Particle Swarm Fitting of Spin Hamiltonians: Magnetic Circular Dichroism of Reduced and NO-bound Flavodiiron Protein

Abstract

A particle swarm optimization (PSO) algorithm is described for the fitting of ground-state spin Hamiltonian parameters from variable temperature/variable field (VTVH) magnetic circular dichroism (MCD) data. This PSO algorithm is employed to define the ground state of two catalytic intermediates from a flavodiiron protein (FDP), a class of enzymes with nitric oxide reductase activity. The bimetallic iron active site of this enzyme proceeds through a biferrous intermediate and a mixed ferrous–{FeNO}⁷ intermediate during the catalytic cycle, and the MCD spectra of these intermediates are presented and analyzed. The fits of the spin Hamiltonians are shown to provide important geometric and electronic insight into these species that is compared and contrasted with previous reports.

1. Introduction

Magnetic circular dichroism (MCD) spectroscopy is one of the most information-dense techniques in the inorganic chemist’s toolbox. Commonly performed in energy ranges from near-infrared (NIR) to X-ray, it can provide detailed insight into electronic structure by simultaneously probing transitions to excited states while providing magnetometry of the ground state. This makes variable-temperature/variable-field (VTVH) MCD invaluable in the study of systems unsuitable for common X- or Q-band EPR studies, such as many non-Kramers ions or systems with fast relaxation. Furthermore, its ability to selectively probe a single transition provides the possibility for analysis of a single component within a complex mixture of compounds, an advantage over bulk techniques like SQuID (superconducting quantum interference device) magnetometry.

Interpretation of VTVH MCD data is complicated by the nonlinear intensity equation and the large number of fit parameters necessary. Several strategies have been used, including linearization in the high-temperature low-field limit,^1–4 fitting of Kramers^3,5,6 and non-Kramers^5,7–9 systems to Brillouin saturation curves, and various approaches informed by ligand field theory.^9–13 These all involve several strong approximations or assumptions, and an attractive alternative is to describe amenable systems using a spin Hamiltonian.¹⁴ Modern technology can easily perform the calculations necessary to fit MCD data within such a spin Hamiltonian formalism.

Herein, we outline our implementation of particle swarm optimization (PSO)^15–17 to fit VTVH MCD data. We then use this algorithm to define the ground state for two relevant forms of the binuclear iron active site of TmFDP (Figure 1), a flavodiiron protein (FDP) from Thermotoga maritima. This gives structural and mechanistic insight into the important FDP class of enzymes, which exhibits scavenging nitric oxide reductase (s-NOR) activity.

Figure 1.

An X-ray crystallographic structure at 2.65 Å resolution of the biferrous form of TmFDP revealed two Fe–His and one Fe–(κ¹-Asp/Glu) interactions at each iron center along with an additional μ-1,3-bridging Asp (PDB 5LLD).¹⁸ An O₂ molecule modeled near one iron site has been omitted. The low resolution leaves questions about the coordination environments.¹⁹

2. Methodology

We begin with a brief overview of the spin Hamiltonian formalism, MCD spectroscopy, and the PSO algorithm.

2.1. The Spin Hamiltonian

A common way to describe the interaction of molecular spin sublevels with an applied magnetic field is to employ a phenomenological spin Hamiltonian. Our treatment here rests on several crucial assumptions about the system under study:

• the system is magnetically isolated,

• its ground state is energetically separated from all electronic excited states,

• it has no unquenched ground-state orbital angular momentum,

• the splitting of magnetic sublevels is significantly smaller than the line width of the transition,

• and the influences²⁰ of Zeeman and spin–orbit coupling (SOC) on the transition dipole moment are small enough to be treated perturbatively.

