J. Phys. Chem. 1995,99, 13094-13102
13094
Density Functional Study of Magnetic Coupling Parameters. Reconciling Theory and Experiment for the TiF3 Complex Paola Belanzoni: Evert Jan Baerends,” Sam van Asselt, and Peter B. Langewen Theoretical Chemistry Department, Free University, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands Received: November 28, 1994; In Final Form: May 1, 1995@
Density functional calculations are reported for the g and A tensors of T i S . The calculated magnetic parameters agreement with experiment. This lends credibility to the are in good (g and Adip) or reasonable (Aisotropic) calculated electronic structure, which however differs markedly from the one deduced originally from the ESR data in that the unpaired electron is predominantly 3d rather than 4s. The approximations that enter the conventional qualitative deductions from ESR data are analyzed and verified by explicit calculations. This highlights the pitfalls of the qualitative treatment. It is traced in detail how the present electronic structure leads to calculated ESR parameters that agree well with experiment and how the usual approximations of the qualitative treatment lead one to infer a different electronic structure.
1. Introduction There is currently considerable interest in the calculation of magnetic coupling parameters using density functional theory. This holds for both NMR1-3 and ESR;4*5see ref 6 for a recent review and references to the large body of literature conceming post-Hartree-Fock methods for these properties. The DFTESR calculations4have concentrated on small organic radicals, as did much of the past ab initio work (see however ref 5 for recent DFT work on transition metal containing systems). In this paper we address the calculation of g and A tensors for the transition metal complex TiF3, with particular emphasis on the relation between the ESR parameters and the electronic structure and on the various factors that affect the accuracy of the calculated ESR parameters and at the same time play a role in linking ESR parameters with electronic structure. The hyperfine structure of TiF3 has been studied experimentally by T. C. DeVore and W. Weltner, Jr.7 The ESR measurements have been conducted by trapping the molecule in neon and argon matrices at 4-10 K, and hence the total hyperfine A tensor consisting of both the isotropic and dipolar contributions could be observed. The interpretation of the ESR parameters led to the conclusion that the unpaired electron is essentially localized in a nonbonding orbital on the titanium, containing 70% 4s character. This result is surprising. Coordinative bonds have been extensively studied, and it is wellknown that transition metal atoms in complexes tend to “valence state” configurations with fewer 4s electrons than in the atomic ground state configuration, (often dn+2sorather than dns2). The reason can be found in the diffuseness of the filled 4s shell, which leads to strong exchange repulsion with donor orbitals on the ligands.8 Therefore, we expect that the same situation would hold in TiF3. In this paper we reinvestigate the composition of the molecular orbital containing the unpaired electron using an LCAO-DFT approach. We find that this MO (4a’l) is actually a predominantly 3d,2 orbital (71% 3d,i 26% 4s). To reconcile this result with the deductions from experiment, we have computed ESR parameters (g and A tensors), both at the LDA (local density approximation) and nonlocal (Becke9 for exchange and Perdew’o,ilfor correlation) levels. The calculated g and A tensors are in fair agreement with
+
’ Permanent address: Dipartimento di Chimica, Universitl di Perugia, via E k e di Sotto 8. 06123 Perugia, Italy. Abstract published in Aduance ACS Abstracts, August 1, 1995. @
0022-365419512099-13094$09.0010
experiment, in spite of the 3d,2 rather than 4s character of the unpaired electron. We analyze in some detail how the calculated electronic structure gives rise to the calculated (and measured) ESR data, revealing important pitfalls that may lead to erroneous deductions from the experimental data. It is shown, for instance, that spin polarization effects of lower occupied shells are essential for the determination of the dipolar term of the hyperfine coupling. For deduction of the 4a’l composition, it appears to be necessary to take account of this and other effects, which are enumerated in the Discussion section (3.3) below. The paper is organized as follows. In section 2 we first briefly review the theoretical methods and the computational details, with emphasis on the points that are important for the present calculations. Section 3 then outlines the main results. We examine the electronic structure of the complex (section 3.1) with the goal of describing the nature of the orbital containingthe odd electron and of checking if the “valence state” argument that led us to expect a 3d,2 unpaired electron can be applied. We then compare (section 3.2) the calculated g and A tensor values to experiment. The effects of various approximations, including neglect of spin polarization in subvalence and core shells, neglect of second-ordercontributions to the A tensor, and neglect of off-center functions and of off-diagonal matrix elements, are quantified by explicit calculation (section 3.3). It is established which of these approximations, which are used in qualitative deductions regarding the electronic structure, are not justified and may lead to erroneous conclusions.
