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Vibronic theory applied to the magnetic properties of tetrahedral mixed-valence clusters
Chemical Physics 179 (1994) 93-104 North-Holland
Vibronic theory applied to the magnetic properties of tetrahedral mixed-valence clusters AlisonJ. Marks and Kosmas School of Chemistry
Prassides
andMolecular Sciences, University of Sussex, Brighton BNI 9QJ. UK
Received 7 June 1993
The influence of vibronic interactions on the magnetic properties of tetrahedral mixed-valence clusters is investigated for a system having one “core” electron at each metal center, and one “excess”, delocalisable electron. The aim of the model is to provide a first step towards an understanding of the magnetic properties of tetranuclear iron-sulfur clusters. The electronic Hamiltonian is based on a generalisation of the Anderson-Hasegawa model Hamiltonian and vibronic interactions are introduced as the interaction of the electronic states with non-totally symmetric combinations of local ligand vibrations. A dynamic vibronic coupling model analogous to that of Piepho, Rrausx and Schatz is developed to obtain the spin-vibronic states of the cluster. In clusters having antiferromagnetic ex&ange interactions, the magnetic moment is found to be significantly influenced by vibronic coupling. When the sign ofthe electron transfer interaction c is negative, vibronic coupling reduces the magnitude of the magnetic moment, but if e > 0 it increases the magnetic moment. When c c 0, the magnitude of the intra-atomic exchange interaction also influences the magnetic moment.
1. Introduction Mixed-valence compounds contain ions of the same element in different oxidation states. Exchangecoupled polynuclear mixed-valence clusters are common in complexes of transition metals [ 11, and studies of biologically important trinuclear and tetranuclear iron-sulfur clusters have revealed some interesting electronic properties [ 1,2-7 1. The clusters can be envisaged as a collection of identical subunits comprising a metal ion and attached ligands, together with one or more “extra” electrons which are distributed amongst the metal sites [ 8,9]. In order to explain the electronic properties of these clusters, the interaction of ligand vibrations with each “extra” electron must be taken into account [ 8,9 1. We have discussed in an earlier paper a semiclassical model to describe the electronic properties of a d’-d’-d’-d* cluster, taking into account exchange interactions, vibronic coupling and electron transfer [ 10 1. The model was derived within the “static” limit of zero nuclear kinetic energy and allows adiabatic potential energy surfaces to be calculated. A similar study has also been carried out by Borshch et al. [ 111.
In this paper, we solve the full “dynamic” (or nonBorn-Oppenheimer) vibronic problem using the same static Hamiltonian, but with the addition of nuclear kinetic energy. The method that we develop to calculate the spin-vibronic states is similar to that derived by Piepho, Krausz and Schatz for the vibronic states of mixed-valence dimers [ 12 1. Once the spinvibronic states have been obtained, a host of electronic, spectroscopic and magnetic properties can be investigated. In this study, a preliminary application of the model to the latter is discussed. Theoretical calculations have shown that vibronic coupling can affect significantly the magnetic properties of mixed-valence dimers [ 8,13- 15 ] and trimers [ 8,161. However, theoretical studies of the magnetic properties of tetranuclear clusters have so far been limited to simple spin models which ignore the effects of vibronic coupling [ 1,17-201. Our aim in this study is to gain insight into the importance of vibronic coupling (J.), intra-atomic (Jo) and interatomic (J) exchange interactions, and electron transfer (c) in determining the magnetic moments of exchange-coupled tetrahedral mixed-valence clusters. We solve the full vibronic problem for various
0301-0104/94/$07.00 Q 1994 Elsevier Science B.V. All rights reserved. SSDI 0301-0104(93)E0343-T
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A.J. Marks, K. Pusides
/ Chemical Physics I 79 (I 994) 93-l 04
values of the parameters 1, e, J,, and J and use the resulting spin-vibronic energy levels to compute the temperature dependence of the effective magnetic moment. The system investigated here is a model for a tetrahedral did’-d’-d* cluster. This is the simplest tetranuclear model having inter-atomic exchange interactions and allows magnetic properties to be studied with the minimum of computational effort. Our hope is that by starting with a simple model we can shed light on the fundamental effects of exchange and vibronic interactions on the magnetic moment. We believe that our model system provides a good starting point for the study of realistic iron-sulfur clusters containing an [Fe&] 3+ unit. These have a d5-d5-d5-d6 configuration comprising four high-spin d5 ion cores (analogous to the four d’ spins in our model) and one extra, delocalisable electron. Ultimately we wish to understand the experimentally measured electronic and magnetic properties of tetranuclear iron-sulfur clusters [ 2 11. These can be investigated by a direct extension of the vibronic model developed in this paper.
2. Model and method
In this study, we consider a four-centre cluster of tetrahedral (Td) symmetry. Each site i (4’=a, b, c, d) has two non-degenerate atomic orbitals, I&) and @I). The orbitals I&) (i=a, b, c, d) are each occupied by one electron and the fifth electron occupies any of the 1q3:) orbitals with equal probability. The orbitals belonging to different sites are assumed to be orthogonal and the extra electron can occupy any of the orbitals I 4: ) . Four equivalent local states I a), Ib) , I c) , I d) are created in which the extra electron is localised at centers a, b, c and d respectively. The four “core” electrons remain fixed, while the extra electron can migrate between sites. We use the Anderson-Hasegawa model Hamiltonian [ 10,22-241 HAHto describe the electronic spectrum of the cluster. For the tetrahedral system considered here, we write
HAH=
tl
El EI
H”H
4
el 4
H&
cl
EI EI ’
(1)
H$ I
where H;, is the Heisenberg Hamiltonian operator for spin states with electron localisation at site i:
-2J,&(i,+&)-2J2$;&.
(2)
& represents the spin operator of the extra electron
(situated at centre i); & 3,, &, & are the operators of the “core” spins associated with local orbitals I q$) , I @j), 1q&) and I qSr)respectively, where i labels the site of electron localisation and j, k, 1 represent the remaining centres a, b, c or d. I is the unit matrix, Jo represents the intra-atomic exchange integral, and J1, J2 are inter-atomic exchange terms arising from d’d’, di-d’ interactions respectively. In our calculations we make the approximation J1 = J2= J. However, we emphasize that differences in J1 and Jz may be important in tetranuclear iron-sulfur clusters [ 201, and that an exploration of this would make a worthwhile study. E is the parameter of inter-centre electron transfer describing the interaction between localised electronic states, and is independent of spin [ 8 1. Typical values for the relative magnitudes of these quantities in transition metal clusters are [ 81 /J/cl N lo-*-lo-’ and IJ,,/~l~1-10’. The matrix HAH (eq. ( 1) ) is written for a basis having a definite local spin and a fixed localisation of the extra electron. By transforming to a spin basis having a definite total spin S and three well-defined intermediate spins for a given localisation of the extra electron, a Hamiltonian H& can be written independently for each S and the sub-matrices of H&, represented by Ht, (i=a, b, c, d) become diagonal matrices whose elements are the eigenvalues of the appropriate Heisenberg Hamiltonian [ 10 1. The spin basis functions that we use are: 1S1., Sk, &cd, s>, I&b, ‘%m&cd, s>, Islo sab, &bd, s>,
ISld,
S,,, S,, S), for the states in which the extra electron s1 is local&d at sites a, b, c and d respectively. The values of the intermediate spins are defined by S, = ]S,+S,-1 1, ee.9]Si-Sj] and S,,= ]S,+S~], srl++-- 1 I, ...) IS,-sk] , where the Sips,, Sk (i, j, k= a, IS*+Sjl,
A.J. Marks, K. Prassides/ ChemicalPhysicsI79 (1994) 93-104
95
Q,=i
b, c or d) represent the spins of the ion core at sites a, b, c and d respectively. Each unit matrix I in eq. ( 1) is replaced in the new basis by the appropriate matrix of transformation coefftcients [lo] and the dimension s of each block of H& is equal to the number of combinations of intermediate spin values that together give S. For the model presented here, the total spins are S= l/2, 3/ 2,5/2 which arise five times, four times and once respectively (see table 1) . Vibronic coupling arises in the cluster when the extra electron interacts with the totally symmetric local displacements Q, (i=a, b, c, d) of the ligands attached to each metal centre [ 8,9]. The matrix elements of the vibronic Hamiltonian H, in the basis of localised states 1i) , i = a, b, c, d are
Q+ transforms as the A, representation of the Td point group and the Qr, Q2, Q3 together transform as the T2 representation. The force constants associated with the symmetric local distortions in different oxidation states are assumed to be equal. This allows the Q+ coordinate to be dropped from the vibronic Hamiltonian since its interaction with each electronic state is identical [ 25 1. In our calculations we use dimensionless symmetrised coordinates
(ilfLlO=~Q,,
1=
(3)
where 1is a vibronic coupling parameter and is inde pendent of spin [ 8 1. We introduce symmetry adapted ligand displacements coordinates:
Q+ =t(Qa+Qb+Qc+Qd
3
Q,=t(Qa-Q,-Qi+Qd
7
(4) (5)
Table 1 Spin basis states used in the model. Subscript label 1 represents the extra electron and i the site of localisation (ka, b, c or d). St, is the local spin of the site i containing the extra electron. Basis states having St, = 0 and S,, = 1 are non-Hund states and Hund states respectively. S,= Isj+skl, Is,+a- 11and S’= 15”+sIl, I S#+q- 1 I are the intermediate spins of the remaining sites (_j, k, La, b, c or d), where s,, sk, s, are the spins of the electrons in core orbitals I@j>, Idk) , I&) . The total spin S of the cluster takes thevahtesS=IS,,+S,~l, IS,,+&,-ll,..., IS,,-S’I.Theintermediate spin combinations giving each S are numbered by k in column one. k
S*,
S,
S,
S
1
1
1
3/2
512
1 2 3 4
1 1 1 0
1 0 1 1
l/2 l/2 312 312
312 312 3/2 3/2
1 2 3 4 5
1 1 1 0 0
0 1 1 0 1
l/2 l/2 312 l/2 l/2
l/2 l/2 l/2 l/2 l/2
(6)
3
Q3=1(Qa-Qb+Qc-Q,>.
