As in previous work, an antiferromagnetic (AF) spin-coupled state within DFT is represented by a ‘‘BS’’ state, where the spin-up () electron density occupies a different region of space from the spin-down () electron density.14,37These two densities are different, but there is usually overlap between them. This idea is broad enough to cover a number of different situations, encompassing metal–radical interactions and spin-coupled dinuclear and polynuclear transition metal com- plexes. (Ligand radicals are most often Sẳ1/2 as in semiquinones, tyrosine, or tryptophan radicals, but molecular oxygen is Sẳ1.) The integrated net spin density may sum to zero (when the ground state is a singlet, total spin Stẳ0), or nonzero (when the ground state has higher spin, for example Stẳ1/2, 3/2, 5/2, etc. or Stẳ1, 2, 3). In some cases, the spin-aligned high-spin (HS) state may be the ground state, and the broken symmetry state lies higher. Further, while typically the transition metal is in a high-spin state for each metal site, intermediate (IS) or low-spin (LS) metal sites are not precluded. One just needs to start with an appropriate guess at the spin density distribution; this is often best done by starting with a parallel spin aligned state, and then interchanging with electron densities on different metal and/or ligand sites as needed. This can usually be quite effectively done with the electron fit densities used in various programs (particularly in the Amsterdam Density Functional codes, ADF). If the broken sym- metry state is not energetically favored (within a given exchange-correlation potential and at any given geometry), then the variational principle requires that the spin density disappears and the self-consistent-field (SCF) solution will converge instead to the ordinary nonbroken symmetry case.
To be concrete, we examine a dinuclear transition metal complex with the same metal ion and spin on each site. Then a BS state is constructed by the spin flip procedure and solution of the SCF problem. We will analyze the spin-coupling energy assuming a Heisenberg spin Hamiltonian of the formHspinẳJS1S2, to determine the Heisenberg coupling constantJ. The pure spin states form a Heisenberg ladder obeying the Lande interval rule, E(S)E(S1)ẳJS for successive states, and with total spins ranging from Sminẳ|S1S2|, Smaxẳ|S1ỵS2| in integer steps. For dominant AF coupling, the spin-coupling interactions give ‘‘spin-bonding’’ for BS, and ‘‘spin-
antibonding’’ for the spin aligned HS state. In this dimer, the energy difference between HS and the singlet ground state is:
EðHSị EðSẳ0ị ẳJSmaxðSmax ỵ 1ị=2 ð20ị whereSmaxis the maximum total system spin. By contrast, the energy difference between HS and BS is
EðHSị EðBSị ẳJSmax2 =2 ð21ị
The BS state is the ‘‘semiclassical analog’’ of the singlet ground state. The BS state is not a pure spin state; instead, it is a specific weighted average of pure spin states, and lies above the pure spin ground state whenE(Sẳ0) is belowE(HS). To see this more clearly, we consider the expansion of (BS) and the corresponding energyE(BS) over pure spin states
ðBSị ẳX
S
CðSị ðSị ð22ị
EðBSị ẳ< ðBSịjHj ðBSị>ẳX
S
CðSị2< ðSịjHj ðSị>ẳX
S
CðSị2EðSị ð23ị o
o (a)
(c) (d)
(b) t
e
t t
e e
Figure 1 The inverted level scheme is shown for iron–sulfur monomer and dimer complexes. (a) Fe(SR)4
complex. (b) Splitting into lower energy majority spin Fe 3dorbitals (spin-up), ligand orbitals (center), and higher energy mainly Fe 3d-based minority spin (ligand) field orbitals, with e below the t2 type.
(c) Fe2S2(SR)4complex. (d) Broken symmetry level structure containing similar spin-dependent and ligand field splitting to (b), but now with left–right spin localization for lower majority spin and higher minority spin Fe 3d levels. Note that the ligand field ordering with e below t2 applies only to the minority spin
orbitals, and the level ordering can be different for deeper majority spin orbitals.
