1. Introduction

Palladium(II) complex K_{2}PdCl_{4} has attracted considerable attention due to the medicinal properties as well as applications in drugs with high antitumoral and antibacterial activity^{1}.Meanwhile, K_{2}PdCl_{4} can act as a catalyst to afford an efficient, mild and environmentally friendly method in the synthesis of ^{2}. This material is also used in an increasing number of applications in chemical, electrical, electronic and glass industries and the manufacture of jewellery^{3}. On the other hand,Cr^{3+} is a widely used dopant in various functional systems such as night-vision materials^{4}, fiberoptic thermometers^{5}, laser hosts^{6}^{,}^{7}, photocatalysis^{8} as well as vivo imaging and drug delivery mediums^{9}. Cr^{3+} is also essential for mammal metabolism with declining concentrations of both glycose and cholesterol in blood^{10}, and its non-specific binding with DNA and other cellular components can possibly result in inhibition of transcription and replication of DNA^{11}. In addition, Cr^{3+} as an environmental contaminant and various industrial and agricultural activities (especially in food and water) has been a focus in the relevant fields^{12}^{,}^{13}.

Usually, the above properties and applications may strongly depend upon the electronic structures and local behaviors of Cr^{3+} in these systems, and the investigations on the local structures and behaviors of Cr^{3+} in K_{2}PdCl_{4} are of scientific and practical significance. It is known that EPR spectroscopy is sensitive to the immediate environment around the paramagnetic impurities and able to reveal important microscopic information (*e.g.*, defect structures such as occupation, strength of crystal-fields and local lattice distortions etc.) in the hosts. Recently, electron paramagnetic resonance (EPR) study was carried out for Cr^{3+} doped K_{2}PdCl_{4} at liquid nitrogen temperature (LNT), and the spin Hamiltonian parameters (SHPs) were measured for the orthorhombic substitutional Cr^{3+} centre on K^{+} site (site I)^{14}. So far, the above EPR experimental results for Cr^{3+} doped K_{2}PdCl_{4} have not been theoretically explained, except that the d-d transition optical spectra were quantitatively analyzed based on the crystal-field theory [14]. Since this orthorhombic Cr^{3+} centre on the host K^{+} site in K_{2}PdCl_{4} is quite different from the conventional tetragonal Cu^{2+} centre on square planar Pd^{2+} site, further theoretical investigations on the SHPs of this system can be helpful to understand the impurity behaviors and the properties of K_{2}PdCl_{4} with transition-metal dopants and are of specific importance. The aim of this article is to propose satisfactory and uniform interpretations of the SHPs for the Cr^{3+} centre in K_{2}PdCl_{4} by involving the local structure of the impurity based on the perturbation calculations of these parameters. The arrangement of this paper is the follows. In Sec. 2, the relevant theoretical formulas for the calculations of the SHPs are provided. The results are discussed in Sec. 3. The conclusion of the article is drawn in the last section.

2. Theoretical calculations

In this section, the theoretical calculations of the SHPs are performed for the Cr^{3+} centre in K_{2}PdCl_{4}. In the treatments, the contributions to the SHPs from both the crystal-field (CF) and charge-transfer (CT) mechanisms are included in view of significant covalency of the system. EPR measurements^{14} indicate that Cr^{3+} in K_{2}PdCl_{4} occupies substitutionally the host K^{+} site and exhibits the local orthorhombic (D_{2h}) site symmetry. For a 3d^{3} (Cr^{3+}) ion under orthorhombically distorted octahedra, its ground orbital singlet A_{2g} may yield ZFSs D and 𝐸 via the combination of the spin-orbit coupling and the orthorhombic CF interactions^{15}.

2.1 Cluster approach treatments involving both the CF and CT contributions

In view of the high valence state of central ion Cr^{3+} and the strong covalency of ligand Cl^{-}, the studied CrCl_{6} cluster may demonstrate strong covalency and CT contributions to the SHPs^{15}^{-}^{17}. Based on the cluster approach^{18}^{,}^{19} involving both the CF and CT contributions, the spin-orbit coupling coefficients and the orbital reduction factors are expressed as follows:

Here ^{3} ion and the ligand in free states. _{h}) irreducible representations T_{2g} and E_{g}, respectively. The subscripts CF and CT represent the related quantities of the CF and CT mechanisms, respectively. A stands for the integral R ^{18}^{,}^{19}.

2.2 Perturbation formulas for an orthorhombically distorted 3d^{3} cluster

For this strongly covalent system, not only the contributions from the conventional CF mechanism (related to the anti-bonding orbitals) but also those from the CT mechanism (related to the bonding and non-bonding orbitals) affect significantly the 𝑔-shifts arising from the second-order perturbation term due to the combination of spin-orbit coupling and orbital angular momentum interactions^{18}^{,}^{19}. As for ZFSs D and E, the CF contributions originate from the third- and fourth- order perturbation terms due to the combination of orthorhombic CF and spin-orbit coupling interactions or the off-diagonal parts of the inter-electronic repulsion interactions, while the CT ones arise mainly from the third order perturbation terms due to the combination of orthorhombic CF and spin-orbit coupling interactions. Utilizing the Macfarlane’s perturbation-loop method^{20}^{,}^{21} and involving the CT contributions to 𝑔 factors and the ZFSs 𝐷 and 𝐸, one can obtain the improved perturbation formulas of the SHPs for an orthorhombically distorted octahedral 3d^{3} cluster as follows:

The quantities D_{s}, Dt, 𝐷 𝜉 and 𝐷 𝜂 are the orthorhombic CF parameters. The corresponding denominators E_{
i
} (I = 1 - 4 and 8)^{20}^{,}^{22} are the energy differences between the ground ^{4}A_{2g} and the excited ^{4}T_{2g} [t ^{2}
_{2}(^{3}T_{1})e], ^{2}T_{2g} (t^{3}
_{2}), ^{2}T_{2g} [t^{2}
_{2} (^{3}T_{1})e], ^{4}T_{1g}[t^{2}
_{2} (^{3}T_{1})e] and ^{2}T_{1g} [t^{2}
_{2} (^{3}T_{1})e] states, respectively. These denominators are expressed in terms of the cubic CF parameter Dq and Racah parameters B and C from the strong field approach^{23}^{,}^{24}:

For an octahedral 3d^{3} cluster, the CT level can be determined as ^{-1})^{15}. Here * _{X}(L)* and

*are the optical electronegativities of ligand and central ions, respectively.*

_{X}(M)3. Calculations for the orthorhombic Cr^{3+} centre in K_{2}PdCl_{4}

The host K^{+} site in K_{2}PdCl_{4} is surrounded by a tetragonally compressed chlorine octahedron, with the cation-anion distances ^{3+} on host K^{+} site can lead to the significant lattice modifications of the local impurity-ligand distances due to the remarkable discrepancies between charge and ionic radii of Cr^{3+} and K^{+}. For convenience, the local structure of the Cr^{3+} centre is described by a relative axial compression ratio 𝜌 and a relative perpendicular bond length variation ratio ^{14}) in K_{2}PdCl_{4} since the ionic radius ri (= 0.615 ^{25}) of the impurity Cr^{3+} is much smaller than the radius r_{h} (=1.38 Å^{25}) of the host K^{+}. The investigations based on experimental superhyperfine constant and extended X-ray absorption fine structure (EXAFS) measurements^{26} reveal that the empirical relationship

Utilizing the local geometry and the superposition model^{27}^{-}^{31} the orthorhombic CF parameters can be determined as:

Here ^{32} are the power-law exponents.
A_{2} (R) and A_{4} (R) are the intrinsic parameters, with the
reference distance R. The relationships A_{4} (R) = (3/4)Dq and ^{32}^{-}^{34} for 3d ions under octahedral CFs based on the
superposition model are reasonably employed here.

In the light of the optical spectral measurements for Cr^{3+} in
K_{2}PdCl_{4}^{14} and similar chlorides^{35}, the cubic field parameter ^{-1} and the effective covalency factor ^{36} and the distance ^{18}^{-}^{19} : ^{3+})^{-1}^{37}
and ^{-}) ^{-1}^{38},
the spin-orbit coupling coefficients and the orbital reduction factors are obtained
for both mechanisms from Eq. (1): ^{-1}, ^{-1}, ^{-1} and kCF≈0.835, _{0} (^{-1}^{37} and
C_{0} (^{-1}^{37} of
Cr^{3+} as: ^{39}. The CT level ^{-})^{3+}) ^{15}. Substituting these values into
Eq. (4) and fitting the calculated SHPs to the observed values, the optimal local
distortion parameters

The corresponding theoretical SHPs (Calc.^{c}) are collected in Table 1. The results (Calc.^{a}) based on the host
structural data of K^{3+} site in K2PdCl4 and inclusion of both the CF and
CT contributions and those (Calc.^{b}) based on the above optimal local
distortion parameters and inclusion of merely the CF contributions are also listed
in Table 1.

^{a}Calculations based on the host structural data of K^{+} site in K_{2}PdCl_{4} and inclusion of both the CF and CT contributions.

^{b}Calculations based on the optimal local distortion parameters

^{c}Calculations based on the optimal local distortion parameters

4. Discussion

Table 1 indicates that the theoretical SHPs (Calc. c) based on the local distortion parameters 𝜌 and 𝜏 show better agreement with the experimental data than those (Calc. a) based on the host structural data of K^{+} site and those (Calc.^{b}) based on omission of the CT contributions. Thus, the measured EPR spectra for the Cr^{3+} centre in K_{2}PdCl_{4} are satisfactorily explained in a uniform way. Meanwhile, the information about the local distortion around the impurity Cr^{3+} is theoretically determined by analyzing the experimental ZFSs. The following points are discussed here.

