J. Phys. Chem. 1995,99, 4935-4940
4935
Density Functional Study of Iron Bound to Ammonia Joice Terra and Diana Guenzburger* Centro Brasileiro de Pesquisas Fisicas, Rua Dr. Xavier Sigaud, 150, 22290-180 Rio de Janeiro, RJ Brazil Received: April 29, 1994; In Final Form: January 17, 1 9 9 9
Density functional calculations were performed to investigate the species formed by the interaction of an Fe atom and ammonia. The discrete variational method was employed, and total energy calculations were performed for several configurations. It was found that the ground state is a 5E, with Fe configuration -3d6.6 4sI.l as obtained in a Mulliken-type population analysis; the Fe-N interatomic distance was determined to be 1.98 A. The hyperfine parameters isomer shift, quadrupole splitting and magnetic hyperfine field were also calculated and compared to reported experimental values obtained by Mossbauer spectroscopy in frozen ammonia.
I. Introduction Transition metals may absorb ammonia strongly on their surfaces, even at room temperature. The heterogeneous catalytic synthesis and decomposition of ammonia is a subject of great technological importance.’ Thus an understanding of the bonding mechanism between transition metals and ammonia is desirable. On the other hand, the technique of isolation of atoms and small molecules in frozen gases allows the use of solidstate experiments such as Mossbauer spectroscopy to probe charge and spin distributions. Accordingly, an investigation of Fe isolated in solid ammonia has been reported; the reaction product FeNH3 was identified, and Mossbauer hyperfine parameters were measured.* However, it became evident that quantum chemical calculations would be needed to better understand the origin of the values obtained. In this work, we report density functional theory (DFT) calculations3 for the species FeNH3. We used the discrete variational method (DVM).4 We determine the ground state by performing total energy calculations for several electronic configurations. The charge and spin distributions are analyzed. Finally, the Mossbauer hyperfine parameters isomer shift (d), quadrupole splitting (AEQ), and components of the magnetic hyperfine field (HF)are calculated and compared to experiment. This paper is organized as follows: in section 2 we briefly describe the theoretical method, in section 3 we discuss the electronic structure, in section 4 we report results for the hyperfine parameters, and in section 5 we summarize our conclusions.
11. Theoretical Method The DVM method employed has been described in the original literat~re,~ here we give a summary and some details of the calculations. The Kohn-Sham equations of DFT3 are solved in a three-dimensional grid of points. The local exchange-correlation potential employed was that derived by von Barth and HedinG5 In these spin-polarized calculations, appropriate for an open-shell molecule, the electron density of spin up is allowed to be different from spin down. The molecular density for each spin is obtained from the sum of the squared amplitudes of the molecular orbitals, which in turn are expansions on a basis of numerical atomic orbitals. According to the DVM scheme, secular equations are obtained,
* Author to whom correspondence should be addressed. @
Abstract published in Advcince ACS Ahsrracrs, March 15, 1995
which are solved self-consistently until a desired criterion is met. In the present calculations, convergence was carried out to < low4in the charge and spin densities. To calculate the Coulomb potential by one-dimensional integrations, the molecular charge density eO is fitted to a multicenter multipolar expansion:6 I
j
v
m
The summation is over a set I of atoms equivalent by symmetry, RNare piecewise parabolic radial functions centered at atoms Y and I distinguishes different basis functions of a given 1 (j = I , 1, A, N). This expansion may be carried out to any degree of accuracy in the fit with the “true” density; in the present calculations, partial waves up to 1 = 2 were employed for Fe and N, and 1 = 1 for H; the least-squares error of the fit of e was -0.04. The total energy E is defined as the expectation vaJue (sum over integration mesh) of the energy density eC;,{R,}). To control numerical errors, the actual computation of E is made by point-by-point subtraction of a reference system of noninteracting (NI) atoms, as in the basis, located at the nuclear sites R,:
-
