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Temperature Dependence of Spin–Phonon Coupling in [VO(acac)2]: A Computational and Spectroscopic Study
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C: Spectroscopy and Dynamics of Nano, Hybrid, and Low-Dimensional Materials

Temperature Dependence of Spin–Phonon Coupling in [VO(acac)2]: A Computational and Spectroscopic Study
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  • Andrea Albino
    Andrea Albino
    Dipartimento di Chimica “Ugo Schiff” & INSTM RU, Universitá degli Studi di Firenze, Via della Lastruccia 3, Sesto Fiorentino, Florence 50019, Italy
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  • Stefano Benci
    Stefano Benci
    European Laboratory for Non-Linear Spectroscopy (LENS), Universitá degli Studi di Firenze, Sesto Fiorentino, Florence 50019, Italy
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  • Matteo Atzori
    Matteo Atzori
    Laboratoire National des Champs Magnétiques Intenses (LNCMI), Univ. Grenoble Alpes, INSA Toulouse, Univ. Toulouse Paul Sabatier, EMFL, CNRS, F38043 Grenoble, France
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    Orcidhttps://orcid.org/0000-0003-1357-6159
  • Laura Chelazzi
    Laura Chelazzi
    Dipartimento di Chimica “Ugo Schiff” & Center of Crystallography, Universitá degli Studi di Firenze, Via della Lastruccia 3, Sesto Fiorentino, Florence 50019, Italy
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  • Samuele Ciattini
    Samuele Ciattini
    Dipartimento di Chimica “Ugo Schiff” & Center of Crystallography, Universitá degli Studi di Firenze, Via della Lastruccia 3, Sesto Fiorentino, Florence 50019, Italy
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  • Andrea Taschin
    Andrea Taschin
    European Laboratory for Non-Linear Spectroscopy (LENS), Universitá degli Studi di Firenze, Sesto Fiorentino, Florence 50019, Italy
    ENEA, Agenzia nazionale per le nuove tecnologie, l’energia e lo sviluppo economico sostenibile, Centro Ricerche Frascati, via Enrico Fermi 45, 00044 Frascati, Roma, Italy
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    Orcidhttps://orcid.org/0000-0003-4088-7964
  • Paolo Bartolini
    Paolo Bartolini
    European Laboratory for Non-Linear Spectroscopy (LENS), Universitá degli Studi di Firenze, Sesto Fiorentino, Florence 50019, Italy
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  • Alessandro Lunghi
    Alessandro Lunghi
    School of Physics, AMBER and CRANN Institute, Trinity College, Dublin 2, Ireland
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    Orcidhttps://orcid.org/0000-0002-1948-4434
  • Roberto Righini
    Roberto Righini
    European Laboratory for Non-Linear Spectroscopy (LENS), Universitá degli Studi di Firenze, Sesto Fiorentino, Florence 50019, Italy
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    Orcidhttps://orcid.org/0000-0003-0260-0248
  • Renato Torre*
    Renato Torre
    Dipartimento di Fisica ed Astronomia, Universitá degli Studi di Firenze, Via G. Sansone 1, Sesto Fiorentino, Florence 50019, Italy
    European Laboratory for Non-Linear Spectroscopy (LENS), Universitá degli Studi di Firenze, Sesto Fiorentino, Florence 50019, Italy
    *Email: [email protected]
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    Orcidhttps://orcid.org/0000-0003-3182-9906
  • Federico Totti*
    Federico Totti
    Dipartimento di Chimica “Ugo Schiff” & INSTM RU, Universitá degli Studi di Firenze, Via della Lastruccia 3, Sesto Fiorentino, Florence 50019, Italy
    *Email: [email protected]
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  • Roberta Sessoli
    Roberta Sessoli
    Dipartimento di Chimica “Ugo Schiff” & INSTM RU, Universitá degli Studi di Firenze, Via della Lastruccia 3, Sesto Fiorentino, Florence 50019, Italy
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The Journal of Physical Chemistry C

Cite this: J. Phys. Chem. C 2021, 125, 40, 22100–22110
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https://doi.org/10.1021/acs.jpcc.1c06916
Published September 30, 2021

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Abstract

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Molecular electronic spins are good candidates as qubits since they are characterized by a large tunability of their electronic and magnetic properties through a rational chemical design. Coordination compounds of light transition metals are promising systems for spin-based quantum information technologies, thanks to their long spin coherence times up to room temperature. Our work aims at presenting an in-depth study on how the spin–phonon coupling in vanadyl-acetylacetonate, [VO(acac)2], can change as a function of temperature using terahertz time-domain spectroscopy and density functional theory (DFT) calculations. Powder THz spectra were recorded between 10 and 300 K. The temperature dependence of vibrational frequencies was then accounted for in the periodic DFT calculations using unit-cell parameters measured at two different temperatures and the optimized ones, as usually reported in the literature. In this way, it was possible to calculate the observed THz anharmonic frequency shift with high accuracy. The overall differences in the spin–phonon coupling magnitudes as a function of temperature were also highlighted showing that the computed trends have to be ascribed to the anisotropic variation of cell parameters.

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Introduction

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The quantum bit or qubit (1) is the basic element of quantum information theory (QIT); (2) it differs from its classical equivalent, the bit, since it can exploit the quantum superposition of the two states 0 and 1. Several systems are under investigation as potential qubits, e.g., electronic defects in diamond, (3) photons, (4) ion traps, (5) superconducting circuits, (6) and spins. (7−9) Molecular electronic spin qubits are good candidates since they are characterized by a large tunability of their electronic and magnetic properties through a rational chemical design. Among them, coordination compounds of light transition metals, such as Ti(III), V(IV), and Cu(II), (10−12) have demonstrated to be promising systems at least from the point of view of the superposition state’s lifetime. This quantity is determined by the spin–spin relaxation time, T2, but it is usually quantified through the phase memory time, Tm, that is the measurable lower limit of T2. (13) Another fundamental parameter in the evaluation of the qubit performance is the spin–lattice relaxation time, T1. (14) If it is too short, it limits Tm, (11) preventing the implementation of more complex algorithms. In solid-state qubits, T1 is closely connected with lattice vibrations, i.e., phonons. Vibrations perturb spin degrees of freedom through the modulation of orbital contributions and the presence of spin–orbit coupling.
Therefore, the knowledge and characterization of phonons in solid-state qubit candidates are of crucial importance. A macroscopic crystal can present a huge number of atomic displacement degrees of freedom and, consequently, of vibrational modes. The use of space-group symmetry and the introduction of the reciprocal lattice allows a fundamental rationalization of this huge number of states. At the center of the Brillouin zone (BZ), the so-called Γ-point, i.e., for wavevector k = 0, the degrees of freedom are reduced to 3N, N being the number of atoms in the crystal unit cell so that the vibrational modes are 3N – 3 (3 are the rigid displacements of the unit cell, corresponding to zero frequency). Only the k = 0 vibrations can be measured with optical techniques at equilibrium; all of the remaining vibrational modes are classified, throughout the entire BZ, in 3N dispersion bands and are optically inaccessible. A peculiar feature of most molecular crystals is that intramolecular forces are much stronger than intermolecular ones so that the intramolecular vibrations in the crystal are almost unchanged from those of the isolated molecule. If the crystal unit cell contains n molecules, there are then 6n – 3 intermolecular phonons at k = 0, corresponding to rigid translations and rotations of the molecules. These “external” phonons have frequencies much lower (typically, less than 200 cm–1) than intramolecular vibrations. However, in some cases, where molecular vibrations of unusually low frequency are present, the separation of “internal” and “external” phonons is impossible and low-frequency phonon modes have mixed character. (15) In the harmonic approximation, phonon modes are orthogonal to each other and their frequencies depend on temperature indirectly, as the thermal expansion of the lattice induces an elastic strain in the solid, which results in a shift (generally negative) of the phonon frequencies. The thermal expansion of crystals is a consequence of the anharmonicity of the crystal potential; however, it has been shown (16,17) that the effect of thermal strain can be dealt with in a quasiharmonic approximation, adopting an effective crystal Hamiltonian that maintains the quadratic dependence on the atomic displacements. On the other hand, the anharmonic terms of the potential energy expansion enter directly into the description of the phonon dynamics as they mix different phonons and make phonon relaxation possible, thus determining the homogeneous linewidths of vibrational spectral lines. Furthermore, the same anharmonic terms cause additional temperature dependence of the frequencies through the temperature-dependent occupation number of phonons.
From a computational point of view, the harmonic treatment of molecular vibrations cannot explain all of the fine details of the vibrational spectra and the physical properties of these crystals that depend on the temperature. (18) Density-functional perturbation theory or molecular dynamics simulations (from several ps to ns) (19) are out-of-reach. The direct method of calculation of phonons (20) here employed can be considered as a cheaper alternative. Unfortunately, even with this approach, the extension of lattice dynamics calculations beyond the second-order of expansion of the potential energy surface (PES) is severely limited by the enormous number of terms appearing in the expression of a large molecule; therefore, it is mandatory to assume some approximations when aiming to calculate higher orders of interaction.
In such a framework, an extensive characterization of the vibrational properties as a function of temperature becomes feasible and it is of paramount importance for the derivation of key chemical insights useful in the engineering of efficient molecular spin qubits. Indeed, as shown in other works, (10,21−27) the study of phonons provides insights into the spin relaxation pathways pointing toward the rationalization of new synthetic strategies to achieve more performing systems.
The aim of this work is an in-depth study of the spin–phonon (hereafter, Sp-Ph for brevity) coupling in vanadyl-acetylacetonate, [VO(acac)2], both from a spectroscopic and theoretical point of view. Thanks to its simplicity (only 60 atoms in the unit cell), [VO(acac)2] has got the physique du rôle as a perfect training ground for extensive DFT simulations to investigate the rich vibrational properties of a molecular crystal. (26) [VO(acac)2], already studied by inelastic neutron scattering, (28) is here investigated by means of terahertz radiation in crystalline powder by using of an in-house built setup performing THz time-domain spectroscopy (THz-TDS). (29−31)
In our computational approach, the temperature dependence of the vibrational frequencies is accounted for using the unit-cell parameters measured at different temperatures. The results of this study confirm the paramount importance of low-energy phonons, where intramolecular and lattice characters are admixed. (15,25,26,32)

Materials and Methods

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Crystal Structure

The [VO(acac)2] metal center comprises a vanadyl ion VO2+ coordinated by two chelating bidentate β-diketonate ligands (Figure 1). The coordination geometry around the penta-coordinated VIV metal center is a distorted square pyramid, as already reported in previous studies. (33) This molecule crystalizes in a P1̅ space group (no. 2), for which the only point symmetry element is inversion. The asymmetric unit contains one molecule and the primitive unit cell contains two molecules, for a total of 60 atoms (see Table 1).

Figure 1

Figure 1. Primitive cell of [VO(acac)2] belonging to the triclinic P1̅ space group. Color code: vanadium, green; oxygen, red; carbon, cyan; and hydrogen, white.

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Table 1. Three Cell Parameter Sets Studied in This Worka
 abcαβγvol.
1300K7.5138.20111.22173.17471.47666.718591.445
1100K7.3008.11711.17872.90172.28267.064569.372
10K,cell–opt7.6627.80211.05670.86971.16264.315549.943
a

Note that the cell parameters of 1100K and 1300K are the experimental ones while 10K,cell–opt are optimized by DFT. Lattice parameters are reported in Å and deg. The volume is in Å3.

The crystallographic structure has been already solved using high-resolution data in 1995  (34) at 294 K. Here, we present new data collected at 300 K, 1300K, and 100 K, 1100K, both on the same sample (Table S1). A good agreement between 300 and 294 K datasets is found, while the shrinking of cell parameters at 100 K is highlighted in Table 1.

Experimental Apparatus and Data Analysis

THz spectra were measured by time-domain spectroscopy using a table-top experimental setup working in a transmission configuration. It allows one to achieve a very high signal-to-noise ratio, higher than what is usually obtained in the standard far-infrared investigation (FTIR). Furthermore, the nontrivial data analysis allows one to decouple the signals arising from the multiple reflections inside the sample, i.e., the Fabry–Pérot effect. The THz-TDS spectrometer is a home-built system based on optical laser pulses (T-light 780 nm fiber laser, MenloSystems) and low-temperature GaAs photoconductive antennas. The emitted THz radiation (pump) is collimated and focused on the sample by a couple of off-axis parabolic mirrors. The signal transmitted through the sample is again collimated and focused on the detector, a second photoconductive antenna, biased by a second optical pulse (probe). It results in a photocurrent that is measured by a lock-in amplifier and an acquisition board. The amplitude of the THz electric field is measured by varying the time delay between the pump and probe pulses so as to reconstruct its time profile. The homemade software in Matlab code was implemented to acquire the processed signal together with the reading of the delay line encoder and retraces the final time-dependent THz field. The whole THz setup is placed in a nitrogen-purged chamber to eliminate the strong absorption of water vapor. To improve the data quality, the effects of external perturbations during the acquisition (e.g., temperature fluctuations) measurements must be taken into account. For this purpose, measurements with and without a sample (reference) are cyclically repeated. We also implemented a new method of data analysis, described in detail in ref (29), that enables the extraction of the index of refraction, n, absorption coefficient, α, and thickness, d, of the material. This method is based on an iterative fitting process of the transmission parameters and it takes into account the Fabry–Pérot effect. More details on the description of the experimental setup and the data analysis procedure are reported elsewhere. (29)

Sample Preparation

THz measurements were performed on two samples: (1a) a pellet of [VO(acac)2] and high-density polyethylene (HDPE) (1:1 mass ratio) prepared by milling vanadyl-acetylacetonate with HDPE powder (Merck CAS 42.901-5), the powder was pressed by means of a manual hydraulic press (about 0.8 GPa); (1b) a pure HDPE pellet on which we deposited a few drops of [VO(acac)2] dichloromethane solution of several microns by the drop-casting method.

Computational Methods

Vibrational Properties

For the density functional theory-based calculations, we used the [VO(acac)2] crystallographic unit-cell coordinates. The temperature effects on the crystal structure were tested exploiting the X-ray data collected at 100 and 300 K. The availability of these data allowed to optimize the molecular structure keeping the experimental cell parameters fixed, (35) in such a way the temperature effects can be indirectly accounted. The simulation of cell parameters at finite temperature can be accomplished only via dynamical DFT methods (ab initio molecular dynamics). Vibrational properties at 0 K were also computed by fully relaxing the electronic degrees of freedom, lattice constants, and atomic positions; the resulting crystal structure is hereafter referred to as 10K,cell–opt. A tight DFT optimization is mandatory to avoid imaginary eigenvalues in the Hessian matrix. The different atoms in the unit cell are displaced by small amounts, and the forces on all other atoms are recorded to calculate the second derivative of the potential energy surface (PES) through numerical differentiation. Any atomic displacement U(m, μ) = ∑iXi(m, μ) – X̅i(m, μ), where X̅i(m, μ) is the equilibrium position, generates forces on all other atoms of the determined cell according to the relationship
(1)
where n and m are the primitive unit-cell indices, and ν and μ are the atomic indices. In the present study, n = 1 because only one primitive cell was included in the computation. This relates the forces generated to the force constant matrix Φ(n, ν, m, μ) and atomic displacement U(m, μ). We imposed a set of constraint equations on the force constant matrix, known as the acoustic sum rule, to ensure that the crystal energy is invariant under the global translation of the whole crystal (ωacoustic). The dynamical matrix is defined as (16,36)
(2)
Here, Mμ and Mν are masses of atoms, and k is the wave vector. The solution of the eigenproblem (eq 3) gives eigenvalues ω2(k, j) and eigenvectors e(k, j) of 3N orthogonal normal modes of vibrations
(3)
The calculation is restricted to one primitive cell, giving access only to 3N – 3 zone-center phonons (at the Γ-point). (37)
The Γ-point approximation comes with no loss of information. The full inclusion of the Brillouin zone integration, important to quantitatively estimate the optical phonon bands, is not part of the main focus of the present work. (25)
A more than fair approximation in the analysis of phonons’ decay mechanisms is achieved limiting the calculation to the DFT Γ-point optical modes, along with the acoustic phonons in the whole Brillouin zone. The latter were already computed by some of us at the force field (FF) level, (33) at the DFT level with the finite displacement method at Γ-point, (25) and in the Brillouin zone. (28)
Periodic ab initio calculations of energy and forces were performed using the Quickstep module (38) present in the CP2K atomistic simulation package. (39)
GTH pseudopotentials (40−42) for the core electrons and valence pseudo-wavefunctions are expanded in Gaussian-type orbitals, and the density is represented in a plane wave auxiliary basis set. In particular, for V, O, C, and H elements, the DZVP-MOLOPT-SR-GTH basis set was used. (38)
The Perdew–Burke–Ernzerhof (PBE) parametrization (43) was chosen for the calculation. A pairwise dispersion correction proposed by Grimme (44−46) was included in DFT calculations. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimizer (47,48) was employed to carry out the structure optimization. The convergence of the plane wave basis cut-off was reached for 10 000 Ry with a convergence threshold of 1.0 × 10–9 au for the SCF energy and 5.0 × 10–8 au/Å for the forces. A cut-off of 10 000 Ry is cumbersome and the best compromise between accuracy and cpu-time consumption was chosen to be 5000 Ry (see Figures S1 and S2). Such a choice was taken also on the basis of the convergence of vibrational energies in the low-energy range (0–100 cm–1). Indeed, the error between the frequency values computed for 10 000 and 5000 Ry is just a few decimal digits of the wavenumber. The high level of accuracy chosen precluded to calculate the phonons’ dispersions at the DFT level for all of the three structures investigated.

Magnetic Properties and the Spin–Phonon Coupling

The method to investigate the Sp-Ph coupling has been developed in previous works (15) and is based on a perturbative formalism. Calculation of the Sp-Ph Hamiltonian parameters was carried out with the ORCA package. (49) The level of theory chosen is DFT and the PBE0 hybrid functional (50) was used throughout the computation. A balanced polarized triple-ζ basis set was adopted (def2-TZVP) for V and S atoms, while a polarized single-ζ basis set (def2-SVP) was used for light elements such as C and H. The geometrical equivalence between the two molecules present in the crystal cell after p-DFT optimization with CP2K was ascertained. An RMSD of 0.12 was computed on V, O, and C atoms (0.94 on V, O, C, and H atoms) indicating that two molecules can be considered equivalent even after a symmetry-unconstrained optimization. For such a reason, only one molecule was used for the Sp-Ph calculations. The unperturbed Hamiltonian is evaluated with respect to external perturbation, sampling six points along each coordinate (see Figure S3). The g vs qα plot is then fitted with a polynomial whose first-order coefficient directly yields the first-order derivative .
The for a S = 1/2 spin system, neglecting the hyperfine term that plays a key role only at the low magnetic field, (11) contains only the Zeeman term , where the main parameter depending on ionic coordinates is the second-order correction to the Landé factor g, μb is the Bohr magneton, and B is the magnetic field.
The Sp-Ph coupling Hamiltonian connects molecular vibrations to the spin via the spin–orbit interaction, (51,52) and, in the weak coupling assumption, it is linear in the ionic displacements and it reads
(4)
The dynamics of the whole system (spin and phonons) can be studied by the Liouville–von Neumann equation for the density operator, within the approximations described by Redfield (53) in the framework of the open quantum system theory. (54) The spin–phonon coupling coefficients constitute the main ingredients to describe the interplay of the principal system (spin degrees of freedom) and the ancillary system (surrounding environment of the phonons), as shown in previous works. (26)

Results and Discussions

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THz Spectra of Powder Samples

(1a) and (1b) have been prepared according to two different procedures to check that the applied pressure, essential to make the pellet, does not induce any phase transition or changes in the crystal structure. The THz spectra of (1a) and (1b) are shown in Figure S4.
The two spectra are identical, except for the intensity that is related to the different concentrations of vanadyl-acetylacetonate in the samples. No evident effects of the applied pressure occur; the same result is observed by PXRD investigation on (1b) compared with the vanadyl-acetylacetonate powder and pure HDPE, as shown in Figure S4. Therefore, hereafter, the discussion will be focused on (1a).
Figure 2 shows the THz spectra measured in the temperature range 10–300 K. At the lowest temperature, five peaks (p1–5) are detected. As the sample temperature is increased, all of them become broader and most of them shift to lower energy. It is worth noting that the lowest phonon detected falls at 43.9 cm–1 (p1), which is exactly the value predicted by the Brons–van Vleck model applied to the spin relaxation rate in our previous work. (21)

Figure 2

Figure 2. THz spectra of (1a) as a function of temperature.

