Comparison of Magnetic Deflection among Neutral Sodium-Doped Clusters: Na(H2O)n, Na(NH3)n, Na(MeOH)n, and Na(DME)n

Using a pulsed Stern–Gerlach deflection experiment, we present the results of a comparative study on the magnetic properties of neutral sodium-doped solvent clusters Na(Sol)n with n = 1–4 (Sol: H2O, NH3, CH3OH, CH3OCH3). Experimental deflection ratios are compared with values calculated from molecular dynamics simulations. NaNH3 and NaH2O are deflected as a spin 1/2 system, consistent with spin transitions occurring on a time scale significantly longer than 100 μs. For all other clusters, reduced deflection is observed. The observed magnetic deflection behavior is correlated to the number of thermally populated rotational states in the clusters. We discuss that spin–rotational couplings allow for avoided crossings and a reduction in the effective magnetic moment of the cluster. This work attempts to understand the evolution of magnetic properties in isolated weakly bound clusters and is relevant to diamagnetic and paramagnetic species expected to exist in solvated electron systems.


INTRODUCTION
Alkali metal-doped solvent clusters provide ideal experimental and theoretical model systems to study properties of excess electrons as a function of system size 1−13 (and references therein).The formation of solvated, spin-paired electrons as a function of alkali metal concentration in sodium-doped nanodroplets 14,15 and liquid bulk solutions 16 is of current interest.Probing spin-pairing effects via photoelectron spectroscopy (PES) is limited in its application due to the similar electron kinetic energies of the involved species.However, investigating differences in the magnetic properties of sodiumdoped clusters might be a favorable approach to probe spinpairing effects.Distinguishing diamagnetic (singlet) and paramagnetic (triplet) states of singly sodium-doped clusters using Stern−Gerlach (SG) deflection is suggested here as a first experimental step toward future studies of more complex systems.
The original SG experiment was designed to determine the quantization of the electron spin in silver atoms. 17,18In atoms, the total angular momentum J is composed of the orbital angular momentum L and its intrinsic spin angular momentum S, neglecting the nuclear spin.In an SG experiment, each Zeeman level corresponds to an individual deflecting beamlet. 19In polyatomic molecules, additional degrees of freedom may contribute to the total angular momentum.The molecular Zeeman effect is analogous to that in atoms, with additional contributions of angular momenta (e.g., rotational and vibrational).Theoretical investigations of the diatomic case were performed by Hill 20 and later Schadee. 21An overview of the theory in diatomic molecules is given by Berdyugina and Solanki, 22 where analytical expressions of the Zeeman levels are given.Gedanken et al. 23,24 showed the importance of spin−rotation interaction in SG deflection experiments of oxygen and nitrogen oxide radicals.
Amirav and Navon 25,26 found that paramagnetic molecules and stable radicals (TEMPO: 2,2,6,6-tetramethylpiperidine-Noxyl and DTBN: di-tert-butyl nitroxide) were deflected less than predicted.They attributed the discrepancy between experiment and prediction to be due to fast intramolecular spin relaxation induced by spin−orbit coupling, causing a loss of orientation of the magnetic moment and reduced deflection.However, Gedanken et al. 23 found that magnetic deflection spectra of TEMPO combined with line-profile calculations did not support the interpretation of spin relaxation processes occurring while traversing the deflector.For large paramagnetic molecular and cluster systems, the density of Zeeman-like levels becomes so high that quantum chemical calculations are prohibitively expensive.Various groups 27−29 developed more simplified theoretical models to explain SG deflection experiments within terms of intramolecular spin relaxation effects caused by several spin transitions while conserving the total angular momentum.De Heer and co-workers 30−33 observed one-sided deflection for Fe m , Co m and Ni m clusters (m = 10− 1000), which was interpreted as a rapid intracluster spin relaxation (ISR) process. 27−41 Here, we employ an SG deflection experiment to study the magnetic properties of neutral sodium-doped solvent cluster Na(Sol) n with 1 ≤ n ≤ 4 (Sol: H 2 O, NH 3 , CH 3 OH, CH 3 OCH 3 ).Interpretation of the SG deflection of molecular clusters is still a challenge.Singly sodium-doped solvent clusters are promising model systems as several characteristic properties can be exploited in order to understand their magnetic properties as a function of system size and type of solvent molecules.In the case of singly sodium-doped clusters with m s = ±1/2, zero field splitting terms are vanishing, and high-symmetry highest occupied molecular orbitals (HOMOs) suggest that contributions of spin−orbit coupling may be minor in some cases. 10,11In the presented work, hyperfine effects from couplings to nuclear spins are neglected, although Fuchs et al. 42 demonstrated that nuclear spin can diminish electron spin coherence in metal clusters.The magnetic deflection behavior of various sodium-doped clusters is discussed in terms of thermally accessible rovibrational states.Spin−rotational couplings can produce avoided crossings between Zeeman-like levels of the same J.We observe that the SG deflection behavior of the studied clusters appears to be linked to thermally accessible rovibrational states and that the measured magnetic deflection can be inversely related to the density of the clusters' rotational state manifolds.

