Enhancement of the Third-Order Nonlinear Optical Properties in Open

Article pubs.acs.org/JPCA

Enhancement of the Third-Order Nonlinear Optical Properties in Open-Shell Singlet Transition-Metal Dinuclear Systems: Effects of the Group, of the Period, and of the Charge of the Metal Atom Hitoshi Fukui,*,† Yudai Inoue,† Taishi Yamada,† Soichi Ito,† Yasuteru Shigeta,† Ryohei Kishi,† Benoît Champagne,¶ and Masayoshi Nakano*,† †

Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan ¶ Laboratoire de Chimie Théorique, Facultés Universitaires Notre-Dame de la Paix (FUNDP), rue de Bruxelles, 61, B-5000 Namur, Belgium S Supporting Information *

ABSTRACT: Metal−metal multiply bonded complexes in their singlet state have been predicted to form a novel class of “σ-dominant” third-order nonlinear optical compounds based on the results of dichromium(II) and dimolybdenum(II) systems (H. Fukui et al. J. Phys. Chem. Lett. 2011, 2, 2063) whose second hyperpolarizabilities (γ) are enhanced by the contribution of the dσ electrons with an intermediate diradical character. In this study, using the spin-unrestricted coupledcluster method with singles and doubles as well as with perturbative triples, we investigate the dependences of γ on the group and on the period of the transition metals as well as on their atomic charges in several open-shell singlet dimetallic systems. A significant enhancement of γ is observed in those dimetallic systems composed of (i) transition metals with a small group number, (ii) transition metals with a large periodic number, and (iii) transition metals with a small positive charge. From the decomposition of the γ values into the contributions of dσ, dπ, and dδ electrons, the γ enhancements are shown to originate from the dσ contribution, because it corresponds to the intermediate diradical character region. Furthermore, the amplitude of dσ contribution turns out to be related to the size of the dz2 atomic orbital of the transition metal, which accounts for the dependence of γ on the group, on the period, and on the charge of the metal atoms. These dependences provide a guideline for an effective molecular design of highly efficient third-order nonlinear optical (NLO) systems based on the metal−metal bonded systems.

1. INTRODUCTION Recent theoretical studies have shown that singlet diradical systems with intermediate diradical characters tend to exhibit larger second hyperpolarizabilities (γ) than pure diradical systems and conventional closed-shell systems of similar size.1−5 The mechanism of the diradical character dependence of γ has been unraveled by resorting to the summation-overstate (SOS) expressions of the static γ6a and of the two-photon absorption cross section6b for a symmetric two-site diradical model (A•−B•) within the valence configuration interaction (VCI) scheme. Ab initio molecular orbital (MO) and density functional theory (DFT) investigations have illustrated these predictions for a broad range of model and real molecular systems.1−4 Experimental evidence of these phenomena have been given by two-photon absorption measurements on sindaceno[1,2,3-cd;5,6,7-c′d′]diphenalene7 and by the thirdharmonic generation spectrum of 1,4-bis-(4,5-diphenylimidazole-2-ylidene)-cyclohexa-2,5-diene,8 while these studies have stimulated experimental and theoretical investigations on thirdorder nonlinear optical (NLO) properties of singlet diradical compounds.9,10 © 2012 American Chemical Society

The third-order NLO properties of singlet transition-metal complexes with metal−metal multiple bonds11 are also of interest because some of them, for example dichromium(II) complexes with Cr(II)−Cr(II) quadruple bonds, were predicted to present multiple diradical characters, which originates in relatively weak d−d orbital interactions constructing dσ, dπ, and dδ orbitals as shown in Scheme 1.12 In a previous study,13 we have investigated the γ values of the singlet dichromium(II) [Cr(II)−Cr(II)] and dimolybdenum(II) [Mo(II)−Mo(II)] model systems along with the diradical characters of the dσ, dπ, and dδ orbitals and the contributions of dσ, dπ, and dδ electrons to γ [γ(dσ), γ(dπ), and γ(dδ), respectively], by varying the metal−metal bond lengths. We found that γ(dσ) is dominant and takes a maximum value [γmax(dσ)] in the intermediate diradical character regions of the dσ orbitals in both model systems, which indicates that open-shell singlet metal−metal bonded systems belong to a novel class of “σdominant” third-order NLO systems. In general, for a symmetric two-site diradical model A•−B•, γ is given as a Received: April 15, 2012 Published: May 16, 2012 5501

dx.doi.org/10.1021/jp303618y | J. Phys. Chem. A 2012, 116, 5501−5509

The Journal of Physical Chemistry A

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Scheme 1. Schematic Orbital Interaction Diagrams of the Cr(II)−Cr(II) Quadruple Bond with Multiple Diradical Character

function of the effective diradical distance (RBA, approximately equal to the A−B bond length), of the effective Coulomb repulsion (U), of the diradical character (y), and of twice the direct exchange integral Kab divided by U (rk ≡ 2Kab/U),6a 4 γ = (RBA /U3)f (y , rk)

