Hund’s Rule, Singlet and Triplet Energies,
and the Pauli Exclusion Principle
Consider two electrons in two orbitals, φA and φB (e.g., two orthogonal 2p orbitals).
py = φA
px = φB
Let’s write a simple wavefunction for this system:
ψ = φA(1)φB(2)
(1)
Now, let’s calculate the probability of finding an electron using this wavefunction:
Electron Density = Probability of finding an electron = ψ2 = φA(1)2φB(2)2
•
•
•
(2)
This corresponds to finding electron 1 localized in orbital φA and electron 2 localized in
orbital φB. This electronic localization is forbidden by quantum mechanics.
Stated another way, electrons must be indistinguishable. If we label the electrons above,
we can clearly distinguish 1 from 2 because, according to our equation, electron 1 will
always be found in orbital φA and electron 2 will always be found in orbital φB.
Our wavefunction is not an acceptable description of the model system.
In order to determine whether or not a wavefunction is written such that electrons are
indistinguishable, we can apply a permutation operator (P12).
• A permutation operator is a mathematical function that switches electron labels. When P12
operates on acceptable wavefunctions, the result is ± the original wavefunction (obtaining
-ψ is acceptable since (-ψ)2 = ψ2).
If ψ is an acceptable wavefunction: P12 ψ = ± ψ
If ψ is an unacceptable wavefunction: P12 ψ ≠ ± ψ
(3)
(4)
Let’s try P12 on our original wavefunction:
P12 ψ = φA(2)φB(1) = ψ′ ≠ ψ
(5)
Just as we thought, ψ is an unacceptable wavefunction. So how can we create an acceptable
wavefunction? Let’s try a linear combination of ψ and ψ′.
ψ+ = 0.707[ψ + ψ′] (0.707 = (2)-1/2 is the normalization factor)
ψ+ = 0.707[φA(1)φB(2) + φA(2)φB(1)]
Supplemental 1
(6)
(7)
Now let’s use P12 to determine if ψ+ is acceptable.
P12 ψ+ = 0.707[φA(2)φB(1) + φA(1)φB(2)] = ψ+
(8)
Upon careful inspection, it is clear that the permuted ψ+ is the same as the original ψ+. Therefore,
ψ+ is an acceptable wavefunction. Now let’s try the other linear combination of ψ and ψ′.
ψ- = 0.707[ψ - ψ′]
ψ- = 0.707[φA(1)φB(2) - φA(2)φB(1)]
(9)
(10)
Again, let’s use P12 to determine if ψ- is acceptable.
P12 ψ- = 0.707[φA(2)φB(1) - φA(1)φB(2)] = -ψ-
(11)
Upon inspection, it is clear that permuted ψ- is -ψ-. Therefore, ψ- is an acceptable wavefunction.
Thus, there are two acceptable wavefunctions for two electrons in two different orbitals.
ψ+ = 0.707[ψ + ψ′]
ψ- = 0.707[ψ - ψ′]
(12)
(13)
Because permuting ψ+ gives ψ+, ψ+ is called a symmetric spatial wavefunction (spatial because
the wavefunction describes the motion of the electrons in a particular region of space). By the
same token, because permuting ψ- gives -ψ-, ψ- is called an antisymmetric spatial wavefunction.
Now, let’s consider the total wavefunction, Ψ, which is a product of spatial (ψ) and spin
wavefunctions (γ).
Ψtot = ψspace γspin
(14)
According to the Pauli exclusion principle, no two electrons in the same atom can have all
quantum numbers the same. Therefore, the total wavefunction, Ψ, must be antisymmetric with
respect to electron exchange.
P12 Ψtot = -Ψtot
(15)
Let’s verify this by examining an antisymmetric total wavefunction comprised of space-and-spin
wavefunctions θA and θB for two electrons.
Ψ = 0.707[θA(1)θB(2) - θA(2) θB(1)]
(16)
Whenever states A and B are taken to be the same in all respects, including spin orientation, the
total wavefunction vanishes. It is easy to convince yourself that the same would not be true for a
symmetric wavefunction.
Supplemental 2
Now, let’s take a closer look at spin wavefunctions. Spin wavefunctions describe the spin
angular momentum of the electrons. For two spins, the allowable wavefunctions are:
γ↑↑+ = α(1)α(2)
(17)
γ↓↓+ = β(1)β(2)
(18)
γ↑↓+ = 0.707[α(1)β(2) + α(2)β(1)]
(19)
γ↑↓- = 0.707[α(1)β(2) - α(2)β(1)]
(20)
•
•
The origins of equations 19 and 20 are similar to their spatial counterparts, and the rules for
creating the spin wavefunctions are the same as those for creating allowable spatial
wavefunctions.