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The extraction of peaks’ parameters, i.e., central frequency and linewidth (Γlw), has been performed by fitting each of them with a pseudo-Voigt function. Moreover, the baseline correction has been taken into account. Figure 3 summarizes the temperature dependence of the five bands. The error analysis on the five peaks’ position led to an average standard deviation of less than 1%. The weak peak at ca. 71 cm–1 (p2) is not detectable above 120 K, as it becomes too broad and the strong peaks at higher frequency get closer and broader. The general trend is a decrease of frequency when the temperature increases and it is ascribed primarily to the effect of the thermal expansion of the lattice, which results in the softening of the intermolecular forces. (16)

Figure 3

Figure 3. (Left) Temperature dependence of the peaks’ frequency. (Right) Temperature dependence of linewidths obtained using a Voigt fit function for each peak and a baseline correction.

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The peak at ca. 77 cm–1 (p3) is the only one that deviates from the general trend: its temperature dependence is very weak and not monotonous. The nature of the computed mode which should correspond to p3 (vide infra) is characterized by an in-phase rigid vibration of the external acac ligands of the two molecules in the cell with the ones belonging to the molecules in the neighbor cell replica (see Figure S7). Therefore, the variation of cell parameters as a function of temperature is expected to have a very limited impact on this mode.
Figure 3 shows how the linewidth of each peak, in terms of pseudo-Voigt, varies by increasing the temperature from 10 to 300 K. A deconvolution process was carried out to separate the Lorentzian contribution (ΓL) and the Gaussian one (ΓG). The first contribution takes into account the homogeneous broadening, connected to the phonon–phonon and phonon–lattice interactions, through the anharmonic terms of the crystal potential expansion, whereas the second one takes into account the inhomogeneous broadening, which is predominantly attributed to the variation of frequency with the phonons wave vector (k) direction of the phonons active in IR. Indeed, in common spectroscopic techniques, e.g., IR and Raman, the conservation of the moment is fulfilled in a small surrounding of the Γ-point (k ≃ 0), making the phonons’ frequency dependent on the k direction.
The results collected in Figure 4 show that, for the temperature range 10–100 K, the homogeneous broadening of peaks p1, p3, and p4 increases almost linearly with temperature, while the inhomogeneous width is practically constant. The relative standard deviation is of the order of 10–20% since ΓG and ΓL are particularly sensitive to the irregularity of the peak tails. In addition, the loss of a well-defined shape of p2 above 70 K and the high noise that affects p5 do not allow to deduce a clear temperature trend for these peaks.

Figure 4

Figure 4. Temperature dependence of (a) Lorentzian component and (b) the Gaussian component of the Voigt profile of each peak.

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Calculated Vibrational Spectra

In the Introduction section, we pointed out that the availability of X-ray data collected at cryogenic and room temperature (100 and 300 K) allowed us to fix the experimental cell parameters in the DFT energy minimization, prompting the estimation of the errors introduced in the vibrational analysis when the temperature effects on the structure are neglected, i.e., when cell parameters are free parameters in the energy minimization procedure. We denote 10K,cell–opt, 1100K,opt, and 1300K,opt as the DFT optimized structures; in 10K,cell–opt, both cell parameters and atomic coordinates are optimized while, in 1100K,opt and 1300K,opt, only atomic coordinates were left to relax keeping the cell parameters fixed to experimental values. Lowering the temperature from 300 to 0 K, a contraction occurred along the b and c axes, while a nonmonotonous expansion along the a-axis was observed. The overall variation of the cell parameters led to a unit-cell volume contraction. 10K,cell–opt was found to be more stable than 1100K,opt of 2.80 kcal/mol, while 1100K,opt was more stable than 1300K,opt of 0.92 kcal/mol, as expected for the most enthalpic favored structure. The full relaxation of 10K,cell–opt led to optimized cell parameters that show a maximum deviation with respect to 1100K of 4.8 and 2.8% for lengths and angles of lattice constants, respectively. The volume underwent a shrinking of ca. 5%. Even more, it is worth highlighting that a, β, and γ parameters computed for 10K,cell–opt do not completely follow the expected trend indicated by the experimental parameters obtained for 1100K and 1300K unit cells. These results have also a general relevance, as they show that neglecting the temperature effects on cells, i.e., using static DFT cell parameters optimized at 0 K, a poorer agreement in reproducing the vibrational spectra at finite temperatures can be attained. See, for instance, the computed p1C, which is shifted to almost 20 cm–1 from 0 to 300 K calculated phonons.
The computed IR spectra for the three cells are reported in Figure 5. The frequencies of the lowest 20 Raman-active (9) and IR-active (11) modes are reported in Table 2 with the trends for the first IR modes gathered in Figure 6. The full spectral data are reported in Table S3.

Figure 5

Figure 5. DFT vibrational analysis of 10K,cell–opt (blue), 1100K,opt (green), and 1300K,opt (red). A direct comparison of experimental and calculated spectra is available in the SI (Figure S5).

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Figure 6

Figure 6. Calculated frequencies (a) and intensities (b) of the lowest 8 IR-active modes.

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Table 2. Frequencies of the Lowest 20 Raman- and IR-Active Modesa
IR normal modes
10K,cell–opt1100K,opt1300K,opt
ανα (cm–1)int. (au)Sp-Ph (au)ανα (cm–1)int. (au)Sp-Ph (au)ανα (cm–1)int. (au)Sp-Ph (au)
655.3230.3112.331 × 10–7442.4110.3033.905 × 10–7438.1370.3353.787 × 10–7
873.2670.5094.399 × 10–7969.5180.3838.549 × 10–7964.1710.4308.199 × 10–7
1183.4610.2202.453 × 10–71076.0130.1404.069 × 10–71069.4670.0722.448 × 10–7
1399.2400.4463.330 × 10–71294.8640.8626.695 × 10–71286.9071.06.289 × 10–7
15112.3460.3955.968 × 10–715108.6850.07693.620 × 10–715102.0720.1092.255 × 10–7
17116.5580.2742.171 × 10–717116.2620.3394.334 × 10–717106.9600.2724.870 × 10–7
18125.1080.09472.842 × 10–718120.4920.3281.231 × 10–718113.7870.2836.313 × 10–8
20145.7450.2348.805 × 10–820125.9330.0872.795 × 10–719115.0770.1494.863 × 10–7
Raman normal modes
10K,cell–opt1100K,opt1300K,opt
ανα (cm–1) Sp-Ph (au)ανα (cm–1) Sp-Ph (au)ανα (cm–1) Sp-Ph (au)
443.313 1.130 × 10–7546.320 1.430 × 10–7540.497 2.427 × 10–8
550.872 1.648 × 10–7648.492 3.037 × 10–7643.222 3.545 × 10–7
763.349 6.260 × 10–7757.788 7.608 × 10–7753.353 5.580 × 10–7
975.290 2.357 × 10–7866.991 3.717 × 10–7861.592 3.906 × 10–7
1081.218 3.927 × 10–71179.998 9.584 × 10–81173.577 1.711 × 10–7
1298.285 4.581 × 10–713100.610 9.038 × 10–71389.691 6.472 × 10–7
14111.392 2.764 × 10–714104.128 4.613 × 10–71496.560 4.950 × 10–7
16114.618 2.050 × 10–716110.094 1.703 × 10–716102.096 3.003 × 10–7
19135.847 2.568 × 10–719124.879 8.979 × 10–820115.669 6.292 × 10–8
a

α is the normal mode index, να is the frequency, int. is the IR intensity, and Sp-Ph is the magnitude of the spin–phonon coupling coefficient.

The computed redshifts of peaks in the range 30–130 cm–1 are in an overall good agreement with those observed for the THz spectra measured at 10, 100, and 300 K. A one-to-one correspondence with the experimental peaks is valid for p1–4; the comparison is impossible for p5, as this band results, according to our calculation, as the superposition of four peaks (p5C–8C). The calculated relative intensities are also in good agreement with the experimental ones with the only exception for p2 and p3, for which an inversion of their intensities can be claimed. This interpretation is more plausible than an energy swap of the two modes because p3C shows a larger shift as a function of the temperature than p2C as, indeed, experimentally observed. The comparison of the calculated linewidths at different temperatures is impracticable, as in our approach the effects of temperature are taken into account only using different cell parameters.
In Figure 7, we report the decomposition procedure outlined by Neto et al. (55) and developed by Lunghi et al. (15) to compute the amplitude of local translation, local rotation, and internal vibrations of a single [VO(acac)2] molecule inside its crystal. The nature of the lowest eight peaks shows that the reticular contributions decrease as the energy of modes increases, as expected. Such a contribution is still detectable up to the 20th mode with a 20–30% of external character. The modes can be sketched as the following:
(1)

in p1C, two [VO(acac)2] molecules rigidly tilt one with respect to the other (Figure S6).

(2)

p2C–3C can be considered quasidegenerate modes where the two acac ligands of a [VO(acac)2] molecule tilt out-of-phase (p2C) or in-phase (p3C) with respect to one acac belonging to a neighboring [VO(acac)2] molecule: in p2C, the neighboring molecule is inside the unit cell; in p3C, in an adjacent one (Figures S6 and S7).

(3)

in p4C–8C, the vanadyl group bending and acac oxygen wagging contributions became significant (Figures S7, S8, and S9)

Figure 7

Figure 7. Total molecular motion associated with (a) IR and (b) Raman normal modes from the 1100K,opt set in the low wavenumbers range (30–130 cm–1) decomposed in intramolecular percentage (blue), rotational percentage (orange), and translational percentage (green). In Figure S10, we show the plots of the first vibrational modes composition for each temperature.

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Phonon Decay Mechanisms from Linewidth Analysis

The analysis of the temperature dependence of the homogeneous linewidth allows the rationalization of the decay mechanism of the phonons and the estimation of the third-order anharmonic coefficients.
The temperature-dependent anharmonic coupling of phonons leads to relaxation via multiphonon processes. (56) In the simplest case (predominant at low temperature), line broadening results from two-photon decay, involving down- and up-conversion mechanisms. In the first case, the optical phonon decays into two phonons of lower frequencies (down-conversion); in the second case, the optical phonon and a phonon of the thermal bath add up to produce a more energetic phonon (up-conversion). (56) Energy and momentum conservation are required for all processes. In the limit kT > hν (where ν is the frequency of the optical phonon), a linear trend of the Lorentzian linewidth indicates that two-phonon decay processes are the dominant relaxation mechanism.
The decay pathway identified for p1, p3, and p4 corresponds to the down- and up-conversion processes. The simplest down-conversion process fulfilling the conservation of energy and momenta requires that the energy of generated phonons is half of the starting one, 2hν = hν0. If we assume, for instance, that the optical phonon ν0 decays into two phonons of the same frequency ν, and the phonons ν1 and ν2 sum up to create a ν0 optical phonon, we can write the following equation (56,57)
(5)
where Γ0 is the residual linewidth at zero temperature, n = [exp(hν/kT) – 1]−1 is the occupation number of the phonon involved in the decay at temperature T, and B is the third-order anharmonic coefficient; the Kronecker δ functions ensure energy conservation.
In principle, the anharmonic lattice dynamics theory provides the recipes for the calculation of phonon linewidth, which requires knowledge of the phonon frequencies in the entire Brillouin zone. This goal goes far beyond the limits of this work; we adopt here a very simplified approach that involves drastic approximations. We verified that the up-conversion mechanism gives a small contribution for p1C, while larger contributions are found for p3C and p4C. In addition, we assume that the down-conversion mechanism in which two phonons of the same frequencies are created can be taken as representative of all the down-conversion processes.
Within these limits, for p1C up-conversion, might be allowed considering the interaction with the Raman-active mode 7 (57 cm–1) to create a Raman-active phonon at 100 cm–1 (mode 13). p3C interacts with p1C and creates a phonon p6C at 116 cm–1. The matching is fulfilled considering that the tail of p6C extends up to 125 cm–1. Phonon p4C can combine with Raman mode 5 (48 cm–1) to generate a phonon at 146 cm–1. Concerning the down-conversion processes, the two lowest optical phonons p1C and p3C can decay only into acoustic phonons, which, as previously reported, (28) extend from 0 to ca. 40 cm–1 at the Brillouin zone boundary. The relaxation of p4C involves two phonons of 48 cm–1, a frequency value at the lowest limit of the optical phonon branches. This schematic description of the relaxation processes is summarized in Figure 8.

Figure 8

Figure 8. Two-phonon up- (a) and down-conversion (b) processes. The optical phonons’ frequencies are chosen among the 1100K,opt set. The blue phonons refer to the ones appearing in the THz spectrum and the red ones are those IR or Raman modes participating in the multiphonon relaxation processes.

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The fitting to the experimental linewidth in the temperature range 10–100 K gives for p1C a third-order anharmonic coefficient B of ca. 2 cm–1, with a negligible contribution from the up-conversion term. Conversely, the B value for p3C and p4C is ca. 2 cm–1 (2.5 and 4 cm–1, respectively, considering only the down-conversion process).

Spin–Phonon Coupling Analysis

A first qualitative fingerprint of interaction between magnetism and vibrations in each molecule requires the calculation of the phonon spectrum and the amplitude of Sp-Ph coupling for all possible modes, , as depicted in the previous section. The computed Sp-Ph couplings for all of the energy range (0–1200 cm–1) for both Raman and IR modes are reported in Figure 9 for all of the three considered cells. Unfortunately, the lack of the inversion center in the molecule did not allow to fully exploit symmetry considerations in the rationalization of the normal modes, as evidenced in previous works. (32)

Figure 9

Figure 9. Comparison between coupled modes at different temperatures: (a) the DFT Sp-Ph coupling interaction in the 0–1200 cm–1 spectral range; (b) zoom on the 30–130 cm–1 spectral range.

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As stated in the previous section, some cell parameters computed for 10K,cell–opt do not completely follow the expected trend indicated by the experimental parameters obtained for 1100K and 1300K unit cells. The different cell parameters led to an alteration of the compositions of the normal modes and, therefore, to their Sp-Ph couplings. A closer similarity of 1100K vs 1300K Sp-Ph coefficients is indeed found, in contrast to 1100K vs 10K,cell–opt and 1300K vs 10K,cell–opt.
The spectrum can be divided into two regions. Above 300 cm–1, the coupled modes are dispersedly grouped for all frequency ranges with very similar distributions among the three different cells. The group of vibrations located at 1000 cm–1 showed the strongest Sp-Ph coefficients. Indeed, the involved normal modes are characterized by a relevant contribution of the characteristic V = Ovanadyl stretching. Strong couplings are also observed for groups centered at 470 cm–1 (Oacac–V–Oacac symmetric stretching) and 365 cm–1 (Oacac–V–Oacac symmetric stretching). The temperature effects on the Sp-Ph coupling magnitudes are not evident in this frequency range. Below 300 cm–1, a denser distribution of coupled modes was computed. The most coupled modes are in the 200–300 cm–1 energy window, and a variation of the magnitude as a function of temperature is now observed. The largest dependence (a factor 5 or higher between 0 and 300 K) was computed for the Raman mode 5 (46.32 cm–1 at 100 K, symmetric translation of molecules) and IR modes 18 (120.49 cm–1 at 100 K, V–Oacac bendings), 45 (260.53 cm–1 at 100 K, Oacac–V = Ovanadyl wagging and twisting), and 47 (264.08 cm–1 at 100 K, Oacac–V–Oacac wagging). For the first three modes, by increasing the temperature, a decrease of the Sp-Ph magnitude was observed. An opposite behavior was observed for the fourth, instead. For several other modes, a decrease or increase in their Sp-Ph coupling was observed though, even if the variation in temperature was not so pronounced. These differences can be ascribed to the modified perturbation of the first coordination sphere caused by the variation of the primitive cell parameters observed for the three considered temperatures (see Table 1). The nonmonotonous (a, β, γ) and monotonous (b, c, α) trends of the cell parameters can differently alter the composition of the normal modes and modify their Sp-Ph coupling efficiency.
Focusing our attention on the IR modes in the experimental THz range (30–130 cm–1), the differences between the computed Sp-Ph dispersions of 10K,cell–opt and 1100K/1300K are more evident.
The computed Sp-Ph coefficients are reported in Figure 9, and only 1100K,opt and 1300K,opt will be discussed for homogeneity.
The Sp-Ph coupling can decrease its magnitude or remain unaffected with the increase of temperature; its variation is not constant for each peak going from a slight variation of the Sp-Ph coupling for p1C (rigid tilting of molecules) to a significant one for p3C and p5C (first coordination sphere distortion).
In a recent paper, some of us have shown for the [Dy(acac)3(H2O)2] (58) complex that a large Sp-Ph coupling is found for those normal modes that can induce large polarization on the atoms of the first coordination sphere. Even in the case of [VO(acac)2], such a statement is confirmed. In detail, the normal modes 45, 62, and 91 (depicted in Figure S11), which involve the largest charges variation, show the largest Sp-Ph coupling (see Figure 10).

Figure 10

Figure 10. Charge variation of one molecule of [VO(acac)2] crystal is analyzed along several normal modes from the 1100K,opt structure. The metal center and first coordination sphere atomic charges are reported. The data are reported in Table S5, while the phonons’ composition is depicted in Figure S10.

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The indications from the computed results are twofold. From a strictly computational point of view, our analysis shows that the use of static approaches for the description of the phonon distribution should be performed when the temperature effects can be indirectly accounted for the use of crystallographic cells collected at different temperatures. We have shown that in such a framework, a quantitative agreement with the THz experiment was obtained, while the ordinary approach of optimizing the geometries along with or without the crystallographic cell parameters can lead only to qualitative results for the vibrational spectrum and for the Sp-Ph coupling. From a structural point of view, it is possible to explicitly show how the Sp-Ph coupling associated with a single normal mode can vary as a function of temperature as a direct consequence of the variation with the temperature of the crystal cell parameters. It is also evident that the Sp-Ph relaxation mechanism can give rise to a more complex behavior than the polynomial or exponential functions usually considered in the literature (59,60) for the T1 temperature dependence.

