P Graczyk
email: piotr.graczyk@univ-angers.fr

P Sawyer
email: psawyer@laurentian.ca

Sharp Estimates of Radial Dunkl and Heat Kernels in the Complex Case A n

Keywords: Punkl 33C67, 43A90, 53C35 MSC (2010) 31B05, 31B25, 60J50, 53C35

In this article, we consider the radial Dunkl geometric case k = 1 corresponding to flat Riemannian symmetric spaces in the complex case and we prove exact estimates for the positive valued Dunkl kernel and for the radial heat kernel.

Dans cet article, nous considérons le cas géométrique radial de Dunkl k = 1 correspondant aux espaces symétriques riemanniens plats dans le cas complexe et nous prouvons des estimations exactes pour le noyau de Dunkl à valeur positive et pour le noyau de chaleur radial.

Introduction and notations

Finding good estimates of Dunkl heat kernels is a challenging and important subject, developed recently in [START_REF] Anker | Harmonic Functions,Conjugate Harmonic Functions and the Hardy Space H 1 in the Rational Dunkl Setting[END_REF]. Establishing estimates of the heat kernels is equivalent to estimating the Dunkl kernel as demonstrated by equation (2.3) below.

In this paper we prove exact estimates in the W -radial Dunkl geometric case of multiplicity k = 1, corresponding to Cartan motion groups and flat Riemannian symmetric spaces with the ambient group complex G, the Weyl group W and the root system A n .

We study for the first time the non-centered heat kernel, denoted p W t (X, Y ), on Riemannian symmetric spaces and we provide its sharp estimates. Exact estimates were obtained in [START_REF] Anker | Heat Kernel and Green Function Estimates on Noncompact Symmetric Spaces[END_REF] in the centered case Y = 0 for all Riemannian symmetric spaces.

We provide exact estimates for the spherical functions ψ λ (X) in the two variables X, λ when λ is real and, consequently, for the heat kernel p W t (X, Y ) in the three variables t, X, Y . We recall here some basic terminology and facts about symmetric spaces associated to Cartan motion groups.

Let G be a semisimple Lie group and let g = k ⊕ p be the Cartan decomposition of G. We recall the definition of the Cartan motion group and the flat symmetric space associated with the semisimple Lie group G with maximal compact subgroup K. The Cartan motion group is the semi-direct product G 0 = K p where the multiplication is defined by (k 1 , X 1 ) • (k 2 , X 2 ) = (k 1 k 2 , Ad(k 1 )(X 2 ) + X 1 ). The associated flat symmetric space is then M = p G 0 /K (the action of G 0 on p is given by (k, X) • Y = Ad(k)(Y ) + X).

The spherical functions for the symmetric space M are then given by ψ λ (X) = K e λ(Ad(k)(X)) dk where λ is a complex linear functional on a ⊂ p, a Cartan subalgebra of the Lie algebra of G. To extend λ to X ∈ Ad(K)a = p, one uses λ(X) = λ(π a (X)) where π a is the orthogonal projection with respect to the Killing form (denoted throughout this paper by •, • ). Note that in [START_REF] Helgason | The bounded spherical functions on the Cartan motion group[END_REF][START_REF] Helgason | Differential Geometry, Lie Groups and Symmetric spaces[END_REF][START_REF] Helgason | Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions[END_REF], λ is replaced by i λ.

Throughout this paper, we usually assume that G is a semisimple complex Lie group. The complex root systems are respectively A n for n ≥ 1 (where p consists of the n × n hermitian matrices with trace 0), B n for n ≥ 2 (where p = i so(2 n + 1)), C n for n ≥ 3 (where p = i sp(n)) and D n for n ≥ 4 (where p = i so(2 n)) for the classical cases and the exceptional root systems

E 6 , E 7 , E 8 , F 4 and G 2 .
The radial heat kernel is considered with respect to the invariant measure µ(dY

) = π 2 (Y ) dY on M , where π(Y ) = α>0 α(Y ).
Note also that in the curved case M 0 = G/K, the spherical functions for the symmetric space M 0 are then given by

φ λ (e X ) = K e (λ-ρ)H(e X k) dk
where ρ is the half-sum of the roots counted with their multiplicities and H(g) is the abelian component in the Iwasawa decomposition of g: g = k e H(g) n.

Estimates of spherical functions and of the heat kernel

We will be developing a sharp estimate for the spherical function ψ λ (X). We introduce the following useful convention. We will write f (t, X, λ) g(t, X, λ) in a given domain of f and g if there exists constants C 1 > 0 and C 2 > 0 independent of t, X and

λ such that C 1 f (t, X, λ) ≤ g(t, X, λ) ≤ C 2 g(t, X, λ) in the domain of consideration.
We conjecture the following global estimate for the spherical function in the complex case.

Conjecture 2.1. On flat Riemannian symmetric spaces with complex group G, we have

ψ λ (X) e λ,X α>0 (1 + α(λ)α(X)) , λ ∈ a + , X ∈ a + .
Remark 2.2. Recall that, denoting δ(X) = α>0 sinh 2 α(X), we have

φ λ (e X ) = π(X) δ 1/2 (X) ψ λ (X). (2.1) Since δ 1/2 (X)
e ρ(X) π(X)/ α>0 (1 + α(X)) in the complex case, Conjecture 2.1 therefore becomes

φ λ (e X ) e (λ-ρ)(X) α>0 1 + α(X) 1 + α(λ)α(X) (2.2)
in the curved complex case. Let us compare the estimate (2.2) we conjecture for φ λ with the one obtained in [START_REF] Narayanana | Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications[END_REF], cf. also [START_REF] Schapira | Contributions to the hypergeometric function theory of Heckman and Opdam: sharp estimates, Schwartz space, heat kernel[END_REF]. The estimates in [START_REF] Narayanana | Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications[END_REF] apply in all the generality of hypergeometric functions of Heckman and Opdam. The authors show that there exists constants C 1 (λ) > 0, C 2 (λ) > 0 such that

C 1 (λ) e (λ-ρ)(X) α>0, α(λ)=0 (1 + α(X)) ≤ φ λ (e X ) ≤ C 2 (λ) e (λ-ρ)(X) α>0, α(λ)=0 (1 + α(X)).
Given (2.1), corresponding estimates clearly also hold in the flat case for ψ λ (X). The interest of our result, in the case A n , lies in the fact that our estimate is universal in both λ and X.

The results of [START_REF] Narayanana | Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications[END_REF][START_REF] Schapira | Contributions to the hypergeometric function theory of Heckman and Opdam: sharp estimates, Schwartz space, heat kernel[END_REF] and our estimates in the A n case strongly suggest that the Conjecture 2.1 is true for any complex root system.

Note that asymptotics of ψ λ (t X) when λ and X are singular and t → ∞ were proven in [START_REF] Graczyk | Integral Kernels on Complex Symmetric Spaces and for the Dyson Brownian Motion[END_REF] for all classical complex root systems and the systems F 4 and G 2 .

Consider the relationship between the Dunkl kernel E k (X, Y ) and the Dunkl heat kernel p t (X, Y ), as given in [10, Lemma 4.5]

p t (X, Y ) = 1 2 γ+d/2 c k t -d 2 -γ e -|X| 2 -|Y | 2 4t E k X, Y 2t , (2.3)
where γ is the number of positive roots and the constant c k is the Macdonald-Mehta-Selberg integral. The formula (2.3) remains true for the W -invariant kernels p W t and E W . In the geometric cases k = 1 2 , 1 and 2, by [START_REF] Jeu | Paley-Wiener theorems for the Dunkl transform[END_REF], the W -invariant formula (2.3) translates in a similar relationship between the spherical function ψ λ and the heat kernel p W t (X, Y ):

p W t (X, Y ) = 1 2 γ+d/2 c k t -d 2 -γ e -|X| 2 -|Y | 2 4t ψ X Y 2t . (2.4)
A simple direct proof of (2.4) for k = 1 is given in [4, Remark 2.9]. Equation (2.4) and Conjecture 2.1 bring us to an equivalent conjecture for the heat kernel p W t (X, Y ). Conjecture 2.3. We have

p W t (X, Y ) t -d 2 e -|X-Y | 2 4t α>0 (t + α(X) α(Y ))
.

Consider also the relationship between the heat kernel p W t (X, Y ) and the heat kernel pW t (X, Y ) in the curved case. We have

pW t (X, Y ) = e -|ρ| 2 t π(X) π(Y ) δ 1/2 (X) δ 1/2 (Y ) p W t (X, Y ). (2.5)
This relation follows directly from the fact that the respective radial Laplacians and radial measures are π -1 L a • π and π(X) dX in the flat case and δ -1/2 (L a -|ρ| 2 ) • δ 1/2 and δ(X) dX in the curved case (L a stands for the Euclidean Laplacian on a).

In the curved complex case, Conjecture 2.3 becomes

pW t (X, Y ) e -|ρ| 2 t t -d 2 e -ρ(X+Y ) α>0 (1 + α(X)) (1 + α(Y )) (t + α(X) α(Y ) e -|X-Y | 2 4t
.

Remark 2.4. In [START_REF] Graczyk | Potential kernels for radial Dunkl Laplacians[END_REF], sharp estimates of W -invariant Poisson and Newton kernels in the complex Dunkl case were obtained, by exploiting the method of construction of these W -invariant kernels by alternating sums. When a root system Σ acts in R d , the sharp estimates of [START_REF] Graczyk | Potential kernels for radial Dunkl Laplacians[END_REF] have the common form

K W (X, Y ) K R d (X, Y ) α>0 (|X -Y | 2 + α(X) α(Y )) , X, Y ∈ a + , (2.6)
where

K W (X, Y ) is the W -invariant kernel in Dunkl setting and K R d (X, Y ) is the classical kernel on R d .
Let us observe a common pattern in the appearance of the classical kernels K R d and of products of roots α(X) α(Y ) in formulas (2.6) and of the Fourier kernel e λ,X and the classical Gaussian heat kernel and of products α(λ)α(X) in the estimates given in Conjecture 2.1 and Conjecture 2.3.

Proof of Conjecture in some cases

We start with a practical result.

Proposition 2.5. Let α i be the simple roots and let A α i be such that X, A α i = α i (X) for X ∈ a. Suppose X ∈ a + and w ∈ W \ {id}. Then we have

Y -w Y = r i=1 2 a w i (Y ) |α i | 2 A α i (2.7)
where a w i is a linear combination of positive simple roots with non-negative integer coefficients for each i.

Proof. Refer to [START_REF] Graczyk | Potential kernels for radial Dunkl Laplacians[END_REF].

Remark 2.6. Note that a w i (Y )/|α i | 2 is bounded by C max k |α k (Y )|
where C is a constant depending only on w ∈ W and, ultimately, on W .

Corollary 2.7. Let Y ∈ a + and w ∈ W . Consider the decomposition (2.7) of Y -wY . If a w k (Y ) = 0 then α k appears in a w k , i.e. a w k = r i=1 n i α i with n k > 0.
Proof. Refer to [START_REF] Graczyk | Potential kernels for radial Dunkl Laplacians[END_REF].

Proposition 2.8. Let δ > 0. Suppose α i (λ) α j (X) ≤ δ for all i, j. Then ψ λ (X) e λ(X) (the constants involved only depend on δ).

Proof. Let K(X, Y ) be the kernel of the Abel transform. Recall that K(X, Y ) dY is a probability measure supported on C(X), the convex envelope of the orbit W • X. Notice that

e w min λ(X) ≤ ψ λ (X) = C(X) e λ(Y ) K(X, Y ) dY ≤ e λ(X) (2.8)
where w min is the element of the Weyl group giving the minimum value of w λ(X). Now, using Proposition 2.5 and Remark 2.6 with Y = λ, we see that for any w ∈ W e λ(X) ≥ e w λ(X) = e wλ-λ,X e λ,X =

r i=1 e -2 a w i (λ) |α i | 2 α i (X) e λ,X ≥ r i=1 e -2 C (max k α k (λ)) α i (X) e λ,X ≥ r i=1 e -2 C δ e λ,X .
Remark 2.9. This case and this method apply for any radial Dunkl case; it suffices to replace K(X, Y ) dY by the so-called Rösler measure µ X (dY ) in the integral in (2.8), see [START_REF] Rösler | Positivity of Dunkl's intertwining operator[END_REF].

Proposition 2.10. A spherical function ψ λ (X) on M is given by the formula

ψ λ (X) = π(ρ) 2 γ π(λ) π(X) w∈W (w)e wλ,X , (2.9)
where ρ = 1

ψ λ (X)
e λ(X) π(λ)π(X) .