(1) Figure 1. Contours of valence dz2 atomic orbitals for period 4 elements V2+, Cr2+, and Mn2+ (a), group 6 elements Cr2+, Mo2+ and W2+ (b), and chromium ions with a different charge Cr+, Cr2+ and Cr3+ (c) at the UHF/SDD level of approximation.

where f (y , rk) = 8(1 − y)4

− {1 + +

1 − (1 − y)2 }2{1 − 2rk + 1/ 1 − (1 − y)2 }3

systems. For these model systems, we investigate the bond length dependence of γ using the spin-unrestricted (U) coupled-cluster method with singles and doubles (UCCSD) and that with perturbative triples [UCCSD(T)] as well as the bond length dependence of the diradical characters of the dσ, dπ, and dδ orbitals using the UCCSD method. The γ value is divided into the contributions of the dσ, dπ, and dδ electrons to reveal the origin of the effects of the group, period, and atomic charge. On the basis of these results, we define design strategies for open-shell singlet metal−metal bonded dimetallic complexes exhibiting large third-order optical nonlinearities.

4(1 − y)2 {1 − 2rk + 1/ 1 − (1 − y)2 }2{1/ 1 − (1 − y)2 } (2)

Usually, rk is close to zero, and f(y,0) attains a maximum in the intermediate y region, which is the origin of the γ enhancement in singlet diradical systems with intermediate y. γ is also strongly affected by the diradical distance RBA: γ is proportional to the fourth power of RBA. Therefore, this equation indicates that an intermediate diradical character with a long A−B bond length leads to large γmax value. This predicts that an intermediate dσ diradical character with a long metal−metal bond length leads to large γmax(dσ) in metal−metal bonded systems, resulting in enhancement of γmax. Because diffuse atomic orbitals can interact with each other at long distance, which leads to an intermediate diradical character with large bond length, metal−metal bonded systems composed of transition metals with a diffuse valence d atomic orbital are expected to satisfy the condition for enhancing γmax. Therefore, the γmax value of a metal−metal bonded system is predicted to depend on the group number, on the period number, and on the charge of the metal atoms because the size of the valence d atomic orbital is related to them. As shown in Figure 1, smaller group number, larger period number, and smaller positive charge of transition metals generally lead to larger size of the valence d atomic orbitals. On the basis of the above prediction, it is expected that a dimetallic system composed of transition metals with smaller group number, larger period number, and/ or smaller positive charge will present an intermediate dσ diradical character along with a longer bond length and will thus exhibit larger γmax. To verify this prediction, we investigate in this paper the effects of the group, of the period, and of the charge of the metal atoms on γ of open-shell singlet dimetallic

2. THEORETICAL AND COMPUTATIONAL ASPECTS 2.1. Model Systems. In the present study, we consider three sets of model systems: (A) dimetallic systems composed of dicationic transition metals of period 4 [V(II)−V(II), Cr(II)−Cr(II) and Mn(II)−Mn(II)], (B) those of dicationic transition metals of group 6 [Cr(II)−Cr(II), Mo(II)−Mo(II) and W(II)−W(II)], and (C) dichromium systems with different charge [Cr(I)−Cr(I), Cr(II)−Cr(II) and Cr(III)− Cr(III)]. Sets A and B are examined to reveal the effects of the group and period of the metal atoms on γ, while set C is examined to illuminate the effect of the charge. Their formal electron configurations are shown in Scheme 2 along with their formal bond orders, where the orbitals having degenerate energies are illustrated by closely lying orbital levels. The d3−d3 orbital interactions in V(II)−V(II) and Cr(III)−Cr(III) lead to the formal triple bond composed of one dσ and two equivalent dπ bonds, while Cr(II)−Cr(II), Mo(II)−Mo(II), and W(II)− W(II) present the formal quadruple bond with one dσ, two equivalent dπ, and one dδ bonds due to d4−d4 orbital interactions. Cr(I)−Cr(I) and Mn(II)−Mn(II) display the formal quintuple bond consisting of one dσ as well as two equivalent dπ and dδ bonds due to d5−d5 orbital interactions, 5502



