A Self-Consistent Exact Diagonalization Approach to the Ground State Magnetic Properties of the Meridional [V(ddpd)₂]³⁺ Complex

Abstract

The present paper presents a thorough study of the ground state magnetic properties of the spin-one mononuclear nanomagnet mer-[V(ddpd)${}_2$][PF${}_6$]${}_3$, with the V${}^{3+}$ center exhibiting a distorted octahedral coordination. The theoretical analysis is based on a multiconfigurational, self-consistent approach that effectively parametrizes the total energy spectrum of the considered coordination complex via exact diagonalization. We provide a comprehensive discussion for the obtained zero-field and field-dependent fine structure of the ground state along with the ensuing crystal field splitting of the 3d orbitals. Furthermore, we report the results for the low-field susceptibility, magnetization and the corresponding reversal dynamics, finding good agreement with the experimental data reported in the literature. The calculations show considerable zero-field splitting and strong field-dependent orbital unquenching underlying the occurrence of a field-induced full profile magnetization reversal barrier.

1. Introduction

As an interdisciplinary research area covering quantum and classical physics, the field of molecular magnetism has crossed the pathways of physics and chemistry for decades, pushing the limits of the design of a variety of 3d to 4f molecule magnets beyond expectation [1,2,3,4,5,6,7,8,9,10,11].

Transition metals are among the most reliable and abundant elements for the synthesis of viable mono- and polynuclear molecule nanomagnets [12,13,14,15,16,17,18,19,20,21]. As one of the first three members in the first row of transition metals with less than a half-filled 3d subshell, the vanadium ion stands as a prospective candidate for the design of cheap, chemically stable, high-spin paramagnetic luminophore single-ion magnets [22,23] with the potential to pave the road to the application of all optically addressable molecular nanomagnets as units of future semiclassical information devices. The property of vanadium cations of having less than four active 3d electrons, however, classifies the associated coordination complexes, which are usually synthesized in a hexacoordinated [24,25,26,27,28] or pentacoordinated [29,30,31] crystal field (CF) as mononuclear nanomagnets with small zero-field splitting (ZFS) and negligible orbital unquenching at the ground state. Accordingly, this poses an inconvenience for the realization of efficient high-temperature optical control of the magnetization dynamics within the relevant band of spin multiplet transitions. The high-spin Kramers species, for example, are expected to have a negligible to zero-energy barrier to magnetization reversal in the absence of the DC magnetic field. Hence, slow relaxation of the magnetization is highly limited. Furthermore, due to the lack of spin singlet states, the probability of observing luminescent properties is negligible. An exception is the very rare case of the high-spin Kramers cation mer-[V(ddpd)${}_2$]${}^{2+}$ of mer-[V(ddpd)${}_2$][BPh${}_4$]${}_2$ (where ddpd stands for $N,N^{\prime }$-dimethyl-$N,N^{\prime }$-dipyridine-2-ylpyridine-2,6-diamine) [32], which shows slow, low-temperature relaxation of magnetization. The latter coordination complex, however, is highly paramagnetic and has a very small ZFS, just as its cis-fac- isomeric counterpart does.

The high-spin non-Kramers vanadium complexes with ideal CF symmetry and a moderate ligand environment are potential candidates for the design of luminophores with a large ZFS. Nevertheless, the synthesis of 3d${}^2$ systems with a highly symmetric CF remains beyond reach. A notable case of the vanadium spin-one complex with interesting luminescent properties is the recently reported, slightly distorted octahedral species mer-[V(ddpd)${}_2$][PF${}_6$]${}_3$ [33]. A tentative evaluation of ZFS was carried out within the effective parametric scheme of the conventional single-ion spin Hamiltonian [34,35,36,37,38] signaling possibly considerable fine splitting and emphasizing the necessity of performing more rigorous calculations. Furthermore, the fine structure related to the ground state (FSG) in the spin-one vanadium complex V(acac)${}_3$ [27] is worthy of thorough study. In this case, the initial analysis pointed to a positive axial parameter of approximately 10 cm${}^{-1}$.