q,=2n(v/k)‘/2Q,,
(7)
i= 1,2, 3,
(8)
and a dimensionless vibronic coupling parameter 1, (kv’)
-‘/2(1/4x)
)
(9)
where v= (21~)-‘& is the frequency of the totally symmetric local ligand vibration and k is the corresponding force constant. The local ligand vibrations are assumed to be harmonic, and the Hamiltonian matrix for the vibrational potential energy in the local&d electronic basis is given by (in units of k v) Hvib=t
1
I= 1,2,3
a?1 7
(10)
where I is the unit matrix. The vibronic Hamiltonian matrix elements expressed in units of h v are (al&
la> =A6
+h2+h3,
(11)
(blH,,Ib)=-Iql+~q,-~q,,
Ic> = -&A
I& =h
(12)
-&?2
-k,
+k3
-b
7
,
(13) (14)
where I i) , i = a, b, c, d, are the localised basis states, and the matrix elements are independent of spin. We write the Hamiltonian matrix for each total spin S as HS= H& + Hvib+ He, ,
(15)
where HL is the Anderson-Hasegawa Hamiltonian matrix for a system having spin S and is expressed in units of kv. A derivation of the matrix elements of HS is given in the Appendix for the case S= l/2. The model described above is “static”, in the sense that the nuclear kinetic energy is not included in the Hamiltonian. Solution of the above problem leads to
A.J. Marks, K. Prassides /Chemical Physics 179 (1994) 93-104
96
a set of adiabatic potential energy surfaces, and nuclear motion is confined to a single potential surface. We wish to solve the full dynamical vibronic problem and therefore must include the nuclear kinetic energy. This has the effect of coupling the nuclear and electronic motion so that the concept of adiabatic potential energy surfaces is no longer valid. The system is described instead by a spectrum of spin-vibronic eigenvalues and wavefunctions. We write the total Hamiltonian for a cluster having spin S as z?;=zP+
&(g)
,
(16)
where AS is the operator of the “static” Hamiltonian described earlier and FN is the nuclear kinetic energy operator for the ligand distortions. The Schrodinger equation to be solved for a system of spin S is I?;@f=E;@$,
(17)
where the 0: and Ef are spin-vibronic eigenfiurctions and eigenvalues respectively for a cluster of spin S. We write the vibronic wavefunctions as @:= ,ck U(k) )x,k,s(!z) 3
(18)
where the Ii(k)) are the localised electronic basis functions described earlier (j= a, b, c, d) and k labels a specific set of values for the intermediate spins. When S= 512 there is only one combination of intermediate spins (k= 1), and for S= 3/2, S= l/2, there are four and five spin combinations respectively (table 1) . The aAv (4) are nuclear wavefunctions, each of which is written as a sum of products of three onedimensional harmonic oscillator wavefunctions [ 26 ] : xi,kJa>
(19) where nl, n2, n3 (ni=O, 1, 2, .... CD) are the vibrational quantum numbers associated with nuclear coordinates ql, q2;q3 respectively. In our model, we limit the size of the three-dimensional vibrational basis to n,andincludeallml,n2,n3suchthatn1+n2+n3~n. Multiplying the vibronic Schr&linger equation by each of the lj( k) ) in turn and integrating over the electronic coordinates, leads to the secular equations
(one equation for each value ofj and k) : j&, [ O’(k) IAS+ TN If (k’ 1)
v= 1, 2, ...) co .
(20)
The electronic basis functions are independent of q and thus (j(k) I rfNIj’ (k’ ) ) ~0, where j#j’ and k#k’. We next substitute the expansion for the nuclear wavefunctions into the above equations. This leads to a large secular determinant (or “dynamic matrix” [ 9,12,26 ] ) which must be diagonalised to obtain the eigenvector coefficients C;$“’ and the eigenvalues E:. The dynamic matrix for S= l/2 is derived as an example in the Appendix. The dimension Ns of the matrix for a total spin S is Ns=nas(n+1)(n+2)(n+3)/6,
(21)
where ilp= 4 is the number of electronic states, s is the number of intermediate spin combinations for a given total spin (s=l, 4, 5 for S=5/2, 3/2, l/2 respeo tively ) and rais the maximum number of vibrational quanta. The dynamic matrix is sparse, real and symmetric and we have used the numerical sparse matrix diagonalisation method of subspace iteration [ 27 ] to obtain the required spin-vibronic eigenvalues for each of the possible spin states of the cluster. These are then used to calculate the effective magnetic moment as a function of temperature, for model systems both with and without vibronic coupling.
3. Results From the spin-vibronic state E:, we calculate the effective magnetic moment p as a function of temperature, using the relationship [ 28 ] p2_g2B2
Gv W+ 1) W+ 1) ew( -EST) &
(2S+ 1) exp( -E:/kT)
’
(22) where g is the Lam% factor, and fi is the Bohr magneton. The summation is over all possible spin states S= l/2,3/2,5/2 and over all the eigenvalues in each of the spin manifolds.
A.J. Marks, R Prassides/ ChemicalPhysicsI79 (1994) 93-104
The exchange interaction parameter J can be positive or negative, corresponding to ferromagnetic and antiferromagnetic exchange respectively. The parameter of inter-centre interaction e can also take positive or negative values, depending on the nature of the orbitals occupied by the five electrons, and on the nature of the bridging ligands through which the extra electron is transferred [ 25,291. The intra-atomic exchange Jo is always positive and is usually large compared with e and J. Jo leads to the enhanced stability of the “Hund states” which have a high local spin of si+ l/2 at the centre of electron localisation i. The spin combinations having a local spin of s, - l/2 are known as non-Hund states [ 8 1. We discuss below the influence of the vibronic interaction 1 on the magnetic moment of clusters having e > 0 or e < 0 and either ferromagnetic or antiferromagnetic exchange interactions. We also consider the contribution of non-Hund states to the magnetic moment, by varying the magnitude of the on-site exchange Jo. To give a physical perspective in terms of electronic structure, we note that systems having no vibronic coupling (A= 0) are those in which the “extra” electron is fully delocalised amongst the four metal centres. Such systems can be represented by a much simplified Hamiltonian, comprising a “double-exchange” or delocalisation term involving t, and a spin-dependent exchange interaction term [ 301. In contrast, clusters with very strong vibronic coupling ( ]e/42*] eO.16) [25] have an “extra” electron which is essentially localised at one of the metal sites. The figures discussed below show the magnetic moment as a function of temperature T expressed as the dimensionless quantity kT/hv, where hv is the frequency of the local, symmetric ligand vibration and k is Boltzmann’s constant. For a typical vibrational frequency of N 500 cm-‘, room temperature corresponds to kT/h v - 0.4. In our calculations we have used a vibrational basis of n=25. This is sufficient for convergence of the magnetic moment in the range of vibronic coupling strengths examined. The size of basis required for convergence increases with vibronic coupling strength. We found that a smaller vibrational basis (e.g., n = 15 ) gives the same qualitative behaviour of the magnetic moment, but overestimates its magnitude with respect to the converged value.