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method.17 In the present study, the diradical character of the dX orbital [y(dX), where X = σ, π, δ] is obtained by using eqs 5 and 6 from the occupation number of the dX bonding and antibonding NO pair. We employ the effective core potential (ECP) of the Stuttgart group with the corresponding valence basis set (SDD) for V, Cr and Mn,18 while for Mo and W, the SDD basis set19 was supplemented with an additional set of f polarization functions20 [referred to as “SDD(f)” in this paper]. This is necessary to achieve balanced basis sets because, contrary to the case of first transition metals, the SDD basis set for second and third transition metals does not include f polarization functions. 2.3. Evaluation and Analysis of Static Second Hyperpolarizability. The longitudinal tensor component γzzzz (≡γ) along the bond axis (z-axis) is calculated using the finite-field (FF) approach,21 which consists in a fourth-order differentiation of the energy with respect to the applied external electric field. The perturbation series expansion convention (called the B convention22) is chosen for defining γ, and the following fourth-order numerical differentiation formula is employed:

Scheme 2. Formal Electron Configurations for Model Systems along with Their Formal Bond Orders

γ=

+ 39E( −F ) − 12E( −2F ) + E( −3F )}

although Mn(II)−Mn(II) is just a model system for the present objective, and no real complexes with the Mn(II)−Mn(II) quintuple bond have been found so far. 2.2. Calculation of Diradical Character. The diradical character is originally defined in the multiconfigurational selfconsistent field (MC-SCF) theory by twice the weight of the doubly excited configuration in the singlet ground state.14a Nevertheless, several other definitions have been proposed. For example, the diradical character can be defined by the occupation number (nLUNO+i) of the lowest unoccupied natural orbital (LUNO)+i (where i = 0, 1, ...):14b yi = nLUNO + i (3)

while for other methods, it is approximately satisfied. Spinunrestricted approaches such as the unrestricted Hartree−Fock (UHF) method are known to suffer from the spin contamination, which leads to an overshot diradical character as compared to that by the MC-SCF method. In such a case, an improved diradical character can be evaluated using a spinprojection scheme,15,16 2Ti 1 + Ti2

(7)

Here, E(F) indicates the total energy in the presence of the static electric field F in the z-direction. We used F values ranging from 0.0010 to 0.0060 au to obtain numerically stable γ values. Moreover, a tight convergence threshold of 10−10 a.u. on the energy is adopted to obtain sufficiently precise γ values. The γ values are given in atomic units (a.u.): 1.0 au of γ is equal to 6.235377 × 10−65 C4 m4 J−3 and 5.0367 × 10−40 esu. For the calculation on open-shell singlet metal−metal bonded complexes, which have nearly degenerate MOs, an inclusion of static electron correlation is indispensable for obtaining correct descriptions of their electronic states. Although multireference methods such as complete-active-space SCF (CASSCF), CAS with second-order perturbation (CASPT2), and multireference second-order Møller−Plesset perturbation (MRMP2) are often used for calculating geometries, energies, properties, and occupation numbers of such systems,16,23 they are hard to be applied to the γ calculations of the present open-shell systems because of the very slow convergence of γ as a function of the size of the active space24 as well as of the very demanding computational resources for extending the active space. An alternative way to approximately treat static correlation effects in open-shell molecular systems is the use of the broken symmetry-based strong correlated methods, e.g., the spinunrestricted-based CC methods, which can include high-order electron correlation effects without suffering from significant spin contaminations.16,25 Therefore, we employed the UCCSD and UCCSD(T) methods for the calculation of γ. In addition, the relativistic effects are taken into account using the ECPs of the SDD and SDD(f) basis sets, which have been shown to reproduce well the Dirac−Hartree−Fock second hyperpolarizability results obtained with an all-electron basis set.26 Like for closed-shell systems,27 extended basis sets with diffuse functions are indispensable for obtaining semiquantitative γ values in open-shell singlet systems.28 The SDD basis set includes one set of diffuse s, two sets of diffuse p, and one set of diffuse d functions. It was confirmed to be sufficiently extended since the addition of one set of d and f diffuse functions to the