We are assuming here that |α i | ≥ 1 for each i.

Proof. Suppose w ∈ W is not the identity. In that case, a w i (λ) is not equal to 0 for some i. By Proposition 2.5 with y = λ and Corollary 2.7, λ(X)

-w λ(X) ≥ 2 a w i (λ) α i (X)/|α i | 2 ≥ 2 α i (λ) α i (X) ≥ log |W |.
Each term e wλ,X in the alternating sum (2.9) corresponding to w = id is bounded by e -log |W | e λ(X) = e λ(X) /|W |. Hence, since only half the terms in the sum are negative,

|W | e λ(X) ≥ w∈W (w)e wλ,X ≥ e λ(X) - |W | 2 e λ(X) /|W | = 1 2 e λ(X) .
3 The conjecture in the case of the root system A n

We will prove the conjecture in the case of the root system of type A.

Theorem 3.1. In the case of the root system of type A n in the complex case, we have

ψ λ (e X ) e λ,X i<j (1 + (λ i -λ j ) (x i -x j )) , λ, X ∈ a + . (3.1) Corollary 3.2. φ λ (e X ) e (λ-ρ)(X) i<j 1 + x i -x j 1 + (x i -x j ) (λ i -λ j ) , p W t (X, Y ) t -d 2 e -|X-Y | 2 4t i<j (t + (x i -x j ) (y i -y j )) , pW t (X, Y ) e -|ρ| 2 t t -d 2 e -ρ(X+Y ) i<j (1 + x i -x j ) (1 + y i -y j ) (t + (x i -x j ) (y i -y j )) e -|X-Y | 2 4t
.

We recall (refer to [START_REF] Sawyer | A Laplace-Type Representation of the Generalized Spherical Functions Associated with the Root Systems of Type A[END_REF]) the following iterative formula for the spherical functions of type A in the complex case. Here we do not assume that the elements of the Lie algebra have trace 0. Here the Cartan subalgebra a for the root system A n-1 is isomorphic to R n . For λ, X ∈ a + ⊂ R n , we have

ψ λ (e X ) = e λ(X) if n = 1 and ψ λ (e X ) = (n -1)! e λn n k=1 x k ( i<j (x i -x j )) -1 x n-1 xn • • • x 1 x 2 ψ λ 0 (e Y ) (3.2) i<j<n (y i -y j ) dy 1 • • • dy n-1
where λ 0 (U ) = n-1 k=1 (λ k -λ n ) u k . Remark 3.3. Formula (3.2) represents the action of the root system A n-1 on R n . If we assume n k=1 x k = 0 = n k=1 λ k , we have then the action of the root system A n-1 on R n-1 . We can also consider the action of A n-1 on any R m with m ≥ n -1 by considering formula (2.9) and deciding on which entries x k , the Weyl group W = S n acts. These considerations do not affect the conclusion of Theorem 3.1.

Approximate factorization for A n

Before proving the conjecture in the case A n , we will prove an interesting "factorization".

Proposition 3.4. For n ≥ 1, consider the root system A n on R n+1 . Let λ, X ∈ a + ⊂ R n+1 and X = [X 1 , . . . , X n ]. Define I (n) = I (n) (λ; X) = xn x n+1 x n-1 xn • • • x 2 x 3 x 1 x 2 e -λ 0 (X -Y ) i<j<n (y i -y j ) (λ i -λ j ) 1 + (y i -y j ) (λ i -λ j ) dy 1 dy 2 • • • dy n .
Then the following approximate factorization holds

I (n) n k=1 I (n) k (3.3)
where

I (n) 1 = x 1
x 2

e -(λ 1 -λ n+1 ) (x 1 -y 1 ) dy 1 and

I (n) k = x k x k+1 e -(λ k -λ n+1 ) (x k -y k ) k-1 j=1 (x j -y k ) (λ j -λ k ) 1 + (x j -y k ) (λ j -λ k ) dy k for 1 < k ≤ n.
Proof. Since u/(1 + u) is an increasing function, we clearly have

I (n) ≤ xn x n+1 x n-1 xn • • • x 2 x 3 x 1 x 2 e -λ 0 (X -Y ) i<j<n (x i -y j ) (λ i -λ j ) 1 + (x i -y j ) (λ i -λ j ) dy 1 dy 2 • • • dy n .
On the other hand,

I (n) ≥ xn (xn+x n+1 )/2
x n-1

(x n-1 +xn)/2 • • • x 2 (x 2 +x 3 )/2 x 1 (x 1 +x 2 )/2 e -λ 0 (X -Y ) i<j<n (y i -y j ) (λ i -λ j ) 1 + (y i -y j ) (λ i -λ j ) dy 1 dy 2 • • • dy n ≥ xn (xn+x n+1 )/2
x n-1

(x n-1 +xn)/2 • • • x 2 (x 2 +x 3 )/2 x 1 (x 1 +x 2 )/2 e -λ 0 (X -Y ) i<j<n ((x i + x i+1 )/2 -y j ) (λ i -λ j ) 1 + ((x i + x i+1 )/2 -y j ) (λ i -λ j ) dy 1 dy 2 • • • dy n xn (xn+x n+1 )/2
x n-1

(x n-1 +xn)/2 • • • x 2 (x 2 +x 3 )/2 x 1 (x 1 +x 2 )/2 e -λ 0 (X -Y ) i<j<n (x i -y j ) (λ i -λ j ) 1 + (x i -y j ) (λ i -λ j ) dy 1 dy 2 • • • dy n = = n k=1 x k (x k +x k+1 )/2 e -(λ k -λ n+1 ) (x k -y k ) k-1 j=1 (x j -y k ) (λ j -λ k ) 1 + (x j -y k ) (λ j -λ k ) dy k = n k=1 A (n) k since ((x i + x i+1 )/2 -y j ) (λ i -λ j ) 1 + ((x i + x i+1 )/2 -y j ) (λ i -λ j ) ≤ (x i -y j ) (λ i -λ j ) 1 + (x i -y j ) (λ i -λ j ) while ((x i + x i+1 )/2 -y j ) (λ i -λ j ) 1 + ((x i + x i+1 )/2 -y j ) (λ i -λ j ) ≥ ((x i -y j )/2 (λ i -λ j ) 1 + (x i -y j )/2 (λ i -λ j ) ≥ 1 2 (x i -y j ) (λ i -λ j ) 1 + (x i -y j ) (λ i -λ j ) . Now, let B (n) k = (x k +x k+1 )/2 x k+1 e -(λ k -λ n+1 ) (x k -y k ) k-1 j=1 (x j -y k ) (λ j -λ k ) 1 + (x j -y k ) (λ j -λ k )
dy k and note that I

(n) k = A (n) k + B (n)
k . Now, using the change of variable 2w = x k -y k , we have

B (n) k = 2 (x k -x k+1 )/2 (x k -x k+1 )/4 e -2 (λ k -λ n+1 ) w k-1 j=1 (x j -x k + 2 w) (λ j -λ k ) 1 + (x j -x k + 2 w) (λ j -λ k ) dw ≤ 4 (x k -x k+1 )/2 (x k -x k+1 )/4 e -2 (λ k -λ n+1 ) w k-1 j=1 (x j -x k + w) (λ j -λ k ) 1 + (x j -x k + w) (λ j -λ k ) dw ≤ 4 (x k -x k+1 )/2 0 e -(λ k -λ n+1 ) w k-1 j=1 (x j -x k + w) (λ j -λ k ) 1 + (x j -x k + w) (λ j -λ k ) dw = 4 A (n) k ,
where the last equality comes from the change of variable w = x k -y k in the expression for

A (n) k . Therefore I (n) k = A (n) k + B (n) k ≤ 5 A (n)
k . The result follows. The next proposition gives an inductive way of estimating I (n+1) , knowing I (n) and I (n-1) . Proposition 3.5. Consider the root system

A n+1 on R n+2 . Let λ, X ∈ a + ⊂ R n+2 . Assume α 1 (X) ≥ α n+1 (X). Then I (n+1) (λ; X) I (n) (λ 1 , . . . , λ n , λ n+2 ; x 1 , . . . , x n+1 ) (x 1 -x n+1 )(λ 1 -λ n+1 ) 1 + (x 1 -x n+1 )(λ 1 -λ n+1 ) I (n) (λ 2 , .
. . , λ n+1 , λ n+2 ; x 2 , . . . , x n+2 ) I (n-1) (λ 2 , . . . , λ n , λ n+2 ; x 2 , . . . , x n+1 ) .

Proof. We start with an outline of the proof.

(i) I (n+1) is estimated by a product of n + 1 factors

I (n+1) k (λ; X).
(ii) The product of the first n factors

I (n+1) 1 (λ; X), . . . , I (n+1) n
(λ; X) give an estimate of the term I (n) (λ 1 , . . . , λ n , λ n+2 ; X ) by Proposition 3.4.

(iii) In the last factor I (n+1) n+1 (λ; X), we "draw off" one term from under the integral, using the additional hypothesis α 1 (X) ≥ α n+1 (X). The remaining integral corresponds to 1) , up to a change of variables (we re-use the idea of (ii)).

I (n) n (λ 2 , . . . , λ n+2 ; x 2 , . . . , x (iv) The last factor I (n) n of I (n) is estimated by I (n) /I (n-
Since x n+2 ≤ y n+1 ≤ x n+1 and x n+1 -

x n+2 ≤ x 1 -x 2 , we get x 1 -x n+1 ≤ x 1 -y n+1 ≤ x 1 -x n+2 ≤ 2(x 1 -x n+1
) and we have

I (n+1) n+1 x n+1 x n+2 e -(λ n+1 -λ n+2 ) (x n+1 -y n+1 ) (x 1 -y n+1 ) (λ 1 -λ n+1 ) 1 + (x 1 -y n+1 ) (λ 1 -λ n+1 ) n j=2 (x j -y n+1 ) (λ j -λ n+1 ) 1 + (x j -y n+1 ) (λ j -λ n+1 ) dy n+1 (x 1 -x n+1 )(λ 1 -λ n+1 ) 1 + (x 1 -x n+1 )(λ 1 -λ n+1 ) x n+1 x n+2 e -(λ n+1 -λ n+2 ) (x n+1 -y n+1 ) n j=2 (x j -y n+1 ) (λ j -λ n+1 ) 1 + (x j -y n+1 ) (λ j -λ n+1 ) dy n+1 .
Hence, noting that

I (n+1) 1 (λ; X) • • • I (n+1) n
(λ; X) I (n) (λ 1 , . . . , λ n , λ n+2 ; X ), we have

I (n+1) (λ; X) I (n) (λ 1 , . . . , λ n , λ n+2 ; X ) (x 1 -x n+1 ) (λ 1 -λ n+1 ) 1 + (x 1 -x n+1 ) (λ 1 -λ n+1 ) x n+1 x n+2 e -(λ n+1 -λ n+2 ) (x n+1 -y n+1 ) n j=2 (x j -y n+1 ) (λ j -λ n+1 ) 1 + (x j -y n+1 ) (λ j -λ n+1 ) dy n+1 .
Finally,

x n+1 x n+2 e -(λ n+1 -λ n+2 ) (x n+1 -y n+1 ) n j=2 (x j -y n+1 ) (λ j -λ n+1 ) 1 + (x j -y n+1 ) (λ j -λ n+1 ) dy n+1 = n k=1
x k+1

x k+2 e -(λ k+1 -λ n+2 ) (x k+1 -y k+1 ) k-1 j=1

(x j+1 -y k+1 ) (λ j+1 -λ k+1 )
1+(x j+1 -y k+1 ) (λ j+1 -λ k+1 ) dy k+1 n-1 k=1

x k+1

x k+2 e -(λ k+1 -λ n+2 ) (x k+1 -y k+1 ) k-1 j=1

(x j+1 -y k+1 ) (λ j+1 -λ k+1 ) 1+(x j+1 -y k+1 ) (λ j+1 -λ k+1 ) dy k+1 = I (n) (λ 2 , . . . , λ n+1 , λ n+2 ; x 2 , . . . , x n+2 ) I (n-1) (λ 2 , . . . , λ n , λ n+2 ; x 2 , . . . , x n+1 ) .
Remark 3.6. When n = 1, the result of Proposition 3.5 remains valid if we set

I (0) = 1.
We now prove our main result.

