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Determining Key Local Vibrations in the Relaxation of Molecular Spin Qubits and Single Molecule Magnets Luis Escalera-Moreno, Nicolas Suaud, Alejandro Gaita-Ariño, and Eugenio Coronado J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.7b00479 • Publication Date (Web): 28 Mar 2017 Downloaded from http://pubs.acs.org on March 29, 2017

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The Journal of Physical Chemistry Letters

Determining Key Local Vibrations in the Relaxation of Molecular Spin Qubits and Single Molecule Magnets L. Escalera-Morenoa, N. Suaudb, A. Gaita-Ariño*,a, E. Coronadoa a

UIMM-ICMol, University of Valencia, c/ José Beltrán 2, 46980, Paterna, Spain

b

LCPQ-IRSAMC, Université de Toulouse, 118 route de Narbonne, 31062, Toulouse, France

Corresponding Author * E-mail: [email protected]

ABSTRACT: To design molecular spin qubits and nanomagnets operating at high temperatures, there is an urgent need to understand the relationship between vibrations and spin relaxation processes. Herein we develop a simple first-principles methodology to determine the modulation that vibrations exert on spin energy levels. This methodology is applied to [Cu(mnt)2]2- (mnt2- = 1,2-dicyanoethylene-1,2-dithiolate), a highly coherent complex. By theoretically identifying the most relevant vibrational modes, we are able to offer general strategies to chemically design more resilient magnetic molecules, where the energy of the spin states is not coupled to vibrations.

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KEYWORDS Molecular Magnetism, Molecular Spin Qubit, Single Molecule Magnet, Coordination Chemistry, Computational Chemistry, Molecular Vibrations Quantum information technologies rely on quantum two-level systems, known as quantum bits or qubits. Among them, molecular spin qubits1-6 hold great promise for the implementation of quantum algorithms,7-10 and even for the scalability and organization.11-16 The challenge is quantum decoherence: uncontrolled interactions with the environment that cause the loss of quantum information.17-21 Decoherence arising from spin-vibration coupling, also critically important for the relaxation of molecular nanomagnets,22-25 is mediated via intramolecular vibrations,26 an ideal circumstance that allows using synthetic chemistry to engineer it. Around the nitrogen boiling point and above, these intramolecular vibrations can compete with other relaxation mechanisms like Raman processes and even dominate the spin-lattice decoherence rate 1/T1.27,28 Thus, owing to the upper limit T2 ≤ 2T1, the characteristic time T2 will end up being controlled by molecular vibrations at high temperature. Note that achieving (i) high-temperature quantum operations –which would require algorithmic cooling29– would facilitate the practical application of quantum technologies, and (ii) long T1 times at high temperature would represent


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a breakthrough in molecular magnetism (for reasons of space, a lengthier discussion about qubits, molecular magnets and temperature is available as SI). A chemical control and modelling of local vibrations is thus necessary to achieve functional spin qubits or molecular nanomagnets that operate at moderately high temperatures. However, both the role and tuning of spin-vibration interactions remain open problems which impede a clear and concise strategy for quenching the relaxation in these systems.15,22-24,30 Vanadyl complexes acting as potential spin qubits have been recently explored at high temperature,31-33 evidencing the importance of molecular stiffness to enhance quantum coherence. In another previous study on a series of Cu(II) complexes, relaxation rates were higher for the flexible molecules than for the rigid ones.27,28 Nonetheless, as some authors point out, a non-empirical ab-initio framework able to encompass both rationalization and prediction of promising candidates based on complex molecular solids is still missing.17,22 The method developed herein focuses on local modes modulating spin energies, such as the qubit transition energy. This relaxation mechanism involving local modes adds an extra term to the decoherence rate 1/T1,27,28 and is the first key step in the dissipation path which, starting in the spin qubit, traverses local and lattice vibrations and finally reaches the heat bath.22 This work aims to quantify the individual effect of each local vibration, allowing to identify the ones that might be most likely to promote spin relaxation in subsequent steps. In particular, it can provide useful information for modelling spin relaxation dynamics in the stage immediately prior to spinphonon bottleneck effects.31-33 It also allows defining a first-principles quantitative measure of molecular rigidity, establishing proper synthetic guidelines for a rational design of highly coherent magnetic molecules. We will use [Cu(mnt)2]2- (1, mnt2- = 1,2-dicyanoethylene-1,2dithiolate, Figure 1) as a model, a spin qubit whose two states are the ground spin doublet of