498 Density Functional Theory
since the electronic Hamiltonian does not connect states of different total spin. In the weak metal–metal interaction regime, valid for most spin-coupled dimers,C(S)ẳC1(S), whereC1(S) is a standard Clebsch-Gordon coefficient C1(S)ẳC(S1S2,S;M1M2), where S1, S2 are the principal spins of the two metal sites, andM1,M2are theirzcomponents, specifying the spin alignment in the broken symmetry state. For nonequivalent spin sites, we need a more general equation for E(HS) –E(BS). LetSAand SBbe spin quantum numbers for A and B subunits. Then using the spin algebra of raising/lowering operators acting on HS and BS,38we find
<SA:SB>HS;BSẳ SASB ð24ị
with theþand signs for HS and BS respectively, and where the right-hand side involves only quantum numbers. Then
EðSmaxị EðBSị ẳ2JSASB ð25ị
This equation can be readily generalized to polynuclear complexes with multiple HeisenbergJ parameters.
If the overlap of the magnetic orbitals (the orbitals that are different for versus spin) is larger, more general methods based on projected-unrestricted-Hartree–Fock (PUHF) methods can be used for transition metal–ligand radical, dinuclear complexes, or organic diradicals.39,40 For organic diradicals, the question of whether symmetry breaks at the proper point along the reaction path for Cope rearrangements (sigmatropic shifts) has been investigated.41,42Depending on the systems studied, the authors concluded either that unrestricted UBPW91 performs better than UB3LYP, or that the two methods perform similarly and bracket the expected enthalpies of transition states and diradical intermediates.
An interesting case is the singlet–triplet problem, where a general solution is obtained valid from the weak coupling regime where E(BS)ẳ[E(Sẳ1)ỵE(Sẳ0)]/2 to the strong coupling regime where the spin density vanishesE(BS)ẳE(Sẳ0).43The general singlet–triplet equation is:
EðBSị ẳ ẵð1ỵ S2abịEðSẳ0ị ỵ ð1S2abịEðSẳ1ị=2 ð26ị whereSab is the spatial overlap of the two nonorthogonal magnetic orbitals (a,b for and spin),Sab!0gives weak coupling,Sab!1 gives strong coupling. This flexibility is particularly valuable when following the potential energy surface (PES) as a bond is broken to produce a diradical. We have recently examined ‘‘twisted stilbene,’’43and others have examined many other organic diradical systems.39
The BS plus spin projection method discussed here is closely connected to the simple open-shell singlet method for optical excitations based on the Slater sum rule andSCF (self-consistent-field total energy difference method). The ‘‘mixed spin excited state’’ is like the BS state, also of mixed spin. The Slater sum rule method44is also quite effective for multiplet problems for excited states of transition metal complexes as shown in the work of Dahl and Baerends.8,45
Proceeding to mixed-valence dimers (as in diiron complexes with high-spin Fe2þ, Fe3þsites), the higher total spin states can gain stabilization energy from a spin-dependent delocalization (SDD) mechanism (also called resonance delocalization or ‘‘double exchange’’)10,46 with a spin Hamiltonian of the formHspinẳJS1S2B(Stotỵ1/2), while for lower total spin states (and for BS), vibronic coupling and solvent effects can quench the resonance term and leave only the Heisenberg spin couplingJS1S2. Calculation of the BS and HS states is again feasible to extract J and B parameters. Further, the extension of these methods to polynuclear complexes (most prominantly FemSn(SR)p) is straightforward.
Both J(x) and B(x) can be geometry dependent, as we have demonstrated, with the most important effects when J is fairly strong, and the E(HS) geometric minimum differs significantly from that ofE(BS).6,7ForB(x), the clearest effects are observed in HS delocalized mixed-valence dimers including Fe2S2and Fe2(OH)3whereStotẳ9/2, and optical spectroscopy allows an assess- ment of 2B(x)(Stotþ1/2). ForJ(x), a more accurate evaluation would include the path dependence, both in calculations and in modeling of experimental magnetic susceptibility. From a theoretical perspective, if a path is traced fromE(HS) toE(BS) in Cartesian steps (and somewhat beyond these limits), the bounds onJpath(x) are |Jadiabatic|<|Jpath|<|Jvert,BS|. The solvent and protein environ- ment can also modulate the Jpath(x) over the path. It seems, however, that presently this is less important than the quality of the exchange-correlation potential in determiningJ.