In the case of spin-unrestricted single determinant schemes, the occupation numbers satisfy the following relationship: nLUNO + i = 2 − nHONO − i (4)

yi = 1 −

1 {E(3F ) − 12E(2F ) + 39E(F ) − 56E(0) 36(F )4

(5)

where Ti, the orbital overlap between the corresponding orbital pairs, is expressed in terms of the occupation numbers of the UHF natural orbitals (UNOs): n − nLUNO + i Ti = HONO − i (6) 2 These spin-projected UHF (PUHF) yi values range between 0 and 1, which correspond to closed-shell and pure diradical states, respectively. Although eqs 5 and 6 are very simple, they closely reproduce the diradical characters calculated by other methods such as the ab initio configuration interaction (CI) 5503



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SDD basis set only provides minor changes in γ of Cr(II)− Cr(II) at the UCCSD and UCCSD(T) levels of approximation (see Supporting Information of ref 13). For a detailed analysis of γ, their dσ, dπ, and dδ electron contributions [γ(dX), where X = σ, π and δ] using a partitioning scheme of the second hyperpolarizability,14b where a given γ(dX) contribution reads γ(dX ) = −

1 3!

∫ rzdzzzdX(3)(r) dr

(8)

ddX(3) zzz (r)

is the third-order derivative of the electron density of the bonding (dX) and antibonding (dX*) NO pair with respect to the electric field in the z-direction, and it reads dX (3) dzzz (r) =

∂3 {ndX ϕd*X (r)ϕdX (r) + ndX *ϕd*X *(r)ϕdX *(r)} ∂Fz3

Fz = 0

(9)

Here, ϕdX(r) and ndX represent the dX NO and its occupation number, respectively. The γ(dX) values were evaluated at the UCCSD level of approximation with the SDD or SDD(f) basis sets. All calculations on y(dX) and γ were performed with the Gaussian 09 program package.29

3. RESULTS AND DISCUSSION 3.1. Group Effect on the Diradical Character and on γ. Using set A composed of group 5, 6, and 7 elements (V, Cr, and Mn, respectively) of the same period, we discuss the influence of the group on the diradical character and γ in dimetallic systems. The diradical characters y(dX) increase with increasing R in all the three systems (see Figure 2 and Tables 1S−3S in the Supporting Information). At a given R, for the three systems, the diradical characters satisfy the following order, dδ > dπ > dσ, except for V(II)−V(II) between 1.6 and 1.8 Å, where y(dσ) > y(dπ). When comparing the three systems at the same bond length, the following orders of the diradical characters are observed: Mn(II)−Mn(II) > Cr(II)−Cr(II) > V(II)−V(II) for y(dσ) and y(dπ), and Cr(II)−Cr(II) > V(II)− V(II) for y(dδ), i.e., the smaller the group number, the smaller the y(dX) values are (for a given period and charge). This behavior is related to the size of the d atomic orbitals, which decreases when going toward higher group number (Figure 1a), because smaller atomic orbitals are associated with smaller d−d overlaps (for a fixed R), i.e., larger diradical character. The bond length dependences of γ and γ(dX) for the systems of set A are displayed in Figure 3 (see also Tables 1S−3S in the Supporting Information), where γ(dX) values are obtained by the UCCSD/SDD method, while the γ values are calculated both by the UCCSD/SDD and UCCSD(T)/SDD methods. The UCCSD method reproduces well the UCCSD(T) γ in all systems, which means that the perturbative triple excitations have little impact on γ, and which allows us to analyze the γ of these systems by referring to the UCCSD method. Moreover, there is a small difference between the UCCSD γ values and the sum of all γ(dX) values. This small difference originates from the contribution of the nearly closed-shell inner-shell electrons, which is negligible as compared to γ in the intermediate diradical character region. For all systems, the dσ contribution [γ(dσ)] is dominant, especially around the bond length giving γmax (referred to as Rmax), while dπ and dδ electrons make small contributions. As shown in Figure 3b, Cr(II)−Cr(II) displays a

Figure 2. Variation in the diradical character of dX orbitals [y(dX), where X = σ, π and/or δ] as a function of the bond length (R) for V(II)−V(II) (a), Cr(II)−Cr(II) (b), and Mn(II)−Mn(II) (c). The y(dX) values are calculated using the PUHF/SDD method.

bell-shape behavior of γ(dσ) with a maximum value [γmax(dσ)] of 1570 au at R = 2.8 Å, at which the dσ diradical character is intermediate [y(dσ) = 0.776]. This result indicates the 5504