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Cu2+, since its high coherence was attributed to the lattice rigidity.34 We are aware that the T1related relaxation dynamics of 1 may be dominated by mechanisms other than local vibrational decoherence at the reported experimental conditions. Nevertheless, our purpose is simply to take 1 as an example-system, for being a spin qubit of interest, in order to illustrate the method herein presented. This method can then be applied to systems where local modes are indeed the dominant source of T1-decoherence at high temperature.27,28

Figure 1. From top to bottom: Lewis structure, upper view and side view of the experimental geometry of 1 at 100 K. Orange: Cu, Yellow: S, Black: C, Blue: N. Note that 1 contains no H atoms. Derivation of the model. The theoretical model starts by considering a property B of the magnetic molecule whose alteration by vibrations is expected to have an important effect in its relaxation. The influence of vibrations on B, modelled as harmonic, is accounted for by assuming that B is a function of the vibrational normal coordinates, Qk, one per each mode: B = B(Q1,…,QR). Then, a Taylor expansion up to second order around Q1 = … = QR = 0 is performed.


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At this point, a single-molecule expectation value for B is calculated under the harmonic approximation, allowing to quantify the property-vibration interaction with any and every vibrational mode. Thus, given a set of harmonic vibrational quantum numbers N = {n1,…,nR}, the harmonic vibrational wave function defined in Equation (1) is expressed as the product of the harmonic wave functions of the vibrational modes:

If the Taylor expansion is used along with Equation (1), the single-molecule expectation value is obtained as a sum of independent contributions, one per mode:

(B)e is the value of B at Q1 = … = QR = 0, νk are the harmonic vibrational frequencies, mk are the reduced masses and the derivatives are evaluated at Q1 = … = QR = 0. Note that one could extend the Taylor expansion up to third order, but because of parity arguments, since we are under the harmonic approximation, odd-order terms in Equation (3) cannot exist. This model is able to incorporate any general discrete lattice or local vibration, once its harmonic frequency, reduced mass and vector displacement of atomic coordinates are known. It is also possible to deal with vibrational wave functions which are not necessarily harmonic by changing Equation (1) and recalculating the expectation value. In particular, anharmonic effects can be included, see SI.


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Finally, temperature is included by considering a Grand Canonical Ensemble, where the probability of each single-molecule expectation value is given in terms of the Grand Partition Function, see SI. The expression obtained for the thermal dependence of the expectation value of B is:

where

and

are expressed as:

Equation (5) defines the zero-point contribution to B, and Equation (6) is the boson number according to the Bose-Einstein statistics. This allows estimating the modulation – and thus the effective coupling – of each vibrational mode in the property B at any given temperature. Proposed methodology. As step (a), one optimizes the geometry of the set of atoms involved in the relevant vibrations. This optimized geometry corresponds to Q1 = … = QR = 0. Step (b) consists in determining the harmonic vibrational spectrum of the optimized geometry: harmonic frequencies νk, reduced masses mk and the 3P dimensional displacement vectors vk of the normal modes, P being the number of atoms. Finally, in step (c), the second derivatives of B respect to Qk are calculated within an ab-initio approach. This is achieved by: (c1) generating a certain number of distorted geometries according to the vector vk around the optimized geometry; (c2) calculating B at each distorted geometry; and (c3) fitting the calculated B values to a polynomial,


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whose second derivative is analytically calculated at Qk = 0, see Figures S2-S21. Each distorted geometry is given by the 3P dimensional vector, vdist,k = veq + Qk·vk, and is generated by giving Qk a proper value, see SI, veq being the 3P-dimensional vector of the equilibrium atomic coordinates.

For 1, the qubit energy is controlled by the Electron Landé g tensor. Because of the availability of experimental data, here we take B as the gz component. Nevertheless, the methodology is entirely analogous for gx and gy; the axial D and rhombic E zero-field splitting parameters, characteristic of molecular magnets; or the tunneling gap ∆ in anisotropic complexes, a crucial parameter for the design of lanthanoid-based spin qubits.35,36 In this latter case, it would also be possible to determine the interplay between magnetic field, vibrations and spin energies, and therefore finding the optimum magnetic field.

Some general considerations. When considering B = gz and the molecular modes as the relevant vibrations, 3P-6 independent Bk contributions arise, one per each vibrational mode:

These contributions should be as small as possible to make the qubit energy transparent to vibrations. Equation (7) is expressed as the product of a factor that characterizes the coupling strength of a given mode – Equation (8) – and a temperature dependent factor: the boson number, which gives the thermal population of the mode. Thus, the total contribution is simply the result of how populated the mode is and how strong it couples with the spin excitation. Effectively, low couplings are achieved both when modes that are significantly populated at the


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working temperature hardly couple and when modes that are strongly coupled are not highly populated.

