Quasi-Restricted Orbital Treatment for the Density Functional Theory

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Quasi-Restricted Orbital Treatment for the DFT Calculations of the Spin–orbit Term of Zero-Field Splitting Tensors Kenji Sugisaki, Kazuo Toyota, Kazunobu Sato, Daisuke Shiomi, and Takeji Takui J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b10253 • Publication Date (Web): 15 Nov 2016 Downloaded from http://pubs.acs.org on November 18, 2016

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The Journal of Physical Chemistry

Quasi-restricted orbital treatment for the DFT calculations of the spin–orbit term of zero-field splitting tensors Kenji Sugisaki,* Kazuo Toyota,* Kazunobu Sato, Daisuke Shiomi, and Takeji Takui* Department of Chemistry and Molecular Materials Science, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

ABSTRACT: A quasi-restricted orbital (QRO) approach for the calculation of the spin–orbit term of zero-field splitting tensors (DSO tensors) by means of DFT importantly features in the fact that it is free from spin contamination problems because it uses spin eigenfunctions for the zeroth order wave functions. In 2011, however, Schmitt and coworkers pointed out that in the originally proposed QRO working equation some possible excitations were not included in their sum-over-states procedure, which causes spurious DSO contributions from closed-shell subsystems located far from the magnetic molecule under study. We have revisited the derivation of the QRO working equation and modified it, making it include all possible types of excitations in the sum-over-states procedure. We have found that the spurious DSO contribution can be eliminated by taking into account contributions from all possible types of singly excited

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configuration state functions. We have also found that only the SOMO(α) → SOMO(β) excited configurations have nonzero contributions to the DSO tensors as long as α and β spin orbitals have the same spatial distributions and orbital energies. For the DSO tensor calculations by using a ground state wavefunction free from spin contamination, we propose a Natural Orbital-Based Pederson–Khanna (NOB-PK) method, which utilizes the single determinant wavefunction consisting of natural orbitals in conjunction with the Pederson–Khanna (PK) type perturbation treatment. Some relevant calculations revealed that the NOB-PK method can afford more accurate DSO tensors than the conventional PK method as well as the QRO approach in MnII complexes and ReIV based single molecule magnets.


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Introduction Zero-field splitting (ZFS) is energy differences of electron spin multiplet sublevels for otherwise degenerate states in the absence of an external magnetic field, and it is an important term in the spin Hamiltonian for open-shell systems having a spin quantum number S larger than 1/2. In the spin Hamiltonian the ZFS term is written as in eq. (1), and characterized by the second rank tensor D. H ZFS = S ⋅ D ⋅ S

(1)

Major physical origins of ZFS are spin–spin dipolar (SS) and spin–orbit (SO) couplings, and the former and latter appear as the first and second order terms, respectively, in the perturbation expansion starting from the non-relativistic Schrödinger equation.1–3 Since the D tensor contains information on spatial distributions of electron spins in molecular systems, quantitative determination and analysis of the D tensor are of essential importance. We emphasize that the second order contributions to ZFS are not necessarily to be smaller than the first order ones. Only for triplet or high-spin genuine hydrocarbons the first order contributions dominate ZFS. In the transition metal complexes of S ≥ 1, magnitude of ZFS strongly depends on metal ions and their coordination scheme, and is governed by the strength of SOC. Even for hybrid molecular systems such as organic open shell entities with transition metals coordination, their SOC enormously contributes to electronic magnetic properties. From the experimental side, rapid development in high-field/high-frequency electron spin resonance (HF-ESR)4,5 techniques enables us to observe allowed transitions among magnetic sublevels arising from sizable ZFS, which are out of range of transition energy at conventional microwave frequencies such as Xband (9.5 GHz) and Q-band (35 GHz). To date, ZFS parameters of hundreds of systems have been experimentally determined by means of HF-ESR.6–10 As the number of experimental


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reports increases, development of theoretical methodologies for D tensor calculations becomes more and more important, not only for understanding the mechanisms of ZFS but also for theoretically designing novel magnetic materials such as single molecule magnets with large magnetic anisotropies. We note that putative effective spin Hamiltonian based analyses for high spins with sizable ZFS parameters and their experimental principal g-values so far documented are misleading from the viewpoint of quantum chemistry. Typical examples are ferric iron complexes in their high spin ground state in low symmetry with important biological implications. High spin CoII complexes in their low symmetry are also the cases. Recently, quantum properties of molecular nanomagnets have attracted attention in the field of quantum computing and quantum information processing, as candidates for molecular electron spin-based quantum bits (molecular spin qubits) and quantum spin memories.11–14 Molecular high spin qubits utilizing high spin molecular systems can afford a counterpart of quadrupolar nuclear qubits in NMR quantum computing. This requires proper tuning of D tensors which allows us to execute quantum control of the qubits by current pulse microwave spin technology. In the regime of strong coupling between microwave photons and molecular spin qubits in ensemble, properties of D tensors are essential, which are underlain by contributions of SOC and SSC. Referred to the implementation of quantum spin memories, molecular devices designing requires the tuning of contributions of SOC to ZFS. The past decade has witnessed rapid progress in quantum chemical approaches for D tensor calculations, in spite of a small number of special software available. There have been significant developments of theoretical frameworks for the SS and SO terms of D tensors (DSS and DSO tensors, respectively) made by several groups,15–28 and some of the methods have been implemented in published quantum chemistry program packages such as Dalton,29 ADF,30 and


