Analysis of vibronic coupling in a 4f molecular magnet with FIRMS

Vibronic coupling, the interaction between molecular vibrations and electronic states, is a fundamental effect that profoundly affects chemical processes. In the case of molecular magnetic materials, vibronic, or spin-phonon, coupling leads to magnetic relaxation, which equates to loss of magnetic memory and loss of phase coherence in molecular magnets and qubits, respectively. The study of vibronic coupling is challenging, and most experimental evidence is indirect. Here we employ far-infrared magnetospectroscopy to directly probe vibronic transitions in [Yb(trensal)] (where H3trensal = 2,2,2-tris(salicylideneimino)trimethylamine). We find intense signals near electronic states, which we show arise due to an “envelope effect” in the vibronic coupling Hamiltonian, which we calculate fully ab initio to simulate the spectra. We subsequently show that vibronic coupling is strongest for vibrational modes that simultaneously distort the first coordination sphere and break the C3 symmetry of the molecule. With this knowledge, vibrational modes could be identified and engineered to shift their energy towards or away from particular electronic states to alter their impact. Hence, these findings provide new insights towards developing general guidelines for the control of vibronic coupling in molecules.


Experimental electronic structure
(black), the CASSCF-SO CFPs X-Ray (XRD) crystallographic structure in Table S4 (light blue), and the CASSCF-CASPT2-SO CFPs of 1 in Table S3 (dark   blue), the CASSCF-SO CFPs of the optimised (1opt) structure in Table S7 (light red), and the CASSCF-CASPT2-SO CFPs of 1opt in Table S8 (dark red).

S9
To obtain ] T U / ,^5 V for each vibrational mode in Equation 9, we distort the molecular geometry along the normal mode coordinate up to some maximum displacement (vide infra) and calculate the electronic structure with CASSCF-SO (differences in CFPs are taken from the electronic states calculated at equilibrium using CASSCF-SO, Table S7). Assuming the harmonic approximation for each vibrational mode, we define the cartesian displacement vector of a given mode in its th harmonic state as:

S10
Where , is the reduced mass of mode , and c⃗ * , is a unit vector containing each atomic displacement of mode . We define the zero-point displacement (ZPD) of a mode as the magnitude of Equation S10 when , = 0: ,,GWX = g c⃗ ,," g = e ℏ , ,

S11
Simply setting the maximal displacement of each mode equal to its ZPD does not accurately capture the fact that, even at low temperatures, there is a non-zero population of excited vibrational states which have larger displacements. Therefore, we calculate the thermal population of excited states using the Boltzmann distribution and thus determine the thermally averaged displacement for each mode at some specific temperature. In this work, we choose a temperature of 4 K which corresponds roughly to that of the experimental FIRMS spectrum and displace the molecule up to ±1.5× the zero-point displacement (ZPD) for all modes.
When distorting along each mode, we calculate the electronic structure with CASSCF-SO at 4 evenly spaced points in both positive and negative directions up to ±1.5× ,,GWX , and fit each crystal field parameter at these points to a cubic polynomial ( Figure S6, Equation S12, where T U 5V is the set of CFPs obtained from the optimised structure using CASSCF-SO). Here, , is the dimensionless fraction of ,,GWX , thus , = 1 corresponds to a physical displacement of the atoms along c⃗ * , up to ,,GWX (Equations S10 and S11).

S12
Defining the equilibrium geometry as , = 0 the first derivative of Equation S12 with respect to , evaluated at equilibrium is simply the linear coefficient: The matrix elements Equation 10 are evaluated in the harmonic basis using the harmonic oscillator raising and lowering operators, which using our ZPD-based coordinate system can be written as m , + 1g p \ g , o = e , + 1 2 S14 As we restrict ourselves to the , = 0, 1 basis, the vibrational part becomes: To calculate the electric dipole moment matrix elements, we first define the electric dipole moment ] for a single particle between a pair of states (where is the particle's charge, and is the position operator): ] () = ⟨^| | _ ⟩ S16 For a system of many particles this becomes: In our case, we calculate the electric dipole moment in the basis of harmonic vibrational states | ⟩. This is typically accomplished by expanding the electric dipole moment as a Taylor series in vibrational mode coordinate , such that for mode : 2 The first term is independent of and is simply the electric dipole moment of the system at equilibrium (i.e. the permanent electric dipole) and so can be ignored. The term linear in is the transition dipole moment for a single quantum transition (e.g. |0⟩ → |1⟩), while higher order terms correspond to overtones (e.g. |0⟩ → |2⟩), which are small by comparison and are therefore excluded. Then, the electric transition dipole moment matrix elements are: is the derivative of the electric dipole moment with respect to the vibrational mode coordinate. We use the results of our DFT optimisation to obtain this derivative for each mode, however Gaussian does not use our ZPD based coordinate system, and instead defines , as displacement along one unit of c⃗ * , , a definition we refer to as z , , where we note that: Therefore, to match the value of k