Here we employed the algorithms of Delley and Ellis’ to calculate E. The variational basis set included all inner orbitals, e.g., no “frozen core” approximation was made. The valence functions included 3d, 4s, and 4p on Fe, 2s, 2p, and 3d on N, and Is, 2s, and 2p on H. The self-consistent process was initiated with basis functions from the neutral atoms. After convergence, the basis was improved by generating atomic functions for atoms with configurations similar to those of the molecule. These are obtained by a Mulliken-type population analysis, in which the overlap population is divided proportionally to the coefficients of the atoms.* This basis optimization process is repeated several times, until the configuration of the atoms in the molecule is approximately the same as in the basis. The three-dimensional grid of points was divided into two regions: the volume inside spheres placed around each nucleus, where precise polynomial integrations are performed on a regular grid,9and elsewhere in space, where the pseudorandom Diophantine point generator is used.4 The spheres had radii
0022-365419512099-4935$09.00/0 0 1995 American Chemical Society
4936 J. Phys. Chem., Vol. 99, No. 14, 1995
Terra and Guenzburger
TABLE 1: Configurations, States, Charges, Magnetic Moments and Populations for FeNH3, for Calculations at the Equilibrium Distance@ configuration
states 'AI
+ 'A2 + 5E
charge Fe f0.16
magnetic moment @a)
N - 1.97
Fe 3.76
Fe populations
N 0.15
3d 6.94 4s 0.84 0.07 4P 5E +0.22 - 1.98 3.77 0.20 3d 6.63 4s 1.12 0.04 4P 'E +0.22 - 1.98 3.78 0.20 3d 6.64 4s 1.11 0.05 4P 'E f0.22 - I .97 3.74 0.23 3d 6.58 4s 1.18 0.04 4P 'A1 +'Az+'E +0.03 -1.81 1.99 -0.05 3d 6.93 4s 1.03 0.02 4P 'A2 f0.12 -1.90 1.95 -0.07 3d 7.02 4s 0.85 0.02 4P +0.17 -2.00 3.76 0.13 3d 6.95 'A2 4s 0.83 0.07 4P ' I Since the apportioning of the overlap term in the definition of Mulliken-type populations is necessarily arbitrary, as is the choice of basis sets, the second digits in the populations given contain a degree of uncertainty
1.8 (Fe), 1.0 (N), and 0.5 au (H). A total of 12 700 points were used inside the spheres and 2200 Diophantine points elsewhere. The computer program used is the discrete variational code developed by Ellis and collaborators at Northwestem University.
-0.5
7
I
5
111. Electronic Structure We performed self-consistent calculations for seven configurations derived by distributing eight electrons among the spinup and spin-down highest-energy valence molecular orbitals, originating mainly from the valence orbitals of Fe. In C3" symmetry, the 3d,(z2), 4s and 4p0(z) orbitals of Fe transform as the al representation, and the 3d,3(x2 - y2, xy), 3d,(xz, yz) and 4pz (x, y) transform as the e representation. Here the symbols 0,n,and 6 are employed in analogy to the linear molecules. For all configurations, the spin polarization resulted in a splitting of the spin-up and spin-down levels of a few electronvolts; thus the spin-up levels, of lower energy, were kept fully occupied. In Table 1 are given the configurations considered, charges, Mulliken-type populations, and magnetic moments (in Bohr magnetons) defined as the difference between the spin-up and spin-down populations. Figure 1 shows the total energies for these configurations (relative to an arbitrary origin) plotted as a function of Fe-N distance. The ammonia N-H interatomic distance and H-N-H angle were kept constant and equal to the experimental values in the gas phase (1 .OO A and 107.2°).i0a Determination of the N-H distance for the free NH3 molecule, employing the same method described here, showed excellent agreement with experiment; in fact, for a fixed HNH angle of 107.2", we obtained a N-H equilibrium distance equal to 1.03 A, in good agreement with a more recent experimental value reported (1.008 A at an angle equal to 107.3°).'0b Details of these calculations will be published elsewhere.' I To understand the relative energies of the different configurations, one must take into account a number of factors that will influence those energies. First, the energies of the asymptotic atomic or ionic states must be considered. Molecular states that tend to excited states of the atom at large interatomic distances will have higher energies. For this type of analysis, Mulliken
...
-2.5
IT-V-T 1.6
20
1 2.4
2.8
3.2
3.6
Fe-N distance (A) Figure 1. Total energies of several configurations of FeNH3 as a function of the Fe-N interatomic distance. Energies were shifted by an arbitrary value. Numbering of configurations as in Table 1.
