Schrodinger equation

Transcription

1 CH1. Atomic Structure orbitals periodicity 1 Schrodinger equation - (h 2 /2p 2 m e2 ) [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 gives quantized energies E = total energy 2 1

2 Y n,l,ml (r,q,f) = R n,l (r) Y l,ml (q,f) R n,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 3 R n,l (r) Orbital n l R n,l for H atom 1s 2s (Z/a o ) 3/2 e -r/ / (2 2) (Z/a o ) 3/2 (2 - ½r) e -r/4 r = 2 Zr / na 0 4 2

3 RDF max is the Bohr radius Radial Distribution Function (RDF) R(r) 2 is a probability function (always positive) The volume increases exponentially with r, and is 0 at nucleus (where r = 0) 4pr 2 R 2 is a radial distribution function (RDF) that takes into account the spherical volume element 5 Y n,l,ml (r,q,f) = R n (r) Y l,ml (q,f) Y l,ml (q,f) is the angular component of Y m l = - l to + l When l = 0 (s orbital), Y is a constant, and Y is spherically symmetric 6 3

4 Some Y 2 functions When l = 1 (p orbitals) m l = 0 (p z orbital) Y = 1.54 cosq, Y 2 cos 2 q, (q = angle between z axis and xy plane) xy is a nodal plane Y positive Y negative 7 Orbitals an atomic orbital is a specific solution for Y, parameters are Z, n, l, and m l Examples: 1s is n = 1, l = 0, m l = 0 2p x is n = 2, l = 1, m l =

5 Example - 3p z orbital From SA Table 1.2 for hydrogenic orbitals; n = 3, l = 1, m l = 0 Y 3pz = R 3pz. Y 3pz Y 3pz = (1/18)(2p) -1/2 (Z/a 0 ) 3/2 (4r - r 2 )e -r/2 cosq where r = 2Zr/na 0 9 Some orbital shapes Atomic orbital viewer 10 5

6 Orbital energies For 1 e - (hydrogenic) orbitals: E = m e e 4 Z 2 / 8h 2 e 02 n 2 E (Z 2 / n 2 ) 11 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 12 6

7 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) 13 Orbital energies 14 7

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

9 Valence Z eff trends s,p 0.65 Z / e - d 0.15 Z / e - f 0.05 Z / e - 17 Electron Spin m s (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) 18 9

10 Electronic Configurations Examples: Ca (Z = 20) ground state config. 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 or just write [Ar]4s 2 N (Z = 7) 1s 2 2s 2 2p 3 [He] 2s 2 2p 3 actually [He] 2s 2 2p x1 2p y1 2p 1 z 19 Multiplicity Hund's rule of maximum multiplicity atom is more stable when electron's correlate with the same m s sign This is a small effect, only important where orbitals have same or very similar energies (ex: 2p x 2p y 2p z, or 4s and 3d) S = max total spin = the sum adding +½ for each unpaired electron multiplicity = 2S

11 1 st row transition metals # unpaired e - multiplicity 3d half-filled 3d filled Sc [Ar]3d 1 4s Ti [Ar]3d 2 4s V [Ar]3d 3 4s Cr [Ar]3d 5 4s Mn [Ar]3d 5 4s Fe [Ar]3d 6 4s Co [Ar]3d 7 4s Ni [Ar]3d 8 4s Cu [Ar]3d 10 4s Zn [Ar]3d 10 4s 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, d n s 0 Fe (Z = 26) Fe is [Ar]3d 6 4s 2 But Fe(III) is [Ar]3d 5 4s

12 Atomic Orbitals - Summary Y (R,Y) RDF and orbital shapes shielding, Z eff, and orbital energies electronic configurations, multiplicity 23 Periodic Trends Ionization Energy ( I ) Electron Affinity (E a ) Electronegativity (c) Atomic Radii Hardness / Softness 24 12

13 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 25 Ionization energy Note the similarity of trends for I and Z eff, 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 26 13

14 Ionization energy I is really difference between two atomic states Example: N(g) N + (g) + e - p x1 p y1 p 1 z 2p x1 2p 1 y mult = 4 mult = 3 vs. O(g) O + (g) + e - p x2 p y1 p 1 z p x1 p y1 p 1 z mult = 3 mult = 4 Trend in I is unusual, but not trend in Z eff 27 Ionization energy Molecular ionization energies can help explain some compounds stabilities. I can be measured for molecules cation NO 893 NO CH O I (kj/mol) OH 1254 N NOAsF 6 NO 2 AsF 6 CH 3 SO 3 CF 3, (CH 3 ) 2 SO 4 O 2 AsF 6 HOAsF 6 N 2 AsF 6 DNE 28 14

15 Electron affinity Energy gained by capturing an electron E a = DH [A(g) + e - A - (g)] Note the negative sign above Example: DH [F(g) + e - F - (g)] = kj/mol E a (F) = kj/mol (or +3.4 ev) notice that I(A) = E a (A + ) I = DH [A(g) A + (g) + e - ] 29 Electron affinity Why aren t sodide A + Na - (s) salts common? Periodic trends similar to those for I, that is, large I means a large E a E a negative for group 2 and group 18 (closed shells), but E a positive for other elements including alkali metals: DH [Na(g) + e - Na - (g)] - 54 kj/mol Some trend anomalies: E a (F) < E a (Cl) and E a (O) < E a (S) these very small atoms have high e - densities that cause greater electronelectron repulsions 30 15

16 Electronegativity Attractive power of atom or group for electrons Pauling's definition (c P ): 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: c M = ½ (I + E a ) 31 Atomic Radii Radii decrease left to right across periods Z eff increases, n is constant Smaller effect for TM due to slower increase Z eff (sp block = 0.65, d block = 0.15 Z / added proton) 32 16

17 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 O 2 solid consists of molecules O=O... O=O P(s) radius depends on allotrope studied 33 Atomic radii - trends Radii increase down a column, since n increases lanthanide contraction: 1 st row TM is smaller, 2nd and 3 rd row TMs in each triad have similar radii (and chemistries) Why? Because 4f electrons are diffuse and don't shield effectively Period 4 5 Group 5 V 1.35 Å Nb 1.47 Group 8 Fe 1.26 Ru 1.34 Group 10 Ni 1.25 Pd Ta 1.47 Os 1.35 Pt

18 Hardness / Softness hardness (h) = ½ (I - E a ) h prop to HOAO LUAO gap large gap = hard, unpolarizable small gap = soft, polarizable polarizability (a) is ability to distort in an electric field 35 18