These criteria are frequently met for dilute (bio)inorganic molecules containing 3d transition metals. For such systems, we can define a ground state spin Hamiltonian as

$$H_{\text{spin}}=\sum _{\alpha }{\mu }_{\text{B}}\overrightarrow{H}{\tilde{g}}_{\alpha }{\overrightarrow{S}}_{\alpha }+\sum _{\alpha }{\overrightarrow{S}}_{\alpha }{\tilde{D}}_{\alpha }{\overrightarrow{S}}_{\alpha }+\sum _{\alpha <\beta }{\overrightarrow{S}}_{\alpha }{\tilde{J}}_{\alpha \beta }{\overrightarrow{S}}_{\beta },$$ (1)

where we use boldface (H) to represent an operator, an arrow ($\overrightarrow{H}$) to represent a vector, and a tilde ($\tilde{g}$) to represent a matrix or tensor. Here, μ_B is the Bohr magneton, $\overrightarrow{H}$ the applied magnetic field, α (β) indexes the paramagnetic spin centers of the system, ${\tilde{g}}_{\alpha }$ is the g matrix for center α, ${\tilde{D}}_{\alpha }$ is the zero-field splitting (ZFS) tensor for center α, and ${\tilde{J}}_{\alpha \beta }$ is the exchange matrix coupling centers α and β. The ${\tilde{J}}_{\alpha \beta }$ matrix coupling centers α and β is often discussed in terms of its isotropic (trace, −2J_αβ), antisymmetric (${\overrightarrow{G}}_{\alpha \beta }$), and anisotropic (symmetric, ${\tilde{D}}_{\alpha \beta }$) components:

$${\overrightarrow{S}}_{\alpha }{\tilde{J}}_{\alpha \beta }{\overrightarrow{S}}_{\beta }=(-2)J_{\alpha \beta }{\overrightarrow{S}}_{\alpha }\cdot {\overrightarrow{S}}_{\beta }+{\overrightarrow{G}}_{\alpha \beta }\cdot {\overrightarrow{S}}_{\alpha }\times {\overrightarrow{S}}_{\beta }+{\overrightarrow{S}}_{\alpha }{\tilde{D}}_{\alpha \beta }{\overrightarrow{S}}_{\beta },$$ (2)

where the (−2) prefactor varies by definition. For S > 3/2 systems, terms of higher order in S can be added.²¹ Any contributions from weaker interactions (hyperfine, nuclear Zeeman, etc.) are generally not resolved in VTVH MCD data so they are not treated here.

2.2. VTVH MCD Theory

Circular dichroism (CD) and MCD measure the differential absorption of left- and right-circularly polarized light through a sample.²² Although nominally similar, the theory of MCD differs greatly from CD and is applicable to a far larger set of molecules. In fact, all matter exhibits MCD. Historical treatments have generally parametrized MCD intensity using the three Faraday terms ${\mathbb{A}}_1$, ${\mathbb{B}}_0$, and ${ℂ}_0$ as^1–4

$$\frac{\Delta ϵ}{E}=\gamma {\mu }_{\text{B}}H\left[-{\mathbb{A}}_1\frac{\partial f}{\partial E}(E)+\left({\mathbb{B}}_0+\frac{{ℂ}_0}{k_{\text{B}}T}\right)f(E)\right].$$ (3)

Here, γ is a proportionality constant, f(E) the bandshape function, k_B the Boltzmann constant, and T the temperature. The Faraday terms ${\mathbb{A}}_1$, ${\mathbb{B}}_0$, and ${ℂ}_0$ are all scalars, and their subscripts indicate the moment of their contribution (ie, zeroth-moment f vs. first-moment ∂f/∂E).

Eq 3 is a linear function of field and of reciprocal temperature, and it is an approximation that can only be used at high temperatures²³ or low fields where magnetic saturation effects are negligible. At low temperatures and high fields, the mechanism underlying ${ℂ}_0$ term intensity deviates from a linear intensity contribution due to Boltzmann (de)population of sublevels. For systems outlined in Section 2.1, this nonlinearity can be well described using a spin Hamiltonian.^14,24,25 We can model the zeroth-moment MCD intensity for an unoriented (powder/glass) sample as (see SI)