populations are useful to determine the atomic configuration of the atom in the molecule. In the case of bonds between transition metals and an element, such as N, containing a lone pair of electrons in the molecular axis, a repulsive factor will be present, due to the Pauli repulsion between the metal electrons and the lone pair. This factor will also influence the metal configuration; for example, promotion of an electron in the diffuse 4s orbital into the more compact 3d will decrease Pauli repulsion and thus lower the energy. In this respect, the energies of the different 3d orbitals could be expected to be in the order dd < d, < do. The da orbitals have the lowest Pauli repulsion since they have no component in the molecular z axis. On the other hand, the 3d(z2) orbital may hybridize with the 4s forming two combinations, (4s - 3d) and (4s 3d). Occupation of the former hybrid will draw electrons
+
J. Phys. Chem., Vol. 99, No. 14, 1995 4937
Density Functional Study of Fe Bound to Ammonia into the plane perpendicular to the bond direction, thus reducing the Pauli rep~lsion.'?.'~ Another form of diminishing repulsion is hybridization with the 4p.I4 Bond formation may involve covalent and ionic factors. Charge transfer may occur between metal and ligands; ionic bonding may occur involving attraction between oppositely charged atoms and between dipoles. Finally, it is the interplay between all these different factors that ultimately determines the relative stability of the different states. It is seen from Figure 1 that the configurationswith the lowest energies are 2, 3, and 3'. These configurations are constructed by doubly occupying the 9a1 orbital, predominantly Fe 3d(z2), and placing one electron in 10alt, predominantly 4s. Thus they have a large number of valence a1 orbitals occupied, favoring sd, hybridization and mixture with N(o). This characterisic is an important cause of the stability of 2,3, and 3'. The orbitals 4et and 5et, almost purely 3d(xz, yz) or 3d(x2 - y2, xy), are also doubly occupied. The subtle difference between these three configurations is in the occupied spin down orbitals of e symmetry. In configuration 2, the singly occupied orbital 4e4 is predominantly 3d(xy, x2 - y2). However, a nonnegligible admixture with 3d(xz, yz) (17.4%) is present. In configuration 3 the singly occupied orbital 5e4 is almost totally 3d(xy, x2 y2) (97%)and in configuration 3' the same singly occuped orbital is 99% 3d(xz, yz). In 3 and 3' the lower-energy 4e4 is unoccupied. The rationale behind the energy ordering 2 < 3 < 3' is the following: configuration 3 has a slightly lower energy than 3' since occupation of the dg orbitals, localized on a plane perpendicular to the Fe-N axis, leads to a smaller Pauli repulsion with the N lone pair on the same axis, as compared to d,. Configuration 2, with lower energy still, presents a mixture in orbital 4e4 of dg and d,, thus lowering the energy still by diminishing the interelectronic Coulomb repulsion inside the 3d orbital. This configuration may be viewed as the density functional counterpart of a configuration interation calculation, formed by a mixture of two configurations, one with dg and the other with d, occupation. It will be seen further on that the mixture of d, and dg in orbital 4e4 will prove essential in the interpretation of the measured quadrupole splitting. All other configurations have significantly higher energies at all distances; thus the ground state of FeNH3 is determined to be a SE with configuration 2. It is interesting to notice the similarity with the 6E ground state of the ion FeNH3+, as obtained with a modified coupled-pair functional (MCPF) ca1c~lation.l~In fact, our ground-state configuration corresponds approximately to that obtained for the positive ion, with the addition of an electron to the 9a14(0)orbital. In proposing the degenerate 5E ground state, we have not taken into account Jahn-Teller distortion as this is expected to be quite small, since bending will adversely affect the sd, hybridization, a major source of bond stabilization. This same argument was evoked to justify several degenerate (E) ground states found in calculations for the ions MNH3+ and M(NH3)2+ (M = transition metal).I4 To understand the relative energies of the states in Figure 1, one first may consider atomic Fe. The difference in energy between the ground state sD(3d64s2)and the lowest state of configuration 3d74s1(SF)is 0.86 eV. The lowest 3d74s1triplet (3F) is at 1.48 eV. Promotion of one more electron from 4s to 3d gives states of much higher energies, the lowest of which, 'F (3d84s0),has an energy 4.08 eV higher than the ground state.I5 Thus molecular configurations leading to this atomic state will be highly unfavorable. In fact, if we analyze the Mulliken populations in Table 1, we observe that all are near 3d74si.Thus we conclude that the promotion of about one 4s electron into
TABLE 2: a,jc, and 6 3d Populations for Configurations of FeNI-13a 3d population configuration 1 2
3 3' 4 5 6
d,,(z2) 1.oo 1.66 1.66 1.62 I .02 1.14 1.02
dZ(XZ,
YZ)
2.95 2.17 2.01 2.99 2.95 2.00 1.99
d,dXY, X? - Y? 2.98 2.81 2.97 1.99 2.96 3.88 3.92
Configurations numbered as in Table 1; calculations at the equilibrium distances. See caption of Table 1.
