J. Phys. Chem. 1992,96,6255-6263
6255
Magnetic Interactions in a Three Center, Four Electron System J. R. Hart, A. K. Rap&* Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523
S . M. Gorun, and T. H. Upton*?+ Corporate Research-Science Laboratories, Exxon Research and Engineering, Annandale, New Jersey 08801 (Received: February 13, 1992)
The results of ab initio electronic structure calculations on model four electron, three center systems are used to develop a general understanding of the effect of bridging ligands on metal-metal magnetic interactions. An expression is obtained for the energy difference between singlet and triplet states as a function of the overlap with the bridging ligand. A linear relation is derived and computationally verified between the singlet-triplet splitting and the fourth power of the overlap between the symmetric combination of magnetic orbitals and the bridge orbitals.
hboduction The theoretical characterization of the magnetic interactions of transition metal atoms in molecules and solids has been the subject of considerable recent interest. In bioinorganic chemistry development of an understanding of the magnetic interactions between metal centers provides useful information about the coordination environment of metal centers, provides information about the nature of bridging ligands, and provides an assessment about the distance between metal centers and thus the potential for cooperativereactivity involving multiple centers. In solid-state inorganic chemistry (magnetism, high T,superconductivity),there is a general interest in developing a model of the distance dependence of the bridging ligand mediated spin coupling of electrons. The most common framework for discussing magnetic interactions in the literature is the model derived by Heisenberg, Dirac, and Van Vleck (HDVV).' The metal centers are assumed to be ions in a local ligand field, and the spins are assumed to interact isotropically to account for the magnetic exchange. While the HDVV spin Hamiltonian provides an accurate description of exchange interactions, it contains no information about the origin of these interactions. The theoretical interpretation of magnetic exchange interactions in bridged transition metal systems has its roots in the investigations of solid-state insulators.2 The metal atoms in the lattices are taken to be positive ions, and the bridging atoms are assumed to be adequately described as being negative anions. Configuration interaction (CI) is then necessary to build covalency into the wave function and to offer a means of describing the magnetic behavior of the system. A major problem with this approach is that orbitals from the crystal wave functions are much different than the free ion functions, and using the free ion functions as a starting point is difficult to justify. Additionally, the free ion wave functions are not orthogonal to each other; the overlap between them is significant at finite distances. An energy expression derived from this approach is guaranteed to be too complex for simple interpretation. While this approach effectively builds covalency into the ionic wave functions, a much simpler approach begins with covalency already in the wave function before any configuration interaction is performed. Anderson' developed a new framework in which to work on the magnetic exchange problem in solid-state systems wherein the overlap problem is eliminated. The first step in the Anderson approach is to solve for a wave function describing the metal center in its local ligand field environment while excluding the effects of the magnetic ions on each other. This can be accomplished by solving for the high spin HartreeFock wave function for the f Products Research Division, Exxon Research and Engineering, Linden, NJ 07036.
0022-365419212096-6255$03.00/0
system. The oneelectron Bloch orbitals consisting of mostly metal character from the high spin wave function are then transformed to an equivalent set of localized Wannier functions. The second step is to describe the magnetic ion interaction. The method Anderson chooses is to use the Wannier functions from this wave function in a perturbational treatment to obtain the energy difference between the ferromagnetic and antiferromagnetic states. While Anderson's method' has been useful for qualitatively explaining some of the trends in magnetic behavior of related compounds, it has been shown to be inadequate for reproducing experimentally reported values. Until the middle of the 197Os, most of the theoretical interest in magnetic coupling constants was confined to the solid state. This changed when Kahn4 and Hay, Thilbeault, and Hoffmann (HTH)S separately published the molecular equivalent of Anderson's solid-state theory. In adapting Anderson's formalism from the solid state to the molecular level, it was again assumed that the relaxation of the orthogonality between the magnetic orbitals was the largest contribution to the observed antiferromagnetic coupling. This methodology has been used to interpret and predict the angular dependence of the magnetic coupling constant in bridged transition metal systems. Since the only quantities calculated in this scheme are the orbital energies, only qualitative insights are gained. In order to begin developing a model of the distance dependence of the bridging ligand mediated spin coupling of electrons, the qualitative electronic aspects of a three center, four electron interaction as a function of internuclear distance are investigated in the present work. The linear molecules H - H e H and H-F-Hare chosen for conceptual simplicity. The effects of varying the nature of the bridge on the Ander~on'*~ superexchange energy are computationally studied. The interactionsof the hydrogen s orbitals with an s orbital (H-He-H) and a p orbital (H-F-H-) on the center atom X are studied over a range of H-H separations ( R ) from 2.5 to 10 A. An expression for the energy difference between the singlet and triplet states is obtained as a function of overlap. It is found that a plot of the singlet-triplet splitting versus the fourth power of the overlap between the symmetric combination of H orbitals and the bridge orbitals is nearly linear.