$$\begin{gathered} \frac{\Delta ϵ}{E}=\gamma \frac{1}{4\pi }∯\left({\mu }_{\text{B}}\overrightarrow{z}\tilde{B}\overrightarrow{H}-\frac{1}{S}\overrightarrow{z}\tilde{C}\langle \overrightarrow{S}\rangle \right)\text{d}\Omega f(E) \\ =\gamma \left(\frac{\text{Tr}\tilde{B}}{3}{\mu }_{\text{B}}H-\frac{1}{S}\frac{1}{4\pi }∯\overrightarrow{z}\tilde{C}\langle \overrightarrow{S}\rangle \text{d}\Omega \right)f(E), \end{gathered}$$ (4)

where dΩ = sin θdθdϕ is the differential solid angle, $\overrightarrow{z}=[\text{sin}\theta \text{cos}\varphi ,\text{sin}\theta \text{sin}\varphi ,\text{cos}\theta ]$ is the direction of light in the molecular frame, $\langle \overrightarrow{S}\rangle$ is the net Boltzmann-weighted spin expectation value (eq 1), and Tr is the matrix trace. Here, we have introduced two 3 × 3 matrices $\tilde{B}$ and $\tilde{C}$ that vary with transition. The former arises from Zeeman field-induced mixing between states, and the latter from spin–orbit coupling (SOC) between states. As implied by the labels, these can be related to the linearized Faraday terms of eq 3 as ${\mathbb{B}}_0=\text{Tr}[\tilde{B}]/3$ and ${ℂ}_0=(S+1)\text{Tr}\left[\tilde{C}{\tilde{g}}^{\top }\right]/9$ at sufficiently high temperatures and low applied fields (see SI).

The entries of the $\tilde{C}$ matrix for a spin-allowed transition from ground state A to excited state J are related to the electronic structure of the molecule through

$$\tilde{C}\propto \sum _{K\ne A}\frac{\left({\overrightarrow{m}}_{JA}\times {\overrightarrow{m}}_{JK}\right){\overrightarrow{\lambda }}_{KA}^{\top }}{E_K-E_A}-\sum _{K\ne J}\frac{\left({\overrightarrow{m}}_{JA}\times {\overrightarrow{m}}_{KA}\right){\overrightarrow{\lambda }}_{JK}^{\top }}{E_J-E_K}$$ (5)

for Born–Oppenheimer transition dipole moments ${\overrightarrow{m}}_{XY}=\langle X|\overrightarrow{m}|Y\rangle$, and reduced spin–orbit coupling elements ${\overrightarrow{\lambda }}_{XY}$ (see SI §S3.1). Here, K indexes all states other than A or J (as indicated). In cases where all spin Hamiltonian tensors are collinear, terms resulting from off-diagonal $\tilde{C}$ matrix elements integrate to zero, so in these cases it is appropriate (necessary) to treat the $\tilde{C}$ matrix as if diagonal, and we often use the labels below for these diagonal entries,

$$\tilde{C}=\left(\begin{array}{ccc}{M}_{yz}& 0& 0\\ 0& M_{zx}& 0\\ 0& 0& M_{xy}\end{array}\right),$$

where the $M_{yz}={\tilde{C}}_{xx}$ (etc.) are defined as in eq. 5. These values can be related to the polarization of the A → J transition in specific cases.¹⁴ Spin-forbidden transitions also follow eq 4, but have a modified analog to eq 5 and do not exhibit ${\mathbb{B}}_0$-term intensity ($\tilde{B}=0$) at this level of theory (see SI).

Eq 5 reveals two important characteristics of the ${ℂ}_0$ term intensity. First, the transition dipole moments ${\overrightarrow{m}}_{JA}$ and ${\overrightarrow{m}}_{JK}\left({\overrightarrow{m}}_{KA}\right)$ and the reduced SOC vector ${\overrightarrow{\lambda }}_{KA}\left({\overrightarrow{\lambda }}_{JK}\right)$ must have mutually orthogonal components, a useful phenomenon for interpretation of the $\tilde{C}$ matrix for molecules in orthorhombic point groups. Second, it is useful to note how the two sums in eq 5 relate to the “sum rule” of MCD intensity.^10,26–28 The latter sum represents SOC among excited states, and MCD intensity integrates to zero when this sum is dominant. The former sum represents SOC with the ground state and its contribution results in nonzero integration, therefore deviating from the sum rule over excited states.