In addition to the above-discussed vanadium complexes, there are a few that exhibit even more highly coordinated 3d${}^2$ and 3d${}^3$ ones with ZFS, magnetic anisotropy (MA) and magnetization reversal dynamics that are yet to be investigated by theoretical means. Some prominent cases are the heptacoordinated species [39] with low-field susceptibility behavior signaling with an intricate FSG and AC susceptibility behavior, which are unexplored so far. The recently designed pentagonal–bipyramidal complexes [40] are yet another prominent case, underlining the need for additional efforts to thoroughly study the fine structure (FS) of such systems.

The present paper presents a comprehensive study of the ground state magnetic properties of the slightly distorted octahedral complex mer-[V(ddpd)${}_2$]${}^{3+}$ [33]. In particular, it elucidates the microscopic mechanisms underpinning FSG and MA behavior. Furthermore, its focus is to provide a complete representation of the underlying electrons’ correlations and reveal the genuine contribution of SO to the FS under the exhibited distortion. To this end, we carry out multiconfigurational, self-consistent quantum computations over the active electrons from the considered system by the method of exact diagonalization [30,41]. We calculate the total energy spectra, the corresponding ZFS, and the evolution of FSG under the action of the externally applied magnetic field. Furthermore, we reproduce the low-field magnetic susceptibility measurements available in the literature, simulate the magnetization dependence on the temperature, and obtain the effective energy barrier for the reversal of magnetization.

The rest of the paper is organized as follows: In Section 2, we outline the theoretical method used to study the ground state magnetic properties of the considered compound and introduce the corresponding Hamiltonian. In Section 3, we analyze the obtained zero-field energy spectrum and the dependence of FSG on the action of the externally applied magnetic field. Furthermore, we report the results for the low-field susceptibility and magnetization. Section 4 summarizes the obtained results.

2. Theoretical Background

2.1. General Considerations

The theoretical analysis of the ground state magnetic properties of mer-[V(ddpd)${}_2$][PF${}_6$]${}_3$, particularly the cation [V(ddpd)${}_2$]${}^{3+}$, is based on exact diagonalization calculations implemented within the multiconfigurational, self-consistent field method [30,41]. The corresponding parameterization scheme is developed particularly to address the magnetic and related spectroscopic properties of mononuclear nanomagnets. It accounts for the electrons’ dynamics within the adiabatic approximation and at the nonrelativistic limit, whereas the 3d electrons are considered to be localized around the transition metal ion.

In the calculations, we take into account all electron–electron, electron–nuclei and nuclei–nuclei interactions from the first coordination sphere of the complex under investigation, where the Coulomb interactions related to the action of CF are calculated according to the principles of the superposition model [42]. The only relativistic contribution taken into account is the SO interaction originating from the reference frames of both 3d electrons. The magnetic dipole–dipole interactions are omitted from the calculations as they were found to have a negligible effect on the FS magnetic properties.

The exact diagonalization is carried out by considering all observables related to the total antisymmetric initial basis states as invariant under the exchange symmetry. For the considered system, these are the effective energy of the Coulomb interactions between both 3d electrons and their total spin. Note that in order to achieve gauge invariance for the relevant direct exchange interactions, we perform calculations with both spherical and tesseral harmonics for the single-electron orbital states. Furthermore, since the effect of all kinetic terms and interactions related to the nuclei and electrons occupying core orbitals on FS vanishes the energy eigenvalues from the final energy spectrum are normalized such that the ground state one equals zero.

In the framework of the outlined approach, the considered complex is viewed as an effective spin-one magnetic system composed of two electrons confined within the spatial domain spanned by seven point-like charges. Six of these charges represent the ligands with nitrogen as a bond agent, and the remaining one is the vanadium center (see Figure 1). The resulting effective magnetic system is characterized by two types of parameters, structural and intrinsic ones. The former represent the ligands’ coordinates and are directly related to CF symmetry. The intrinsic parameters include the vanadium charge number Z, the ligands’ charge numbers $Z_k$, and the orbital reduction factor $\kappa$, where $k=1,\dots ,6$.

