Molecular transitions

Chapter II: Molecular Structure

2.9 Molecular transitions

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.

(a transitions between different rovibronic states) while the latter induces rotational transitions within a single vibronic state. By observing either the absorption or emission of photons when these transitions are driven, we can determine the tran- sition frequencies and ultimately the energies of the states involved. In addition to providing a means to observe and measure molecules, molecular transitions can also be utilized to control and manipulate molecules (laser cooling, coherent state preparation, etc.).

We are interested in understanding electric dipole transitions 20, how the electric dipole operator (first term in the multipole expansion of the electric field of the laser or microwaves) connects two different molecular states. For two given molecular states, the transition intensity or strength is given by the expectation value of the electric dipole operator, ˆ𝜇, between the two states

𝐼 =|⟨𝛼|𝜇ˆ|𝛽⟩|². (2.79) Here𝛼and𝛽describe all the quantum numbers of the ground and excited states.

Vibrational transitions

If we only consider vibronic transitions (e.g., the laser is broad enough that transitions from many rotational/fine/hyperfine states are excited at once so that the rotational, fine and hyperfine structure is unresolved) then the intensity is given by [81]

𝐼𝑣𝑖 𝑏𝑟 𝑜𝑛𝑖 𝑐 =|⟨𝜂^′′, 𝜈^′′|𝜇ˆ|𝜂^′, 𝜈^′⟩|²= |⟨𝜈^′′|𝜈^′⟩⟨𝜂^′′|𝜇ˆ|𝜂^′⟩|²=𝑞_𝜈′′,𝜈^′

𝜇_𝜂′′,𝜂^′

2

, (2.80) where here we use 𝜂 and 𝜈 to describe the electronic vibrational wavefuctions respectively, and we have used the spectroscopic notation of the double prime and prime reffering to the ground and excited states respectively. In Eq. 2.80 we have used the fact ˆ𝜇 does not operate on the vibrational coordinates. 𝑞_𝜈′′,𝜈^′ is known as the Frank Condon factor (FCF) which is the overlap integral of the vibrational wavefunctions

𝑞_𝜈′′,𝜈 =|⟨𝜈^′′|𝜈^′⟩|²=

∫

𝜓_𝜈′′𝜓^∗

𝜈^′𝑑 𝜏

2

. (2.81)

If the electronic PES is known for both the ground and excited states,𝑞_𝜈′′,𝜈^′ can be easily calculated. 𝜇_𝜂′′,𝜂^′is the transition dipole moment (TDM) and is dependent on the specific nature of the electronic wavefunctions. While it is possible to calculate

20There are also magnetic dipole, electric quadrupole, and higher-order transitions, however they are largely suppressed compared to electric dipole transitions and, for the transitions of interest here, are not discussed.

the TDM using numerical methods, it is often a measured quantity. For the set of different vibrational transitions between two electronic states,𝜇_𝜂′′,𝜂^′ is constant and the relative transition intensities are only dependent on the FCFs.

Following excitation to an excited state |𝜂^′, 𝜈^′⟩the molecule will decay back down to the ground state (or some other lower state). The probability that the molecule will decay to a given vibrational state |𝜂^′′, 𝜈^′′⟩is given by the vibrational branching ratio [91]

𝑏_𝜈′→𝜈^′′ =

𝑞_𝜈′′,𝜈^′𝜔³

𝜈^′′,𝜈^′

Í

𝜈^′′𝑞_𝜈′′,𝜈^′𝜔³

𝜈^′′,𝜈^′

, (2.82)

where𝜔_𝜈′′,𝜈^′ is the frequency of the transitions from |𝜂^′′, 𝜈^′′⟩to |𝜂^′, 𝜈^′⟩. Therefore the probability of a vibrational decay is dependent on the FCF. This fact becomes important when laser cooling molecules.

Vibrational selection rules

Whether or not a vibrational transition is allowed is determined by 𝑞_𝜈′′,𝜈^′. From Eq. 2.81 we know that𝑞_𝜈′′,𝜈^′ = 0 if the product𝜓_𝜈′′𝜓^∗

𝜈^′ is odd with respect to the origin of the coordinate system. This occurs when one of the wavefunctions is even and the other is odd. For a linear molecule, all stretching mode wavefunctions are even with respect to the origin and therefore nothing inherently prevents vibrational transitions between different stretching modes. However, for bending vibrations the situation is different. All bending wavefunctions with odd values of𝑙 are odd while even values of 𝑙 are even. This results in the following vibrational selection rules for linear molecules: Δ𝑙 =0 and thereforeΔ𝜈_{𝑏 𝑒𝑛𝑑} =0,±2,±4, ...More intuitively, this can be though of in the following way. Since the photon only interacts with the valence electron’s charge or the molecule’s dipole moment, it can not cause a change in the bending angular momentum (similar to how a photon can’t change the spin), andΔ𝑙 =0. For a more rigorous description of vibrational selection rules with respect to the molecular symmetry, see Ref. [81, 82].