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ORCA.31 For ab initio approaches, the multi-configurational self-consistent field (MCSCF)based approaches are widely used for both DSS and DSO tensor calculations.32–55 In 2009, we proposed a hybrid complete active space self-consistent field (CASSCF)/multi-reference secondorder Møller–Presset (MRMP256–58) approach25 for the DSO tensor calculations, which utilizes CASSCF SOC integrals and MRMP2 excitation energies in conjunction with the sum-over-states (SOS) equation, to take into account both static and dynamical electron correlation effects effectively. Note that, the second order N-electron valence state perturbation theory (NEVPT2),59 spectroscopy oriented

configuration

interaction

(SORCI),60

and difference dedicated

configuration interaction (DDCI)61 methods are frequently used for DSO tensor calculations. For DFT-based DSO tensor calculations, the two component noncolinear spin density functional method,23 the Pederson–Khanna (PK),20 the quasi-restricted orbital (QRO),21 and the coupled-perturbed (CP)24 methods have been widely used. Among these four approaches, the QRO method utilizes spin eigenfunctions as the zeroth order wavefunctions in the perturbation treatment, and therefore it is free from spin contamination problem. In 2011, however, Schmitt and coworkers pointed out that the closed shell subsystems located far from magnetic center (enough to regard to have no interactions between them) give a spurious contribution to the DSO tensor in the QRO approach, because only few classes of excitations have been selected for the SOS procedure.62 In fact, they reported QRO calculations of the DSO tensor of Mn(acac)3 (acac = acetylacetonate) and a supersystem composed of Mn(acac)3 and a PdCl2(NH3)2 molecule 100 Bohr away from the magnetic center, and revealed that inclusion of diamagnetic Pd cluster results in the change of the absolute sign of the D values (D = −1.61 cm−1 for Mn(acac)3, and +1.64 cm−1 for the Mn(acac)3 + PdCl2(NH3)2 supersystem). This fact encouraged us to revisit the


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derivation of the QRO working equation, aiming to obtain a new approach logically taking into account all classes of excitations and to examine the performance of it. This paper is organized as follows: In the section of Theory, the derivation of the QRO working equation is revisited and the derived expressions are given. Importantly, it is shown that no spurious DSO contribution occurs if all possible types of one-electron excitations are considered in the SOS procedure. Furthermore, we show that only SOMO(α) → SOMO(β) excitations have nonzero DSO contributions if α and β spin orbitals have the same spatial distributions and orbital energies. Then, we propose a natural orbital-based Pederson–Khanna (NOB-PK) approach for the DSO tensor calculations by means of DFT. Some significant results of the DSO tensor calculations of the spin-sextet MnII complexes and the spin-quartet ReIV single molecule magnets, by using the NOB-PK method, are given in the section of Results and Discussion.

Theory In the configuration interaction-type wavefunction theories such as CIS, CISD, and CASSCF, the DSO tensors can be calculated by means of a SOS equation described in eq. (2).15

DklSO

= ∑ C (σ ) × n ,σ

Ψ0, S , S H kSO Ψn, S +σ , S +σ

Ψn, S + σ , S +σ H lSO Ψ0, S , S

E n − E0

(2)

Here, HSO is a Breit–Pauli spin–orbit Hamiltonian and C(σ) is a prefactor depending not only on the spin quantum number but also on the magnetic quantum number of the excited states to be summed over. The prefactors, C(σ)’s are given as follows: C (σ = +1) = −

1 1 1 , C (σ = 0 ) = − 2 , C (σ = −1) = − (S + 1)(2S + 1) S (2 S − 1) S

(3)


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It should be noted that C(σ)'s in eq. (3) are different from those in our previous publications,38– 40

because the selection of the spin sublevels in the SOS equation is different. In eq. (2) MS = S

states are chosen for convenience of the derivation. To apply the SOS equation within a DFT framework, Neese proposed the quasi-restricted orbital (QRO) approach.22 In the QRO approach, the single determinant consisting of natural orbitals is used for the ground state wavefunction Ψ0. To construct the single determinant wavefunction, natural orbitals with their eigenvalues (occupation numbers) close to 2.0, 0.0, and equal to 1.0 are taken to be doubly occupied (DOMO), unoccupied (VIRTUAL), and singly occupied (SOMO), respectively. Orbital energies are obtained by the canonicalization within the invariant subspaces: The doubly occupied orbitals are transformed to diagonalize spin-down Fock operator Fβ, unoccupied orbitals to diagonalize spin-up Fock operator Fα, and the singly occupied orbitals to diagonalize their average (Fα + Fβ)/2, and then orbital energies are obtained as an expectation value of the Fock operator. For the excited states wavefunctions Ψn, the following four types of excitations are taken into account: (1) the DOMO(β) → SOMO(β) excitations of σ = 0, (2) the SOMO(α) → VIRTUAL(α) excitations of σ = 0, (3) the SOMO(α) → SOMO(β) excitations of σ = −1, and (4) the DOMO(β) → VIRTUAL(α) excitations of σ = +1. By using relevant orbital energy differences as approximation of the excitation energies, the following working equation is derived.

DklSO

ϕiβ hkSO ϕ pβ ϕ pβ hlSO ϕiβ 1 =− 2 ∑ 4S i, p ε βp − ε i

ϕ pα hkSO ϕ aα ϕaα hlSO ϕ pα 1 − 2 ∑ 4S p , a ε a − ε αp


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ϕ pα hkSO ϕ qβ ϕ qβ hlSO ϕ pα 1 + ∑ 2 S (2 S − 1) p , q ε qβ − ε αp ϕiβ hkSO ϕaα ϕ aα hlSO ϕiβ 1 + ∑ 2(S + 1)(2 S + 1) i , a εa − εi

(4)

Unless otherwise specified, we use subscripts i for doubly occupied orbitals, p and q for singly occupied orbitals, and a for virtual orbitals. It should be noted that the prefactor of the third term of the right hand side in eq. (4) is 1/4S(2S − 1) in the Neese's original paper,22 but it should be modified, because the factor of −1/S(2S − 1) came from the SOS equation (eq. (3)) and −1/2 came from the matrix element of s+1 and s−1 operators: H SO = ∑ (− 1) h− m sm = h0 s0 − h+1s−1 − h−1s+1 m

m

1

s+1 = −

s−1 =

2

1 2

(s

(s x

x

)

+ is y = −

)

− is y =

1 2

1 2

s+

s−

(5)

