1.3 Review of atomic orbitals and their shapes
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Atomic orbitals describe the electron distribution as a standing matter wave in the
potential field of a nucleus:
⎡
⎤
⎢
⎥
2
2
⎢ − h
⎥
Ze
2
⎢ 2 ∇ −
⎥ Ψ = EΨ
2
2
2
8π!#
m!
4πε 0 x + y + z ⎥
" $
⎢ $
!!!#!!!
"
tic
⎢ kine
⎥
potential
energy
energy
⎣
⎦
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Derived by solving the Schrödinger equation for the hydrogen atom.
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Because every Ψ matches an atomic orbital, there is no limit to the number of
solutions to the Schrödinger equation!
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Each Ψ describes the wave properties of a given electron in a particular orbital.
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Since 2-body problems can be solved, we can fully solve for the H atom in its ground
and all its excited states. (NOTE: 3-body problems cannot be solved explicitly!)
Ψ ( r, Θ, Φ ) = R( r )Θ(θ )Φ (φ ) = R( r )Y (θ , φ )
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Mathematical solutions for Ψ are of the general form shown above (in polar spherical
coordinates, NOT Cartesian coordinates x,y,z)
R = radial factors; Y = angular factors
Section 1.3 - 1
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Each and every orbital is uniquely characterized by its set of quantum numbers:
Source: Tarr & Miessler, Inorganic Chemistry, 2nd Edition.
Four quantum numbers are required for a full description of an electron:
n = 1, 2, 3, 4, …
-
principal quantum number
n – 1 = total # of nodes in the orbital wavefunction
therefore, fundamental role in determining energy
recall n from particle-in-a-box example
l = 0, 1, 2, 3, …
= s, p, d, f, …
- l = # of planar nodes
- determines angular shape of the orbital wavefunction
- orbital angular momentum
ml = 0, ±1, ±2, …, ±l - determines spatial orientation of the orbital wavefunction
- e.g., l = 1
therefore ml = +1, 0, -1
p-orbital set
px, py, pz
ms = +½ , -½
- determines the spatial orientation of the electron spin vector
- note that there are only two options for an electron
However, these are not the only quantum numbers that can be use.
s=½
-
spin angular momentum
only one value of s is possible for a single electron
s is ALWAYS POSITIVE!!
s is related to ms in the same way l is related to ml
j=l+s
-
total angular momentum
this is a vector sum of l and s, thus there may be more than one
possible value for j given a value of l and s.
mj = j, j-1, j-2, …, -j -
determines the spatial orientation of the total angular mom.
Section 1.3 - 2
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Angular functions Y(θ,φ) describe the shape of the orbital
Source: Purcell + Kotz, Inorganic Chemistry, 1977
These are determined by quantum numbers l and ml.
Resulting orbital shapes:
Note that the orbitals are 3D waves!
All atomic orbitals of a given atom must be
orthogonal to one another.
The surface is typically the 75% probability
boundary.
Section 1.3 - 3
Radial functions R(r) describe the radial electron distribution
and occurrence of nodal planes
Source: Tarr & Miessler, Inorganic Chemistry, 2nd Edition.
These are determined by quantum numbers n and l.
Section 1.3 - 4
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The shape of the radial functions and their nodal planes are:
Source: Tarr & Miessler, Inorganic Chemistry, 2nd Edition.
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Note that according to the Copenhagen Interpretation it is the square of the
wavefunction that has a physical meaning (which one?) and hence is relevant for
chemistry.
Section 1.3 - 5
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These are 2D “sections” of the 3D orbitals (imagine slicing through an onion):
Section 1.3 - 6
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In general, we list the orbitals in order of energy as: 1s, 2s, 2p, 3s, 3p, 4s, 3d, etc.
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Where does this energy ordering come from?
Section 1.3 - 7
The energies of the orbitals in many electron atoms are, in part, a function of nuclear
charge:
Source: Shriver & Atkins, Inorganic Chemistry, 3rd Edition.
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These energies cannot be calculated exactly as for the H-atom, as no closed form or
analytical solution exists for the Schrödinger equation for three or more particles
(same is true in classical mechanics – n-body problem with n > 2)
→ must use approximations !
Section 1.3 - 8
1.4 Penetration and shielding
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The subtle dependence of the AO energies can be rationalized through the concepts of
penetration and shielding leading to an effective nuclear charge Z*.
Z* = Z - S
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S = shielding parameter
There are two effects that result in an effective nuclear charge:
1) Shielding of the outer electrons from the nuclear charge by the “inner” electrons.
2) Penetration of “inner” electron density by “outer” electron density.
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NOTE: the effective nuclear charge felt by an electron depends on its radial density
function!
e.g. 2s vs. 2p electron:
The 2s electron experiences a greater effective
nuclear charge because it has a higher probability
density closer to the nucleus.
Similar comparisons can be made for np vs. nd and
nd vs. nf orbitals.
The net effect of shielding and penetration is the
energetic ordering
ns < np < nd < nf
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Shielding becomes less effective for larger quantum numbers n and l.
- Orbital energies increase as (n+l) increases.
- For two orbitals with the same (n+l), the one with the smaller n lies lower in energy.
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This results in the filling order:
n+ l =
1
2
3
4
5
Orbital
1s
2s
2p, 3s
3p, 4s
3d, 4p, 5s
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This is the AUFBAU Principle, which together with Pauli’s Exclusion Principle
(unique set of quantum numbers n, l, ml, ms for every electron) and Hund’s Rule
(maximum multiplicity) determines the electronic ground state configuration.
Section 1.3 - 9