the 3d orbital, at a not so large energy expense, stabilizes the molecule by decreasing the Pauli repulsion between the 4s electrons and the N lone pair in the bond direction. However, further 4s 3d transfer to form Fe 3d8 would cost too much energy. It may be noticed in Table 1 and Figure 1 that the three configurations of lower energy 2,3, and 3' correspond to Fe 3d populations somewhat lower than the others (3d66,as opposed to 3d69 or 3d7" in configurations 1, 4, 5 , and 6). This demonstrates that the energy difference between Fe 3d64s2and 3d74s' is still high enough that despite Pauli repulsion, configurations with a 3d population somewhat smaller than 7 are more stable. It is noticeable that the ground-state 2 has a configuration on Fe (3d6634s'124p004)very similar to that found for Fe in the molecule FeN, as obtained with a different theoretical method.I3 In all states considered, Fe has a small, positive charge. However, in configurations 2,3, and 3' more electronic density has been transferred from Fe to NH3, resulting in a charge on Fe (f0.22) somewhat higher than for the other configurations. Thus we may infer that these states are stabilized also by a higher ionic character in the bond. The actual magnetic moment on Fe in the quintuplet ground state is somewhat smaller than 4 p since, ~ as may be seen in Table 1, a small portion of the spin is delocalized toward the nitrogen atom. For the two configurations corresponding to triplet states (4 and 5 ) , the 9a14orbital (partly 4s) is occupied, resulting in negative (antiferromagnetic) 4s magnetic moments on Fe. We observed that only the loa'? orbital represents s-d hybridization of the favorable kind, since in this orbital the 4s and 3d(z2) coefficients have opposite signs. As mentioned earlier, this type of hybridization is expected to stabilize the bond.I2.l3 Although the 4p coefficients are not negligible in the extended loa]?orbitals, the overall 4p populations are very small, and so hybridization with 4p must play a minor role. Configurations 5 and 6 have higher energies than the others due to the presence of two electrons in the SeJ(xy, x2 - y2) orbital (and none in (xz, yz)). Thus Coulomb repulsion between these electrons raises the energy. The counterpart configurations with two electrons in (xz, yz) could not be converged. In analyzing the energy difference between ground-state 5E and the triplet states, one must also take into account that the extra unpaired electron in 5Eis responsible for a larger exchange coupling energy with respect to the triplets. For comparison, the energy difference between the 5F lowest state of configuration 3d74s1 of Fe and the lowest triplet 3F of the same configuration is 0.62 eV.I5 To obtain a molecular triplet state more stable than a quintuplet, there would have to be a mechanism allowing to at least surpass this energy barrier. As seen in Figure 1, the equilibrium distance for the 5Eground state is 1.98 A. For all configurations except 4 and 5, the minimum lies at distances slightly smaller than 2.0 A. In the
-
4938 J. Phys. Chem., Vol. 99, No. 14, 1995
Terra and Guenzburger -3.00-
TABLE 3: Population Analysis for the Valence Orbitals of the 5EGround State (Configuration 2) of FeNH3 (Fe-N Interatomic Distance = 1.98 A) orbital
energy (eV)
occupation
- 10.72 - 10.22 -6.32 -6.28 -5.85 -3.62 -3.59 -3.56 -3.44
population analysis" (in % of one electron) 13.0 Fe d(z2), 5.5 N 2s, 87.9 N 2p(z) 6.6 Fe d(z2),6.6 N 2s. 94.6 N 2p(z) 97.9 Fe d(xz, yz) 98.6 Fe d(xy, x2 - y2) 79.1 Fe d(z?),8.4 N 2p(z), 5.2 Fe 4s 28.7 Fe 4s. 62.4 Fe d(z?) 17.4 Fe d(xz, yz), 8 1.9 Fe d(xy, x2 - y 2 ) 8 1.4 Fe d(xz, yz), 17.7 Fe d(xy, x2 - y 2 ) 91.9 Fe 4s, 5.9 Fe 4p(z), 5.9 Fe d(z2)
-4.00
-
-5'00
-
z
W
E C
w
" Only populations >5% were included. Orbitals adding > 100% include small negative H 2p populations, due to strongly antibonding nature. TABLE 4: Population Analysis for the Valence Orbitals of the 5E States Pertaining to Configurations 3 and 3' (Fe-N Distance = 1.98 A) orbital
energy (eV)
4et 5et 9alt 9ali 4el 5ei loait
-6.32 -6.04 -5.90 -3.61 -3.53 -3.43 -3.43
4et 5et 9alt 4el 5el 9ali IOalt
-7.50 -6.31 -5.57 -4.33 -3.73 -3.65 -3.42
occupation
population analysis"
Configuration 3 2 99.7 Fe d(xz, yz) 2 100.0 Fe d(xy, x2 - y 2 ) 1 4.3 Fe 4s, 79.5 Fe d(z2),8.9 N 2p(z) 1 28.0 Fe 4s, 62.9 Fe d(z2) 0 96.7 Fe d(xz, yz) 1 97.4 Fe d(xy, x? - y 2 ) 1 92.4 Fe 4s, 6.1 Fe 4p(z), 5.2 Fe d(z2) 2 2 1 0 1
1 I
Configuration 3' 100.0 Fe d(xy, x2 - y 2 ) 99.8 Fe d(xz, yz) 12.1 Fe 4s, 75.6 F e d (z?), 5.7 N 2p(z) 99.9 Fe d(xy, xz - y?) 98.9 Fe d(xz, yz) 34.7 Fe 4s, 57.8 Fe d(z2) 87.5 Fe 4s. 10.6 Fe d(z?)