2. Computational and Methodological Details Calculations were performed on the TiF3 system using the Amsterdam density functional (ADF)I2-l4 and the companion GATENQ program packages.l5-l7 The approximate selfconsistent field (SCF) Kohn-Sham one-electron equations are solved by employing an expansion of the molecular orbitals in a basis set of Slater type orbitals (STOs).Is Use is made of accurate and efficient numerical techniques19.*0 to calculate the effective one-electron Hamiltonian matrix elements. The parametrization of electron gas data by Vosko, Wilk, and Nusair2I was used for the local density approximation (LDA) calculations. The generalized gradient corrections (GGAs) of Becke9 for exchange and of Perdewlo.” for correlation were employed in the present calculations. Spin polarization effects have been investigated by spin-unrestricted calculations. The Coulomb and exchange potentials were generated in each SCF 0 1995 American Chemical Society
DFT Study of Magnetic Coupling Parameters
J. Phys. Chem., Vol. 99, No. 35, 1995 13095
cycle by means of a density fitting procedure,22and the frozen core approximation allowed for the evaluation of valence orbitals which are orthogonalized onto the core by augmenting the valence set with a single-5 STO for each core type orbital. The molecular orbitals were expanded.in terms of a double-5 STO basis set for F 2s (exponents 1.92,3.22) and 2p (exp 1.48, 3.52) and for 3s (exp 3.10, 4.75) and 3p (exp 2.50, 4.05) Ti orbitals. The Ti 3d (exp 1.04, 2.30, 4.95) and 4s (exp 0.80, 1.20, 1.90) were of triple-5 quality, and only one 4p STO (exp 1.20) was employed as polarization function for Ti and one 3d STO (exp 2.00) for F. Core orbitals were frozen for Ti 1s-2p (exp 1s 17.35, exp 2s 7.50, exp 2p 8.95) and F 1s (exp 8.33). More extensive basis sets were used, also in all-electron calculations, in order to investigate the effect of the basis and of the frozen core approximation on the values of the A tensor terms (contact and dipolar) and of the g tensor for the molecule. The implementation of the analytical gradient of the energy in ADP3 allowed for a geometry optimization of the molecule: the molecule adopts D3h symmetry, and the Ti-F distance turned out to be 3.32 au (1.756 8,) (3.37 au or 1.784 8, if we include Becke and Perdew nonlocal corrections). Calculations of the magnetic properties at various Ti-F bond distances have served to give insight into the metal-F bonding mechanism and to get information about the dependence of the A and g values on this parameter. The ESR g and A tensors of the considered system TiF3, containing one unpaired electron and two nuclei with a magnetic moment, have been calculated by means of second-order perturbation theory.’5-24-30 We assume an approximation of the ground and excited states by determinantal wave functions, with excitation energies approximated by orbital energy differences. This is a better approximation in the Kohn-Sham approach of DFT than in Hartree-Fock calculations since the virtual KohnSham orbitals feel an effective field of N - 1 electrons and are not shifted upward like the Hartree-Fock virtuals. The expressions for the g and A tensors may be cast in the following (spin-restricted case):
element (&8l~&) is to be understood with 4: as that part of the molecular orbital n whose basis functions have their origin on nucleus k. The MO coefficients are included in &,-and 3 is an operator with the origin defined on nucleus k. If Ok = ek(rk)Lf,where &(lk) represents the radial factor of the spinorbit coupling operator, the integrals are known as spin-orbit matrix elements. If @ is the jth component of the electron angular momentum about nucleus k’, L f , the integrals are called orbital Zeeman matrix elements. Note that in the matrix elements of the Lf operator the orbital qmis not restricted to the nucleus k’, which implies that here two-center contributions are taken into account. This is not the case in the matrix elements of &( rk)Lf since this operator weighs the region close to nucleus k very heavily due to the 11s behavior of &(lk). The excited states which can contribute to the gq are the states with one electron excited into the MO of the unpaired electron, the states with the unpaired electron excited into an initially empty MO, and the states with one electron excited from a doubly occupied MO into an empty one, giving rise to two doublets and one quartet state. Excited states with two or more excited electrons do not contribute because only one-electron operators are involved. The formula for the g tensor is gauge invariant only together with a first-order term, but expression 1 corresponds to a particular gauge (named the “natural gauge”) in which the first-order term is negligible.’5.30 In the expression for the hyperfine coupling tensor Ai,, referred to the nucleus k, we can distinguish first-order contributions and second-order ones. At first order a traceless, symmetrical tensor is obtained from the elements
where
c.