MCD intensity of an exchange-coupled system behaves similarly.^29,30 Assuming the transitions are primarily localized on a single spin center n within the ensemble, the intensity varies with the spin projection at that spin center on the exchange coupled dimer wavefunction, $\Delta ϵ/E~\overrightarrow{z}\tilde{C}\langle {\overrightarrow{S}}_n\rangle$. In these cases, the same machinery used to analyze MCD data of a single spin can be applied to exchange-coupled systems with only minor modification. When the approximation of a localized transition breaks down, a modified approach is needed.^31,32

2.3. Particle Swarm Optimization

The nonlinear nature of the optimization problem makes fitting of VTVH MCD data difficult. Beyond the ability to numerically integrate eq 4, any fitting program must have an algorithm to judge goodness-of-fit and move accordingly in the space of floating variables (“search space”). Many algorithms exist for nonlinear optimization problems: grid search, simplex (Nelder–Mead), Levenberg–Marquardt, gradient descent, genetic algorithms, simulated annealing, pattern search optimization, and more.^14,33

The particle swarm optimization (PSO) algorithm is designed to mimic the natural behavior of swarms of animals (birds, fish, bees, etc.), as it employs a large number of individuals (“particles”) that individually traverse the search space, reporting back to the “swarm” information about their local environment.^15,16 The swarm therefore samples several local minima simultaneously to help to identify the global minimum.

In PSO, a user-specified number of particles are randomly initiated within the search space at iteration n = 0. These particles each take a large number of steps (N), where each particle’s position ($\overrightarrow{x}$) and velocity ($\overrightarrow{v}$) are continually updated as

$$\begin{gathered} {\overrightarrow{x}}_i(n+1)={\overrightarrow{x}}_i(n)+{\overrightarrow{v}}_i(n+1) \\ {\overrightarrow{v}}_i(n+1)=\omega {\overrightarrow{v}}_i(n)+{\varphi }_p{r}_1(n)\left[{\overrightarrow{p}}_i(n)-{\overrightarrow{x}}_i(n)\right]+{\varphi }_g{r}_2(n)\left[\overrightarrow{g}(n)-{\overrightarrow{x}}_i(n)\right], \end{gathered}$$

where ω is the inertial parameter, ϕ_p is the cognitive (personal) parameter, ϕ_g is the social (group) parameter, r_1,2(n) are random numbers between 0 and 1, ${\overrightarrow{p}}_i(n)$ is the particle’s personal best up to iteration n, and $\overrightarrow{g}(n)$ is the group’s best up to iteration n. In this way, the particles’ behaviors in the (n+1)^th iteration are informed by the n^th. By changing the values of the inertial, cognitive, and social parameters, the behavior of the particles changes. Some settings give more independent particles that leave and explore other local minima, and some settings give more social particles that congregate within a single local minimum to find its deepest point.

In our implementation, the coordinates of a particle within the search space correlates with user-requested spin Hamiltonian parameters. Each particle’s position corresponds to a particular tensor component (g values, D and E/D values, exchange coupling, etc.). This allows us to search multiple disparate parts of the search space simultaneously, and has yielded consistently good results when employed with sufficiently many particles and steps. We have provided the program online in the Github repository.

3. Results & Discussion

Flavodiiron proteins (FDPs) are a class of enzyme known for their scavenging nitric oxide reductase (s-NOR) and/or scavenging oxygen reductase (s-O₂R) activities. The relative s-NOR and s-O₂R activities vary among FDPs and among organisms, and these differences have led to many questions about the catalytic mechanism and identity of intermediates. A unifying feature is the presence of a non-heme/non-sulfur binuclear iron active site, which has been defined crystallographically in the biferrous state only at low resolution (2.65 Å, Figure 1).^18,19 Two important intermediates in the s-NOR catalytic cycle are known to be the fully reduced biferrous (Fe^II₂) state and the ferrous–{FeNO}⁷ (using Enemark–Feltham notation³⁶) state. We have prepared samples of both of these intermediates and have characterized their electronic structures using MCD spectroscopy. Their ground state spin Hamiltonian parameters give important mechanistic insight.