Conclusions

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The harmonic approximation of molecular vibrations is highly valuable for interpretation of the main features of the vibrational spectra but it cannot explain all their fine structures, which depend on the temperature. At finite temperatures, anharmonic motions of the molecules must be considered since they allow for the possibility of interactions among the harmonic phonons, resulting in slight changes in the energy and composition of characteristic vibrations. The most straightforward parameters to be considered as modified by anharmonicity are the cell parameters. The crystal structures obtained from X-ray diffraction collected at cryogenic and room temperatures allowed us to calculate the observed THz anharmonic frequency shift with high accuracy.
The overall differences in the spin–coupling magnitudes as a function of temperature were highlighted. It turned out that Sp-Ph coupling can decrease or increase its magnitude with the increase of temperature. The reason for such trends has to be ascribed to variations of primitive vectors and unit-cell angles observed for the two considered temperatures. Indeed, the cell volume tunes the normal modes modifying the perturbation to the first coordination sphere of the vanadium ion. The overall results presented in this paper have also a general relevance allowing the quantification of the errors introduced by neglecting the temperature effects on the cell parameters. The cell parameters obtained by full optimization at 0 K do not follow the trends outlined by 100 and 300 K experimental structures, suggesting the possibility that a constrained optimization employing the experimental cells could yield more accurate results.
Moreover, the presented results can have a general relevance regarding the calculation of the T1 values, too. Indeed, even if significant results have been recently achieved in its computation at the ab initio level, (25,26) the T1 temperature dependence was only due to the phonon thermal distribution, keeping the Sp-Ph coupling as temperature-independent. However, our calculations suggest that even though the computed Sp-Ph coupling variations are small, the overall calculated variation becomes mandatory for a more rigorous reproduction of T1 along a wide temperature range.
Therefore, the results of this study represent further pieces of the puzzle of the role played by vibrations in determining the relaxation properties of potential molecular qubits at the quantitative level. In particular, the deeper knowledge of the Sp-Ph coupling might allow one to exploit phonons for the accurate control, e.g., initialization and readout of the qubits. This can be performed by using THz pulses by means of time-resolved ultrafast spectroscopy.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpcc.1c06916.

  • X-ray crystallographic data; convergence test; cartesian derivatives comparison; full spectral data; normal-mode composition; atomic charges (PDF)

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Author Information

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  • Corresponding Authors
    • Renato Torre - Dipartimento di Fisica ed Astronomia, Universitá degli Studi di Firenze, Via G. Sansone 1, Sesto Fiorentino, Florence 50019, Italy;  European Laboratory for Non-Linear Spectroscopy (LENS), Universitá degli Studi di Firenze, Sesto Fiorentino, Florence 50019, Italy;  Orcidhttps://orcid.org/0000-0003-3182-9906;  Email: [email protected]
    • Federico Totti - Dipartimento di Chimica “Ugo Schiff” & INSTM RU, Universitá degli Studi di Firenze, Via della Lastruccia 3, Sesto Fiorentino, Florence 50019, Italy;  Orcidhttps://orcid.org/0000-0003-4752-0495;  Email: [email protected]
  • Authors
    • Andrea Albino - Dipartimento di Chimica “Ugo Schiff” & INSTM RU, Universitá degli Studi di Firenze, Via della Lastruccia 3, Sesto Fiorentino, Florence 50019, Italy
    • Stefano Benci - European Laboratory for Non-Linear Spectroscopy (LENS), Universitá degli Studi di Firenze, Sesto Fiorentino, Florence 50019, Italy
    • Matteo Atzori - Laboratoire National des Champs Magnétiques Intenses (LNCMI), Univ. Grenoble Alpes, INSA Toulouse, Univ. Toulouse Paul Sabatier, EMFL, CNRS, F38043 Grenoble, France;  Orcidhttps://orcid.org/0000-0003-1357-6159
    • Laura Chelazzi - Dipartimento di Chimica “Ugo Schiff” & Center of Crystallography, Universitá degli Studi di Firenze, Via della Lastruccia 3, Sesto Fiorentino, Florence 50019, Italy
    • Samuele Ciattini - Dipartimento di Chimica “Ugo Schiff” & Center of Crystallography, Universitá degli Studi di Firenze, Via della Lastruccia 3, Sesto Fiorentino, Florence 50019, Italy
    • Andrea Taschin - European Laboratory for Non-Linear Spectroscopy (LENS), Universitá degli Studi di Firenze, Sesto Fiorentino, Florence 50019, Italy;  ENEA, Agenzia nazionale per le nuove tecnologie, l’energia e lo sviluppo economico sostenibile, Centro Ricerche Frascati, via Enrico Fermi 45, 00044 Frascati, Roma, Italy;  Orcidhttps://orcid.org/0000-0003-4088-7964
    • Paolo Bartolini - European Laboratory for Non-Linear Spectroscopy (LENS), Universitá degli Studi di Firenze, Sesto Fiorentino, Florence 50019, Italy
    • Alessandro Lunghi - School of Physics, AMBER and CRANN Institute, Trinity College, Dublin 2, Ireland;  Orcidhttps://orcid.org/0000-0002-1948-4434
    • Roberto Righini - European Laboratory for Non-Linear Spectroscopy (LENS), Universitá degli Studi di Firenze, Sesto Fiorentino, Florence 50019, Italy;  Orcidhttps://orcid.org/0000-0003-0260-0248
    • Roberta Sessoli - Dipartimento di Chimica “Ugo Schiff” & INSTM RU, Universitá degli Studi di Firenze, Via della Lastruccia 3, Sesto Fiorentino, Florence 50019, Italy;  Orcidhttps://orcid.org/0000-0003-3783-2700
  • Notes
    The authors declare no competing financial interest.

Acknowledgments

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Financial support by the EU Commission through the QuantERA Project SUMO by the FETOPEN project FATMOLS (GA 862893), by Ente Cassa di Risparmio firenze (prog. 2018.1042), Ministero dell’Istruzione dell’Universitá e della Ricerca Italiano (PRIN2017-2017Z55KCW), and by European Union’s Horizon 2020 research and innovation program under grant agreement no. 871124 Laserlab-Europe are acknowledged. The computing resources and the related technical support used for this work have been provided by CRESCO/ENEAGRID High Performance Computing infrastructure and its staff. (61) CRESCO/ENEAGRID High Performance Computing infrastructure is funded by ENEA, the Italian National Agency for New Technologies, Energy and Sustainable Economic Development and by the Italian and European research programs, see http://www.cresco.enea.it/english for information.