97
3.1. E-CO,J
When no vibronic interaction is present (A= 0 ) , the spin of the ground state can take any of the values S= l/2, S= 312, S= 512, depending on the relative magnitudes of J and e [ IO]. A large antiferromagnetic interaction favours a low-spin ground state, whereas a large electron transfer interaction 1c 1 favours a high-spin ground state. In fg la we show an example of the temperature dependence of the effective magnetic moment of a cluster which has an S= 3/2 ground state when 1= 0. When there is no vibronic coupling, the magnetic moment shows little variation as a function of temperature. The effect of 1 is to reduce the spin of the ground state to S= l/2, thus reducing the low-temperature value of the magnetic moment. At higher temperatures the magnetic moment remains lower than that of the system with no vibronic coupling. That is, I strengthens the antiferromagnetic properties of the cluster, and vibronic states of low spin are stabilised more than those of higher spin. Thus, if the vibronic coupling is strong enough, a reduction in the ground state spin and the magnetic moment can be achieved. For weaker vibronic coupling there is no change in the ground state spin and the magnetic moment is reduced to a lesser extent. We note that the magnetic moment illustrated in lig. 1a for the case of strong vibronic coupling corresponds to a situation in which the extra electron is highly localised ( I s/4A2I =0.03). In this case energy minima obtained from potential-energy surfaces within the “static” model [ 10,251 give an almost identical description of the magnetic moment curve. This suggests that a semi-classical model comprising a modified Heisenberg Hamiltonian similar to that described by Blondin et al. [ 30 ] for trimeric clusters, will also give a good description of the magnetic moment in the limit of very strong vibronic coupling. 3.2. eO The cluster in this case has a ferromagnetic, highspin (S=5/2) ground state, both with and without vibronic coupling. The main effect of A is to cause a slight reduction in the magnetic moment at high temperatures. That is, states having S= l/2 and S= 3/2
A.J. Marks, K. Prassides/ ChemicalPhysicsI79 (1994) 93-104
98
b
l.SC
0.4c
c.30
0 60
0.80
I
5.20-L 0 00
1 OC
0.20
kT
0.40
0.60
0.80
1
kT
4.50
-----__-_-_____d_ l/r---
g
iii,
4 00./
?i h
\
3 oo3.50.
& z
2 53.
%
2.00-
1.50+
0.40
o.co
0 20
0 40
0.60
0.80
4
1.00
kT
kT
4.00-
0)
3 503.002.502.001.5OT 0.00
0.20
0.40 kT
0.60
0.80
1 1.00
0.4c kT
Fig. 1. Temperature dependence of the effective magnek moment for a tetrahedral mixed-valence cluster, showing the influence of vibronic coupling L The magnetic moment is expressed in Bohr magneton (B.M.) units and the effective temperature kTis expressed in units of hv, the symmetricligand vibration frequency. Jo, Jand t are also given in units of hv. A vibrational basis of dimension n= 25 is used. (a) C-CO,JO,JO, JO. 6=5.0,&=50.0, J= +0.5. (i) rl=O. (ii)1=5.0.
A.J. Marks, K. Prassides/ ChemicalPhysics179 (I 994) 93-l 04
are preferentially reduced in energy by 1, and become populated as the temperature increases. An example is shown in fig. lb and shows that the qualitative behaviour of the magnetic moment does not change when vibronic coupling is included. In both cases fl decreases slowly as the S= l/2 and S= 31 2 states become populated. 3.3. e>O, J-CO When there is no vibronic interaction the ground state is antiferromagnetic (SE l/2), irrespective of the relative magnitudes of Eand J [ IO]. The effect of vibronic coupling on the magnetic moment is shown in fig. lc for a typical case where e = 5.0 and J= - 0.1. The magnetic moment is increased slightly by 1 but the ground state remains S= l/2 and the curves show similar qualitative behaviour. Thus, in this case J stabiliscs the S= l/2 states substantially more than the S= 3/2 and S= 5/ 2 states. Addition of vibronic coupling results in a slight enhancement of the magnetic moment at high temperatures, with the ground state remaining S= 1 / 2. The observed increase in magnetic moment shows that states of spin S= 3/2 and S= 5/2 are stabilised more by vibronic coupling than those of spin S= 1 / 2. We note that this effect is the opposite of that observed when E< 0. For the case of weak antiferromagnetic coupling (or small IJ/C) ), a change of ground state spin from S= l/2 to S= 3/2 or S= 5/2 can arise in the presence of vibronic coupling. This gives a large increase in the low temperature magnetic moment, as shown in figs. 1d and 1e for the stabilisation of an S= 3/2 and S= 5/2 ground state respectively. Vibronic coupling has the greatest effect on the magnetic moment (inducing a change in ground state spin from S= l/2 to S= 5/2) when 1J/C 1 is very small (fig. le). The change in ground state spin induced by vibionic coupling when the exchange interaction J is weak can be rational&d as follows: If there is no inter-atomic exchange interaction or vibronic coupling, the low-energy (Hund) states of different spin are degenerate. On adding a very small antiferromagnetic exchange interaction (J-CO), an S= l/2 ground state results. However, because the exchange interaction is small, the low-energy states of different spin lie very close together. Introduction of vibronic
99
coupling (which in this case stabilises high-spin states more than low-spin states), then results in the lowest S= 312 or S= 512 states being reduced in energy below the S= l/2 electronic ground state. When k0 and JO, J>O The cluster has a ferromagnetic (S= 5/2) ground state, irrespective of the magnitudes of Eand 1. It can be seen from fg If that when 1= 0 there is very little variation in the magnetic moment as a function of T. S= 312 and S= l/2 states become populated at high temperatures, leading to a small decrease in the magnetic moment. Vibronic coupling has very little effect on the magnetic moment, although at high temperatures it is slightly enhanced with respect to the ,J= 0 case. Thus, when E> 0, vibronic coupling appears to stabilisc the low-lying S= 5/2 states more than states of low spin. 3.5. The contribution of non-Hundstates to the magnetic moment Previous vibronic models for calculating the effeo tive magnetic moments of d’-d2 mixed-valence dimers and d’-d’-d2 trimers have excluded the nonHund basis states on the grounds that they are high in energy and make little contribution to the magnetic moment [ 8,14,16 1. We examine here the possibility that non-Hund basis states are important in determining the magnetic moment of tetrahedral mixed-valence clusters. The magnitude of the intraatomic exchange interaction Jo determines the stability of the non-Hund basis states which exist for S= I/ 2 and S= 3 /2 (see table 1) . We have investigated the effect of a relatively large IJo/e ) (which destabilises the non-Hund states) and a small IJo/e I, on the magnetic moment. We first consider a cluster having e < 0 and antifer-
100
A. J. Ma&,
K. Prawides / Chemical Physics I 79 (I 994) 93-104
romagnetic exchange J-z 0. An example of the effect on the magnetic moment is given in fig. 2a and shows that when I = 0 the magnetic moments differ significantly. The system with small Jo in general has a smaller magnetic moment than that with large Jo although at very low temperatures the magnetic moments are identical, corresponding to an S=5/2 ground state. The magnetic moment of the large Jo cluster decreases slowly as a function of T whereas that of the small Jo cluster decreases sharply before levelling off. The differences between the two magnetic moments in fig. 2a (curves (i) and (ii) with 1= 0) can be explained in terms of the relative stabilities of the
S= l/2 and S= 3/2 non-Hund states. When IJo/t I is small, the non-Hund basis states mix into the lowest S= l/2 and S= 3/2 vibronic levels and stabilise them, thus reducing the magnetic moment. These low-spin states are also stabilised by the antiferromagnetic exchange interaction J. Similar arguments explain the difference between the magnetic moments for large and small Jo when 13# 0 (fig. 2a, curves (iii) and (iv) ) . However, since vibronic coupling tends to reduce the magnetic moment for a given e < 0, variations in Jo have less effect on the magnetic moment and the curves lie closer together. In clusters with ferromagnetic exchange and e < 0, the magnetic moment is slightly smaller for small Jo than for large Jo. An example is shown in fg 2b for a system with no vibronic interaction. As in the previous example, non-Hund states stabilise the lowest S= l/2 and S= 3/2 vibronic levels when J,-,is small, leading to a reduction in the magnetic moment. Finally, we note that when e > 0, the magnetic moments for large and small J,-,are identical over the range of temperatures studied. This implies that in contrast to the situation when E< 0, non-Hund basis states do not mix into the low-energy S= l/2 and S= 3/2 vibronic states.
4. Discussion kT 6.00,
I
b
0 20
0.40
0.60
0 80
1 00
kT
As fig, 1, illustrating the influence of non-Hund states on the magnetic moment. (a) tO. E=-3.0, J= -0.1. (i) 1=0, Jo=200. (ii) Fig.
2.
1=0, Jp3.