3.1. Ferrous–{FeNO}⁷ Intermediate

We begin with the Fe^II{FeNO}⁷ intermediate, obtained through treatment of flavinated Fe^II₂-TmFDP with NO (~1 equiv). This intermediate shows a broad and poorly resolved feature spanning the visible region by UV–vis spectroscopy (Figure 2a). Combination of these data with CD and MCD spectra emphasizes the strength of polarized spectroscopies, revealing the presence of several transitions by simultaneous Gaussian fitting (Figure 2a–c). Through comparison with literature examples, the bands shown in the CD and MCD spectra in the 12–23000 cm⁻¹ region can be assigned as NO⁻ → Fe^III ligand-to-metal charge transfer (LMCT) {FeNO}⁷ transitions.^34,35,37 The prominent feature at ~5500 cm⁻¹ in the NIR region of the MCD spectrum also stands out as a ligand-field transition characteristic of a five-coordinate ferrous center.¹³ While the flavin centers are chromophoric, they are diamagnetic when fully reduced so they contribute negligible MCD intensity at low temperature.³⁸

Figure 2.

Electronic transitions of Fe^II{FeNO}⁷-TmFDP are revealed by fitting Gaussian curves simultaneously to (a) absorption, (b) CD, and (c) MCD spectra. Note how the {FeNO}⁷ LMCT transitions are inverted compared to the MCD spectrum of [{FeNO}⁷(EDTA)]²⁻,^34,35 demonstrating antiferromagnetic Fe^II{FeNO}⁷ coupling. (d) Fits (lines) to VTVH MCD data (points) taken at 5710 and 21550 cm⁻¹ (see arrows in (c)) confirm the coupling between the centers.

Additionally noteworthy is the opposite sign of the MCD intensity for the {FeNO}⁷ LMCT transition at 21550 cm⁻¹ in comparison with mononuclear {FeNO}⁷ complexes.^34,35 Figure 2c includes an overlay of the MCD spectra of [{FeNO}⁷(EDTA)]²⁻ scaled down by 0.2, clearly showing the sign reversal of the Fe^II{FeNO}⁷-TmFDP MCD signal along with a decrease in its absolute intensity. Both the MCD sign inversion and the decrease in intensity reflect the internal field imposed by the Fe^II at the {FeNO}⁷ center, and are a direct experimental indication of antiferromagnetic coupling between the centers. Figure 3a demonstrates how the relative intensity of C-term MCD signal changes as J becomes increasingly negative (i.e., antiferromagnetic). Comparison of experimental MCD spectra is complicated by B-term intensity, but the experimental scaling of −0.2 is similar to the C-term scaling of −0.3 predicted by Figure 3a. This is a reflection of the change in composition of the ground state wavefunction, as its largest contributor swaps M_S({FeNO}⁷) = −3/2 → +3/2 once $|-2J|\gtrsim \left|D_{{\text{Fe}}^{\text{II}}}\right|$ (antiferromagnetic, left side of Figure 3b).

Figure 3.

{FeNO}⁷ LMCT transitions are largely z-polarized so should have strongest intensity when the applied field is in the xy plane. (a) Using best-fit values from Table 1, the relative sign of C-term MCD intensity (2 K, 7 T) for a purely M_yz-polarized transition flips from that of an uncoupled (or mononuclear) {FeNO}⁷ once the Fe^II{FeNO}⁷ system is sufficiently antiferromagnetic. (b) In a field along the x molecular axis, the projection of the lowest energy wavefunction onto the {FeNO}⁷ sublevels shows how its composition changes with J (M_S quantized along $\overrightarrow{H}$). The discontinuity (J ~ −1.5 cm⁻¹) is caused by a level crossing. Note that the major contributor switches −3/2 ↔ +3/2 near J ~ −4 cm⁻¹ or $|-2J|~\left|D_{{\text{Fe}}^{\text{II}}}\right|$, paralleling the changes in MCD intensity.