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    Asaad, Serwan; Mourik, Vincent; Joecker, Benjamin; Johnson, Mark A. I.; Baczewski, Andrew D.; Firgau, Hannes R.; Madzik, Mateusz T.; Schmitt, Vivien; Pla, Jarryd J.; Hudson, Fay E.; Itoh, Kohei M.; McCallum, Jeffrey C.; Dzurak, Andrew S.; Laucht, Arne; Morello, Andrea
    Nature (London, United Kingdom) (2020), 579 (7798), 205-209CODEN: NATUAS; ISSN:0028-0836. (Nature Research)
    Nuclear spins are highly coherent quantum objects. In large ensembles, their control and detection via magnetic resonance is widely exploited, for example, in chem., medicine, materials science and mining. Nuclear spins also featured in early proposals for solid-state quantum computers and demonstrations of quantum search and factoring algorithms. Scaling up such concepts requires controlling individual nuclei, which can be detected when coupled to an electron. However, the need to address the nuclei via oscillating magnetic fields complicates their integration in multi-spin nanoscale devices, because the field cannot be localized or screened. Control via elec. fields would resolve this problem, but previous methods relied on transducing elec. signals into magnetic fields via the electron-nuclear hyperfine interaction, which severely affects nuclear coherence. Here we demonstrate the coherent quantum control of a single 123Sb (spin-7/2) nucleus using localized elec. fields produced within a silicon nanoelectronic device. The method exploits an idea proposed in 1961 but not previously realized exptl. with a single nucleus. Our results are quant. supported by a microscopic theor. model that reveals how the purely elec. modulation of the nuclear elec. quadrupole interaction results in coherent nuclear spin transitions that are uniquely addressable owing to lattice strain. The spin dephasing time, 0.1 s, is orders of magnitude longer than those obtained by methods that require a coupled electron spin to achieve elec. driving. These results show that high-spin quadrupolar nuclei could be deployed as chaotic models, strain sensors and hybrid spin-mech. quantum systems using all-elec. controls. Integrating elec. controllable nuclei with quantum dots could pave the way to scalable, nuclear- and electron-spin-based quantum computers in silicon that operate without the need for oscillating magnetic fields.
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  10. 10
    de Camargo, L. C.; Briganti, M.; Santana, F. S.; Stinghen, D.; Ribeiro, R. R.; Nunes, G. G.; Soares, J. F.; Salvadori, E.; Chiesa, M.; Benci, S. Exploring the Organometallic Route to Molecular Spin Qubits: the [CpTi(cot)] case. Angew. Chem., Int. Ed. 2021, 60, 2588– 2593,  DOI: 10.1002/anie.202009634
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    10
    Exploring the Organometallic Route to Molecular Spin Qubits: The [CpTi(cot)] Case
    de Camargo, Luana C.; Briganti, Matteo; Santana, Francielli S.; Stinghen, Danilo; Ribeiro, Ronny R.; Nunes, Giovana G.; Soares, Jaisa F.; Salvadori, Enrico; Chiesa, Mario; Benci, Stefano; Torre, Renato; Sorace, Lorenzo; Totti, Federico; Sessoli, Roberta
    Angewandte Chemie, International Edition (2021), 60 (5), 2588-2593CODEN: ACIEF5; ISSN:1433-7851. (Wiley-VCH Verlag GmbH & Co. KGaA)
    The coherence time of the 17-electron, mixed sandwich complex [CpTi(cot)], (η8-cyclooctatetraene)(η5-cyclopentadienyl)titanium, reaches 34 μs at 4.5 K in a frozen deuterated toluene soln. This is a remarkable coherence time for a highly protonated mol. The intramol. distances between the Ti and H atoms provide a good compromise between instantaneous and spin diffusion sources of decoherence. Ab initio calcns. at the mol. and crystal packing levels reveal that the characteristic low-energy ring rotations of the sandwich framework do not yield a too detrimental spin-lattice relaxation because of their small spin-phonon coupling. The volatility of [CpTi(cot)] and the accessibility of the semi-occupied, non-bonding dz2 orbital make this neutral compd. an ideal candidate for single-qubit addressing on surface and quantum sensing in combination with scanning probe microscopy.
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  11. 11
    Atzori, M.; Morra, E.; Tesi, L.; Albino, A.; Chiesa, M.; Sorace, L.; Sessoli, R. Quantum Coherence Times Enhancement in Vanadium(IV)-based Potential Molecular Qubits: the Key Role of the Vanadyl Moiety. J. Am. Chem. Soc. 2016, 138, 11234– 11244,  DOI: 10.1021/jacs.6b05574
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    11
    Quantum Coherence Times Enhancement in Vanadium(IV)-based Potential Molecular Qubits: the Key Role of the Vanadyl Moiety
    Atzori, Matteo; Morra, Elena; Tesi, Lorenzo; Albino, Andrea; Chiesa, Mario; Sorace, Lorenzo; Sessoli, Roberta
    Journal of the American Chemical Society (2016), 138 (35), 11234-11244CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
    In the search for long-lived quantum coherence in spin systems, vanadium(IV) complexes have shown record phase memory times among mol. systems. When nuclear spin-free ligands are employed, vanadium(IV) complexes can show at low temp. sufficiently long quantum coherence times, Tm, to perform quantum operations, but their use in real devices operating at room temp. is still hampered by the rapid decrease of T1 caused by the efficient spin-phonon coupling. In this work we have investigated the effect of different coordination environments on the magnetization dynamics and the quantum coherence of two vanadium(IV)-based potential mol. spin qubits in the solid state by introducing a unique structural difference, i.e., an oxovanadium(IV) in a square pyramidal vs. a vanadium(IV) in an octahedral environment featuring the same coordinating ligand, namely, the 1,3-dithiole-2-thione-4,5-dithiolate. This investigation, performed by a combined approach of alternate current (ac) susceptibility measurements and continuous wave (CW) and pulsed ESR spectroscopies revealed that the effectiveness of the vanadyl moiety in enhancing quantum coherence up to room temp. is related to a less effective mechanism of spin-lattice relaxation that can be quant. evaluated by the exponent n (ca. 3) of the temp. dependence of the relaxation rate. A more rapid collapse is obsd. for the non-oxo counterpart (n = 4) hampering the observation of quantum coherence at room temp. Record coherence time at room temp. (1.04 μs) and Rabi oscillations are also obsd. for the vanadyl deriv. in a very high concd. material (5 ± 1%) as a result of the addnl. benefit provided by the use of a nuclear spin-free ligand.
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    Bader, K.; Dengler, D.; Lenz, S.; Endeward, B.; Jiang, S.-D.; Neugebauer, P.; van Slageren, J. Room temperature quantum coherence in a potential molecular qubit. Nat. Commun. 2014, 5, 5304  DOI: 10.1038/ncomms6304
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    12
    Room temperature quantum coherence in a potential molecular qubit
    Bader, Katharina; Dengler, Dominik; Lenz, Samuel; Endeward, Burkhard; Jiang, Shang-Da; Neugebauer, Petr; van Slageren, Joris
    Nature Communications (2014), 5 (), 5304CODEN: NCAOBW; ISSN:2041-1723. (Nature Publishing Group)
    The successful development of a quantum computer would change the world, and current internet encryption methods would cease to function. However, no working quantum computer that even begins to rival conventional computers has been developed yet, which is due to the lack of suitable quantum bits. A key characteristic of a quantum bit is the coherence time. Transition metal complexes are very promising quantum bits, owing to their facile surface deposition and their chem. tunability. However, reported quantum coherence times have been unimpressive. Here we report very long quantum coherence times for a transition metal complex of 68 μs at low temp. (qubit figure of merit QM=3,400) and 1 μs at room temp., much higher than previously reported values for such systems. We show that this achievement is because of the rigidity of the lattice as well as removal of nuclear spins from the vicinity of the magnetic ion.
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    Schweiger, A.; Jeschke, G. Principles of Pulse Electron Paramagnetic Resonance; Oxford University Press: Oxford, UK., 2001.
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    Mirzoyan, R.; Hadt, R. G. The dynamic ligand field of a molecular qubit: Decoherence through spin-phonon coupling. Phys. Chem. Chem. Phys. 2020, 22, 11249– 11265,  DOI: 10.1039/D0CP00852D
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    14
    The dynamic ligand field of a molecular qubit: decoherence through spin-phonon coupling
    Mirzoyan, Ruben; Hadt, Ryan G.
    Physical Chemistry Chemical Physics (2020), 22 (20), 11249-11265CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
    Quantum coherence of S = 1/2 transition metal-based quantum bits (qubits) is strongly influenced by the magnitude of spin-phonon coupling. While this coupling is recognized as deriving from dynamic distortions about the first coordination sphere of the metal, a general model for understanding and quantifying ligand field contributions has not been established. Here we derive a general ligand field theory model to describe and quantify the nature of spin-phonon coupling terms in S = 1/2 transition metal complexes. We show that the coupling term for a given vibrational mode is governed by: (1) the magnitude of the metal-based spin-orbit coupling const., (2) the magnitude and gradient in the ligand field excited state energy, which dets. the magnitude of ground state orbital angular momentum, and (3) dynamic relativistic nephelauxetic contributions reflecting the magnitude and gradient in the covalency of the ligand-metal bonds. From an extensive series of d. functional theory (DFT) and time-dependent DFT (TDDFT) calcns. calibrated to a range of exptl. data, spin-phonon coupling terms describing minimalistic D4h/D2d [CuCl4]2- and C4v [VOCl4]2- complexes translate to and correlate with exptl. quantum coherence properties obsd. for Cu(II)- and V(IV)-based mol. qubits with different ligand sets, geometries, and coordination nos. While providing a fundamental framework and means to benchmark current qubits, the model and methodol. described herein can be used to screen any S = 1/2 mol. qubit candidate and guide the discovery of room temp. coherent materials for quantum information processing.
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    Lunghi, A.; Totti, F.; Sanvito, S.; Sessoli, R. Intra-molecular origin of the spin-phonon coupling in slow-relaxing molecular magnets. Chem. Sci. 2017, 8, 6051– 6059,  DOI: 10.1039/C7SC02832F
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    15
    Intra-molecular origin of the spin-phonon coupling in slow-relaxing molecular magnets
    Lunghi, Alessandro; Totti, Federico; Sanvito, Stefano; Sessoli, Roberta
    Chemical Science (2017), 8 (9), 6051-6059CODEN: CSHCCN; ISSN:2041-6520. (Royal Society of Chemistry)
    We perform a systematic investigation of the spin-phonon coupling leading to spin relaxation in the prototypical mononuclear single mol. magnet [(tpaPh)Fe]-. In particular we analyze in detail the nature of the most relevant vibrational modes giving rise to the relaxation. Our fully ab initio calcns., where the phonon modes are evaluated at the level of d. functional theory and the spin-phonon coupling by mapping post-Hartree-Fock electronic structures onto an effective spin Hamiltonian, reveal that acoustic phonons are not active in the spin-phonon relaxation process of dil. SMMs crystals. Furthermore, we find that intra-mol. vibrational modes produce anisotropy tensor modulations orders of magnitude higher than those assocd. to rotations. In light of these results we are able to suggest new designing rules for spin-long-living SMMs which go beyond the tailoring of static mol. features but fully take into account dynamical features of the vibrational thermal bath evidencing those internal mol. distortions more relevant to the spin dynamics.
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    Califano, S.; Schettino, V.; Neto, N. Lattice Dynamics of Molecular Crystals; Springer-Verlag: Berlin, Heidelberg, 1981.
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    Califano, S. Vibrational States; Wiley, 1976.
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    Qian, K.; Baldoví, J. J.; Jiang, S.-D.; Gaita-Ariño, A.; Zhang, Y.-Q.; Overgaard, J.; Wang, B.-W.; Coronado, E.; Gao, S. Does the thermal evolution of molecular structures critically affect the magnetic anisotropy?. Chem. Sci. 2015, 6, 4587– 4593,  DOI: 10.1039/C5SC01245G
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    Does the thermal evolution of molecular structures critically affect the magnetic anisotropy?
    Qian, Kang; Baldovi, Jose J.; Jiang, Shang-Da; Gaita-Arino, Alejandro; Zhang, Yi-Quan; Overgaard, Jacob; Wang, Bing-Wu; Coronado, Eugenio; Gao, Song
    Chemical Science (2015), 6 (8), 4587-4593CODEN: CSHCCN; ISSN:2041-6520. (Royal Society of Chemistry)
    A dysprosium based single-ion magnet is synthesized and characterized by the angular dependence of the single-crystal magnetic susceptibility. Ab initio and effective electrostatic analyses are performed using the mol. structures detd. from single crystal X-ray diffraction at 20 K, 100 K and 300 K. Contrary to the common assumption, the results reveal that the structural thermal effects that may affect the energy level scheme and magnetic anisotropy below 100 K are negligible.
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    Lawler, H. M.; Chang, E. K.; Shirley, E. L. Dynamical Matrices and Interatomic-Force Constants from Wave-Commensurate Supercells. arXiv preprint cond-mat/0407221, 2004.
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    Wang, Y.; Shang, S. L.; Fang, H.; Liu, Z. K.; Chen, L. Q. First-principles calculations of lattice dynamics and thermal properties of polar solids. Npj Comput. Mater. 2016, 2, 16006  DOI: 10.1038/npjcompumats.2016.6
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    Atzori, M.; Tesi, L.; Benci, S.; Lunghi, A.; Righini, R.; Taschin, A.; Torre, R.; Sorace, L.; Sessoli, R. Spin Dynamics and Low Energy Vibrations: Insights from Vanadyl-Based Potential Molecular Qubits. J. Am. Chem. Soc. 2017, 139, 4338– 4341,  DOI: 10.1021/jacs.7b01266
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    21
    Spin Dynamics and Low Energy Vibrations: Insights from Vanadyl-Based Potential Molecular Qubits
    Atzori, Matteo; Tesi, Lorenzo; Benci, Stefano; Lunghi, Alessandro; Righini, Roberto; Taschin, Andrea; Torre, Renato; Sorace, Lorenzo; Sessoli, Roberta
    Journal of the American Chemical Society (2017), 139 (12), 4338-4341CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
    Here the authors report the study of the magnetization dynamics of a vanadyl complex with diethyldithiocarbamate (Et2dtc-) ligands, [VO(Et2dtc)2] (1), in both solid-state and frozen soln. This showed an anomalous and unprecedentedly obsd. field dependence of the relaxation time, which was modeled with three contributions to the relaxation mechanism. The temp. dependence of the wt. of the two processes dominating at low fields was found to well correlate with the low energy vibrations as detd. by THz spectroscopy. This detailed exptl. comparative study represents a fundamental step to understand the spin dynamics of potential mol. quantum bits, and enriches the guidelines to design mol.-based systems with enhanced quantum coherence.
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  22. 22
    Atzori, M.; Benci, S.; Morra, E.; Tesi, L.; Chiesa, M.; Torre, R.; Sorace, L.; Sessoli, R. Structural Effects on the Spin Dynamics of Potential Molecular Qubits. Inorg. Chem. 2018, 57, 731– 740,  DOI: 10.1021/acs.inorgchem.7b02616
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    22
    Structural Effects on the Spin Dynamics of Potential Molecular Qubits
    Atzori, Matteo; Benci, Stefano; Morra, Elena; Tesi, Lorenzo; Chiesa, Mario; Torre, Renato; Sorace, Lorenzo; Sessoli, Roberta
    Inorganic Chemistry (2018), 57 (2), 731-740CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
    Control of spin-lattice magnetic relaxation is crucial to observe long quantum coherence in spin systems at reasonable temps. Such a control is most often extremely difficult to achieve, because of the coexistence of several relaxation mechanisms, that is direct, Raman, and Orbach. These are not always easy to relate to the energy states of the investigated system, because of the contribution to the relaxation of addnl. spin-phonon coupling phenomena mediated by intramol. vibrations. In this work, we have investigated the effect of slight changes on the mol. structure of four vanadium(IV)-based potential spin qubits on their spin dynamics, studied by alternate current (AC) susceptometry. The anal. of the magnetic field dependence of the relaxation time correlates well with the low-energy vibrational modes exptl. detected by time-domain THz spectroscopy. This confirms and extends our preliminary observations on the role played by spin-vibration coupling in detg. the fine structure of the spin-lattice relaxation time as a function of the magnetic field, for S = 1/2 potential spin qubits. This study represents a step forward in the use of low-energy vibrational spectroscopy as a prediction tool for the design of mol. spin qubits with long-lived quantum coherence. Indeed, quantum coherence times of ca. 4.0-6.0 μs in the 4-100 K range are obsd. for the best performing vanadyl derivs. identified through this multitechnique approach.
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  23. 23
    Yamabayashi, T.; Atzori, M.; Tesi, L.; Cosquer, G.; Santanni, F.; Boulon, M. E.; Morra, E.; Benci, S.; Torre, R.; Chiesa, M. Scaling Up Electronic Spin Qubits into a Three-Dimensional Metal-Organic Framework. J. Am. Chem. Soc. 2018, 140, 12090– 12101,  DOI: 10.1021/jacs.8b06733
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    23
    Scaling Up Electronic Spin Qubits into a Three-Dimensional Metal-Organic Framework
    Yamabayashi, Tsutomu; Atzori, Matteo; Tesi, Lorenzo; Cosquer, Goulven; Santanni, Fabio; Boulon, Marie-Emmanuelle; Morra, Elena; Benci, Stefano; Torre, Renato; Chiesa, Mario; Sorace, Lorenzo; Sessoli, Roberta; Yamashita, Masahiro
    Journal of the American Chemical Society (2018), 140 (38), 12090-12101CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
    Practical implementation of highly coherent mol. spin qubits for challenging technol. applications, such as quantum information processing or quantum sensing, requires precise organization of electronic qubit mol. components into extended frameworks. Realization of spatial control over qubit-qubit distances can be achieved by coordination chem. approaches through an appropriate choice of the mol. building blocks. However, translating single qubit mol. building units into extended arrays does not guarantee a priori retention of long quantum coherence and spin-lattice relaxation times due to the introduced modifications over qubit-qubit reciprocal distances and mol. crystal lattice phonon structure. The authors report the prepn. of a three-dimensional (3D) metal-org. framework (MOF) based on vanadyl qubits, [VO(TCPP-Zn2-bpy)] (1; TCPP = tetracarboxylphenylporphyrinate; bpy = 4,4'-bipyridyl) and the study of how such structural modifications influence qubits' performances. This was done through a multitechnique approach where the structure and properties of a representative mol. building block [VO(TPP)] (TPP = tetraphenylporphyrinate) (2) were compared with those of 3-dimensional 1. Pulsed ESR measurements on magnetically dild. samples in titanyl isostructural analogs revealed that coherence times are retained almost unchanged for 1 with respect to 2 up to room temp., while the temp. dependence of the spin-lattice relaxation time revealed insights into the role of low-energy vibrations, detected through terahertz spectroscopy, on the spin dynamics.
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  24. 24
    Albino, A.; Benci, S.; Tesi, L.; Atzori, M.; Torre, R.; Sanvito, S.; Sessoli, R.; Lunghi, A. First-Principles Investigation of Spin-Phonon Coupling in Vanadium-Based Molecular Spin Quantum Bits. Inorg. Chem. 2019, 58, 10260– 10268,  DOI: 10.1021/acs.inorgchem.9b01407
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    24
    First-Principles Investigation of Spin-Phonon Coupling in Vanadium-Based Molecular Spin Quantum Bits
    Albino, Andrea; Benci, Stefano; Tesi, Lorenzo; Atzori, Matteo; Torre, Renato; Sanvito, Stefano; Sessoli, Roberta; Lunghi, Alessandro
    Inorganic Chemistry (2019), 58 (15), 10260-10268CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
    Paramagnetic mols. can show long spin-coherence times, which make them good candidates as quantum bits (qubits). Reducing the efficiency of the spin-phonon interaction is the primary challenge toward achieving long coherence times over a wide temp. range in soft mol. lattices. The lack of a microscopic understanding about the role of vibrations in spin relaxation strongly undermines the possibility of chem. designing better-performing mol. qubits. Here we report a first-principles characterization of the main mechanism contributing to the spin-phonon coupling for a class of vanadium(IV) mol. qubits. Post-Hartree-Fock and d. functional theory methods are used to det. the effect of both intermol. and intramol. vibrations on modulation of the Zeeman energy for four mols. showing different coordination geometries and ligands. This comparative study provides the first insight into the role played by coordination geometry and ligand-field strength in detg. the spin-lattice relaxation time of mol. qubits, opening an avenue to the rational design of new compds.
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    Lunghi, A.; Sanvito, S. How do phonons relax molecular spins?. Sci. Adv. 2019, 5, eaax7163  DOI: 10.1126/sciadv.aax7163
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    Lunghi, A.; Sanvito, S. The Limit of Spin Lifetime in Solid-State Electronic Spins. J. Phys. Chem. Lett. 2020, 11, 6273– 6278,  DOI: 10.1021/acs.jpclett.0c01681
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    26
    The Limit of Spin Lifetime in Solid-State Electronic Spins
    Lunghi, Alessandro; Sanvito, Stefano
    Journal of Physical Chemistry Letters (2020), 11 (15), 6273-6278CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
    The development of spin qubits for quantum technologies requires their protection from the main source of finite-temp. decoherence: at. vibrations. Here the authors eliminate one of the main barriers to the progress in this field by providing a complete 1st-principles picture of spin relaxation that includes up to 2-phonon processes. Method is based on machine learning and electronic structure theory and makes the prediction of spin lifetime in realistic systems feasible. The authors study a prototypical V-based mol. qubit and reveal that the spin lifetime at high temp. is limited by Raman processes due to a small no. of THz intramol. vibrations. These findings effectively change the conventional understanding of spin relaxation in this class of materials and open new avenues for the rational design of long-living spin systems.
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    Ullah, A.; Baldoví, J. J.; Gaita-Ariño, A.; Coronado, E. Insights on the coupling between vibronically active molecular vibrations and lattice phonons in molecular nanomagnets. Dalton Trans. 2021, 50, 11071– 11076,  DOI: 10.1039/D1DT01832A
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    27
    Insights on the coupling between vibronically active molecular vibrations and lattice phonons in molecular nanomagnets
    Ullah, Aman; Baldovi, Jose J.; Gaita-Arino, Alejandro; Coronado, Eugenio
    Dalton Transactions (2021), 50 (32), 11071-11076CODEN: DTARAF; ISSN:1477-9226. (Royal Society of Chemistry)
    Spin-lattice relaxation is a key open problem to understand the spin dynamics of single-mol. magnets and mol. spin qubits. While modeling the coupling between spin states and local vibrations allows to det. the more relevant mol. vibrations for spin relaxation, this is not sufficient to explain how energy is dissipated towards the thermal bath. Herein, we employ a simple and efficient model to examine the coupling of local vibrational modes with long-wavelength longitudinal and transverse phonons in the clock-like spin qubit [Ho(W5O18)2]9-. We find that in crystals of this polyoxometalate the vibrational mode previously found to be vibronically active at low temp. does not couple significantly to lattice phonons. This means that further intramol. energy transfer via anharmonic vibrations is necessary for spin relaxation in this system. Finally, we discuss implications for the spin-phonon coupling of [Ho(W5O18)2]9- deposited on a MgO (001) substrate, offering a simple methodol. that can be extrapolated to est. the effects on spin relaxation of different surfaces, including 2D materials.
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  28. 28
    Garlatti, E.; Tesi, L.; Lunghi, A.; Atzori, M.; Voneshen, D. J.; Santini, P.; Sanvito, S.; Guidi, T.; Sessoli, R.; Carretta, S. Unveiling phonons in a molecular qubit with four-dimensional inelastic neutron scattering and density functional theory. Nat. Commun. 2020, 11, 1751  DOI: 10.1038/s41467-020-15475-7
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    28
    Unveiling phonons in a molecular qubit with four-dimensional inelastic neutron scattering and density functional theory
    Garlatti, E.; Tesi, L.; Lunghi, A.; Atzori, M.; Voneshen, D. J.; Santini, P.; Sanvito, S.; Guidi, T.; Sessoli, R.; Carretta, S.
    Nature Communications (2020), 11 (1), 1751CODEN: NCAOBW; ISSN:2041-1723. (Nature Research)