Experimental Setup.
The experimental setup to study size-dependent magnetic properties of neutral sodiumdoped solvent clusters has been previously described in detail. 43Nevertheless, a short description of the experimental and theoretical methods is given here.A sketch of the experimental setup is shown in Figure 1.
Molecular clusters are generated via supersonic expansion of either neat gas or mixtures with He or N 2 into a vacuum (A).The cluster formation conditions of the individual substances are summarized in Table S1 of the Supporting Information (SI).The expansion is skimmed at S1, and the resulting cluster beam was doped with sodium as it traverses the temperaturecontrolled oven in the chamber (B).See Table S1 for the Naoven temperatures used in the presented work (SI).
The sodium-doped solvent clusters pass a 1.5 mm diameter skimmer (S2) to enter the deflection chamber (C) which houses the pulsed SG deflector.The magnetic field gradient dB/dz in the deflector flight channel acts on the paramagnetic clusters, causing the cluster beam to diverge.We have included a description of the magnetic field pulses in the SI.The magnetic field gradient obtained from COMSOL simulations at 1000 A deflector current is shown in Figure 1d in ref 43 and in the SI.Selection of clusters after the SG deflector is performed by a 2 mm diameter skimmer (S3) at the entrance of the detection chamber (D).With the existing experimental setup, we are able to measure clusters that are transmitted only through S3.
Upon reaching the ionization region inside the extraction optics, the neutral clusters are ionized with a pulsed (∼7 ns) 266 nm (4.66 eV) Nd:YAG laser.The resulting photoions are extracted perpendicular to the molecular beam axis and detected by either time-of-flight (TOF) mass spectrometry or velocity map imaging (VMI).Ion-resolved VMI was achieved by gating the microchannel plates of the imaging detector to only be high when the ions of interest arrived at the detector.The perpendicular configuration of the molecular beam axis and TOF axis allows us to determine molecular beam velocities v y .The displacement from the image center r is related to the neutral clusters' velocity v y by where V R is the voltage on the repeller, m is the cluster mass, and C is a setup-dependent calibration constant.Velocity distributions of the sodium-doped solvent clusters obtained in this study are listed in Section 3. Highest possible deflection in our pulsed SG setup is achieved by synchronization of magnetic field pulse trigger timings of the three coils in the deflector setup t 1 , t 2 , and t 3 to the laser timing t L .These relative time delays are optimized to the velocity distribution of each cluster beam studied (see ref 43) in order to achieve optimal deflection.Extracting photoions in VMI conditions and mass-gating allow us to record velocity-dependent and cluster-resolved deflection data.The relative signal of the photoions θ rel is defined as the ratio The Journal of Physical Chemistry A of the signal with the deflector "on", θ on , and the deflector "off", θ off .
where γ d = 1 indicates that 100% of particles are deflected and corresponds to θ on = 0. We do not expect fragmentation or monomer evaporation of these weakly bound clusters to be significant upon photoionization.We are ionizing the Na atom and not the monomer units of the cluster, and the outgoing photoelectron is carrying away the excess energy of the photon as the kinetic energy.Nadoping and UV ionization has been established as an ultrasoft method for cluster size determination. 44,45Furthermore, as an experimental verification that the magnetic deflector did not alter the cluster signal, we collect mass spectra and VMI's with the deflector "off" and with the deflector "on" but with the ionizing laser pulse not synchronized to interest the clusters that experienced the magnetic field.The signals showed no significant deviations.