We have developed a dynamic spin-vibronic coupling model and illustrated by application to a simple system that vibronic interactions can play an important role in determining the magnetic properties of tetrahedral mixed-valence clusters. The most significant effect of vibronic coupling on the magnetic moment arises in the presence of antiferromagnetic exchange interactions. In particular, when the inter-centre interaction e is negative, vibronic coupling decreases the magnetic moment and if strong enough, causes a change from a high-spin to a lower-spin ground state with a corresponding decrease in the low-temperature magnetic moment. In contrast, when e > 0, vibronic coupling increases the magnetic moment and when the antiferromagnetic interaction is very small, can induce a change from an S= l/2 to an S= 3/2 or S= 5/2 ground state. When E< 0, the intra-atomic exchange interaction
A.J. Marks, K. Prmides
/ Chemical Physics I79 (1994) 93-104
J,, also influences the magnetic moment. If the ratio 1Jo/c 1- 1 (i.e. Jo is relatively small), then the S= 1/ 2 and S=3/2 non-Hund basis states mix into the lowest vibronic levels and reduce the magnetic moment. Thus, when E< 0, the greatest reduction in the magnetic moment arises when the intra-atomic exchange is small and the vibronic interaction is strong. As noted in section 3.1, systems having a strongly local&d “extra” electron can be described using the “static” model, suggesting that a simple modified Heisenberg Hamiltonian may be sufficient to describe the properties of clusters with strong vibronic coupling. This observation is consistent with the conclusions of Blondin et al. [ 301 who have developed an effective Heisenberg model for describing strongly vibronically coupled mixed-valence trimers. The usual interpretation of experimentally measured magnetic moments in mixed-valence clusters is based on the assumption of a simple spin Hamiltonian comprising an exchange interaction J and a term in e allowing electron delocalisation [ 1,17- 19,2 11. An important message from our results is that the interpretation of the magnetic moment in this way can lead to incorrect estimates of inter-atomic exchange and electron transfer interactions. The inclusion of vibronic coupling, and to a lesser extent the intra-atomic exchange interaction, is necessary for the correct evaluation of e and J in antiferromagnetic systems. This may be achieved through the use of the dynamical model described here, or, when vibronic coupling is strong, with a simpler semi-classical model [ 10,301. The magnitude of the vibronic interaction 1 in tetranuclear mixed-valence clusters can in principle be extracted from intervalence charge-transfer spectra [ 8,9,12 1. However, the optical spectra are extremely complicated and further experiments are required
101
before our vibronic coupling model can be used to simulate the intervalence bands of exchange-coupled clusters. Nevertheless, we believe that this preliminary study provides some new insights into the magnetic properties of vibronically coupled mixed-valence clusters, and we aim to extend the model so that realistic systems such as tetranuclear iron-sulfur clusters can be investigated. Finally, we emphasise that although the current study focuses on the calculation of magnetic properties and utilises only the spin-vibronic eigenvalues, the dynamic spin-vibronic coupling model can also be used to investigate electronic properties. For example, the spin-vibronic eigenvector coefficients can be used to compute probability distributions [ 91 describing the delocalisation properties of the “extra” electron as a function of vibronic coupling strength [ 10,251. The calculations in this paper pertain to systems in which the “extra” electron is either fully delocalised or localised (i.e., weak and strong vibronic coupling respectively). However, the range of vibronic coupling strengths required for an investigation of interesting delocalisation patterns is an intermediate regime at the borderline between electron localisation and full delocalisation [ 10 1. Work in this direction complementing our earlier study [lo] is under way.
Acknowledgements This work is supported by the Science and Engineering Research Council and NATO travel grant RG0146/87. We thank Professor Paul Schatz and Professor Susan Piepho for useful discussions and the University of Virginia for provision of computing facilities.
Appendix. Derivation of the S= l/2 dynamic matrix
We illustrate first the derivation of the “static” matrix HS in the chosen electronic and spin basis for the case S=1/2.Thisisa20x20matrixandhastheform (seeeq. (15)): H”2=Hj$+Htib+H
en*
Using the spin basis functions given in table 1, we obtain
(23)
102
A.J. Marks, K. Prassides/ ChemicalPhysics179 (1994) 93-104
(24)
(25)
is the Heisenberg Hamiltonian matrix for electron localisation at site i, and J (I), J@) are the exchange energies for Hund states and non-Hund states respectively (obtained from eq. ( 2 ) and table 1) J”‘=-i(J,,-75))
(26)
J”‘=$(J,,+J).
(27)
Each matrix T”=Tj’ is a matrix of transformation coefficients relating the different spin representations for electron localisation at sites i and j. There are five S= l/2 intermediate spin combinations k arising for each localisation of the extra electron (table 1) , so each T’j has dimension 5 X 5. The elements are of the form: T&=(i(k)lj(k’)),
(28)
where k, k’ refer to the S= l/2 basis functions given in table 1. Each of these elements is a transformation coefficient between two coupling schemes (i and j) for five electrons. The vibronic matrix H,, is given by
(alfLla)l Hen=
0 0 0
0
0
0
0
0
(ClKnlC)~ 0
(29)
0 (dlfLld>l
where
i=l
[~~(qi)+fq:l+(jlH,lj)+J’e’-E,
>
&,k,v-e c
f#Jk”l
i
T&G,~,~=O,
where j=a, b, c or d is the centre of electron local&-&on and k= 1,2, 3,4 or 5. The exchange energy Jce)= J(l) (eq. (26) ) when k= 1, 2, 3 (Hund basis states) and J @)=J(‘) (eq. (27)) when k=4, 5 (non-Hund basis states). We substitute the expansion (31)
A.J.
103
Marks,K. Prassides / Chemical Physics 179 (I 994) 93-104
into the above secular equations. We multiply each equation by the product of harmonic oscillator functions x,,,, ( q1)x,, ( q2)xms( q3) (for all allowed combinations of ml, mz, m3 in the vibrational basis), and integrate over the nuclear coordinates ql, q2, q3. The non-zero terms can be identified using the following properties of harmonic oscillator wavefunctions (expressed in units of hv): (32)
tx,I~~(4,)+14:lX~)=n,+1/2 and
(33)
This leads to the following four sets of secular equauons (one for each of the local&d electronic basis functions withj=a, b, c, d, respectively),
where Hg,q!,=(m,+mz+m3+3/2+J(“)-E,)6
with J(‘)=J(l) H!%=&/~&n,x,-I
m,,n*6m2,nt6?nwl,9
when k= 1,2, 3 and Jtc)=J(*)
(38)
when k=4, 5.
+&&,,+~l&,&m,.r
(P, 4, r)=l, 2>3;q>MP)
.
(39)
In each of the four sets of eqs. (34)-( 37), k= 1,2,3,4 or 5 and the vibrational quantum numbers are chosen such that m, + m2 + m3 < II where n is the selected basis size. A total of IV,,equations results (eq. (2 1) ) , and the dynamic matrix can be diagonalised to obtain the eigenvalues Es and the corresponding eigenvector coefficients C$JY3. The procedure for deriving the S= 3/2 and S= S/2 secular equations is identical to that given here for s= l/2. Iteferences
[l] G. Blondin and J.-J. Girerd, Chem. Rev. 90 (1990) 1359, and references therein. [ 2 ] V. Papaeibymiou, J.-J. Girerd, I. Moura, J.J.G. Moura and E. Mtlnck, J. Am. Chem. See. 109 (1987) 4703. [ 31 J.A. Weigel, K.K.P. Srivastava, E.P. Day and E. Mtlnek, J. Am. Chem. See. 112 (1990) 8015. [4] D. Coucouvanis, S.A. Al-Ahmad, A. Salifoglou, V. Papaefthymiou, A. Kostikas and A. Simopoulos, J. Am. Chem.Soe. 114 (1992) 2472. [ 5 ] V. Papaeibymiou, M.M. Millar and E. Milnck, Inorg. Chem. 25 (1986) 3010. [6] G. Rius and B. Lamotte, J. Am. Chem. See. 111 ( 1989) 2464.
[7]S.A.BorshehandL.F.Chibotaru,Chem.Phys. 135 (1989) 375. [ 81 I.B. Bersuker and S.A. Borshch, Advan. Chem. Phys. 81 ( 1992) 704, and references therein. [ 91 KY. Wang and P.N. Sehatz, Progr. Inorg. Chem. 28 ( 198 1) 369. [lo] A.J. Marks and K. Prassides, J. Chem. Phys. 98 (1993) 4805. [ 111 SA. Borsheh, E.L. Bominaar and J.-J. Girerd, New J. Chem. 17 (1993) 39. [ 121 S.B. Piepho, E.R. Krausz and P.N. S&ate, J. Am. Chem. See. 100 (1978) 2996. [13] J.-J. Girerd, J. Chem. Phys. 79 (1983) 1766. [ 141 S.A. Borsheh, I.N. Kotov and LB. Bersuker, Chem. Phys. Letters 111 (1984) 264. [15] B.S. Tsulcerblat, AV. Palii, H.M. Kishinevsky, V.Ya. Gamurar and AS. Berengol’ts, Mol. Phys. 76 ( 1992) 1103.
104
A.J. Marks, K. Prassides/ ChemicalPhysics179 (I 994) 93- IO4
[ 161 LB. Bersuker, S.A. Borshch and LN. Kotov, Rev. Roumaine Chem. 32 (1987) 1075. [ 171 L. Noodleman, Inorg. Chem. 27 (1988) 3677, and references therein. [ 181 L. Noodleman, Inorg. Chem. 30 (1991) 246. [ 191 L. Noodleman, Inorg. Chem. 30 (1991) 256. [20] M. Belinskii, Chem. Phys. 172 (1993) 213. [21] J. Jordanov, E.K.H. Roth, P.H. Fries and L. Noodleman, Inorg. Chem. 29 (1990) 4288. [22] P.W. Anderson and H. Hasegawa, Phys. Rev. 100 (1955) 675.