Proof of Theorem 3.1. We use induction on the rank. In the case of A 1 , we have

ψ λ (e X ) = e λ 2 (x 1 +x 2 ) (x 1 -x 2 ) -1 x 1 x 2 e (λ 1 -λ 2 ) y dy = e λ 2 (x 1 +x 2 ) (x 1 -x 2 ) -1 e (λ 1 -λ 2 ) x 1 -e (λ 1 -λ 2 ) x 2 λ 1 -λ 2 = e λ 1 x 1 +λ 2 x 2 1 -e -(λ 1 -λ 2 ) (x 1 -x 2 ) (λ 1 -λ 2 ) (x 1 -x 2 ) e λ 1 x 1 +λ 2 x 2 1 1 + (λ 1 -λ 2 ) (x 1 -x 2 ) since 1 -e -u u/(1 + u) for u ≥ 0.
Assume that the result is true for A r , 1 ≤ r ≤ n, n ≥ 1. Using (3.2) and the induction hypothesis, we have for r = 1, . . . , n + 1 and λ, X in positive Weyl chamber in R r+1 π(X) π(λ ) e -λ(X) ψ λ ([x 1 , . . . , x r , x r+1 ]) = r! π(λ ) e -λ(X) e λ r+1 r+1 Still using the induction hypothesis, we have

k=1 x k xr x r+1 • • • x 1 x 2 ψ λ 0 (e Y ) i<j<r+1 (y i -y j ) dy 1 • • • dy r xr x r+1 x r-1 xr • • • x 2 x 3 x 1 x 2 e -λ 0 (X -Y ) i<j<r+1 (y i -y j ) (λ i -λ j ) 1 + (y i -y j ) (λ i -λ j ) dy 1 dy 2 • • •
π(X) π(λ ) e -λ(X) ψ [λ 1 ,...,λ r+1 ] ([x 1 , . . . , x r+1 ] = r! I (r) (λ 1 , . . . , λ r+1 ; x 1 , . . . , x r+1 ) π(X) π(λ ) i<j≤r+1 (1 + (λ i -λ j ) (x i -x j )) (3.4)
for r = 1, . . . , n.

It remains to show that (3.4) holds for r = n + 1, i.e. that

I (n+1) (λ 1 , . . . , λ n+2 ; x 1 , . . . , x n+2 ) π(X) π(λ ) i<j≤n+2 (1 + (λ i -λ j ) (x i -x j ))
.

It is sufficient to prove the last formula under the hypothesis that α 1 (X) ≥ α n+1 (X) since the case α 1 (X) ≤ α n+1 (X) is symmetric. Now, according to Proposition 3.5 and (3.4),

I (n+1) (λ; X) (x 1 -x n+1 )(λ 1 -λ n+1 ) 1 + (x 1 -x n+1 )(λ 1 -λ n+1 ) I (n) (λ 1 , . . . , λ n , λ n+2 ; x 1 , . . . , x n+1 ) I (n) (λ 2 , .
. . , λ n+1 , λ n+2 ; x 2 , . . . , x n+2 ) I (n-1) (λ 2 , . . . , λ n , λ n+2 ; x 2 , . . . , x n+1 )

-1

(x 1 -x n+1 )(λ 1 -λ n+1 ) 1 + (x 1 -x n+1 )(λ 1 -λ n+1 ) i<j≤n+1 (x i -x j ) i<j<n+1 (λ i -λ j ) i<j≤n (1 + (x i -x j ) (λ i -λ j )) n i=1 (1 + (x i -x n+1 ) (λ i -λ n+2 )) 1<i<j≤n+2 (x i -x j ) 1<i<j≤n+1 (λ i -λ j ) 1<i<j≤n+2 (1 + (x i -x j ) (λ i -λ j )) 1<i<j≤n (1 + (x i -x j ) (λ i -λ j )) n i=2 (1 + (x i -x n+1 ) (λ i -λ n+2 )) 1<i<j≤n+1 (x i -x j ) 1<i<j<n+1 (λ i -λ j ) = x 1 -x n+1 x 1 -x n+2 1 + (x 1 -x n+1 ) (λ 1 -λ 2 ) 1 + (x 1 -x n+1 ) (λ 1 -λ 2 ) i<j≤n+2 (x i -x j ) i<j≤n+1 (λ i -λ j ) i<j≤n+2 (1 + (λ i -λ j ) (x i -x j ))
.

The result follows since x 1 -x n+1 x 1 -x n+2 given that x 1 -x 2 ≥ x n+1 -x n+2 .

∈ a + C 1 e -c 1 |X-Y | 2 /t min{w(B(X, √ t)), w(B(Y, √ t))} ≤ p t (X, Y ) ≤ C 2 e -c 2 |X-Y | 2 /t max{w(B(X, √ t)), w(B(Y, √ t))} (4.1)
where w is the W -invariant reference measure (in our paper w = π(X) 2 dX) and the w-volume of a ball satisfies the estimate ([1, p. 2365])

w(B(X, r)) r n α>0 (r + α(X)) 2k(α) .
The same estimates follow for p W t (X, Y ). Our sharp estimates in Corollary 3.2 for k(α) = 1 in the W -radial case A n suggest that c 1 = c 2 = 1/4 in (4.1) and that products of terms (t+α(X)α(Y )) k(α) are natural in place of separate terms w(B(X, √ t)) and w(B(Y, √ t)). On the other hand, estimates (4.1) and in Corollary 3.2 suggest that the following conjecture is true in the Dunkl setting.

Conjecture 4.1. The Weyl-invariant heat kernel for a root system Σ in R d satisfies the following estimates

p W t (X, Y ) t -d 2 e -|X-Y | 2 4t α>0 (t + α(X)α(Y )) k(α) . (4.2)
Formula (2.3) then implies that the W -invariant Dunkl kernel satisfies the estimate

E W k (X, Y ) e λ(X) α>0 (1 + α(X) α(λ)) k(α) .
5 Additional formulas for p W t (X, Y )

Let us finish by giving formulas relating the heat kernel p W t (X, Y ) with the spherical functions ψ iλ and φ iλ . These formulas can be useful in further study of the kernel p W t (X, Y ). Proposition 5.1. (a) In the flat Riemannian symmetric case, the following formula holds: where c(λ) is the Harish-Chandra c-function (refer to [START_REF] Helgason | Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions[END_REF] for details). The constant C can be given explicitly.

p W t (X, Y ) = C a e -|λ| 2 t ψ i λ (X) ψ -i λ (Y ) π(λ) 2 dλ, C > 0. ( 5
Proof. We will prove (b). We show that the right hand side of equation ( 5 Remark 5.2. The heat kernel estimates of h W t (X) = p W t (X, 0) on symmetric spaces ([2] and references therein) are based on the inverse spherical Fourier transform formula which is a special case of (5.2) when Y = 0. Thus one may hope that estimates of p W t (X, Y ) can be deduced from (5.2).

Remark 5.3. The passage from h W t (X) to p W t (X, Y ) is well understood at the group level:

p W t (g, h) = h W t (h -1 g),
which is equivalent to

p W t (X, Y ) = K h W t (e -Y k -1 e X ) dk
and to

p W t (X, Y ) = a h W t (H) k(H, -Y, X) π(H) dH, (5.3)
where the last formula contains the product formula kernel k which is defined by

2 α∈Σ

 2 + m α α = α∈Σ + α and γ = |Σ + | is the number of positive roots (refer to [8, Chap. IV, Proposition 4.8 and Chap. II, Theorem 5.35]). Proposition 2.11. Suppose α(λ) α(X) ≥ (log |W |)/2 for all α > 0. Then

  dy r where X = diag[x 1 , . . . , x r ] and λ = [λ 1 , . . . , λ r ]. Using the notation introduced in Proposition 3.4, we have π(X) π(λ ) e -λ(X) ψ [λ 1 ,...,λ r+1 ] ([x 1 , . . . , x r+1 ]) = r! I (r) (λ 1 , . . . , λ r+1 , x 1 , . . . , x r+1 ).

. 1 )

 1 (b) In the curved non-compact Riemannian symmetric case the following formula holdsp W t (X, Y ) = C a * e -(|λ| 2 +|ρ| 2 ) t φ i λ (X) φ -i λ (Y ) dλ |c(λ)| 2(5.2)

. 1 )

 1 satisfies the definition of the heat kernel. For a test function f , consideru(X, t) = C a a e -(|λ| 2 +|ρ| 2 ) t φ i λ (X) φ -i λ (Y ) K |c(λ)| -2 dλ f (Y ) |c(λ)| -2 dY where K |c(λ)| -2 dλ is Plancherel measure.The fact that ∆ u(X, t) = ∂ ∂t u(X, t) where ∆ is the radial Laplacian follows easily from the fact that ∆φ i λ (X) = -(|λ| 2 + |ρ| 2 ) φ i λ (X) and ∂ ∂t e -(|λ| 2 +|ρ| 2 )] t = -(|λ| 2 + |ρ| 2 ) e -|λ| 2 t . Now, using Fubini's theorem, u(X, t) = C K a e -(|λ| 2 +|ρ| 2 ) t a φ -i λ (Y ) f (Y ) |c(λ)| -2 dY φ i λ (X) π(λ) 2 dλ = C K a e -(|λ| 2 +|ρ| 2 ) t f (λ) φ i λ (X) π(λ) 2 dλwhich tends to f (X) as t → 0 by the dominated convergence theorem.

a

  ψ λ (e H ) k(H, X, Y ) π(H) dH = ψ λ (e X ) ψ λ (e Y ) = K ψ λ (e X k e Y ) dk. Similarly, pW t (X, Y ) = a hW t (H) k(H, -Y, X) δ(H) dH,(5.4)where the last formula contains the product formula kernel k which is defined bya φ λ (e H ) k(H, X, Y ) δ(H) dH = φ λ (e X ) φ λ (e Y ) = K φ λ (e X k e Y ) dk.

4

  Comparison with the estimates of Anker et al. in[START_REF] Anker | Harmonic Functions,Conjugate Harmonic Functions and the Hardy Space H 1 in the Rational Dunkl Setting[END_REF].Conjecture for Dunkl settingIn [1, Theorems 4.1 p. 2372 and 4.4, p. 2377] the following estimates were proven for the heat kernel p t (X, Y ) in the Dunkl setting on R n . There exists positive constants c 1 , c 2 , C 1 and C 2 such that for all X, Y


Subrata Ghosh

Sujit Kamilya

Mayurika Das

Sakshi Mehta

Marie-Emmanuelle Boulon

Ivan Nemec

Mathieu Rouzieres

Radovan Herchel

Abhishake Mondal

Effect of Coordination Geometry on Magnetic Properties in a Series of Cobalt(II) Complexes and Structural Transformation in Mother Liquor

The three Co(II) complexes [Co(bbp) 2 ][Co(NCS) 4 ]•4DMF (1), [Co(bbp)(NCS) 2 (DMF)]•2DMF (2), and [Co(bbp)(NCS) 2 ] (3) have been synthesized and characterized by single-crystal X-ray diffraction, magnetic, and various spectroscopic techniques. Complexes 1 and 3 are obtained by the reaction of Co(NCS) 2 with 2,6-bis(1H -benzo[d]imidazol-2yl)pyridine (bbp), and complex 1 undergoes a structural transformation to form complex 2. A single-crystal X-ray study revealed that complex 1 is comprised of two Co(II) centers, a cationic octahedral Co(II) unit and an anionic tetrahedral Co(II) unit, while the Co(II) ion is in a distorted-octahedral environment in 2. Moreover, in complex 3, the Co(II) ion is in a distorted-square-pyramidal geometry. The effect of coordination geometry on the magnetic properties was studied by both static and dynamic magnetic measurements. Direct current (dc) magnetic susceptibility measurements showed that all of the Co(II) ions are in high-spin state in these complexes. Alternating current (ac) magnetic susceptibility measurements indicated that complexes 2 and 3 display slow relaxation of magnetization in an external dc magnetic field, while complex 1 displayed no such property. EPR experiments and theoretical calculations were consistent with the above findings.