Modeling of the Deflection.
A detailed description of the molecular dynamics (MD) approach to model the deflection process has been given previously. 43Here, we summarize the main aspects of our approach and point out improvements from earlier work.Cluster trajectories are simulated from the entry of the deflector to the ionization region.The initial coordinates (x, y, z) and momentum (p x , p y , p z ) of each particle are defined in the cluster beam, where the mass is defined as a given cluster mass.The velocity in the molecular beam direction (y-axis; see Figure 1) is sampled from experimentally determined velocity distributions.The velocity components in the x, z-direction are calculated from the beam divergence angle and v y .More than 10 5 particle trajectories are typically simulated.The experimental dimensions of the deflector, flight distances, skimmer diameters, and positions, as well as the ionization region, are used in our model.For particles which reach the ionization region simultaneously at time t L , an output is written.The simulation output consists of coordinates (x i , y i , z i ) at ionization and corresponding momentum (p x , p y , p z ) for particles of spin states −1/2, 0, and +1/2.Simulations of S = 0 produce results equivalent to those of deflector "off".Instead of taking the topology of the rovibrational Zeeman diagram into account to simulate the magnetic deflection behavior of the clusters, we have introduced an optional additional scaling factor of the magnetic moment to account for possible ISR effects.We scale the magnetic moment μ 0 of a single m s = ±1/2 particle with an exponential decay, defined by the interaction time with the magnetic field t m and a characteristic relaxation time τ.
where t m = s def /v y is defined by the deflector length s def and molecular beam velocity v y .By sampling various values of τ and comparing simulations to experimental deflection results, we estimate the characteristic spin relaxation times τ for the investigated clusters.With modeled effective magnetic moments μ eff , we quantitatively compare the deflection behavior of the clusters.2a shows the NaH 2 O velocity distributions (obtained from VMIs) for deflector "off" and "on" measurements at an operating current of 700 A and optimized deflector coil timings 43 for the sampled velocity distribution (center velocity v c = 2000 m/s, full width at half-maximum (fwhm) = 150 m/s) to achieve maximal possible deflection.Under these conditions, we measured a deflection ratio, γ d , of 0.66 (10) as shown in Figure 2b.Agreement between the experiment and simulations is found across the velocity distribution within experimental errors.Figure 2b summarizes the experimental γ d values for all sampled I d .Deflection (γ d > 0) is observed for I d ≥ 200 A and agrees with the MD simulations for each sampled deflector current.We note that the experimental γ d is slightly and systematically higher than the simulations for a spin 1/2 particle and suspect collisions of the cluster beam with background gas molecules (e.g., outgassing from the deflector) while the deflector is switched "on" to be a possible The Journal of Physical Chemistry A explanation for the systematic deviations.The heating of the deflector arises from thermal loads generated by the in-vacuum coils that cannot be completely dissipated with our cooling system.A total loss of the signal occurs at a chamber pressure above 5 × 10 −4 mbar, which we attribute to obstruction of the flight channel (see the SI of ref 43).The additional signal depletion contributes to measured deflection due to the comparison to deflector "off" measurements.Within experimental error, the deflection behavior of NaH 2 O is equal to that of an m s = ±1/2 particle.As we are only sensitive to processes occurring on time scales that are either faster or similar to t m , we have the condition that τ > t m must hold in order for a cluster to behave like an m s = ±1/2 particle in our experiment.For the data shown in Figure 2, t m is ∼100 μs, which is consistent with τ > 100 μs for NaH 2 O.

Magnetic Deflection of Na(H
In a previous study, we observed deflection behavior of an m s = ±1/2 particle for NaNH 3 and reduced deflection for cluster sizes of n > 1. 43 This raises the question whether Na(H 2 O) n clusters with n > 1 exhibit a similar cluster size dependence in magnetic deflection.
The deflection results for the highest possible magnetic field gradient (I d = 700 A) and a broad velocity distribution of the cluster beam (∼800−1500 m/s) are shown in Figure 3 for (a) Na(H 2 O) 2 , (b) Na(H 2 O) 3 , and (c) Na(H 2 O) 4 .The distinct peaks of the velocity profiles are caused by mechanical recoils of the pulsed Even−Lavie valve 46 used to generate the beam of water clusters.The simulations predict full deflection (θ rel = 0) for velocities <1100 m/s.This, however, is not observed in the experiment and means that these clusters do not interact with the magnetic field as free spin 1/2 systems.The results for Na(H 2 O) 2 , Na(H 2 O) 3 , and Na(H 2 O) 4 are consistent with spin transitions taking place on significantly faster time scales than the experiment.In the simulated data for a spin 1/2 particle (red traces) of Figure 3, the residual signal to the fast side of the velocity distribution results from the magnetic field pulses being too short in time to act on all of the velocities present in the molecular beam.The pulses of the magnetic field were optimized to deflect the slower velocities but were not long enough in time to act on all velocity components present in the cluster beam.This can be seen by the simulated data for a spin 1/2 particle being fully deflected for the slower velocities, partially deflected for the intermediate, and nondeflected for the fastest velocities of the profile in Figure 3. 4a displays velocity-calibrated photoion VMIs of NaMeOH for deflector "off" and deflector "on" measurements at I d = 700 A. NaMeOH exhibits deflection with θ rel = 0.5(1) over the full velocity distribution, and minimal relative signal (θ rel = 0.2(1)) in the velocity range of 800−1100 m/s.The residual signals to the slow and fast side of the velocity distribution are a result of the applied magnetic field pulses, which are too short in time to ensure significant magnetic field gradients across the sampled velocity distribution.For the data shown in Figure 4a, the magnetic field pulse timings were optimized for deflection of velocities near the center of the molecular beam's velocity profile.The slowest and fastest particles of the molecular beam did not experience a significant magnetic field gradient, and their trajectories were left unperturbed.Although NaMeOH exhibits clear deflection, it is less than the simulations predicted for an m s = ±1/2 system.Figure 4b  The deflector "off" trace and deflector "on" traces are the same within experimental error.The obtained simulations, however, predict deflection for all three cluster sizes.These deviations of experiment and simulation again show that the magnetic