    Abstr.: Phonons are the main source of relaxation in mol. nanomagnets, and different mechanisms have been proposed in order to explain the wealth of exptl. findings. However, very limited exptl. investigations on phonons in these systems have been performed so far, yielding no information about their dispersions. Here we exploit state-of-the-art single-crystal inelastic neutron scattering to directly measure for the first time phonon dispersions in a prototypical mol. qubit. Both acoustic and optical branches are detected in crystals of [VO(acac)2] along different directions in the reciprocal space. Using energies and polarisation vectors calcd. with state-of-the-art D. Functional Theory, we reproduce important qual. features of [VO(acac)2] phonon modes, such as the presence of low-lying optical branches. Moreover, we evidence phonon anti-crossings involving acoustic and optical branches, yielding significant transfers of the spin-phonon coupling strength between the different modes.
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  29. 29
    Taschin, A.; Bartolini, P.; Tasseva, J.; Torre, R. THz time-domain spectroscopic investigations of thin films. Measurement 2018, 118, 282– 288,  DOI: 10.1016/j.measurement.2017.05.074
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    Tasseva, J.; Taschin, A.; Bartolini, P.; Striova, J.; Fontana, R.; Torre, R. Thin layered drawing media probed by THz time-domain spectroscopy. Analyst 2017, 142, 42– 47,  DOI: 10.1039/C6AN02113A
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    Thin layered drawing media probed by THz time-domain spectroscopy
    Tasseva, J.; Taschin, A.; Bartolini, P.; Striova, J.; Fontana, R.; Torre, R.
    Analyst (Cambridge, United Kingdom) (2017), 142 (1), 42-47CODEN: ANALAO; ISSN:0003-2654. (Royal Society of Chemistry)
    Dry and wet drawing materials were investigated by THz time-domain spectroscopy in transmission mode. Carbon-based and iron-gall inks have been studied, some prepd. following ancient recipes and others using current synthetic materials; a com. ink was studied as well. We measured the THz signals on the thin films of liq. inks deposited on polyethylene pellicles, comparing the results with the thick pellets of dried inks blended with polyethylene powder. This study required the implementation of an accurate exptl. method and data anal. procedure able to provide a reliable extn. of the material transmission parameters from a structured sample composed of thin layers, down to a thickness of a few tens of micrometers. THz measurements on thin ink layers enabled the detn. of both the absorption and the refractive index in an abs. scale in the 0.1-3 THz range, as well as the layer thickness. THz spectroscopic features of a paper sheet dyed by using one of the iron-gall inks were also investigated. Our results showed that THz time-domain spectroscopy enables the discrimination of various inks on different supports, including the application on paper, together with the proper detn. of the absorption coeffs. and indexes of refraction.
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    Taschin, A.; Bartolini, P.; Tasseva, J.; Striova, J.; Fontana, R.; Riminesi, C.; Torre, R. Drawing materials studied by THz spectroscopy. Acta Imeko 2017, 6, 12– 17,  DOI: 10.21014/acta_imeko.v6i3.447
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    Santanni, F.; Albino, A.; Atzori, M.; Ranieri, D.; Salvadori, E.; Chiesa, M.; Lunghi, A.; Bencini, A.; Sorace, L.; Totti, F. Probing Vibrational Symmetry Effects and Nuclear Spin Economy Principles in Molecular Spin Qubits. Inorg. Chem. 2021, 60, 140– 151,  DOI: 10.1021/acs.inorgchem.0c02573
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    Probing Vibrational Symmetry Effects and Nuclear Spin Economy Principles in Molecular Spin Qubits
    Santanni, Fabio; Albino, Andrea; Atzori, Matteo; Ranieri, Davide; Salvadori, Enrico; Chiesa, Mario; Lunghi, Alessandro; Bencini, Andrea; Sorace, Lorenzo; Totti, Federico; Sessoli, Roberta
    Inorganic Chemistry (2021), 60 (1), 140-151CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
    The selection of mol. spins qubits with long coherence time, Tm, is a central task for implementing mol.-based quantum technologies. Even if sufficiently long Tm can be achieved through an efficient synthetic strategy and ad hoc exptl. measurement procedures, many factors contributing to the loss of coherence still need to be thoroughly investigated and understood. Vibrational properties and nuclear spins of hydrogens are two of them. The formers play a paramount role, but detailed theor. investigation aimed at studying their effects on the spin dynamics of mol. complexes such as the benchmark phthalocyanine (Pc) is still missing, whereas the effect of the latter deserves to be examd. in detail for such a class of compds. In this work, authors adopted a combined theor. and exptl. approach to investigate the relaxation properties of classical [Cu(Pc)] and a CuII complex based on the ligand tetrakis-thiadiazoleporphyrazine (H2TTDPz), characterized by a hydrogen-free mol. structure. Systematic calcns. of mol. vibrations exemplify the effect of normal modes on the spin-lattice relaxation process, unveiling a different contribution to T1 depending on the symmetry of normal modes. Moreover, they obsd. that an appreciable Tm enhancement could be achieved by removing hydrogens from the ligand. The spin dynamics of [Cu(Pc)] and of the hydrogen-free analog [Cu(TTDPz)] (H2TTDPz = tetrakis-thiadiazoleporphyrazine) were compared through a combined theor. and exptl. approach. The theor. anal. of mol. vibrations highlights the crucial effect of normal modes symmetry on the spin-lattice relaxation process. Furthermore, an appreciable enhancement of the coherence time is obsd. when hydrogen nuclei are removed from the ligand.
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    Tesi, L.; Lunghi, A.; Atzori, M.; Lucaccini, E.; Sorace, L.; Totti, F.; Sessoli, R. Giant spin-phonon bottleneck effects in evaporable vanadyl-based molecules with long spin coherence. Dalton Trans. 2016, 45, 16635– 16643,  DOI: 10.1039/C6DT02559E
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    Giant spin-phonon bottleneck effects in evaporable vanadyl-based molecules with long spin coherence
    Tesi, L.; Lunghi, A.; Atzori, M.; Lucaccini, E.; Sorace, L.; Totti, F.; Sessoli, R.
    Dalton Transactions (2016), 45 (42), 16635-16643CODEN: DTARAF; ISSN:1477-9226. (Royal Society of Chemistry)
    Vanadium(IV) complexes have recently shown record quantum spin coherence times that in several circumstances are limited by spin-lattice relaxation. The role of the environment and vibronic properties in the low temp. dynamics is here investigated by a comparative study of the magnetization dynamics as a function of crystallite size and the steric hindrance of the β-diketonate ligands in VO(acac)2 (1), VO(dpm)2 (2) and VO(dbm)2 (3) evaporable complexes (acac- = acetylacetonate, dpm- = dipivaloylmethanate, and dbm- = dibenzoylmethanate). A pronounced crystallite size dependence of the relaxation time is obsd. at unusually high temps. (up to 40 K), which is assocd. with a giant spin-phonon bottleneck effect. The authors modeled this behavior by an ad hoc force field approach derived from d. functional theory calcns., which evidences a correlation of the intensity of the phenomenon with ligand dimensions and the unit cell size.
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    Shuter, E.; Rettig, S. J.; Orvig, C. Oxobis(2,4-pentanedionato)vanadium(IV), a Redetermination. Acta Crystallogr., Sect. C: Cryst. Struct. Commun. 1995, 51, 12– 14,  DOI: 10.1107/S0108270194010462
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    King, M. D.; Korter, T. M. Application of London-type dispersion corrections in solid-state density functional theory for predicting the temperature-dependence of crystal structures and terahertz spectra. Cryst. Growth Des. 2011, 11, 2006– 2010,  DOI: 10.1021/cg200211x
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    Application of London-Type Dispersion Corrections in Solid-State Density Functional Theory for Predicting the Temperature-Dependence of Crystal Structures and Terahertz Spectra
    King, Matthew D.; Korter, Timothy M.
    Crystal Growth & Design (2011), 11 (5), 2006-2010CODEN: CGDEFU; ISSN:1528-7483. (American Chemical Society)
    Solid-state d. functional theory (DFT) has been shown to be a valuable tool for the simulation of low-frequency vibrational motions in mol. crystals. While it is typically required that an exptl. known crystal structure be used as the initial input for these types of calcns., it is sometimes found that suitable crystallog. data are not easily obtainable. In this study, the low-temp. unit cell structure of the β polymorph of deuterated oxalic acid dihydrate, for which a structure has only been reported at room temp., was predicted using solid-state DFT augmented with a modified empirical correction for weak long-range dispersive interactions. The dispersion correction parameters were optimized against the known 100 K crystal structure of the α polymorph, and then used for full-geometry optimization of the β crystal structure. Using this predicted structure for the β polymorph, the obsd. cryogenic THz spectrum of a mixt. of deuterated oxalic acid dihydrate polymorphs was simulated.
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    Parlinski, K. Phonons calculated from first-principles. École thématique de la Société Française de la Neutronique 2011, 12, 161– 166,  DOI: 10.1051/sfn/201112008
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    Baroni, S.; de Gironcoli, S.; Dal Corso, A. Phonons and related crystal properties from density-functional perturbation theory. Rev. Mod. Phys. 2001, 73, 515– 567,  DOI: 10.1103/RevModPhys.73.515
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    Phonons and related crystal properties from density-functional perturbation theory
    Baroni, Stefano; De Gironcoli, Stefano; Dal Corso, Andrea; Giannozzi, Paolo
    Reviews of Modern Physics (2001), 73 (2), 515-562CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)
    This article reviews with many refs. the current status of lattice-dynamical calcns. in crystals, using d.-functional perturbation theory, with emphasis on the plane-wave pseudopotential method. Several specialized topics are treated, including the implementation for metals, the calcn. of the response to macroscopic elec. fields and their relevance to long-wavelength vibrations in polar materials, the response to strain deformations, and higher-order responses. The success of this methodol. is demonstrated with a no. of applications existing in the literature.
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    VandeVondele, J.; Krack, M.; Mohamed, F.; Parrinello, M.; Chassaing, T.; Hutter, J. QUICKSTEP: Fast and accurate density functional calculations using a mixed Gaussian and plane waves approach. Comput. Phys. Commun. 2005, 167, 103– 128,  DOI: 10.1016/j.cpc.2004.12.014
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    QUICKSTEP: fast and accurate density functional calculations using a mixed Gaussian and plane waves approach
    VandeVondele, Joost; Krack, Matthias; Mohamed, Fawzi; Parrinello, Michele; Chassaing, Thomas; Hutter, Juerg
    Computer Physics Communications (2005), 167 (2), 103-128CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)
    We present the Gaussian and plane waves (GPW) method and its implementation in which is part of the freely available program package CP2K. The GPW method allows for accurate d. functional calcns. in gas and condensed phases and can be effectively used for mol. dynamics simulations. We show how derivs. of the GPW energy functional, namely ionic forces and the Kohn-Sham matrix, can be computed in a consistent way. The computational cost of computing the total energy and the Kohn-Sham matrix is scaling linearly with the system size, even for condensed phase systems of just a few tens of atoms. The efficiency of the method allows for the use of large Gaussian basis sets for systems up to 3000 atoms, and we illustrate the accuracy of the method for various basis sets in gas and condensed phases. Agreement with basis set free calcns. for single mols. and plane wave based calcns. in the condensed phase is excellent. Wave function optimization with the orbital transformation technique leads to good parallel performance, and outperforms traditional diagonalisation methods. Energy conserving Born-Oppenheimer dynamics can be performed, and a highly efficient scheme is obtained using an extrapolation of the d. matrix. We illustrate these findings with calcns. using commodity PCs as well as supercomputers.
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    Kühne, T. D.; Iannuzzi, M.; Del Ben, M.; Rybkin, V. V.; Seewald, P.; Stein, F.; Laino, T.; Khaliullin, R. Z.; Schütt, O.; Schiffmann, F. CP2K: An electronic structure and molecular dynamics software package -Quickstep: Efficient and accurate electronic structure calculations. J. Chem. Phys. 2020, 152, 194103  DOI: 10.1063/5.0007045
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    CP2K: An electronic structure and molecular dynamics software package - Quickstep: Efficient and accurate electronic structure calculations
    Kuehne, Thomas D.; Iannuzzi, Marcella; Del Ben, Mauro; Rybkin, Vladimir V.; Seewald, Patrick; Stein, Frederick; Laino, Teodoro; Khaliullin, Rustam Z.; Schuett, Ole; Schiffmann, Florian; Golze, Dorothea; Wilhelm, Jan; Chulkov, Sergey; Bani-Hashemian, Mohammad Hossein; Weber, Valery; Borstnik, Urban; Taillefumier, Mathieu; Jakobovits, Alice Shoshana; Lazzaro, Alfio; Pabst, Hans; Mueller, Tiziano; Schade, Robert; Guidon, Manuel; Andermatt, Samuel; Holmberg, Nico; Schenter, Gregory K.; Hehn, Anna; Bussy, Augustin; Belleflamme, Fabian; Tabacchi, Gloria; Gloess, Andreas; Lass, Michael; Bethune, Iain; Mundy, Christopher J.; Plessl, Christian; Watkins, Matt; VandeVondele, Joost; Krack, Matthias; Hutter, Juerg
    Journal of Chemical Physics (2020), 152 (19), 194103CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    A review. CP2K is an open source electronic structure and mol. dynamics software package to perform atomistic simulations of solid-state, liq., mol., and biol. systems. It is esp. aimed at massively parallel and linear-scaling electronic structure methods and state-of-the-art ab initio mol. dynamics simulations. Excellent performance for electronic structure calcns. is achieved using novel algorithms implemented for modern high-performance computing systems. This review revisits the main capabilities of CP2K to perform efficient and accurate electronic structure simulations. The emphasis is put on d. functional theory and multiple post-Hartree-Fock methods using the Gaussian and plane wave approach and its augmented all-electron extension. (c) 2020 American Institute of Physics.
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    Goedecker, S.; Teter, M.; Hutter, J. Separable dual-space Gaussian pseudopotentials. Phys. Rev. B 1996, 54, 1703– 1710,  DOI: 10.1103/PhysRevB.54.1703
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    Separable dual-space Gaussian pseudopotentials
    Goedecker, S.; Teter, M.; Hutter, J.
    Physical Review B: Condensed Matter (1996), 54 (3), 1703-1710CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
    We present pseudopotential coeffs. for the first two rows of the Periodic Table. The pseudopotential is of an analytic form that gives optimal efficiency in numerical calculations using plane waves as a basis set. At most, even coeffs. are necessary to specify its analytic form. It is separable and has optimal decay properties in both real and Fourier space. Because of this property, the application of the nonlocal part of the pseudopotential to a wave function can be done efficiently on a grid in real space. Real space integration is much faster for large systems than ordinary multiplication in Fourier space, since it shows only quadratic scaling with respect to the size of the system. We systematically verify the high accuracy of these pseudopotentials by extensive at. and mol. test calcns.
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    Hartwigsen, C.; Goedecker, S.; Hutter, J. Relativistic separable dual-space Gaussian pseudopotentials from H to Rn. Phys. Rev. B 1998, 58, 3641– 3662,  DOI: 10.1103/PhysRevB.58.3641
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    Relativistic separable dual-space Gaussian pseudopotentials from H to Rn
    Hartwigsen, C.; Goedecker, S.; Hutter, J.
    Physical Review B: Condensed Matter and Materials Physics (1998), 58 (7), 3641-3662CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
    We generalize the concept of separable dual-space Gaussian pseudopotentials to the relativistic case. This allows us to construct this type of pseudopotential for the whole Periodic Table, and we present a complete table of pseudopotential parameters for all the elements from H to Rn. The relativistic version of this pseudopotential retains all the advantages of its nonrelativistic version. It is separable by construction, it is optimal for integration on a real-space grid, it is highly accurate, and, due to its analytic form, it can be specified by a very small no. of parameters. The accuracy of the pseudopotential is illustrated by an extensive series of mol. calcns.
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    Krack, M. Pseudopotentials for H to Kr optimized for gradient-corrected exchange-correlation functionals. Theor. Chem. Acc. 2005, 114, 145– 152,  DOI: 10.1007/s00214-005-0655-y
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    Pseudopotentials for H to Kr optimized for gradient-corrected exchange-correlation functionals
    Krack, M.
    Theoretical Chemistry Accounts (2005), 114 (1-3), 145-152CODEN: TCACFW; ISSN:1432-881X. (Springer GmbH)
    Pseudopotential parameter sets for the elements from H to Kr using the relativistic, norm-conserving, separable, dual-space Gaussian-type pseudopotentials of Goedecker, Teter, and Hutter (GTH) are presented as optimized for the gradient-cor. exchange-correlation functionals of Becke, Lee, Yang, and Parr (BLYP), Becke and Perdew (BP), and Perdew, Burke, and Ernzerhof (PBE). The accuracy and reliability of the GTH pseudopotentials is shown by calcns. for a series of small mols.
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    Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77, 3865,  DOI: 10.1103/PhysRevLett.77.3865
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    Generalized gradient approximation made simple
    Perdew, John P.; Burke, Kieron; Ernzerhof, Matthias
    Physical Review Letters (1996), 77 (18), 3865-3868CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    Generalized gradient approxns. (GGA's) for the exchange-correlation energy improve upon the local spin d. (LSD) description of atoms, mols., and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental consts. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential.
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    Grimme, S. Accurate description of van der Waals complexes by density functional theory including empirical corrections. J. Comput. Chem. 2004, 25, 1463– 1473,  DOI: 10.1002/jcc.20078
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    Accurate description of van der Waals complexes by density functional theory including empirical corrections
    Grimme, Stefan
    Journal of Computational Chemistry (2004), 25 (12), 1463-1473CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
    An empirical method to account for van der Waals interactions in practical calcns. in the framework of the d. functional theory (termed DFT-D) was tested for a wide variety of mol. complexes. As in previous schemes, the dispersive energy was described by damped interat. potentials of the form C6R-6. The use of pure, gradient-cor. d. functionals (BLYP and PBE), together with the resoln.-of-the-identity (RI) approxn. for the Coulomb operator, allows very efficient computations for large systems. In contrast to the previous work, extended AO basis sets of polarized TZV or QZV quality were employed, which reduced the basis set superposition error to a negligible extend. By using a global scaling factor for the at. C6 coeffs., the functional dependence of the results could be strongly reduced. The "double counting" of correlation effects for strongly bound complexes was found to be insignificant if steep damping functions were employed. The method was applied to a total of 29 complexes of atoms and small mols. (Ne, CH4, NH3, H2O, CH3F, N2, F2, formic acid, ethene, and ethine) with each other and with benzene, to benzene, naphthalene, pyrene, and coronene dimers, the naphthalene trimer, coronene·H2O and four H-bonded and stacked DNA base pairs (AT and GC). In almost all cases, very good agreement with reliable theor. or exptl. results for binding energies and intermol. distances is obtained. For stacked arom. systems and the important base pairs, the DFT-D-BLYP model seems to be even superior to std. MP2 treatments that systematically over-bind. The good results obtained suggest the approach as a practical tool to describe the properties of many important van der Waals systems in chem. Furthermore, the DFT-D data may either be used to calibrate much simpler (e.g., force-field) potentials or the optimized structures can be used as input for more accurate ab initio calcns. of the interaction energies.
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    Grimme, S. Semiempirical GGA-type density functional constructed with a long-range dispersion correction. J. Comput. Chem. 2006, 27, 1787– 1799,  DOI: 10.1002/jcc.20495
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    Semiempirical GGA-type density functional constructed with a long-range dispersion correction
    Grimme, Stefan
    Journal of Computational Chemistry (2006), 27 (15), 1787-1799CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
    A new d. functional (DF) of the generalized gradient approxn. (GGA) type for general chem. applications termed B97-D is proposed. It is based on Becke's power-series ansatz from 1997 and is explicitly parameterized by including damped atom-pairwise dispersion corrections of the form C6·R-6. A general computational scheme for the parameters used in this correction has been established and parameters for elements up to xenon and a scaling factor for the dispersion part for several common d. functionals (BLYP, PBE, TPSS, B3LYP) are reported. The new functional is tested in comparison with other GGAs and the B3LYP hybrid functional on std. thermochem. benchmark sets, for 40 noncovalently bound complexes, including large stacked arom. mols. and group II element clusters, and for the computation of mol. geometries. Further cross-validation tests were performed for organometallic reactions and other difficult problems for std. functionals. In summary, it is found that B97-D belongs to one of the most accurate general purpose GGAs, reaching, for example for the G97/2 set of heat of formations, a mean abs. deviation of only 3.8 kcal mol-1. The performance for noncovalently bound systems including many pure van der Waals complexes is exceptionally good, reaching on the av. CCSD(T) accuracy. The basic strategy in the development to restrict the d. functional description to shorter electron correlation lengths scales and to describe situations with medium to large interat. distances by damped C6·R-6 terms seems to be very successful, as demonstrated for some notoriously difficult reactions. As an example, for the isomerization of larger branched to linear alkanes, B97-D is the only DF available that yields the right sign for the energy difference. From a practical point of view, the new functional seems to be quite robust and it is thus suggested as an efficient and accurate quantum chem. method for large systems where dispersion forces are of general importance.
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    Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J. Chem. Phys. 2010, 132, 154104  DOI: 10.1063/1.3382344
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    A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu
    Grimme, Stefan; Antony, Jens; Ehrlich, Stephan; Krieg, Helge
    Journal of Chemical Physics (2010), 132 (15), 154104/1-154104/19CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The method of dispersion correction as an add-on to std. Kohn-Sham d. functional theory (DFT-D) has been refined regarding higher accuracy, broader range of applicability, and less empiricism. The main new ingredients are atom-pairwise specific dispersion coeffs. and cutoff radii that are both computed from first principles. The coeffs. for new eighth-order dispersion terms are computed using established recursion relations. System (geometry) dependent information is used for the first time in a DFT-D type approach by employing the new concept of fractional coordination nos. (CN). They are used to interpolate between dispersion coeffs. of atoms in different chem. environments. The method only requires adjustment of two global parameters for each d. functional, is asymptotically exact for a gas of weakly interacting neutral atoms, and easily allows the computation of at. forces. Three-body nonadditivity terms are considered. The method has been assessed on std. benchmark sets for inter- and intramol. noncovalent interactions with a particular emphasis on a consistent description of light and heavy element systems. The mean abs. deviations for the S22 benchmark set of noncovalent interactions for 11 std. d. functionals decrease by 15%-40% compared to the previous (already accurate) DFT-D version. Spectacular improvements are found for a tripeptide-folding model and all tested metallic systems. The rectification of the long-range behavior and the use of more accurate C6 coeffs. also lead to a much better description of large (infinite) systems as shown for graphene sheets and the adsorption of benzene on an Ag(111) surface. For graphene it is found that the inclusion of three-body terms substantially (by about 10%) weakens the interlayer binding. We propose the revised DFT-D method as a general tool for the computation of the dispersion energy in mols. and solids of any kind with DFT and related (low-cost) electronic structure methods for large systems. (c) 2010 American Institute of Physics.
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  • Abstract