Only populations >5% were included. Orbitals adding > 100% include small negative H 2p populations, due to strongly antibonding nature.
case of 4 and 5, smaller equilibrium distances are obtained (-1.88 A). Interatomic distances obtained with density functional calculations compare well with experiment, being usually more accurate than Hartree-Fock.I6 However, the local density approximation in the exchange and correlation potential results in equilibrium distances for bonds involving transition metal atoms which are usually underestimated by roughly 0.05 In Table 2 are displayed the do, d,, and ds populations. We may notice the d,-da mixture, present in configuration 2 and practically absent in all others. Another interesting feature is that configurations 2,3, and 3', with three electrons in 9a1 and 10a1, have only d, populations of 1.6-1.7. This is the result of mixing with the 2s and 2p orbitals of N and consequent charge transfer toward the latter. In Table 3 is given the valence orbitals population analysis for the 5E(configuration 2) ground state. It may be observed that the orbitals of al symmetry are responsible for the 3d-4s hybridization on Fe, as well as mixture with N 2s and 2p. Table 4 shows the composition of the valence orbitals of configurations 3 and 3'. Analyzing Tables 3 (configuration 2) and 4, we may perceive that what essentially distinguishes these three configurations from each other is the composition of the occupied ei orbital: in configuration 3, it is almost totally Fe d(xy, x2 - y2), in configuration 3', Fe d(xz, y z ) , and in 2 it is a mixture of these two orbitals, with predominance of 3d(xy, x2 - Y2).
1.60
2.00
1.80
2.20
2.40
Fe-N distance (A) Figure 2. Energies of the valence molecular orbitals of FeNH3 as a function of the Fe-N interatomic distance, for the 5E ground state (configuration 2). Orbitals were labeled !ccording to the major 3d or 4s contribution. Fe-N distance = 1.98 A.
In Figure 2 are plotted the one-electron energy levels of predominantly Fe 3d nature, for the 5Eground state (configuration 2), as a function of the Fe-N distance. The spin-up and spin-down e-symmetry orbitals of predominantly d(xy, x2 - y 2 ) character become more stable than their counterparts at short distances, due to increased Pauli repulsion along the z axis and resulting increased antibonding character of the d orbitals with z components. For the same reason, the diffuse 4s (loa,?)orbital has the highest energy at short distances but is stabilized rather steeply as the Fe-N distance is increased. We must keep in mind that DFT applies variationally to the lowest-lying state of each irreducible repre~entation.~There are still unresolved issues of the applicability, in principle, of DFT to a set of states of common symmetry. However, the success obtained in several such calculation^'^.'^ and the coherent picture obtained here lend confidence to the present results. As is well-known, in density functional theory one may obtain a mixture of states (multiplets) pertaining to a given configuration; this is the case for configurations 1 and 4 (see Table l), which include three states each. In going beyond the present method to treat the multiplets energies, we cannot entirely rule out the possibility that the splitting of the energies of the states from configurations 1 or 4 would produce a level lower in energy than our proposed ground-state 5E;however, we consider this unlikely since configurations 1 and 4 are constructed from 2, 3, or 3' by transferring one electron from an orbital of a1 symmetry into an e orbital. As orbitals of a l symmetry are the ones where 4s-3d hybridization takes place, as well as most of the interaction with NH3, this electron transfer would destabilize the molecule. To summarize, some of the factors leading to the 5Eground state found for FeNH3 are higher exchange coupling energy with respect to the triplet states, occupation of orbital lOal? with (4s3d) hybridization, larger Fe NH3 charge transfer resulting in more ionic character in the Fe-N bond, configuration on Fe between 3d64s2 and 3d74s' (more stable than -3d74s' found for the other states), lower Coulomb repulsion due to maximized electron distribution among the orbitals available and higher occupation of the 3dh orbital relative to 3d,.