The Journal of Physical Chemistry A
properties of the investigated clusters cannot be described by the magnetic moment of m s = ±1/2 particles.
For Na(NH 3 ) n (see ref 43) and Na(H 2 O) n (see Section 3.1), we noted that an increase in cluster size, n, was correlated with a decrease in the experimentally observed deflection, γ d .Similar cluster-size-dependent deflection behavior is observed for Na(MeOH) n .The presented results can be explained by ISR processes taking place on significantly faster time scales than the experiment.In order to discuss possible ISR processes of partially and nondeflecting clusters, we analyze thermally accessible rovibrational states as a function of cluster size, n see (Section 3.5).6a for deflector "off" and "on" measurements at I d = 700 A. We determined θ rel = 0.49 (5) for the sampled velocity distribution.The corresponding simulated data exhibits full deflection (θ rel = 0) for velocities between 600 and 900 m/s, but incomplete deflection (θ rel ≈ 0.20) for slower ≤600 m/s and faster ≥900 m/s particles.However, the simulation did not match our experimental observation.Figure 4b shows γ d as a function of deflector current, where deflection is observed for I d > 100 A. Quantitative agreement between experiment and simulation is not found for any of the studied deflector currents, although deflection increases with increasing deflector current in both.

Magnetic Deflection of
For NaDME, the deviation between simulations and experiment may be due to ISR processes, which occur on similar time scales to the interaction time with the magnetic field, causing a reduction of the cluster's magnetic moment.The Journal of Physical Chemistry A 3.3.2.Na(DME) 2 .Magnetic deflection data of Na(DME) 2 , analogous to that of NaDME, is presented in Figure 7.At I d = 700 A (Figure 7a), the deflector "on" signal is slightly decreased when compared to the deflector "off" signal, which results in an overall θ rel = 0.82 (7).According to the simulations, full deflection is expected in the velocity range of 600−800 m/s and residual signals for slower ≤600 m/s and faster ≥800 m/s particles.We classify Na(DME) 2 as partially deflected at I d = 700 A, whereas at lower deflector currents, no significant deflection was observed (see Figure 7b).
The reduced deflection observed for Na(DME) 2 may be due spin transitions occurring on a time scale similar to the interaction time with the magnetic field gradient.It is interesting to note that deflection of Na(DME) 2 becomes observable at I d > 600 A. The higher deflector current would produce a stronger magnetic field gradient and a larger Zeeman splitting, which could make more avoided crossings of Zeeman-like states.For I d > 600 A, it appears that the magnetic field gradient is large enough to deflect some Na(DME) 2 clusters, even though we essentially determine μ eff = 0 at lower currents.
3.3.3.Na(DME) 3 .Calibrated photoion VMIs of Na(DME) 3 at I d = 700 A are shown in Figure 8a.We classify Na(DME) 3 as nondeflected, although slight deflection might be visible in the region of the center velocities (650−850 m/s).We needed increased sensitivity to make a definitive statement.The simulated signal in contrast reduces to θ rel = 0.25 and as in the case of previous clusters exhibiting reduced deflection.Again, the deflection process is not described by the interaction of m s = ±1/2 particles with the inhomogeneous magnetic field.Measurements at lower deflector currents confirm this deflection behavior since γ d = 0 was measured (see Figure 8b).
The differences between experiment and simulation found for Na(DME) n (n = 1−3) are discussed in terms of possible ISR processes as a function of cluster size n in Section 3.5.All other clusters can be described with μ eff < μ 0 .As previously mentioned, it is expected that ISR processes would result in a reduction of μ 0 .By introducing an attenuation factor a, we describe the simulated deflection with μ eff = μ 0 •a.It has been discussed 27,29,47 that ISR processes are determined by the topology of the Zeeman diagram, the thermal population of spin−rotational-coupled Zeeman-like levels, spin−rotational coupling constants, and the magnetic field gradient ∇B a cluster experiences while traversing the deflector. 27,29,47An exact description considering all relevant interactions in detail is a complex task, and we therefore choose a simplified model to account for spin relaxation effects.Modeling the effective magnetic moment via the Curl formula 48 may be a valid approach for cluster systems which behave as rigid rotors with only a few vibrational states populated.Recently, Rivic et al. 49 have elegantly demonstrated this.