is the dipolar operator, defined by
e=3rirj - r2dq r
g , = g A , + ge
x cx
m(tn)
k
k’
(6;I6k(2>Lbl 4:>(wm IL: 14;) En
- Em
and P is (1)
Pe, the Bohr magneton (eW2mc);Pn, the nuclear magneton (eW2Mc); and gk, the g value of the nucleus k given by gk = pk/Ik. pk is the nuclear magnetic moment of nucleus k in units of Pe, and Ik is the total nuclear spin for that nucleus. The summations over p and q extend over the Cartesian coordinates x , y , and z, and crpq (Levi-Civita symbol) is an alternating tensor which is defined as
po is the magnetic permeability of free space;
I
0 if any two of i, p , q are equal = +1 if ipq is an even permutation of xyz - 1 if ipq is an odd permutation of xyz
where i, j = x, y , z, C k extends over the atoms (nuclei), and lk is the distance to nucleus k, F - Rkl. In the expression for the g tensor, g e is the free electron g value (2.002 319 31), q,, is the singly occupied MO, en is the one-electron energy of the nth MO, and the general matrix
The second and third terms in the A t expression ( 2 ) are firstorder two-center terms taking into account the two-center contributions which might be important in some cases. There are two kinds of such contributions: one arises when the operator and one of the basis functions have their origin on center A, and the other basis function has its origin on center B; the other, when the operator has its origin on center A, and both basis functions have their origin on center B. Three-center contributions (operator and basis functions on three different centers) are not considered. In the second order, the fourth and
Belanzoni et al.
13096 J. Phys. Chem., Vol. 99, No. 35, 1995 fifth terms in the Afj expression yield a nontraceless tensor. Their contributions are due to the mixing of the excited states with one electron excited into the MO of the unpaired electron and those with the unpaired electron excited into an initially empty MO. Since all operators in the expression for a general A tensor element depend on 1/13, it is expected that multicenter integrals can be neglected. Summing the first- and second-order contributions to the A&tensor, we obtain the A tensor without, however, the Fermi contact contribution. For the so-called axial molecules, i.e. molecules with one axis of trigonal or higher symmetry, this part of the A tensor may be written as3]
to
0
where the anisotropic part is characterized by
+W”o
I
Aaniso,
5“ is the spin-orbit parameter for the considered atom for the orbital la),32and frs becomes the partial spin-orbit parameter used in the calculation for the basis functions r and s of the A 0 la). We have used for the atomic spin-orbit parameters 5“ not the empirical spin-orbit splittings but calculated splittings, using fully relativistic numerical (Le. basis-free) atomic calculations. The proportionality constant a can, of course, be determined from various atomic configurations, e.g. from a A value for Ti3+ dl in connection with the corresponding Ti3+ 3d atomic orbital la) or from a A value for Ti s2 d2 with its Ti 3d atomic orbital la). Our spin-orbit calculation proves to be consistent in that we find the same value for a. We have used for Ti and F spin-orbit coupling constants theoretical values for Ti3+dl (A = 208.2 cm-I) and F (A = 346.0 cm-I), calculated employing fully relativistic numerical atomic calculation^.^^ The nuclear g value for 47Tiwas taken as gk = -0.315 32 and for I9F as g k = +5.2576.34 Finally, we observe that when we use spin-unrestricted calculations, the determinantal wave function constructed from the Kohn-Sham orbitals does not exactly correspond to a doublet spin eigenstate. A spin-projection procedure might be applied, although its status within density functional theory is still somewhat unclear, but the differences between a and p spin orbitals turned out to be quite small, and we have not applied spin projection. 3. Results and Discussion