I. Review Magnetic interactions between transition metal ion centers are commonly described in terms of the HDVV Hamiltonian which is of the form where S,and Sj are the spin operators for magnetically interacting atoms i and j, and Jii is the isotropic magnetic coupling constant between two ions. J j is positive for a system where the spins couple ferromagnetically, and it is negative for antiferromagnetic in@ 1992 American Chemical Society
6256 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 teractions. For a transition metal dimer with atoms a and b, the HDVV Hamiltonian reduces to
H = -2JSa.Sb
(2)
It can be shown6that the energy levels obtained from this Hamiltonian are E(S’) = -J[S’(S’ + 1) - Sa(Sa+ 1) - Sb(Sb + I)] (3) where Saand Sb are the spin quantum numbers for the ions, and S‘ is the overall spin state of the dimer. For a dimeric system with spin S’,with equivalent ions Sa= Sb = S, the energy resulting from applying the HDVV Hamiltonian is given by E(S’) = -J[S’(S’+ 1) - 2S(S +I)]
(4)
Before the spin Hamiltonian is applied, all spin states are degenerate. The HDVV spin Hamiltonian is then turned on to perturb the system. The resulting diagram is a ladder of spin states which relates J to the energy differences between spin states. If the energies of two different spin states, E ( S 1 )and E(S2),are known for a given dimer, these values can be substituted into eq 4, and, upon rearrangement, an expression for J is obtained: J = -
W
l
)
-W 2 ) - SZ(S2 + 1)
(5)
SIVI + 1) Equation 5 is important because it is useful for calculating J values from the results of theoretical calculations. For example, in a molecule with two interacting d’ metal ions, there are two possible spin states arising from the coupling of the d electrons, a triplet (ferromagneticcoupling) and a singlet (antiferromagnetic coupling). If the energies of wave functions representing these spin states can be accurately calculated, a J value can be obtained for comparison to experimental J values obtained from magnetic susceptibility measurements. Anderson3developed a framework in which to work on the magnetic exchange problem in solid-state systems. The fvst step is to solve for a wave function describing the metal center in its local ligand field environment while excluding the effects of the magnetic ions on each other. This can be accomplished by solving for the high spin Hartree-Fock wave function for the system. The one-electron Bloch orbitals consisting of mostly metal character from the high spin wave function are then transformed to an equivalent set of localized Wannier functions. The second step is to describe the magnetic ion interadon. The method Anderson chose is to use the Wannier functions from this wave function in a perturbational treatment to obtain the energy difference between the ferromagnetic and antiferromagnetic states. The form of Anderson’s result for two magnetically interacting electrons is
whereK12= . f . f W * d I >W*2(2) (l/r12)%(1) r 1 ( 2 )drl dr2, K is an orthogonal Wannier function from the ferromagnetic HartreeFock wave function, bI2is the “transfer integral”, and U is given in terms of coulombic integrals as J I 1- J12.The term 2KI2is a positive contribution to 2J and is called *potential exchange” by Anderson. It arises from the favorable exchange energy of electrons with like spins. The second term, 4 ( b 1 J 2 / U , is called “kinetic exchange”, and it is the energy gained from relaxing the orthogonality between the magnetic orbitals. In other words, when neighboring spins in a solid are parallel, the spatial parts of the orbitals must be orthogonal. However, when the spins on two different centers are coupled antiferromagnetically, the spatial parts of the orbitals may ~ v e r l a p .The ~ kinetic exchange term in eq 6 reflects this effect. In order to qualitatively illustrate what determines the size of the transfer integral bI2, the simple example of the interaction of two du orbitals with the p orbitals of a bridging ligand is presented. Kanamori’ and Goodenough* each derived general qualitative rules for interpreting the kinetic exchange part of Anderson’s model. The analysis is based on the symmetry properties of the metal orbitals in a ligand field interacting with
Hart et al. the orbitals on a bridging ligand. The two limiting cases are for M-X-M angles of 180’ and 90’. For 180°, each metal atomic orbital can interact with the same bridging ligand orbital. In this case, the transfer integral is expected to be large, and, therefore, a large antiferromagnetic contribution to the coupling is expected. For the 90’ case, the metal orbitals interact through two orthogonal bridging orbitals. The transfer integral is small in this case, and the K12term is expected to give rise to a slightly ferromagnetic contribution. When there is more than one singly occupied magnetic orbital per center, the different contributions can be summed up to arrive at an educated guess about the magnetic behavior of the system. In molecules with competing contributions to J, a positive or negative coupling constant can be the overall result. In adapting Anderson’s formalism from the solid state to the molecular level Hay, Thilbeault, and Hoffmand assumed that the relaxation of the orthogonality between the magnetic orbitals was the largest contribution to the observed antiferromagnetic coupling. Specifically, the HTH model for the d9A9case begins with a high spin self-consistent field (SCF) calculation for the triplet state. Following Anderson’s formalism, the doubly oocupied molecular orbitals (core orbitals) are assumed to contribute little to the magnetic coupling; the interaction of the two half-filled orbitals of mainly metal character should account for the dominant contribution to the singlet-triplet energy splitting. The possible states using these orbitals are
T:
I4l&JZ4
S1:
I4ldJlSI 142a4281
Sz: 1
- (I4I&J2SI - I4IB424)
S3:
fi
(7)
The energy of the triplet state is given by
ET = hll + h22 + Jl2 - Kl2
(8)
where hij is the one-electron operator (4ilhlt$j),J I 2is the Coulomb integral, and K12is the exchange integral for orbitals i and j. The lowest energy singlet state is an approximately equal mixture of the wave functions SIand S2. The energy is obtained by considering interaction between the two wave functions. The result of solving the 2 X 2 configuration interaction (CI) matrix yields the following expression for the energy:
E, = hll
1
+ h22 + $511 + 5 2 2 ) 1
-[(2hl1 + J11 - 2h22 - J ~ z+) 4K122]1/2 ~ (9) 2 An expression for the coupling constant is then obtained by substituting the energies for the singlet and triplet states into eq 5: 1
+ Kl2 + z(J1l + J22) 1 -[(2hIl + Jll - 2h22 - J22)’ + 4K122]1’2(IO) 2
2J = (Es - ET) -J12
To this point, the only approximationsmade are that the H D W Hamiltonian can be used to relate the differences in energy of two spin states to the coupling constant, that the triplet is adequately approximated by the high spin restricted Hartrce-Fock (RHF) wave function, and that the singlet is described by the two configuration CI. Equation 10 is important because it represents the contribution to the magnetic coupling constant which arises from the mechanism proposed by Ander~on.~ In order to arrive at a workable expression, more approximations were made by HTH.5 Inside the square root in (91,l J l l - 5221 was neglected, and K I z 2was assumed to be much larger than (hll - hZ21. Expanding this term in the expression (using the approximation ( 1 + x ) I / ~ 1 + x / 2 + ...), the following expression for J was obtained:
=
The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6251
Three Center, Four Electron Magnetic Interactions
By taking linear combinations of the singly occupied molecular orbitals, two localized molecular orbitals are defined as the magnetic orbitals. They will resemble the atomic orbitals of the free metal ion in the ligand field, but they will interact slightly with each other and contain some ligand character. They also differ from atomic orbitals because they are orthogonal to each other. A new basis consisting of these localized molecular orbitals are then defined:
0 sx
@ SI
@ s2
b
Figure 1. Symmetry interactions of three s orbitals. (bb
=
1
- 42)
(12)
X..