(1)As compared with the axial compression ratio (^{+} site in K_{2}PdCl_{4}, the impurity Cr^{3+} centre exhibits smaller axial compression (^{a}) based on the structural data of host K^{+} site can be understood. The microscopic mechanisms of the local lattice deformations are illustrated as the much stronger Cr^{3+}-Cl^{-} interaction than the host K^{+}-Cl^{-} combination due to the much higher impurity valence state. Meanwhile, the long reference impurity-ligand distance ^{3+}-Cl^{-} bonds also account for the above remarkable Cr^{3+}-Cl^{-} bond length variations with respect to the host K^{+}-Cl^{-} bonds. As a result, the axial and perpendicular distortions are considerably decreased and increased, respectively, from host K^{+} site to the impurity centre. As one might deliberately assign the measured EPR signals to Cr^{3+} on the square planar Pd^{2+} site, the calculated D would be much (two or three order in magnitude) larger than the experimental value, and the above remarkable discrepancy cannot be removed by adjusting the related spectral parameters or local distortion parameters. So, Cr^{3+} may tend to occupy the octahedral K^{+} site instead of the square planar Pd^{2+} site and is quite different from the conventional tetragonal Cu^{2+} centre on Pd^{2+} site in K^{2}PdCl_{4}^{40}. This point can be illustrated by the dissimilar electron configurations of Cu^{2+} (3d^{9}) and Cr^{3+} (3d^{3}), which tend to exhibit square planar [CuX_{4}] and octahedral [CrX_{6}] groups by means of dsp^{2} and d^{2}sp^{3} hybrid orbitals^{41}^{,}^{42}. Interestingly, similar octahedral trivalent Fe^{3+} (3d^{5}) centre on K^{+} site was also reported for iron doped K_{2}PdCl_{4}^{43}, which can also be ascribed to the octahedral [FeCl_{6}] group by means of sp^{3}d^{2} hybrid orbitals^{41}^{,}^{42}. It seems that electronic configuration plays a more important role than size or charge mismatch in occupation of transition-metal impurities in K_{2}PdCl_{4}. So, the impurity Cr^{3+} occupying K^{+} site rather than Pd^{2+} site is understandable in physics.

(2)When the CT contributions are ignored, the calculated g factors and ZFSs (Calc. ^{b}) are smaller and much larger than the experimental data. For the studied [CrCl] group, the system demonstrates strong covalency due to the covalent ligand Cl^{-} and central ion Cr^{3+} with high valence state and much larger ζp0(Cl^{-}) ^{-1}^{38} than ^{3+}) ^{-1}^{37}. Substantially, the host K_{2}PdCl_{4} may exhibit strong covalency due to the optical electronegativity ^{15}) of Pd^{2+} comparable with that ^{15}) for Cl^{-}. This can be illustrated by the effective covalency factor N (_{CT} and E_{CT} are opposite in sign and almost the same (characterized by the ratios |D_{CT}/D_{CF} and /E_{CF}_{CF} and E_{CF}. Thus, the CT contributions are found to be important and should be included in the EPR analysis.

(3)The errors of the present theoretical calculations may be analyzed as follows. First, the final results may be brought forward some errors due to the approximations of the theoretical model and formulas based on the spin Hamiltonian theory, in which only the contributions from the nearest neighbor ligands are considered whereas the influences of the ions in the outer ligand spheres are omitted. For the present impurity Cr^{3+} in tetragonal K_{2}PdCl_{4} lattice, the electric fields from the rest ions of the crystal could be anisotropic and bring forward some influences on the total crystal fields acting upon Cr^{3+}. To some extent, the opposite contributions to the crystal fields arising from the relevant ions (e.g., Pd^{2+}, Cl^{-}; K^{+}, Cl^{-}; etc.) may partially cancel each other due to opposite charges and lead to insignificant effects on the total crystal fields of Cr^{3+} because of their successively farther distances ^{26}. Fortunately, the influences of ^{+} by Cr^{3+} with two extra positive charge may induce some means of charge compensators (e.g., one Pd^{2+} vacancy or two K^{+} vacancies) and might slightly modify the local structure of the impurity centre. However, the non-axial charge compensators would lead to considerable perpendicular orthorhombic distortion and significant E, which was obviously inconsistent with the experimental tiny Evalue^{14}. On the other hand, the possible axial compensators (e.g., Pd^{2+} and K^{+} vacancies) might bring forward important effect on the charge distribution of the impurity centre. However, the negative effective charges of these possible compensators would repel the axial ligand(s) Cl^{-} towards the central Cr^{3+}, inducing much more axial compression of the impurity centre and hence larger magnitude of D. Nevertheless, this is actually in disagreement with the observed very small D^{14}. So, the possible charge compensators may occur far from the impurity and its influence on the local structure is very small and negligible. Finally, the empirical superposition model and ratio

5**. Conclusions**

The SHPs and local structure for Cr^{3+} in K_{2}PdCl_{4} are theoretically analyzed from the perturbation calculations for an orthorhombically distorted 3d^{3} cluster. The impurity Cr^{3+} is found not to occupy the host Pd^{2+} site but to locate at the octahedral K^{+} site, associated with the relative axial compression ratio ^{+} site in K_{2}PdCl_{4} and hence a more regular ligand octahedron, corresponding to the very small D and E. The contributions to g-shifts ^{3+} centre on octahedral K^{+} site, which is quite dissimilar to the conventional tetragonal Cu^{2+} centre on square planar Pd^{2+} site, is illustrated in view of the d^{2}sp^{3} hybrid orbitals.