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Density Functional Study of Fe Bound to Ammonia
J. Phys. Chem., Vol. 99, No. 14, 1995 4939
IV. Mcissbauer Hyperfine Parameters Density functional methods have been applied with success to the theoretical investigation of Mossbauer hyperfine parameters. Mossbauer isomer shifts 6 correlate well with electronic densities at the nucleus obtained with the multiple scattering method for covalent Fe complexes.ls Isomer shifts and quadrupole splittings of Au(1) complexes were calculated with the DVM method.I9 Hyperfine parameters for the Fe dimer were also obtained with this method,20 as well as for the linear molecules FeC12, FeBr2, and EuC122’and for F ~ ( C O ) SIsomer .~~ shifts and quadrupole splitting of several Sn compounds were obtained with DVM cluster calculations, correlating well with e~periment.*~-~j Finally, Mossbauer hyperfine parameters have been also calculated for Fe metals, alloys, and intermetallic compounds with density functional luster^^-^^ and band structure methods.30 The isomer shift 6 is defined as3]
d = 2/3e2nZS’(z)A(r2)[eA(0) - es(0)] aAe(0)
~~
6 configuration
1 2 3 3’ 4 5 6
Hc
AEQ (mds)
(mds)
HD (kOe)
(kOe)
HF (kOe)
f0.74 f0.42 f0.43 f0.32
experimental
f0.16 +I224 +94 -1.66 $539 $29 -0.59 +552 $66 -7.27 f438 -167 +OS0 f0.64 -810 -7 +0.66 +6.40 -1088 +248 +0.77 +631 +1193 +333 f0.67” -2.0“ +0.60(5)b I 1.90(5)lb
+1318 +568 1-618 +271 -817 -840 f1526 f 8 0 0 or
-900“
From ref 2;6 relative to Fe metal at room temperature. From ref 35; 6 relative to Fe metal. Theoretical values of HF= HC f HD. (I
200
-
1.00
4
(3)
where S’(Z) is a correction for relativistic effects, A(3) is the variation of the mean-square radius of the nucleus between excited and ground states of the Mossbauer transition, and the electronic term in brackets is the difference between the density at the nucleus in the absorber A and source S (in other words, between a given compound and a standard system). In a nonrelativistic calculation, only electrons pertaining to the totally symmetric representation contribute to e(0). The quadrupole splitting AEQ for 57Fein axial symmetry is given by3’
AEQ = ‘I2e2qQ
TABLE 5: Calculated and Experimental Mcissbauer Hyperfine Parameters of FeNH3, for Calculations at the Equilibrium Distances
‘\
I
E E E
-
3de4s’(Fe+’)
00
\
\
(4)
3de4sZ(Fe0)*\
where Q is the quadrupole moment of the nucleus with spin I = 3 / 2 in the excited state of the 14.4 keV Mossbauer transition and q is the electric field gradient:21
-1.00
I 136.00
\
I 140.00
I
I
144.00
148.00
I
152.00
Figure 3. Isomer shifts versus electron density at the nucleus (3sand 4s)for Fe atom and ions. Values of 6 from ref 33,relative to Fe metal.
The first term is the electronic contribution, calculated in the DVM method as a sum over the three-dimensional grid, and the second term is the point-charge contribution of the neighbor N and H nuclei. The contributions to the magnetic hyperfine field32 are the contact or Fermi H,, given by
H,= ( 8 n I j ) g & ~ [ @ t ( O-) @ko>lX
‘12
(6)
where g, is the electronic spectroscopic factor and ,UB the Bohr magneton. The dipolar field HD is defined as20
where the integral is calculated as a sum over the threedimensional grid. Finally, the orbital contribution HLis defined as
so that the total measured hyperfine field HFis
HF= H,+ HD + HL In Table 5 are shown the results obtained for the Mossbauer parameters described above at the Fe nucleus in FeNH3. To derive 6 , we first performed calculations for free Fe atoms and
ions, for which 6 was measured in frozen gas matrices,33also with a density functional self-consistent method. The results are plotted in Figure 3 (only 3s and 4s contributions are considered, since Q(0) for the inner electrons do not show significant changes for different configurations and thus cancel out). The negative slope is due to the negative sign of A(+) for the 14.4 keV transition of 57Fe(see eq 3). From this plot, the value a = -0.228 “/s af3 is obtained, well within the range of “acceptable” values for Fe.26.34Employing the equation derived for the line in Figure 3 (6 = -0.228~(0) 33.638), the numbers for 6 given in Table 5 were obtained, using the molecular values of e(0). For all configurations, 6 is much higher than would be obtained for Fe 3d64s2, as may be seen from Figure 3. The values are clustered slightly above the one for Fef 3d64s1.This is coherent with the Fe configurations in Table 1 (-3d66-7.” 4s’): the smaller charge on Fe, due to larger 3d population, decreases e(0) (increases 6) due to shielding of the 3s and 4s electrons by the 3d. The values of AEQ are also shown in Table 5 . Since the sign of the experimental value was also determined,2 this constitutes a valuable test for the 5E ground state found. In fact, electrons in 3d orbitals pertaining to different spherical harmonics differ largely in both sign and magnitude of the electric field gradient q.31 This may be verified in the large spread of calculated AEQ values for the different configurations, which differ among themselves in the occupation of 3d orbitals.