Hi
0 H-
where da is localized on metal A, and (bb is localized on metal B. These orbitals are the molecular equivalent of Anderson's Wannier functions. Equation 11 then becomes, in terms of the integrals for the localized molecular orbitals,
4hat w Z 2K,b - Jaa - Jab
(13)
which is analogous to Anderson's expression (eq 6). Using the facts that 4hd = hl - h2 and that the Hartree-Fock operator for the triplet orbitals are € 1 = hi J12 - K12 €2
+ h2 + Jl2 - K12
(14)
the fmal expression relating the coupling constant to the difference in orbital energies is obtained:
0
Figure 2. Symmetry interactions of a bridging p orbital with two s
orbitals. where H I and H, are the atomic s orbitals on the left and right hydrogens, respectively. The two lowest states are the singlet and triplet covalent states, which can be represented in terms of the molecular orbitals:
'a = cll((bH1)21 + cZl((bH~)~1 = J(@H,)((~HJI Equation 15 has been used to interpret and predict the angular dependence of the magnetic coupling constant in bridged transition metal systems. The orbital energies are obtained from extended Hiickel calculations, the term 2Kab is assumed to be small and is therefore ignored, and the denominator Jaa - Jab is assumed to be constant for closely related compounds. Therefore, since the only quantities calculated are the orbital energies, only qualitative insights can be gained using this method. This method has been used to rationalize the observed bridge angle dependence of the coupling constants in related Fe(II1)X-Fe(II1) complexes, where X is either S or 0.9There was no attempt to quantitatively duplicate the experimental results. A magnetostructural relationship has been p r o m for bridged iron(II1) complexes in which varying the Fe-bridge distance is found to be more important than varying the Fe-bridgeFe angle.l0 The coupling constant is related exponentially to the "average superexchange pathway-, defined as an average of Fe-bridge distances. No relationship could be found relating the FebridgeFe angle or the Fe-Fe distance to the coupling constant. The idea of an exponential dependence of the magnetic coupling constant has been suggested previously." Configuration interaction calculations on the four electron system VZtF-V2+ were performed in an attempt to describe the distance dependence of J.Ilb Several of the leading terms in the expression were found to vary as (Le., the fourth power of the overlap between a d orbital on the metal and a bridging p orbital), which behaves as an exponential function of the V-F distance. 11. Effect of Varying the Nature of the Bridge
d ~ =, Hi + Hr 4~~= HI - Hr
(16) (17)
(19)
For the interaction of the two s orbitals with an s orbital in the center (Figure l), three molecular orbitals can be formed from the three atomic s orbitals: @SI
=H 1 + Hr (bsz
u s
= HI - Hr
(20) (21)
(bsx Xs + &'[HI + Hr1 (22) where X, is a measure of the degree of mixing of X , with the positive combination of the hydrogen s orbitals. At R = 10.0 A, X, is close to zero, and the following orbitals can be defined at this distance:
dSIO= (bsl(lO.O)= HI
+ Hr
(23)
(bsxo = (bsx(lO.0) = xs (24) The quantity X, varies as a function of R and the energy separation between bsIoand (bSxo. It is therefore desirable to vary R and the energy of (bSxoand observe the resulting changes in the orbital energy difference and the singlet-triplet energy gap. For the interaction of the two hydrogen s orbitals with a p orbital on the bridge atom (Figure 2), three molecular orbitals can also be formed, but, due to symmetry considerations, the center p orbital now interacts with the negative combination of the hydrogen orbitals: 4pI
(bpz
We begin with a simple reference calculation on the H2 molecule. There are two molecular orbitals which are formed when two hydrogen atoms are brought together (normalition ignored):
(18)
= HI + Hr
= H I - Hr
+ Adu,
(25) (26)
(bpx Xp - HI - Hr1 (27) Once again at R = 10.0 A, A, is approximately zero, and (bplo = (b,,(lO.O) 4,XO
= HI - Hr
= (bpx(10.0) = x,
(28) (29)
6258 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 TABLE I: H, Qm~~titie~ R(H-H) OE(4H1)(au) OE(&H,) (au) AOEH, (cm-I) 2.5 2.75 3.0 3.25 3.5 3.75 4.0 4.25 4.5 4.75 5.0 5.25 5.5 5.75 6.0 6.25 6.5 7.0 7.5 8.0 10.0
-0.52632 -0.51790 -0.51 1 97 -0.507 83 -0.504 99 -0.503 05 -0.501 75 -0.500 89 -0.500 32 -0.499 94 -0.499 70 -0.499 54 -0.499 44 -0.499 37 -0.499 33 -0.499 3 1 -0.499 29 -0.499 28 -0.499 28 -0.499 28 -0.499 28
-0.47277 -0.480 95 -0.486 73 -0.49078 -0.493 59 -0.495 51 -0,496 80 -0.497 67 -0.498 24 -0,498 61 -0.498 86 -0.499 02 -0.499 12 -4.991 8 -4.992 2 -4.992 5 -4.992 6 -4.992 7 -4.992 8 -4.992 8 -4.992 8
-1 1851.6 -8 120.6 -5486.9 -3731.1 -24 14.2 -1656.4 -1086.6 -707.4 -456.5 -29 1.9 -184.4 -1 14.6 -69.7 -41.7 -24.1 -13.6 -7.5 -2.0 -4.4 -2.2
0
Hart et al. 100
J (cm-I) -766.8 -360.2 -166.3 -75.6 -33.9 -15.0 -6.6 -2.9 -1.2 -5.5 -2.2 -1.1 0 0 0 0
50
i
0 0
