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CH1. Atomic Structure
orbitals
periodicity
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Schrodinger equation
- (h2/2p2me2) [d2Y/dx2+d2Y/dy2+d2Y/dz2] + V Y = E Y
gives quantized
energies
h = constant
me = electron mass
V = potential E
E = total energy
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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
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Rn,l (r)
Orbital n l Rn,l for H atom
1s 1 0 2 (Z/ao)3/2 e-r/2
2s 2 0 1 / (2√2) (Z/ao)3/2 (2 - ½r) e-r/4
r = 2 Zr / na0
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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)
• 4pr2R2 is a radial distribution function (RDF) that takes into account the spherical volume element
RDF max
is the Bohr
radius
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Yn,l,ml (r,q,f) = Rn(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
Y positive Y negative
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
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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(4r - r2)e-r/2cosq
where r = 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 / 8h2e0
2n2
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
• 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
shielding parameter
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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 e-
per 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 2py
1 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
# 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
3d half-filled
3d filled
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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)
• Atomic Radii
• 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) + e-
px1py
1pz1 → 2px
12py1
mult = 4 → mult = 3
vs. O(g) → O+(g) + e-
px2py
1pz1 → px
1py1pz
1
mult = 3 → mult = 4
Trend in I is unusual, but not trend in Zeff
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Ionization energy
I can be measured for molecules
cation I (kJ/mol)
NO 893
NO2 940
CH3 950
O2 1165
OH 1254
N2 1503
HOAsF6
N2AsF6
CH3SO3CF3, (CH3)2SO4
NOAsF6
NO2AsF6
O2AsF6
Molecular
ionization
energies can
help explain
some
compounds’
stabilities.
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
• 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 ?
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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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Atomic Radii
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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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
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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)
Pt 1.39Os 1.35Ta 1.476
Pd 1.37Ru 1.34Nb 1.475
Ni 1.25Fe 1.26 V 1.35 Å4
Group 10Group 8Group 5PeriodWhy?
Because 4f
electrons are
diffuse and
don't shield
effectively
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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