^{1,a)}and Juan E. Peralta

^{2}

### Abstract

We present a method for calculating magnetic coupling parameters from a single spin-configuration via analytic derivatives of the electronic energy with respect to the local spin direction. This method does not introduce new approximations beyond those found in the Heisenberg-Dirac Hamiltonian and a standard Kohn-Sham Density Functional Theory calculation, and in the limit of an ideal Heisenberg system it reproduces the coupling as determined from spin-projected energy-differences. Our method employs a generalized perturbative approach to constrained density functional theory, where exact expressions for the energy to second order in the constraints are obtained by analytic derivatives from coupled-perturbed theory. When the relative angle between magnetization vectors of metal atoms enters as a constraint, this allows us to calculate all the magnetic exchange couplings of a system from derivatives with respect to local spin directions from the high-spin configuration. Because of the favorable computational scaling of our method with respect to the number of spin-centers, as compared to the broken-symmetry energy-differences approach, this opens the possibility for the blackbox exploration of magnetic properties in large polynuclear transition-metal complexes. In this work we outline the motivation, theory, and implementation of this method, and present results for several model systems and transition-metal complexes with a variety of density functional approximations and Hartree-Fock.

The authors acknowledge support from National Science Foundation (NSF) (Grant No. DMR-1206920).

I. INTRODUCTION

II. GENERALIZED PERTURBATIVE APPROACH TO CONSTRAINED KS-DFT

III. APPLICATION TO MAGNETIC EXCHANGE COUPLING PARAMETERS

A. Dinuclear systems

B. Polynuclear systems

IV. CONCLUSIONS

### Key Topics

- Density functional theory
- 19.0
- Laser Doppler velocimetry
- 13.0
- Torque
- 13.0
- Perturbation theory
- 9.0
- Antiferromagnetism
- 8.0

##### H01F13/00

## Figures

(Left) Molecular scheme of [Fe2OCl6]2 −, where Cl, O, and Fe atoms are colored green, red, and metallic gray, respectively. (Right) Löwdin spin populations for the zeroth-order z-component , first-order x-component , and the first-order rotation defined as . These results were obtained using LDA.

(Left) Molecular scheme of [Fe2OCl6]2 −, where Cl, O, and Fe atoms are colored green, red, and metallic gray, respectively. (Right) Löwdin spin populations for the zeroth-order z-component , first-order x-component , and the first-order rotation defined as . These results were obtained using LDA.

Schematic representation of blackbox approach to calculating couplings in polynuclear TM complexes, for the hypothetical case of a pentanuclear system. Here (a) indicates that the off-diagonal elements of the constraint Hessian grant directly the “off-terms” of the couplings, while (b) indicates that the remaining N − 1 couplings are obtained by summing the respective columns of derivatives,

Schematic representation of blackbox approach to calculating couplings in polynuclear TM complexes, for the hypothetical case of a pentanuclear system. Here (a) indicates that the off-diagonal elements of the constraint Hessian grant directly the “off-terms” of the couplings, while (b) indicates that the remaining N − 1 couplings are obtained by summing the respective columns of derivatives,

(a) Molecular scheme for the H–He “snowflake” model system, with H–He distance of 1.5 Å. By symmetry there are two unique magnetic coupling, J 12 and J 13. (b) Zeroth-order spin-density. (c) First-order spin-density from a torque perturbation between H-1 and H-2. (d) First-order spin-density from a torque perturbation between H-1 and H-3. All spin-densities visualized here were obtained with LDA.

(a) Molecular scheme for the H–He “snowflake” model system, with H–He distance of 1.5 Å. By symmetry there are two unique magnetic coupling, J 12 and J 13. (b) Zeroth-order spin-density. (c) First-order spin-density from a torque perturbation between H-1 and H-2. (d) First-order spin-density from a torque perturbation between H-1 and H-3. All spin-densities visualized here were obtained with LDA.

(Left) Molecular structure of the biomimetic trinuclear MnIV complex. 74–77 (Right-Top) perturbation Hessian, in units of Hartrees. (Right-Bottom) Constraint Hessian, in units of cm−1. These results were obtained using LDA+15%HFX.