[23] .%A.Borshch, Soviet Phys. Solid State 29 (1987) 896. [24] M.I. Belinskii, Soviet Phys. Solid State 27 (1985) 1057. [25] A.J. Marks and K. Prassides, New J. Chem. 17 (1993) 59. [ 261 K. Prassides and P.N. Schatz, J. Phys. Chem. 93 (1989) 83. [ 271 B.N. Parlett, The symmetric eigenvalue problem (PrenticeHall, Englewood Cliffs, 1980). [28] A. Earnshaw, Introduction to magnetochemistry (Academic Press, London, 1968) [29] J.P. Launay and F. Babonneau, Chem. Phys. 67 (1982) 295. [ 301 G. Blondin, S. Borshch and J.-J. Girerd, Comments Inorg. Chem. 12 (1992) 315.
Vibronic theory applied to the magnetic properties of tetrahedral mixed-valence clusters AlisonJ. Marks and Kosmas School of Chemistry
Prassides
andMolecular Sciences, University of Sussex, Brighton BNI 9QJ. UK
Received 7 June 1993
The influence of vibronic interactions on the magnetic properties of tetrahedral mixed-valence clusters is investigated for a system having one “core” electron at each metal center, and one “excess”, delocalisable electron. The aim of the model is to provide a first step towards an understanding of the magnetic properties of tetranuclear iron-sulfur clusters. The electronic Hamiltonian is based on a generalisation of the Anderson-Hasegawa model Hamiltonian and vibronic interactions are introduced as the interaction of the electronic states with non-totally symmetric combinations of local ligand vibrations. A dynamic vibronic coupling model analogous to that of Piepho, Rrausx and Schatz is developed to obtain the spin-vibronic states of the cluster. In clusters having antiferromagnetic ex&ange interactions, the magnetic moment is found to be significantly influenced by vibronic coupling. When the sign ofthe electron transfer interaction c is negative, vibronic coupling reduces the magnitude of the magnetic moment, but if e > 0 it increases the magnetic moment. When c c 0, the magnitude of the intra-atomic exchange interaction also influences the magnetic moment.
1. Introduction Mixed-valence compounds contain ions of the same element in different oxidation states. Exchangecoupled polynuclear mixed-valence clusters are common in complexes of transition metals [ 11, and studies of biologically important trinuclear and tetranuclear iron-sulfur clusters have revealed some interesting electronic properties [ 1,2-7 1. The clusters can be envisaged as a collection of identical subunits comprising a metal ion and attached ligands, together with one or more “extra” electrons which are distributed amongst the metal sites [ 8,9]. In order to explain the electronic properties of these clusters, the interaction of ligand vibrations with each “extra” electron must be taken into account [ 8,9 1. We have discussed in an earlier paper a semiclassical model to describe the electronic properties of a d’-d’-d’-d* cluster, taking into account exchange interactions, vibronic coupling and electron transfer [ 10 1. The model was derived within the “static” limit of zero nuclear kinetic energy and allows adiabatic potential energy surfaces to be calculated. A similar study has also been carried out by Borshch et al. [ 111.
In this paper, we solve the full “dynamic” (or nonBorn-Oppenheimer) vibronic problem using the same static Hamiltonian, but with the addition of nuclear kinetic energy. The method that we develop to calculate the spin-vibronic states is similar to that derived by Piepho, Krausz and Schatz for the vibronic states of mixed-valence dimers [ 12 1. Once the spinvibronic states have been obtained, a host of electronic, spectroscopic and magnetic properties can be investigated. In this study, a preliminary application of the model to the latter is discussed. Theoretical calculations have shown that vibronic coupling can affect significantly the magnetic properties of mixed-valence dimers [ 8,13- 15 ] and trimers [ 8,161. However, theoretical studies of the magnetic properties of tetranuclear clusters have so far been limited to simple spin models which ignore the effects of vibronic coupling [ 1,17-201. Our aim in this study is to gain insight into the importance of vibronic coupling (J.), intra-atomic (Jo) and interatomic (J) exchange interactions, and electron transfer (c) in determining the magnetic moments of exchange-coupled tetrahedral mixed-valence clusters. We solve the full vibronic problem for various
0301-0104/94/$07.00 Q 1994 Elsevier Science B.V. All rights reserved. SSDI 0301-0104(93)E0343-T
94
A.J. Marks, K. Pusides
/ Chemical Physics I 79 (I 994) 93-l 04
values of the parameters 1, e, J,, and J and use the resulting spin-vibronic energy levels to compute the temperature dependence of the effective magnetic moment. The system investigated here is a model for a tetrahedral did’-d’-d* cluster. This is the simplest tetranuclear model having inter-atomic exchange interactions and allows magnetic properties to be studied with the minimum of computational effort. Our hope is that by starting with a simple model we can shed light on the fundamental effects of exchange and vibronic interactions on the magnetic moment. We believe that our model system provides a good starting point for the study of realistic iron-sulfur clusters containing an [Fe&] 3+ unit. These have a d5-d5-d5-d6 configuration comprising four high-spin d5 ion cores (analogous to the four d’ spins in our model) and one extra, delocalisable electron. Ultimately we wish to understand the experimentally measured electronic and magnetic properties of tetranuclear iron-sulfur clusters [ 2 11. These can be investigated by a direct extension of the vibronic model developed in this paper.
2. Model and method
In this study, we consider a four-centre cluster of tetrahedral (Td) symmetry. Each site i (4’=a, b, c, d) has two non-degenerate atomic orbitals, I&) and @I). The orbitals I&) (i=a, b, c, d) are each occupied by one electron and the fifth electron occupies any of the 1q3:) orbitals with equal probability. The orbitals belonging to different sites are assumed to be orthogonal and the extra electron can occupy any of the orbitals I 4: ) . Four equivalent local states I a), Ib) , I c) , I d) are created in which the extra electron is localised at centers a, b, c and d respectively. The four “core” electrons remain fixed, while the extra electron can migrate between sites. We use the Anderson-Hasegawa model Hamiltonian [ 10,22-241 HAHto describe the electronic spectrum of the cluster. For the tetrahedral system considered here, we write
HAH=
tl
El EI
H”H
4
el 4
H&
cl
EI EI ’
(1)
H$ I
where H;, is the Heisenberg Hamiltonian operator for spin states with electron localisation at site i:
-2J,&(i,+&)-2J2$;&.
(2)
& represents the spin operator of the extra electron
(situated at centre i); & 3,, &, & are the operators of the “core” spins associated with local orbitals I q$) , I @j), 1q&) and I qSr)respectively, where i labels the site of electron localisation and j, k, 1 represent the remaining centres a, b, c or d. I is the unit matrix, Jo represents the intra-atomic exchange integral, and J1, J2 are inter-atomic exchange terms arising from d’d’, di-d’ interactions respectively. In our calculations we make the approximation J1 = J2= J. However, we emphasize that differences in J1 and Jz may be important in tetranuclear iron-sulfur clusters [ 201, and that an exploration of this would make a worthwhile study. E is the parameter of inter-centre electron transfer describing the interaction between localised electronic states, and is independent of spin [ 8 1. Typical values for the relative magnitudes of these quantities in transition metal clusters are [ 81 /J/cl N lo-*-lo-’ and IJ,,/~l~1-10’. The matrix HAH (eq. ( 1) ) is written for a basis having a definite local spin and a fixed localisation of the extra electron. By transforming to a spin basis having a definite total spin S and three well-defined intermediate spins for a given localisation of the extra electron, a Hamiltonian H& can be written independently for each S and the sub-matrices of H&, represented by Ht, (i=a, b, c, d) become diagonal matrices whose elements are the eigenvalues of the appropriate Heisenberg Hamiltonian [ 10 1. The spin basis functions that we use are: 1S1., Sk, &cd, s>, I&b, ‘%m&cd, s>, Islo sab, &bd, s>,
ISld,
S,,, S,, S), for the states in which the extra electron s1 is local&d at sites a, b, c and d respectively. The values of the intermediate spins are defined by S, = ]S,+S,-1 1, ee.9]Si-Sj] and S,,= ]S,+S~], srl++-- 1 I, ...) IS,-sk] , where the Sips,, Sk (i, j, k= a, IS*+Sjl,
A.J. Marks, K. Prassides/ ChemicalPhysicsI79 (1994) 93-104
95
Q,=i
b, c or d) represent the spins of the ion core at sites a, b, c and d respectively. Each unit matrix I in eq. ( 1) is replaced in the new basis by the appropriate matrix of transformation coefftcients [lo] and the dimension s of each block of H& is equal to the number of combinations of intermediate spin values that together give S. For the model presented here, the total spins are S= l/2, 3/ 2,5/2 which arise five times, four times and once respectively (see table 1) . Vibronic coupling arises in the cluster when the extra electron interacts with the totally symmetric local displacements Q, (i=a, b, c, d) of the ligands attached to each metal centre [ 8,9]. The matrix elements of the vibronic Hamiltonian H, in the basis of localised states 1i) , i = a, b, c, d are
Q+ transforms as the A, representation of the Td point group and the Qr, Q2, Q3 together transform as the T2 representation. The force constants associated with the symmetric local distortions in different oxidation states are assumed to be equal. This allows the Q+ coordinate to be dropped from the vibronic Hamiltonian since its interaction with each electronic state is identical [ 25 1. In our calculations we use dimensionless symmetrised coordinates
(ilfLlO=~Q,,
1=
(3)
where 1is a vibronic coupling parameter and is inde pendent of spin [ 8 1. We introduce symmetry adapted ligand displacements coordinates:
Q+ =t(Qa+Qb+Qc+Qd
3
Q,=t(Qa-Q,-Qi+Qd
7
(4) (5)
Table 1 Spin basis states used in the model. Subscript label 1 represents the extra electron and i the site of localisation (ka, b, c or d). St, is the local spin of the site i containing the extra electron. Basis states having St, = 0 and S,, = 1 are non-Hund states and Hund states respectively. S,= Isj+skl, Is,+a- 11and S’= 15”+sIl, I S#+q- 1 I are the intermediate spins of the remaining sites (_j, k, La, b, c or d), where s,, sk, s, are the spins of the electrons in core orbitals I@j>, Idk) , I&) . The total spin S of the cluster takes thevahtesS=IS,,+S,~l, IS,,+&,-ll,..., IS,,-S’I.Theintermediate spin combinations giving each S are numbered by k in column one. k
S*,
S,
S,
S
1
1
1
3/2
512
1 2 3 4
1 1 1 0
1 0 1 1
l/2 l/2 312 312
312 312 3/2 3/2
1 2 3 4 5
1 1 1 0 0
0 1 1 0 1
l/2 l/2 312 l/2 l/2
l/2 l/2 l/2 l/2 l/2
(6)
3
Q3=1(Qa-Qb+Qc-Q,>.