We will follow ref 7 in denoting the anisotropic part also as Adip, although it should be noted that in our case this is not just the first-order dipolar term: the second-order anisotropic contribution is included. A,, is the isotropic pseudocontact contribution obtained (in second order) from the dipolar operator,
Apart from this isotropic contribution, there is of course also the Fermi contact contribution, A F or~ Acontact, ~ ~ in ~good approximation equal to
A: = (8x/3)Plq,(0)12 The isotropic (pseudocontact) contribution due to the secondorder terms in At, must then be added to the Fermi contact term in order to obtain the total AiSOterm: Ais0
= ‘pseudocontact + ‘Fermi
Of course AF~,,,,~ also should be added to A1 and All to obtain the total A tensor. The evaluation of the matrix elements in the gij and AS tensor formula is performed by expanding the orbitals 4: in terms of basis functions and by separating the radial and the angular part. For each atom the radial spin-orbit integrals are calculated between basis functions on the same nucleus. The radial parts of the spin-orbit matrix elements are evaluated by scaling r-3 matrix elements in the given set of basis functions in such a way that atomic spin-orbit splittings would be reproduced. So we write
where the proportionality constant a is the same as in the expression
3.1. Electronic Structure Calculations. The results from the restricted ADF calculation on TiF3 show that the Ti-F bonds are highly ionic, and the titanium can be considered to be a Ti3+ ion in a trigonal, planar field. In this D3h symmetry, the Ti3+ has an orbital of A’I symmetry, which can be described as an sd,? hybrid (4s 26% and 3d,2 71%) and which contains also a small percentage (3%) of a normalized A’I combination of F px and p? (the radial p,) orbitals. The unpaired electron resides in this orbital. Higher in energy we find empty 3d orbitals of E” symmetry (87% 3dx,, 3dY, and 11% of a normalized E” combination of F p: (pn) orbitals). The highest 3d level is of E’ symmetry (72% 3d,,, 3dx2-,,2, 5% 4px, 4p,, 11% 9% normalized combinations of F px and pJ (p, and pn in-plane) orbitals), as is shown in the orbital correlation diagram of Figure 1. The F3 fragment levels, at the right side of the diagram, lie very low in energy with respect to the metal d’s and are slightly destabilizedby charging. As may be inferred from the Mulliken gross populations of the fragment orbitals, the complex may be considered to be built up from charged F33- and Ti3+ fragments. The non-negligible 4s population (0.27 electron) is mainly coming from F p a orbitals (population 1.84 electrons in F3 pa a’I orbital), but also 3d,z contributes, lowering its population from 1.0 to 0.85. These populations fit in with a 3d’4s0Ti “valence configuration”. As is well-known, a change from a ground state dns2to a “valence state” dn+*s0configuration of the metal atom frequently occurs in transition metal-ligand interactions, in particular when the metal atom is surrounded by several ligands with bulky lone pairs directed toward the metal. The driving force is the reduction of the large exchange repulsion of an occupied metal 4s orbital with the ligand lone pair orbitals8 Therefore, we expect that a Ti ground state configuration 4s23d2is not suitable for interaction with F3, as it would result in a large closed shell repulsion between the F a’s and the diffuse 4s2 shell. To investigate this point more closely in TiF3, we compare the interaction of the F33- ion with the Ti2+ ion in either the ( 4 ~ or ) ~the (3d,2)2 configuration (Table 1). The interaction energy is decomposed into various terms.35 The steric repulsion Ai? is defined following Ziegler35 as the energy difference
+
J. Phys. Chem., Vol. 99, No. 35, 1995 13097
DFT Study of Magnetic Coupling Parameters -0 -1
3.3 eV. We note that the Pauli repulsion, although significant and responsible for the low 4s population, is not so dramatic as in other transition metal complexes, for instance in Cr(CO),j.s This result fits in with a relatively low overlap of the A’, combination of F p,’s with the Ti 4s and 3d9 (0.20 and 0.097, respectively). This overlap is considerably less than the 0.81 between metal 4s and CO 5a in the Cr(C0)6 complex. Having established that, according to the electronic stmcture calculation, the unpaired electron in TiF3 is not likely to be a 4s electron, we now focus our attention on the composition of the singly occupied 4a’l orbital, which we write in the explicit form