It is important to note that forbidden Δ𝑙 ≠ 0 transitions due occur and are due to mixing from vibronic perturbations such as the Renner-Teller effect. For example consider the case of a ²Σ⁺(𝜈_{𝑏 𝑒𝑛𝑑} = 1) →² Π_1/2(𝜈_{𝑏 𝑒𝑛𝑑} = 0) transition which is normally forbidden. Vibronic and spin orbit interactions can mix the²Π_1/2(𝜈_{𝑏 𝑒𝑛𝑑} = 0)state with the bending mode of an excitedΣ⁻state so that the true wavefunction of the ²Π₁/2(𝜈_{𝑏 𝑒𝑛𝑑} = 0) state is now 𝜓 = |²Π₁/2(𝜈_{𝑏 𝑒𝑛𝑑=0}⟩ +𝜖|²Σ⁻(𝜈_{𝑏 𝑒𝑛𝑑} = 1)⟩. The²Σ⁺(𝜈_{𝑏 𝑒𝑛𝑑} =1) →²Σ⁻(𝜈_{𝑏 𝑒𝑛𝑑} =1) transitions is not forbidden which will now

allow the²Σ⁺(𝜈_{𝑏 𝑒𝑛𝑑} =1) →²Π_1/2(𝜈_{𝑏 𝑒𝑛𝑑} =0) to occur via intensity borrowing. A thorough calculation and measurement of this “forbidden” vibrational branching for CaOH, SrOH, and YbOH was done in Ref. [95].

Rotational and rovibronic transitions

If the rotational, fine, and hyperfine structure are resolved, then the transition inten- sity will also depend on the specifics of the rotational/fine/hyperfine states involved.

Taking this into account, the transition intensity is given by 𝐼𝑟 𝑜𝑣𝑖 𝑏𝑟 𝑜𝑛𝑖 𝑐 = |⟨𝜂^′′, 𝜈^′′, 𝜅^′′|𝜇ˆ|𝜂^′, 𝜈^′, 𝜅^′⟩|²=𝑞_𝜈′′,𝜈

𝜇_𝜂′′,𝜂^′

2𝑆_𝜅′′,𝜅^′ (2.83) where𝑆_𝜅′′,𝜅^′is the Hönl-London factor and𝜅denotes the quantum numbers which de- scribe the rotational/fine/ hyperfine state. In a case (a) basis

|𝜂, 𝜈, 𝜅⟩ = |𝜂,Λ;𝜈, 𝑙;𝑆,Σ;𝐽 , 𝑃⟩. In reality the state |𝜂, 𝜅⟩ will be some linear combination of the basis functions which will be determined by diagonalizing ˆ𝐻_{𝑒 𝑓 𝑓} and therefore the values of𝑆_𝜅′′,𝜅^′will depend on the specifics of the effective Hamil- tonian.

Rotational and rovibronic transitions have the following selection rules for the total angular momentum and parity; Δ𝐽 = 0,±1 (Δ𝐹 = 0,±1 when there is hyperfine structure), and parity+ ↔ −[96]. There are also the following approximately good selection rules; Δ𝑆 = 0, ΔΣ = 0, ΔΛ = ΔΩ = 0,±1. The value of Δ𝐽 is used to label different rotational branches. Δ𝐽 =−1 transitions form the P branch,Δ𝐽 =0 transitions form the Q branch, andΔ𝐽 =+1 transitions form the R branch.

More specifically, each transition is labeled with the following branch designation,

Δ𝑁

Δ𝐽_𝐹′ 𝑖,𝐹^′′

𝑖 (𝑁^′′) where Δ𝑁=O,P,Q,R,S for Δ𝑁 = −2,−1,0,1,2 and Δ𝐽=P,Q,R for -1,0,1. Here𝐹_𝑖does not refer to the angular momentumFbut denotes the spin orbit and spin rotation components of the ground and excited states. For a ²Σ⁺ →² Π transitions𝐹^′

𝑖 = 1 or 2 if the transition is to theΩ =1/2 or 3/2 spin orbit components respectively and 𝐹^′′

𝑖 = 1 or 2 if the transitions comes from the 𝐽 = 𝑁 +1/2 or 𝐽 =𝑁−1/2 spin rotation components respectively [97]. An example of a²Σ⁺ → ²Π transitions with the branch designations is shown in Fig. 2.2.