For this goal, harmonic frequencies νk can be increased by restricting the movement of the atoms involved in the relevant modes, as in porphyrines or phtalocyanines; or by encapsulating the metallic ion inside a tight cage, as in fullerenes hosting atoms,37-40 “stapled” bisphthalocyanines,41 or polyoxometalates.42,43 Moreover, reduced masses mk can be maximized by removing light atoms, or by replacing them by other heavier atoms with low spin-orbit coupling. In particular, one can replace hydrogen by deuterium or fluorine as done in ref. (5). For the second derivative in Equation (8) to be as small as possible, ideally one should achieve a horizontal gz evolution with the distortion coordinate Qk. This means having a highly isotropic electron spin (a very small spin-orbit coupling), like in organic radicals or, among S > ½ metals, in Mn2+, Fe3+ or Gd3+ spin qubits.42,43 Often, magnetic anisotropy is desirable since it facilitates qubit addressing or slows down the magnetization relaxation in nanomagnets.44 In this case, there is no general strategy. Effectively, harmless vibrations will exist when vibrational modes and molecular orbitals are of different symmetry,45,46 or, for high-symmetry molecules, in symmetric modes (see below). A case study: [Cu(mnt)2]2-. Let us consider for illustration the modes 4 and 12 of 1 which are the most relevant ones, see Figure 2. Mode 4, which is a low frequency mode and thus expected to populate from low temperature, is an out-of-plane twisting vibration. This mode alters the dihedral angle between the two ligands, which behave almost as rigid planes in this case. Mode


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12, which has a much higher frequency, can be seen as an out-of-plane wagging vibration. In this mode, the ligands are twisted and the orientation of the CuS4 moiety is altered while its square planar structure is maintained (see SI for mpg animations).

Figure 2. Out-of-plane vibrational modes 4 (top) and 12 (bottom) of 1. a-labeled atoms go toward the reader while b-labeled atoms go away from the reader and vice versa.

Figure 3 depicts the coupling constant Ck of each one of the first 25 vibrational modes of 1. There are five of them which stand out from the rest: (a) modes 4 and 13, giving a positive coupling and (b) modes 12, 23 and 24, with a negative coupling. Note that the sign of Ck is the sign of the second derivative of gz respect to Qk. Thus, positive coupling constants increase gz from its reference value at T = 0 K, while negative ones decrease it as temperature rises.


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Figure 3. Top: gz evolution with Q24 (left) and Q25 (right) between -0.110 Å and +0.110 Å. The near-linear dependence of gz with Q25, a breathing vibration, gives a rather small second derivative. Bottom: Spin-vibration coupling constants Ck of the first 25 vibrational modes of 1, Equation (8).

From Figures 2, 3, S22, the rationalization and prediction power of the above model can now be sketched. Using only chemical intuition, one would state that the most detrimental modes are precisely those that significantly distort the coordination environment of the metal. Often, this is true, but not strictly always. Indeed, we identify modes which largely distort the coordination sphere in 1 but do not couple very much, and also modes with a significant coupling which alter


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the coordination sphere very little. The first case is illustrated by the mode 25 when compared to the modes 13, 23 and 24; while the modes 4 and 12 exemplify the second case. The mode 25 is a breathing vibration in which gz evolves near-linearly along the distortion range, see Figure 3. Despite comparable displacements of the coordinating atoms, its second derivative is much smaller than those of the modes 13, 23 and 24. This is clear evidence of the crucial role of symmetric modes. On the other hand, the effect of the metal environment distortion is sometimes supplemented by the motion of mobile parts in the ligands (modes 21, 23, 24). Thus, ligands with mobile parts may promote vibrational decoherence and should be engineered consequently. The capability of calculating individual coupling constants offers the possibility to establish priorities on a given set of vibrations, an order that could result rather counterintuitive a priori. In particular, it allows deciding which modes must be engineered first, whenever they seem to couple similarly.

As stated above, to obtain the spin energy modulation Bk of each mode, one has to include not only its coupling strength but also its thermal population via the boson number. Figure 4 gathers these individual modulations with temperature.


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Figure 4. gz individual thermal modulations of the first 25 vibrational modes of 1, Equation (4). Inset: zoom-in at the range 10-30 K. Note that B1 and B3 have very similar thermal evolutions, as their slopes only differ in less than 1%.