0 0 0 2c
The energy difference between and GpXo, and also the internuclear distance, still determine the magnitude of the mixing of XV Varying R is straightforward, but altering the energy of the orbitals on X must be achieved somewhat artificially by manip ulation of the basis set and potential used in the calculations. A minimum basis set contraction of four Gaussian functions is used on the hydrogen atoms for all cases, the contraction being optimized for atomic hydrogen. For the bridging atom s orbital case, a minimum basis set is used for the He atom, with the contraction coefficients of the four Gaussians obtained from a calculation on the isolated atom. The energy of q5sxo is then varied in two ways. With the nuclear charge held constant at 2 = 2, the basis set is scaled with the factors t = 1.05, 1.10, 1.15, 1.20, and 1.25. As {increases, the energy of +,xo is raised. Alternatively, the charge of the bridging atom is varied, and the exponents of the Gaussians in the basis set are scaled accordingly. Increasing Z effectively lowers the energy of +,xo. Charges of 2.25 and 2.50, 2.75, and 3.0 were used. In order to study the interaction of a p orbital with the two hydrogen 1s orbitals, the ion H 2 F is used. For the Z = 9.0 case, the basis set for the F atom is a minimum basis set derived from the Dunning12(9s5p/3s2p) contraction of the Huzinaga basis.I3 To vary the energy of q5pxo, the basis is scaled for Z values of 9.5, 10.0, and 10.5. The wave functions used for the singlet and triplet states are kept simple in this study to permit development of qualitative insights. For both spin states, the bridging orbitals are treated as doubly occupied, closed shell orbitals. The energy of the triplet spin state is determined by performing an open shell restricted HartreeFock calculation at each internuclear distance for both H2 and H-X-H. The singlet state is described by a GVB(l/Z)-PP wave function. By using these wave functions, the calculation of Anderson's superexchange is possible. The resulting energies from these two wave functions can be used to calculate the magnetic coupling constant J from the Heisenberg-DiracVan Vleck (HDVV) Hamiltonian, eq 1. For the present two state case, the HDVV Hamiltonian leads to the following simple relationship: ET - Es = -2J
(30)
By using eq 30, it is possible to calculate J directly from the results of the two calculations. A negative J value indicates the singlet state is lower in energy, and a positive J value means the triplet state is lower in energy. The orbital energies of the triplet state have been connected to the Anderson superexchange mechanism. Recalling the HTH final approximate expression, eq 15, the behavior of the orbital energy difference el - c2 as a function of internuclear distance and
30
40
50
60
7c
(A) Figure 3. Coupling constant (J) energy and orbital energy difference (AOE) as functions of the internuclear distance in H1. R(H-H)
TABLE II: Bridge s Orbital Energies (OE)
i
OE,X
i
OESX
1.o 1.05 1.10
-0.914 13 -0.88488 -0.848496
1.15 1.20 1.25
-0.804972 -0.770087 -0.696511
charge
0E.w
charge
0E.y
2.25 2.50
-1.301 35 -1.749 23
2.75 3.0
-2.257 77 -2.826 97
~~
as a function of the bridge can be compared to the dependence of the coupling constant on these factors. Results for H2. The results for Hz are as expected (Table I). They provide a reference point for the three center qualitative results. At R = 10.0 A, the two hydrogen atom system behaves as two separated hydrogen atoms, with E(&J = E(&J = -0.499 au. As expected, the symmetric orbital +HI is stabilized as the distance between the two atoms is decreased, while the energy of the antibonding orbital h2 increases. The orbital energy gap, AOE, = OE(&J - OE(&J, is zero at R = 10.0 A and becomes negative with its magnitude increasing smoothly and quickly as R decreases (Figure 3). The singlet-triplet energy gap, or W , behaves in a similar manner as a function of R. However, whereas AOEH2begins its sharp decrease at R r 6.0 A, J does not begin to get more ne ative until R r 4.0 A. In the region of interest, 2.5 IR I4.0 ,the direct interaction between the two hydrogen atoms stabilizes the singlet state relative to the triplet. Results for H-x-H. At R = 10.0 A, the three center, four electron system consists of essentially three separated atoms. The singlet-triplet gap is zero, and the orbitals and 4,,, which correspond to hland hlin the H2 molecule, are degenerate with an energy of -0.499 au. The doubly occupied orbital +,xo is localized on the bridging atom located at the midpoint of the l i e joining the two hydrogen atoms. The energy of this orbital, E(+,xo), ranges from -0.697 au for Z = 2, ( = 1.25, to -2.83 au for Z = 3.0, l = 1.5. Consider first the neutral case for HzHewhere the parameters for the bridge are Z = 2.0, { = 1.0 (helium bridge). On decreasing R from 10.0 A (He-H distance = 5.0 A), both the singlet and triplet states show a rapid increase in energy as R becomes less than 6.0 A (Figure 4). There is no distance where the energy of either spin state is lower than that of the separated atoms-the present qualitative calculations do not predict a stable molecule to be formed from these atoms. For R I4.5 A, the destabilization of the singlet and triplet spin states becomes differential. Adding