+
4940 J. Phys. Chem., Vol. 99, No. 14, 1995
Terra and Guenzburger
The value of Q utilized to calculate AEQ was 0.20b.31 Although there is always a degree of uncertainty in any chosen value, one may expect this not to be far from the exact one. It may be thus seen that the 5E ground-state configuration 2 is indeed the one that gives the calculated AEQ closest to experiment. Configuration 2, as discussed previously, is a mixture of 3 and 3', for which AEQ has the correct sign but is too small and too large in magnitude, respectively. All other configurations give AEQ very far from the experimental value. The reasonably good accord between calculated and theoretical values of AEQ represents evidence against a significant Jahn-Teller distortion in FeNH3. In fact, the original experimental data were successfully simulated with asymmetry parameter3' 7 = 0, which is valid for the C3,, geometry adopted. Significant Jahn-Teller distortion would have caused a considerable deviation from this result. As for the magnetic hyperfine field, unfortunately it was not possible to determine the sign experimentally, the two possible values being +800 or -900 kOe.* The values calculated and displayed in Table 5 are the sum of the contact field Hc and the dipolar field HD. They are all very large in absolute value. Positive fields Hc correspond to configurations where the loa'? (4s) orbital is occupied. Since the counterpart 10ali is not occupied, the positive field generated is very large (see eq 6). The core field H, of Fe (Is, 2s, and 3s) is negative'* but not large enough to counterbalance the positive contribution of loa,?. The large negative values of H,for configurations 4 and 5 stem from the negative magnetic moments on the 4s orbital (larger 4s4 population) which results in negative valence contriutions to H,, added to the negative core fields. The orbital field H L (eq 8) could not be obtained with the present calculations, since it requires the inclusion of_the spin-orbit interaction. Without this term, the direction of 1 relative to the spin remains undetermined. We calculated the maximum magnitude of EL,utilizing eq 8, for the 5E ground state; this corresponds to 1 and 3 being aligned. The number found was f 4 3 6 kOe; if the positive value is added to +568 kOe (Hc HD),the result would not be far from the experimental +800 kOe. For configuration 5 and 6, HL= 0 due to the nondegenerate ground state.32 It must be kept in mind when comparing theoretical values with experiment, that the calculations did not take into account the effect of the environment of solid ammonia upon the measured Mossbauer hyperfine interactions.
+
V. Conclusions The theoretical investigation of FeNH3, performed with the density functional discrete variational method, demonstrated that the ground state is a 5E.The pertaining configuration presents a small mixture of Fe 3d(xz, y z ) with 3d(xy, x2 - y2);this proves to be essential in understanding the value of the measured quadrupole splitting. The configuration on Fe for the ground state is -3d6.64s'.'. The value for the isomer shift found is not far from the experimental value. The calculated quadrupole splitting compares well with experiment; in fact, the groundstate 5E is the only one among the configurations considered that gives a theoretical AEQ reasonably close to experiment. The contact magnetic hyperfine field for the ground state is large
and positive, due to the unbalanced occupation of a spin-up molecular orbital of predominantly 4s character. The set of calculated hyperfine parameters 6 , AEQ, and HFare consistent with MSssbauer spectroscopy measurements, thus giving support to the 5Eground state found.