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    Figure 1

    Figure 1. Primitive cell of [VO(acac)2] belonging to the triclinic P1̅ space group. Color code: vanadium, green; oxygen, red; carbon, cyan; and hydrogen, white.

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    Figure 2

    Figure 2. THz spectra of (1a) as a function of temperature.

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    Figure 3

    Figure 3. (Left) Temperature dependence of the peaks’ frequency. (Right) Temperature dependence of linewidths obtained using a Voigt fit function for each peak and a baseline correction.

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    Figure 4

    Figure 4. Temperature dependence of (a) Lorentzian component and (b) the Gaussian component of the Voigt profile of each peak.

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    Figure 5

    Figure 5. DFT vibrational analysis of 10K,cell–opt (blue), 1100K,opt (green), and 1300K,opt (red). A direct comparison of experimental and calculated spectra is available in the SI (Figure S5).

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    Figure 6

    Figure 6. Calculated frequencies (a) and intensities (b) of the lowest 8 IR-active modes.

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    Figure 7

    Figure 7. Total molecular motion associated with (a) IR and (b) Raman normal modes from the 1100K,opt set in the low wavenumbers range (30–130 cm–1) decomposed in intramolecular percentage (blue), rotational percentage (orange), and translational percentage (green). In Figure S10, we show the plots of the first vibrational modes composition for each temperature.

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    Figure 8

    Figure 8. Two-phonon up- (a) and down-conversion (b) processes. The optical phonons’ frequencies are chosen among the 1100K,opt set. The blue phonons refer to the ones appearing in the THz spectrum and the red ones are those IR or Raman modes participating in the multiphonon relaxation processes.

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    Figure 9

    Figure 9. Comparison between coupled modes at different temperatures: (a) the DFT Sp-Ph coupling interaction in the 0–1200 cm–1 spectral range; (b) zoom on the 30–130 cm–1 spectral range.

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    Figure 10

    Figure 10. Charge variation of one molecule of [VO(acac)2] crystal is analyzed along several normal modes from the 1100K,opt structure. The metal center and first coordination sphere atomic charges are reported. The data are reported in Table S5, while the phonons’ composition is depicted in Figure S10.