x
Three Center, Four Electron Magnetic Interactions
The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6259
T
E
i+ 9
6ool 0 X
400
-
200 0
4.5
4
50 1
Figure 4. Energies of the singlet and triplet states for HzHeas a function
of H-H distance. a helium atom to the midpoint of the H2 molecule does significantly increase the magnitude of the magnetic interaction between the outer two electrons. The orbital energy gap, hE12= E(&,) - E(&), has been proposed as an indicator of the sign and magnitude of J (Figure 5). For Z = 2.0, f = 1.0, the bridging atom interacts with the molecular orbital formed from the symmetric combination of hydrogen atomic orbitals, to form 41s. The interaction is antibonding, and the effect is large enough to push E(dS,)above E(&) for all R I7.0 A. This effect dominates any lowering of E(qQ resulting from a direct bonding interaction between the atomic orbitals HIand H,. The plots of X, as a function of R (Figure 6a,b) shows that the amount of center orbital character built into 4Iis significant. Comparing hE12to MH,, at any given R, hEI2is opposite in sign, and its magnitude is much larger than
b,
5
5.5 6 6.5 R(H-H) A
7
7.5
8
X
Z=2.5 ' - 3
I
t
,,
300
-
100
+
A
. ;
E s
W
9 -100 -
1 I
-300 -JUU
UHz.
The effect of scaling the basis set on the bridging atom while holding the charge constant at Z = 2.0 is to raise E($,x0) from the optimum minimum basis set value of -0.914 au (f = 1.0) to -0.697 au (5 = 1.25). One would intuitively expect the antibonding interaction to increase as E(&O) increases and approaches E(&), and this is found to be true. For a given R, hElz(Figure Sa) and X, (Figure 6a), both smoothly increase as increases. These trends in the orbital energy difference and the degree of mixing of X,with HIand H,are reflected in the singlet-triplet energy gap (Figure 7a). The singlet state becomes lower in energy relative to the triplet as hEI2grows. For example, the graph of J versus R at R = 3.25 A shows a steady decline from -183 to -351 cm-I for fvalues of 1.0 and 1.25. As a reference point, the H2 molecule has a calculated J value of -76 cm-I. By varying the charge of the bridging atom from the neutral value of Z = 2.0 to 3.0, the effect on hEI2of lowering the orbital energy of E(C$,~O) is determined (Figure 5b). Just as bringing E(&xo) up in energy increases hEI2,lowering E(4,xo) reduces the degree of the antibonding interaction in &, and hElzshrinks. For the extreme case of a nuclear charge of Z = 3.0, E(4,xo)is -2.827 au, the coefficient X,(R)(Figure 6b) is much smaller than the value at Z = 2.0, and E(4,xo)is small enough for the direct overlap of H I and H, to dominate for 3.25 IR I6.0 A. The orbital is more stable than +sz over this range, and the orbitals are degenerate for R somewhere between 3.0 and 3.25 A. As is shown in the graphs, the curves for the intermediate charges provide a smooth transition between the two limiting cases. The singlet-triplet splitting (or 2J) as a function of R contains a sign change for the higher nuclear charge (Figure 7b). For Z = 2.5, there exists a region where the triplet is slightly more stable than the singlet. The region expands for Z = 2.75, and when Z
-
Zeta 1.05 Zeta = 1.10 Zeta I 1.15 + Zeta = 1.20 A Zeta = 1.25 0
-500
I 2.5
':
.!
.'
, i
+'
2~2.75
I
f I
3.5
A H2 I
4.5
5.5
6.5
7.5
R (H-H) A
Figwe 5. (a) Orbital energy differences (AOE) for HzHe as a function
of H-H distance for various orbital exponents. (b) Orbital energy differences (AOE) for H2He as a function of H-H distance for various nuclear charges. TABLE IIk Bridge Orbital J h e r i e s
charge
O b
charge
9.0 9.5
-0.127
10.0 10.5
-0.584
0E.x
-1.090 -1.642
= 3.0, J becomes positive by more than 100 cm-I over an even larger interval of R. The sign change of J can be rationalized by recalling HTH's finding^.^ The final approximate equation for a two electron system is
W = 2K,b - hE12/2KIZ
(31) where Kab is the exchange integral in terms of localized orbitals and K12is the exchange integral in terms of and &. As MI2 shrinks, i.e., 4sland qjs2approach degeneracy, the negative (antiferromagnetic) contribution to J vanishes, and the triplet state is favored over the singlet. Results for H-X+ As described in the calculational details section, a p orbital on the bridging atom interacts with the two terminal atoms differently than an s orbital. At T = 10.0 A,the system does resemble the three separated atoms. The entire range of energies covered by the pu orbital (Table 111),using nuclear
6260 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 0.8-r
Hart et al.