-
-2 -3
-
-4 -
-5 -
-7 -8 -9 -6
4a’l = +0.51(Ti)4s
resulting from the “restricted” calculation, and 4a’, = +0.49(Ti)4s
-13
+ 0.84(Ti)3d2, + 0.15(F)o
1
Figure 1. Orbital correlation diagram for TiFj (restricted). Energy units on the vertical axis are electrovolts. TABLE 1: Steric Repulsion (U) of Ti2+ (d,z)2and Ti2+ ( 4 ~ Valence )~ States with Three F-’s at R(Ti-F) = 3.32 bohr (Energies in eV) Ti2+( ~ l r 2 ) ~ Ti2+( 4 ~ ) ~ AEe~staF -53.1 -61.0 A E ~ ~ +17.9 ~ ~ ~ +30.1 AEO -35.1 -30.9 a Electrostatic interaction between the fragments. Exchange repulsion.
between separate noninteracting fragments and the composite system described by the determinantal wavefunction vO, constructed as the antisymmetrized and renormalized product of the overlapping fragment orbitals. A P is the sum of the electrostatic interaction energy and a remainder, the exchange repulsion, as in the Morokuma p r o c e d ~ r e . ~ ~The . ~ ’ former component represents the classical electrostatic energy AEelStat between the unmodiJied interpenetrating SCF charge distributions of the fragments when placed at their final positions. It is usually negative, i.e. stabilizing. The Pauli- (or overlap-, or exchange-) repulsion component AEpaulior AEXREP is generally positive. This contribution is essentially due to the effect of the antisymmetry requirement on Q0 and is well-known as the closed shell-closed shell (four electron-two orbital) destabilizing interaction. It is basically a kinetic energy e f f e ~ t . There ~~,~~ is finally the energy lowering due to mixing of virtual orbitals (charge transfer and polarization), leading to the fully converged ground state wavefunction of the total molecule but we will not be concerned with this last energy term. The purely electrostatic interaction is, of course, strongly attractive. The exchange repulsion, which is in the configuration 4s03d,2* mostly due to F pa’s overlapping the Ti 3s and 3p AOs, increases strongly in the configuration change from 4s03d,z2to 4s23d,20 on account of the added repulsion of the ligands with the closed 4s2 shell. A modest increase in the electrostatic attraction also appears, but the total effect is a less attractive AI?, -30.9 eV for ( 4 ~compared )~ to -35.7 eV for (3d;~)~. The 4s2 configuration is thus unfavorable by 4.8 eV. For comparison, the energetic cost to promote neutral Ti from the d2s2 ground state configuration to the “valence configuration” d4 is
vSCF,
+ 0.85(Ti)3dz2+ 0.17(F)a
resulting from a corresponding “unrestricted” calculation, both performed using the “general” basis set we described in the previous section and the optimized geometry with a Ti-F bond length of 1.76 A. This MO has indeed predominantly 3d character. Since the phase convention is such that the Ti3d? has positive lobes around the z-axis and a negative ring in the xy-plane and the F a is a combination of F 2p, orbitals pointing with positive lobes toward the metal, this orbital represents an antibonding combination of Ti 3dZ2with F u,stabilized by the mixing of 4s, which interacts in a bonding fashion with F u. One may wonder if the nature of the 4a’l is so sensitive to the Ti-F distance that small deviations from our calculated Ti-F distance, due for instance to crystal packing or a slight discrepancy between theoretical and experimental distances, might invalidatethis conclusion. We have therefore investigated the dependency on Ti-F bond distance. The percentage contribution of the Ti 4s atomic orbital ranges from 34 to 28, 24, 17, and 13% on passing from Ti-F = 1.61, 1.69, 1.76, 1.91, and 2.06 A, respectively. The 3d,2 percentage contribution follows the trend 61, 67, 71, 79, and 83%, in the above order. Finally, the u percentage contribution of fluorine does not change much, varying from 4.3 to 