q,=2n(v/k)‘/2Q,,
(7)
i= 1,2, 3,
(8)
and a dimensionless vibronic coupling parameter 1, (kv’)
-‘/2(1/4x)
)
(9)
where v= (21~)-‘& is the frequency of the totally symmetric local ligand vibration and k is the corresponding force constant. The local ligand vibrations are assumed to be harmonic, and the Hamiltonian matrix for the vibrational potential energy in the local&d electronic basis is given by (in units of k v) Hvib=t
1
I= 1,2,3
a?1 7
(10)
where I is the unit matrix. The vibronic Hamiltonian matrix elements expressed in units of h v are (al&
la> =A6
+h2+h3,
(11)
(blH,,Ib)=-Iql+~q,-~q,,
Ic> = -&A
I& =h
(12)
-&?2
-k,
+k3
-b
7
,
(13) (14)
where I i) , i = a, b, c, d, are the localised basis states, and the matrix elements are independent of spin. We write the Hamiltonian matrix for each total spin S as HS= H& + Hvib+ He, ,
(15)
where HL is the Anderson-Hasegawa Hamiltonian matrix for a system having spin S and is expressed in units of kv. A derivation of the matrix elements of HS is given in the Appendix for the case S= l/2. The model described above is “static”, in the sense that the nuclear kinetic energy is not included in the Hamiltonian. Solution of the above problem leads to
A.J. Marks, K. Prassides /Chemical Physics 179 (1994) 93-104
96
a set of adiabatic potential energy surfaces, and nuclear motion is confined to a single potential surface. We wish to solve the full dynamical vibronic problem and therefore must include the nuclear kinetic energy. This has the effect of coupling the nuclear and electronic motion so that the concept of adiabatic potential energy surfaces is no longer valid. The system is described instead by a spectrum of spin-vibronic eigenvalues and wavefunctions. We write the total Hamiltonian for a cluster having spin S as z?;=zP+
&(g)
,
(16)
where AS is the operator of the “static” Hamiltonian described earlier and FN is the nuclear kinetic energy operator for the ligand distortions. The Schrodinger equation to be solved for a system of spin S is I?;@f=E;@$,
(17)
where the 0: and Ef are spin-vibronic eigenfiurctions and eigenvalues respectively for a cluster of spin S. We write the vibronic wavefunctions as @:= ,ck U(k) )x,k,s(!z) 3
(18)
where the Ii(k)) are the localised electronic basis functions described earlier (j= a, b, c, d) and k labels a specific set of values for the intermediate spins. When S= 512 there is only one combination of intermediate spins (k= 1), and for S= 3/2, S= l/2, there are four and five spin combinations respectively (table 1) . The aAv (4) are nuclear wavefunctions, each of which is written as a sum of products of three onedimensional harmonic oscillator wavefunctions [ 26 ] : xi,kJa>
(19) where nl, n2, n3 (ni=O, 1, 2, .... CD) are the vibrational quantum numbers associated with nuclear coordinates ql, q2;q3 respectively. In our model, we limit the size of the three-dimensional vibrational basis to n,andincludeallml,n2,n3suchthatn1+n2+n3~n. Multiplying the vibronic Schr&linger equation by each of the lj( k) ) in turn and integrating over the electronic coordinates, leads to the secular equations
(one equation for each value ofj and k) : j&, [ O’(k) IAS+ TN If (k’ 1)
v= 1, 2, ...) co .
(20)
The electronic basis functions are independent of q and thus (j(k) I rfNIj’ (k’ ) ) ~0, where j#j’ and k#k’. We next substitute the expansion for the nuclear wavefunctions into the above equations. This leads to a large secular determinant (or “dynamic matrix” [ 9,12,26 ] ) which must be diagonalised to obtain the eigenvector coefficients C;$“’ and the eigenvalues E:. The dynamic matrix for S= l/2 is derived as an example in the Appendix. The dimension Ns of the matrix for a total spin S is Ns=nas(n+1)(n+2)(n+3)/6,
(21)
where ilp= 4 is the number of electronic states, s is the number of intermediate spin combinations for a given total spin (s=l, 4, 5 for S=5/2, 3/2, l/2 respeo tively ) and rais the maximum number of vibrational quanta. The dynamic matrix is sparse, real and symmetric and we have used the numerical sparse matrix diagonalisation method of subspace iteration [ 27 ] to obtain the required spin-vibronic eigenvalues for each of the possible spin states of the cluster. These are then used to calculate the effective magnetic moment as a function of temperature, for model systems both with and without vibronic coupling.
3. Results From the spin-vibronic state E:, we calculate the effective magnetic moment p as a function of temperature, using the relationship [ 28 ] p2_g2B2
Gv W+ 1) W+ 1) ew( -EST) &
(2S+ 1) exp( -E:/kT)
’
(22) where g is the Lam% factor, and fi is the Bohr magneton. The summation is over all possible spin states S= l/2,3/2,5/2 and over all the eigenvalues in each of the spin manifolds.
A.J. Marks, R Prassides/ ChemicalPhysicsI79 (1994) 93-104
The exchange interaction parameter J can be positive or negative, corresponding to ferromagnetic and antiferromagnetic exchange respectively. The parameter of inter-centre interaction e can also take positive or negative values, depending on the nature of the orbitals occupied by the five electrons, and on the nature of the bridging ligands through which the extra electron is transferred [ 25,291. The intra-atomic exchange Jo is always positive and is usually large compared with e and J. Jo leads to the enhanced stability of the “Hund states” which have a high local spin of si+ l/2 at the centre of electron localisation i. The spin combinations having a local spin of s, - l/2 are known as non-Hund states [ 8 1. We discuss below the influence of the vibronic interaction 1 on the magnetic moment of clusters having e > 0 or e < 0 and either ferromagnetic or antiferromagnetic exchange interactions. We also consider the contribution of non-Hund states to the magnetic moment, by varying the magnitude of the on-site exchange Jo. To give a physical perspective in terms of electronic structure, we note that systems having no vibronic coupling (A= 0) are those in which the “extra” electron is fully delocalised amongst the four metal centres. Such systems can be represented by a much simplified Hamiltonian, comprising a “double-exchange” or delocalisation term involving t, and a spin-dependent exchange interaction term [ 301. In contrast, clusters with very strong vibronic coupling ( ]e/42*] eO.16) [25] have an “extra” electron which is essentially localised at one of the metal sites. The figures discussed below show the magnetic moment as a function of temperature T expressed as the dimensionless quantity kT/hv, where hv is the frequency of the local, symmetric ligand vibration and k is Boltzmann’s constant. For a typical vibrational frequency of N 500 cm-‘, room temperature corresponds to kT/h v - 0.4. In our calculations we have used a vibrational basis of n=25. This is sufficient for convergence of the magnetic moment in the range of vibronic coupling strengths examined. The size of basis required for convergence increases with vibronic coupling strength. We found that a smaller vibrational basis (e.g., n = 15 ) gives the same qualitative behaviour of the magnetic moment, but overestimates its magnitude with respect to the converged value.