i
0.6
dm
-
0.4
-
1.20
-150
7-
E
Zeta-1.0 Zeta-1.05 x Zeta-1.15 + Zeta-1.20 + Zeta-1.25 0
2 -3 -350
0
0.2 -550
0.0 2.5
3.5
4.5
5.5
6.5
7.5
__
0.5
-750
2.50
E
b)
Charge
250
3.00
3.50 4.00 R (H-H) A
4.50
5.00
1 0 0 0
X
2-2.0 2-2.25 2-2.50 2-2.75
+ 2-3.0
R (H-H)
Figure 6. (a) Degree of mixing (&) in H2He as a function of H-H distance for various orbital exponents. (b) Degree of mixing (&) in H2He as a function of H-H distance for various nuclear charges.
charge variation to change the energy, is from -0.127 to -1.64 au. The 2 = 9 case, a minimum basis description of H 2 F , is considered. For both the singlet and triplet states, the plots of the energies of the wave functions versus the internuclear distance possess a slight minimum at a H-H separation between 5.25 and 5.5 A. For distances less than 5.25 A, the energies of both wave functions increase, with the triplet increasing more quickly, leading to a negative J value (Figure 8). The orbital containing the interaction of the bridging p orbital and the hydrogen s orbitals is $h, analogous to dH2in the H2 molecule. The measure of the antibonding character contained in this orbital is represented by A,, and it increases rapidly when R I 6.5 A (Figure 9). In this case, $ is destabilized by this effect in addition to the direct antibon%inginteraction between the two outer centers. The other singly occupied molecular orbital, qi,,, cannot interact with the p orbital, and remains below $h in energy. The result of adding an atom with a pu orbital to the center of the H-H bond is to increase the energy gap (Figure 10) between the two singly occupied orbitals and subsequently give a more negative number for J. Lowering the energy of the bridge orbital by manipulating the fluorine basis set decreases the amount of antibonding interaction in the antisymmetric orbital; Le., A, decreases, and the behavior
I
-750
(A)
2.5
I
3.5 4 R (H-H) A
3
I
I
4.5
5
Figure 7. (a) Coupling constants (J)for H2He as a function of H-H distance for various orbital exponents. (b) Coupling constants (J)for H2He as a function of H-H distance for various nuclear charges.
2oo
1
'0°1
-100
-
7
-200 -300
-400
-500 25
30
35
40
45
50
R (ti-H) (A)
Figure 8. Coupling constants (J) for H I F as a function of H-H distance for various nuclear charges.
The Journal of Physical Chemistry, Vol. 96, NO.15, 1992 6261
Three Center, Four Electron Magnetic Interactions
III. Distance Dependence
OE
By considering a simple three center, four electron system, the overlap dependence of the Anderson3 superexchange coupling constant is found. The physics of this model is the same as for transition metal containing complexes, so the results from an investigation into the distance dependence of the magnetic coupling in this system should offer some insight into the distance dependence of magnetic coupling in general. It will be shown that the energy expression for the coupling constant is nearly linear as a function of the fourth power of the overlap between the bridge atomic orbital and the magnetic atomic orbital. The model used consists of three centers, each atom possessing a single orbital-the one on the left designated as a, the one on the right as 6, and the bridging center as x . From these atomic orbitals, the following molecular orbitals can be defined:
0.5
-
0.4
0.3
dm
100
0.2
0.1
4c =x 00
45
35
25
55
75
65
R (H-H) (A) F v 9. Degree of mixing (;I) in H 2 F as a function of H-H distance
for various nuclear charges.
where S o b = (alb) = (bla). In this molecular orbital basis, the orbital & overlaps with The orbitals @1 and 42are orthogonal, as are 4cand c $ ~Therefore . (112) = (21c) = 0; (llc) = SIC
--
(33)
These orbitals are used in an analysis analogous to HTH.5 The triplet wave function used is
-400-
I
E
= Ab#c'4c#'$l'21 (34) The triplet energy in terms of the molecular orbitals defined above is simply *T
-500-
u w -600-
v
0
'
-700-
ET = [1/(1 - Si,Z)IPhCc + hi1 + h22 + Ji2 - K12 + Jcc + 2J1, - K1, + 2J2, - KzC- SlC[2hlc+ ~ ( c c I ~+ c )2(~1122)(C2112)l - SI,Z[hcc + h22 + J2c11 (35)
10.5 H2
-800-900I
-1ooo! 3
.
!'+',
4
5
.
1, 6
. 7
.
I
0
.
