Enhancement of the second hyperpolarizability by dσ electrons in one-dimensional tetrametallic transition-metal systems

Chemical Physics Letters 527 (2012) 11–15

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Enhancement of the second hyperpolarizability by dr electrons in one-dimensional tetrametallic transition-metal systems Hitoshi Fukui a,⇑, Masayoshi Nakano a,⇑, Benoît Champagne b a b

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

a r t i c l e

i n f o

Article history: Received 14 December 2011 In final form 10 January 2012 Available online 18 January 2012

a b s t r a c t The second hyperpolarizabilities (c) of the one-dimensional Cr(II) and Mo(II) tetrametallic systems are investigated by focusing on their dependences on the open-shell singlet characters and by comparing with those of their dimetallic analogs. Significant enhancements of c are observed in tetrametallic systems, which are attributed to the dr electrons with an intermediate open-shell character. The present results enable us to extend the concept of ‘r-dominant third-order NLO’ obtained in dimetallic transition-metal systems to general one-dimensional polymetallic systems. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Open-shell singlet molecules were recently theoretically proposed as a novel class of nonlinear optical (NLO) systems. In particular, the second hyperpolarizability c (the third-order NLO property at the molecular scale) of singlet diradical systems with intermediate diradical characters is enhanced with respect to those of pure diradical and closed-shell systems of similar size [1–7]. The mechanism of this structure–property relationship has been unraveled by analyzing the summation-over-states (SOS) expressions of the static c and of the two-photon absorption cross section within the valence configuration interaction (VCI) scheme [4,5]. This relationship has also been verified by ab initio molecular orbital and density functional theory studies on several model and real molecular systems [2,3,6,7]. Afterwards, these theoretical predictions have been experimentally confirmed by two-photon absorption [8] and third-harmonic generation [9] measurements, while they have stimulated other experimental and theoretical investigations on the third-order NLO properties of singlet diradical compounds [10–14]. On the other hand, transition-metal dinuclear complexes with metal–metal bonds have attracted great attention for decades because of their multiple dr, dp, and dd bonds resulting from the dd orbital interactions [15–19]. These systems also present an interesting multiple diradical character emerging from the metal–metal bond, the degree of which has been theoretically predicted to be controllable through the metal–metal bond length [20,21]. Therefore, transition metal complexes with multiple metal–metal bonds are expected to be promising open-shell singlet ⇑ Corresponding authors. Fax: +81 6 6850 6268. E-mail addresses: [email protected] (H. Fukui), [email protected]. osaka-u.ac.jp (M. Nakano). 0009-2614/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2012.01.020

third-order NLO systems. Although the NLO properties of metal– metal bonded compounds have already been studied by several groups [22–26], few have addressed the relationship between the NLO properties and the open-shell characters. In a previous study [27], an investigation of the bond length dependence of c in open-shell singlet dichromium(II) and dimolybdenum(II) systems has evidenced their potential as a new class of ‘r-dominant third-order NLO’ compounds. In singlet diradical systems, the key factors for enhancing c are the intermediate diradical character and the distance between the unpaired or diradical electrons [4]. The former represents the sensitivity to the applied electric field, while the latter corresponds to the distance of the field-induced charge transfer. These two factors, however, are generally antagonistic because a long distance between diradical electrons leads to a large diradical character whereas an intermediate diradical character often imposes a rather small distance between the unpaired electrons. One way to attenuate this trade-off relationship consists in building one-dimensional (1D) systems with tuned multiple diradical characters (1D multiradicalization). Indeed, if the radical sites are aligned with an appropriate distance as shown in Scheme 1, the distance between the ends can be increased while keeping intermediate interactions between the radicals, and that should give rise to a nonlinear enhancement of c with the size. This exaltation of c was theoretically evidenced in a model hydrogen chain (Hn) [28]. Moreover, transition-metal atoms are known to form chains, often called extended metal-atom chains (EMACs), displaying specific electronic and magnetic properties [15,29–34]. The emergence of open-shell singlet character has been predicted in 1D singlet tetrametallic systems composed of Cr(II) and Mo(II) on the basis of spinunrestricted (U) Hartree–Fock (UHF) and density functional theory calculations [20,21]. Therefore, chains of transition-metal atoms are expected to exhibit large c. To assess this speculation, we here