97
3.1. E-CO,J
When no vibronic interaction is present (A= 0 ) , the spin of the ground state can take any of the values S= l/2, S= 312, S= 512, depending on the relative magnitudes of J and e [ IO]. A large antiferromagnetic interaction favours a low-spin ground state, whereas a large electron transfer interaction 1c 1 favours a high-spin ground state. In fg la we show an example of the temperature dependence of the effective magnetic moment of a cluster which has an S= 3/2 ground state when 1= 0. When there is no vibronic coupling, the magnetic moment shows little variation as a function of temperature. The effect of 1 is to reduce the spin of the ground state to S= l/2, thus reducing the low-temperature value of the magnetic moment. At higher temperatures the magnetic moment remains lower than that of the system with no vibronic coupling. That is, I strengthens the antiferromagnetic properties of the cluster, and vibronic states of low spin are stabilised more than those of higher spin. Thus, if the vibronic coupling is strong enough, a reduction in the ground state spin and the magnetic moment can be achieved. For weaker vibronic coupling there is no change in the ground state spin and the magnetic moment is reduced to a lesser extent. We note that the magnetic moment illustrated in lig. 1a for the case of strong vibronic coupling corresponds to a situation in which the extra electron is highly localised ( I s/4A2I =0.03). In this case energy minima obtained from potential-energy surfaces within the “static” model [ 10,251 give an almost identical description of the magnetic moment curve. This suggests that a semi-classical model comprising a modified Heisenberg Hamiltonian similar to that described by Blondin et al. [ 30 ] for trimeric clusters, will also give a good description of the magnetic moment in the limit of very strong vibronic coupling. 3.2. e
A.J. Marks, K. Prassides/ ChemicalPhysicsI79 (1994) 93-104
98
b
l.SC
0.4c
c.30
0 60
0.80
I
5.20-L 0 00
1 OC
0.20
kT
0.40
0.60
0.80
1
kT
4.50
-----__-_-_____d_ l/r---
g
iii,
4 00./
?i h
\
3 oo3.50.
& z
2 53.
%
2.00-
1.50+
0.40
o.co
0 20
0 40
0.60
0.80
4
1.00
kT
kT
4.00-
0)
3 503.002.502.001.5OT 0.00
0.20
0.40 kT
0.60
0.80
1 1.00
0.4c kT
Fig. 1. Temperature dependence of the effective magnek moment for a tetrahedral mixed-valence cluster, showing the influence of vibronic coupling L The magnetic moment is expressed in Bohr magneton (B.M.) units and the effective temperature kTis expressed in units of hv, the symmetricligand vibration frequency. Jo, Jand t are also given in units of hv. A vibrational basis of dimension n= 25 is used. (a) C-CO,J
A.J. Marks, K. Prassides/ ChemicalPhysics179 (I 994) 93-l 04
are preferentially reduced in energy by 1, and become populated as the temperature increases. An example is shown in fig. lb and shows that the qualitative behaviour of the magnetic moment does not change when vibronic coupling is included. In both cases fl decreases slowly as the S= l/2 and S= 31 2 states become populated. 3.3. e>O, J-CO When there is no vibronic interaction the ground state is antiferromagnetic (SE l/2), irrespective of the relative magnitudes of Eand J [ IO]. The effect of vibronic coupling on the magnetic moment is shown in fig. lc for a typical case where e = 5.0 and J= - 0.1. The magnetic moment is increased slightly by 1 but the ground state remains S= l/2 and the curves show similar qualitative behaviour. Thus, in this case J stabiliscs the S= l/2 states substantially more than the S= 3/2 and S= 5/ 2 states. Addition of vibronic coupling results in a slight enhancement of the magnetic moment at high temperatures, with the ground state remaining S= 1 / 2. The observed increase in magnetic moment shows that states of spin S= 3/2 and S= 5/2 are stabilised more by vibronic coupling than those of spin S= 1 / 2. We note that this effect is the opposite of that observed when E< 0. For the case of weak antiferromagnetic coupling (or small IJ/C) ), a change of ground state spin from S= l/2 to S= 3/2 or S= 5/2 can arise in the presence of vibronic coupling. This gives a large increase in the low temperature magnetic moment, as shown in figs. 1d and 1e for the stabilisation of an S= 3/2 and S= 5/2 ground state respectively. Vibronic coupling has the greatest effect on the magnetic moment (inducing a change in ground state spin from S= l/2 to S= 5/2) when 1J/C 1 is very small (fig. le). The change in ground state spin induced by vibionic coupling when the exchange interaction J is weak can be rational&d as follows: If there is no inter-atomic exchange interaction or vibronic coupling, the low-energy (Hund) states of different spin are degenerate. On adding a very small antiferromagnetic exchange interaction (J-CO), an S= l/2 ground state results. However, because the exchange interaction is small, the low-energy states of different spin lie very close together. Introduction of vibronic
99
coupling (which in this case stabilises high-spin states more than low-spin states), then results in the lowest S= 312 or S= 512 states being reduced in energy below the S= l/2 electronic ground state. When k0 and J
100
A. J. Ma&,
K. Prawides / Chemical Physics I 79 (I 994) 93-104
romagnetic exchange J-z 0. An example of the effect on the magnetic moment is given in fig. 2a and shows that when I = 0 the magnetic moments differ significantly. The system with small Jo in general has a smaller magnetic moment than that with large Jo although at very low temperatures the magnetic moments are identical, corresponding to an S=5/2 ground state. The magnetic moment of the large Jo cluster decreases slowly as a function of T whereas that of the small Jo cluster decreases sharply before levelling off. The differences between the two magnetic moments in fig. 2a (curves (i) and (ii) with 1= 0) can be explained in terms of the relative stabilities of the
S= l/2 and S= 3/2 non-Hund states. When IJo/t I is small, the non-Hund basis states mix into the lowest S= l/2 and S= 3/2 vibronic levels and stabilise them, thus reducing the magnetic moment. These low-spin states are also stabilised by the antiferromagnetic exchange interaction J. Similar arguments explain the difference between the magnetic moments for large and small Jo when 13# 0 (fig. 2a, curves (iii) and (iv) ) . However, since vibronic coupling tends to reduce the magnetic moment for a given e < 0, variations in Jo have less effect on the magnetic moment and the curves lie closer together. In clusters with ferromagnetic exchange and e < 0, the magnetic moment is slightly smaller for small Jo than for large Jo. An example is shown in fg 2b for a system with no vibronic interaction. As in the previous example, non-Hund states stabilise the lowest S= l/2 and S= 3/2 vibronic levels when J,-,is small, leading to a reduction in the magnetic moment. Finally, we note that when e > 0, the magnetic moments for large and small J,-,are identical over the range of temperatures studied. This implies that in contrast to the situation when E< 0, non-Hund basis states do not mix into the low-energy S= l/2 and S= 3/2 vibronic states.
4. Discussion kT 6.00,
I
b
0 20
0.40
0.60
0 80
1 00
kT
As fig, 1, illustrating the influence of non-Hund states on the magnetic moment. (a) t
2.
1=0, Jp3.
We have developed a dynamic spin-vibronic coupling model and illustrated by application to a simple system that vibronic interactions can play an important role in determining the magnetic properties of tetrahedral mixed-valence clusters. The most significant effect of vibronic coupling on the magnetic moment arises in the presence of antiferromagnetic exchange interactions. In particular, when the inter-centre interaction e is negative, vibronic coupling decreases the magnetic moment and if strong enough, causes a change from a high-spin to a lower-spin ground state with a corresponding decrease in the low-temperature magnetic moment. In contrast, when e > 0, vibronic coupling increases the magnetic moment and when the antiferromagnetic interaction is very small, can induce a change from an S= l/2 to an S= 3/2 or S= 5/2 ground state. When E< 0, the intra-atomic exchange interaction
A.J. Marks, K. Prmides
/ Chemical Physics I79 (1994) 93-104
J,, also influences the magnetic moment. If the ratio 1Jo/c 1- 1 (i.e. Jo is relatively small), then the S= 1/ 2 and S=3/2 non-Hund basis states mix into the lowest vibronic levels and reduce the magnetic moment. Thus, when E< 0, the greatest reduction in the magnetic moment arises when the intra-atomic exchange is small and the vibronic interaction is strong. As noted in section 3.1, systems having a strongly local&d “extra” electron can be described using the “static” model, suggesting that a simple modified Heisenberg Hamiltonian may be sufficient to describe the properties of clusters with strong vibronic coupling. This observation is consistent with the conclusions of Blondin et al. [ 301 who have developed an effective Heisenberg model for describing strongly vibronically coupled mixed-valence trimers. The usual interpretation of experimentally measured magnetic moments in mixed-valence clusters is based on the assumption of a simple spin Hamiltonian comprising an exchange interaction J and a term in e allowing electron delocalisation [ 1,17- 19,2 11. An important message from our results is that the interpretation of the magnetic moment in this way can lead to incorrect estimates of inter-atomic exchange and electron transfer interactions. The inclusion of vibronic coupling, and to a lesser extent the intra-atomic exchange interaction, is necessary for the correct evaluation of e and J in antiferromagnetic systems. This may be achieved through the use of the dynamical model described here, or, when vibronic coupling is strong, with a simpler semi-classical model [ 10,301. The magnitude of the vibronic interaction 1 in tetranuclear mixed-valence clusters can in principle be extracted from intervalence charge-transfer spectra [ 8,9,12 1. However, the optical spectra are extremely complicated and further experiments are required
101
before our vibronic coupling model can be used to simulate the intervalence bands of exchange-coupled clusters. Nevertheless, we believe that this preliminary study provides some new insights into the magnetic properties of vibronically coupled mixed-valence clusters, and we aim to extend the model so that realistic systems such as tetranuclear iron-sulfur clusters can be investigated. Finally, we emphasise that although the current study focuses on the calculation of magnetic properties and utilises only the spin-vibronic eigenvalues, the dynamic spin-vibronic coupling model can also be used to investigate electronic properties. For example, the spin-vibronic eigenvector coefficients can be used to compute probability distributions [ 91 describing the delocalisation properties of the “extra” electron as a function of vibronic coupling strength [ 10,251. The calculations in this paper pertain to systems in which the “extra” electron is either fully delocalised or localised (i.e., weak and strong vibronic coupling respectively). However, the range of vibronic coupling strengths required for an investigation of interesting delocalisation patterns is an intermediate regime at the borderline between electron localisation and full delocalisation [ 10 1. Work in this direction complementing our earlier study [lo] is under way.