The Journal of Physical Chemistry A
As stated in eq 4, we assume an exponential decay of μ 0 during the transit time t m with a characteristic relaxation time τ.Performing MD simulations with μ eff for various characteristic relaxation times τ (25, 50, 75, 100, 125, and 150 μs) allows us to match the simulations to our experimental observations.For each τ value, the squared deviations (γ d,sim − γ d,exp ) 2 are evaluated by their χ 2 value given by Here, σ i is the experimental error of each sampled velocity v i over a region of interest chosen from the experimental velocity distribution.The evaluated χ 2 (τ) values are fitted quadratically as a function of τ to obtain the characteristic relaxation times.The evaluation of τ is carried out for each deflector current individually in order to be sensitive to any possible magnetic field dependence.For nondeflected clusters, the fitting procedure is not applicable, and we therefore report an upper limit on τ for such cases.The results are summarized in Table 1.
With the characteristic relaxation times τ, we determine where a i is the unitless average attenuation factor of the exponential term for a single particle in eq 4 during the interaction time t m while traversing the deflector, defined by the deflector length s def and molecular beam velocity v i .i k j j j j j i k j j j j j y In Figure 9, the effective magnetic moments for the partially deflecting clusters and the upper limit for the nondeflected clusters (black line) are shown as a function of the cluster velocity.Within our current experimental setup, we are not able to distinguish between clusters with effective magnetic moments smaller than the upper limit (black line in Figure 9) for a given molecular beam velocity.This black line represents the detection limit for deflection in our current experimental setup.
The interaction with the magnetic field gradient is defined by v i and the deflector length dictates the observable μ eff within the applied model.A comparison of different μ eff values is therefore only meaningful when compared at the same velocity or interaction time with a magnetic field gradient.For a constant velocity v i at any value (see Figure 9), we determine the following experimental deflection trend   The experimentally observed deflection trends can be rationalized by using the density and thermal population of rovibrational states within the Zeeman energy splitting.Harmonic frequency analysis of density functional theory (DFT)-optimized cluster geometries were carried out with the Gaussian program package. 50In a first step, the dispersion-corrected ωB97XD density functional with a 6-31+G* basis set was used to optimize cluster geometries.In the second step, harmonic frequencies and rigid rotor rotational constants were evaluated for various geometric isomers.In a last step, the converged minimal energy structures were reoptimized with MP2 calculations with an aug-cc-PVDZ basis set, and additional harmonic frequency calculations were carried out.
The Zeeman diagram of an m s = ±1/2 system is split into two eigenstates according to their spin orientation.In the case of m s = ±1/2, zero field splitting of spin microstates can be neglected as only one unpaired electron is present.Available rotational states increase the number of diabatic Zeeman-like levels to 2J + 1 = (2S + 1)•(2R + 1), where S is the spin, R the rotational, and J is the total angular momentum (neglecting contributions from additional angular momenta).If S and R are uncoupled, then degeneracies of the Zeeman-like states are allowed.However, if S and R are even weakly coupled, then crossings of the Zeeman-like states are avoided for adiabatic states with equal J.At an avoided crossing, m s changes sign in both of the associated Zeeman-like states, while the total angular momentum J = R + S is conserved.For m s = ±1/2 systems, this leads to Δm s = 1, which only allows for spin transitions between adiabatic Zeeman-like states with ΔM R = 1.In terms of the avoided crossing model, 29 clusters with a higher density of populated rovibrational states are expected to undergo more spin transitions and on average exhibit less deflection, when compared to clusters with lower densities of thermally accessible adiabatic Zeeman-like levels.
In order to discuss our experimental findings and relative deflection trends, we analyze rovibrational eigenstates E n,R .We choose to describe E n,R with respect to the vibrational energies of a harmonic oscillator and the rotational energies of a rigid rotor.
where n is the vibrational quantum number, υ 0 is the vibrational eigenfrequency, R is the rotational quantum number, and B rot is the rotational constant.With estimated vibrational T vib and rotational T rot temperatures, we express the average number of thermally accessible rovibrational states in terms of the partition function.The density of rovibrational states is expressed as the average number of populated states within the maximum achievable Zeeman energy splitting ΔE Zeeman = 2μ B B max ≈ 2 cm −1 , where μ B is the Bohr magneton and B max ≈ 2T.The following discussions are based on the assumption of thermally equilibrated clusters, where the population of states is described by a Maxwell−Boltzmann distribution at estimated thermal energies k B T.
In the case of S and R-coupled adiabatic Zeeman-like levels, states with m s = ±1/2 contributions are able to undergo avoided crossings if neighboring states with ΔM R = 1 have an energy difference smaller than the Zeeman energy splitting.We chose the maximal Zeeman energy splitting for a spin 1/2 system (ΔE Zeeman ≈ 2 cm −1 ) as an upper limit for all investigated clusters.Harmonic frequency calculations of the investigated clusters and their structural isomers (see the SI) show that differences in vibrational energies of thermally accessible modes are larger than the Zeeman energy splitting.We find that rotational energy level differences, for a rigid rotor approximated by E rot = ⟨B⟩ rot J(J + 1) with the average rotational constant ⟨B⟩ rot = (A rot + B rot + C rot )/3 (see A, B, C, and ⟨B⟩ rot in Tables S2−S5), are less than ΔE Zeeman for all studied clusters except NaH 2 O and NaNH 3 , which are the two clusters deflected as spin 1/2 particles.Thus, multiple rotational levels within ΔE Zeeman are to be expected in the cases in which we measure reduced deflection.We take the number of avoided crossings a cluster undergoes to be governed by the density of rotational energy levels within ΔE Zeeman and rather independent of the population of vibrational modes, as they are spaced by more than ΔE Zeeman .We, therefore, discuss avoided crossings in terms of rotational states, assuming no additional contributions from excited vibrational states.