(6)

(7)

As previously pointed out by Schmitt and coworkers, the above-mentioned derivation of the QRO equation has a serious problem due to incomplete selections of the excited states. In fact, the following five types of the excitations are omitted in the derivation of the original QRO working equation: (I) DOMO(α) → SOMO(β), (II) SOMO(α) → VIRTUAL(β), (III) DOMO(α) → VIRTUAL(α), (IV) DOMO(β) → VIRTUAL(β), and (V) DOMO(α) → VIRTUAL(β). The single Slater determinants stemming from these types of the electron excitation include unpaired electrons of spin-β, and therefore they are not eigenfunctions of the S2 operator. Thus, we have to construct spin symmetry-adapted configuration state functions (CSFs) by making use of the


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proper linear combinations of the Slater determinants, to take into account the contributions to DSO from these excited single determinants. According to the structure of spin eigenfunctions, the contributions to each spin multiplicity from spin-mixed single determinants can be derived analytically, and expansion coefficients are described as the function of the spin quantum number S. For example, DOMO(α) → SOMO(β) excitations generate single determinants with a β-spin in the DOMO region. The resultant singly excited determinants are expressed as the linear combinations of the eigenfunctions of the spin quantum number S and S − 1, with their expansion coefficients of 1 2 S and

(2 S − 1) 2 S

, respectively. Thus, the DSO contributions

from one-electron excitation of the type (I) can be written as follows:

DklSO

ϕ iα hkSO ϕ pβ ϕ pβ hlSO ϕ iα  1  1  1 (iα → pβ ) =   ×   ×  −  ∑ ε βp − ε i  2 S   S   2 i , p

SO SO   1  ϕiα hk ϕ pβ ϕ pβ hl ϕiα 1  2S − 1    ×  −  ∑ +  ×  − ε pβ − ε i  2S   S (2 S − 1)   2 i , p

(8)

Here, the first and the second terms in the right hand side correspond to the spin quantum number S and S − 1 contributions, respectively. The expansion coefficient in each term consists of three factors: the expansion coefficient of CSF, the prefactor from the SOS equation (see eq. (3) and Supporting Information), and the matrix element of the spin operator. However, we only have to add the second term to eq. (4), because the SOS expansion of eq. (2) contains the states the spin quantum number of which is equal to the magnetic quantum number. In other words, DSO contributions from the first term of eq. (8) are already included in eq. (4) via (iβ → pβ) excitations and C(σ = 0). As a result, the DOMO(α) → SOMO(β) contribution becomes as follows:


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DklSO

ϕiα hkSO ϕ pβ ϕ pβ hlSO ϕiα 1 (iα → pβ ) = 2 ∑ 4S i , p ε pβ − ε i

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(9)

In a similar way, the SOMO(α) → VIRTUAL(β) contributions (type (II)) can be calculated.

DklSO

ϕ pα hkSO ϕaβ ϕaβ hlSO ϕ pα 1 ( pα → aβ ) = 2 ∑ 4S p , a ε a − ε αp

(10)

For the types (III) and (IV), the DOMO(α) → VIRTUAL(α) and DOMO(β) → VIRTUAL(β) excited determinants span the basis of the same CSF. Therefore, the (iα → aα) and (iβ → aβ) crossing terms (the fifth and sixth terms in the right hand side of eq. (11)) appear in the working equation: DklSO (iα → aα ) + DklSO (iβ → aβ ) SO SO  1   1   1  ϕiα hk ϕ aα ϕ aα hl ϕiα = ×  ×  ∑ ε a − εi  2 S + 2   2S + 1   4 i , a

SO SO  2 S + 1   1   1  ϕiα hk ϕ aα ϕ aα hl ϕiα +  ×  − 2  ×  ∑ ε a − εi  2 S + 2   S   4 i , a

SO SO  1   1   1  ϕiβ hk ϕ aβ ϕ aβ hl ϕiβ + ×  ×  ∑ ε a − εi  2 S + 2   2 S + 1   4 i , a

SO SO  2 S + 1   1   1  ϕiβ hk ϕ aβ ϕ aβ hl ϕiβ +  ×  − 2  ×  ∑ εa − εi  2 S + 2   S   4 i , a

SO SO  1   1   1  ϕiα hk ϕ aα ϕ aβ hl ϕiβ −  ×  − 2  ×  ∑ ε a − εi  2 S + 2   S   4 i , a

SO SO  1   1   1  ϕiβ hk ϕ aβ ϕ aα hl ϕiα −  ×  − 2  ×  ∑ ε a − εi  2 S + 2   S   4 i , a

(11)


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Here, the first and the third terms represent the spin quantum number S + 1 contributions and should be omitted, because these contributions are already included via DOMO(β) → VIRTUAL(α) excitations and C(σ = +1). The DOMO(α) → VIRTUAL(β) excited determinants of the type (V) carry two β-spins and therefore they are expressed as the linear combinations of the eigenfunctions of three different spin multiplicities (σ = +1, 0, and −1). DklSO (iα → aβ ) SO SO   1  1  ϕiα hk ϕ aβ ϕ aβ hl ϕiα  × (− 1) ×  −  ∑ =  ε a − εi  2 i , a  (S + 1)(2S + 1) 

SO SO  1   1   1  ϕiα hk ϕ aβ ϕ aβ hl ϕiα +  ×   ×  − ∑ ε a − εi  S + 1   S   2 i , a

SO SO   1  ϕiα hk ϕ aβ ϕ aβ hl ϕiα 1  2S − 1    ×  −  ∑ +  ×  − ε a − εi  2S + 1   S (2 S − 1)   2 i , a

(12)

The first, second, and third terms in the right hand side of eq. (12) correspond to the spin quantum number S + 1, S, and S − 1 terms, respectively. The first term is already taken into account in the QRO approach via DOMO(β) → VIRTUAL(α) excitations, and the second term via DOMO(α) → VIRTUAL(α) and DOMO(β) → VIRTUAL(β) excitations, and therefore only the third term of σ = −1 contribution should be added. By summing up all independent DSO contributions, the following working equation can be derived:

DklSO

ϕiβ hkSO ϕ pβ ϕ pβ hlSO ϕiβ 1 =− 2 ∑ 4S i, p ε βp − ε i


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ϕipα hkSO ϕaα ϕaα hlSO ϕ pα 1 − 2 ∑ 4S p , a ε a − ε αp ϕ pα hkSO ϕ qβ ϕ qβ hlSO ϕ pα 1 + ∑ 2 S (2 S − 1) p , q ε qβ − ε αp ϕiβ hkSO ϕaα ϕ aα hlSO ϕiβ 1 + ∑ 2(S + 1)(2 S + 1) i , a εa − εi ϕiα hkSO ϕ pβ ϕ pβ hlSO ϕiα 1 + 2∑ 4S i , p ε pβ − ε i ϕ pα hkSO ϕaβ ϕaβ hlSO ϕ pα 1 + 2 ∑ 4S p , a ε a − ε αp ϕiα hkSO ϕaα ϕaα hlSO ϕiα 2S + 1 − 2 ∑ ε a − εi 8S (S + 1) i , a ϕiβ hkSO ϕaβ ϕaβ hlSO ϕiβ 2S + 1 − 2 ∑ εa − εi 8S (S + 1) i , a ϕiα hkSO ϕaα ϕaβ hlSO ϕiβ 1 + 2 ∑ ε a − εi 8S (S + 1) i , a ϕiβ hkSO ϕaβ ϕaα hlSO ϕiα 1 + 2 ∑ ε a − εi 8S (S + 1) i , a ϕiα hkSO ϕaβ ϕaβ hlSO ϕiα 1 + ∑ 2S (2 S + 1) i , a ε a − εi

(13)

The first four terms in the right hand side of eq. (13) are included in the conventional QRO working equation. The others (from the fifth to eleventh terms in eq. (13)) should be included for the "complete" SOS treatment. In case of the QRO treatment, spin-α and β electrons occupy the


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same spatial orbital, and therefore we can collect all into four (DOMO → SOMO, SOMO → VIRTUAL, SOMO → SOMO, and DOMO → VIRTUAL) terms. The result is very interesting. DklSO (i → p ) = 0

(14)

DklSO ( p → a ) = 0

(15)

DklSO (i → a ) = 0

(16)

Thus, only SOMO → SOMO terms can afford nonzero DSO contribution. This result is indicative of both positive and negative aspects of the contribution. It obviously guarantees that isolated closed-shell singlet subsystems do not contribute to the DSO tensor. On the other hand, it implies that the DSO contributions from the excited configurations involving DOMO and VIRTUAL cannot be evaluated even if they are physically important, as long as occupied and virtual orbitals have the same spatial distributions and orbital energies between spin-α and β. The latter gives rise to a serious drawback of the approach when applied to some open shell systems, e.g., those containing halogen atoms. It is worthwhile to compare the QRO approach with an exact, CSF-based counterpart to clarify the aforementioned behavior. The most significant difference from an exact SOS expansion over CSFs is that in the QRO approach excitation energies are approximated by orbital energy differences. In general, CSFs of different spin multiplicities (e.g., singlet and triplet) should have different excitation energies. However, such energy differences cannot be represented in the QRO approach, because α and β spin orbitals have the same orbital energies. Thus, the DSO contributions from the spatially identical excitations of different spin multiplicities are strictly cancelled out.


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To circumvent this problem, we diagonalize spin-α and β Fock operators independently, assuming the block diagonal structure between DOMO/SOMO/VIRTUAL subspaces (i.e., ignoring off-diagonal blocks of the Fock operators in natural orbital basis). Such transformation does not change the ground-state determinant itself, and therefore it still remains a spin eigenfunction. However, because the transformed natural orbitals have different spatial distributions and orbital energies between spin-α and β electrons, the generation of CSFs becomes non-trivial. Therefore, we adopt the perturbation treatment with the determinant basis. The determinant-based perturbation treatment for DSO tensors is well known as the Pederson– Khanna approach and the working equation is given in eq. (17).20

DklSO

ϕiα hkSO ϕaα ϕaα hlSO ϕiα 1 =− ∑ 2S (2S − 1) i, a ε aα − ε iα

ϕiβ hkSO ϕaβ ϕaβ hlSO ϕiβ 1 − ∑ 2 S (2 S − 1) i , a ε aβ − ε iβ ϕiα hkSO ϕaβ ϕaβ hlSO ϕiα 1 + ∑ 2 S (2S − 1) i , a ε aβ − ε iα ϕiβ hkSO ϕaα ϕaα hlSO ϕiβ 1 + ∑ 2 S (2S − 1) i , a ε aα − ε iβ

(17)

Here, i and a represent occupied and unoccupied orbitals, respectively. We used the prefactors proposed by van Wüllen,63 instead of the original proposal of Pederson and Khanna (1/4S2). These prefactors are derived to take into account quantum natures of electron spins, and the difference originates from the incommutability of sx and sz operators. Using van Wüllen's prefactors is also important to ensure that no ZFS occurs in spin-doublet (S = 1/2) systems. The main difference between our approach and the conventional PK method is that we use the single