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H. Fukui et al. / Chemical Physics Letters 527 (2012) 11–15

the odd electron number is obtained according to Head-Gordon [36]: : Radical electron : Radical site

nodd  minð2  nk ; nk Þ; k

Scheme 1. Schematic drawing of one-dimensional open-shell singlet systems with an intermediate open-shell (multiradical) character. The arrow of radical electron indicates the spin direction.

investigate the c of 1D singlet tetrametallic systems composed of Cr(II) and Mo(II) in relation with their open-shell characters. 2. Model systems and calculation methods The model systems consist of 1D singlet tetrachromiun(II) [Cr(II)4] and tetramolybdenum(II) [Mo(II)4] with identical bond lengths. Since for a given bond length these two systems are expected to show different open-shell characters [20,21], their comparison should provide useful information on the relationship between c and the open-shell characters. The orbital interactions between the valence d atomic orbitals lead to four dr, dp, dp0 , dd and dd0 orbitals (totally 20 orbitals). The dp0 orbitals are rotated by 90° with respect to the corresponding dp orbitals around the bond axis (defined as the z-axis), while the dd and dd’ orbitals sets are rotated by 45° with respect to each other. The 16 d-electrons formally occupy the lowest two orbitals of each dr, dp, dp0 and dd symmetry, though in the UHF formalism the two highest unoccupied natural orbitals (NOs) of each symmetry possess fractional occupation numbers smaller than 1. The UHF spin density distributions of Cr(II)4 and Mo(II)4 with R = 3.0 Å are shown in Figure 1 for each type of occupied d MOs. Note that the spin polarization, which results from the brokensymmetry solution, is an artifact but that it helps to analyze the spatial spin correlation. For both systems, the a and b spin densities display an alternating pattern consistent with Scheme 1, demonstrating that these systems meet our strategy of 1D multiradicalization. Since the dp and dp0 orbitals present the same open-shell character and the same contributions to c, we will limit our discussion to the dp orbitals. The open-shell character of the dX orbitals [yodd(dX), where X = r, p, and d] is defined as the total odd electron number [ N odd ðdXÞ ] of the four dX NOs divided by the maximum odd electron number (N odd max ) (=4, in the present case) [35]:

yodd ðdXÞ ¼

P4

odd k¼1 nk ðdXÞ odd Nmax

¼

Nodd ðdXÞ Nodd max

;

ð1Þ

ð2Þ

where nk is the occupation number of kth NO in a set of the lowest two occupied and the highest two unoccupied NOs with dX symmetry. yodd(dX) ranges from 0 to 1, which represent the closed-shell and pure open-shell states, respectively. Thus, yodd(dX) can also be regarded as the radical character of the dX orbitals. In the present work, the occupation numbers are calculated at the spin-unrestricted coupled cluster singles and doubles (UCCSD) level. The longitudinal czzzz (c) values were calculated by the finitefield (FF) approach [37], which consists in a fourth-order differentiation of the energy with respect to the applied electric field. Note that we employed the perturbation series expansion convention (called B convention [38]) for defining c. Electric field amplitudes ranging from 0.0010 to 0.0040 a.u. together with a tight convergence threshold of 1010 a.u. on the energy were used to obtain precise c values. The c values were calculated by the UCCSD and that with perturbative triples [UCCSD(T)] methods. In all calculations, the effective core potential (ECP) of the Stuttgart group was employed together with the corresponding valence basis sets (SDD) for Cr [39]. For Mo, the SDD basis set [40] was supplemented with an additional set of f polarization functions [41] (referred to as ‘SDD(f)’ in this Letter). This is required to get a balanced basis sets because, contrary to Cr, the SDD basis set for Mo does not include f polarization functions. These ECP has been shown to closely reproduce the DiracHartree–Fock results of c using an all-electron basis set [42]. As previously recommended for closed-shell [43] and open-shell singlet systems [44,45], extended basis sets (with diffuse functions) are indispensable for obtaining quantitative c values. The SDD basis set already includes one set of diffuse s, two sets of diffuse p and one set of diffuse d functions, which should be sufficient to describe the nonlinear electric field effects. Indeed, further inclusion of d and f diffuse functions to the SDD basis set only provides slight change in c of Cr(II)Cr(II) at the UCCSD and UCCSD(T) levels of approximation (see Supporting information of Ref. [27]). For a detailed analysis of c, the contributions of the dr, dp, and dd electrons [c(dX), where X = r, p and d] were calculated at the UCCSD level along with the c(dX) density by adopting a partitioning scheme of the second hyperpolarizability [35]. The positive and negative values of the c(dX) density multiplied by the cube of the applied electric field (F 3z ) represent, respectively, the field-induced increase and decrease in the dX electron density in proportion to F 3z , which cause the third-order dipole moment in the direction from positive to negative c(dX) densities. All calculations were performed with the GAUSSIAN 09 program package [46]. 3. Results and discussion 3.1. Open-shell character