Again, by inspection, spin wavefunctions 17, 18 and 19 are symmetric and spin
wavefunction 20 is antisymmetric.
Because the Ψtot must be antisymmetric, the allowable total wavefunctions are:
Ψtot = ψ-γ↑↑+
Ψtot = ψ-γ↓↓+
Ψtot = ψ-γ↑↓+
Ψtot = ψ+γ↑↓-
(antisymmetric spatial and symmetric spin)
(antisymmetric spatial and symmetric spin)
(antisymmetric spatial and symmetric spin)
(symmetric spatial and antisymmetric spin)
(21)
(22)
(23)
(24)
The first three total wavefunctions (21 - 23) correspond to the three microstates of the triplet
manifold, while the final total wavefunction (24) corresponds to an open-shell singlet state.
•
•
•
Most notable is that the triplet states have antisymmetric spatial wavefunctions while the
singlet state has a symmetric spatial wavefunction.
The reason that this is noteworthy is that it is the spatial wavefunctions that are used to
determine energies.
In fact, the energies we wish to calculate are the electron-electron repulsion energies since
these are the essence of the energy difference between the singlet and the triplet states.
Let’s calculate the electron-electron repulsion energy of the singlet and triplet states using the
coulomb operator e2/r12 (where e is the charge on the electron and r12 is the interelectron
distance).
Esinglet = ∫∫[ψ+⎜e2/r12⎜ψ+]dτ1dτ2
= ∫∫{0.707[φA(1)φB(2) + φA(2)φB(1)] ⎜e2/r12⎜0.707[φA(1)φB(2) + φA(2)φB(1)]}dτ1dτ2
= 0.5∫∫{[φA(1)φB(2) + φA(2)φB(1)] ⎜e2/r12⎜ [φA(1)φB(2) + φA(2)φB(1)]}dτ1dτ2
= 0.5{∫∫φA(1)φB(2) ⎜e2/r12⎜φA(1)φB(2)dτ1dτ2 + ∫∫φA(1)φB(2) ⎜e2/r12⎜φA(2)φB(1)dτ1dτ2 +
∫∫φA(2)φB(1) ⎜e2/r12⎜φA(1)φB(2)dτ1dτ2 + ∫∫φA(2)φB(1) ⎜e2/r12⎜φA(2)φB(1)dτ1dτ2}
Now, within each double integral, collect terms according to electron labels:
Esinglet = 0.5{∫∫[φA(1)φA(1) ⎜e2/r12⎜φB(2)φB(2)]dτ1dτ2 +
∫∫[φA(1)φB(1) ⎜e2/r12⎜φB(2)φA(2)]dτ1dτ2 +
∫∫[φB(1)φA(1) ⎜e2/r12⎜φA(2)φB(2)]dτ1dτ2 +
∫∫[φB(1)φB(1) ⎜e2/r12⎜φA(2)φA(2)]dτ1dτ2}
Supplemental 3
(25)
(26)
There are four double integrals, but only two different double integrals: the first and fourth are
identical and the second and third are identical, but different from the first and fourth.
Combining like integrals gives:
Esinglet = ∫∫[φA(1)φA(1) ⎜e2/r12⎜φB(2)φB(2)]dτ1dτ2 + ∫∫[φA(1)φB(1) ⎜e2/r12⎜φB(2)φA(2)]dτ1dτ2
(27)
•
The first term in the right side of the equation 27 expresses the electron-electron repulsion
between the electron density of electron 1 in φA (i.e., φA(1)2) and the electron density of
electron 2 in φB (i.e., φB(2)2), and is a straightforward coulomb integral, jAB.
•
The second term in the right side of the equation 27 expresses the repulsion between
electrons 1 and 2 in the overlap region of φA and φB (i.e., φAφB), and is called the
exchange integral, kAB.