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determinants consisting of the natural orbitals transformed in a spin-dependent manner, instead of unrestricted Kohn–Sham orbitals. Since in our approach spin-α and β electrons occupy different spatial orbitals, the word "quasi-restricted orbital" is not appropriate and this approach should be termed a natural orbital-based Pederson–Khanna (NOB-PK) approach. The NOB-PK method is not completely free from the spin contamination problem, because singly electronically excited Slater determinants are not always spin eigenfunctions even if the groundstate wavefunction is a spin eigenfunction. Nevertheless, to exploit the S2 eigenfunction for the ground-state wavefunction is important because there occur nonzero DSO contributions from spin contaminant states (for example, DSO contributions from the spin-quintet and higher spinmultiplet states in the DSO tensor calculations of spin-triplet systems), if the spin contaminated wavefunction is taken for the ground state. The choice of the S2 eigenfunction for the groundstate wavefunction practically avoids this rather general pitfall for open shell systems. We note that the NOB-PK approach cannot remove all "spurious" DSO contributions because only oneelectron excitations from the ground-state wavefunction are taken into account. For example, the NOB-PK gives a nonzero DSO tensor for a particular spin-triplet system which is composed of two well-separated spin-doublet molecules. This spurious contribution also occurs in conventional PK and QRO methods. Elimination of this spurious contribution, which seems intrinsic to the DFT-based framework, remains as an open problem. Also, we note that the NOBPK method cannot be generalized to exchange–correlation functionals which contain Hartree– Fock exchange terms, because the Pederson–Khanna approach is valid only for pure exchange– correlation functionals.

Computational Details


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For preliminary applications of the NOB-PK method, we have selected three series of transition metal complexes as illustrated in Figure 1, including two series of MnII complexes 1X: [MnII(terpy)X2] (terpy = 2,2':6,2''-terpyridine, X = NCS, Cl, Br, and I) and 2-X: [MnII(tpa)X2] (tpa = tris-2-picolylamine, X = Cl, Br, and I), and one series of ReIV single molecule magnets 3X: (NBu4)2[ReIVX4(ox)] (ox = oxalate, NBu4 = tetra-n-butylammonium cation, X = Cl and Br). These systems are suitable for the testing ground of the theory for the following reasons: All the X-ray crystallographic data is available for molecular geometry.10,11,53,64 The reliable ZFS parameters of the complexes are available.10,11,53 Observed heavy atom effects on the ZFS parameters are significant.

Figure 1. Representation of the MnII complexes and the ReIV based single molecule magnets studied in this work. Color code: Re (orange); Mn (light blue); S (yellow); C (green); N (blue); O (red); H (silver); Cl (purple); Br (brown); I (gray).


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Unless otherwise specified, we used the molecular structures obtained from X-ray crystallography for the single point calculations. We used the TPSS exchange–correlation functional65 and Sapporo-DZP basis set.66–71 For Re single molecule magnet 3-X, we used Sapporo-DKH3-DZP-201272 and Sapporo-DZP-2012 basis sets73 for ReX4(ox)2− magnetic core, and Sapporo-DZP for NBu4+ counter ions. For the calculations of 3-X systems we used the second-order Douglas–Kroll–Hess Hamiltonian74 to include relativistic effects in the SCF procedure. The conventional PK and QRO calculations were carried out by using the ORCA program package, and the NOB-PK calculations carried out with the GAMESS-US program suite and our own-coded program.75 In the evaluation of the SOC integrals, we used one-electron SOC Hamiltonian with effective nuclear charges for PK, QRO, and NOB-PK calculations, with the following Zeff parameters: Zeff(H) = 1.0, Zeff(C) = 3.6, Zeff(N) = 4.55, Zeff(O) = 5.6, Zeff(S) = 13.6, Zeff(Cl) = 14.2375, Zeff(Mn) = 12.75, Zeff(Br) = 30.1, Zeff(I) = 53.0, and Zeff(Re) = 75.0. In the PK calculations we used the prefactors proposed by van Wüllen by using the relationship DSO(van Wüllen) = DSO(PK) × 2S/(2S − 1).

Results and Discussion MnII(terpy)X2 systems 1-X Halide-containing transition metal complexes are suitable for analyzing the behavior of theoretical DSO tensors. In typical MnII(ligand)X2 systems (e.g., ligand = terpy,9 (OPPh3)2,76 (pic)4,77 (phen)2,77 (pyr)4,78 and (N2H4)2,79 with the abbreviations of OPPh3 = triphenylphosphine oxide, pic = γ-picoline, phen = o-phenanthroline, and pyr = pyridine), their experimental D value


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increases as the heavier halides are introduced. This tendency (DI > DBr > DCl) is explained as the order in the strength of the halide ligand field (∆I < ∆Br < ∆Cl). The theoretical and experimental ZFS parameters D (D = DZZ − (DXX + DYY)/2) and |E/D| (E = (DXX − DYY)/2) of MnII(terpy)X2 systems (X = NCS, Cl, Br, and I) are summarized in Table 1. Here, we choose the DZZ axis so that the relationship of |E/D| ≤ 1/3 is fulfilled. The DZZ axis is approximately parallel to the N(terpy)–Mn–N(terpy) coordination bond in 1-NCS and 1-Cl, and it is roughly parallel to the direction connecting the two halides in 1-Br and 1-I. Note that the DSS values calculated by using natural orbitals constructed from TPSS/Sapporo-DZP level of calculations are −0.046 cm−1, −0.042 cm−1, −0.038 cm−1, and −0.029 cm−1, for 1-NCS, 1-Cl, 1Br, and 1-I, respectively. The conventional QRO method gives their DSO values close to the experimental ones for 1-NCS and 1-Cl, but discrepancy between theory and experiment is large in 1-Br and 1-I. Particularly, the QRO method fails to predict correct the absolute sign of DSO in 1-I. By contrast, both the PK and NOB-PK methods can predict the DSO values in a qualitatively correct manner. Compare with the conventional PK approach, the NOB-PK method gives smaller |DSO| values in 1-Br and 1-I, and the deviations from the experimental values are smaller.

