1 3 Nodes Nodes Places where the Probability of finding the - - PDF document

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A Closer Look at A Closer Look at ψ ψ

• Contains Information about the Probability
• f finding the Quantum Mechanical Entity

in a Certain State

– For atom, know energy so ψ is related to probability of finding electron at a certain point in space

• The Probability is not ψ, rather ψ2

– Actually this is ψ*ψ

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More on More on Orbitals Orbitals

• Wavefunctions for Atomic Orbitals can be

divided into Two Parts

– Radial (depends on distance from nucleus) – Angular (depends on angles φ and θ)

• For Chemistry Angular Part is (most) Important

– Molecular shape – Bonding

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Nodes Nodes

• Places where the Probability of finding the

Electron is Zero (ψ = 0 so ψ2 = 0 )

• When ψradial is zero, called a radial (or

spherical) node

– There are n - l - 1 radial nodes

• When ψangular is zero, called an angular

node (or a nodal plane)

– There are l angular nodes

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1s Radial 1s Radial Wavefunction Wavefunction

• 0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 2 4 6 8 10 12 14 16

Distance from Nucleus (arbitrary units) ψ (arbitrary units)

There are n -

• 1 = 1 - 0 - 1 = 0 radial nodes.

l

Note that ψ ≠ 0 at x = 0.

/ 2 / 3

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a Zr radial

e a Z

−

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ψ

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1s Orbital 1s Orbital

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• 0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 2 4 6 8 10 12 14 16

Distance from Nucleus (arbitrary units) ψ (arbitrary units)

2s Radial Wavefunction 2s Radial Wavefunction

Note that ψ ≠ 0 at x = 0. There are n -

• 1 = 2 - 0 - 1 = 1 radial node.

l

2 / 2 / 3

2 2 2 1

a Zr radial

e r a Z a Z

−

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ψ

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2s Orbital 2s Orbital

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3s Radial Wavefunction 3s Radial Wavefunction

• 0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 2 4 6 8 10 12 14 16

Distance from Nucleus (arbitrary units) ψ (arbitrary units)

Note that ψ ≠ 0 at x = 0. There are n -

• 1 = 3 - 0 - 1 = 2 radial nodes.

l

3 / 2 2 2 / 3

3 2 4 6 3 9 1

a Zr radial

e r a Z r a Z a Z

−

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ψ

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3s Orbital 3s Orbital

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Probability of Finding an Electron Probability of Finding an Electron

• Remember ψ2, not ψ, is Probability

0.0 0.2 0.4 0.6 0.8 1.0 1.2 2 4 6 8 10 12 14 16

Distance from the Nucleus (arbitrary units) ψ2 (arbitrary units)

ψ2 for 1s, 2s and 3s orbitals Note s orbitals have a non-zero probability of being found at the nucleus.

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Probability of Finding an Electron Probability of Finding an Electron

0.00 0.02 0.04 0.06 0.08 0.10 2 4 6 8 10 12 14 16

Distance from the Nucleus (arbitrary units) Ψ

2 (arbitrary units)

1s 2s 3s Note wherever there was a node ψ2 = 0, there is no probability that the electron can be found there. If the electron can’t be in a certain place, how does it get across?

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Radial Distribution Function Radial Distribution Function

• Problem with ψ2, it over estimates Probability

Close to Nucleus and under estimates it Further Out

• Correct by multiplying ψ2 by 4πr2

– Takes into account that a wedge is smaller toward the center than ends – This correction only works for s orbitals

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Radial Distribution Function Radial Distribution Function

1 2 3 4 5 6 2 4 6 8 10 12 14 16

Distance from Nucleus (arbitrary units) 4 π r2 ψ2 (arbitrary units)

1s 2s 3s Electrons in orbitals with higher n are usually found further from nucleus. Note that 2s and 3s electrons have probability of being closer to nucleus than 1s!