•
Thus
Esinglet = jAB + kAB
(28)
py = φA
px = φB
j = coulomb integral = electron-electron repulsion
between white and black; white and grey; black and
grey; white and white; black and black
k = exchange integral = electron-electron repulsion
within the red volume element
Overlap density = Red
Now let’s calculate the energy of the triplet state in the exact same way as we did the singlet
state:
Etriplet = ∫∫[ψ-⎜e2/r12⎜ψ-]dτ1dτ2
(29)
2
= ∫∫{0.707[φA(1)φB(2) - φA(2)φB(1)] ⎜e /r12⎜0.707[φA(1)φB(2) - φA(2)φB(1)]}dτ1dτ2
= 0.5∫∫{[φA(1)φB(2) - φA(2)φB(1)] ⎜e2/r12⎜ [φA(1)φB(2) - φA(2)φB(1)]}dτ1dτ2
= 0.5{∫∫φA(1)φB(2) ⎜e2/r12⎜φA(1)φB(2)dτ1dτ2 - ∫∫φA(1)φB(2) ⎜e2/r12⎜φA(2)φB(1)dτ1dτ2 ∫∫φA(2)φB(1) ⎜e2/r12⎜φA(1)φB(2)dτ1dτ2 + ∫∫φA(2)φB(1) ⎜e2/r12⎜φA(2)φB(1)dτ1dτ2}
Now, within the double integral, collect all terms according to electron labels:
Supplemental 4
Etriplet = 0.5{∫∫[φA(1)φA(1) ⎜e2/r12⎜φB(2)φB(2)]dτ1dτ2 ∫∫[φA(1)φB(1) ⎜e2/r12⎜φB(2)φA(2)]dτ1dτ2 ∫∫[φB(1)φA(1) ⎜e2/r12⎜φA(2)φB(2)]dτ1dτ2 +
∫∫[φB(1)φB(1) ⎜e2/r12⎜φA(2)φA(2)]dτ1dτ2}
(30)
Again, there are four double integrals but only two different double integrals. Combining like
integrals gives:
Etriplet = ∫∫[φA(1)φA(1) ⎜e2/r12⎜φB(2)φB(2)]dτ1dτ2 - ∫∫[φA(1)φB(1) ⎜e2/r12⎜φB(2)φA(2)]dτ1dτ2
Thus:
Etriplet = jAB - kAB
(31)
(32)
Comparing equation 32 with equation 28, we see that the triplet energy is lower than the singlet
energy by
Esinglet -Etriplet = 2kAB
(33)
•
•
Remember that the exchange integral is the electron-electron repulsion in the overlap region.
This is the amount of energy the system “saves” by aligning its electrons in a parallel,
high-spin fashion.
The greater the overlap region, the more stable the triplet state. This is the essence of Hund’s
Rule: place unpaired electrons into each degenerate orbital (to maximize MS) before
doubly occupying an orbital.
Another way of stating our result is that electrons of the same spin avoid each other.
• Remember that the region of space where two different electrons co-exist is the overlap
region.
• Thus, when two orbitals share a common volume element, spin-aligned electrons avoid this
region of shared space and lower the energy.
• It is worth noting that an overlap region can exist for two orthogonal orbitals (e.g., two 2p
orbitals on the same atom) even though the overlap integral for orthogonal orbitals is
zero.
Yet another way of expressing the correlated motion of spin-aligned electrons is to examine the
spatial wavefunctions that describe their motion.
ψ+ = 0.707[φA(1)φB(2) + φA(2)φB(1)]
ψ- = 0.707[φA(1)φB(2) - φA(2)φB(1)]
(7)
(10)
When two electrons approach one another, their coordinates become nearly identical. In this
case, consider the limit of each spatial wavefunction when the coordinates of electron 1 approach
those of electron 2:
limψ- = lim{0.707[φA(1)φB(2) - φA(2)φB(1)]}
≈ 0.707 [φA(2)φB(2) - φA(2)φB(2)]
=0
Supplemental 5
(34)
limψ+ = lim{0.707[φA(1)φB(2) + φA(2)φB(1)]}
≈ 0.707 [φA(2)φB(2) + φA(2)φB(2)]
≠0
(35)
In a sense, spin-aligned electrons “avoid” each other to prevent the wavefunction from
vanishing, while spin-paired electrons do not have to “avoid” each other. This last example is a
nice description supplied by Professor David Shultz at North Carolina State University.
What then can cause a small singlet-triplet gap, or an open-shell singlet ground state?
• Obviously, if the exchange integral is small (caused by small overlap density) then the
singlet-triplet gap will be small.
• This is often the case when φA and φB are multi-atom orbitals and the electron density is
shared by many atoms.
• There is another important reason. When we first considered the spatial wavefunctions, we
considered only open-shell configurations. There are other closed-shell wavefunctions
that are also acceptable:
ψzA = φA(1)φA(2)
ψzB = φB(1)φB(2)
(36)
(37)
Typically, zwitterionic singlet states described by these wavefunctions are higher in energy than
the open-shell states, but they can mix with the open-shell singlet through configuration
interaction and lower the latter state below the triplet state.
Configuration interaction lowering of the singlet state becomes more important when the
zwitterionic states are low-lying. This is, mixing is strongest when the energy gap between the
states that are mixing is small (second-order perturbation).
Supplemental 6