3.6, 2.8, 2.9, and 3.0 in the same order. So the 4a’l has predominantly 3d character throughout the (large) range of investigated Ti-F distances. In summary then, the unpaired electron appears to have 3d rather than 4s character both from qualitative electronic structure considerations and from actual calculations. We will next explicitly calculate the magnetic coupling parameters (A and g tensors) to investigate whether an unpaired 3d electron can also lead to the observed magnetic data. 3.2. g and A Tensors. The g Tensor. The results from our g and A tensor calculations (restricted and unrestricted) are listed in Table 2, together with the experimental A and g values. It is known that the deviation of the g value from g, is determined by the spin-orbit coupling, which endows the unpaired electron with a small orbital angular momentum and alters its effective magnetic moment. Interpretation of the g tensor depends on the properties of the excited states.27 First the results from the restricted calculations will be considered. The calculated g tensor agrees very well with the observed spectrum. The deviations of the principal values from g, are
Agxx= Agvv= -0.0956 Agzz= 0.0003 and by far the largest departure from the free spin value is
Belanzoni et al.
13098 J. Phys. Chem., Vol. 99, No. 35, 1995 TABLE 2: Experimental (Ref 7) and Theoetical Magnetic g and A Parameters for TiFf spin-orbit parametersb nuclear spin‘ nuclear magnetic momentc I(Ti) 2.5 p(Ti) -0.7883 &Ti) 208.2 cm-l I(F) 0.5 p(F) 2.6288 A(F) 346.0 cm-’ experiment
g Tensor 1.8910(2) grr = 1.9912(2) neon (site a) 1.8808(2) 1.9912(2) neon (site b) 1.8786(2) 1.9986(1) argon 1.9067 2.0020 (1.9245) (2.0026) 1.8816 2.0026 (1.9048) (2.0028)
g , = g,, =
restricted unrestricted
A Tensor
along normal to along AI(Ti)d AII(Ti)d bond(F)d bond(F)d z-axis(F)d Restricted first order 11.0 -22.0 33.5 -18.4 -14.9 2.7 second order 7.5 -1.0 -2.5 8.2 total 18.5 -23.0 31.0 -10.2 -12.2 pseudocontact 4.7 2.9 A~eml -288.0 28.1 -283.3 31.0 total A,,,e Unrestricted first order 8.6 -17.3 32.0 6.0 -38.0 second order 4.9 -0.6 -1.7 5.7 1.8 total 13.6 -17.9 30.3 11.7 -36.2 pseudocontact 3.1 1.9 AF~-~ -292.4 -44.6 -289.3 -42.7 total A,,,‘ experiment restricted unrestricted
(neon) -6.6 (argon) -8.1 -13.8 (-13.4) -10.5 (-8.9)
All A tensor values are expressed in MHz. The theoretical values in parentheses are obtained by including the nonlocal Becke and Perdew (i
corrections. The spin-orbit parameters have been obtained from numerical relativistic atomic calculations. The nuclear spin and the nuclear magnetic moment (expressed in nuclear magneton units) are taken from ref 34. The following calculated A values represent the first order, the second order, and the sum of these two terms (total) (eq 2), without the Fermi contact contribution but including an isotropic “pseudocontact” contribution that is given separately. e The contributions to A,,, are the pseudocontact second-order term and the Fermi contact term ( A F ~ ~ ~ ~ ) . observed when the magnetic field is in a perpendicular position to the principal molecular C3 axis. The negative sign clearly indicates that the main contribution to the g tensor comes from the excitation of the unpaired electron into higher, empty MOs. This is in agreement with the calculated level spectrum (Figure l), which indicates that the lowest transition will be excitation of the unpaired electron into the 2e”l. In the unrestricted calculations, we note a clearly distinguishible shift of -0.025 in gl, bringing the calculated values in closer agreement with experiment. Since the eigenvectors do not change much (see 4a’l composition), the shift in the unrestricted calculation has to be due to changes in the excitation energies that enter the denominator of the expression for the g tensor. We conclude that the simple model used here to calculate the g tensor yields quite reasonable agreement with experiment. The fact that in our calculation the unpaired electron is mostly 3d, and not 4s, apparently does not inhibit good agreement with experiment.