Average Number of Rotational
States within ΔE Zeeman .We represent the average number of rotational eigenstates within ΔE Zeeman through the rotational partition function Q rot .We use Q rot to represent the average number of populated states.Then, Q rot divided by the average rotational energy k B T rot and multiplied by ΔE Zeeman corresponds to the average number of states within ΔE Zeeman .Our analysis is based upon a DFT-optimized cluster with various isomers (ωB97XD/6-31+G*).In the high-temperature approximation, the rotational partition function of a nonlinear rotor is expressed by where σ sym is the symmetry number, T rot the rotational temperature, and A rot , B rot , and C rot are the rotational constants of the rigid rotor.Since the rotational temperatures of free clusters are not well-defined, we assume a temperature range 10 K ≤ T rot ≤ 50 K.This range comes from the comparison of the vibrational and rotational energy transfer cross sections σ vib and σ rot , which follow σ vib ≪ σ rot . 51,5251,52In an adiabatic expansion, the differences in effective energy transfer lead to T rot < T vib .Fuchs et al. 53 determined T rot < T vib with 5 K ≤ T rot ≤ 20 K for small metal clusters generated in a supersonic expansion at a nozzle temperature of 16 K.In our experiments, nozzle temperatures are significantly higher (10 °C ≤ T nozzle ≤ 150 °C) and collisions with sodium atoms in the oven are expected to affect the cluster temperatures.Although it is unclear how much this influences the cluster temperatures, as The Journal of Physical Chemistry A the cluster can evaporate monomer units to dissipate the acquired energy, we take a rotational temperature range of 10 K ≤ T rot ≤ 50 K for the analysis.With estimated rotational temperatures and rotational constants, we determine the rotational partition functions Q rot via eq 10.The results are shown in Figure 10 for 1 ≤ n ≤ 4 where the letters (e.g., a and b) refer to different isomers of a cluster with a given number of monomer units (n) (see Figures S3−S6 in the SI).This analysis was performed to see if the average number of rotational states within ΔE Zeeman (represented by Q rot ) correlated with the trend in the magnetic deflection behavior we measured in this study.8][39][40][41]47 Comparison of Q rot in the cluster systems in the temperature range 10 K ≤ T rot ≤ 50 K shows the lowest values for NaH 2 O and NaNH 3 . Furermore, the rotational state population is only weakly dependent on rotational temperature for NaH 2 O and NaNH 3 .This is consistent with our deflection measurements having a similar behavior to that of a spin 1/2 particle or an effective magnetic moment similar to μ 0 .NaMeOH and NaDME exhibit a similarly low Q rot at 10 K but a more pronounced increase of Q rot with temperature and experimentally only reduced deflection compared to a simulated spin 1/2 particle of the same mass.
Estimates of thermal temperatures showed that Na(DME) n clusters are expected to be colder than the other clusters studied due to the relatively low nozzle temperature used for the DME expansion (see Table S1) and relatively weak intermolecular forces binding DME molecules.With T rot (Na-(NH 3 ) n ) > T rot (Na(DME) n ), we estimate fewer populated rotational states for NaDME than for Na(NH 3 ) 2 and confirm the experimental observation of μ eff (NaDME) > μ eff (Na-(NH 3 ) 2 ).Slight deflection of Na(DME) 2 is consistent with it having a lower rotational temperature, whereas Na(MeOH) 2 shows an increase of populated rotational states, which could explain its lack of deflection.At the lowest temperatures considered here, the change in the rotational state density between NaH 2 O and Na(H 2 O) 2 is unlikely to be the reason for the absence of deflection for Na(H 2 O) 2 .This leads us to the idea that the Na(H 2 O) n clusters may be hotter than the other sodium-doped solvent clusters studied.
The larger clusters of Na(MeOH) 3,4 and Na(DME) 3 exhibit significant population in many rotational states, which supports the observation of no significant deflection.The absence of deflection for Na(H 2 O) 3,4 is difficult to explain by only comparing thermally accessible rotational states.Possible explanations include a higher rotational temperature or contributions other than spin−rotational couplings.The structural isomers of Na(NH 3 ) 3,4 show substantially different populations of rotational states.For Na(NH 3 ) 3 , isomer 3b is 83 meV lower in energy than isomer 3a.The experimental observation of partial deflection for Na(NH 3 ) 3 may imply a preferential production of isomer 3b and its lower rotational state density.Similar arguments hold for Na(NH 3 ) 4 which shows partial deflection, which may be explained by preferential production of the tetrahedral isomer 4b with its significantly reduced population of rotational states compared to isomer 4a.Previous work of our group 10 found n = 4 to be a magic number for the photoelectron anisotropy in sodiumdoped ammonia clusters.These complementary experiments support the idea that Na(NH 3 ) 4 is produced and deflects a highly symmetric tetrahedral structure with substantially lower rotational state density.For the T d structure σ sym = 12, less symmetric isomers with lower σ sym (IB in ref 10) would cause an increase in rotational state density and may lead to reduced deflection.This hints at the possibility of cluster-symmetry selective deflection measurements.With a combination of magnetic deflection of neutral clusters and angle-resolved photoelectron spectroscopy, one could characterize isomerdependent magnetic and electronic properties.In such an experiment, magnetic deflection of highly symmetric cluster structures could be demonstrated by a decrease in the photoelectron anisotropy parameter β (measured in the nondeflected cluster beam), which can be related to the orbital angular momentum of the electron ejected upon photoionization. 10,11,54

