#### Transcript R n,l

```CH1. Atomic Structure
 orbitals
 periodicity
1
Schrodinger equation
- (h2/2p2me2) [d2Y/dx2+d2Y/dy2+d2Y/dz2] + V Y = E Y
h = constant
me = electron mass
V = potential E
gives quantized
energies
E = total energy
2
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
3
Rn,l (r)
Orbital n l Rn,l for H atom
1s
1 0 2 (Z/ao)3/2 e-/2
2s
2 0 1 / (2√2) (Z/ao)3/2 (2 - ½) e-/4
 = 2 Zr / na0
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RDF max
(RDF)
is the Bohr
• 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
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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
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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
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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
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Example - 3pz orbital
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(4 - 2)e-/2cosq
where  = 2Zr/na0
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Some orbital shapes
Atomic orbital viewer
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Orbital energies
For 1 e- (hydrogenic) orbitals:
E = – mee4Z2 / 8h2e02n2
E  – (Z2 / n2)
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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
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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)
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Orbital energies
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Effective Nuclear Charge
shielding parameter
•
•
•
•
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
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Table 1.2
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Valence Zeff trends
s,p 0.65 Z / e-
d 0.15 Z / e-
f 0.05 Z / e-
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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 eper orbital)
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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
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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
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1st row transition metals
3d half-filled
3d filled
# unpaired e-
multiplicity
Sc
[Ar]3d14s2
1
2
Ti
[Ar]3d24s2
2
3
V
[Ar]3d34s2
3
4
Cr
[Ar]3d54s1
6
7
Mn
[Ar]3d54s2
5
6
Fe
[Ar]3d64s2
4
5
Co
[Ar]3d74s2
3
4
Ni
[Ar]3d84s2
2
3
Cu
[Ar]3d104s1
1
2
Zn
[Ar]3d104s2
0
1
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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
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Atomic Orbitals - Summary
Y (R,Y)
• RDF and orbital shapes
• shielding, Zeff, and orbital
energies
• electronic configurations,
multiplicity
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Periodic Trends
•
•
•
•
•
Ionization Energy ( I )
Electron Affinity (Ea)
Electronegativity (c)
Hardness / Softness
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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
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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…
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Ionization energy
•
I is really difference between two
atomic states
• Example:
N(g)
→ N+(g) + epx1py1pz1
→ 2px12py1
mult = 4
→ mult = 3
vs. O(g)
→ O+(g) + epx2py1pz1
→ px1py1pz1
mult = 3
→ mult = 4
Trend in I is unusual, but not trend in Zeff
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Ionization energy
Molecular
ionization
energies can
help explain
some
compounds’
stabilities.
I can be measured for molecules
cation
NO
NO2
I (kJ/mol)
893
NOAsF6
940
NO2AsF6
CH3
O2
OH
N2
950
1165
1254
1503
CH3SO3CF3, (CH3)2SO4
O2AsF6
HOAsF6
N2AsF6
DNE
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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-]
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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 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 edensities that cause greater electronelectron repulsions
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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)
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Radii decrease left to right across periods
• Zeff increases, n is constant
• Smaller effect for TM due to slower increase Zeff
• (sp block = 0.65, d block = 0.15 Z / added proton)
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•
•
X-ray diffraction gives very precise
distances between nuclei in solids
BUT difficulties remain in tabulating atomic
•
•
•
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
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• Radii increase down a column, since n
increases
 lanthanide contraction: 1st row TM is
smaller, 2nd and 3rd row TMs in each triad
Why?
Because 4f
electrons are
diffuse and
don't shield
effectively
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
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Hardness / Softness
•
•
•
hardness (h) = ½ (I - Ea)
h prop to HOAO – LUAO gap
large gap = hard,
unpolarizable
small gap = soft, polarizable
polarizability (a) is ability to
distort in an electric field
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