2.2. The Hamiltonian

The Hamiltonian describing the dynamics of both 3d electrons within the considered approximations read

$$\begin{array}{cc}\hfill \widehat{H}=& \frac{e^2}{4\pi {\epsilon }_o|{\mathbf{r}}_2-{\mathbf{r}}_1|}+\sum _{i=1}^2\sum _{k=1}^6\frac{e^2{Z}_k}{4\pi {\epsilon }_o|{\mathbf{r}}_i-{\mathbf{d}}_k|}\hfill \\ \hfill & +\frac{g_e{\mu }_o{\mu }_{\mathrm{B}}^2}{2\pi }\sum _{i=1}^2\frac{Z}{|{\mathbf{r}}_i{|}^3}{\widehat{\mathit{l}}}_i·{\widehat{\mathit{s}}}_i-{\mu }_{\mathrm{B}}\sum _i\mathbf{B}·\left({\widehat{\mathit{l}}}_i+g_e{\widehat{\mathit{s}}}_i\right),\hfill \end{array}$$

(1)

where ${\mathbf{r}}_i$ is the position vector of the i-th electron, ${\widehat{\mathit{s}}}_i$ is the associated spin operator, ${\widehat{\mathit{l}}}_i$ the orbital angular momentum operator, ${\mathbf{d}}_k$ is the k-th ligand position vector, $\mathbf{B}$ is the external magnetic field, $g_e$ is the electrons’ g-factor, ${\mu }_{\mathrm{B}}$ is the Bohr magneton, and ${\epsilon }_o$ and ${\mu }_o$ are the electric and magnetic constants, respectively.

The first two interaction terms are expanded in series with the aid of Legendre polynomials [30]. The effective matrices of all interaction terms in (1) are calculated within the total antisymmetric two 3d electrons’ states given in the Supplementary Material Equation (S1). For more details about the corresponding matrix elements, the reader may consult the Supplementary Material (Section S4).

2.3. Computational Details

The calculations were carried out using Wolfram Mathematica 12 along with Python, yielding identical results for all computed quantities.

The python implementation was introduced to complement, at first, the Wolfram Mathematica calculations with the primary goal being to have a fully functional package to compute all characteristic quantities of arbitrary magnetic molecules. The package is under active development, and the details will be published elsewhere. It comprises a custom software package that was developed using analytical calculations with the aid of SymPy [43] at its core and takes advantage of multithreading to increase the efficiency. The calculation is compartmentalized into a number of modules for various tasks and managed by a master program that implements low-level multithreading (see Figure S1 in Supplementary Material). The following components were implemented:

• A module for defining and accessing basis functions under SymPy. Using the tesseral spherical harmonics for the representation of single-electron 3d orbitals, we constructed all antisymmetric two-electron basis states relevant to the considered system (see Supplementary Material, Equation (S1)) to be accessed when computing the corresponding energy spectrum.
• A kernel module for computing the matrix elements $\langle {\varphi }_{i,s,m}\left|\widehat{H}\right|{\varphi }_{j,s^{\prime },m^{\prime }}\rangle$, with the Hamiltonian presented in Equation (1) and state functions given in the Supplementary Material Equation (S1). Each matrix element was computed along with the appropriate parameterization (see

  Table 1. The radial distance $\varrho$, azimuthal angle $\phi$, and the polar one $\vartheta$ of the six reactive nonmetals bonded to the vanadium ion from the distorted octahedral coordination complex depicted in Figure 1. In addition to the given structural parameters, we have the orbital reduction factor and charge numbers of all ions. Their values were obtained after reproducing the low-field susceptibility measurements reported in Ref. [33].

  	N1	N2	N3	N4	N5	N6	V
  $\phi$ [deg]	270	270	$0.59$	$90.07$	$180.83$	270	–
  $\vartheta$ [deg]	$0.5$	179	$94.45$	$84.62$	$96.3$	$85.19$	–
  $\varrho$ [Å]	$2.0577$	$2.0623$	$2.0789$	$2.0742$	$2.0675$	$2.0813$	–
  Z	$1.095$	$1.095$	$1.065$	$1.065$	$1.065$	$1.065$	$7.5$
  $\kappa$							$0.85$

  ).
• An analysis module, which was used to substitute numerical values for the parameters of the model, perform direct diagonalization, and make predictions, such as the energy eigenstates and the corresponding 3d orbital configurations.
• A master module, which was used to control and schedule the execution of all previously-mentioned modules.