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      A single atom is the prototypical quantum system, and a natural candidate for a quantum bit, or qubit-the elementary unit of a quantum computer. Atoms were successfully used to store and process quantum information in electromagnetic traps, as well as in diamond through the use of the N-vacancy-center point defect. Solid-state elec. devices possess great potential to scale up such demonstrations from few-qubit control to larger-scale quantum processors. Coherent control of spin qubits was achieved in lithog. defined double quantum dots in both GaAs and Si. However, it is a formidable challenge to combine the elec. measurement capabilities of engineered nanostructures with the benefits inherent in at. spin qubits. Here we demonstrate the coherent manipulation of an individual electron spin qubit bound to a P donor atom in natural Si, measured elec. via single-shot read-out. We use ESR to drive Rabi oscillations, and a Hahn echo pulse sequence reveals a spin coherence time exceeding 200 μs. This time should be even longer in isotopically enriched 28Si samples. Combined with a device architecture that is compatible with modern integrated circuit technol., the electron spin of a single P atom in Si should be an excellent platform on which to build a scalable quantum computer.
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    8. 8
      Warner, M.; Din, S.; Tupitsyn, I. S.; Morley, G. W.; Stoneham, A. M.; Gardener, J. A.; Wu, Z.; Fisher, A. J.; Heutz, S.; Kay, C. W. Potential for spin-based information processing in a thin-film molecular semiconductor. Nature 2013, 503, 504– 508,  DOI: 10.1038/nature12597
      8
      Potential for spin-based information processing in a thin-film molecular semiconductor
      Warner, Marc; Din, Salahud; Tupitsyn, Igor S.; Morley, Gavin W.; Stoneham, A. Marshall; Gardener, Jules A.; Wu, Zhenlin; Fisher, Andrew J.; Heutz, Sandrine; Kay, Christopher W. M.; Aeppli, Gabriel
      Nature (London, United Kingdom) (2013), 503 (7477), 504-508CODEN: NATUAS; ISSN:0028-0836. (Nature Publishing Group)
      Org. semiconductors were studied intensively for applications in electronics and optics, and even spin-based information technol., or spintronics. Fundamental quantities in spintronics are the population relaxation time (T1) and the phase memory time (T2): T1 measures the lifetime of a classical bit, in this case embodied by a spin oriented either parallel or antiparallel to an external magnetic field, and T2 measures the corresponding lifetime of a quantum bit, encoded in the phase of the quantum state. These times are surprisingly long for a common, low-cost and chem. modifiable org. semiconductor, the blue pigment Cu phthalocyanine, in easily processed thin-film form used for device fabrication. At 5 K, a temp. reachable using inexpensive closed-cycle refrigerators, T1 and T2 are resp. 59 ms and 2.6 μs, and at 80 K, which is just above the b.p. of liq. N, they are resp. 10 μs and 1 μs, demonstrating that the performance of thin-film Cu phthalocyanine is superior to that of single-mol. magnets over the same temp. range. T2 is more than two orders of magnitude greater than the duration of the spin manipulation pulses, which suggests that Cu phthalocyanine holds promise for quantum information processing, and the long T1 indicates possibilities for medium-term storage of classical bits in all-org. devices on plastic substrates.
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    9. 9
      Asaad, S.; Mourik, V.; Joecker, B.; Johnson, M. A.; Baczewski, A. D.; Firgau, H. R.; Ma̧dzik, M. T.; Schmitt, V.; Pla, J. J.; Hudson, F. E. Coherent electrical control of a single high-spin nucleus in silicon. Nature 2020, 579, 205– 209,  DOI: 10.1038/s41586-020-2057-7
      9
      Coherent electrical control of a single high-spin nucleus in silicon
      Asaad, Serwan; Mourik, Vincent; Joecker, Benjamin; Johnson, Mark A. I.; Baczewski, Andrew D.; Firgau, Hannes R.; Madzik, Mateusz T.; Schmitt, Vivien; Pla, Jarryd J.; Hudson, Fay E.; Itoh, Kohei M.; McCallum, Jeffrey C.; Dzurak, Andrew S.; Laucht, Arne; Morello, Andrea
      Nature (London, United Kingdom) (2020), 579 (7798), 205-209CODEN: NATUAS; ISSN:0028-0836. (Nature Research)
      Nuclear spins are highly coherent quantum objects. In large ensembles, their control and detection via magnetic resonance is widely exploited, for example, in chem., medicine, materials science and mining. Nuclear spins also featured in early proposals for solid-state quantum computers and demonstrations of quantum search and factoring algorithms. Scaling up such concepts requires controlling individual nuclei, which can be detected when coupled to an electron. However, the need to address the nuclei via oscillating magnetic fields complicates their integration in multi-spin nanoscale devices, because the field cannot be localized or screened. Control via elec. fields would resolve this problem, but previous methods relied on transducing elec. signals into magnetic fields via the electron-nuclear hyperfine interaction, which severely affects nuclear coherence. Here we demonstrate the coherent quantum control of a single 123Sb (spin-7/2) nucleus using localized elec. fields produced within a silicon nanoelectronic device. The method exploits an idea proposed in 1961 but not previously realized exptl. with a single nucleus. Our results are quant. supported by a microscopic theor. model that reveals how the purely elec. modulation of the nuclear elec. quadrupole interaction results in coherent nuclear spin transitions that are uniquely addressable owing to lattice strain. The spin dephasing time, 0.1 s, is orders of magnitude longer than those obtained by methods that require a coupled electron spin to achieve elec. driving. These results show that high-spin quadrupolar nuclei could be deployed as chaotic models, strain sensors and hybrid spin-mech. quantum systems using all-elec. controls. Integrating elec. controllable nuclei with quantum dots could pave the way to scalable, nuclear- and electron-spin-based quantum computers in silicon that operate without the need for oscillating magnetic fields.
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    10. 10
      de Camargo, L. C.; Briganti, M.; Santana, F. S.; Stinghen, D.; Ribeiro, R. R.; Nunes, G. G.; Soares, J. F.; Salvadori, E.; Chiesa, M.; Benci, S. Exploring the Organometallic Route to Molecular Spin Qubits: the [CpTi(cot)] case. Angew. Chem., Int. Ed. 2021, 60, 2588– 2593,  DOI: 10.1002/anie.202009634
      10
      Exploring the Organometallic Route to Molecular Spin Qubits: The [CpTi(cot)] Case
      de Camargo, Luana C.; Briganti, Matteo; Santana, Francielli S.; Stinghen, Danilo; Ribeiro, Ronny R.; Nunes, Giovana G.; Soares, Jaisa F.; Salvadori, Enrico; Chiesa, Mario; Benci, Stefano; Torre, Renato; Sorace, Lorenzo; Totti, Federico; Sessoli, Roberta
      Angewandte Chemie, International Edition (2021), 60 (5), 2588-2593CODEN: ACIEF5; ISSN:1433-7851. (Wiley-VCH Verlag GmbH & Co. KGaA)
      The coherence time of the 17-electron, mixed sandwich complex [CpTi(cot)], (η8-cyclooctatetraene)(η5-cyclopentadienyl)titanium, reaches 34 μs at 4.5 K in a frozen deuterated toluene soln. This is a remarkable coherence time for a highly protonated mol. The intramol. distances between the Ti and H atoms provide a good compromise between instantaneous and spin diffusion sources of decoherence. Ab initio calcns. at the mol. and crystal packing levels reveal that the characteristic low-energy ring rotations of the sandwich framework do not yield a too detrimental spin-lattice relaxation because of their small spin-phonon coupling. The volatility of [CpTi(cot)] and the accessibility of the semi-occupied, non-bonding dz2 orbital make this neutral compd. an ideal candidate for single-qubit addressing on surface and quantum sensing in combination with scanning probe microscopy.
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    11. 11
      Atzori, M.; Morra, E.; Tesi, L.; Albino, A.; Chiesa, M.; Sorace, L.; Sessoli, R. Quantum Coherence Times Enhancement in Vanadium(IV)-based Potential Molecular Qubits: the Key Role of the Vanadyl Moiety. J. Am. Chem. Soc. 2016, 138, 11234– 11244,  DOI: 10.1021/jacs.6b05574
      11
      Quantum Coherence Times Enhancement in Vanadium(IV)-based Potential Molecular Qubits: the Key Role of the Vanadyl Moiety
      Atzori, Matteo; Morra, Elena; Tesi, Lorenzo; Albino, Andrea; Chiesa, Mario; Sorace, Lorenzo; Sessoli, Roberta
      Journal of the American Chemical Society (2016), 138 (35), 11234-11244CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
      In the search for long-lived quantum coherence in spin systems, vanadium(IV) complexes have shown record phase memory times among mol. systems. When nuclear spin-free ligands are employed, vanadium(IV) complexes can show at low temp. sufficiently long quantum coherence times, Tm, to perform quantum operations, but their use in real devices operating at room temp. is still hampered by the rapid decrease of T1 caused by the efficient spin-phonon coupling. In this work we have investigated the effect of different coordination environments on the magnetization dynamics and the quantum coherence of two vanadium(IV)-based potential mol. spin qubits in the solid state by introducing a unique structural difference, i.e., an oxovanadium(IV) in a square pyramidal vs. a vanadium(IV) in an octahedral environment featuring the same coordinating ligand, namely, the 1,3-dithiole-2-thione-4,5-dithiolate. This investigation, performed by a combined approach of alternate current (ac) susceptibility measurements and continuous wave (CW) and pulsed ESR spectroscopies revealed that the effectiveness of the vanadyl moiety in enhancing quantum coherence up to room temp. is related to a less effective mechanism of spin-lattice relaxation that can be quant. evaluated by the exponent n (ca. 3) of the temp. dependence of the relaxation rate. A more rapid collapse is obsd. for the non-oxo counterpart (n = 4) hampering the observation of quantum coherence at room temp. Record coherence time at room temp. (1.04 μs) and Rabi oscillations are also obsd. for the vanadyl deriv. in a very high concd. material (5 ± 1%) as a result of the addnl. benefit provided by the use of a nuclear spin-free ligand.
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    12. 12
      Bader, K.; Dengler, D.; Lenz, S.; Endeward, B.; Jiang, S.-D.; Neugebauer, P.; van Slageren, J. Room temperature quantum coherence in a potential molecular qubit. Nat. Commun. 2014, 5, 5304  DOI: 10.1038/ncomms6304
      12
      Room temperature quantum coherence in a potential molecular qubit
      Bader, Katharina; Dengler, Dominik; Lenz, Samuel; Endeward, Burkhard; Jiang, Shang-Da; Neugebauer, Petr; van Slageren, Joris
      Nature Communications (2014), 5 (), 5304CODEN: NCAOBW; ISSN:2041-1723. (Nature Publishing Group)
      The successful development of a quantum computer would change the world, and current internet encryption methods would cease to function. However, no working quantum computer that even begins to rival conventional computers has been developed yet, which is due to the lack of suitable quantum bits. A key characteristic of a quantum bit is the coherence time. Transition metal complexes are very promising quantum bits, owing to their facile surface deposition and their chem. tunability. However, reported quantum coherence times have been unimpressive. Here we report very long quantum coherence times for a transition metal complex of 68 μs at low temp. (qubit figure of merit QM=3,400) and 1 μs at room temp., much higher than previously reported values for such systems. We show that this achievement is because of the rigidity of the lattice as well as removal of nuclear spins from the vicinity of the magnetic ion.
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    13. 13
      Schweiger, A.; Jeschke, G. Principles of Pulse Electron Paramagnetic Resonance; Oxford University Press: Oxford, UK., 2001.
      There is no corresponding record for this reference.
    14. 14
      Mirzoyan, R.; Hadt, R. G. The dynamic ligand field of a molecular qubit: Decoherence through spin-phonon coupling. Phys. Chem. Chem. Phys. 2020, 22, 11249– 11265,  DOI: 10.1039/D0CP00852D
      14
      The dynamic ligand field of a molecular qubit: decoherence through spin-phonon coupling
      Mirzoyan, Ruben; Hadt, Ryan G.
      Physical Chemistry Chemical Physics (2020), 22 (20), 11249-11265CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
      Quantum coherence of S = 1/2 transition metal-based quantum bits (qubits) is strongly influenced by the magnitude of spin-phonon coupling. While this coupling is recognized as deriving from dynamic distortions about the first coordination sphere of the metal, a general model for understanding and quantifying ligand field contributions has not been established. Here we derive a general ligand field theory model to describe and quantify the nature of spin-phonon coupling terms in S = 1/2 transition metal complexes. We show that the coupling term for a given vibrational mode is governed by: (1) the magnitude of the metal-based spin-orbit coupling const., (2) the magnitude and gradient in the ligand field excited state energy, which dets. the magnitude of ground state orbital angular momentum, and (3) dynamic relativistic nephelauxetic contributions reflecting the magnitude and gradient in the covalency of the ligand-metal bonds. From an extensive series of d. functional theory (DFT) and time-dependent DFT (TDDFT) calcns. calibrated to a range of exptl. data, spin-phonon coupling terms describing minimalistic D4h/D2d [CuCl4]2- and C4v [VOCl4]2- complexes translate to and correlate with exptl. quantum coherence properties obsd. for Cu(II)- and V(IV)-based mol. qubits with different ligand sets, geometries, and coordination nos. While providing a fundamental framework and means to benchmark current qubits, the model and methodol. described herein can be used to screen any S = 1/2 mol. qubit candidate and guide the discovery of room temp. coherent materials for quantum information processing.
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    15. 15
      Lunghi, A.; Totti, F.; Sanvito, S.; Sessoli, R. Intra-molecular origin of the spin-phonon coupling in slow-relaxing molecular magnets. Chem. Sci. 2017, 8, 6051– 6059,  DOI: 10.1039/C7SC02832F
      15
      Intra-molecular origin of the spin-phonon coupling in slow-relaxing molecular magnets
      Lunghi, Alessandro; Totti, Federico; Sanvito, Stefano; Sessoli, Roberta
      Chemical Science (2017), 8 (9), 6051-6059CODEN: CSHCCN; ISSN:2041-6520. (Royal Society of Chemistry)
      We perform a systematic investigation of the spin-phonon coupling leading to spin relaxation in the prototypical mononuclear single mol. magnet [(tpaPh)Fe]-. In particular we analyze in detail the nature of the most relevant vibrational modes giving rise to the relaxation. Our fully ab initio calcns., where the phonon modes are evaluated at the level of d. functional theory and the spin-phonon coupling by mapping post-Hartree-Fock electronic structures onto an effective spin Hamiltonian, reveal that acoustic phonons are not active in the spin-phonon relaxation process of dil. SMMs crystals. Furthermore, we find that intra-mol. vibrational modes produce anisotropy tensor modulations orders of magnitude higher than those assocd. to rotations. In light of these results we are able to suggest new designing rules for spin-long-living SMMs which go beyond the tailoring of static mol. features but fully take into account dynamical features of the vibrational thermal bath evidencing those internal mol. distortions more relevant to the spin dynamics.
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    16. 16
      Califano, S.; Schettino, V.; Neto, N. Lattice Dynamics of Molecular Crystals; Springer-Verlag: Berlin, Heidelberg, 1981.
      There is no corresponding record for this reference.
    17. 17
      Califano, S. Vibrational States; Wiley, 1976.
      There is no corresponding record for this reference.
    18. 18
      Qian, K.; Baldoví, J. J.; Jiang, S.-D.; Gaita-Ariño, A.; Zhang, Y.-Q.; Overgaard, J.; Wang, B.-W.; Coronado, E.; Gao, S. Does the thermal evolution of molecular structures critically affect the magnetic anisotropy?. Chem. Sci. 2015, 6, 4587– 4593,  DOI: 10.1039/C5SC01245G
      18
      Does the thermal evolution of molecular structures critically affect the magnetic anisotropy?
      Qian, Kang; Baldovi, Jose J.; Jiang, Shang-Da; Gaita-Arino, Alejandro; Zhang, Yi-Quan; Overgaard, Jacob; Wang, Bing-Wu; Coronado, Eugenio; Gao, Song
      Chemical Science (2015), 6 (8), 4587-4593CODEN: CSHCCN; ISSN:2041-6520. (Royal Society of Chemistry)
      A dysprosium based single-ion magnet is synthesized and characterized by the angular dependence of the single-crystal magnetic susceptibility. Ab initio and effective electrostatic analyses are performed using the mol. structures detd. from single crystal X-ray diffraction at 20 K, 100 K and 300 K. Contrary to the common assumption, the results reveal that the structural thermal effects that may affect the energy level scheme and magnetic anisotropy below 100 K are negligible.
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    19. 19
      Lawler, H. M.; Chang, E. K.; Shirley, E. L. Dynamical Matrices and Interatomic-Force Constants from Wave-Commensurate Supercells. arXiv preprint cond-mat/0407221, 2004.
      There is no corresponding record for this reference.
    20. 20
      Wang, Y.; Shang, S. L.; Fang, H.; Liu, Z. K.; Chen, L. Q. First-principles calculations of lattice dynamics and thermal properties of polar solids. Npj Comput. Mater. 2016, 2, 16006  DOI: 10.1038/npjcompumats.2016.6
      There is no corresponding record for this reference.
    21. 21
      Atzori, M.; Tesi, L.; Benci, S.; Lunghi, A.; Righini, R.; Taschin, A.; Torre, R.; Sorace, L.; Sessoli, R. Spin Dynamics and Low Energy Vibrations: Insights from Vanadyl-Based Potential Molecular Qubits. J. Am. Chem. Soc. 2017, 139, 4338– 4341,  DOI: 10.1021/jacs.7b01266
      21
      Spin Dynamics and Low Energy Vibrations: Insights from Vanadyl-Based Potential Molecular Qubits
      Atzori, Matteo; Tesi, Lorenzo; Benci, Stefano; Lunghi, Alessandro; Righini, Roberto; Taschin, Andrea; Torre, Renato; Sorace, Lorenzo; Sessoli, Roberta
      Journal of the American Chemical Society (2017), 139 (12), 4338-4341CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
      Here the authors report the study of the magnetization dynamics of a vanadyl complex with diethyldithiocarbamate (Et2dtc-) ligands, [VO(Et2dtc)2] (1), in both solid-state and frozen soln. This showed an anomalous and unprecedentedly obsd. field dependence of the relaxation time, which was modeled with three contributions to the relaxation mechanism. The temp. dependence of the wt. of the two processes dominating at low fields was found to well correlate with the low energy vibrations as detd. by THz spectroscopy. This detailed exptl. comparative study represents a fundamental step to understand the spin dynamics of potential mol. quantum bits, and enriches the guidelines to design mol.-based systems with enhanced quantum coherence.
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    22. 22
      Atzori, M.; Benci, S.; Morra, E.; Tesi, L.; Chiesa, M.; Torre, R.; Sorace, L.; Sessoli, R. Structural Effects on the Spin Dynamics of Potential Molecular Qubits. Inorg. Chem. 2018, 57, 731– 740,  DOI: 10.1021/acs.inorgchem.7b02616
      22
      Structural Effects on the Spin Dynamics of Potential Molecular Qubits
      Atzori, Matteo; Benci, Stefano; Morra, Elena; Tesi, Lorenzo; Chiesa, Mario; Torre, Renato; Sorace, Lorenzo; Sessoli, Roberta
      Inorganic Chemistry (2018), 57 (2), 731-740CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
      Control of spin-lattice magnetic relaxation is crucial to observe long quantum coherence in spin systems at reasonable temps. Such a control is most often extremely difficult to achieve, because of the coexistence of several relaxation mechanisms, that is direct, Raman, and Orbach. These are not always easy to relate to the energy states of the investigated system, because of the contribution to the relaxation of addnl. spin-phonon coupling phenomena mediated by intramol. vibrations. In this work, we have investigated the effect of slight changes on the mol. structure of four vanadium(IV)-based potential spin qubits on their spin dynamics, studied by alternate current (AC) susceptometry. The anal. of the magnetic field dependence of the relaxation time correlates well with the low-energy vibrational modes exptl. detected by time-domain THz spectroscopy. This confirms and extends our preliminary observations on the role played by spin-vibration coupling in detg. the fine structure of the spin-lattice relaxation time as a function of the magnetic field, for S = 1/2 potential spin qubits. This study represents a step forward in the use of low-energy vibrational spectroscopy as a prediction tool for the design of mol. spin qubits with long-lived quantum coherence. Indeed, quantum coherence times of ca. 4.0-6.0 μs in the 4-100 K range are obsd. for the best performing vanadyl derivs. identified through this multitechnique approach.
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    23. 23
      Yamabayashi, T.; Atzori, M.; Tesi, L.; Cosquer, G.; Santanni, F.; Boulon, M. E.; Morra, E.; Benci, S.; Torre, R.; Chiesa, M. Scaling Up Electronic Spin Qubits into a Three-Dimensional Metal-Organic Framework. J. Am. Chem. Soc. 2018, 140, 12090– 12101,  DOI: 10.1021/jacs.8b06733
      23
      Scaling Up Electronic Spin Qubits into a Three-Dimensional Metal-Organic Framework
      Yamabayashi, Tsutomu; Atzori, Matteo; Tesi, Lorenzo; Cosquer, Goulven; Santanni, Fabio; Boulon, Marie-Emmanuelle; Morra, Elena; Benci, Stefano; Torre, Renato; Chiesa, Mario; Sorace, Lorenzo; Sessoli, Roberta; Yamashita, Masahiro
      Journal of the American Chemical Society (2018), 140 (38), 12090-12101CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
      Practical implementation of highly coherent mol. spin qubits for challenging technol. applications, such as quantum information processing or quantum sensing, requires precise organization of electronic qubit mol. components into extended frameworks. Realization of spatial control over qubit-qubit distances can be achieved by coordination chem. approaches through an appropriate choice of the mol. building blocks. However, translating single qubit mol. building units into extended arrays does not guarantee a priori retention of long quantum coherence and spin-lattice relaxation times due to the introduced modifications over qubit-qubit reciprocal distances and mol. crystal lattice phonon structure. The authors report the prepn. of a three-dimensional (3D) metal-org. framework (MOF) based on vanadyl qubits, [VO(TCPP-Zn2-bpy)] (1; TCPP = tetracarboxylphenylporphyrinate; bpy = 4,4'-bipyridyl) and the study of how such structural modifications influence qubits' performances. This was done through a multitechnique approach where the structure and properties of a representative mol. building block [VO(TPP)] (TPP = tetraphenylporphyrinate) (2) were compared with those of 3-dimensional 1. Pulsed ESR measurements on magnetically dild. samples in titanyl isostructural analogs revealed that coherence times are retained almost unchanged for 1 with respect to 2 up to room temp., while the temp. dependence of the spin-lattice relaxation time revealed insights into the role of low-energy vibrations, detected through terahertz spectroscopy, on the spin dynamics.
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    24. 24
      Albino, A.; Benci, S.; Tesi, L.; Atzori, M.; Torre, R.; Sanvito, S.; Sessoli, R.; Lunghi, A. First-Principles Investigation of Spin-Phonon Coupling in Vanadium-Based Molecular Spin Quantum Bits. Inorg. Chem. 2019, 58, 10260– 10268,  DOI: 10.1021/acs.inorgchem.9b01407
      24
      First-Principles Investigation of Spin-Phonon Coupling in Vanadium-Based Molecular Spin Quantum Bits
      Albino, Andrea; Benci, Stefano; Tesi, Lorenzo; Atzori, Matteo; Torre, Renato; Sanvito, Stefano; Sessoli, Roberta; Lunghi, Alessandro
      Inorganic Chemistry (2019), 58 (15), 10260-10268CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
      Paramagnetic mols. can show long spin-coherence times, which make them good candidates as quantum bits (qubits). Reducing the efficiency of the spin-phonon interaction is the primary challenge toward achieving long coherence times over a wide temp. range in soft mol. lattices. The lack of a microscopic understanding about the role of vibrations in spin relaxation strongly undermines the possibility of chem. designing better-performing mol. qubits. Here we report a first-principles characterization of the main mechanism contributing to the spin-phonon coupling for a class of vanadium(IV) mol. qubits. Post-Hartree-Fock and d. functional theory methods are used to det. the effect of both intermol. and intramol. vibrations on modulation of the Zeeman energy for four mols. showing different coordination geometries and ligands. This comparative study provides the first insight into the role played by coordination geometry and ligand-field strength in detg. the spin-lattice relaxation time of mol. qubits, opening an avenue to the rational design of new compds.
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    25. 25
      Lunghi, A.; Sanvito, S. How do phonons relax molecular spins?. Sci. Adv. 2019, 5, eaax7163  DOI: 10.1126/sciadv.aax7163
      There is no corresponding record for this reference.
    26. 26
      Lunghi, A.; Sanvito, S. The Limit of Spin Lifetime in Solid-State Electronic Spins. J. Phys. Chem. Lett. 2020, 11, 6273– 6278,  DOI: 10.1021/acs.jpclett.0c01681
      26
      The Limit of Spin Lifetime in Solid-State Electronic Spins
      Lunghi, Alessandro; Sanvito, Stefano
      Journal of Physical Chemistry Letters (2020), 11 (15), 6273-6278CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
      The development of spin qubits for quantum technologies requires their protection from the main source of finite-temp. decoherence: at. vibrations. Here the authors eliminate one of the main barriers to the progress in this field by providing a complete 1st-principles picture of spin relaxation that includes up to 2-phonon processes. Method is based on machine learning and electronic structure theory and makes the prediction of spin lifetime in realistic systems feasible. The authors study a prototypical V-based mol. qubit and reveal that the spin lifetime at high temp. is limited by Raman processes due to a small no. of THz intramol. vibrations. These findings effectively change the conventional understanding of spin relaxation in this class of materials and open new avenues for the rational design of long-living spin systems.
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      https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhtl2rsrfP&md5=3c6e646b3b537f46e159933629c0b465
    27. 27
      Ullah, A.; Baldoví, J. J.; Gaita-Ariño, A.; Coronado, E. Insights on the coupling between vibronically active molecular vibrations and lattice phonons in molecular nanomagnets. Dalton Trans. 2021, 50, 11071– 11076,  DOI: 10.1039/D1DT01832A
      27
      Insights on the coupling between vibronically active molecular vibrations and lattice phonons in molecular nanomagnets
      Ullah, Aman; Baldovi, Jose J.; Gaita-Arino, Alejandro; Coronado, Eugenio
      Dalton Transactions (2021), 50 (32), 11071-11076CODEN: DTARAF; ISSN:1477-9226. (Royal Society of Chemistry)
      Spin-lattice relaxation is a key open problem to understand the spin dynamics of single-mol. magnets and mol. spin qubits. While modeling the coupling between spin states and local vibrations allows to det. the more relevant mol. vibrations for spin relaxation, this is not sufficient to explain how energy is dissipated towards the thermal bath. Herein, we employ a simple and efficient model to examine the coupling of local vibrational modes with long-wavelength longitudinal and transverse phonons in the clock-like spin qubit [Ho(W5O18)2]9-. We find that in crystals of this polyoxometalate the vibrational mode previously found to be vibronically active at low temp. does not couple significantly to lattice phonons. This means that further intramol. energy transfer via anharmonic vibrations is necessary for spin relaxation in this system. Finally, we discuss implications for the spin-phonon coupling of [Ho(W5O18)2]9- deposited on a MgO (001) substrate, offering a simple methodol. that can be extrapolated to est. the effects on spin relaxation of different surfaces, including 2D materials.
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    28. 28
      Garlatti, E.; Tesi, L.; Lunghi, A.; Atzori, M.; Voneshen, D. J.; Santini, P.; Sanvito, S.; Guidi, T.; Sessoli, R.; Carretta, S. Unveiling phonons in a molecular qubit with four-dimensional inelastic neutron scattering and density functional theory. Nat. Commun. 2020, 11, 1751  DOI: 10.1038/s41467-020-15475-7
      28
      Unveiling phonons in a molecular qubit with four-dimensional inelastic neutron scattering and density functional theory
      Garlatti, E.; Tesi, L.; Lunghi, A.; Atzori, M.; Voneshen, D. J.; Santini, P.; Sanvito, S.; Guidi, T.; Sessoli, R.; Carretta, S.
      Nature Communications (2020), 11 (1), 1751CODEN: NCAOBW; ISSN:2041-1723. (Nature Research)
      Abstr.: Phonons are the main source of relaxation in mol. nanomagnets, and different mechanisms have been proposed in order to explain the wealth of exptl. findings. However, very limited exptl. investigations on phonons in these systems have been performed so far, yielding no information about their dispersions. Here we exploit state-of-the-art single-crystal inelastic neutron scattering to directly measure for the first time phonon dispersions in a prototypical mol. qubit. Both acoustic and optical branches are detected in crystals of [VO(acac)2] along different directions in the reciprocal space. Using energies and polarisation vectors calcd. with state-of-the-art D. Functional Theory, we reproduce important qual. features of [VO(acac)2] phonon modes, such as the presence of low-lying optical branches. Moreover, we evidence phonon anti-crossings involving acoustic and optical branches, yielding significant transfers of the spin-phonon coupling strength between the different modes.
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    29. 29
      Taschin, A.; Bartolini, P.; Tasseva, J.; Torre, R. THz time-domain spectroscopic investigations of thin films. Measurement 2018, 118, 282– 288,  DOI: 10.1016/j.measurement.2017.05.074
      There is no corresponding record for this reference.
    30. 30
      Tasseva, J.; Taschin, A.; Bartolini, P.; Striova, J.; Fontana, R.; Torre, R. Thin layered drawing media probed by THz time-domain spectroscopy. Analyst 2017, 142, 42– 47,  DOI: 10.1039/C6AN02113A
      30
      Thin layered drawing media probed by THz time-domain spectroscopy
      Tasseva, J.; Taschin, A.; Bartolini, P.; Striova, J.; Fontana, R.; Torre, R.
      Analyst (Cambridge, United Kingdom) (2017), 142 (1), 42-47CODEN: ANALAO; ISSN:0003-2654. (Royal Society of Chemistry)
      Dry and wet drawing materials were investigated by THz time-domain spectroscopy in transmission mode. Carbon-based and iron-gall inks have been studied, some prepd. following ancient recipes and others using current synthetic materials; a com. ink was studied as well. We measured the THz signals on the thin films of liq. inks deposited on polyethylene pellicles, comparing the results with the thick pellets of dried inks blended with polyethylene powder. This study required the implementation of an accurate exptl. method and data anal. procedure able to provide a reliable extn. of the material transmission parameters from a structured sample composed of thin layers, down to a thickness of a few tens of micrometers. THz measurements on thin ink layers enabled the detn. of both the absorption and the refractive index in an abs. scale in the 0.1-3 THz range, as well as the layer thickness. THz spectroscopic features of a paper sheet dyed by using one of the iron-gall inks were also investigated. Our results showed that THz time-domain spectroscopy enables the discrimination of various inks on different supports, including the application on paper, together with the proper detn. of the absorption coeffs. and indexes of refraction.
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    31. 31
      Taschin, A.; Bartolini, P.; Tasseva, J.; Striova, J.; Fontana, R.; Riminesi, C.; Torre, R. Drawing materials studied by THz spectroscopy. Acta Imeko 2017, 6, 12– 17,  DOI: 10.21014/acta_imeko.v6i3.447
      There is no corresponding record for this reference.
    32. 32
      Santanni, F.; Albino, A.; Atzori, M.; Ranieri, D.; Salvadori, E.; Chiesa, M.; Lunghi, A.; Bencini, A.; Sorace, L.; Totti, F. Probing Vibrational Symmetry Effects and Nuclear Spin Economy Principles in Molecular Spin Qubits. Inorg. Chem. 2021, 60, 140– 151,  DOI: 10.1021/acs.inorgchem.0c02573
      32
      Probing Vibrational Symmetry Effects and Nuclear Spin Economy Principles in Molecular Spin Qubits
      Santanni, Fabio; Albino, Andrea; Atzori, Matteo; Ranieri, Davide; Salvadori, Enrico; Chiesa, Mario; Lunghi, Alessandro; Bencini, Andrea; Sorace, Lorenzo; Totti, Federico; Sessoli, Roberta
      Inorganic Chemistry (2021), 60 (1), 140-151CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
