Hyperfine structure

Chapter II: Molecular Structure

2.8 Hyperfine structure

accounted for as well. Oftentimes, and especially for low-rotational states, these corrections will not make any significant or measurable changes to the energy levels, and they do not need to be included in the effective Hamiltonian.

overlap the electron wavefunction has with the nucleus. Comparison of determined 𝑏_𝐹 values to atomic𝑏_𝐹 values can provide experimental information on the atomic orbital composition of the MO.

The second magnetic dipole interaction is referred to as the anisotropic dipole- dipole interaction and describes the magnetic dipole-dipole interaction between the electron and nuclear magnetic dipole moments. The effective Hamiltionian for the dipole-dipole interaction is [89, 94]

ˆ 𝐻_𝑐 = 1

3𝑐(3𝐼_𝑧𝑆_𝑧−I·S), (2.68) where𝑐is the dipole-dipole coupling parameter and is defined as

𝑐= 𝜇₀ 4𝜋 ℎ

3

2Σ𝑔_𝑒𝑔_𝑁𝜇_𝐵𝜇_𝑁⟨𝜂,Λ;𝑆,Σ|∑︁

𝑖

s𝑖

(3𝑐𝑜 𝑠²𝜃_𝑖−1) 𝑟³

𝑖

|𝜂,Λ;𝑆Σ⟩. (2.69) 𝑐is again dependent on the valence electron’s wavefunction and therefore measure- ments of it can provide information about the valence MOs.

In Eq. 2.68 and 2.69, we only considered the terms from the magnetic dipole- dipole interaction which were diagonal inΛ. There are also terms off-diagonal in Λ, specifically those which connect states of |Λ = ±1⟩ to states with |Λ = ∓1⟩16. These off-diagonal terms provide a hyperfine contribution to theΛ-doubling so that one component of a Λ-doublet obtains a different hyperfine shift than the other.

Therefore, if the state in question has a non-zero value of Λ, the parity-dependent hyperfine interaction must be included. The effective Hamiltonian for this interaction is given by [89, 94]

ˆ 𝐻_𝑑= 1

2𝑑(𝑆₊𝐼₋𝑒^{−2𝑖 𝜙}+𝑆₋𝐼₊𝑒^{2𝑖 𝜙}), (2.70) where𝑑is the parity-dependent hyperfine parameter and is given by

𝑑 = 𝜇₀ 4𝜋 ℎ

3 2

𝑔_𝑒𝑔_𝑁𝜇_𝐵𝜇_𝑁

√︁

𝑆(𝑆+1) −Σ(Σ+1)

⟨𝜂,Λ = −1;𝑆,Σ+1|∑︁

𝑖

s⁺_𝑖 (𝑠𝑖𝑛²𝜃_𝑖𝑒^{−2𝑖 𝜙𝐼}) 𝑟³

𝑖

|𝜂,Λ =+1;𝑆,Σ⟩

(2.71)

where s_𝑖⁺ is the spin-raising operator for a single electron and 𝜙_𝑖 is the azimuthal angle for the𝑖th electron.

16There are also terms which changeΛby 1 and therefore would mix electronic state. These effects would only show up at second-order in the effective Hamiltonian and provide energy shifts much below any measurable limit so they are generally not included. These second order terms only become measurable when there are nearly degenerate electronic states [94].

Finally, if the molecular state has Λ ≥ 1, there will be an additional magnetic hyperfine interaction describing the magnetic dipole interaction between the nuclear spin and the magnetic moment generated by the orbiting electron. This is analogous to the spin orbit interaction. The effective Hamiltonian for the nuclear spin orbit interaction to first order17is given by [94]

ˆ

𝐻_𝑎=𝑎 𝐼_𝑧𝐿_𝑧, (2.72)

where𝑎is the nuclear spin orbit parameter and is given by 𝑎 = 𝜇₀

4𝜋 ℎ

𝑔_𝑒𝑔_𝑁𝜇_𝐵𝜇_𝑁1

Λ⟨𝜂,Λ;𝑆,Σ|∑︁

𝑖

𝑙_𝑧𝑖 𝑟³

𝑖

|𝜂,Λ;𝑆,Σ⟩. (2.73) In the above equation𝑙_𝑧𝑖is the z-component of the single electron angular momentum operator of the𝑖th electron. Note in the above discussion we have assumed that there is only one nucleus with non-zero spin. If the molecule contains multiple nuclei with non-zero spin then the above interactions with each nucleus must be accounted for separately. Therefore, if the molecule has 𝑗 nuclei, the total magnetic dipole hyperfine effective Hamiltonian is give by18