S24
Therefore, in terms of z , A brief summary of the assignment of the main features of the FIRMS map is given in the main text, but here we give a more detailed account.
We begin with signal A ( Zd" = 393 cm -1 ) which moves to lower energy with increasing field; this is consistent with an electronically hot intra-KD transition |1 4 , 0⟩ → |1 $ , 1⟩. This signal intersects with a band moving to higher energy in the field range 8 -10 T and therefore could show some avoided crossing type behaviour; unfortunately, due to the low intensity of signal A, such behaviour is not seen within our current data. The nearby signal B ( Zd" = 407 cm -1 ) is likely the same as side-band 2a in the luminescence data (Figure 2), and is consistent with a cold intra-KD vibronic transition |1 $ , 0⟩ → |1 4 , 1⟩. Both A and B emanate from a region of low IR transmittance that shows only small shifts at its edges due to vibronic coupling, hence their apparent field-independence near zero-field is likely an artefact of the normalisation scheme.
Our  (Table S5, Supplementary Videos 14 and 15). Considering the excellent agreement with the zero-field FTIR spectrum ( Figure S2) and that there are no other vibrational modes within ±25 cm -1 of the A and B signals, we are confident with these assignments. Simulations of FIRMS maps obtained from coupling to modes 34-36 simultaneously ( Figure S7) gives further weight to our assignment, allowing us to reproduce the broad nature of the vibronic signals seen in the experiment, specifically the dominant band at ~410 cm -1 that moves to higher energy with increasing field.  Signals D ( Zd" = 468 cm -1 ) and E ( Zd" = 474 cm -1 ) are composed of many features at zero field which are poorly resolved, though we clearly see both hot and cold electronic bands moving as a function of field. Further, we note that signal D intersects with a band from signal C giving the appearance of an avoided crossing, but, similar to signal A, we do not see any interaction between the two, perhaps due to the low intensity of the peak that moves to low energy with field. Signals D and E are complicated as they are very close to the purely electronic transition at 474 cm -1 , and, hence, could be electronic (g1 ± , o → g2 ± , o), intra-KD cold (g1 ± , 0o → g1 ± , 1o, ℏ ~ 470 cm -1 ) or inter-KD hot (g1 ± , 0o → g2 ± , 1o, ℏ < 20 cm -1 ) bands.
Our DFT calculations do not clarify matters, as they reveal a mode at 473 cm   (Figure 3), coupling to modes 43-45 does yield filed dependent signal at the right energy. Thus, we suggest that signal F is a cold intra-KD transition g1 ± , 0o → |1 ∓ , 1⟩ coupling to vibrational modes 43-45.  one-another, all will couple to the electronic structure simultaneously, and so we see only a single purely electronic signal (g1 ± o → g2 ± o, Zd" ≈ 470-480 cm -1 ) in our 9-mode simulation which is very close to the experimental value ( Zd" ≈ 474 cm -1 ) ( Figure S12).   In the higher energy range near the second excited KD, there is a very intense signal G ( Zd" ≈ 775 cm -1 ) in the experimental FIRMS map which seems almost field independent at high (> 10 T) and low (< 6 T) fields, but more field dependent at intermediate fields ( Figure S17).
The shape of this signal is similar to that of the avoided crossings observed by Xue et al., in which a purely electronic and field dependent transition interacts with a nearby purely vibrational transition. 5 Unfortunately, we are unable to see any signal corresponding to the nearby purely electronic g1 ± , o → g3 ± , o transition at zero field, as it lies in a region of low IR transmission. However, due to the relatively strong field dependence of signal G and its distance from the expected g1 ± , o → g3 ± , o transition energy (ca. 745 cm -1 ), signal G is unlikely to be the result of an avoided crossing. Rather, we believe signal G is caused by an intra-Kramers vibronic transition and will have an increased signal intensity due to its proximity to the nearby electronic transition. To account for its unusual shape, we note that, much like signals A and B, signal G emanates from a region of high IR absorption (~770 cm -1 ) which effectively masks any less intense field dependent signals. Indeed, tracing backwards from the field-dependent region at 8 T (785 cm -1 ) to 0 T gives Zd" ≈ 765 cm -1 , which coincides   In the lower energy range, there are weak signals in the experimental FIRMS map at 167 cm -1 , 238 cm -1 , 553 cm -1 , 581 cm -1 , all of which are distant from any electronic transitions ( Figures   S19 and S20). Of these, the former two are clearly present in the luminescence measurements (Peaks 1a and 1c, Figure 2) and are intra-KD vibronic transitions at energies which match those of vibrational modes from our DFT calculations (Table S6). The energies of the latter two signals match well with vibrational modes from DFT and could be intra-KD signals.
However, as they are also above the energy of the first excited electronic state they could also be inter-KD transitions involving low energy modes which are also present in our DFT results.