Table 1. The experimental and theoretical ZFS parameters of MnII(terpy)X2 systems 1-X. Experimental

PK

QRO

NOB-PK

Molecule

D/cm−1

|E/D|

DSO/cm−1

|ESO/DSO|

DSO/cm−1

|ESO/DSO|

DSO/cm−1

|ESO/DSO|

1-NCS

−0.30

0.17

−0.272

0.117

−0.270

0.190

−0.254

0.136

1-Cl

−0.26

0.29

−0.299

0.317

−0.352

0.288

−0.344

0.198

1-Br

+0.605

0.26

+0.901

0.100

+0.095

0.281

+0.750

0.230

1-I

+1.000

0.19

+2.385

0.001

−9.114

0.278

+1.618

0.123


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To analyze the obtained theoretical DSO tensors in detail, an orbital region partitioning technique (ORPT)40 is useful. In ORPT, the "most similar" pairs of α and β spin orbitals having the largest || overlap are defined, and generated orbital pairs are assigned to three regions; the doubly occupied region (DOR), singly occupied region (SOR), and unoccupied region (UOR). Then the DSO tensors are decomposed into four contributions: DOR → SOR, SOR → UOR, SOR → SOR, and DOR → UOR. Such decomposition of theoretical DSO tensors can provide the more chemist's intuition-friendly physical picture of the DSO tensors compared with the conventional spin configuration-based analysis (α → α, β → β, α → β, and β → α), because in the spin configuration-based analysis the decomposed DSO tensors are generally much larger in magnitude than the DSO tensor itself and large amount of DSO contributions are cancelled out. Since ORPT is proposed for the analysis of the PK-based DSO tensors, we can straightforwardly adopt ORPT for the analysis of the DSO tensors calculated at NOB-PK. The decomposed DSO(NOB-PK) tensors of 1-X based on ORPT in the coordinate systems fixed to the molecular frame are summarized in Table 2. Here, the terpyridine ligand is approximately located on the xy plane, with the pseudo-Cs plane on the yz plane. The DZZ principal axis is parallel to the x axis in 1-NCS and 1-Cl, and to the z axis in 1-Br and 1-I.


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Table 2. The decomposed DSO(NOB-PK) principal values of 1-X on the basis of ORPT.a 1-NCS

1-Cl

1-Br

1-I

Dxx

Dyy

Dzz

Dxx

Dyy

Dzz

Dxx

Dyy

Dzz

Dxx

Dyy

Dzz

/cm−1

/cm−1

/cm−1

/cm−1

/cm−1

/cm−1

/cm−1

/cm−1

/cm−1

/cm−1

/cm−1

/cm−1

DOR → SOR

−0.002

0.002

0.000

−0.010

0.001

0.009

−0.065

−0.015

0.080

−0.268

−0.083

0.351

SOR → UOR

−0.003

0.003

0.000

−0.002

0.000

0.002

−0.002

−0.001

0.003

−0.009

0.000

0.009

SOR → SOR

−0.165

0.115

0.050

−0.221

0.048

0.173

−0.356

−0.061

0.417

−0.458

−0.217

0.675

DOR → UOR

0.001

−0.001

0.000

0.003

−0.002

−0.001

0.000

0.000

0.000

−0.003

−0.041

0.044

Total

−0.169

0.119

0.050

−0.230

0.047

0.183

−0.423

−0.077

0.500

−0.738

−0.341

1.079

a

The terpyridine ligand is approximately on the xy plane, with the pseudo-Cs plane on the yz plane. ORPT: Orbital Region Partitioning Technique.40 The tensors are traceless.

From Table 2, when the axial ligand consists of light atoms, the SOR → SOR excitation dominantly contributes to the DSO principal values with positive signs. As the axial ligands are substituted to heavier halides, the negative DSO contributions from DOR → SOR become significant, especially in the x and y directions. This trend can be explained by the presence of the excited configurations of valence lone pair orbitals of halides → SOMO. Heavier halide atoms have lone pair orbitals higher in energy, which results in the increase of DSO contributions through diminishment of orbital energy differences between the lone pairs and SOMOs. An analysis similar to ORPT is available in ORCA software for DSO(QRO) tensors. The decomposed DSO values of 1-I in the DSO principal axes system are DSO(SOMO(α) → VIRTUAL(α)) = +0.374 cm−1, DSO(DOMO(β) → SOMO(β)) = +2.568 cm−1, DSO(SOMO(α) →


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SOMO(β)) = +1.090 cm−1, and DSO(DOMO(β) → VIRTUAL(α)) = −13.146 cm−1. Clearly, the DOMO → VIRTUAL excitations are responsible for the large negative DSO value in 1-I. The DOMO → SOMO and SOMO → VIRTUAL excited configurations also have larger DSO contributions compared with NOB-PK. As for the SOMO → SOMO excitations, the DSO contributions are comparable between QRO and NOB-PK. This result is quite reasonable, because as discussed above, the excited configurations except for SOMO → SOMO transitions suffer from the incomplete summation in the QRO method. The serious defect originating from the incomplete summation in the SOS procedure in QRO also leads to divergence of the DSO value against the size of basis sets. The DSO(QRO) value calculated by using Sapporo-DZP + diffuse (s + p), and Sapporo-TZP in conjunction with TPSS functional are −7.858 cm−1 and −5.699 cm−1, respectively, in contrast to the DSO(PK) (+2.404 cm−1 and +2.428 cm−1 for Sapporo-DZP + diffuse (s + p) and Sapporo-TZP, respectively) and DSO(NOB-PK) values (+1.651 cm−1 and +1.713 cm−1, for Sapporo-DZP + diffuse (s + p) and Sapporo-TZP, respectively).

MnII(tpa)X2 systems 2-X In contrast to MnII(terpy)X2 and other several MnII(ligand)X2 complexes, the DSO value decreases as the heavier halides are introduced in the MnII(tpa)X2 systems 2-X (D(Exptl.) = +0.115 cm−1, −0.360 cm−1, and −0.600 cm−1 for 2-Cl, 2-Br, and 2-I, respectively).10 The PKBP86/TZVP calculations of the DSS and DSO tensors of 2-X using the DFT optimized geometries were also reported, and the theoretical D values agreed well with the experimental ones (DSS+SO = +0.155 cm−1, −0.510 cm−1, and −0.868 cm−1, for 2-Cl, 2-Br, and 2-I, respectively). The theoretical ZFS parameters of 2-X of this work are summarized in Table 3. The NOB-PK method


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gives the best agreement with the experimental D values among three DFT-based approaches examined. However, a negative DSO value is obtained for 2-Cl, which contradicts the experimental finding. We also performed D tensor calculations of 2-Cl with the UTPSS/Sapporo-DZP optimized geometry. The obtained ZFS parameters are DSS+SO = +0.152 cm−1, ESS+SO = −0.046 cm−1.