(Left) Molecular structure of the biomimetic trinuclear MnIV complex. 74–77 (Right-Top) perturbation Hessian, in units of Hartrees. (Right-Bottom) Constraint Hessian, in units of cm−1. These results were obtained using LDA+15%HFX.

(Left) Molecular structure of the ferromagnetic trinuclear MnIII complex. 78 (Right-Top) Perturbation Hessian, in units of Hartrees. (Right-Bottom) Constraint Hessian, in units of cm−1. These results were obtained using LDA+15%HFX.

(Left) Molecular structure of the ferromagnetic trinuclear MnIII complex. 78 (Right-Top) Perturbation Hessian, in units of Hartrees. (Right-Bottom) Constraint Hessian, in units of cm−1. These results were obtained using LDA+15%HFX.

(Left) Molecular structure of the tetranuclear FeIII “ferric-star” complex. 79 (Right-Top) Perturbation Hessian, in units of Hartrees. (Right-Bottom) Constraint Hessian, in units of cm−1. These results were obtained using LDA+15%HFX.

(Left) Molecular structure of the tetranuclear FeIII “ferric-star” complex. 79 (Right-Top) Perturbation Hessian, in units of Hartrees. (Right-Bottom) Constraint Hessian, in units of cm−1. These results were obtained using LDA+15%HFX.

## Tables

Comparison of magnetic coupling parameters calculated by spin-projected energy-differences (J ΔE ), and as calculated by our noncollinear coupled-perturbed method utilizing derivatives with respect to spin rotations (J HS and J BS ). All couplings are in cm−1.

Comparison of magnetic coupling parameters calculated by spin-projected energy-differences (J ΔE ), and as calculated by our noncollinear coupled-perturbed method utilizing derivatives with respect to spin rotations (J HS and J BS ). All couplings are in cm−1.

Comparison of magnetic coupling parameters calculated by energy differences (J ΔE ), and as calculated by our noncollinear coupled-perturbed method utilizing derivatives with respect to spin rotations (J HS ), for the Hydrogen-Helium “snowflake” system (shown in Figure 3 ). All couplings are in cm−1.

Comparison of magnetic coupling parameters calculated by energy differences (J ΔE ), and as calculated by our noncollinear coupled-perturbed method utilizing derivatives with respect to spin rotations (J HS ), for the Hydrogen-Helium “snowflake” system (shown in Figure 3 ). All couplings are in cm−1.

Comparison of magnetic coupling parameters calculated by energy differences (J ΔE ), and as calculated by our noncollinear coupled-perturbed method utilizing derivatives with respect to spin rotations (J HS ), for the biomimetic trinuclear MnIV complex 74–77 (shown in Figure 4 ). All couplings are in cm−1.

Comparison of magnetic coupling parameters calculated by energy differences (J ΔE ), and as calculated by our noncollinear coupled-perturbed method utilizing derivatives with respect to spin rotations (J HS ), for the biomimetic trinuclear MnIV complex 74–77 (shown in Figure 4 ). All couplings are in cm−1.

Comparison of magnetic coupling parameters calculated by energy differences (J ΔE ), and as calculated by our noncollinear coupled-perturbed method utilizing derivatives with respect to spin rotations (J HS ), for the ferromagnetic trinuclear MnIII complex 78 (shown in Figure 5 ). All couplings are in cm−1.

Comparison of magnetic coupling parameters calculated by energy differences (J ΔE ), and as calculated by our noncollinear coupled-perturbed method utilizing derivatives with respect to spin rotations (J HS ), for the ferromagnetic trinuclear MnIII complex 78 (shown in Figure 5 ). All couplings are in cm−1.

Comparison of magnetic coupling parameters calculated by energy differences (J ΔE ), and as calculated by our noncollinear coupled-perturbed method utilizing derivatives with respect to spin rotations (J HS ), for the tetranuclear FeIII “ferric-star” complex 79 (shown in Figure 6 ). All couplings are in cm−1.

Comparison of magnetic coupling parameters calculated by energy differences (J ΔE ), and as calculated by our noncollinear coupled-perturbed method utilizing derivatives with respect to spin rotations (J HS ), for the tetranuclear FeIII “ferric-star” complex 79 (shown in Figure 6 ). All couplings are in cm−1.

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