The Journal of Physical Chemistry A
In general, we find a correlation between experimental deflection trends and rotational level spacings and populations.The most significant deviations of the experiment and model are possibly due to differences in cluster rotational temperatures, with a proposed order in our experiments being T rot (Na(DME) n ) < T rot (Na(NH 3 ) n ) < T rot (Na(MeOH) n ) < T rot (Na(H 2 O) n ).Although influences from contributions other than spin−rotational coupling may also be important, further and more refined studies are needed.

CONCLUSIONS
The magnetic deflection behavior of Na(H 2 O) n (n = 1−4), Na(NH 3 ) n (n = 1−4), Na(MeOH) n (n = 1−4), and Na(DME) n (n = 1−3) clusters was characterized.We showed that the fully deflecting m s = ±1/2 character of NaH 2 O and NaNH 3 is consistent with these clusters having large rotational level splitting compared with ΔE Zeeman .In all other clusters studied, the observation of reduced deflection was found to be consistent with an increased population of closely spaced, excited rovibrational states.Magnetic deflection observed to be less than that of a free m s = ±1/2 system was modeled by the attenuation of the magnetic moment μ 0 .With this approach, we were able to retrieve characteristic spin relaxation times τ.The relative experimental deflection trends were discussed in terms of thermally accessible rovibrational states in an attempt to understand trends in their deflection behavior.Thermal energies of the studied clusters were estimated using evaporative ensemble theory and quantum chemical calculations of vaporization enthalpies.The rovibrational states were analyzed in the harmonic oscillator and rigid rotor approximations of geometry-optimized cluster geometries.
The magnetic deflection behavior observed in the weakly bound cluster systems studied here does not have an obvious explanation and will need further investigations to give a clear understanding.For the moment, we point to spin−rotational couplings and avoided crossings as a possible cause for a reduction in the effective magnetic moment and reduced deflection.More refined experimental and theoretical work is needed in order to understand other contributions to the attenuation of the magnetic moment while traversing a magnetic field gradient.In particular, well-defined and low temperatures for vibrational T vib and rotational T rot degrees of freedom would simplify the interpretation of measured trends in magnetic deflection.Additional contributions, such as spin− orbit and hyperfine couplings, may perturb the Zeeman-like levels and influence the deflection behavior.Methods to calculate the cluster Zeeman effect with further angular momenta contributions have been summarized by Sears. 55erdyugina and Kuzmychov 56 determined the energy level structure of CrH in the presence of a magnetic field via quantum chemical calculations and predicted magnetic deflection behavior.A similar approach seems feasible for small clusters (e.g., NaNH 3 and NaH 2 O); however, for larger systems, calculations will be challenging.Calculations of the Zeeman diagram for small sodium-doped clusters are of interest to evaluate contributions from other angular momenta in magnetic deflection experiments.
Cluster generation conditions; explanation of magnetic field pulses; velocity dependence of magnetic deflection; calculated cluster geometries, symmetries and rotational constants; vibrational level analysis of studied clusters; and population of vibrational levels as a function of temperature (PDF)