■ INTRODUCTION

Bistable magnetic materials based on transition metals have achieved enormous attention over the past few decades, particularly materials exhibiting slow relaxation of magnetization, 1-3 e.g. single-molecule magnets (SMMs) and singlechain magnets (SCMs), due to their potential applications in the fields of quantum computing, information storage, and molecular spintronics. [START_REF] Sessoli | Nanoscience: Single-atom data storage[END_REF][START_REF] Natterer | Reading and writing single-atom magnets[END_REF][START_REF] Baumann | Electron paramagnetic resonance of individual atoms on a surface[END_REF] Magnetic anisotropy (D) and ground state spin (S) produce energy differences to reverse the magnetization, which is the origin of slow magnetic relaxation. The effective energy barrier is defined by Δ eff = S 2 (D) or (S 2 -1/4)(D) for molecules with integer or noninteger ground spin states, respectively. In order to achieve a significant energy barrier, researchers sought giant spin systems. [START_REF] Ako | A Ferromagnetically Coupled Mn19 Aggregate with a Record S = 83/2 Ground Spin State[END_REF][START_REF] Sessoli | High-spin molecules: [Mn12O12(O2CR)16(H2O)4[END_REF][START_REF] Sessoli | Magnetic bistability in a metal-ion cluster[END_REF] However, the desired value of energy barrier was not obtained effectively, as the value of D varied inversely with S. Recently interest has shifted to enhancing the anisotropy barrier associated with the single metal center. [START_REF] Frost | The rise of 3-d single-ion magnets in molecular magnetism: towards materials from molecules?[END_REF][START_REF] Craig | 3d single-ion magnets[END_REF] After the discovery of lanthanide complex based mononuclear SMMs, [START_REF] Guo | Magnetic hysteresis up to 80 K in a dysprosium metallocene single-molecule magnet[END_REF][START_REF] Guo | A Dysprosium Metallocene Single-Molecule Magnet Functioning at the Axial Limit[END_REF][START_REF] Goodwin | Molecular magnetic hysteresis at 60 K in dysprosocenium[END_REF][START_REF] Ishikawa | Lanthanide Double-Decker Complexes Functioning as Magnets at the Single-Molecular Level[END_REF] several mononuclear complexes with the 3d transition metals V, Cr, Mn, Fe, Ni, and Cu have been studied. [START_REF] Craig | Probing the origin of the giant magnetic anisotropy in trigonal bipyramidal Ni(II) under high pressure[END_REF][START_REF] Bhowmick | Slow magnetic relaxation in octahedral low-spin Ni(III) complexes[END_REF][START_REF] Lin | Counterion influence on dynamic spin properties in a V(IV) complex[END_REF][START_REF] Rajnaḱ | Field-Induced Slow Magnetic Relaxation in a Mononuclear Manganese(II) Complex[END_REF][START_REF] Bar | Magnetic anisotropy in twoto eight-coordinated transition-metal complexes: Recent developments in molecular magnetism[END_REF] Great interest has been devoted to Co(II) due to its high magnetic anisotropy, strong first-order spin-orbit coupling, and noninteger ground spin state (S = 3/ 2), which conceals quantum tunneling of magnetization (QTM). [START_REF] Frost | The rise of 3-d single-ion magnets in molecular magnetism: towards materials from molecules?[END_REF][START_REF] Craig | 3d single-ion magnets[END_REF][START_REF] Bar | Magnetic anisotropy in twoto eight-coordinated transition-metal complexes: Recent developments in molecular magnetism[END_REF] In this aspect, mononuclear Co(II) complexes exhibiting slow relaxation of magnetization have been observed in numerous coordination geometries and coordination numbers such as two, three, four, five, six, seven, and eight. [START_REF] Bar | Magnetic anisotropy in twoto eight-coordinated transition-metal complexes: Recent developments in molecular magnetism[END_REF][START_REF] Drahos | Impact of Halogenido Coligands on Magnetic Anisotropy in Seven-Coordinate Co(II) Complexes[END_REF][START_REF] Vahovska | New Cobalt(II) Field-Induced Single-Molecule Magnet and the First Example of a Cobalt(III) Complex with Tridentate Binding of a Deprotonated 4-Amino-3,5-bis(pyridin-2-yl)-1,2,4-Triazole Ligand: New Cobalt(II) Field-Induced Single-Molecule Magnet and the First Example of a Cobalt(III) Complex with Tridentate Binding of a Deprotonated 4-Amino-3[END_REF][START_REF] Palion-Gazda | Field-induced slow magnetic relaxation in pseudooctahedral cobalt(II) complexes with positive axial and large rhombic anisotropy[END_REF][START_REF] Nemec | Two polymorphic Co(II) field-induced single-ion magnets with enormous angular distortion from the ideal octahedron[END_REF][START_REF] Shao | Tuning the Ising-type anisotropy in trigonal bipyramidal Co(II) complexes[END_REF][START_REF] Bunting | A linear cobalt(II) complex with maximal orbital angular momentum from a non-Aufbau ground state[END_REF][START_REF] Jin | Slow Magnetic Relaxation in a Series of Mononuclear 8-Coordinate Fe(II) and Co(II) Complexes[END_REF][START_REF] Mitsuhashi | Hydrogen-bonding interactions and magnetic relaxation dynamics in tetracoordinated cobalt(II) single-ion magnets[END_REF][START_REF] Vallejo | Reversible solvatomagnetic switching in a single-ion magnet from an entatic state[END_REF][START_REF] Rechkemmer | A four-coordinate cobalt(II) single-ion magnet with coercivity and a very high energy barrier[END_REF][START_REF] Sertphon | Slow relaxation of Inorganic Chemistry pubs[END_REF] Despite major improvements in mononuclear Co(II)-based SMMs over the past few years, an appropriate relationship between the structural parameters and magnetic properties in Co complexes is still challenging. Intense research is needed to extend our understanding for the rational design of desired magnetic materials.

To explore such aspects, herein we disclose a detailed investigation of magneto-structural chemistry in a series of mononuclear cobalt(II) complexes based on the tridentate bbp ligand (bbp = 2,6-bis(1H-benzo[d]imidazol-2-yl)pyridine) through preparation, detailed physical characterization including X-ray structure analyses, magnetic measurements, and EPR studies, and theoretical calculations. The studies reveal the effect of the coordination geometry on the magnetic properties of Co(II) complexes.

■ EXPERIMENTAL SECTION

Materials and Physical Measurements. All manipulations were carried out under an argon atmosphere using standard Schlenk techniques unless otherwise stated. Solvents were dried by standard methods and freshly distilled prior to use. All chemicals were used as purchased from chemical sources without further purification. The ligand bbp was synthesized according to a literature procedure. [START_REF] Hong | Rhodium-catalyzed selective C-H functionalization of NNN tridentate chelating compounds via a rollover pathway[END_REF] The crystals of the complexes were removed from the mother liquor and dried on filter paper to remove any adhering solvent, prior to measurement. The elemental analyses of C, H, and N were performed with a Thermo Scientific Flash 2000 Organic Elemental Analyzer. Infrared (IR) spectra were recorded in the range of 4000-400 cm -1 on a Bruker Tensor 27 spectrometer. UV/vis/NIR spectra were carried out in the region of 250-2000 nm on a Lambda 750 UV/vis/ NIR spectrometer. The UV/vis spectroscopic measurements in solution were done in quartz cuvettes with a path length of 1 cm. Solid-state measurements were carried out by taking a 5% sample by weight in KBr. Thermogravimetric analysis (TGA) was done on a Mettler Toledo TGA/SDTA851 analyzer with a heating rate of 5 °C min -1 ranging from room temperature to 300 °C under a nitrogen atmosphere. Differential scanning calorimetry (DSC) measurements were performed using a Mettler Toledo DSC 823 e differential scanning calorimeter with a scan rate of 5 K min -1 under a nitrogen atmosphere. Powder X-ray diffraction (PXRD) measurements were carried out on a PANalytical Empyrean diffractometer at 45 kV and 30 mA, under Cu Kα radiation (λ = 1.54059 Å). PXRD data analyses were done using PANalytical X'Pert HighScore Plus software. [START_REF] Degen | The HighScore suite[END_REF] Cyclic voltammetry (CV) measurements were performed with a Metrohm Autolab PGSTAT101 instrument using platinum as the working electrode in DMF (for the bbp ligand and complexes 1-3) and DCM (for complex 3) with 0.1 M n Bu 4 NPF 6 as the supporting electrolyte with a scan rate of 0.1 V/s. The concentration of the complexes used is around 1 mM. Ferrocene was used as an internal reference. EPR analyses were done using an EMX Bruker spectrometer equipped with a dielectric resonator. The temperature was monitored using an Intelligent Temperature Controller from Oxford Instrumentation employing a 4 He flow cryostat. For measurement of powders, the sample was ground to avoid microcrystallinity and eventual preferential orientation effects were checked by rotating the tube in the cavity and double-measuring the spectra. The powdered sample was installed in 4 mm diameter quartz tubes. The EPR data were analyzed with the help of EasySpin software. [START_REF] Stoll | a comprehensive software package for spectral simulation and analysis in EPR[END_REF] Synthesis Magnetic Measurements. The magnetic susceptibility measurements were performed with a Quantum Design MPMS-XL SQUID magnetometer, between 1.8 and 300 K for dc applied fields ranging from -7 to 7 T for 1 and 3. For complex 2, magnetic susceptibility measurements were carried out with a Quantum Design MPMS-XL SQUID magnetometer and a Quantum Design PPMS EverCool-II apparatus, between 1.8 and 300 K and between 2.5 and 280 K, respectively, for dc applied fields ranging from -7 to 7 T in mother liquor. Polycrystalline samples of 1-3 (11.10, 15.27, and 5.71 mg, respectively) placed in a polyethylene bag (2.4 × 0.6 × 0.02 cm) were subjected to measurements. The temperature-dependent data were measured using a 2500 Oe dc field for 1 and a 1000 Oe dc field for 2 and 3. The isothermal magnetization data were acquired at 3 and 5 K for 1, at 2.5, 3, 5, and 8 K for 2, and at 4 and 6 K for 3. The ac susceptibility measurements were carried out with an oscillating ac field of 3 Oe with frequency ranging from 1 to 1500 Hz (for 1 and 3) and from 1 to 10000 Hz (for 2). The frequency dependence was measured at different external dc magnetic fields ranging from 500 to 6000 Oe at 1.8 K (for 1 and 3) and from 0 to 10000 Oe at 2 K (for 2). M vs H measurements were performed at 100 K to check for the presence of ferromagnetic impurities, which were found to be absent. The magnetic data were corrected for the sample holder and the diamagnetic contribution.

X-ray Crystallography. Single-crystal X-ray structure analysis data of complexes 1-3 were collected with a Bruker SMART APEX CCD diffractometer equipped with graphite-monochromated Mo Kα radiation (λ = 0.71073 Å). The single crystals were mounted on crystal mounting loops with the help of Paratone oil at 240 K and slowly cooled to 120 K at a 2 K/min ramping rate, followed by data collection at 120 K for 1 and 3. For complex 2, data collection was done at 293 K. Data integration and reduction were performed with the help of SAINT software, and empirical absorption corrections were applied with the SADABS program. [START_REF] Sheldrick | Bruker Analytical X-Ray Systems[END_REF] Structures were solved using direct methods and refined with a full-matrix least-squares method on F 2 using SHELXL-2014. [START_REF] Sheldrick | Crystal structure refinement with SHELXL[END_REF] All non-hydrogen atoms were refined using anisotropic displacement parameters. Hydrogen atoms were located in ideal positions and refined isotropically using a riding model. Crystal data and refinement parameters are given in Table 1. Selected bond lengths and bond angles are given in Table S1. Nonroutine aspects of structure refinement are as follows. One of the NCS -ligands on the [Co(NCS) 4 ] 2-anion in 1 is dynamically disordered. We modeled this issue as a disorder over three positions with fixed occupation factors (0.3:0.5:0.2). The values of the occupation factors were determined using the SUMP instruction in the refinement. One of the noncoordinated DMF molecules exhibits large thermal movement, but attempts to model it as a disorder over two positions did not result in any improvements, and therefore the potential disorder of this molecule was left unresolved in the final refinement.