This approach allows the input of various CF configurations with different type of ligands and the use of both cubic and spherical single-electron orbital states for comparison. With improved efficiency, it could be useful for future investigations, such as the current one.

2.4. The Effective Zero-Field Single-Ion Spin Hamiltonian

In addition to the numerical results for ZFS obtained from the exact diagonalization, we took into account the conventional effective zero-field single-ion Hamiltonian conforming to the quantum perturbation method [34,35,36,37,38],

$$\widehat{\mathcal{H}}=\frac{D}{3}\left(3{\widehat{S}}_z^2-S(S+1)\right)+E\left({\widehat{S}}_x^2-{\widehat{S}}_y^2\right),$$

(2)

where $\widehat{\mathbf{S}}=\left({\widehat{S}}_{\alpha }\right)$, $\alpha =x,y,z$, is the corresponding spin-one operator, D is the axial ZFS parameter, and E is the rhombic one, with rhombicity obeying the inequality $\lambda \le \pm \frac{1}{3}$.

3. Magnetic Properties

3.1. Zero-Field Properties

The total zero-field energy spectrum of the studied complex is depicted in Figure 2a. Its numerical representation is given in the Supplementary Material (Section S2). The corresponding energy level sequence was obtained for the parameter values given in Table 1 with the intrinsic ones fitted in accordance with the low-field susceptibility data from Ref. [33].

As a result of the distorted geometry, $d_{xy}$ has the lowest energy among the remaining cubic orbitals (see the orbital splitting diagram shown as an inset in Figure 2b). Accordingly, the ground state is represented as a superposition of the configurations $d_{xz}^1{d}_{yz}{d}_{xy}^1{d}_{x^2-y^2}{d}_{z^2}$ and $d_{xz}{d}_{yz}^1{d}_{xy}^1{d}_{x^2-y^2}{d}_{z^2}$ with the corresponding eigenstate given in the Supplementary Material (Section S3). Although the octahedral geometry is generally distorted, we identified a symmetry axis parallel to z (see Figure 1), that keeps the ground state energy-invariant under ${180}^{\circ }$ rotation. Consequently, the obtained CF splitting between the above-given configurations is very large. It is about 20 meV. Such a large CF energy gap leads to a considerable ZFS (see Figure 2b), with the first and second energy gaps being approximately equal to $0.167$ meV and $1.348$ meV, respectively. The corresponding eigenstates are provided in the Supplementary Material (Section S3). These results are underlined by the strong localization of 3d electrons around the vanadium center, as indicated by the obtained values of Z and $\kappa$ shown in Table 1.

The effective axial and rhombic parameters of the Hamiltonian (2) quantify the ZFS depicted in Figure 2a in the standard scheme, taking the values $D\approx -10.9$ cm${}^{-1}$ and $\left|E\right|\approx 1.35$ cm${}^{-1}$. Note that we obtained a larger absolute value for the axial parameter than that reported in Ref. [33] and nearly equivalent to that calculated for the counterpart V(III) species [27]. In contrast to the results reported in Refs. [27,33], we obtained a negative D value.

Interestingly, the complex was magnetic at the ground state, where the expectation value of the total magnetic moment was ${\mathit{\mu }}_1=(0.1165,0.1490,-0.3030)$, with the respective spin and orbital components ${\mathit{\mu }}_{1,s}=(0.1087,0.1633,-0.4650)$ and ${\mathit{\mu }}_{1,l}=(0.0078,-0.0143,0.1620)$. The expectation values of the total magnetic moments associated with the excited states of FSG were ${\mathit{\mu }}_2=(-0.2249,0.1959,0.3676)$ and ${\mathit{\mu }}_3=(0.0795,-0.3657,-0.0281)$. Therefore, along with the unexpectedly large ZFS, we observed unquenching of the orbital angular momentum. The contribution of the latter rapidly increased under the action of the externally applied magnetic field.