This antiferromagnetic coupling is quantified by rigorous VTVH MCD fitting. The application to transitions centered on both iron centers (arrows in Figure 2c) increased the precision with which spin Hamiltonian parameters could be fit from VTVH MCD data (Figure 2d), as the spin expectation values on both centers could be simultaneously monitored. Table 1 summarizes the fits obtained with data from transitions on only the Fe^II or only the {FeNO}⁷ (first and second data columns, respectively) in comparison with a fit obtained by simultaneously operating on both with the PSO program (third column). As the fitting routine works with the dimer wavefunctions, simultaneous fitting of VTVH MCD data from both spin centers is clearly important to the accuracy of the parameters obtained. Our final spin Hamiltonian parameters here match well with those previously reported from magnetic Mößbauer and EPR spectroscopies.³⁹

Table 1.

PSO Fits of Fe^II{FeNO}⁷-TmFDP VTVH MCD data using the exchange-coupled dimer spin Hamiltonian applied to only transitions on Fe(II) or only transitions on {FeNO}⁷ or simultaneously fitting transitions on both

		VTVH Data Sets
		FeIIf	{FeNO}7 g	Bothfg
FeII	g x a	2.0(3)	2.0(2)	2.1(1)
	g y a	2.1(3)	2.0(2)	2.1(1)
	g z a	2.3(3)	2.2(2)	2.2(1)
	D b	−15(3)	−6(3)	−9(3)
	E/D	0.1(2)	0.1(1)	0.11(5)
{FeNO}7	g c	2.0023	2.0023	2.0023
	D b	16(5)	6(5)	12(4)
	E/Dc	0	0	0
J b d		−8(2)	−7(2)	−8(2)
WRSSe		0.0050	0.0025	0.0021

^aOne g value was floated and the others were calculated using a ligand field approximation $\tilde{g}\approx g+(2\kappa /\lambda )\tilde{D}$ assuming κ = 0.8, λ = −100 cm⁻¹

^bUnits of cm⁻¹

^cConstrained

^dUsing a $-2J{\overrightarrow{S}}_1\cdot {\overrightarrow{S}}_2$ formalism

^eWeighted residual-squared sum (WRSS) reported here for all data points, even if the fit was performed on a restricted data set

^fUsing data at 1750 nm

^gUsing data at 514 and 464 nm

The magnitude of the antiferromagnetic coupling between the spin centers indicates a strong superexchange pathway, demonstrating that a bridging hydroxide must be present. Our MCD data reveal a five-coordinate geometry of the Fe^II center and four ligands are crystallographically characterized as FeHis₂(Asp/Glu)₂ (Figure 1), so this hydroxide ligand must be the final ligand. Thus, we have a full definition of the primary coordination sphere. Previous work has suggested that the adjacent metal nitrosyl may bend toward this ferrous site;⁴⁰ our data reveal that any such interaction between the Fe^II center and the O-terminus of the {FeNO}⁷ must be weak. This leaves an open sixth coordination site at the Fe^II center for further NO binding, a crucial step in the catalytic mechanism for binding of a second NO, which spectroscopy of trapped intermediates indicates is a crucial step in the catalytic mechanism.^41,42

3.2. Biferrous Intermediate

Treatment of biferric Fe^III₂-TmFDP in its flavinated state with sodium dithionite (Na₂S₂O₄, 4.6 equiv) over >5 h under anaerobic conditions yielded a sample of Fe^II₂-TmFDP. Its NIR CD and MCD spectra (Figure 4a,b) unambiguously revealed the presence of two inequivalent ferrous centers, as each Fe^II may provide only two ligand-field transitions in this energy region. The uniformly positive sign of MCD intensity is a result of the strong coupling between the ground state and low-lying ligand-field excited states, giving deviation from the sum-rule as predicted by eq 5.

Figure 4.

Ligand-field electronic transitions of Fe^II₂-TmFDP in the NIR region are shown in the (a) CD and (b) MCD spectra. (c) The VTVH MCD data (points) and fits (lines).