Theoretical Calculations. The ORCA 4.1 computational package was used for quantum chemical calculations. [START_REF] Neese | Software update: the ORCA program system, version 4.0[END_REF][START_REF] Neese | The ORCA program system[END_REF] First, the molecular structures of [Co(bbp) 2 ] 2+ and [Co(NCS) 4 ] 2-of 1, [Co(bbp)-(NCS) 2 (DMF)] of 2, and [Co(bbp)(NCS) 2 ] of 3 were extracted from X-ray data and the positions of hydrogen atoms were optimized with the DFT method using the PBE functional. Next, calculations of the zero-field splitting (ZFS) parameters were carried out using state average complete active space self-consistent field (SA-CASSCF) [START_REF] Malmqvist | The CASSCF state interaction method[END_REF] wave functions complemented by N-electron valence second-order perturbation theory (NEVPT2) [START_REF] Angeli | n-electron valence state perturbation theory: A spinless formulation and an efficient implementation of the strongly contracted and of the partially contracted variants[END_REF][START_REF] Angeli | Introduction of n-electron valence states for multireference perturbation theory[END_REF] using the triple-ζ basis set def2-TZV [START_REF] Weigend | Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy[END_REF] for all atoms. In the state-averaged approach all multiplets for a given electron configuration were equally weighted. The ZFS parameters, based on dominant spin-orbit coupling contributions from excited states, were calculated through quasi-degenerate perturbation theory (QDPT), [START_REF] Ganyushin | First-principles calculations of zerofield splitting parameters[END_REF] in which an approximation to the Breit-Pauli form of the spin-orbit coupling operator (SOMF

a R1 = ∑||F o | -|F c ||/∑|F o | and wR2= [∑w(|F o | 2 -|F c | 2 )|/∑|w(F o ) 2 |] 1/2 .
approximation) [START_REF] Neese | Efficient and accurate approximations to the molecular spin-orbit coupling operator and their use in molecular gtensor calculations[END_REF] and the effective Hamiltonian theory [START_REF] Maurice | Universal Theoretical Approach to Extract Anisotropic Spin Hamiltonians[END_REF] were utilized. The calculations utilized the RIJCOSX approximation with the auxiliary basis sets def2/J 46 and def2-TZVP/C. [START_REF] Hellweg | Optimized accurate auxiliary basis sets for RI-MP2 and RI-CC2 calculations for the atoms Rb to Rn[END_REF] Increased integration grids (Grid6 and GridX6 in ORCA convention) and tight SCF convergence criteria were used in all calculations. Ab initio ligand field theory (AILFT) was utilized for energy calculations of d orbitals. [START_REF] Singh | Covalency and chemical bonding in transition metal complexes: An ab initio based ligand field perspective[END_REF][START_REF] Mingos | Molecular Electronic Structures of Transition Metal Complexes II[END_REF] Moreover, recently introduced dynamic correlation dressed CAS with a second-order treatment (DCD-CAS( 2)) was also utilized to calculate ZFS parameters as well, where the spin-orbit and the spin-spin interactions were included. [START_REF] Pathak | A dynamic correlation dressed complete active space method: Theory, implementation, and preliminary applications[END_REF] ■ RESULTS AND DISCUSSION Synthesis. Complexes 1 and 3 were prepared by the reaction of Co(NCS) 2 and the bbp ligand in a mixture of DCM and MeOH (Scheme 2). Slow diffusion of diethyl ether into the solution of the reaction precipitate in a mixture of DMF and acetone gave green crystals of 1. Pink crystals of 3 were obtained from slow evaporation of the reaction filtrate. Green crystals of complex 1 on prolonged standing in mother solution (DMF/acetone) converted to red crystals of 2 through a structural transformation. In summary, the observed transformations can be described best by a simple reaction equation with two kinetically different steps in the crystallization of the reaction precipitate from DMF/acetone (Scheme 2), whereas a slower step involves solvolysis of the

[Co(bbp) 2 ] 2+ and [Co(NCS) 4 ] 2-molecules, resulting in the formation of [Co(bbp)(DMF)(NCS) 2 ].
The purity of all complexes was confirmed by elemental analyses and PXRD measurements (Figures S1-S3). The thermal stabilities of all complexes were studied by TGA up to 300 °C (Figure S4). Complexes 1 lost two crystallized DMF molecules around 125 °C and one crystallized DMF molecule around 185 °C, whereas complex 2 started to lose a crystallized DMF molecule above 80 °C. However, complex 3 is stable in the measured temperature range. The complexes have been well characterized by IR, UV/vis/NIR, and electrochemistry (Figures S6-S14).

Structure Analysis. Single-crystal X-ray diffraction analysis was performed on suitable single crystals of 1 and 3 at 120 K and 2 at 293 K. Complex 1 crystallizes in triclinic space group P1. Its crystal structure consists of a cationic [Co(bbp) 2 ] 2+ unit, an anionic [Co(NCS) 4 ] 2-unit, and four molecules of crystallized DMF (Figure 1). The overall coordination geometry of the cationic [Co(bbp) 2 ] 2+ unit of 1 is best described as a distorted octahedron with a CoN 6 environment using the continuous shape measures (CShM) program 51 (Table S2), where the cobalt center is coordinated to six nitrogen donor atoms of two mutually perpendicular bbp ligands. An angular distortion of [Co(bbp S3). In addition, the crystal structure of 1 is stabilized by intermolecular π-π interactions between the benzimidazolyl rings as well as pyridinyl rings of the bbp ligands, as shown in Figure 2 An X-ray structure analysis shows that complex 2 crystallizes in triclinic space group P1. Its crystal structure consists of a mononuclear neutral [Co(bbp)(NCS) 2 (DMF)] unit and two molecules of crystallized DMF (Figure 3). The overall coordination geometry of 2 is best described as a distorted octahedron with a CoN 5 O environment (Table S2). In the complex, the cobalt center is coordinated to three nitrogen donor atoms of the bbp ligand, two nitrogen donor atoms of two thiocyanate anions, and one oxygen donor atom from a DMF molecule. The trans N( 1 unit (Figure 5). The overall coordination geometry of 3 is best described as a distorted square pyramid with a CoN 5 environment (Table S2), where the cobalt center is coordinated by three nitrogen donor atoms of the bbp ligand and two nitrogen donor atoms of the two thiocyanate ligands. The nitrogen atoms of the bbp ligand and one of the thiocyanate nitrogen atoms occupy equatorial positions with two trans-positioned N-Co-N angles of 151.3(1)°for (N(1)-Co-N(4) and 134.3(1)°for N(3)-Co-N(7), while the other thiocyanate nitrogen atom occupies an apical position. These angles deviate significantly from the ideal values in an perfect square-pyramidal geometry, and thus the coordination geometry in 3 can be classified using the Addison parameter (τ) to be closer to square pyramidal (τ = 0.28, where τ = 0 for ideal square pyramidal and τ = 1 for ideal trigonal bipyramidal). [START_REF] Addison | structure, and spectroscopic properties of copper(II) compounds containing nitrogen-sulphur donor ligands; the crystal and molecular structure of aqua[1,7-bis(N-methylbenzimidazol-2′yl)-2,6-dithiaheptane]copper(II) perchlorate[END_REF] S3).

Magnetic Studies. The temperature dependence of χT (χ is the magnetic susceptibility, equal to M/H per two Co II ions) was studied on 1 in the temperature range of 2-300 K at 2500 Oe (Figure 6). At 300 K, the measured χT value is 4.5 cm 3 mol -1 K, which lies in the expected range for two HS d [START_REF] Ako | A Ferromagnetically Coupled Mn19 Aggregate with a Record S = 83/2 Ground Spin State[END_REF] Co II ions (S = 2 × 3/2, g = 2.0; χT = 2 × 1.875 cm 3 mol -1 K) (contribution from one pseudo-octahedral and another tetrahedral Co(II) coordination polyhedron). [START_REF] Shankar | Hydroxide-Bridged Mixed-Valence Tetranuclear Cobalt 4-Nitrophenol Inclusion Complex Showing Single Molecule Magnet Property[END_REF][START_REF] Khakhlary | Structural and magnetic properties of oxyquinolinate clusters of cobalt(ii) and manganese(ii) and serendipitous intake of carbonate during synthesis[END_REF][START_REF] Mondal | On/Off Photoswitching in a Cyanide-Bridged {Fe2Co2} Magnetic Molecular Square[END_REF][START_REF] Mondal | A cyanide and hydroxo-bridged nanocage: a new generation of coordination clusters[END_REF][START_REF] Mondal | W-Co Discrete Complex Exhibiting Photo-and Thermo-Induced Magnetisation[END_REF][START_REF] Lloret | Magnetic properties of six-coordinated high-spin cobalt(II) complexes: Theoretical background and its application[END_REF] The χT value decreases gradually with a decrease in the temperature to reach a value of 4.1 cm 3 mol -1 K at 50 K and then decreases abruptly, reaching a minimum of 2.6 cm 3 mol -1 K at 2 K. In addition, Co(II) complexes in a similar type of Co(N3) 2 coordination environment exhibit spin crossover behavior depending on the appropriate ligand field. [START_REF] Drath | Switchable cobalt coordination polymers: Spin crossover and valence tautomerism[END_REF][START_REF] Mcpherson | Predictable Substituent Control of CoIII/II Redox Potential and Spin Crossover in Bis(dipyridylpyrrolide)cobalt Complexes[END_REF][START_REF] Miller | Spin crossover with thermal hysteresis in cobalt(ii) complexes and the importance of scan rate[END_REF][START_REF] Habib | Influence of the Ligand Field on Slow Magnetization Relaxation versus Spin Crossover in Mononuclear Cobalt Complexes[END_REF][START_REF] Krivokapic | Spin-crossover in cobalt(II) imine complexes[END_REF] However, for complex 1, we have not observed any such property in the measured temperature range from 2 to 300 K, which is also confirmed by DSC measurements (Figure S30). The magnetic properties of 2 were investigated by measuring the temperature dependence of χT in the temperature range of 2.5-280 K at 1000 Oe (Figure 6). The measured χT value at 280 K is 2.57 cm 3 mol -1 K, which is significantly larger than the spin-only value for one HS d [START_REF] Ako | A Ferromagnetically Coupled Mn19 Aggregate with a Record S = 83/2 Ground Spin State[END_REF] Co II ion (S = 3/2, g = 2.0; χT = 1.875 cm 3 mol -1 K), indicating the presence of an orbital angular momentum contribution. [START_REF] Shankar | Hydroxide-Bridged Mixed-Valence Tetranuclear Cobalt 4-Nitrophenol Inclusion Complex Showing Single Molecule Magnet Property[END_REF][START_REF] Khakhlary | Structural and magnetic properties of oxyquinolinate clusters of cobalt(ii) and manganese(ii) and serendipitous intake of carbonate during synthesis[END_REF][START_REF] Mondal | On/Off Photoswitching in a Cyanide-Bridged {Fe2Co2} Magnetic Molecular Square[END_REF][START_REF] Mondal | A cyanide and hydroxo-bridged nanocage: a new generation of coordination clusters[END_REF][START_REF] Mondal | W-Co Discrete Complex Exhibiting Photo-and Thermo-Induced Magnetisation[END_REF][START_REF] Lloret | Magnetic properties of six-coordinated high-spin cobalt(II) complexes: Theoretical background and its application[END_REF] Upon a decrease in the temperature, the χT value decreases slowly to reach a value of 2.39 cm 3 mol -1 K at 100 K. On a further decrease in temperature, the χT value decreases abruptly to a value of 1.68 cm 3 mol -1 K at 2.5 K. DSC measurements confirm the absence of phase transition at the measured temperature range from 295 to 130 K (Figure S31) and thus the absence of spin crossover properties.

Magnetic investigations of 3 were carried out by measuring the temperature dependence of χT in the temperature range of 2-300 K at 1000 Oe (Figure 6). The measured χT value at 300 K is 2.21 cm 3 mol -1 K, which is slightly larger than the spin-only value for one HS d [START_REF] Ako | A Ferromagnetically Coupled Mn19 Aggregate with a Record S = 83/2 Ground Spin State[END_REF] Co II ion (S = 3/2, g = 2.0; χT = 1.875 cm 3 mol -1 K), suggesting the presence of an orbital angular momentum contribution. [START_REF] Shankar | Hydroxide-Bridged Mixed-Valence Tetranuclear Cobalt 4-Nitrophenol Inclusion Complex Showing Single Molecule Magnet Property[END_REF][START_REF] Khakhlary | Structural and magnetic properties of oxyquinolinate clusters of cobalt(ii) and manganese(ii) and serendipitous intake of carbonate during synthesis[END_REF][START_REF] Mondal | On/Off Photoswitching in a Cyanide-Bridged {Fe2Co2} Magnetic Molecular Square[END_REF][START_REF] Mondal | A cyanide and hydroxo-bridged nanocage: a new generation of coordination clusters[END_REF][START_REF] Mondal | W-Co Discrete Complex Exhibiting Photo-and Thermo-Induced Magnetisation[END_REF][START_REF] Lloret | Magnetic properties of six-coordinated high-spin cobalt(II) complexes: Theoretical background and its application[END_REF] The χT value decreases gradually with a decrease in the temperature to reach a value of 1.97 cm 3 mol -1 K at 90 K and then decreases abruptly, reaching a minimum of 1.14 cm 3 mol -1 K at 2 K.