1
9
. I
I
1
0
R (H-H) ( A ) Figure 10. Orbital energy differences (AOE) for H 2 F as a function of
H-H distance for various nuclear charges. of the system tends toward that of H2. The orbital energy difference decreases, approaching the orbital energy difference for H2. The coupling constant J also becomes smaller, and, as a function of internuclear separation, it closely resembles the behavior of the coupling constant for H2. conchrsiaoa By comparing the systems H-&-H and H-X,-H, two major differences are exposed. The most obvious is that the ordering of the symmetric and antisymmetric orbital energies is reversed for the two cases. The antibonding interaction between symmetric combination of the two hydrogen s orbitals with an s orbital on the bridge in H - H e H pushes the symmetric orbital above the antisymmetric orbital in energy. In [H-F-HI-, the molecular orbital derived from the antisymmetric combination of hydrogen s orbitals is raised in energy by its interaction with the bridging atomic p orbital. The second major difference is that while H-X,-H contains interactions of the same type, H-X,-H contains two competing interactions. Depending on the nature of the bridge, the sign of the coupling constant can be either positive or negative when there is only an s orbital on the bridging ligand. However, the p orbital interaction is dominant, and for oxygen or halide bridged transition metal systems, the magnetic behavior will resemble that of [H-F-HI-. The point to be made here is that the qualitative dependence of the coupling constant on the nature of the bridge is small. The energy of the bridge orbitals can be changed substantially without causing a major change in the magnetic interactions of the system.
where hij = (ilhb); Jij= (iibj);and Kij = (ijlij). While the triplet state is assumed to be adequately described by the single determinant, the wave function for the singlet state is determined through configuration interaction between the following two states q s , = Al4c4c4I4II
(36)
qs, = Al4c4c4242l
(37)
The lowest energy solution is found by diagonalizing the resulting 2 X 2 matrix:
The lowest solution to the energy expression is simply expressed as 1 Es = -[(HI1 + H22) - [(HI, - H22I2- 4H12211/21 (39) 2 The matrix elements Hij are expressed in terms of their molecular energetic quantities as HI1
= H / ( l -s,,2)21 x [2h, 2hll Jcc J I I + 4J1, - 2K1, - S1,[4hlC+ ~ ( c c ( ~ c )4(cl)ll)] - Sl:[hcc + hi1 + 251, - 6Klc]] (40)
+ + + +
H22
HI2
= 2hcc + 2h22
+ Jcc + 5 2 2 + 432, - 2K2,
(41)
= [1/(1 - Sl,2)1[&2 - 2SlC(C2112) + Sl,ZK2cl (42)
The expression for the energy difference Es - ET = 2J in this molecular orbital basis is given by eq 43.
6262 The Journal of PhysicaI Chemistry, VoI. 96, No. 15, 1992
25 = 1 / ( 1 -SI,2)(-512 J22)
+ Kl2) + j1( 1 / ( 1
Hart et al.
+
-SI,Z)’JII
-Si,Z)’-2/(1 -S~,Z))hcc+ (1/(1 1 / ( 1 -Si,Z))hii + ( 1 - 1 / ( 1 -S1,2))h22 +
+ (1 + 1/(1
[j ( 1 + 1
1/(1
3000
1 !
SI:)^) - 1/(1 - SI:)]J, +
1/ ( I - S I , ~ ) ~ ( W-IK CI A + 2J2c - K2, Slc/[2(1 - S1,2)2][4hlc+ ~ ( c c I ~+ c )4 ( ~ l l l l )-] Sl,2/[2(1 1 - 2) x Sl,Z)21[hcc+ A l l + Wlc- 6Klcl - p / ( 1 (hcc + J C A + 2 / ( 1 - Si,2)2hii - 2h22 + 1/(1 - SI,Z)’JII+ 1/( 1 - Si,2)2(4J1c- 2K1,) - 4J2, + 2K2, - SI>/(1 SI,Z)2(4hlc+ ~(CCIIC) + 4 ( ~ l ) l l )-Sl,2/(1 ) Si,Z)’(hcc + hi1 + 251, - 6KiC)l2 4 [ 1 / ( 1 - Si,Z)(K12 - ~iC(c2112)+ S I , Z K ~ C ) I ~(43) I~/~
522
0 000
It should be noted that the terms dependent on c disappear if there are no electrons on the center atom, and eq 43 reduces cleanly to eq 10. Terms involving the center atom appear because 41 must be orthogonal to the pair on c. The final expression for the energy difference ET - Es = -2J is preferably described in terms of atomic orbital quantities. The terms in the above expressions are over the molecular orbital basis. A transformation was necessary for each of the terms. For example, K12becomes
0 15
0010
0 005
4
cHIHe> Figure 11. Coupling constant for H2He plotted as a function of the fourth power of the overlap between the hydrogen s orbital and the bridging helium s orbital.
3000
L
Kl2 = [ 1 / ( 4 ( 1 - Sab2))l((a+ b)(a - b)l(a + b)(a - b)) = [1/(4(1 - sab2))](aa- bb - ab
+ balas - bb - ab + ba)
= [ 1 / ( 2 ( 1 - Sab2))l(Jaa - Jab)
(44)
Similar transformations are made for the rest of the terms. Several approximations are also made in order to arrive at the final expression. Following HTH, we assume 12H12l>>lH11-H22l. Terms which depend on powers of SIC of 5 or greater are neglected, along with terms of Sa, of power 3 or greater. (Note: SIC is approximately the same order of magnitude as Sa,, the overlap between atomic orbitals a and x. For identical atoms a, b, and x, S , = Sd2.For x different than a and 6, S , should be of the same order of magnitude as Sob2. Therefore, Sicshould also be of the same order of magnitude as Sa;.) The following mathematical expansion is used to handle expressions with 1 Snterms in the denominator:
+
(45)
where the expansion is limited to ST/, n I4, and Sax”,n I2 as above. After much algebra invoking the above approximations, the energy difference between the singlet and triplet states is obtained:
+
ET - & = -25 = -2Kab 4Sab(UUlUb)- Sa2(Jab+ Jaa) SIc[4(ablax)- 2Sab((ox(bb)+ (aalax))]/21/2 + S,,2[-2(UblXX) - 2(UXlbX) - 2(Ub(UU)- 2Kax Sab(Jaa+ Jab + 2Jax)]+ S1:[4(axlxx) 4- Z(axl6b) + 2(aa(ax)]/21’2+ S1?[-2Jxx - 24Jax - 3Jaa - 35ab1/4 + (HII- Hd2/8H12 (46)
+
The above expression will now be evaluated according to its explicit dependence on SIC by applying the Mulliken approximation and recalling that SI: = Sob.