Figure 1. Spin density distributions of dr, (dp + dp0 ) and dd electrons for Cr(II)4 (a) and Mo(II)4 (b) with the bond length (R) of 3.0 Å obtained from UHF molecular orbitals. The red and blue surfaces represent a and b spin densities with contour values of ±0.05 a.u., respectively. (For interpretation of reference to colors in this figure legend, the reader is referred to the web version of this article.)

Figure 2 shows the bond length (R) dependence of yodd(dX) for Cr(II)4 and Mo(II)4 obtained by the UCCSD method. For both systems yodd(dX) with X = r, p and d increases with R, as a result of decreasing dd orbital overlaps. For any given R, the yodd(dd) > yodd(dp) > yodd(dr) order is observed in both Cr(II)4 and Mo(II)4, though yodd(dr) is slightly larger than yodd(dp) at R = 1.6 Å for Cr(II)4 and at R = 2.0 Å for Mo(II)4. This tendency originates from the dd overlaps, which decrease from dxy (forming the dd MOs) to dxz and dyz (dp MOs), and to dz2 at the origin of the dr MOs. In a similar way, Cr(II)4 presents larger yodd(dX) than Mo(II)4 at any R [for example, at R = 2.5 Å, Cr(II)4 and Mo(II)4 show

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H. Fukui et al. / Chemical Physics Letters 527 (2012) 11–15

0.8

0.8

y odd(dX) [-]

(b) 1.0

y odd(dX) [-]

(a) 1.0

0.6

0.4

yodd(d ) yodd(d ) yodd(d )

0.2

0.0 1.5

2.0

2.5

3.0

0.6

0.4

yodd(d ) yodd(d ) yodd(d )

0.2

0.0 2.0

3.5

2.5

R [Å]

3.0

3.5

4.0

R [Å]

Figure 2. Variations in the open-shell characters of dX orbitals [yodd(dX)] for X = r, p and d as a function of the bond length (R) in Cr(II)4 (a) and Mo(II)4 (b) obtained using the UCCSD method.

50

300 250

40

200

(d ):UCCSD (d ):UCCSD (d ):UCCSD :UCCSD :UCCSD(T)

3

3

(dX) [x 10 a.u.]

60

(b)

(d ):UCCSD (d ):UCCSD (d ):UCCSD :UCCSD :UCCSD(T)

(dX) [x 10 a.u.]

(a) 70

30 20 10

150 100 50

0 0 1.5

2.0

2.5

3.0

3.5

R [Å]

(c)

3

2.5

3.0

3.5

4.0

R [Å]

30 25

(dX) [x 10 a.u.]

2.0

20

(d ):UCCSD (d ):UCCSD (d ):UCCSD :UCCSD :UCCSD(T)

15 10 5 0 -5 2.0

2.2

2.4

2.6

2.8

3.0

R [Å] Figure 3. Bond length (R) dependence of c calculated at the UCCSD and UCCSD(T) levels of approximation as well as those of its dX contributions [c(dX)] for X = r, p and d calculated using the UCCSD method in Cr(II)4 (a) and in Mo(II)4 (b). The R dependences of c and c(dX) for Mo(II)4 in the region from 2.0 to 3.0 Å are highlighted in (c).