In all complexes 1-3, the decrease in χT value at low temperature is probably due to the combined effect of the magnetic anisotropy and spin-orbit coupling coming from a HS Co(II) ion and also to the weak antiferromagnetic intermolecular interactions.

In the case of compound 1, there are two distinct paramagnetic centers and the spin Hamiltonian was formulated as

∑ μ ̂= ̂-̂+ ̂-̂+ - ⟨ ̂⟩ = H D S S E S S Bg S zj S S ( / 3 ) ( ) i i z i i i x i y i i a i a a 1 2 , 2 2 , 2 , 2 B , (1)
and incorporates the single-ion axial (D) and the single-ion rhombic (E) ZFS parameters, the Zeeman term defined for a direction of the magnetic field as B a = B(sin θ cos φ, sin θ sin φ, cos θ) using the spherical coordinates and also zj, the molecular field correction. Next, the molar magnetization for any a direction of the magnetic field (M a ) was computed by the formula

ε ε = - ∑ ∑ ∑ - ∑ - + M N C Z C kT kT ( ( ) ) exp( / ) exp( / ) a i k l ik a kl li a i i a i A , , (2)
in which the matrix elements of the Zeeman term (Z a ) and the eigenvectors (C) of the complete spin Hamiltonian matrix computed along the a direction of the magnetic field are exploited. Moreover, the occurrence of a molecular field correction implies that the iterative technique must be employed. [START_REF] Bocǎ | A Handbook of Magnetochemical Formulae[END_REF] Finally, an integral averaging was applied to compute the powder average of the molar magnetization using eq 3.

∫ ∫

π θ θ ϕ = π π M M 1 4 sin d d a mol 0 2 0 (3)
According to the theoretical CASSCF/NEVPT2 calculations (vide infra), the D parameter of the [Co(NCS) 4 ] 2-complex anion of 1 should be negligibly small in comparison with the value of D for the [Co(bbp) 2 ] 2+ complex cation (Table 2). Therefore, D anion was fixed to zero to prevent overparametrization of the model. As a result, the best-fit parameters were found to be D cation = 39.0 cm -1 , E/D = 0.09, g = 2.16, and χ TIP = 3.8 × 10 -4 cm 3 mol -1 with zj = -0.18 cm -1 , where χ TIP stands for the temperature-independent paramagnetism (Figure 6).

Next, the experimental magnetic data of 2 were treated with a spin Hamiltonian for the mononuclear system with zero-field splitting

μ ̂= ̂-̂+ ̂-̂+ Ĥ D S S E S S BgS ( /3 ) ( ) z x y a 2 2 2 2 B (4)
which resulted in D = 69.5 cm -1 , E/D = 0.12, and g xy = 2.46 with the fixed parameter g z = 2.00 (Figure 6). However, the theoretical calculations for this complex suggests that there are low-lying excited states (vide inf ra), which makes the application of the spin Hamiltonian formalism questionable. The 4 F atomic term fragments into the 4 T 1g ground state and excited [START_REF] Sessoli | Nanoscience: Single-atom data storage[END_REF] T 2g and 4 A 2g states upon formation of Co(II) (pseudo)octahedral complexes. Thus, the ground state possesses an unquenched orbital angular momentum and the formalism of the spin Hamiltonian is forbidden. [START_REF] Boca | Magnetic functions beyond the spin-Hamiltonian[END_REF][START_REF] Titis | Magnetostructural D Correlations in Hexacoordinated Cobalt(II) Complexes[END_REF] Such a situation was treated by Griffith and Figgis, [START_REF] Figgis | The magnetic behaviour of cubic-field 4T1g terms in lower symmetry[END_REF][START_REF] Griffith | The theory of transition-metal ions[END_REF] who described the splitting of energy levels arising from the 4 T 1g state with the L-S Hamiltonian using this equation

α λ μ α ̂= -• ⃗ • ⃗ + Δ ̂-̂+ Δ ̂- + ⃗ ⃗ -⃗ H S L L L L L B g S L ( ) ( /3) ( ) ( ) z x y ax 2 2 rh 2 2 B e (5)
where the parameters Δ ax and Δ rh measure splitting of the 4 T 1g ground term produced by lowering of the symmetry, α is an orbital reduction factor, λ is the spin-orbit coupling, and g e is the free-electron g factor (2.0023). This Hamiltonian exploits T 1 -P isomorphism; consequently, the angular orbital momentum L is equal to 1 with the effective Lande g factor g L = -α and is applied to |S,L,M S ,M L ⟩ functions with M L = 0, ±1 and M S = ±1/2, ±3/2. [START_REF] Boca | A handbook of magnetochemical formulae[END_REF] The orbital reduction factor α equals Aκ; thus, it assimilates two parameters, namely A, which is the Figgis coefficient of the configuration interaction resulting from the admixture of the excited terms reflecting the ligand field strength (1 < A < 3/2), and κ defines the lowering orbital contribution caused by covalency of the metal-ligand bond and is typically in the range 0.70 < κ < 1. Furthermore, the spin-orbit coupling parameter λ can be also reduced in comparison with its free-ion value λ 0 = -180 cm -1 due to the covalent character of the donor-acceptor bond. Thus, the magnetic data of 2 were also treated with the L-S Hamiltonian (eq 5), which resulted in the following parameters: α = 0.93, λ = -172 cm -1 , Δ ax = -733 cm -1 , Δ rh = -105 cm -1 (Figure S36). These parameters are in accordance with the parameters obtained from the evaluation of CASSCF/NEVPT2 calculations (vide infra).

The last complex 3 was again analyzed with the spin Hamiltonian in eq 4, and the best fit was obtained with D = 38.1 cm -1 , E/D = 0.0, g = 2.04, and χ TIP = 9.0 × 10 -4 cm 3 mol -1 , confirming the large magnetic anisotropy of this complex.

The field dependence of magnetization studies for 1 was carried out from 0 to 7 T at 3 and 5 K (Figure 7 and Figure S15). At 3 K and 7 T, the magnetization value is 5.38 μ B , which is slightly lower than the expected value for two S = 3/2 systems. In addition, M vs H/T measurements display nonsuperposition of the data on a single master curve (Figure S15), suggesting the presence of magnetic anisotropy in this system. The field-dependent magnetization measurements for 2 were performed from 0 to 9 T at 2.5, 3, 5, and 8 K (Figure 7 and Figure S16). The magnetization value at 2.5 K and 9 T is 2.22 μ B , which is lower than the expected value of an S = 3/2 system. Moreover, M vs H/T measurements show nonsuperposition of the data on a single master curve (Figure S16). The nonsaturation and nonsuperposition of the magnetization curves suggest the presence of magnetic anisotropy and/or low-lying excited states. The field dependence of magnetization studies for 3 was performed from 0 to 7 T at 4 and 6 K (Figure 7 and Figure S17). At 4 K and 7 T, the magnetization value is 1.82 μ B , which is lower than the expected value of a S = 3/2 system. M vs H/T measurements display nonsuperposition of the data on a single master curve (Figure . S17). M vs H and M vs H/T curves suggest the presence of magnetic anisotropy. No hysteresis was observed in M vs H at 2 K from -70000 to +70000 Oe with a sweep rate of 100-600 Oe (Figure S18).

In order to understand the slow dynamics of magnetization, ac susceptibility measurements were carried out on 1-3. The real (χ′) and imaginary (χ″) components of the ac susceptibility under zero dc field were measured at low temperature at driving frequencies up to 1500 Hz (for 1 and 3) and 10000 Hz (for 2) (Figure 8 and, Figures S19-S21). No out-of-phase χ″ signal under zero dc field was observed for any of the three complexes, indicating that effective quantum tunneling of magnetization (QTM) occurred at zero dc field.

It was demonstrated that the external magnetic field strongly affected the QTM. Thus, in order to minimize the probability of the quantum relaxation pathway in zero dc field and to estimate the energy barrier accurately, the ac susceptibility was measured under different dc fields. It is worth mentioning that no slow relaxation of magnetization was detected in the case of complex 1 from ac susceptibility measurements even by using a 6000 Oe dc magnetic field (Figure S21), although slow relaxation of magnetization has recently been reported in [Co(N3) 2 ][Co(NCS) 4 ] (N3 = tridentate N-donor ligand) types of complexes as well as in the [Co(NCS) 4 ] 2-unit with different metal cation complexes. [70][START_REF] Kalinke | From Paramagnetic to Single-Molecule Magnet Behaviour in Heterobimetallic Compounds Containing the Tetrakis(thiocyanato-κN)cobaltate-(II) Anion[END_REF][START_REF] Zhu | Slow Magnetic Relaxation in Weak Easy-Plane Anisotropy: the Case of a Combined Magnetic and HFEPR Study[END_REF][START_REF] Shao | Slow Magnetic Relaxation and Spin-Crossover Behavior in a Bicomponent Ion-Pair Cobalt(II) Complex[END_REF] For complex 2, the dynamics of the magnetization was studied under different applied dc fields up to 10000 Oe at 2 K to reduce the possibility of a quantum relaxation pathway (Figure 8). A generalized Debye model [START_REF] Cole | Dispersion and Absorption in Dielectrics I. Alternating Current Characteristics[END_REF] was employed to fit χ′ vs ν and χ″ vs ν to obtain the characteristic frequency of relaxation (Figure S22). A plot of the characteristic frequency (maxima of χ″ vs ν plot) vs applied different dc fields gives an optimum dc field of 1000 Oe, where the dynamics is the slowest (Figure S23).

Accordingly, ac susceptibility measurements were performed using a 1000 Oe external dc field at different temperatures and ac frequencies (Figure 9).

A clear maximum is detected in the χ″ vs ν plot at 1000 Oe, allowing us to deduce the relaxation time. A linear fit of the experimental τ vs 1/T plot using the Arrhenius equation allows an estimation of the energy barrier Δ eff /k B = 35.4 K and the pre-exponential factor τ 0 = 4.03 × 10 -8 s (Figure 10). Moreover, the whole temperature dependence of the relation times was fitted to a combination of direct, Raman, and Orbach relaxation processes using

τ τ = + + - Δ AT CT kT 1 1 exp n 0 eff i k j j j j y { z z z z which resulted in A = 128 K -1 s -1 , C = 39.4 K -n s -1
, n = 4.01, τ 0 = 8.59 × 10 -9 s -1 , and Δ eff = 53.2 K (Figure 10). Alternatively, the only slightly poor fit was achieved by the combination of direct and Raman relaxation processes with A = 284 K -1 s -1 , C = 13.6 K -n s -1 , and n = 4.77 (Figure 10), suggesting that at such low temperatures the first excited Kramers doublet is high in energy and thus its low Boltzmann population makes the Orbach relaxation pathway inefficient. The Cole-Cole plots were fitted using a generalized Debye model (Figure S24 and Table S4); details of this fitting are given in the Supporting Information. From the Cole-Cole plots, we obtained different values of α, which lie in the range of 0.04-0.36, suggesting a narrow distribution of the relaxation times (τ) (Figure S25).

Under an applied dc field, complex 3 shows frequencydependent signals in both components of the ac susceptibility, which suggest the slow relaxation of the magnetization. Similar to the case for complex 2, ac susceptibility measurements were carried out at different applied dc fields up to 6000 Oe at 1.8 K (Figure S26). From the ac measurements and the field variation of the characteristic frequency (Figure S27), quantum effects are evidently minimized around 1500 Oe, as displayed by the minimum of the characteristic frequency at this dc field. Therefore, ac susceptibility measurements were carried out under 1500 Oe external dc field at different temperatures and ac frequencies (Figure 11 and Figure S28) to calculate the temperature dependence of the relaxation time.

A relaxation mode is detected on χ″ vs ν data. From the ac data (Figure 11), an Arrhenius plot was produced, as shown in Figure 12. The relaxation shows a thermally activated behavior with the energy barrier Δ eff /k B = 6.9 K and the pre-exponential factor τ 0 = 1.8 × 10 -5 s. From the Cole-Cole plots, we obtained different values of α, which are in the range of 0.41, also indicating a narrow distribution of the relaxation times (τ) (Figure S29 and Table S5).