Below 10-15 K, any modulation is practically equal to zero, in concordance with the fact that vibrational decoherence is negligible at very low temperature. At high temperature, meaning kBT>>νk, each curve acquires a near linear behavior, since the boson number tends to a straight line with slope kB/νk as temperature rises, see SI. Thus, Bk becomes a straight line with slope proportional to (gz)kk/µkνκ2, (gz)kk being the second derivative of gz respect to Qk. From 20 K on, where kBT is a sizeable fraction of νk, modes begin to be populated appreciably and may contribute to the relaxation. The twisting mode 4 clearly gives the largest positive contribution in the whole temperature range: besides its remarkable coupling, it is a low-frequency mode, ν4 = 34.76 cm-1, so it becomes populated already from low T. On the contrary, the modes 23 and 24, despite their large couplings, are high-frequency modes, ν23 = 278.70 cm-1 and ν24 = 298.81 cm-1, so they become significantly populated only at high T and thus give a much weaker contribution, confirming the validity of our simplification of neglecting modes 26 and beyond.

Applying Equation (4) allows us to check the joint effect of the molecular vibrations on gz and compare it against the experimental thermal evolution. The gz experimental values at low temperature (5K) and room temperature (~294K) are 2.0932 and 2.0910, resp.; while the theoretical values (Equation (4) up to the mode 25) at the same temperatures are 2.1380 and 2.1410, resp. The accuracy in the calculation of g, which is satisfactory compared to the state-ofthe-art quantum-chemical calculations,47-49 can be improved using a procedure adapted from that reported in ref. (50), (51) followed by a DDCI (Difference Dedicated Configuration Interaction


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Method) evaluation of the spin-free spectrum, resulting in a more accurate value of 2.0990, but at a prohibitive cost, see SI. Most importantly, our calculations recover the experimental stability of gz with temperature.

In sum, both in Molecular Spin Qubits and in Single Ion Magnets, the high-temperature relaxation behavior is governed by spin-vibration interactions. Therefore, it is necessary to clarify how chemical structures determine these interactions, firstly in order to rationalize the latest experimental advances but also to predict and design new molecular complexes displaying long coherence times and slow relaxation at high temperature. We have presented a straightforward scheme to quantify, from first principles, the influence of vibrations on spin excitations. This theoretical framework allows us to quantitatively identify any vibration that modulates the relevant spin energy levels, e.g. those altering the magnetic anisotropy of the magnetic complex. When these vibrations are molecular modes, they will mediate in the energy dissipation pathway between the spin and the thermal bath. Practically, we find that the focus needs to be put on the least energetic vibrations and, among those, on the ones with high values of the coupling constant C. In particular, it is now possible to quantify the benefit of impeding the most detrimental vibrations among those that destabilize the qubit energy, e.g. the result of deuteration or fluorination. The presented methodology is rather general: if in the first stage one uses standard programs to calculate lattice vibrations in extended systems, these can also be incorporated along with molecular vibrations, thus including intermolecular effects on these local modes. Therefore, this method can constitute a widely applicable tool and offer a first step toward understanding and controlling the whole vibrational decoherence process.


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ASSOCIATED CONTENT

Supporting Information. The following files are available free of charge. -Supplementary information (PDF): Computational details on the geometry optimization, the vibrational spectrum calculation and the wavefunction-based calculations to evaluate the magnetic anisotropy. Evolution of the gz component along the first 25 normal vibrational coordinates. Pictures of some molecular vibrational modes: 1, 3, 13, 16, 21, 23, 24, 25. -Animations (mpg format) of the first 25 vibrational modes. Output of the geometry optimization and the vibrational spectrum calculation.

AUTHOR INFORMATION Notes The authors declare no competing financial interests.

ACKNOWLEDGMENT We thank Profs. Nathalie Guihéry and Jean-Paul Malrieu for fruitful discussions. The research reported here was supported by the Spanish MINECO (grants MAT 2014-56143-R, CTQ 201452758-P, and Excellence Unit María de Maeztu MDM-2015-0538), the European Union (ERCCoG DECRESIM 647301, and COST 15128 Molecular Spintronics Project) and the Generalitat Valenciana (Prometeo Program of Excellence). A.G.-A. thanks the Spanish MINECO for a Ramón y Cajal Fellowship. L.E.-M. acknowledges the Generalitat Valenciana for a VALi+D predoctoral contract. N.S. thanks Université Toulouse III Paul Sabatier and CNRS for fundings. This work was performed using HPC resources from CALMIP (Grant 2015-1144).


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