References and Notes ( I ) Tamaru, K. Acc. Chem. Res. 1988, 21, 88 and references therein. (2) Baggio Saitovitch, E. M.; Terra, J.; Litterst, F. J. Phys. Rev. B 1989, 39, 6403. (3) Parr. R. G.; Yang, W. Densiv-Functional T h e o p of Atoms and Molecules; Oxford University Press: New York, 1989. (4) Ellis, D. E.; Painter, G. S. Phyx Rev. 1970, 52, 2887. Ellis, D. E. Int. J . Quantum Chem. 1968, S2, 35. Baerends, E. J.; Ellis, D. E.; Ros, P. Chem. Phys. 1973, 2 , 41. ( 5 ) von Barth. U.; Hedin, L. J. Phys. C 1972, 5, 1629. (6) Delley, B.; Ellis, D. E. J . Chem. Phys. 1982, 76, 1949. (7) Delley, B.; Ellis, D. E.; Freeman, A. J.; Baerends, E. J.; Post, D. Phys. Rev B 1983, 27, 2132. (8) Umrigar, C.; Ellis, D. E. Phys. Rev. B 1980, 21, 852. (9) Stroud, A. H. Approximate calculation of multiple integrals; Prentice-Hall: Englewood Cliffs, NJ, 197 I. (IO) (a) Herzberg, G. Electronic Spectra ofPolyatomic Molecules; Van Nostrand Reinhold: New York, 1966. Cited in: Bauschlicher, C. W. J . Chem. Phys. 1986, 84, 260. (b) Handbook of Chemistv and Physics, 57th ed.; Weast, R. C., Ed.; CRC Press: Cleveland 1976-1977. ( I 1) Terra, J. Unpublished. (12) Blomberg. M. R. A.; Brandemark, U. B.; Siegbahn, P. E. M.; Mathisen, K. B.; Karlstrom. G. J . Phys. Chem. 1985, 89, 2171. (13) Siegbahn, P. E. M.; Blomberg, M. R. A. Chem. Phys. 1984, 87, 189. (14) Langhoff, S. R.; Bauschlicher, C. W., Jr.; Partridge, H.; Sodupe, M. J . Phys. Chem. 1991, 95, 10677. (15) Moore, C. E. Atomic Energy Levels; US Natl. Bur. Stand. Circular 467, 1952. (16) Ziegler, T. Chem. Rev. 1991, 91, 651. (17) PBpai, I.; Mink, J.; Foumier, R.; Salahub, D. R. J . Phys. Chem. 1993, 97, 9986 and references therein. (18) Guenzburger, D.; Esquivel, D. M. S . ; Danon, J. Phys. Rev. E 1978, 18, 4561. (19) Guenzburger, D.; Ellis, D. E. Phys. Rev. B 1980, 22, 4203. (20) Guenzburger, D.; Saitovitch, E. M. B. Phys. Rev. E 1981,24,2368. (21) Ellis, D. E.; Guenzburger, D.; Jansen, H. B. Phys. Rev. 5 1983, 28, 3697. (22) Guenzburger, D.; Saitovitch, E. M. B.; de Paoli, M. A,; Manela, H . J . Chem. Phys. 1984, 80,735. (23) Terra, J.; Guenzburger, D. Phys. Rev. B 1989, 39, 50. (24) Terra, J.; Guenzburger, D. J . Phys.: Condensed Matter 1991, 3, 6763. (25) Terra, J.; Guenzburger, D. Phys. Rev. 5 1991, 44, 8584. (26) Guenzburger, D.; Ellis, D. E. Phys. Rev. B 1985, 31, 93. (27) Chacham, H.; Galvio da Silva, E.; Guenzburger, D.; Ellis, D. E. Phys. Rev. B 1987, 35, 1602. (28) Guenzburger, D. First-principles calculations of Miisshauer h y perfne interactions in metals, alloys and inorganic compounds; BaggioSaitovitch. E.. Galvlo da Silva, E., Rechenberg, H. R., Eds.; World Scientific: Singapore, 1990; p 132. (29) Guenzburger, D.;Sobral, R. R.; Guimaries, A. P. J . Phys: Condensed Matter 1994, 6 , 2385. (30) See, for example: de Mello, L. A,; Petrilli, H. M.; Frota-PZssoa, S . J . Phys.: Condensed Matter 1993, 5 , 1. (3 1) Greenwood, N. N.; Gibb. T. C. Miissbauer Spectroscopy; Chapman and Hall: London, 1971. (32) Abragam, A. The principles qf nuclear magnetism; Oxford University Press: Oxford, 1961. (33) McNab, T. K.; Micklitz, H.: Barrett, P. H. Phys. Rev. E 1971, 4 , 3787. Montano. P. A.; Barrett, P. H.; Shanfield, Z. J . Chem. Phys. 1976, 6 4 , 2896. Micklitz, H.; Barrett, P. H. Phys. Rev. Lett. 1972, 28, 1547. (34) Miissbauer Isomer Shifts; Shenoy, G. K., Wagner, F. E., Eds.; North-Holland: Amsterdam, 1978. (35) Barrett. P. H.: Pastemak, M. J . Chem. Phys. 1979, 71, 3837.
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