Very recently Mondal [START_REF] Mondal | Probing the Effects of Ligand Field and Coordination Geometry on Magnetic Anisotropy of Pentacoordinate Cobalt(II) Single-Ion Magnets[END_REF] et al. reported the magnetic properties of the similar complex [(Co(bbp)(NCS) 2 ], where they demonstrated the magnetic anisotropic parameter D to be 10.7(4) cm -1 , which is significantly lower than the expected value of around 30-70 cm -1 for Co(II) with a squarepyramidal coordination geometry. [START_REF] Nemec | Magnetic anisotropy in pentacoordinate 2,6-bis(arylazanylidene-1chloromethyl)pyridine cobalt(II) complexes with chlorido co-ligands[END_REF] The authors further reported an effective energy barrier of 9.7 K and a relaxation time of 4.6 × 10 -5 s under a static magnetic field of 1000 Oe. Indeed, the analysis reported herein ended with D ≈ 40 cm -1 and the results are also supported by CASSCF/NEVPT2 calculations (Table 2). In addition, the analysis reported herein also ended up with the optimization of the applied dc field of 1500 Oe as well as the deduction of an effective energy barrier of 6.9 K and a relaxation time of 1.8 × 10 -5 s under the optimized applied dc field.

X-band EPR experiments were performed on powdered samples of 1 and 3 in the temperature range of 5-50 K (Figure 13). The signal intensity for both complexes decreases with an increase in the temperature, which indicates the presence of a positive D value. Simulation of the spectra of 1 and 3 at 5 K are given in Figures S33 andS34 in the Supporting Information.

Theoretical Calculations. In order to get a better insight into the electronic structure and ZFS parameters, theoretical calculations on the complexes extracted from the experimental X-ray structures of 1-3 were applied for the ab initio, DFT, and semiempirical SCF-MO package ORCA 4.1. First, the post-Hartree-Fock multireference calculations built on the state-averaged complete active space self-consistent field method (SA-CASSCF) were accomplished using the def2-TZVP basis set. Such calculations resulted in the spin Hamiltonian parameters: particularly, the zero-field splitting parameters D and E. Due to the 3d 7 electronic configuration of Co II ions, the active space CAS (7,5) was defined by seven electrons in five d orbitals. In addition to the standard Nelectron valence state perturbation theory (NEVPT) used, we have exploited dynamic correlation dressed CAS with the second-order treatment (DCD-CAS(2)) to treat dynamic electron correlation effects. For the calculations of the energy of the d orbitals, ab initio ligand field theory (AILFT) was applied (Figure 14). The increasing coordination number from four to six is reflected in increasing overall splitting of d orbitals from the tetracoordinate [Co(NCS) 4 ] 2-complex of 1 through the pentacoordinate [Co(bbp)(NCS) 2 ] complex of 3 to the hexacoordinate [Co(bbp) 2 ] 2+ and [Co(bbp)(NCS) 2 (DMF)] complexes of 1 and 2, respectively. In the case of ideal symmetry, we could expect two sets of e and t 2 orbitals for T d point group symmetry, four sets of e, b 2 , a 1 , and b 1 orbitals for C 4v point group symmetry, and finally two sets of t 2g and e g orbitals for O h point group symmetry. However, in all complexes the degeneracy of d orbitals is removed due to the nonhomogenous ligand field and lower symmetry of the coordination polyhedra (Figure 14, left). Calculated quartet and doublet ligand-field terms originating from the 3d 7 electron configuration are shown in Figure 14 (middle), where in all cases the quartet is the ground state, in agreement with the high-spin state of complexes under study. The subsequent spin-orbit interaction leads to the ligand-field multiplets, hence the zero-field splitting of S = 3/2, which is clearly visible in Figure 14 (right). Large and positive D parameters of ∼40-50 cm -1 were calculated for the [Co-(bbp) 2 ] 2+ complex of 1 and the [Co(bbp)(NCS) 2 ] complex of 3, whereas a minute and negative D parameter is found for the [Co(NCS) 4 ] 2-complex of 1. The largest D parameter was derived for the [Co(bbp)(NCS) 2 (DMF)] complex of 2, reaching a value of ∼100 cm -1 (Table 2). However, the calculated g tensor and D tensor parameters for this complex may not be fully valid due to the existence of the low-lying excited states (Figure 14), which limit the applicability of the spin Hamiltonian approach. Therefore, we have analyzed the CASSCF/NEVPT2 zero-field multiplet energy levels arising from the splitting of the 4 T 1g ground term with the L-S Hamiltonian (eq 5). [START_REF] Nemec | Two polymorphic Co(II) field-induced single-ion magnets with enormous angular distortion from the ideal octahedron[END_REF] Fitting the energies of the six lowest Kramers doublets resulted in the following parameters: α•λ = -245 cm -1 , Δ ax = -714 cm -1 , and Δ rh = -151 cm -1 (Figure S35). To summarize, both methods dealing with the dynamic electron correlation provided similar ZFS parameters and, in general, there is good quantitative agreement with the parameters fitted from the experimental magnetic data of 1-3.

■ DISCUSSION

To discuss the relationship between the geometry of the coordination polyhedron and the magnetic anisotropy, we start with tetracoordinate Co II complexes. The structural parameter δ proposed by Bocǎ et al. was based on two angles, ∠(X-Co-X) (α) and ∠(Y-Co-Y) (β) for [CoX 2 Y 2 ] complexes, as δ = 2α Td -(α + β), where α Td = 109.5°, and it was suggested that this should directly correlate with D. [START_REF] Titis | Magnetostructural study of tetracoordinate cobalt(II) complexes[END_REF] A later study by Nemec et al. showed that such a relationship between δ and D is not linear in general. [START_REF] Nemec | Magnetic Anisotropy and Field-induced Slow Relaxation of Magnetization in Tetracoordinate CoII Compound[END_REF] In case of tetracoordinate [Co(NCS) 4 ] 2-of 1, the homoleptic nature of this complex made it difficult to calculate δ; however, the symmetry is close to ideal T d symmetry for which D = 0 and, indeed, the CASSCF/ NEVPT2 calculation resulted in a very small D value of -1.23 cm -1 (Table 2). Also, other homoleptic tetrahedral complexes of Co II were studied, [START_REF] Sottini | Magnetic Anisotropy of Tetrahedral CoII Single-Ion Magnets: Solid-State Effects[END_REF] and an attempt to correlate D with the continuous symmetry measure (CShM) of SHAPE was reported. [START_REF] Wu | Correlating magnetic anisotropy with the subtle coordination geometry variation of a series of cobalt(ii)-sulfonamide complexes[END_REF] The pentacoordinate complexes usually varied between trigonal-bipyramidal and square-planar arrangements, where the latter can possess a large contribution of the angular orbital momentum to the ground state. Recently, the magnetostructural correlation of D based on the Addison parameter τ was reported, showing that the square-pyramidal Co II complexes adopt large and negative D values and, moreover, that the sign of D is changed to positive on a transition to the chromophore geometries closer to the trigonal-bipyramidal arrangement, with the crossing point at τ ≈ 0.3. [START_REF] Nemec | Magnetic anisotropy in pentacoordinate 2,6-bis(arylazanylidene-1chloromethyl)pyridine cobalt(II) complexes with chlorido co-ligands[END_REF] Another correlation was later reported for complexes close to a trigonal- bipyramidal arrangement, for which small values of |D| are expected. [START_REF] Massoud | Pentacoordinate cobalt(II) complexes with neutral tripodal N-donor ligands: Zero-field splitting for a distorted trigonal bipyramidal geometry[END_REF] Thus, for a value of τ = 0.28 in 3 we could expect |D| to be roughly around 40 cm -1 , which was also found experimentally with D = 39.4 cm -1 . Similar pentacoordinate complexes were also studied recently, and extensive modeling based on the CASSCF/DCD-CAS(2) method confirmed the huge potential of such molecular species for achieving systems with large axial magnetic anisotropy. [START_REF] Brachnǎkova | Stereochemistry of coordination polyhedra vs. single ion magnetism in penta-and hexacoordinated Co(ii) complexes with tridentate rigid ligands[END_REF] The hexacoordinate Co II complexes are well-known to possess a 4 T 1g ground state in ideal O h symmetry, which, however, is prohibited by the Jahn-Teller theorem, leading to geometrical distortions. The foundations of the magnetism were originally laid down by Figgis [START_REF] Figgis | Ligand field theory and its applications[END_REF] and Griffith, [START_REF] Griffith | The theory of transition-metal ions[END_REF] as already mentioned during the analysis of compound 2. A lowering of the ligand field symmetry can lead to [START_REF] Sessoli | Nanoscience: Single-atom data storage[END_REF] A or [START_REF] Sessoli | Nanoscience: Single-atom data storage[END_REF] B well-isolated ground states, for which a spin Hamiltonian is an adequate model for the magnetism, and Bocǎ et al. [START_REF] Titis | Magnetostructural D Correlations in Hexacoordinated Cobalt(II) Complexes[END_REF] reported a nonlinear magneto-structural correlation for the axial and rhombic specifically defined structural parameters. Importantly, both positive and negative large D values can be found in distorted-octahedral Co II complexes e.g. by synthetic modification of the tripodal ligands [START_REF] Ding | Probing the influence of molecular symmetry on the magnetic anisotropy of octahedral cobalt(ii) complexes[END_REF] or by modification of the ligands in the axial positions, where in addition to CASSCF ab initio methods also the AOM (angular overlap model) was successfully employed. [START_REF] Herchel | Slow Magnetic Relaxation in Octahedral Cobalt(II) Field-Induced Single-Ion Magnet with Positive Axial and Large Rhombic Anisotropy[END_REF][START_REF] Vahovska | Effect of linear and non-linear pseudohalides on the structural and magnetic properties of Co(ii) hexacoordinate singlemolecule magnets[END_REF][START_REF] Vahovska | New Cobalt(II) Field-Induced Single-Molecule Magnet and the First Example of a Cobalt(III) Complex with Tridentate Binding of a Deprotonated 4-Amino-3,5-bis(pyridin-2-yl)-1,2,4-Triazole Ligand[END_REF] Especially for these systems the theoretical methods are important to guide the interpretation of the magnetic data and the choice of ) coordination geometries, respectively. These structural differences affect the magnetic anisotropy of these compounds significantly. In 1, the octahedral unit exhibits a large and positive parameter of axial magnetic anisotropy (D = +39.0 cm -1 ), whereas the tetracoordinate counteranion possesses only very small anisotropy, which was even neglected in a fitting of the magnetic data. Complex 2 has a ZFS parameter D of +69.5 cm -1 and exhibits a field-induced slow relaxation of magnetization under 1000 Oe dc field with a relaxation time of 1.9 × 10 -6 s, whereas complex 3 has a ZFS parameter D of +38.1 cm -1 and also shows a field-induced magnetic relaxation under a 1500 Oe dc field with a relaxation time of 1.8 × 10 -5 s. X-band EPR measurements and CASSCF/NEVPT2 calculations confirmed the stated results.

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PXRD, TGA, DSC, infrared, and electronic spectra, magnetic and EPR simulation data, and crystallographic tables (PDF)

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  of Complexes: [Co(bbp) 2 ][Co(NCS) 4 ]•4DMF (1), [Co(bbp)(NCS) 2 (DMF)]•2DMF (2), and [Co(bbp)(NCS) 2 ] (3). Under an argon atmosphere, Co(NCS) 2 (35 mg, 0.2 mmol) was suspended in dry DCM (10 mL). A solution of bbp (62 mg, 0.2 mmol) in a mixture of dry MeOH and dry DCM (1/2) (18 mL) was added. The resulting reaction mixture was stirred overnight. The yellow-green precipitate was filtered off, giving a pink filtrate. Green crystals of 1 were obtained in 47% yield from slow diffusion of diethyl ether into a green solution of the obtained precipitate in a mixture of DMF and acetone (7/1). Red crystals of 2 were obtained after a long standing (approximately 20 days) of green crystals of 1 in the mother solution. Slow evaporation of the pink filtrate gave analytically pure pink crystals of 3 in 35% yield. Scheme 1 shows schematic presentations and pictures of crystals of complexes 1-3. Data for 1, C 54 H 54 Co 2 N 18 O 4 S 4 (1265.25), are as follows. Anal. Calcd: C, 51.26; H, 4.30; N, 19.93. Found: C, 50.85; H, 4.69; N, 19.87. ATR-IR (only intense bands): ν (cm

Scheme 2 . 3 Figure 1 .