Table 3. The experimental and theoretical ZFS parameters of MnII(tpa)X2 systems 2-X. Experimental

PK

QRO

NOB-PK

Molecule

D/cm−1

|E/D|

DSO/cm−1

|ESO/DSO|

DSO/cm−1

|ESO/DSO|

DSO/cm−1

|ESO/DSO|

2-Cl

+0.115

0.174

−0.150

0.283

−0.123

0.320

−0.113

0.267

2-Br

−0.360

0.203

−0.496

0.251

−0.364

0.075

−0.338

0.193

2-I

−0.600

0.158

−0.976

0.207

+6.355

0.317

−0.675

0.176

Again, the QRO approach gives a too large DSO value in iodide complex 2-I, which originates from large DSO contributions from DOR → UOR excitations. Except for the absolute sign of DSO in 2-Cl, the PK and NOB-PK methods predict qualitatively correct DSO tensors (the directions of the DSO principal axes nearly coincide with each other), and the PK method shows a tendency to overestimate the absolute value of DSO to some extent. The DZZ axis of DSO(NOB-PK) is perpendicular to the pseudo-Cs plane in 2-X, although that of DSO(QRO) strongly depends on the kind of halides: the DSOZZ(QRO) of 2-Br is roughly parallel to the Br–Mn–N(pyridine) coordination bond, and that of 2-I is approximately parallel to the direction of I–Mn–N(CH2Py)3 coordination. We emphasize that the theoretical DSO values of 2-Br calculated by QRO and NOB-PK are similar, but the structures of the DSO tensors are qualitatively different. Namely, the


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direction of the DZZ axis with respect to the molecular frame is different between those of QRO and NOB-PK. The decomposed DSO(NOB-PK) tensors of 2-X based on ORPT are given in Table 4. Regardless of the kind of halides, the SOR → SOR excited configurations dominantly contribute to their DSO values in 2-X. Contributions from the DOR → SOR excitations grow as the heavier halides are introduced, but the decomposed DSO tensors are smaller in magnitude and have more isotropic structures compared with those of 1-X. Such differences can be explained by the cis/trans configurations of the two halides, as discussed by Duboc et al.10

Table 4. The decomposed DSO(NOB-PK) principal values of 2-X on the basis of ORPT.a 2-Cl

2-Br

2-I

Dxx

Dyy

Dzz

Dxx

Dyy

Dzz

Dxx

Dyy

Dzz

/cm−1

/cm−1

/cm−1

/cm−1

/cm−1

/cm−1

/cm−1

/cm−1

/cm−1

DOR → SOR

0.000

0.002

−0.002

0.004

0.005

−0.009

0.016

0.027

−0.043

SOR → UOR

0.000

0.000

0.000

0.000

0.000

0.000

0.001

0.000

−0.001

SOR → SOR

0.007

0.066

−0.073

0.043

0.173

−0.216

0.290

0.083

−0.373

DOR → UOR

0.000

0.000

0.000

0.000

0.000

0.000

0.037

−0.004

−0.033

Total

0.007

0.068

−0.075

0.047

0.178

−0.225

0.344

0.106

−0.450

a

The X–Mn–N(CH2Py)3 coordination is approximately parallel to the x axis, and the N(Py)– Mn–N(Py) coordination is roughly parallel to the z axis. ORPT: Orbital Region Partitioning Technique.40 The tensors are traceless.

Single Molecule Magnets, (NBu4)2[ReIVX4(ox)] systems 3-X Single molecule magnets are one of the most important targets of the DFT-based D tensor calculations. (NBu4)2[ReIVX4(ox)] systems 3-X (X = Cl and Br) were recently synthesized and their magnetic parameters were determined experimentally.53 They have highly anisotropic D


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tensors with large negative D values: D = −53 cm−1 and E/D = 0.26 for 3-Cl, D = −73 cm−1 and E/D = 0.205 for 3-Br. The CASSCF and DFT calculations of their D tensors were also reported. These systems consist of ReIVX4(ox)2− magnetic core and surrounding NBu4+ counter ions. In the present study, we have carried out DFT calculations in the presence of two counter ions (denoted as 3-X) as well as ReIVX4(ox)2− core-only structures (specified as [3-X]2− core). The results are summarized in Table 5. The conventional QRO method gives positive DSO values in both 3-Cl and 3-Br, failing to predict the correct absolute sign of DSO. By contrast, both the PK and NOBPK methods give negative DSO values. It should be mentioned that the calculated values of |ESO/DSO| are close to the limiting value (1/3) and therefore the sign of the DSO values may change by small amounts of difference in the individual tensor elements. The theoretical DSO values calculated by using the ReIVX4(ox)2− core structures are slightly overestimated in magnitude, and inclusion of the counter ions in the calculations improves theoretical DSO values. The PK method predicts tendency of the D values against halide ligands, although the magnitude of the heavy atom effect is rather small compared to the experiments. By contrast, the NOB-PK method exhibits more stronger heavy atom effects on the DSO value, in consistent with the experimental findings.