SI: term: -2Kab
(UUlUb)
= Sab(Joa+ Jab)/2
-
-
SI?
SI> and Sa$i>
E
5 1000
0 0000
0002
0004
4
Since each of the separate terms varies at SI$,the entire Si,“term varies as SI$. The same relationship is found for the rest of the equation: Slcl term:
SlC[4(ablax)- 2Sab((UXlbb)
(UUlUX))]
(abIax) = S a $ ~ c ( J x x + 2Jax + Jaa) (axlbb) = S I c ( J a b + Jab)/2 (aalbx) = s,,(J,b + Jab)/2
-
+
+
/21’2
Sle3
SICand S a b s ~ c SI;
-
sdiC
sicand
+
Si:
(48)
Each of these terms is multiplied by SIC, and the overall dependence is SI?. SIc2term: S1,2[-2(ob(xx)- 2(ax(bx)- 2(ab)aa)- 2Ka, + Sab(Jaa + Jab + W a x ) ]
-
SI:
+ 2Jax + 5&)/4
(ab(aa) = Sab(Jaa + Jab)/2
SI? (47)
0010
Flgme 12. Coupling constant for H 2 Fplotted as a function of the fourth power of the overlap between the hydrogen s orbital and the bridging fluorine p orbital.
(axlbx) = s l > ( J x x
-
0008
0006
(ablxx) = S a d a x
+ 4Sab(UU(Ub)- Sab2(Jab+ Jaa)
Kab = Sab2(Jaa + Jab)/2
2000
-
-
SI;
SI?
Kax = SlC2(Jxx + 2J1c+ Jaa)/4 -.* SI:
(49)
The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6263
Since each of the terms in H I ,- H22 varies as SI?, (HII- Ha)' varies as S I:. The last part of the equation for the triplet-singlet energy difference is the denominator:
+
1/2(Jaa- 5.6) + SlC[(axlbb)-(UUlUX)] Sab[(UXlbb)+ ( U U l U X ) ] + Sl,2[(UXlbX)+ 1/2(Jaa- J d )
8Hl2
+
(56) In eq 56, the leading term contains no explicit overlap dependence, and the rest of the terms are multiplied by very small numbers. Therefore, 8Hl2 is expected to be fairly constant. The tripletsinglet difference can now be written in the following form: ET - Es = -2J = S1:(constant) (57) &XI
Therefore, a plot of J v s SI: should yield a straight line. The overlap is varied by changing the bond lengths. Such plots for two different systems are shown in Figure 1 1 (H-HeH) and Figure 12 ([H-F-HI-). The details of the calculations are described e1~ewhere.I~
Conclusioas Although there are many equations involved, an interesting relationship between the overlap of atomic orbitals and the coupling constant has been found. The magnetic coupling constant J should vary linearly as a function of the fourth power of the overlap between the bridge and magnetic orbitals. This is found to be true for calculated examples. References and Notes (1) Murray, K. S. Coord. Chem. Rev. 1974, 12, 1. (2) (a) Anderson, P. W. Phys. Rev. 1950, 79, 350. (b) Yamashita, J.; Kondo, J. Phys. Rev. 1958, 109, 730. (c) Anderson, P. W. Phys. Rev. 1959, 115, 2. (3) Anderson, P. W. Solid State Phys. 1963, 14, 99. (4) Kahn, 0.;Briat, B. J. Chem. SOC.,Faraday Tram. 2 1976,72, 268. (5) Hay, P. J.; Thibeault, J. C.; Hoffmann, R. J . Am. Chem. Soc. 1975, 97, 4884. (6) Ball, P. W. Coord. Chem. Rev. 1969, 4, 361. (7) Kanamori, J. J. Phys. Chem. Solids 1959, IO, 87. (8) G d e n o u g h , J. B. J. Phys. Chem. Solids 1958,6, 287. (9) Mukherjee, R. N.; Stack, T. D. P.; Holm, R. H. J. Am. Chem. Soc. 1988, 110, 1850. (10) Gorun, S. M.; Lippard, S.J. Inorg. Chem. 1991, 30, 1625. (11) (a) Johnson, K. C.; Severs, A. J. Phys. Rev. B 1974,10, 1027. (b) Shrivastava, K.N.; Jaccarino, V. Phys. Rev. B 1976, 13, 299. (12) Dunning, T. H.; Hay, P. J. In Methods of Electronic Structure Theory; Schaefer, H. F., Ed.; Plenum Press: New York, 1977; Vol. 4, Chapter 1. (13) Huzinaga, L. J . Chem. Phys. 1965, 42, 1293. (14) Hart, J. R.; Rap@, A. K.; Gorun, S. M.;Upton, T. H.J . Phys. Chem., following paper in this issue.