The A Tensor. The experimentally derived and theoretically calculated values for Adip for the metal and fluorine nuclei in TiF3 are also given in Table 2. We first focus on the restricted results. The hyperfine splittings due to the F nuclei are small if compared to the calculated atomic fluorine values of AcOntact = 47 910 MHz and Adip = 1515 MHze40 The A tensor on F is small because there is little participation of the fluorine 2p orbitals in the MO containing the unpaired electron so that very little spin is on the ligands. The A tensor on F has a component along the Ti-F bond, a smaller component along the z-axis (parallel to the C3 axis), and an even smaller component in the plane and normal to the bond direction. The experimental fluorine A tensor has been determined assuming an axial spin Hamiltonian (eq 1, ref 7). It is not possible then to compare directly the experimental fluorine A tensor with the calculated one, and we just note that the orders of magnitude agree. As we are not so interested in the ligand hyperfine splitting, we will not investigate this point further. For the remainder of this paper we will be dealing with the Ti hyperfine splitting. The dipolar part of the A tensor on Ti is small. The difference between the two experimental values when TiF3 is trapped in a neon matrix or in an argon matrix indicates that the calculated value is reasonable, but it is not yet very close. The effect of the inclusion of the nonlocal Becke and Perdew corrections is also displayed in the tables. In this case we can notice that the dipolar hf couplings are relatively insensitive to the nonlocal functional form (for a counterexample see ref 4). The unrestricted results are not qualitatively different, but they do yield somewhat better quantitative agreement with experiment, in particular when the nonlocal corrections are used. This may be somewhat fortuitous, since it is well-known that the BeckePerdew exchange-correlation potential, which enters the KohnSham one-electron equations and determines the electron density and unpaired spin distribution, is not improved over the LDA p ~ t e n t i a l . ~It’ .has ~ ~ an erroneous Coulombic singularity at the nucleus and might therefore be expected to yield poorer results than the LDA potential. Obviously our calculated spin distribution leads to a dipolar A tensor that is in quite reasonable agreement with experiment, in spite of the predominantly 3d character, rather than 4s, of the unpaired electron in the calculated electronic structure. The calculated Aiso is quite a bit larger than experiment (- 185 MHz). This is surprising in the light of the smaller calculated 4s character than deduced from the experimental data, which would lead us to expect a smaller calculated Aiso. We will analyze in the next section a number of factors that influence the accuracy of calculations of magnetic data and therefore might explain remaining differences between theory and experiment. The same factors play a role in qualitative inferences from the experimental data and may therefore help us understand the discrepancy between the original deductions from the experimental data and our calculated electronic structure. 3.3. Discussion. In general, for a molecule with axial symmetry, only the knowledge of the contact term and Adip is required to characterize the hyperfine coupling. We will first identify some approximations that are made in the conventional treatment of these terms and then investigate to what extent they are justified. For this purpose, we write the Fermi contact term at the magnetic nucleus k in the following form:
or, explicitly,
DFT Study of Magnetic Coupling Parameters
J. Phys. Chem., Vol. 99, No. 35, 1995 13099 If all of these approximations are applied, one obtains the following simplified expression for Acontact:
For the first-order contribution to the dipolar part of the TiF3 A tensor on Ti one obtains
J where m refers to the occupied spin-orbitals, nz indicates the occupation number, and contributions with both basis functions on other nuclei (k’ and k”) are not taken into account. Using STOs, only 1s functions are responsible for the electronic density on the nucleus; therefore, only terms of the form ~ $ 0 )&O) = lsklsk and $(O)xky’(O) = lsk$(O) give contributions in the above expression. We write the first-order dipolar term at the magnetic nucleus k as
We notice that the integrals involved in these expressions are characteristic of the Ti atom, so that the unpaired electron spin distribution over 4s and 3d can be obtained from
and from Pn
where =