The energies of these transitions are directly linked to the coordination geometry: six-coordinate sites exhibit two transitions in the 8–12000 cm⁻¹ region, four-coordinate sites in the 3–7000 cm⁻¹ region, and five-coordinate sites exhibit one around ~6000 and one around ~10000 cm⁻¹.¹³ The manner in which the four transitions of Figure 4 pair thus provides structural information about the Fe^II₂ active site. Using the labels in Figure 4a, a 13/24 pairing scheme would be consistent with a five/five-coordinate system, a 12/34 pairing scheme with a four/six-coordinate system, and a 14/23 pairing scheme does not match particularly well as the 23 splitting is small and in a low-energy region. While we cannot rigorously exclude alternate geometries from our VTVH MCD data alone, the energies of the Fe^II₂-TmFDP transitions lead us to suggest a five/five-coordinate system (13/24 pairing). This is also supported by the predominant s-O₂R activity of TmFDP,⁴³ as two five-coordinate centers would retain an open coordination site at each iron required for efficient binding and reduction of O₂.⁴⁴

The VTVH MCD data we report here are intriguing as their analysis contrasts with that of a previous magnetic Mößbauer study.⁴⁵ The Mößbauer study revealed a single quadrupole doublet, suggesting the Fe^II₂ site to comprise two strongly antiferromagnetically-coupled (J = −16 cm⁻¹) and electronically-equivalent ferrous centers. The magnitude of this J led the authors to propose a μ-OH bridge between iron centers. However, the spin Hamiltonian fit from Mößbauer data do not describe our VTVH MCD data well (see SI). The distinct excited states we observed for each Fe^II site in Figure 4b suggest variations in electronic structure at each metal. Additionally, our VTVH MCD data at multiple wavelengths are best fit using the PSO program with the two Fe^II ions experiencing weak antiferromagnetic coupling of J = −3 ± 1 cm⁻¹ (Table 2).

Table 2.

Best-fit spin Hamiltonian parameters for the reduced Fe^II₂-TmFDP species from simultaneous fitting of VTVH MCD data assuming 13/24 pairing

	FeIIA	FeIIB
g iso a	2.1	2.1
D	−10.2(7) cm−1	−14(1) cm−1
E/D	0.26(2)	0.18(2)
J b	−3.5(5) cm−1

^aConstrained to be isotropic and 2.1 to reduce strongly correlated variables

^bUsing a $-2J{\overrightarrow{S}}_1\cdot {\overrightarrow{S}}_2$ formalism ·

The causes of the differences between the MCD and Mößbauer studies will need further study. Our MCD samples were prepared with deuterated buffer (50 mM MOPS, pD 7.3) to eliminate O–H stretch overtones in the NIR region, and they required addition of a glassing agent (glycerol-d₈, 50% v/v) to achieve optical transparency; whereas, Mößbauer samples were prepared in protiated buffer without glycerol. The reductant also differs between the studies: our MCD samples were prepared with sodium dithionite and the Mößbauer samples with NADH and reductases (NADH–rubredoxin oxidoreductase and rubredoxin). We observed no significant perturbations to the ligand field in the CD spectra of Fe^II₂-TmFDP with/without glycerol, or when reduced using NADH/reductases (see SI).

The MCD/Mößbauer differences underscore a strength of the joint ground state–excited state interrogation provided here by MCD spectroscopy: weakly overlapping transitions have given us insight into the spin expectation value at each center. Our best-fit value for the J coupling also calls into question the presence of an intermetallic hydroxide bridge in the Fe^II₂-TmFDP state, as stronger antiferromagnetic coupling would generally be expected.⁴⁶

3.3. Performance of PSO

To allow us to judge the results of PSO fitting of our VTVH MCD data, we also undertook a comprehensive grid search across reasonable spin Hamiltonian parameters for Fe^II₂ and Fe^II{FeNO}⁷ systems. When a sufficiently fine grid is used, such a grid search is a reliable tool to locate the lowest minimum within the search space, as every permutation of parameters is tested by the computer. Identification of minimum grid points allowed us to use BFGS (Broyden–Fletcher–Goldfarb–Shanno) optimization to find all local minima (see SI section S2.2), and the global minimum of each data set is tabulated in Table 3 alongside the best PSO fits.