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1s, 2s, and 3s 1s, 2s, and 3s orbitals

• rbitals

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• Every Time ψ goes through a node Sign of

Wavefunction changes

– s orbital has same angular sign throughout – p orbital lobes have different signs – Lobes alternate signs in a d orbital

• Difference in Phase

p orbital d orbital dz2 orbital

Angular Part of Angular Part of Wavefunction Wavefunction

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p Orbitals p Orbitals

When n = 2, then l = 0 and 1 Therefore, in n = 2 shell there are 2 types of orbitals (2 subshells) For l = 0 ml = 0 this is a s subshell For l = 1 ml = -1, 0, +1 this is a p subshell with 3 orbitals When n = 2, then When n = 2, then l l = 0 and 1 = 0 and 1 Therefore, in n = 2 shell there Therefore, in n = 2 shell there are 2 types of orbitals (2 are 2 types of orbitals (2 subshells subshells) ) For For l l = 0 = 0 m ml

l = 0

= 0 this is a s subshell this is a s subshell For For l l = 1 m = 1 ml

l =

= -

• 1, 0, +1

1, 0, +1 this is a p subshell this is a p subshell with 3 orbitals with 3 orbitals

planar node Typical p orbital planar node Typical p orbital

When When l l = 1, there is a = 1, there is a PLANAR NODE PLANAR NODE thru thru the nucleus. the nucleus.

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p p Orbitals Orbitals

A p orbital A p orbital The three p The three p

• rbitals lie 90
• rbitals lie 90o
• apart in space

apart in space

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2p Radial Wavefunction 2p Radial Wavefunction

• 0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 2 4 6 8 10 12 14 16

Distance from Nucleus (arbitrary units) ψ (arbitrary units)

Note that ψ = 0 at x = 0. Only s

• rbitals have any probability

density at the nucleus. There are n -

• 1 = 2 - 1 - 1 = 0 radial nodes.

l

2 / 2 / 3

2 6 4 1

a Zr radial

e r a Z a Z

−

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ψ

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2p 2px

x Orbital

Orbital

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2p 2py

y Orbital

Orbital

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2p 2pz

z Orbital

Orbital

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Degenerate 2p Degenerate 2p Orbitals Orbitals

• All 3 orbitals have the same energy (n and l

l), but differ in orientation (ml

l)

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23 / 2 / 3

2 3 2 4 6 27 1

a Zr radial

e r a Z r a Z a Z

−

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ψ

Number of radial nodes = n -

• 1

Number of radial nodes = 3 - 1 - 1 = 1

• 0.04
• 0.02

0.00 0.02 0.04 0.06 0.08 0.10 5 10 15 20 25 30

Distance from Nucleus (Zr/a0) Ψradial ((Z/a0)3/2)

l

3p Radial 3p Radial Wavefunction Wavefunction

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3p 3px

x Orbital

Orbital

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3p 3py

y Orbital

Orbital

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3p 3pz

z Orbital

Orbital

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d d Orbitals Orbitals

When n = 3, what are the values of l? l = 0, 1, 2 so there are 3 subshells in the shell. For l = 0, ml = 0

• --> s subshell with single orbital

For l = 1, ml = -1, 0, +1

• --> p subshell with 3 orbitals

For l = 2, ml = -2, -1, 0, +1, +2

• --> d subshell with 5 orbitals

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d d Orbitals Orbitals

s orbitals have no planar nodes (l = 0) and are spherical. p orbitals have l = 1, have 1 planar node, and are “dumbbell” shaped. This means d orbitals (l = 2) have 2 planar nodes

typical d orbital planar node planar node

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3d Radial 3d Radial Wavefunction Wavefunction

3 / 2 2 / 3

2 30 81 1

a Zr radial

e a Zr a Z

−

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ψ

Number of radial nodes = 0

0.00 0.01 0.02 0.03 0.04 0.05 5 10 15 20 25 30 35

Distance from Nucleus (Zr/a0) Ψradial ((Z/a0)3/2) 30

3d 3dxy

xy Orbital

Orbital

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3d 3dxz

xz Orbital

Orbital

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3d 3dyz

yz Orbital

Orbital

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3d 3dz

z2

2 Orbital

Orbital

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3d 3dx

x2

2-

• y

y2

2 Orbital

Orbital

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d d Orbitals Orbitals

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f f orbitals

• rbitals

When n = 4, l = 0, 1, 2, 3 so there are 4 subshells in the shell. For l = 0, ml = 0 → s subshell with single orbital For l = 1, ml = -1, 0, +1 → p subshell with 3 orbitals For l = 2, ml = -2, -1, 0, +1, +2 → d subshell with 5 orbitals For l = 3, ml = -3, -2, -1, 0, +1, +2, +3

→ f subshell with 7 orbitals

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f f orbitals

• rbitals