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enhancement of γ(dσ) in the intermediate diradical character region of dσ orbitals. The dπ contribution of Cr(II)−Cr(II) is also enhanced in the intermediate dπ diradical character region [γmax(dπ) = 77.3 au and y(dπ) = 0.576 at R = 2.0 Å]. The origin of the γmax(dX) difference between dσ and dπ was discussed in detail on the basis of eqs 1 and 2 in our previous study.13 Similar features are observed for the γ(dX) of V(II)−V(II) and Mn(II)−Mn(II) (see Figure 3a,c). Table 1 summarizes the Table 1. Maximum UCCSD γ(dX) [γmax(dX)], PUHF Diradical Character [ymax(dX)] and Bond Length [Rmax(dX)] Corresponding to γmax(dX) for V(II)−V(II), Cr(II)−Cr(II), and Mn(II)−Mn(II) as Well as Their γmax and Rmax Obtained by the UCCSD and UCCSD(T) Methodsa V(II)−V(II)

Cr(II)−Cr(II)

Mn(II)−Mn(II)

γmax(dσ) [a.u.] ymax(dσ) [-] Rmax(dσ) [Å]

4600 0.744 3.0

1570 0.776 2.8

535 0.797 2.6

γmax(dπ) [a.u.] ymax(dπ) [-] Rmax(dπ) [Å]

80.6 0.434 2.0

77.3 0.576 2.0

58.9 0.531 1.8

4390 (4480) 3.1 (3.1)

1570 (1650) 2.8 (2.8)

568 (599) 2.6 (2.6)

γmax [a.u.] Rmax [Å] a

The values in parentheses correspond to UCCSD(T) results.

γmax(dX), the diradical characters [ymax(dX)], and the bond length [Rmax(dX)] corresponding to γmax(dX) along with γmax and Rmax. Note that, in Table 1, the results of γ(dδ) are not presented because the dδ electrons of Cr(II)−Cr(II) and Mn(II)−Mn(II) do not show maximum contributions within the R region studied here, while V(II)−V(II) has no dδ electron. The global picture points out an increase of γmax(dX) and Rmax(dX) for X = σ and π (see Table 1) in smaller group number as well as the domination of the γ(dσ) contribution, owing to the larger valence d atomic orbitals. 3.2. Period Effect on the Diradical Character and on γ. To reveal the impact of the period on the diradical character and γ, we examine set B of dimetallic systems, which consists of the elements of periods 4, 5, and 6 and of group 6 (Cr, Mo and W, respectively). The diradical characters of dX orbitals for Mo(II)−Mo(II) and W(II)−W(II) obtained by the PUHF method are plotted as a function of R in Figure 4 [see also Tables 4S and 5S in the Supporting Information, and see Figure 2b and Table 2S for Cr(II)−Cr(II)]. First, a rather similar behavior is observed in Mo(II)−Mo(II) and W(II)−W(II): (i) y(dX) increases with the increase in R for X = σ, π and δ, (ii) at a given R, y(dX) decreases in the order, δ > π > σ, except for the reverse ordering [y(dσ) > y(dπ)] at small R [R = 1.6 to 1.8 Å in Mo(II)−Mo(II) and R = 1.6 to 2.0 Å in W(II)−W(II)]. Then, Cr(II)−Cr(II) presents the largest y(dX) for X = σ, π, and δ, while W(II)−W(II) presents the smallest, which indicates a correlation between y(dX) and the period number of metal atoms. Again, the size of the d atomic orbitals is the key parameter to explain these variations: when going to higher period, the valence d atomic orbitals of transition metals expand (see UHF dz2 atomic orbitals of Cr2+, Mo2+, and W2+ in Figure 1b) because of the increase in the principal quantum number and of the relativistic effect, which explains the largest/smallest y(dX) in Cr(II)−Cr(II)/W(II)−W(II) due to the smallest/ largest overlap (for a fixed R).

