals; n = 3, l = 1, ml = 0 Y3pz = R3pz .Y3pz Y3pz = (1/18)(2p)-1/2(Z/a0)3/2(4r - r2)e-r/2cosq where r = 2Zr/na0 Example - 3pz orbital

  10. Some orbital shapes Atomic orbital viewer

  11. Orbital energies For 1 e- (hydrogenic) orbitals: E = – mee4Z2 / 8h2e02n2 E – (Z2 / n2)

  12. Many electron atoms • with three or more interacting bodies (nucleus and 2 or more e-) we can’t solve Y or E directly • common to use a numerative self-consistent field (SCF) • starting point is usually hydrogen atom orbitals • E primarily depends on effective Z and n, but now also quantum number l

  13. Shielding • e- -e- interactions (shielding, penetration, screening) increase orbital energies • there is differential shielding related to radial and angular distributions of orbitals • example - if 1s electron is present then E(2s) < E(2p)

  14. Orbital energies

  15. shielding parameter Effective Nuclear Charge • Zeff = Z* = Z - s • SCF calculations for Zeff have been tabulated (see text) • Zeff is calculated for each orbital of each element • E approximately proportional to -(Zeff)2 / n2

  16. Table 1.2

  17. f 0.05 Z / e- Valence Zeff trends s,p 0.65 Z / e- d 0.15 Z / e-

  18. Electron Spin • ms (spin quantum number) with 2 possible values (+ ½ or – ½). • Pauli exclusion principal - no two electrons in atom have the same 4 quantum numbers (thus only two e- per orbital)

  19. Electronic Configurations Examples: • Ca (Z = 20) ground state config. 1s2 2s2 2p6 3s2 3p6 4s2 or just write [Ar]4s2 • N (Z = 7) 1s2 2s2 2p3 [He] 2s22p3 actually [He] 2s22px1 2py1 2pz1

  20. Multiplicity • Hund's rule of maximum multiplicity – atom is more stable when electron's correlate with the same ms sign • This is a small effect, only important where orbitals have same or very similar energies (ex: 2px 2py 2pz, or 4s and 3d) • S = max total spin = the sum adding +½ for each unpaired electron • multiplicity = 2S + 1

  21. 1st row transition metals 3d half-filled 3d filled

  22. Ionic configurations • Less shielding, so orbital E’s are ordered more like hydrogenic case, example: 3d is lower in E than 4s • TM ions usually have only d-orbital valence electrons, dns0 Fe (Z = 26) Fe is [Ar]3d64s2 But Fe(III) is [Ar]3d5 4s0

  23. Atomic Orbitals - Summary Y (R,Y) • RDF and orbital shapes • shielding, Zeff, and orbital energies • electronic configurations, multiplicity

  24. Periodic Trends • Ionization Energy ( I ) • Electron Affinity (Ea) • Electronegativity (c) • Atomic Radii • Hardness / Softness

  25. Ionization energy • Energy required to remove an electron from an atom, molecule, or ion • I = DH [A(g) → A(g)+ + e- ] • Always endothermic (DH > 0), so I is always positive

  26. Ionization energy • Note the similarity of trends for I and Zeff, both increase left to right across a row, more rapidly in sp block than d block • Advantage of looking at I trend is that many data are experimentally determined via gas-phase XPS • But, we have to be a little careful, I doesn't correspond only to valence orbital energy…

  27. Ionization energy • I is really difference between two atomic states • Example:N(g) → N+(g) + e-px1py1pz1 → 2px12py1mult = 4 → mult = 3 vs. O(g) → O+(g) + e- px2py1pz1 → px1py1pz1mult = 3 → mult = 4 Trend in I is unusual, but not trend in Zeff

  28. cation I (kJ/mol) NO 893   NO2 940  CH3 950  O2 1165  NOAsF6 OH 1254  NO2AsF6 N2 1503  CH3SO3CF3, (CH3)2SO4 O2AsF6 HOAsF6 N2AsF6 Ionization energy I can be measured for molecules Molecular ionization energies can help explain some compounds’ stabilities. DNE

  29. Electron affinity • Energy gained by capturing an electron • Ea = –DH [A(g) + e- → A-(g)] • Note the negative sign above • Example: DH [F(g) + e- → F- (g)] = - 330 kJ/mol Ea(F) = + 330 kJ/mol (or +3.4 eV) • notice that I(A) = Ea(A+) I = DH [A(g) → A+(g) + e-]

  30. Electron affinity • Periodic trends similar to those for I, that is, large I means a large Ea • Ea negative for group 2 and group 18 (closed shells), but Ea positive for other elements including alkali metals: DH [Na(g) + e- → Na-(g)] ≈ - 54 kJ/mol • Some trend anomalies: Ea (F) < Ea (Cl) and Ea (O) < Ea (S) these very small atoms have high e- densities that cause greater electron-electron repulsions Why aren’t sodide A+ Na- (s) salts common ?

  31. Electronegativity Attractive power of atom or group for electrons Pauling's definition (cP): A-A bond enthalpy = AA (known) B-B bond enthalpy = BB (known) A-B bond enthalpy = AB (known) If DH(AB) < 0 then AB > ½ (AA + BB) AB – ½ (AA + BB) = const [c(A) - c(B)]y Mulliken’s definition: cM = ½ (I + Ea)

  32. Atomic Radii Radii decrease left to right across periods • Zeffincreases, n is constant • Smaller effect for TM due to slower increase Zeff • (sp block = 0.65, d block = 0.15 Z / added proton)

  33. Atomic Radii • X-ray diffraction gives very precise distances between nuclei in solids • BUT difficulties remain in tabulating atomic or ionic radii. For example: • He is only solid at low T or high P, but all atomic radii change with P,T • O2 solid consists of molecules O=O........O=O • P(s) radius depends on allotrope studied

  34. Period Group 5 Group 8 Group 10 4 V 1.35 Å Fe 1.26 Ni 1.25 5 Nb 1.47 Ru 1.34 Pd 1.37 6 Ta 1.47 Os 1.35 Pt 1.39 Atomic radii - trends • Radii increase down a column, since n increases • lanthanide contraction: 1st row TM is smaller, 2nd and 3rd row TMs in each triad have similar radii (and chemistries) Why? Because 4f electrons are diffuse and don't shield effectively

  35. Hardness / Softness • hardness (h) = ½ (I - Ea) h prop toHOAO – LUAO gap • large gap = hard, unpolarizable small gap = soft, polarizable • polarizability (a) is ability to distort in an electric field

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