      The selection of mol. spins qubits with long coherence time, Tm, is a central task for implementing mol.-based quantum technologies. Even if sufficiently long Tm can be achieved through an efficient synthetic strategy and ad hoc exptl. measurement procedures, many factors contributing to the loss of coherence still need to be thoroughly investigated and understood. Vibrational properties and nuclear spins of hydrogens are two of them. The formers play a paramount role, but detailed theor. investigation aimed at studying their effects on the spin dynamics of mol. complexes such as the benchmark phthalocyanine (Pc) is still missing, whereas the effect of the latter deserves to be examd. in detail for such a class of compds. In this work, authors adopted a combined theor. and exptl. approach to investigate the relaxation properties of classical [Cu(Pc)] and a CuII complex based on the ligand tetrakis-thiadiazoleporphyrazine (H2TTDPz), characterized by a hydrogen-free mol. structure. Systematic calcns. of mol. vibrations exemplify the effect of normal modes on the spin-lattice relaxation process, unveiling a different contribution to T1 depending on the symmetry of normal modes. Moreover, they obsd. that an appreciable Tm enhancement could be achieved by removing hydrogens from the ligand. The spin dynamics of [Cu(Pc)] and of the hydrogen-free analog [Cu(TTDPz)] (H2TTDPz = tetrakis-thiadiazoleporphyrazine) were compared through a combined theor. and exptl. approach. The theor. anal. of mol. vibrations highlights the crucial effect of normal modes symmetry on the spin-lattice relaxation process. Furthermore, an appreciable enhancement of the coherence time is obsd. when hydrogen nuclei are removed from the ligand.
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    33. 33
      Tesi, L.; Lunghi, A.; Atzori, M.; Lucaccini, E.; Sorace, L.; Totti, F.; Sessoli, R. Giant spin-phonon bottleneck effects in evaporable vanadyl-based molecules with long spin coherence. Dalton Trans. 2016, 45, 16635– 16643,  DOI: 10.1039/C6DT02559E
      33
      Giant spin-phonon bottleneck effects in evaporable vanadyl-based molecules with long spin coherence
      Tesi, L.; Lunghi, A.; Atzori, M.; Lucaccini, E.; Sorace, L.; Totti, F.; Sessoli, R.
      Dalton Transactions (2016), 45 (42), 16635-16643CODEN: DTARAF; ISSN:1477-9226. (Royal Society of Chemistry)
      Vanadium(IV) complexes have recently shown record quantum spin coherence times that in several circumstances are limited by spin-lattice relaxation. The role of the environment and vibronic properties in the low temp. dynamics is here investigated by a comparative study of the magnetization dynamics as a function of crystallite size and the steric hindrance of the β-diketonate ligands in VO(acac)2 (1), VO(dpm)2 (2) and VO(dbm)2 (3) evaporable complexes (acac- = acetylacetonate, dpm- = dipivaloylmethanate, and dbm- = dibenzoylmethanate). A pronounced crystallite size dependence of the relaxation time is obsd. at unusually high temps. (up to 40 K), which is assocd. with a giant spin-phonon bottleneck effect. The authors modeled this behavior by an ad hoc force field approach derived from d. functional theory calcns., which evidences a correlation of the intensity of the phenomenon with ligand dimensions and the unit cell size.
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    34. 34
      Shuter, E.; Rettig, S. J.; Orvig, C. Oxobis(2,4-pentanedionato)vanadium(IV), a Redetermination. Acta Crystallogr., Sect. C: Cryst. Struct. Commun. 1995, 51, 12– 14,  DOI: 10.1107/S0108270194010462
      There is no corresponding record for this reference.
    35. 35
      King, M. D.; Korter, T. M. Application of London-type dispersion corrections in solid-state density functional theory for predicting the temperature-dependence of crystal structures and terahertz spectra. Cryst. Growth Des. 2011, 11, 2006– 2010,  DOI: 10.1021/cg200211x
      35
      Application of London-Type Dispersion Corrections in Solid-State Density Functional Theory for Predicting the Temperature-Dependence of Crystal Structures and Terahertz Spectra
      King, Matthew D.; Korter, Timothy M.
      Crystal Growth & Design (2011), 11 (5), 2006-2010CODEN: CGDEFU; ISSN:1528-7483. (American Chemical Society)
      Solid-state d. functional theory (DFT) has been shown to be a valuable tool for the simulation of low-frequency vibrational motions in mol. crystals. While it is typically required that an exptl. known crystal structure be used as the initial input for these types of calcns., it is sometimes found that suitable crystallog. data are not easily obtainable. In this study, the low-temp. unit cell structure of the β polymorph of deuterated oxalic acid dihydrate, for which a structure has only been reported at room temp., was predicted using solid-state DFT augmented with a modified empirical correction for weak long-range dispersive interactions. The dispersion correction parameters were optimized against the known 100 K crystal structure of the α polymorph, and then used for full-geometry optimization of the β crystal structure. Using this predicted structure for the β polymorph, the obsd. cryogenic THz spectrum of a mixt. of deuterated oxalic acid dihydrate polymorphs was simulated.
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    36. 36
      Parlinski, K. Phonons calculated from first-principles. École thématique de la Société Française de la Neutronique 2011, 12, 161– 166,  DOI: 10.1051/sfn/201112008
      There is no corresponding record for this reference.
    37. 37
      Baroni, S.; de Gironcoli, S.; Dal Corso, A. Phonons and related crystal properties from density-functional perturbation theory. Rev. Mod. Phys. 2001, 73, 515– 567,  DOI: 10.1103/RevModPhys.73.515
      37
      Phonons and related crystal properties from density-functional perturbation theory
      Baroni, Stefano; De Gironcoli, Stefano; Dal Corso, Andrea; Giannozzi, Paolo
      Reviews of Modern Physics (2001), 73 (2), 515-562CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)
      This article reviews with many refs. the current status of lattice-dynamical calcns. in crystals, using d.-functional perturbation theory, with emphasis on the plane-wave pseudopotential method. Several specialized topics are treated, including the implementation for metals, the calcn. of the response to macroscopic elec. fields and their relevance to long-wavelength vibrations in polar materials, the response to strain deformations, and higher-order responses. The success of this methodol. is demonstrated with a no. of applications existing in the literature.
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    38. 38
      VandeVondele, J.; Krack, M.; Mohamed, F.; Parrinello, M.; Chassaing, T.; Hutter, J. QUICKSTEP: Fast and accurate density functional calculations using a mixed Gaussian and plane waves approach. Comput. Phys. Commun. 2005, 167, 103– 128,  DOI: 10.1016/j.cpc.2004.12.014
      38
      QUICKSTEP: fast and accurate density functional calculations using a mixed Gaussian and plane waves approach
      VandeVondele, Joost; Krack, Matthias; Mohamed, Fawzi; Parrinello, Michele; Chassaing, Thomas; Hutter, Juerg
      Computer Physics Communications (2005), 167 (2), 103-128CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)
      We present the Gaussian and plane waves (GPW) method and its implementation in which is part of the freely available program package CP2K. The GPW method allows for accurate d. functional calcns. in gas and condensed phases and can be effectively used for mol. dynamics simulations. We show how derivs. of the GPW energy functional, namely ionic forces and the Kohn-Sham matrix, can be computed in a consistent way. The computational cost of computing the total energy and the Kohn-Sham matrix is scaling linearly with the system size, even for condensed phase systems of just a few tens of atoms. The efficiency of the method allows for the use of large Gaussian basis sets for systems up to 3000 atoms, and we illustrate the accuracy of the method for various basis sets in gas and condensed phases. Agreement with basis set free calcns. for single mols. and plane wave based calcns. in the condensed phase is excellent. Wave function optimization with the orbital transformation technique leads to good parallel performance, and outperforms traditional diagonalisation methods. Energy conserving Born-Oppenheimer dynamics can be performed, and a highly efficient scheme is obtained using an extrapolation of the d. matrix. We illustrate these findings with calcns. using commodity PCs as well as supercomputers.
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    39. 39
      Kühne, T. D.; Iannuzzi, M.; Del Ben, M.; Rybkin, V. V.; Seewald, P.; Stein, F.; Laino, T.; Khaliullin, R. Z.; Schütt, O.; Schiffmann, F. CP2K: An electronic structure and molecular dynamics software package -Quickstep: Efficient and accurate electronic structure calculations. J. Chem. Phys. 2020, 152, 194103  DOI: 10.1063/5.0007045
      39
      CP2K: An electronic structure and molecular dynamics software package - Quickstep: Efficient and accurate electronic structure calculations
      Kuehne, Thomas D.; Iannuzzi, Marcella; Del Ben, Mauro; Rybkin, Vladimir V.; Seewald, Patrick; Stein, Frederick; Laino, Teodoro; Khaliullin, Rustam Z.; Schuett, Ole; Schiffmann, Florian; Golze, Dorothea; Wilhelm, Jan; Chulkov, Sergey; Bani-Hashemian, Mohammad Hossein; Weber, Valery; Borstnik, Urban; Taillefumier, Mathieu; Jakobovits, Alice Shoshana; Lazzaro, Alfio; Pabst, Hans; Mueller, Tiziano; Schade, Robert; Guidon, Manuel; Andermatt, Samuel; Holmberg, Nico; Schenter, Gregory K.; Hehn, Anna; Bussy, Augustin; Belleflamme, Fabian; Tabacchi, Gloria; Gloess, Andreas; Lass, Michael; Bethune, Iain; Mundy, Christopher J.; Plessl, Christian; Watkins, Matt; VandeVondele, Joost; Krack, Matthias; Hutter, Juerg
      Journal of Chemical Physics (2020), 152 (19), 194103CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      A review. CP2K is an open source electronic structure and mol. dynamics software package to perform atomistic simulations of solid-state, liq., mol., and biol. systems. It is esp. aimed at massively parallel and linear-scaling electronic structure methods and state-of-the-art ab initio mol. dynamics simulations. Excellent performance for electronic structure calcns. is achieved using novel algorithms implemented for modern high-performance computing systems. This review revisits the main capabilities of CP2K to perform efficient and accurate electronic structure simulations. The emphasis is put on d. functional theory and multiple post-Hartree-Fock methods using the Gaussian and plane wave approach and its augmented all-electron extension. (c) 2020 American Institute of Physics.
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    40. 40
      Goedecker, S.; Teter, M.; Hutter, J. Separable dual-space Gaussian pseudopotentials. Phys. Rev. B 1996, 54, 1703– 1710,  DOI: 10.1103/PhysRevB.54.1703
      40
      Separable dual-space Gaussian pseudopotentials
      Goedecker, S.; Teter, M.; Hutter, J.
      Physical Review B: Condensed Matter (1996), 54 (3), 1703-1710CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
      We present pseudopotential coeffs. for the first two rows of the Periodic Table. The pseudopotential is of an analytic form that gives optimal efficiency in numerical calculations using plane waves as a basis set. At most, even coeffs. are necessary to specify its analytic form. It is separable and has optimal decay properties in both real and Fourier space. Because of this property, the application of the nonlocal part of the pseudopotential to a wave function can be done efficiently on a grid in real space. Real space integration is much faster for large systems than ordinary multiplication in Fourier space, since it shows only quadratic scaling with respect to the size of the system. We systematically verify the high accuracy of these pseudopotentials by extensive at. and mol. test calcns.
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    41. 41
      Hartwigsen, C.; Goedecker, S.; Hutter, J. Relativistic separable dual-space Gaussian pseudopotentials from H to Rn. Phys. Rev. B 1998, 58, 3641– 3662,  DOI: 10.1103/PhysRevB.58.3641
      41
      Relativistic separable dual-space Gaussian pseudopotentials from H to Rn
      Hartwigsen, C.; Goedecker, S.; Hutter, J.
      Physical Review B: Condensed Matter and Materials Physics (1998), 58 (7), 3641-3662CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
      We generalize the concept of separable dual-space Gaussian pseudopotentials to the relativistic case. This allows us to construct this type of pseudopotential for the whole Periodic Table, and we present a complete table of pseudopotential parameters for all the elements from H to Rn. The relativistic version of this pseudopotential retains all the advantages of its nonrelativistic version. It is separable by construction, it is optimal for integration on a real-space grid, it is highly accurate, and, due to its analytic form, it can be specified by a very small no. of parameters. The accuracy of the pseudopotential is illustrated by an extensive series of mol. calcns.
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    42. 42
      Krack, M. Pseudopotentials for H to Kr optimized for gradient-corrected exchange-correlation functionals. Theor. Chem. Acc. 2005, 114, 145– 152,  DOI: 10.1007/s00214-005-0655-y
      42
      Pseudopotentials for H to Kr optimized for gradient-corrected exchange-correlation functionals
      Krack, M.
      Theoretical Chemistry Accounts (2005), 114 (1-3), 145-152CODEN: TCACFW; ISSN:1432-881X. (Springer GmbH)
      Pseudopotential parameter sets for the elements from H to Kr using the relativistic, norm-conserving, separable, dual-space Gaussian-type pseudopotentials of Goedecker, Teter, and Hutter (GTH) are presented as optimized for the gradient-cor. exchange-correlation functionals of Becke, Lee, Yang, and Parr (BLYP), Becke and Perdew (BP), and Perdew, Burke, and Ernzerhof (PBE). The accuracy and reliability of the GTH pseudopotentials is shown by calcns. for a series of small mols.
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    43. 43
      Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77, 3865,  DOI: 10.1103/PhysRevLett.77.3865
      43
      Generalized gradient approximation made simple
      Perdew, John P.; Burke, Kieron; Ernzerhof, Matthias
      Physical Review Letters (1996), 77 (18), 3865-3868CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      Generalized gradient approxns. (GGA's) for the exchange-correlation energy improve upon the local spin d. (LSD) description of atoms, mols., and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental consts. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential.
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    44. 44
      Grimme, S. Accurate description of van der Waals complexes by density functional theory including empirical corrections. J. Comput. Chem. 2004, 25, 1463– 1473,  DOI: 10.1002/jcc.20078
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      Accurate description of van der Waals complexes by density functional theory including empirical corrections
      Grimme, Stefan
      Journal of Computational Chemistry (2004), 25 (12), 1463-1473CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
      An empirical method to account for van der Waals interactions in practical calcns. in the framework of the d. functional theory (termed DFT-D) was tested for a wide variety of mol. complexes. As in previous schemes, the dispersive energy was described by damped interat. potentials of the form C6R-6. The use of pure, gradient-cor. d. functionals (BLYP and PBE), together with the resoln.-of-the-identity (RI) approxn. for the Coulomb operator, allows very efficient computations for large systems. In contrast to the previous work, extended AO basis sets of polarized TZV or QZV quality were employed, which reduced the basis set superposition error to a negligible extend. By using a global scaling factor for the at. C6 coeffs., the functional dependence of the results could be strongly reduced. The "double counting" of correlation effects for strongly bound complexes was found to be insignificant if steep damping functions were employed. The method was applied to a total of 29 complexes of atoms and small mols. (Ne, CH4, NH3, H2O, CH3F, N2, F2, formic acid, ethene, and ethine) with each other and with benzene, to benzene, naphthalene, pyrene, and coronene dimers, the naphthalene trimer, coronene·H2O and four H-bonded and stacked DNA base pairs (AT and GC). In almost all cases, very good agreement with reliable theor. or exptl. results for binding energies and intermol. distances is obtained. For stacked arom. systems and the important base pairs, the DFT-D-BLYP model seems to be even superior to std. MP2 treatments that systematically over-bind. The good results obtained suggest the approach as a practical tool to describe the properties of many important van der Waals systems in chem. Furthermore, the DFT-D data may either be used to calibrate much simpler (e.g., force-field) potentials or the optimized structures can be used as input for more accurate ab initio calcns. of the interaction energies.
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    45. 45
      Grimme, S. Semiempirical GGA-type density functional constructed with a long-range dispersion correction. J. Comput. Chem. 2006, 27, 1787– 1799,  DOI: 10.1002/jcc.20495
      45
      Semiempirical GGA-type density functional constructed with a long-range dispersion correction
      Grimme, Stefan
      Journal of Computational Chemistry (2006), 27 (15), 1787-1799CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
      A new d. functional (DF) of the generalized gradient approxn. (GGA) type for general chem. applications termed B97-D is proposed. It is based on Becke's power-series ansatz from 1997 and is explicitly parameterized by including damped atom-pairwise dispersion corrections of the form C6·R-6. A general computational scheme for the parameters used in this correction has been established and parameters for elements up to xenon and a scaling factor for the dispersion part for several common d. functionals (BLYP, PBE, TPSS, B3LYP) are reported. The new functional is tested in comparison with other GGAs and the B3LYP hybrid functional on std. thermochem. benchmark sets, for 40 noncovalently bound complexes, including large stacked arom. mols. and group II element clusters, and for the computation of mol. geometries. Further cross-validation tests were performed for organometallic reactions and other difficult problems for std. functionals. In summary, it is found that B97-D belongs to one of the most accurate general purpose GGAs, reaching, for example for the G97/2 set of heat of formations, a mean abs. deviation of only 3.8 kcal mol-1. The performance for noncovalently bound systems including many pure van der Waals complexes is exceptionally good, reaching on the av. CCSD(T) accuracy. The basic strategy in the development to restrict the d. functional description to shorter electron correlation lengths scales and to describe situations with medium to large interat. distances by damped C6·R-6 terms seems to be very successful, as demonstrated for some notoriously difficult reactions. As an example, for the isomerization of larger branched to linear alkanes, B97-D is the only DF available that yields the right sign for the energy difference. From a practical point of view, the new functional seems to be quite robust and it is thus suggested as an efficient and accurate quantum chem. method for large systems where dispersion forces are of general importance.
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    46. 46
      Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J. Chem. Phys. 2010, 132, 154104  DOI: 10.1063/1.3382344
      46
      A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu
      Grimme, Stefan; Antony, Jens; Ehrlich, Stephan; Krieg, Helge
      Journal of Chemical Physics (2010), 132 (15), 154104/1-154104/19CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The method of dispersion correction as an add-on to std. Kohn-Sham d. functional theory (DFT-D) has been refined regarding higher accuracy, broader range of applicability, and less empiricism. The main new ingredients are atom-pairwise specific dispersion coeffs. and cutoff radii that are both computed from first principles. The coeffs. for new eighth-order dispersion terms are computed using established recursion relations. System (geometry) dependent information is used for the first time in a DFT-D type approach by employing the new concept of fractional coordination nos. (CN). They are used to interpolate between dispersion coeffs. of atoms in different chem. environments. The method only requires adjustment of two global parameters for each d. functional, is asymptotically exact for a gas of weakly interacting neutral atoms, and easily allows the computation of at. forces. Three-body nonadditivity terms are considered. The method has been assessed on std. benchmark sets for inter- and intramol. noncovalent interactions with a particular emphasis on a consistent description of light and heavy element systems. The mean abs. deviations for the S22 benchmark set of noncovalent interactions for 11 std. d. functionals decrease by 15%-40% compared to the previous (already accurate) DFT-D version. Spectacular improvements are found for a tripeptide-folding model and all tested metallic systems. The rectification of the long-range behavior and the use of more accurate C6 coeffs. also lead to a much better description of large (infinite) systems as shown for graphene sheets and the adsorption of benzene on an Ag(111) surface. For graphene it is found that the inclusion of three-body terms substantially (by about 10%) weakens the interlayer binding. We propose the revised DFT-D method as a general tool for the computation of the dispersion energy in mols. and solids of any kind with DFT and related (low-cost) electronic structure methods for large systems. (c) 2010 American Institute of Physics.
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    47. 47
      Shanno, D. F. Conjugate Gradient Methods with Inexact Searches. Math. Oper. Res. 1978, 3, 244– 256,  DOI: 10.1287/moor.3.3.244
      There is no corresponding record for this reference.
    48. 48
      Pfrommer, B. G.; Cote, M.; Louie, S. G.; Cohen, M. L. Relaxation of Crystals with the Quasi-Newton Method. J. Comput. Phys. 1997, 131, 233– 240,  DOI: 10.1006/jcph.1996.5612
      48
      Relaxation of crystals with the quasi-Newton method
      Pfrommer, Bernd G.; Cote, Michel; Louie, Steven G.; Cohen, Marvin L.
      Journal of Computational Physics (1997), 131 (1), 233-240CODEN: JCTPAH; ISSN:0021-9991. (Academic)
      A quasi-Newton method was used to simultaneously relax the internal coordinates and lattice parameters of crystals under pressure. The symmetry of the crystal structure is preserved during the relaxation. From the inverse of the Hessian matrix, elastic properties, and some optical phonon frequencies at the Brillouin zone center can be estd. The efficiency of the method is demonstrated for Si test systems.
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    49. 49
      Neese, F. Software update: the ORCA program system, version 4.0. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2018, 8, 1– 6,  DOI: 10.1002/wcms.1327
      There is no corresponding record for this reference.
    50. 50
      Adamo, C.; Barone, V. Toward reliable density functional methods without adjustable parameters: The PBE0 model. J. Chem. Phys. 1999, 110, 6158– 6170,  DOI: 10.1063/1.478522
      50
      Toward reliable density functional methods without adjustable parameters: the PBE0 model
      Adamo, Carlo; Barone, Vincenzo
      Journal of Chemical Physics (1999), 110 (13), 6158-6170CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We present an anal. of the performances of a parameter free d. functional model (PBE0) obtained combining the so called PBE generalized gradient functional with a predefined amt. of exact exchange. The results obtained for structural, thermodn., kinetic and spectroscopic (magnetic, IR and electronic) properties are satisfactory and not far from those delivered by the most reliable functionals including heavy parameterization. The way in which the functional is derived and the lack of empirical parameters fitted to specific properties make the PBE0 model a widely applicable method for both quantum chem. and condensed matter physics.
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    51. 51
      Lunghi, A. Ligand-Field Contributions to Spin-phonon Coupling in a Family of Vanadium Molecular Qubits from Multi-Reference Electronic Structure Theory. arXiv preprint arXiv:1912.04545, 2019.
      There is no corresponding record for this reference.
    52. 52
      Escalera-Moreno, L.; Suaud, N.; Gaita-Ariño, A.; Coronado, E. Determining Key Local Vibrations in the Relaxation of Molecular Spin Qubits and Single-Molecule Magnets. J. Phys. Chem. Lett. 2017, 8, 1695– 1700,  DOI: 10.1021/acs.jpclett.7b00479
      52
      Determining Key Local Vibrations in the Relaxation of Molecular Spin Qubits and Single-Molecule Magnets
      Escalera-Moreno, L.; Suaud, N.; Gaita-Arino, A.; Coronado, E.
      Journal of Physical Chemistry Letters (2017), 8 (7), 1695-1700CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
      To design mol. spin qubits and nanomagnets operating at high temps., there is an urgent need to understand the relationship between vibrations and spin relaxation processes. Herein we develop a simple first-principles methodol. to det. the modulation that vibrations exert on spin energy levels. This methodol. is applied to [Cu(mnt)2]2- (mnt2- = 1,2-dicyanoethylene-1,2-dithiolate), a highly coherent complex. By theor. identifying the most relevant vibrational modes, we are able to offer general strategies to chem. design more resilient magnetic mols., where the energy of the spin states is not coupled to vibrations.
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    53. 53
      Redfield, A. G. Advances in Magnetic and Optical Resonance; Academic Press, 1965; Vol. 1, pp 1– 32.
      There is no corresponding record for this reference.
    54. 54
      Breuer, H. P.; Petruccione, F. The Theory of Open Quantum Systems; Oxford University Press: New York, 2007.
      There is no corresponding record for this reference.
    55. 55
      Neto, N.; Bellucci, L. A new algorithm for rigid body molecular dynamics. Chem. Phys. 2006, 328, 259– 268,  DOI: 10.1016/j.chemphys.2006.07.009
      55
      A new algorithm for rigid body molecular dynamics
      Neto, Natale; Bellucci, Luca
      Chemical Physics (2006), 328 (1-3), 259-268CODEN: CMPHC2; ISSN:0301-0104. (Elsevier B.V.)
      The mol. dynamics of a completely rigid mol. is described in terms of external coordinates, namely translations and rotations, and a new algorithm is proposed, which is faster than other known methods and satisfies the constraints up to a desired accuracy. The procedure dispenses with the adoption of Lagrange multipliers and it is derived from an expression previously proposed for the motion of a semirigid mol., when constraints are imposed to any selected no. of intramol. parameters. The latter need not to be specified for a rigid body but cannot be altogether ignored since it is necessary to guarantee that internal and external coordinates form a complete set of independent variables. This requirement is met by the familiar Eckart-Sayvetz conditions which provide with an iterative procedure for the evaluation, through sym. orthogonalization, of a matrix of rotation. It turns out that only a first approxn. of this matrix is necessary, therefore a final algorithm is proposed, based on the definition of infinitesimal angles of rotation about the mass center.
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    56. 56
      Califano, S.; Schettino, V. Vibrational relaxation in molecular crystals. Int. Rev. Phys. Chem. 1988, 7, 19– 57,  DOI: 10.1080/01442358809353204
      56
      Vibrational relaxation in molecular crystals
      Califano, Salvatore; Schettino, Vincenzo
      International Reviews in Physical Chemistry (1988), 7 (1), 19-57CODEN: IRPCDL; ISSN:0144-235X.
      A review with 97 refs. discussing mechanisms of vibrational relaxation in mol. crystals. The theory of anharmonic processes in mol. crystals and the exptl. and theor. results reported for individual crystals are summarized.
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    57. 57
      Bini, R.; Foggi, P.; Salvi, P. R.; Schettino, V. FTIR study of vibrational relaxation in potassium perchlorate crystal. J. Phys. Chem. A 1990, 94, 6653– 6658,  DOI: 10.1021/j100380a025
      57
      FTIR study of vibrational relaxation in potassium perchlorate crystal
      Bini, R.; Foggi, P.; Salvi, P. R.; Schettino, V.
      Journal of Physical Chemistry (1990), 94 (17), 6653-8CODEN: JPCHAX; ISSN:0022-3654.
      The IR spectrum of KClO4 single crystal was studied in the region of weak IR ν1, ν2, and ν1 + 2ν2 vibrational excitons at low temp. The ν2 dispersion is ≃10 cm-1. The IR and Raman spectroscopy in the ν1 region, in close coincidence with the first overtone of ν2, is discussed in terms of Fermi resonance in crystals. The dependence on temp. of the bandwidths of the ν1 and ν1 + 2ν2 modes was measured in the 20-160 K temp. range. These bandwidth data are interpreted on the basis of current theories on relaxation mechanisms. In particular, the main decay processes in the ν1 and ν1 + 2N2 case involve the cubic anharmonicity and are detd. only by depopulation. The calcd. decays compare well with exptl. results. On the whole, the decay data are also in agreement with recently reported picosecond coherent anti-Stokes Raman spectroscopy (CARS) bandwidth measurements.
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    58. 58
      Briganti, M.; Santanni, F.; Tesi, L.; Totti, F.; Sessoli, R.; Lunghi, A. A Complete Ab Initio View of Orbach and Raman Spin-Lattice Relaxation in a Dysprosium Coordination Compound. J. Am. Chem. Soc. 2021, 143, 13633– 13645,  DOI: 10.1021/jacs.1c05068
      58
      A Complete Ab Initio View of Orbach and Raman Spin-Lattice Relaxation in a Dysprosium Coordination Compound
      Briganti, Matteo; Santanni, Fabio; Tesi, Lorenzo; Totti, Federico; Sessoli, Roberta; Lunghi, Alessandro
      Journal of the American Chemical Society (2021), 143 (34), 13633-13645CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
      The unique electronic and magnetic properties of lanthanide mol. complexes place them at the forefront of the race toward high-temp. single-mol. magnets and magnetic quantum bits. The design of compds. of this class has so far being almost exclusively driven by static crystal field considerations, with an emphasis on increasing the magnetic anisotropy barrier. Now that this guideline has reached its max. potential, a deeper understanding of spin-phonon relaxation mechanisms presents itself as key in order to drive synthetic chem. beyond simple intuition. In this work, we compute relaxation times fully ab initio and unveil the nature of all spin-phonon relaxation mechanisms, namely Orbach and Raman pathways, in a prototypical Dy single-mol. magnet. Computational predictions are in agreement with the exptl. detn. of spin relaxation time and crystal field anisotropy, and show that Raman relaxation, dominating at low temp., is triggered by low-energy phonons and little affected by further engineering of crystal field axiality. A comprehensive anal. of spin-phonon coupling mechanism reveals that mol. vibrations beyond the ion's first coordination shell can also assume a prominent role in spin relaxation through an electrostatic polarization effect. Therefore, this work shows the way forward in the field by delivering a novel and complete set of chem. sound design rules tackling every aspect of spin relaxation at any temp.
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    59. 59
      Mirzoyan, R.; Kazmierczak, N. P.; Hadt, R. G. Deconvolving Contributions to Decoherence in Molecular Electron Spin Qubits: A Dynamic Ligand Field Approach. Chem. Eur. J. 2021, 27, 9482– 9494,  DOI: 10.1002/chem.202100845
      59
      Deconvolving Contributions to Decoherence in Molecular Electron Spin Qubits: A Dynamic Ligand Field Approach
      Mirzoyan, Ruben; Kazmierczak, Nathanael P.; Hadt, Ryan G.
      Chemistry - A European Journal (2021), 27 (37), 9482-9494CODEN: CEUJED; ISSN:0947-6539. (Wiley-VCH Verlag GmbH & Co. KGaA)
      A review. In the past decade, transition metal complexes have gained momentum as electron spin-based quantum bit (qubit) candidates due to their synthetic tunability and long achievable coherence times. The decoherence of magnetic quantum states imposes a limit on the use of these qubits for quantum information technologies, such as quantum computing, sensing, and communication. With rapid recent development in the field of mol. quantum information science, a variety of chem. design principles for prolonging coherence in mol. transition metal qubits have been proposed. Here the spin-spin, motional, and spin-phonon regimes of decoherence are delineated, outlining design principles for each. It is shown how dynamic ligand field models can provide insights into the intramol. vibrational contributions in the spin-phonon decoherence regime. This minireview aims to inform the development of mol. quantum technologies tailored for different environments and conditions.
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    60. 60
      Shrivastava, K. N. Theory of Spin–Lattice Relaxation. Phys. Status Solidi B 1983, 117, 437– 458,  DOI: 10.1002/pssb.2221170202
      60
      Theory of spin-lattice relaxation
      Shrivastava, K. N.
      Physica Status Solidi B: Basic Research (1983), 117 (2), 437-58CODEN: PSSBBD; ISSN:0370-1972.
      A review with 46 refs. Topics include: the spin-lattice interaction, the direct process, the Raman process, the sum process, the Orbach process, the 3-phonon process, the local mode process, and the collision process.
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    61. 61
      Iannone, F.; Ambrosino, F.; Bracco, G.; De Rosa, M.; Funel, A.; Guarnieri, G.; Migliori, S.; Palombi, F.; Ponti, G.; Santomauro, G.; CRESCO ENEA HPC Clusters: A Working Example of a Multifabric GPFS Spectrum Scale Layout. In Proceedings of the 2019 International Conference on High Performance Computing Simulation (HPCS) , 2019; pp 1051– 1052.
      There is no corresponding record for this reference.

  • Supporting Information

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    • X-ray crystallographic data; convergence test; cartesian derivatives comparison; full spectral data; normal-mode composition; atomic charges (PDF)


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