Figure 1 .
Figure 1.Sketch of the experimental setup consisting of the source chamber (A), sodium oven chamber (B), deflection chamber (C), and detection chamber (D).For a detailed description see text below and ref 43.
θ rel as a function of the relative deflector delay t d is characterized by a distinct signal minimum.The signal minimum is quantified as the deflection ratio γ d for various deflector currents I d .

Figure 2 .
Figure 2. (a) Experimental NaH 2 O velocity distribution retrieved from photoion VMI for deflector "off" (black circles) and deflector "on" (green dots) at I d = 700 A. The corresponding simulated NaH 2 O velocity distribution for m s = ±1/2 particles are shown as red traces.(b) γ d as a function of I d for NaH 2 O with experimental data in green and simulation data in red.
depicts γ d as a function of the deflector current, where partial deflection is observed for I d > 500 A. Again, the MD simulations are not in agreement with the experimental observations.3.2.2.Na(MeOH) 2−4 .The deflection results for I d = 700 A are shown in Figure 5 for the clusters (a) Na(MeOH) 2 , (b) Na(MeOH) 3 , and (c) Na(MeOH) 4 .None of these clusters exhibits deflection across their sampled velocity distributions.

Figure 4 .
Figure 4. (a) Experimental NaMeOH velocity distributions retrieved from photoion VMIs for deflector "off" (black circles) and deflector "on" (yellow dots) with I d = 700 A. The corresponding simulated velocity distributions for m s = ±1/2 particles obtained from simulations are shown as red traces.(b) γ d as a function of I d for NaMeOH with experimental data in yellow and simulation data in red.
Effective Magnetic Moment.The effective magnetic moments μ eff of NaH 2 O (see Section 3.1.1)and NaNH 3 (see ref 43) are equal to the magnetic moment μ 0 of an m s = ±1/2 particle within our experimental uncertainty.

Figure 6 .
Figure 6.(a) Experimental NaDME velocity distribution retrieved from photoion VMI for deflector "off" (black circles) and deflector "on" (blue dots) at I d = 700 A. The corresponding simulated NaDME velocity distribution for m s = ±1/2 particles is shown by the red trace.(b) γ d as a function of I d for NaDME with experimental data in blue and simulation data in red.

Figure 7 .
Figure 7. (a) Experimental Na(DME) 2 velocity distribution retrieved from photoion VMIs for deflector "off" (black circles) and deflector "on" (blue dots) at I d = 700 A. The corresponding simulated Na(DME) 2 velocity distribution for m s = ±1/2 particles is shown as red traces.(b) γ d as a function of I d for Na(DME) 2 with experimental data in blue and simulation data in red.

Figure 8 .
Figure 8.(a) Experimental Na(DME) 3 velocity distribution retrieved from photoion VMI for deflector "off" (black circles) and deflector "on" (blue dots) at I d = 700 A. The corresponding simulated Na(DME) 3 velocity distribution for m s = ±1/2 particles is shown with the red trace.(b) γ d as a function of I d for Na(DME) 3 with experimental data in blue and simulation data in red.

Figure 9 .
Figure 9. Effective magnetic moment μ eff as a function of molecular beam velocity for the partially deflecting clusters and limiting values for the nondeflected clusters (black line).

Figure 10 .
Figure 10.Rotational partition function Q rot of (a) Na(H 2 O) n , (b) Na(NH 3 ) n , (c) Na(MeOH) n , and (d) Na(DME) n for various rotational temperatures T rot .The clusters are evaluated from n = 1 to 4 (n = 3 for Na(DME) n ).Letters (e.g., "a" and "b") denote isomers of a given cluster size.The different structures are described in Tables S2−S5, and geometries are shown in the SI.
Published as part of The Journal of Physical Chemistry virtual special issue "Marsha I. Lester Festschrift".