Acknowledgements This work is supported by the Science and Engineering Research Council and NATO travel grant RG0146/87. We thank Professor Paul Schatz and Professor Susan Piepho for useful discussions and the University of Virginia for provision of computing facilities.
Appendix. Derivation of the S= l/2 dynamic matrix
We illustrate first the derivation of the “static” matrix HS in the chosen electronic and spin basis for the case S=1/2.Thisisa20x20matrixandhastheform (seeeq. (15)): H”2=Hj$+Htib+H
en*
Using the spin basis functions given in table 1, we obtain
(23)
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A.J. Marks, K. Prassides/ ChemicalPhysics179 (1994) 93-104
(24)
(25)
is the Heisenberg Hamiltonian matrix for electron localisation at site i, and J (I), J@) are the exchange energies for Hund states and non-Hund states respectively (obtained from eq. ( 2 ) and table 1) J”‘=-i(J,,-75))
(26)
J”‘=$(J,,+J).
(27)
Each matrix T”=Tj’ is a matrix of transformation coefficients relating the different spin representations for electron localisation at sites i and j. There are five S= l/2 intermediate spin combinations k arising for each localisation of the extra electron (table 1) , so each T’j has dimension 5 X 5. The elements are of the form: T&=(i(k)lj(k’)),
(28)
where k, k’ refer to the S= l/2 basis functions given in table 1. Each of these elements is a transformation coefficient between two coupling schemes (i and j) for five electrons. The vibronic matrix H,, is given by
(alfLla)l Hen=
0 0 0
0
0
0
0
0
(ClKnlC)~ 0
(29)
0 (dlfLld>l
where
i=l
[~~(qi)+fq:l+(jlH,lj)+J’e’-E,
>
&,k,v-e c
f#Jk”l
i
T&G,~,~=O,
where j=a, b, c or d is the centre of electron local&-&on and k= 1,2, 3,4 or 5. The exchange energy Jce)= J(l) (eq. (26) ) when k= 1, 2, 3 (Hund basis states) and J @)=J(‘) (eq. (27)) when k=4, 5 (non-Hund basis states). We substitute the expansion (31)
A.J.
103
Marks,K. Prassides / Chemical Physics 179 (I 994) 93-104
into the above secular equations. We multiply each equation by the product of harmonic oscillator functions x,,,, ( q1)x,, ( q2)xms( q3) (for all allowed combinations of ml, mz, m3 in the vibrational basis), and integrate over the nuclear coordinates ql, q2, q3. The non-zero terms can be identified using the following properties of harmonic oscillator wavefunctions (expressed in units of hv): (32)
tx,I~~(4,)+14:lX~)=n,+1/2 and
(33)
This leads to the following four sets of secular equauons (one for each of the local&d electronic basis functions withj=a, b, c, d, respectively),
where Hg,q!,=(m,+mz+m3+3/2+J(“)-E,)6
with J(‘)=J(l) H!%=&/~&n,x,-I
m,,n*6m2,nt6?nwl,9
when k= 1,2, 3 and Jtc)=J(*)
(38)
when k=4, 5.
+&&,,+~l&,&m,.r
(P, 4, r)=l, 2>3;q>MP)
.
(39)
In each of the four sets of eqs. (34)-( 37), k= 1,2,3,4 or 5 and the vibrational quantum numbers are chosen such that m, + m2 + m3 < II where n is the selected basis size. A total of IV,,equations results (eq. (2 1) ) , and the dynamic matrix can be diagonalised to obtain the eigenvalues Es and the corresponding eigenvector coefficients C$JY3. The procedure for deriving the S= 3/2 and S= S/2 secular equations is identical to that given here for s= l/2. Iteferences
[l] G. Blondin and J.-J. Girerd, Chem. Rev. 90 (1990) 1359, and references therein. [ 2 ] V. Papaeibymiou, J.-J. Girerd, I. Moura, J.J.G. Moura and E. Mtlnck, J. Am. Chem. See. 109 (1987) 4703. [ 31 J.A. Weigel, K.K.P. Srivastava, E.P. Day and E. Mtlnek, J. Am. Chem. See. 112 (1990) 8015. [4] D. Coucouvanis, S.A. Al-Ahmad, A. Salifoglou, V. Papaefthymiou, A. Kostikas and A. Simopoulos, J. Am. Chem.Soe. 114 (1992) 2472. [ 5 ] V. Papaeibymiou, M.M. Millar and E. Milnck, Inorg. Chem. 25 (1986) 3010. [6] G. Rius and B. Lamotte, J. Am. Chem. See. 111 ( 1989) 2464.
[7]S.A.BorshehandL.F.Chibotaru,Chem.Phys. 135 (1989) 375. [ 81 I.B. Bersuker and S.A. Borshch, Advan. Chem. Phys. 81 ( 1992) 704, and references therein. [ 91 KY. Wang and P.N. Sehatz, Progr. Inorg. Chem. 28 ( 198 1) 369. [lo] A.J. Marks and K. Prassides, J. Chem. Phys. 98 (1993) 4805. [ 111 SA. Borsheh, E.L. Bominaar and J.-J. Girerd, New J. Chem. 17 (1993) 39. [ 121 S.B. Piepho, E.R. Krausz and P.N. S&ate, J. Am. Chem. See. 100 (1978) 2996. [13] J.-J. Girerd, J. Chem. Phys. 79 (1983) 1766. [ 141 S.A. Borsheh, I.N. Kotov and LB. Bersuker, Chem. Phys. Letters 111 (1984) 264. [15] B.S. Tsulcerblat, AV. Palii, H.M. Kishinevsky, V.Ya. Gamurar and AS. Berengol’ts, Mol. Phys. 76 ( 1992) 1103.
104
A.J. Marks, K. Prassides/ ChemicalPhysics179 (I 994) 93- IO4
[ 161 LB. Bersuker, S.A. Borshch and LN. Kotov, Rev. Roumaine Chem. 32 (1987) 1075. [ 171 L. Noodleman, Inorg. Chem. 27 (1988) 3677, and references therein. [ 181 L. Noodleman, Inorg. Chem. 30 (1991) 246. [ 191 L. Noodleman, Inorg. Chem. 30 (1991) 256. [20] M. Belinskii, Chem. Phys. 172 (1993) 213. [21] J. Jordanov, E.K.H. Roth, P.H. Fries and L. Noodleman, Inorg. Chem. 29 (1990) 4288. [22] P.W. Anderson and H. Hasegawa, Phys. Rev. 100 (1955) 675.
[23] .%A.Borshch, Soviet Phys. Solid State 29 (1987) 896. [24] M.I. Belinskii, Soviet Phys. Solid State 27 (1985) 1057. [25] A.J. Marks and K. Prassides, New J. Chem. 17 (1993) 59. [ 261 K. Prassides and P.N. Schatz, J. Phys. Chem. 93 (1989) 83. [ 271 B.N. Parlett, The symmetric eigenvalue problem (PrenticeHall, Englewood Cliffs, 1980). [28] A. Earnshaw, Introduction to magnetochemistry (Academic Press, London, 1968) [29] J.P. Launay and F. Babonneau, Chem. Phys. 67 (1982) 295. [ 301 G. Blondin, S. Borshch and J.-J. Girerd, Comments Inorg. Chem. 12 (1992) 315.