# 1.3 Review of atomic orbitals and their shapes

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1_3_OrbitalsReviewSection 1.3 - 1

1.3 Review of atomic orbitals and their shapes • Atomic orbitals describe the electron distribution as a standing matter wave in the

potential field of a nucleus:

• Derived by solving the Schrödinger equation for the hydrogen atom. • Because every Ψ matches an atomic orbital, there is no limit to the number of

solutions to the Schrödinger equation! • Each Ψ describes the wave properties of a given electron in a particular orbital. • 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!)

• 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

Ψ=Ψ

Section 1.3 - 2

• 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, … - l = # of planar nodes = s, p, d, f, … - 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 - 3

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 - 4

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 - 5

• The shape of the radial functions and their nodal planes are: Source: Tarr & Miessler, Inorganic Chemistry, 2nd Edition. • 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 - 6

• These are 2D “sections” of the 3D orbitals (imagine slicing through an onion):

Section 1.3 - 7

• In general, we list the orbitals in order of energy as: 1s, 2s, 2p, 3s, 3p, 4s, 3d, etc. • Where does this energy ordering come from?

Section 1.3 - 8

The energies of the orbitals in many electron atoms are, in part, a function of nuclear charge:

Source: Shriver & Atkins, Inorganic Chemistry, 3rd Edition. • 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 - 9

1.4 Penetration and shielding

• 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 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.

• 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

• 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.

• This results in the filling order:

n+ l = 1 2 3 4 5

Orbital 1s 2s 2p, 3s 3p, 4s 3d, 4p, 5s

1.3 Review of atomic orbitals and their shapes • Atomic orbitals describe the electron distribution as a standing matter wave in the

potential field of a nucleus:

• Derived by solving the Schrödinger equation for the hydrogen atom. • Because every Ψ matches an atomic orbital, there is no limit to the number of

solutions to the Schrödinger equation! • Each Ψ describes the wave properties of a given electron in a particular orbital. • 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!)

• 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

Ψ=Ψ

Section 1.3 - 2

• 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, … - l = # of planar nodes = s, p, d, f, … - 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 - 3

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 - 4

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 - 5

• The shape of the radial functions and their nodal planes are: Source: Tarr & Miessler, Inorganic Chemistry, 2nd Edition. • 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 - 6

• These are 2D “sections” of the 3D orbitals (imagine slicing through an onion):

Section 1.3 - 7

• In general, we list the orbitals in order of energy as: 1s, 2s, 2p, 3s, 3p, 4s, 3d, etc. • Where does this energy ordering come from?

Section 1.3 - 8

The energies of the orbitals in many electron atoms are, in part, a function of nuclear charge:

Source: Shriver & Atkins, Inorganic Chemistry, 3rd Edition. • 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 - 9

1.4 Penetration and shielding

• 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 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.

• 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

• 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.

• This results in the filling order:

n+ l = 1 2 3 4 5

Orbital 1s 2s 2p, 3s 3p, 4s 3d, 4p, 5s