Introduction
The layered trigonal antiferromagnets VX_{2} (X = Cl, Br, I) have the
crystal structure of CdI_{2} with space group PmL,
where the halide anions form hexagonal closepacked layers and the cations occupy
half the octahedral holes. Recently, VX_{2} have attracted extensive
interest due to the properties of quantum fluctuations (Bondarenko et al.,
1996; Rastelli and Tassi, 1996; Watabe et al., 1995), critical behaviours
(Kawamura, 1988) and Raman scattering (Guntherodt et al., 1979; Bauhofer
et al., 1980). In addition, Electron Paramagnetic Resonance (EPR) experiments
have been carried out on these systems and the anisotropic g factors g_{//}
and g_{⊥} were measured at 300 K (Yamada et al., 1984).
In order to interpret these EPR experimental results, inspiring investigations have been made on VCl_{2} and VBr_{2} from the perturbation formulas of the g factors for a 3d^{3} ion in trigonal symmetry based on the cluster approach (Du and Li, 1994). In these formulas, not only the Spinorbit (SO) coupling coefficient of the V^{2+} ion, but also that (as well as the p orbitals) of the ligands is taken into account (Du and Li, 1994). The theoretical results indeed show improvement compared with those based on only the SO coupling coefficient of the central ion. However, the trigonal crystalfield parameters in their calculations were taken as the adjustable parameters, without correlating with structural data of the studied systems. In addition, the experimental g factors of VI_{2} have not been explained uniformly. Since the S.O. coupling coefficient of the ligand I¯ is much larger than that of the V^{2+}, the sorbitals of the ligand may be important and lead to some contributions to the g factors. To investigate the g factors of VX_{2} to a better extent, in this study, the sorbitals of the ligand are introduced to the singleelectron wave functions of the 3d^{3} octahedral clusters. Then the previous theoretical formulas are also modified and applied to the studied systems.
Theory and Calculations
In VX_{2}, V^{2+} locates on the octahedral site with slight
trigonal (D_{3d}) distortion (Wyckoff, 1951). For a V^{2+}(3d^{3})
ion in trigonal symmetry, its ground ^{4}A_{2} state would be
split into two Kramers doublets due to the combination effect of the SO coupling
and the trigonal crystalfield interactions. Thus, the perturbation formulas
of the gshifts Δg_{i} (=g_{i } g_{s}, where g_{s}
= 2.0023 is the spin only value and I = // and ⊥) based on the cluster
approach can be expressed as (Du and Li, 1994):
Here the energy denominators E_{i} (I = 1 ~ 5 ) stand for the separations between the excited states ^{4}T_{2}, ^{2}T_{2a}, ^{2}T_{2b}, ^{4}T_{1a} and ^{4}T_{1b} and the ground state ^{4}A_{2} in terms of the cubic field parameter Dq and the Racah parameters B and C for the studied 3d^{3} clusters (Du and Li, 1994). V and V ’ are the trigonal field parameters. ζ and ζ' are the SO coupling coefficients, while k and k' are the orbital reduction factors. In the cluster approach of the previous treatments (Du and Li, 1994; Du and Rudowicz, 1992), the contributions of the sorbitals of ligands were usually neglected for simplicity. Unlikely, the above contributions are considered here. Thus, the total single electron wave function including the ligand sorbital contributions may be written as
where φ_{γ} (γ = e and t denote the irreducible representations of the O_{h} group) are the d orbitals of the central metal ion. χ_{pγ} and χ_{s} stand for the p and s orbitals of the ligands. N_{γ} and λ_{γ} (or λ_{s}) are, respectively, the normalization factors and the orbital mixing coefficients. From the semiempirical method similar to those in the previous treatments (Du and Li, 1994; Du and Rudowicz, 1992), we have the approximate relationships
and the normalization conditions
Here S_{dpγ} (and S_{ds}) are the group overlap integrals. Usually, the mixing coefficients increase with increasing the group overlap integrals and one can approximately adopt the proportional relationship between the mixing coefficients and the related group overlap integrals, i.e., λ_{e}/S_{dpe} ≈ λ_{s}/S_{ds} within the same irreducible representation e_{g}. In general, the covalency factors f_{t} and f_{e} can be determined from the ratio of the Racah parameters for the 3d^{3} ion in a crystal to those in free state, i.e., f_{t} ≈ f_{e} ≈ C/C_{0}.
According to the cluster approach containing the ligand sorbital contributions, the SO coupling coefficients ζ, ζ' and the orbital reduction factors k, k' for the 3d^{3} octahedral clusters may be expressed as
where ζ_{d}^{0} and ζ_{p}^{0} are
the SO coupling coefficients of the free 3d^{3} and the ligand ions,
respectively. A denotes the integral, where
R is the metalligand distance of the studied systems. Obviously, when taking
S_{ds} = λ_{s} = 0 and A = 0, the above formulas are reduced
to those in the absence of the ligand sorbital contributions (Du and Li, 1994;
Du and Rudowicz, 1992).
From the superposition model (Newman and Ng, 1989), the trigonal field parameters V and V’ for the studied systems are written as follows:
where β is the angle between the metalligand bond and the C_{3} axis. The magnitude and nature (elongation or compression along the C_{3} axis) of trigonal distortion can be characterized by the value and sign of the angular difference δβ (= β β_{0}, where β_{0} ≈ 54.74° is the bonding angle in cubic symmetry). }(R) and }_{4} (R) are the intrinsic parameters with the reference bonding length R. For 3d^{n} ions in octahedra, the relationships }_{4} (R) ≈(3/4)Dq and }_{2} (R)/}_{4} (R) ≈ 9 ~ 12 have been proved to be valid in many crystals (Newman and Ng, 1989; Yu et al., 1994; Edgar, 1976). Here we take }_{2}(R) ≈ 9}_{4} (R). Therefore, the trigonal distortion (or local structure) is related to the low symmetrical parameters V and V’ and hence to the g factors (particularly the anisotropy g_{//}  g_{⊥}) of the studied systems.