(yodd(dr), yodd(dp), yodd(dd)) of (0.618, 0.874, 0.983) and (0.248, 0.518, 0.914), respectively], which is caused by the smaller size of valence d atomic orbitals in Cr(II) than in Mo(II), and therefore by weaker dd orbital interactions in Cr(II)4. These open-shell characteristics are consistent with those observed in the related dimetallic systems, Cr(II)2 and Mo(II)2 [27]. 3.2. Bond length dependences of c and c(dX) The R dependence of c is shown in Figure 3, where the R ranging from 2.0 to 3.0 Å is focused in Figure 3c to highlight the reductions in c and c(dr) of Mo(II)4 around 2.5 Å. The UCCSD(T) c value of

Cr(II)4 decreases from R = 1.6 to 2.0 Å, then, significantly increases, attains a maximum (cmax = 68 000 a.u.) at R = Rmax = 2.7 Å, and then decreases again (Figure 3a). The UCCSD method provides similar results with slightly smaller Rmax (2.6 Å). The UCCSD(T) c variations are similar for Mo(II)4 (Figure 3b and c), though the minimum and maximum c values move to longer R region as compared with Cr(II)4. The UCCSD(T) cmax of Mo(II)4 (257 000 a.u.) is about 4 times as large as that of Cr(II)4 and the UCCSD method is also appropriate for describing the variations in c in the whole R region, allowing the UCCSD c(dX)-based analysis. Table 1 compares the cmax per M(II)2 unit, cunit max , for the dimetallic and tetrametallic systems as well as their Rmax values. Though Rmax varies little with the system

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H. Fukui et al. / Chemical Physics Letters 527 (2012) 11–15

Table 1 Second hyperpolarizabilities (cunit max ) per unit [M(II)M(II)] (where M = Cr, Mo) and the bond length (Rmax) giving cunit max unit for Cr(II)2, Cr(II)4, Mo(II)2 and Mo(II)4.

a cunit max [a.u.]

Rmaxa [Å]

Cr(II)2

Cr(II)4

Mo(II)2

1650

34 000

8000

Mo(II)4 129 000

2.8

2.7

3.4

3.4

a Obtained using the UCCSD(T) method with the SDD basis set for Cr, and the SDD(f) basis set for Mo.

size, cunit max increases strongly. So, the concept of multiradicalization in open-shell singlet metal–metal bonded systems leads to exaltation of cunit max by factors of 20 and 16 for Cr(II)4 and Mo(II)4, respectively. Figure 3 also describes the evolution with R of the dX contributions to c, as obtained with the UCCSD method. The major features are (i) c(dr) gives the dominant contribution to c in the bond length region where c is enhanced, while the dp and dd electrons provide negligible contributions, (ii) the bell-shape behavior of c(dr) corresponds to the intermediate open-shell character region of the dr orbitals (see Figures 2 and 3) and is characterized by a cmax(dr) of 63 100 a.u. at R = 2.6 Å for Cr(II)4 and a cmax(dr) of 231 000 a.u. at R = 3.3 Å for Mo(II)4, and (iii) qualitatively, these characteristics are similar to those observed in Cr(II)2 and Mo(II)2. So, both openshell singlet dimetallic and 1D tetrametallic systems are ‘r-dominant’ third-order NLO systems. Since the enhancement of c is caused by the dr electrons with an intermediate open-shell character, the difference in cmax between Cr(II)4 and Mo(II)4 results from that in cmax(dr). As mentioned above, in 1D multiradical systems, the large c is achieved by the intermediate open-shell character, which makes radical electrons intermediately interact with each other and thus more sensitive to the applied electric field, as well as by the larger distance between the radical electrons on the both-end atoms. Although the c(dr) values of both Cr(II)4 and Mo(II)4 are maximized in the intermediate yodd(dr) region, the corresponding bond length regions are different. Indeed, yodd(dr) of 0.5 lies between R = 2.2 and 2.3 Å for Cr(II)4, and between R = 3.0 and 3.1 Å for Mo(II)4, which makes a significant difference in cmax(dr). Since Mo(II) possesses a bigger valence dz2 atomic orbital than Cr(II), Mo(II)4 needs longer bond length than Cr(II)4 to achieve an intermediate yodd(dr). This is the fundamental reason for the larger cmax of Mo(II)4 with respect to Cr(II)4. The difference in the c values in the intermediate yodd(dr) region between Cr(II)4 and Mo(II)4 can be understood from the general form of the SOS expression of c for symmetric singlet diradical systems [4],

c ¼ f ðy; rk ÞR4BA =U 3 ;

ð3Þ

where

f ðy; r k Þ ¼ 

þ

8ð1  yÞ4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f1 þ 1  ð1  yÞ2 g2 f1  2rk þ 1= 1  ð1  yÞ2 g3 4ð1  yÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : f1  2r k þ 1= 1  ð1  yÞ2 g2 f1= 1  ð1  yÞ2 g ð4Þ