 231 Scheme 2. Synthesis of 1-3

  ) 2 ] 2+ is observed in the N-Co-N angles. The N-Co-N angles for the nitrogen atoms in cis arrangements are in the range of 75.7-116.2°, while those in a trans disposition are in the range of 150.7-168.1°. The Co-N bond lengths to the central pyridyl rings (2.092(3) and 2.090(4) Å) are slightly shorter than those involving the outer benzimidazolyl rings (2.140(3)-2.177(3) Å). The average Co-N bond length of 2.138 Å suggested a HS state of the central Co(II) atom. The coordination geometry of the [Co(NCS) 4 ] 2-complex anion in 1 is best described as a distorted-tetrahedral geometry (Table S2) with an average N-Co-N angle of 109.9°and average Co-N bond length of 1.946 Å. The coordinated thiocyanate anions are almost linear with an average N-C-S angle of 178.5°; however, the Co-N-C angles are slightly bent with values ranging from 164.4(9)°to 174.3(5)°. The DMF molecules in the structure of 1 are participating in hydrogen bonds. The NH group of the uncoordinated benzimidazolyl ring of the bbp ligand is hydrogen-bonded to the oxygen atom of the DMF molecule with N•••O distances of 2.673-2.767 Å (Figure 2 and Table

  . The shortest Co•••Co distance between [Co(bbp) 2 ] 2+ and [Co-(NCS) 4 ] 2-is 8.392(9) Å, while the shortest Co•••Co distance between the two [Co(bbp) 2 ] 2+ cations is 9.3513(8) Å.

  )-Co-N(4), N(3)-Co-N(6) and N(7)-Co-O(1) angles are 150.5(1), 174.2(1), and 174.3(1)°respectively, whereas cis N-Co-N and N-Co-O angles are in the range of 75.4(1)-106.9(1)°. The Co-N bbp bonds with an average distance of 2.146 Å are longer than Co-N NCS bonds with an average value of 2.050 Å, while the Co-O DMF bond distance is 2.272(3) Å. These values are similar to the expected HS Co(II) complexes in an octahedral geometry. The coordinated thiocyanate anions are almost linear with N-C-S angles of 178.9(4) ((N(7)-C(20)-S(1)) and 179.3(4)°( N(6)-C(21)-S(2)); however, Co-N(6)-C(21) is almost linear with a value of 173.1(4)°and Co-N(7)-C(20) is significantly bent with a value of 142.8(3)°. In the complex 2, the DMF molecules form N-H•••O hydrogen bonds with the NH groups of the benzimidazolyl ring of the bbp ligand with N•••O distances of 2.746(5) and 2.712(5) Å and N-H•••O angles of 176.15 and 178.29°( Figure 4). Moreover, the crystal structure of 2 is stabilized by intermolecular π-π interactions between pyridinyl rings of the bbp ligands and intermolecular C-H•••N and C-H•••C interactions, as shown in Figure 4. Short intra-and intermolecular contacts are collected in Table S3. The shortest Co•••Co distance between two [Co(bbp)(NCS) 2 (DMF)] units is 8.556(1) Å. Complex 3 crystallizes in triclinic space group P1. Its crystal structure consists of a mononuclear neutral [Co(bbp)(NCS) 2 ]

Figure 2 .

 2 Figure 2. Perspective views of the molecular structure of 1 displaying N-H•••O hydrogen bonds (violet dotted lines) (left) and the supramolecular double chain formed by weak intermolecular π•••π interactions (red and green dotted lines) (right). Hydrogen atoms, solvent molecules, and counteranions are omitted for clarity.

Figure 3 .

 3 Figure 3. Perspective view of complex 2. Hydrogen atoms and solvent molecules are omitted for clarity (Co, pink; C, gray; N, blue; O, red; S, orange).

  In the complex, the average Co-N bond length of 2.067 Å indicates a HS state of the central Co(II) atom. The coordinated thiocyanate anions are almost linear with N-C-S angles of 177.8(2) and 177.2(3)°; however, the Co-N-C angles are 161.9(3) and 162.6(3)°. In addition, the complex molecules are assembled in supramolecular chains by N-H•••S intermolecular hydrogen bonds between the sulfur atom of a thiocyanate anion and an uncoordinated nitrogen atom of the bbp ligand with N•••S distances of 3.313(2) and 3.271(2) Å (Figure S5 and Table

Figure 4 .

 4 Figure 4. Perspective views of the molecular structure of 2 displaying N-H•••O hydrogen bonds (violet dotted lines) (left) and the fragment of the supramolecular structure formed by weak intermolecular π•••π (red dotted lines), C-H•••N (green dotted lines), and C-H•••C (cyan and purple dotted lines) interactions (right).

Figure 5 .

 5 Figure 5. Perspective view of complex 3. Hydrogen atoms and solvent molecules are omitted for clarity (Co, pink; C, gray; N, blue; S, orange).

Figure 6 .

 6 Figure 6. Temperature dependence of χT for 1-3. The empty symbols denote experimental data and full lines calculated data with the parameters in the text.

Figure 7 . 3 .

 73 Figure 7. Field dependence of the magnetization as M vs B plots of 1-3. The empty symbols denote experimental data and the full lines calculated data with the parameters in the text.

Figure 8 .

 8 Figure 8. Frequency dependence of the real (χ′, left) and imaginary (χ″, right) components of the ac susceptibility at different ac frequencies from 1 to 10000 Hz and different external dc field from 10 to 10000 Oe, respectively, with a 3 Oe ac field for 2 at 2 K.

Figure 9 .

 9 Figure 9. Frequency dependence of the real (χ′, left) and imaginary (χ″, right) components of the ac susceptibility at different ac frequencies from 10 to 10000 Hz under a 1000 Oe external dc field, with a 3 Oe ac field for 2 at 2 K.

Figure 10 .

 10 Figure 10. Temperature dependence of the magnetization relaxation time as τ versus T -1 plot for 2 in a 1000 Oe dc field. The full lines correspond to various combinations of the relaxation processes as described in the text.

Figure 11 .

 11 Figure 11. Frequency dependence of the real (χ′, top) and imaginary (χ″, bottom) components of the ac susceptibility at different ac frequencies from 1 to 1500 Hz and different temperatures from 1.8 to 5.4 K, respectively, with a 3 Oe ac field for 3 at 1500 Oe external dc field.

Figure 12 .

 12 Figure 12. Temperature dependence of the magnetization relaxation time as τ versus T -1 plot for 3 in 1500 Oe dc field.

Figure 13 .

 13 Figure 13. Variable-temperature X-band EPR spectra of the polycrystalline powder of 1 (top) and 3 (bottom).

Figure 14 .

 14 Figure 14. Graphical representation of the CASSCF/NEVPT with CAS(7,5) calculations for the molecular fragments [Co(bbp) 2 ] 2+ and [Co(NCS) 4 ] 2-of 1, [Co(bbp)(NCS) 2 (DMF)] of 2, and [Co(bbp)(NCS) 2 ] of 3. Plot of the d orbital splitting calculated by ab initio ligand field theory (AILFT) (left), low-lying ligand-field terms with various multiplicities (middle), and ligand-field multiplets showing the zero-field splitting (right).

  the proper Hamiltonian model to use. This was exemplified in this work for [Co(bbp) 2 ] 2+ of 1 and [Co(bbp)(NCS) 2 (DMF)] of 2, in which a larger structural deviation from O h symmetry in the former made a spin Hamiltonian the right model to apply, whereas the latter compound was treated by a Figgis-Griffith Hamiltonian model. ■ CONCLUSION Utilization of two different crystallization procedures for the products of the reaction between Co(NCS) 2 and the bbp ligand resulted in the preparation of three distinct compounds: [ C o ( b b p ) 2 ] [ C o ( N C S ) 4 ] • 4 D M F ( 1 ) , [ C o ( b b p ) -(NCS) 2 (DMF)]•2DMF (2), and [Co(bbp)(NCS) 2 ] (3). Complex 1 shows a remarkably interesting structural transformation to form complex 2. Complex 1 contains two different Co(II) ions with different coordination geometries: octahedral ([Co(bbp) 2 ] 2+ ) and tetrahedral ([Co(NCS) 4 ] 2-), whereas the metal centers in 2 and 3 have distorted-octahedral ([Co(bbp)(NCS) 2 (DMF)]) and distorted-square-pyramidal ([Co(bbp)(NCS) 2 ]

  Scheme 1. Schematic Presentations (Top) and Pictures of Crystals (Bottom) of Complexes 1-3

	Inorganic Chemistry
	1789, 1642, 1606, 1493, 1460, 1385, 1317, 1253, 1235, 1160, 1146,
	1105, 1064, 993, 967, 938, 905, 846, 817, 753, 694, 666, 628, 584,
	536, 480, 459, 433. Data for 3, C 21 H 13 CoN 7 S 2 (486.45), are as
	follows. Anal. Calcd: C, 51.85; H, 2.69; N, 20.16. Found C, 51.62; H, 2.97; N, 20.01. ATR-IR (only intense bands): ν (cm -1 ) 3088, 2103,
	2077, 1625, 1590, 1571, 1454, 1398, 1312, 1234, 1145, 1116, 1086,
	1015, 993, 947, 931, 909, 895, 812, 762, 733, 655, 581, 567, 468, 425.

-1 ) 3084, 2100, 2059, 1671, 1638, 1591, 1574, 1491, 1458, 1414, 1374, 1315, 1251, 1234, 1160, 1105, 1060, 995, 967, 902, 846, 817, 802, 745, 665, 583, 477, 413. Data for 2, C 30 H 34 CoN 10 O 3 S 2 (705.72), are as follows. Anal. Calcd: C, 51.06; H, 4.86; N, 19.85. Found: C, 50.81; H, 5.00; N, 19.76. ATR-IR (only intense bands): ν (cm -1 ) 3056, 2991, 2875, 2072, 2052, 1913, 1834,

Table 1 .

 1 X-ray Crystallographic Data for Complexes 1-3

	1	2	3

Table 2 .

 2 Ab Initio Calculated and Fitted Spin Hamiltonian Parameters for Compounds 1-3

		[Co(bbp) 2 ] 2+ of 1	[Co(NCS) 4 ] 2-of 1	[Co(bbp)(NCS) 2 (DMF)] of 2	[Co(bbp)(NCS) 2 ] of 3
			ZFS and g Values Based on CASSCF/NEVPT2 Calculations
	D (cm -1 )	46.7	-1.23	96.6	39.4
	E/D	0.093	0.209	0.274	0.137
	g x	2.429	2.230	2.223	2.364
	g y	2.569	2.224	2.962	2.477
	g z	2.051	2.241	1.900	2.041
			ZFS Values Based on CASSCF/DCD-CAS(2) Calculations
	D (cm -1 )	49.0	-1.32	99.3	41.0
	E/D	0.093	0.211	0.278	0.113
			Magnetic Analysis of the Experimental Data
	D (cm -1 )	39.0	0.0	69.5	38.1
	E/D	0.09	0.0	0.12	0.0
	g iso	2.16	2.16	g xy = 2.46, g z = 2.00	2.04

  AccessionCodesCCDC 1916563 and 1916572-1916573 contain the supplementary crystallographic data for this paper. These data can be obtained free of charge via www.ccdc.cam.ac.uk/data_request/ cif, or by emailing data_request@ccdc.cam.ac.uk, or by contacting The Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge CB2 1EZ, UK; fax: +44 1223 336033.

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We are thankful to the IISc and Ministry of Human Resource Development (MHRD), Government of India, for funding the Start-up Research Grant (SG/MHRD-17-0014 and SR/ MHRD-17-0011). S.G., M.D., and S.M. thank the IISc and S.K. thanks the Council of Scientific & Industrial Research (CSIR), Government of India, for their fellowships. R.H. and I.N. acknowledge financial support from institutional sources of the Department of Inorganic Chemistry, PalackýUniversity Olomouc, Czech Republic. I.N. acknowledges the project CEITEC 2020 (LQ1601) with financial support from the Ministry of Education, Youth and Sports of the Czech Republic under the National Sustainability Programme II. Dilip Kumar Maity is thanked for useful discussion. M.-E.B. thanks the Engineering and Physical Sciences Research Council (EPSRC) for funding her postdoctoral position through grant EP/L018470/1 and the EPSRC National Electron Paramagnetic Resonance Facility and Service at the University of Manchester, where the experiments were held.

This research work was supported by the Indian Institute of Science (IISc), Bangalore, India, and the Science & Engineering Research Board (SERB) (Project No. SRG/2019/000317).

Notes

The authors declare no competing financial interest.

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