Section 6.4 - 1

6.4 Exchange Coupling, a different perspective • So far, we’ve only been looking at the effects of J on the magnetic susceptibility but

haven’t said anything about how one might predict the sign and magnitude of J. • Exchange coupling can frequently be understood using a simple molecular orbital

approach. • We all know that the lowest energy configuration for two electrons in two orthogonal

orbitals is such that the individual spins are the same. (Look back at our intro lecture). This is “Hund’s Rule” of maximum multiplicity.

So, if our unpaired electrons are in two orbitals that are:

1. orthogonal 2. have significant overlap such that interaction occurs 3. have negligible contributions from excited state singlets through configuration

interaction then we observe FM coupling

pz dz2 • If our unpaired electrons are in two non-orthogonal orbitals (non-zero overlap

integral), then they are governed by the Pauli principle and we see AFM coupling

pz dxz THUS, by inspection of the geometry of the magnetic orbitals, we can predict the spin ground state and therefore the sign of J.

Section 6.4 - 2

This model works well even if the exchange coupling is mediated by diamagnetic (filled or empty) orbitals. We then refer to the coupling mechanism as superexchange. It is often the case that two or more paramagnetic metal ions are coordinated in a molecule by a diamagnetic (i.e., closed shell) ligand. Equally, exchange coupling between two paramagnetic ligands (e.g., semiquinones) can be mediated by a diamagnetic metal center. Examples of superexchange coupling mechanisms:

This superexchange model invokes the principle of spin polarization. • If you have ever performed Gaussian DFT calculations on an open shell system, you

will know that you have to use an “unrestricted” density functional and you will notice that the resulting calculations generate a unique spatial molecular orbital with a unique energy for each INDIVIDUAL electron in the system.

• The output file lists these as α and β orbitals to denote the spin of the occupying (or

virtual) electron. By default, there are more α spins than β spins, meaning that if you are calculating a system that is a ground state doublet (S = ½), the spin of the unpaired electron is defined as being α.

• You can generate a map of the spin density on the molecular system. You’ll see that

there is an uneven spatial distribution of α and β spin (usually called positive and negative spin density). This is a result of spatial polarization of the underlying filled orbitals according the electron spin interactions.

Section 6.4 - 3

Spin Polarization and Molecular Diradicals • The concept of double (or dynamic) spin polarization (DSP) can be used in order to

understand electron-electron repulsion in open-shell species for which S > ½. • This model requires that we use different orbitals for different spins.

i.e., each MO can only contain 1 electron ! • The rationale behind this model is the following:

Ø An unpaired electron in a frontier MO has a specific spatial distribution. Ø The magnetic moment of this electron interacts with the magnetic moments of the

paired electrons in lower-lying filled MOs and thus influences their relative energies and spatial distributions.

Ø The influence of the unpaired electron spin is such that the α spin and β spin electrons of a pair will be differently affected and will therefore have different energies and different spatial distributions. We can no longer consider them to be in the same MO.

For example:

DSP results in two possible scenarios:

1. Additive DSP occurs when multiple unpaired electrons polarize filled orbitals in the same way. This lowers the energy (lowers electron-electron repulsions).

2. Competative DSP arises when an unpaired electron polarizes filled shells in the

opposite fashion from the previous electron. This raises the energy.

Section 6.4 - 4

Let’s look at an example: square planar C4H4

Are the two electrons in the NBMOs high-spin or low-spin coupled? • If we place an α electron into the first NBMO (Ψ2), the Ψ1 will be polarized as

follows…

This polarization reduces the Coulombic repulsions between Ψ1(β) and Ψ2(α) and enhances exchange repulsions between Ψ1(α) and Ψ2(α). Both of these effects lower the total energy.

Section 6.4 - 5

• Now, we introduce the second electron…let’s choose a β spin and see what happens.

Introducting a β spin into Ψ3 polarizes Ψ1 in exactly the same way. The DSP is additive! If we had introduced a second α spin electron into Ψ3, the polarization caused by this second electron would negate the polarization caused by the first α spin electron. There would have been no overall gain.

Conclusion: Compared to the singlet configuration, the triplet experiences greater electron-electron repulsion, and therefore has a greater total energy. THEREFORE: The singlet is the ground state. NOTE: These results can be generalized to any disjoint diradical system, i.e., any diradical system in which there are no atomic orbitals that contribute non-zero coefficients to both of the singly occupied molecular orbitals (…so everywhere that there are coefficients on one, there are nodes on the other, and vice versa).

Ψ1(β)

Ψ2(α)

Ψ1(α)

Ψ3(β)

Ψ1(β)

Ψ2(α)

Ψ1(α)

Ψ3(α)

polarizationfrom 1st e-

polarizationfrom 2nd e-

Section 6.4 - 6

Let’s look at another example: trimethylene methane (TTM) C3H6

(Note: These are the MOs from the 2pz atomic orbitals of carbon.)

Clearly, the triplet is lower in energy. Therefore the map of spin polarization looks like this: NOTE: The is β spin density at the central C atom! There is a node in the doubly degenerate SOMO at this atom. Large β spin density is very common at atomic sites that are nodal in the SOMO. An EPR active nucleus with β spin density will give rise to coupling (i.e., hyperfine structure).

Ψ1

Ψ2 Ψ3

H H

H

H

H

H

D3h

Ψ4

Ψ2(α) Ψ3(α)

SPIN POLARIZED TRIPLET SPIN POLARIZED SINGLET

Ψ1(α)

Ψ1(β)

Ψ2(α) Ψ3(β)

Ψ1(α)

Ψ1(β)

H H

H

H

H

H

αα

α

β

Section 6.4 - 7

IN GENERAL: Spin polarization in conjugate carbon π systems generally gives an alternating spin polarization scheme.

SPIN POLARIZATION of the σ framework … Example: a σ-bond (C-H) …how spin density “leaks” onto the H atom …EPR HYPERFINE COUPLING to H atoms!

CH2

α

βα

β

αβ

α

CH2

α

βα

β

αβ β

αβ

α

αβ

αβ

α

Ψ1

Ψ2

Ψ3H

(σ)

(2p NBMO)look at terminalH atom

(σ∗)

Spin density at the nucleus of C is zero (angular node of π orbital), so how does the H atom "see" spin density?

Section 6.4 - 8

Inclusion of the spin polarization effects lowers the total energy and elucidates the origin of hyperfine coupling to the 1H nuclei in EPR experiments.

The negative spin density is “seen” by the H nucleus. McConnell equation: aH = QρC • States that the hyperfine coupling constant at H (aH) is directly proportional to the

spin density at the carbon to which it is attached. • The proportionality constant Q is a measure of spin polarization. Topological coupling Given the coupling mechanisms described above, we can derive some simple topological models to predict the nature of intramolecular coupling in various molecular species with more than one unpaired electron. • Doubly occupied, doubly degenerate nondisjoint MOs are characteristic of ground

state triplet molecules (large S-T gap) • Doubly occupied, doubly degenerate disjoint MOs are characteristic of singlet (or

small S-T gap) molecules

Ψ1(β)

Ψ2(α) (2p NBMO)

Ψ1(α)

C H

radical

σ−bond polarizes H e- density to give (-)ve spin density at H.

hyperfine coupling to 1H(negative spin density on H)