Table 3.

Agreement between spin Hamiltonian parameters fit by PSO versus those from a comprehensive grid search

FeII{FeNO}7-TmFDP			FeII2-TmFDP
	PSO	Grid		PSO	Grid
g x1 a	2.12(8)	2.1(1)	D 1 c	−10.2(7)	−11.3(7)
D 1 b,c	−9(3)	−6(2)	(E/D)1	0.26(2)	0.24(2)
(E/D)1b	0.11(5)	0.12(8)	D 2 c	−14(1)	−15(1)
D 2 b,c	12(4)	16(4)	(E/D)2	0.18(2)	0.17(2)
J c	−8(2)	−10(2)	J c	−3.5(5)	−3(1)

^aOther g values are calculated as described in Table 1

^bFe^II is center 1 and {FeNO}⁷ is center 2

^cUnits of cm⁻¹

The agreement between PSO and grid search parameters highlights the potential of PSO to allow for rapid exploring of a multidimensional search space. The speed of PSO is one of its major advantages, and it stands in stark contrast with the grid search procedure. The nature of a grid search necessarily causes the grid size to grow exponentially with the number of fitted parameters, and this places limits on the fineness and the dimension of the grid. For example, 32 particles taking 128 steps requires just over 4000 calculations; whereas, a five-dimensional grid with ten steps in each direction requires 10⁵ or 100,000 calculations. PSO additionally has the potential to avoid falling into a single local minimum, which is demonstrated here by its avoidance of the other local minima identified in the grid search. However, one must also be aware that PSO, like all nonlinear optimization algorithms, cannot ensure that a global minimum has truly been found. It is thus good practice to run multiple fits of the same data set or to use bootstrapping methods to ensure a good understanding of local minima is established. Another useful tool is to check whether the WRSS values of the particles appear to be converging during the runs.

4. Conclusion

We have found the PSO algorithm to be an invaluable tool in performing the multidimensional fits demanded by VTVH MCD spectroscopy. Especially in cases where well-resolved transitions are observed on each metal center, simultaneous fitting provides ideal circumstances for obtaining exchange-coupled spin Hamiltonian parameters.

Analysis of the Fe^II{FeNO}⁷-TmFDP intermediate demonstrated this clearly, providing spin Hamiltonian parameters that matched well with previous studies³⁹ and confirmed the presence of a μ-hydroxide ligand between the metals. This antiferromagnetic coupling resulted in the dramatic change in sign and magnitude of the MCD signal in Figure 2c for the NO⁻ → Fe^III LMCT transition relative to a mononuclear {FeNO}⁷ complex. Such changes in sign and magnitude are thus useful diagnostics for exchange coupling, and should be broadly applicable among multispin systems. The MCD spectrum also showed a NIR ligand-field transition revealing the electronic structure of the Fe^II center to reflect a five-coordinate geometry. Importantly, this leaves an open coordination site for a second NO in the next step of the catalytic cycle, enabling N⋯N coupling (Scheme 1).

Scheme 1.

Core active-site structures of several important intermediates in the s-NOR activity of TmFDP. The gain/loss of H⁺ is not explicitly indicated. The ferrous sites in Fe^II₂ and Fe^II{FeNO}⁷-TmFDP are five-coordinate, and the Fe^II₂-TmFDP is shown without a μ-OH due to the small J fit from VTVH MCD data.

Our MCD data on the reduced Fe^II₂-TmFDP intermediate have also been intriguing, as they call into question the presence of a hydroxide bridge between the irons. The small J coupling found in our data could arise simply from the μ-1,3-bridging carboxylate resolved in the X-ray crystallographic study (Figure 1), and this is a notable difference from earlier Mößbauer studies.⁴⁵ Whether this arises from a change in reductant, from the presence/absence of glycerol, or some other variable remains unclear. Further studies are needed to reconcile the two pictures of the Fe^II₂-TmFDP active site defined by Mößbauer and MCD spectroscopies, and to determine if a five/five-coordinate biferrous structure with small J coupling is a general feature of FDPs.