For VX_{2} (Wyckoff, 1951), the metalligand bonding lengths R and the angles β between R and the C_{3} axis are collected in Table 1. From the distances R and the Slatertype SCF functions (Clementi and Raimondi, 1963), the group overlap integrals S_{dpγ} (as well as the integrals S_{ds} and A) are obtained and shown in Table 1.
Table 1: 
The metalligand distances and angles, group overlap integrals,
the spectral parameters Dq, B and C (cm^{1}), N_{γ}
and λ_{γ} (and λ_{s}), SO coupling coefficients
(cm^{1}), orbital reduction factors and the trigonal field parameters
V and V’ (in cm^{1}) for VX_{2} (X = Cl, Br, I) 

Table 2: 
The gyromagnetic factors for VX_{2} at 300 K 

^{a} Calculations based on neglecting of the contributions
from the s orbitals of the ligands, i.e., similar to the treatments in
Du and Li (1994); Du and Rudowicz (1992).^{ b} Calculations based
on inclusion of the contributions from the sorbitals of the ligands in
this work. ^{c} Yamada et al. (1984) 
The spectral parameters Dq, B and C are acquired from the optical spectra for these systems (Erk and Hass, 1975) and collected in Table 1. By using Eq. 3 and 4 and the freeion parameters B_{0 }≈ 766 cm^{1} and C_{0 }≈ 2855 cm^{1} (Griffith, 1964) for V^{2+}, the covalency factor f_{γ} and hence the molecular orbital coefficients N_{γ} and λ_{γ} can be calculated. From the freeion values ζ_{d}^{0} ≈ 167 cm^{1} (Griffith, 1964) for V^{2+} and ζ_{p}^{0} ≈ 587, 2460 and 5060 cm^{1} for X = Cl, Br and I (McPerson et al., 1974), the parameters ζ, ζ’, k and k' are obtained from Eq. 5 and also shown in Table 1.
Substituting the related parameters into Eq. 1, the theoretical g factors are obtained and compared with the observed values in Table 2. For comparisons, the calculated results by neglecting the contributions from the sorbitals of the ligands (i.e., S_{ds} = λ_{s} = 0 and A = 0, corresponding to the previous treatments (Du and Li, 1994; Du and Rudowicz, 1992) ) are also given in Table 2.
Discussion
From Table 2, one can find that the theoretical g factors including the ligand sorbital contributions are in good agreement with the observed values. This means that the ligand sorbital contributions seem not negligible in the analyses of the EPR g factors for VX_{2} systems, especially for the larger ligands Br and I.
For X = Cl, the magnitudes of the theoretical Δg_{i} by including the ligand sorbital contributions differ little (no more than 1%) from the results on neglecting the above contributions. So the previous treatments (Du and Li, 1994; Du and Rudowicz, 1992) can be regarded as good approximations for this ligand. Nevertheless, for VBr_{2} and VI_{2} the calculated Δg_{i} in the absence of the ligand sorbital contributions are about 39% smaller and 28% larger than the experimental values, respectively. This means that the contributions from the SO coupling (which are much larger than that of V^{2+}) of the ligands Br¯ and I¯ are somewhat overestimated, if only the contributions from the porbtials of the ligands are considered. In fact, inclusion of the sorbitals of the ligands can modify the parameters N_{e} and λ_{e}, then change the magnitudes of k’ and ζ’ and finally lead to more reasonable Δg_{i}. For X = Br, the significantly larger Δg_{i} in magnitude by considering the ligand sorbital contributions compared with those by neglecting the above contributions can be attributed to the noticeable (twice) increase in the positive ζ’ related to the very small value (about 13 cm^{1}) in the absence of the contributions. Therefore, the useful assumption that the contributions of the s orbitals of the ligands may be negligible for 3d^{n} ions in octahedra (e.g., KNiF_{3}) (Du and Li, 1994) seems not always valid for ligands having much larger SO coupling coefficient (e.g., I¯) and so the ligand sorbital contributions should be considered in the studies of the g factors for VX_{2} here.
In the above calculations, the trigonal field parameters V and V’ are
determined from the structural data of the systems under study and the superposition
model, instead of taking as adjustable parameters. The calculated anisotropies
of the g factors are also comparable with experiment. The negative g_{//}g_{⊥}
for VCl_{2} is in consistence with the fact that the ligand octahedron
is slightly compressed (i.e., δβ ≈ 0.22°>0). For VBr_{2}
and VI_{2}, the small calculated anisotropies agree largely with the
nearly isotropic g factors (the observed anisotropies are almost zero within
the experimental errors (Yamada et al., 1984). This point may be interpreted
as the small magnitudes of ζ’ (Table 1) due to the
larger ζ_{p}^{0} for both ligands (Eq. 1
and 5) and the very slightly elongated ligand octahedra (i.e.,
δβ ≈ 0.12° and 0.57°<0 for X = Br and I, respectively).
Thus, the trigonal field parameters obtained from the superposition model in
this work can also be regarded as reasonable.
Conclusions
The gyromagnetic factors for the layered antiferromagnets VX_{2} are theoretically studied by using the perturbation formulas of the g factors including the contributions from the sorbitals of the ligands in this study. The above investigations seem to be useful to the experimentalists working on magnetic properties of these materials by mean of EPR technique.
Acknowledgement
This research was supported by the Youth Fundation of Science and Technology of UESTC under grant No. JX04022.