ˆ

𝐻_{dipole𝐻 𝐹 𝑆} =∑︁

𝑗

ˆ

𝐻_{𝑎 𝑗} +𝐻ˆ_𝑏

𝐹𝑗 +𝐻ˆ_{𝑐 𝑗}+𝐻ˆ_{𝑑 𝑗}. (2.74) Electric Quadrupole terms

If a nucleus has a nuclear spin ofI ≥ 1 then the nucleus could have an observable quadrupole moment, a non-spherical distribution of mass and charge so that the nucleus resembles an ellipsoid. This non-spherical charge distribution will result in the nucleus having an electric quadrupole moment. This electric quadrupole moment will interact with the electric field gradient created between the positively charged nucleus and the negativly charged electrons. This will lead to different orientations of the quadrupole deformed nucleus in the electric field gradient having different energies. Since the orientation of the nucleus is given by the orientation of the nuclear spin, this will result in an interaction that is dependent on both the nuclear spin orientation and the magnitude of the electric field gradient. The effective Hamiltonian that describes this electric quadrupole interaction is [89]

ˆ

𝐻_{𝐸 𝑄0}=𝑒²𝑄 𝑞₀

(3𝐼_𝑧−I²)

4𝐼(2𝐼−1), (2.75)

17Technically ˆ𝐻_𝑎 = 𝑎I·Lbut, just as with the spin orbit interaction, the effective operator is given by only the elements diagonal in𝜂,𝐼_𝑧𝐿_𝑧.

18We have not included the nuclear spin rotation (analogous to fine structure spin rotation) terms in the effective dipole hyperfine Hamiltonian as they are not relevant for describing the structure of YbOH. The nuclear spin rotation effective Hamiltonian is ˆ𝐻_𝑁𝑆 𝑅=𝐶_𝐼I·J.

where 𝑒 is the charge of the electron and 𝑄 is the quadrupole moment of the nucleus. 𝑞₀is the expectation value of the electric field gradient over the electronic wavefunction

𝑞₀ =−2⟨𝜂,Λ|𝑇²

𝑞=0(∇𝐸) |𝜂,Λ⟩= −1 4𝜋𝜖₀ℎ

⟨𝜂,Λ|∑︁

𝑖

(3𝑐𝑜 𝑠²𝜃_𝑖−1) 𝑟³

𝑖

|𝜂,Λ⟩, (2.76) where𝜖₀ is the electric constant,𝑇²

𝑞=0(∇𝐸) is the𝑞 =0 component of the second- rank spherical tensor describing the electric field gradient19, and the sum over𝑖 is now taken over all electrons.

In Eq. 2.75 we have only included terms diagonal in Λ; however, the electric quadrupole interaction also has terms off diagonal inΛ. As with the dipolar interac- tions, the terms which mix states withΔΛ =±1 will cause electronic states to mix and will only become relevant at higher order. These terms are not included in the effective Hamiltionian. The off diagonal terms which connect states withΔΛ =±2 are included in the non-axial electric quadrupole effective Hamiltonian

ˆ

𝐻_{𝐸 𝑄2}=−𝑒²𝑄 𝑞₂

(𝑒^{−2𝑖 𝜃}𝐼₊²+𝑒^{2𝑖 𝜃}𝐼₋²)

4𝐼(2𝐼−1) . (2.77)

𝑞₂is the non-axial electric quadrupole coupling parameter and is determined by the non-diagonal effects of the electric field gradient

𝑞₂ =−2

√ 6 ∑︁

𝑞=±2

⟨𝜂,Λ = ±1|𝑇²

𝑞(∇𝐸) |𝜂,Λ =∓1⟩, (2.78) where now the𝑇²

𝑞=±2(∇𝐸)are the𝑞 =±2 components of the second-rank spherical tensor describing the electric field gradient. ˆ𝐻_{𝐸 𝑄2} has the selection rulesΔΣ = 0 and ΔΛ = 2 and therefore will only connect states with ΔΩ = 2. Therefore, in a

2Π state, ˆ𝐻_{𝐸 𝑄2} will only connect states from different spin orbit components, the Ω =1/2 andΩ =3/2 states.