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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