Figure 3. Evolution of γ and of its dX electron contributions [γ(dX), where X = σ, π and/or δ] as a function of the bond length (R) for V(II)−V(II) (a), Cr(II)−Cr(II) (b), and Mn(II)−Mn(II) (c). The γ(dX) values are calculated by the UCCSD/SDD method, while the γ values are done using both the UCCSD/SDD and UCCSD(T)/SDD methods. Note that γ(dπ) for the three systems indicates one of the two equivalent dπ orbital contributions to γ, while γ(dδ) for Mn(II)− Mn(II) indicates one of the two equivalent dδ orbital contributions. 5505



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Figure 5. Evolution of γ and of its dX electron contributions [γ(dX), where X = σ, π and δ] as a function of the bond length (R) for Mo(II)−Mo(II) (a) and W(II)−W(II) (b). The γ(dX) values are calculated using the UCCSD/SDD(f) method, while the γ values are done using both the UCCSD/SDD(f) and UCCSD(T)/SDD methods. Note that γ(dπ) indicates one of the two equivalent dπ orbital contributions to γ.

Figure 4. Evolution of γ and of its dX electron contributions [y(dX), where X = σ, π and δ] as a function of the bond length (R) for Mo(II)−Mo(II) (a) and W(II)−W(II) (b). The y(dX) values are calculated by the PUHF/SDD(f) method.

The contributions of dX electrons to the γ for Mo(II)− Mo(II) and W(II)−W(II) calculated by the UCCSD method are plotted as a function of R in Figure 5 (see also Tables 4S and 5S in Supporting Information) together with the γ calculated by the UCCSD and UCCSD(T) methods. The dX contributions of the systems of set B show similar tendencies to those of set A: (i) the dσ contributions are dominant, and (ii) the γ(dX) values are enhanced in the intermediate y(dX) regions. Table 2 lists the γmax(dX), ymax(dX), and Rmax(dX) values as well as the γmax and Rmax values for the systems of set B. The γmax(dX) and Rmax(dX) values for X = σ, π, and δ increase as a function of the period number of the metal atoms. Since γ(dσ) is the dominant contribution to γ at Rmax, the γmax value shows the same period dependence as γmax(dσ): W(II)− W(II) (17600 au) > Mo(II)−Mo(II) (8000 au) > Cr(II)− Cr(II) (1650 au), which is also in agreement with our prediction and confirms the guiding principle that the larger the size of the valence d atomic orbital, the larger the γmax values in open-shell singlet transition-metal dimers. γ of W(II)−W(II) presents a specific behavior around R = 3.2 Å, which originates from the dσ contribution as shown in

Figure 4b. In fact, in symmetric diradical systems, γ could be negative in the small diradical character region due to the negative type II term (γII) in the SOS γ expression,6a the amplitude of which is enhanced by the strong decrease in the first excitation energy in the small diradical region (see Figure 2 in ref 6a). Indeed, for R ≤ 3.2 Å, y(dσ) ≤ 0.2. To a lower extent, this is also visible for V(II)−V(II) (around R = 1.8 Å) and for Mo(II)−Mo(II) (around R = 2.2 Å). 3.3. Charge Effect on the Diradical Character and on γ. Figure 6 shows the variation in y(dX) with respect to the bond length for Cr(I)−Cr(I) and Cr(III)−Cr(III) obtained at the PUHF/SDD level of approximation [see also Tables 6S and 7S in the Supporting Information, and see Figure 2b and Table 2S for Cr(II)−Cr(II)]. First, the same tendencies are observed: the y(dX) values increase with the increase in R, and, for a given R, y(dX) decreases in the order, dδ > dπ > dσ, although y(dσ) of Cr(I)−Cr(I) is larger than y(dπ) between 1.6 to 2.0 Å. Within set C, for a given R, y(dX) increases with the charge, which can also be interpreted by the degree of expansion of the d atomic orbitals as in the case of the group and period 5506



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Table 2. Maximum UCCSD γ(dX) [γmax(dX)], PUHF Diradical Character [ymax(dX)], and Bond Length [Rmax(dX)] Corresponding to γmax(dX) for Cr(II)−Cr(II), Mo(II)−Mo(II), and Mn(II)−Mn(II) as Well as Their γmax and Rmax Obtained by the UCCSD and UCCSD(T) Methodsa Cr(II)−Cr(II)

Mo(II)−Mo(II)

W(II)−W(II)


1570 0.776 2.8

7800 0.662 3.4

17400 0.776 4.1


77.3 0.576 2.0

246 0.362 2.4

435 0.431 2.6

γmax(dδ) [a.u.] ymax(dδ) [-] Rmax(dδ) [Å]

b

159 0.659 2.0

306 0.528 2.0

γmax [a.u.] Rmax [Å]

1570 (1650) 2.8 (2.8)

7630 (8000) 3.4 (3.4)