3.2. Susceptibility and Magnetization

The theoretical results for the low-field susceptibility of [V(ddpd)${}_2$]${}^{3+}$ are depicted in Figure 3. The calculations were carried out under the assumption of a powder sample. The lack of a stable demagnetization plateau typical of paramagnetic complexes was not observed, since the first and second energy gaps (see Figure 2b) were large enough to ensure the slow and continuous population of the corresponding excited energy levels in the temperature domain $12\le T\le 300$ K. The population rates of the three energy levels from FSG are depicted in Figure 4. Thus, the ground state and first exited one were almost equally populated at about 130 K. Even at $T\to 300$ K, however, the population of the second excited level, corresponding to a zero net magnetic moment, remained less than that of the ground state and first excited one. Consequently, a non-temperature-dependent plateau in the case of $T{\chi }_{\mathrm{m}}$ was not observed, even at room temperature.

The corresponding magnetization as a function of the temperature is depicted in the inset of Figure 3. It is worth pinpointing the noticeable extent of orbital unquenching, which reduces the level of saturation for every single complex from $2{\mu }_{\mathrm{B}}$ to approximately $1.6{\mu }_{\mathrm{B}}$ at low temperatures. Although, the occurrence of an orbital magnetic moment is not expected for distorted geometries, in this case, the unquenching results from the negligible energy difference between $d_{xz}$ and $d_{yz}$ orbitals. The energy gap separating both orbitals is about $4.25$ meV, which is barely visible on the orbital splitting diagram shown in Figure 2b.

3.3. Magnetization Reversal

Although the complex exhibits considerable ZFS, regardless of its distorted octahedral coordination, the energy barrier to the reversal of its magnetization vanished at an applied external magnetic field of zero. A barrier was not observed, since in $99\%$ of all attempts, the absolute value of the system’s spin magnetic moment remained at a good quantum number, even at high temperatures (see the superposition representing the FSG eigenstates given in the Supplementary Material (Section S3)). Within the remaining $1\%$ of all reversal attempts, there was a negligible barrier height along the z axis of about ${10}^{-4}$ meV. At $0.1$ K, it lay between two minima with the expectation values of the total magnetic moment being ${\mathit{\mu }}_1=(0.1165,0.1490,-0.3030)$ and ${\mathit{\mu }}_1^{\prime }=(0.1417,0.1140,0.2350)$.

The field-induced magnetization reversal barrier is depicted in Figure 5. It was calculated under the assumption of a powder sample, and it represents the internal energy of the system for each single vanadium center as a function of the directional angle between the corresponding magnetization and the z axis in the absence of phonon/photon scattering. Note that FSG remains invariant under $n\pi$, $n\in \mathbb{Z}$, rotation around z axis. For more details on the inter-relationship between FSG, the energy barrier to the reversal of magnetization, and the anisotropy energy, the reader may consult Ref. [44] and the references therein. As we pointed out above, since the axial anisotropy is not strong enough, the probability of observing components of the complex’s total magnetic moment perpendicular to the magnetic field direction is not zero. As a result, we observed an almost-complete barrier profile that was visible at small field-sweep rates (magenta colored circles in Figure 5a).

4. Summary

We presented a thorough investigation of the ground state magnetic properties of the spin-one mononuclear nanomagnet mer-[V(ddpd)${}_2$][PF${}_6$]${}_3$ exhibiting a distorted octahedral geometry with respect to the V${}^{3+}$ center. The research focused on the exact characterization of FSG and its representation within the conventional effective single-ion parameterization scheme (see Section 2.4).

The theoretical analysis of the system’s low-field susceptibility uncovered results that were previously out of sight. The calculations pointed out slightly delocalized electrons (see the last two rows in Table 1) and hence the occurrence of a large, with respect to the considered geometry, ZFS, corresponding to a negative axial parameter and associated with significant orbital unquenching. These features lead to the low saturation level of the magnetization and an intrinsic field induced magnetization reversal barrier with a full profile. The details are provided in Section 3.

The calculations were carried out using a self-consistent microscopic method based on the exact diagonalization technique and particularly developed to address the magnetic properties of mononuclear nanomagnets. The applied method works within the adiabatic approximation in the nonrelativistic limit, conforming to the localized electrons approach and giving stationary solutions (see Section 2). It has the potential to provide a close perspective of the underlying electron dynamics, along with the originating magnetic and spectroscopic properties for a variety of coordination complexes.