Here, y, RBA, U and rk are the diradical character, effective diradical distance, effective Coulomb repulsion and twice the direct exchange integral divided by U, respectively. As indicated by this equation, c evolves as the fourth power of the diradical distance (nearly equal to the bond length R) and the corresponding (3.05/ 2.25)4  3 factor is close to the cmax ratio between Mo(II)4 and Cr(II)4. On the other hand, c(dp) and c(dd) of both compounds are negative for those bond lengths where c(dr) is enhanced. This is probably related to the strong third-order polarization of the dr electrons, which induces the third-order polarizations of dp and dd electrons in the reverse direction due to the drdp and drdd electron repulsions. Similar antagonistic effects, e.g., positive pelectron vs. negative r electron contributions to c, are also observed in organic p-conjugated molecules [3,35,47]. As shown in Figure 3c, the reduction of c observed for Mo(II)4 before its enhancement is caused by the dr contribution. Note that for Cr(II)4 this reduction cannot be explained by the UCCSD c(dr) since the UCCSD method does not reproduce the reduction of the UCCSD(T) c. In symmetric singlet diradical systems, c could be negative in the small diradical character region due to the negative type II term of the SOS c expression [48], which is enhanced when the first excitation energy decreases (see Figure 2 in Ref. [4]). A similar mechanism is predicted for the singlet multiradical Cr(II)4 and Mo(II)4 systems. 3.3. c(dr) density analysis The UCCSD c(dr) density distributions are plotted in Figure 4a and 4b for the situations corresponding to cmax(dr), i.e. bond lengths of 2.6 Å and 3.3 Å for Cr(II)4 and Mo(II)4, respectively. Large positive and negative c(dr) densities are observed at the extremities, indicating that the dr radical electrons on both-end atoms provide a large positive contribution to c(dr). In contrast, the middle region gives negative contributions to c, but of smaller amplitudes. Moreover, the c(dr) densities are larger in Mo(II)4 than in Cr(II)4, which, as explained above, originates from the larger Rmax in Mo(II)4 while keeping similar intermediate yodd(dr). 4. Summary We have investigated the bond length dependence of c in the 1D tetrametallic Cr(II) and Mo(II) systems with open-shell singlet characters using the UCCSD and UCCSD(T) methods. Both Cr(II)4 and Mo(II)4 show significant enhancement of c, which is caused by the dr electron contributions with an intermediate open-shell character. This result demonstrates that open-shell singlet 1D transition-metal tetrametallic systems belong to the ‘r-dominant’ third-order NLO systems like their open-shell singlet dimetallic analogs. Then, Mo(II)4 exhibits larger maximum c value than Cr(II)4, which originates from the difference in the size of the valence dz2 atomic orbitals. Indeed, for a fixed dr open-shell character, the larger the dz2 atomic orbitals, the longer the distance between the radical electrons on the extremities. The c(dr) densities further reveal that such c enhancement is dominated by the dr electrons mainly located on the both-end atoms whereas those on the middle atoms give negative but smaller contributions to c.

Figure 4. c(dr) density distribution for Cr(II)4 with R = 2.6 Å (a) and for Mo(II)4 with R = 3.3 Å (b) obtained at the UCCSD level. The yellow and blue regions represent positive and negative densities with contour values of ±1000 a.u. The UCCSD c(dr) values are also shown. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

H. Fukui et al. / Chemical Physics Letters 527 (2012) 11–15

The present results enable to extend the new concept of ‘r-dominant third-order NLO’ obtained in open-shell singlet transition-metal dimetallic systems to their tetrametallic analogs, which realizes further enhancements of c as well as a significant size dependence. Further investigations on longer metallic chains are in progress in our laboratories while such extended metal atom chains have already been synthesized within complex systems [15,29–34]. Acknowledgements This work is supported by Grant-in-Aid for Scientific Research (No. 21350011) from 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 JSPS Research Fellowship for Young Scientists. This work has also been supported by the Academy Louvain (ARC ‘Extended p-Conjugated Molecular Tinkertoys for Optoelectronics, and Spintronics’) and by the Belgian Government (IUAP program No. P06-27 ‘Functional Supramolecular Systems’). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

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