16500 (17600) 4.1 (4.1)

b b

a The values in parentheses correspond to UCCSD(T) results. bThe maximum dδ contribution is not observed for Cr(II)−Cr(II) within the bond length region studied here.

dependences. Indeed, Figure 1c evidences the contraction of the valence dz2 atomic orbital in the Cr+, Cr2+ and Cr3+ series, due to the reduction of the repulsion between the valence electrons caused by their decreasing number. The smallest d atomic orbitals of Cr(III)−Cr(III) lead to weak d−d orbital interactions, while the d−d orbital interactions in Cr(I)−Cr(I) are the strongest within set C. The bond length dependence of γ(dX) for Cr(III)−Cr(III) estimated from UCCSD/SDD calculations is displayed in Figure 7b along with γ at the UCCSD/SDD and UCCSD(T)/ SDD levels of approximation (see also Table 7S in Supporting Information). We did not calculate the UCCSD γ(dX) for Cr(I)−Cr(I) because the UCCSD method provides quite different γ behavior from the UCCSD(T) method (see Figure 7a as well as Table 6S in the Supporting Information) [this also shows that Cr(I)−Cr(I) is a nice test case to assess highly correlated methods]. In a previous study,30 the long-range corrected UBLYP (LC-UBLYP) method with range separating parameter (μ) ranging from 0.7 to 0.9 was shown to provide a reliable bond length dependence of γ around Rmax in metal− metal bonded systems. Instead of CCSD, we thus calculated the γ values of Cr(I)−Cr(I) using the LC-UBLYP functional with μ = 0.75, which is optimized to reproduce the UCCSD(T) Rmax of Cr(I)−Cr(I). Although the LC-UBLYP γ values undershoot the UCCSD(T) ones around Rmax (Figure 7a), it succeeds in reproducing the bell-shape behavior of UCCSD(T) γ with the same Rmax (γmax of 32700 au along with Rmax of 3.2 Å). Therefore, we calculated the dX contributions of Cr(I)−Cr(I) by using the LC-UBLYP method with μ = 0.75 (Figure 7a) and found that the dσ electrons make the dominant contribution to γ in Cr(I)−Cr(I) and the dσ contribution is enhanced to reach a maximum [γmax(dσ) = 37000 au] at R = 3.2 Å, where the diradical character of the dσ orbitals is intermediate [y(dσ) = 0.790]. Although the γ(dπ) of Cr(I)−Cr(I) obtained by the LC-UBLYP method is also enhanced at the intermediate diradical region of dπ orbitals, quantitative discussion on γmax(dπ) should be avoided because μ = 0.75, which was

Figure 6. Evolution of γ and of its dX electron contributions [y(dX), where X = σ, π and/or δ] as a function of the bond length (R) for Cr(I)−Cr(I) (a) and Cr(III)−Cr(III) (b). The y(dX) values are calculated using the PUHF/SDD method.

optimized to reproduce the UCCSD(T) Rmax, does not necessarily provide reliable γ(dX) values in the bond length region far from Rmax, even though the γmax(dπ) and γmax(dδ) values are much smaller than γmax(dσ). On the other hand, both γ(dσ) and γ(dπ) of Cr(III)−Cr(III) calculated by the UCCSD/ SDD method present the typical bell-shape behavior with maximum values of 304 au and 59.0 au at R = 2.2 and 1.8 Å, respectively (see Figure 7b). At these bond lengths, the diradical characters of dσ and dπ orbitals are intermediate, which substantiates the enhancements of γ(dσ) and γ(dπ) in the intermediate diradical character region. Table 3 summarizes the γmax(dX), ymax(dX) and Rmax(dX) values along with the γmax and Rmax values for the systems of set C. As expected, the γmax value reduces as a function of the atomic charge: Cr(I)−Cr(I) (66100 au) > Cr(II)−Cr(II) (1650 au) > Cr(III)−Cr(III) (439 au), confirming our prediction of the relation between the size of the valence d atomic orbital and the γmax value in open-shell singlet metal− metal bonded systems. Surprisingly, Cr(I)−Cr(I) exhibits 150 times enhanced γmax as compared to Cr(III)−Cr(III). Although 5507



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the atomic charge dependence of γmax is explained on the basis of the size of valence d atomic orbitals, the remarkable enhancement of γmax in Cr(I)−Cr(I) as compared with the γmax of Cr(II)−Cr(II) suggests the existence of additional effects of the charge on γ.