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Table 5. The experimental and theoretical ZFS parameters of single molecule magnets, (NBu4)2[ReIVX4(ox)] systems 3-X Experimental

PK

QRO

NOB-PK

Molecule

D/cm−1

|E/D|

DSO/cm−1

|ESO/DSO|

DSO/cm−1

|ESO/DSO|

DSO/cm−1

|ESO/DSO|

3-Cl

−53

0.26

−63.051

0.299

+56.982

0.272

−64.122

0.316

3-Br

−73

0.205

−67.274

0.188

+68.468

0.285

−72.023

0.241

[3-Cl]2− core

−53

0.26

−75.407

0.306

+72.958

0.294

−74.734

0.332

[3-Br]2− core

−73

0.205

−77.877

0.279

+82.796

0.247

−84.496

0.274

Conclusion As pointed out by Schmitt and coworkers in 2011, the QRO method suffers from spurious DSO contributions from closed shell subsystems at a long distance from magnetic molecules, because only a few types of excitations are taken into account in the sum-over-states procedure. We have revisited the derivation of the QRO working equation, proving that spurious DSO contribution does not occur if all possible excitations are included in the sum-over-states procedure. At the same time, we showed that only SOMO → SOMO excited configurations have nonzero contributions to the DSO tensor, as long as α and β spin orbitals have the same spatial distributions and orbital energies. The complete cancellation of DSO contributions from, e.g., DOMO(α) → SOMO(β) and DOMO(β) → SOMO(β), originates from the fact that in the QRO approach excitation energies are approximated by the orbital energy differences of quasirestricted orbitals. To circumvent the problem that we cannot evaluate DSO contributions from the excitations involving DOMO and VIRTUAL, e.g., lone pair → SOMO often very important in halogen-containing molecules, we have adopted the perturbation expansion in the Slater determinant basis in conjunction with the natural orbitals transformed in a spin-dependent


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manner. In this approach the working equation becomes identical to that of the Pederson–Khanna method. The calculations of the DSO tensors of MnII complexes and ReIV based single molecule magnets revealed that this natural orbital-based Pederson–Khanna (NOB-PK) approach can reproduce correct trends of the DSO values against halide ligand heavy atom effects, and deviations from the experimentally determined D value are smaller than the DSO values calculated by the conventional Pederson–Khanna approach. The DSO tensor analysis based on the orbital region partitioning technique (ORPT) is also carried out, which can provide a more intuitively understandable interpretation of the theoretical DSO tensors. In the NOB-PK method, we use a spin eigenfunction for the ground state wavefunction. However, since we used Slater determinant-based perturbation theory for the DSO tensor calculations, the NOB-PK method is not completely free from the spin contamination problem. Nevertheless, the spin contamination effect on the ground state wavefunction in the DSO tensor calculations is an interesting issue, because in the DSS tensor calculations the restricted openshell (RO)-DFT method gives much more reliable results than the unrestricted (U)-DFT method, and the relationship between theoretical DSS tensors and spin contamination problems has been discussed.39,80 More extensive investigations of the DSO tensors for metal ion based magnetic molecules in terms of the NOB-PK method are underway.

ASSOCIATED CONTENT Supporting Information


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Derivation of the spin quantum number-dependent prefactors in the sum-over-states equation, Cartesian coordinates of 2-Cl optimized at the UTPSS/Sapporo-DZP level. This material is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION Corresponding Author [email protected]; [email protected]; [email protected] Author Contributions The authors contributed equally. K.Su., K.T. and T.T. wrote the paper. Notes The authors declare no competing financial interests. ACKNOWLEDGMENT This work was supported by AOARD Scientific Project on "Quantum Properties of Molecular Nanomagnets" (Award No. FA2386-13-1-4030). The support by Grants-in-Aid for Scientific Research (B) from Ministry of Education, Culture, Sports, Science and Technology (Japan) is also acknowledged. The work was partially supported by Grants-in-Aid for Scientific Research on Innovative Areas (Quantum Cybernetics) from Ministry of Education, Culture, Sports, Science and Technology (Japan), and also by FIRST Quantum Information Processing Project. REFERENCES 1. Harriman, J. E. Theoretical Foundations of Electron Spin Resonance; Academic Press: New York, 1978.


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2. Kaupp, M.; Bühl, M.; Malkin, V. G. Calculation of NMR and EPR Parameters: Theory and Applications; Wiley-VCH: Weinheim, 2005. 3. Sugisaki, K.; Toyota, K.; Sato, K.; Shiomi, D.; Kitagawa, M.; Takui, T. Quantum chemical calculations of the zero-field splitting tensors for organic spin multiplets. In EPR of Free Radicals in Solids I Progresses in Theoretical Chemistry and Physics A24; Rund, A.; Shiotani, M. Eds.; Springer: Dordrecht, 2013. 4. Very High Frequency (VHF) ESR/EPR, Biological Magnetic Resonance 22; Grinberg, O.; Berlinear, L. J., Eds.; Kluwer Academic/Plenum Publishers: New York, 2004. 5. Gilbert, B. C.; Davies, M. J.; Murphy, D. M.; Riedi, P. C.; Smith, G. M. Progress in highfield EPR. In Electron Paramagnetic Resonance Vol. 18; Gilbert, B. C.; Davies, M. J.; Murphy, D. M., Eds.; Royal Society of Chemistry, Cambridge, UK, 2002, 18, 254–303. 6. Hagen, W. R. High-frequency EPR of transition ion complexes and metalloproteins. Coord. Chem. Rev. 1999, 190–192, 209–229. 7. Boča, R. Zero-field splitting in metal complexes. Coord. Chem. Rev. 2004, 248, 757–815. 8. Kryzstek, J.; Ozarowski, A.; Telser, J. Multi-frequency, high-field EPR as a powerful tool to accurately determine zero-field splitting in high-spin transition metal coordination complexes. Coord. Chem. Rev. 2006, 250, 2308–2324. 9. Mantel, C.; Baffert, C.; Romero, I.; Deronzier, A.; Pécaut, J.; Collomb, M.-N.; Duboc, C. Structural characterization and electronic properties determination by high-field and highfrequency EPR of a series of five-coordinated Mn(II) complexes. Inorg. Chem. 2004, 43, 6455– 6463.


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Quasi-Restricted Orbital Treatment for the Density Functional Theory

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