4. CONCLUSIONS On the basis of the analytical formula of the second hyperpolarizabilities (γ) for a symmetric two-site diradical model (A•−B•) system (eqs 1 and 2), we have predicted the relation between the size of the valence d atomic orbital and the γmax value, that is, the larger the d atomic orbital, the larger the γmax value. To verify this prediction, the dependences of γ with respect to the group, the period, and the atomic charge of transition-metal atoms have been theoretically investigated in open-shell singlet dimetallic systems with metal−metal multiple bond because the size of the valence d atomic orbital depends on the group, on the period, and on the atomic charge. It is found that, for all studied systems, γ is enhanced and takes a maximum value (γmax) in the intermediate diradical character region of the dσ orbitals. The γmax value of a dimetallic system turns out to depend on the group and on the period: small group number and/or large period number leads to large γmax value. The atomic charge has also a significant effect on γmax, i.e., the smaller the positive charge, the larger the γmax value is. These results are in agreement with our prediction, although the significant enhancement of γmax in Cr(I)−Cr(I) suggests the existence of additional charging effects on γ. From the present results, metal−metal bonded diniobium, ditantalum, and ditungsten complexes with small charge can be proposed as compounds exhibiting significantly large thirdorder optical nonlinearly provided their metal−metal bond lengths can be tuned such as to lead to an intermediate diradical character of the dσ orbitals. The charge of the metal atoms could be controlled not only by the oxidation state but also by donor atoms of both axial and equatorial ligands. However, optimal bond lengths for enhancing γ in these dimetallic complexes would be fairly large as compared with bond lengths of real complexes synthesized so far, although a singlet dichromium(II) complex with a bond length (2.69 Å) close to the optimal one (2.8 Å) was reported.31 The present theoretical guiding principle for achieving γmax in these dimetallic complexes can stimulate the design of new ligands favoring elongated metal−metal bonds, which would also be a challenging study from the experimental side.

Figure 7. Evolution of γ and of its dX electron contributions [γ(dX), where X = σ, π and/or δ] as a function of the bond length (R) for Cr(I)−Cr(I) (a) and Cr(III)−Cr(III) (b). The γ(dX) values are calculated using the LC-UBLYP method with μ = 0.75 for Cr(I)− Cr(I) or using the UCCSD/SDD method for Cr(III)−Cr(III). Note that γ(dX) (X = π and δ) of Cr(I)−Cr(I) indicates one of the two equivalent dX orbital contributions to γ.

Table 3. Maximum UCCSD γ(dX) [γmax(dX)], PUHF Diradical Character [ymax(dX)] and Bond Length [Rmax(dX)] Corresponding to γmax(dX) for Cr(I)−Cr(I), Cr(II)−Cr(II), and Cr(III)−Cr(III) as Well as Their γmax and Rmax Obtained by UCCSD and UCCSD(T) Methodsa Cr(I)−Cr(I)

Cr(II)−Cr(II)

Cr(III)−Cr(III)


37000 0.790 3.2b

1570 0.776 2.8

304 0.602 2.2


100b 0.307 1.8b

77.3 0.576 2.0

59.0 0.505 1.8

32700b (66100) 3.2b (3.2)

1570 (1650) 2.8 (2.8)

364 (439) 2.2 (2.1)

γmax [a.u.] Rmax [Å]

b

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ASSOCIATED CONTENT

S Supporting Information *

The diradical characters y(dX), second hyperpolarizabilities γ, and dX contributions to γ at each bond length for all systems. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Fax: +81-6-6850-6268. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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a

The values in parentheses correspond to UCCSD(T) results. bThese values are obtained by the LC-UBLYP method with μ = 0.75.

ACKNOWLEDGMENTS This work is supported by a Grant-in-Aid for Scientific Research (No. 21350011) and the “Japan-Belgium Cooperative 5508



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Program” (J091102006) from the Japan Society for the Promotion of Science (JSPS), and the global COE (center of excellence) program “Global Education and Research Center for Bio-Environmental Chemistry” of Osaka University. H.F. expresses his special thanks for a JSPS Research Fellowship for Young Scientists. This work has also been supported by the Academy Louvain (ARC “Extended π-Conjugated Molecular Tinkertoys for Optoelectronics, and Spintronics”) and by the Belgian Government (IUAP program No. P06-27 “Functional Supramolecular Systems”).

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Enhancement of the Third-Order Nonlinear Optical Properties in Open

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