Dynamic Behaviors of Interactions: Application of Normal Coordinates

Jun 14, 2010 - Dynamic Behaviors of Interactions: Application of Normal Coordinates of Internal Vibrations to AIM Dual Functional Analysis. Waro Nakan...
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J. Phys. Chem. A 2010, 114, 7423–7430

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Dynamic Behaviors of Interactions: Application of Normal Coordinates of Internal Vibrations to AIM Dual Functional Analysis Waro Nakanishi* and Satoko Hayashi Department of Material Science and Chemistry, Faculty of Systems Engineering, Wakayama UniVersity, 930 Sakaedani, Wakayama, 640-8510 Japan ReceiVed: May 11, 2010

A method to evaluate the dynamic nature of interactions is proposed based on the AIM dual functional analysis. Normal coordinates of internal vibrations (NIV) are employed to generate the perturbed structures necessary for the analysis. Hb(rc) are plotted versus Hb(rc) - Vb(rc)/2 [) (p2/8m)32Fb(rc)] at bond critical points for the purpose. The plots are represented by the polar (R, θ) coordinate. Each plot for an interaction shows a specific curve, which is expressed by (θp, κp): θp corresponds to the tangent line for the plot from the y-direction, and κp is the curvature. Although (R, θ) values correspond to the static nature of interactions, (θp, κp) values show the dynamic nature. The applicability of NIV is examined exemplified by the charge-transfer interactions as the first step to analyze the dynamic behaviors of interactions with NIV. The (θp, κp) values evaluated with NIV are very close to those obtained by the partial-optimization method (POM), where the distances or angles in question are fixed suitably, if the internal vibrations are substantially located on the interactions in question. The magnitudes of differences in θp and κp between those evaluated with NIV and POM are e2° and e2 au-1, respectively, for usual interactions. The treatment is demonstrated to be applicable to a wide range of interactions. Introduction The nature of chemical bonds and interactions will be elucidated by the functions of AIM (atoms-in-molecules) at bond critical points (BCPs: rc, *).1-9 We have recently proposed the AIM dual functional analysis that can classify and evaluate weak to strong interactions.10,11 Hb(rc) are plotted versus Hb(rc) Vb(rc)/2 () (p2/8m)32Fb(rc)) at BCPs in the treatment, where Hb(rc), Vb(rc), Fb(rc), and 32Fb(rc) are total electron energy densities, potential energy densities, electron densities, and Laplacian Fb(rc) at BCPs, respectively. Data for full-optimized structures and those near them (perturbed structures) are employed for the analysis (see Figure 1).10,11 The plots turn to the right drawing helical curves, as a whole. Hb(rc) must be a more appropriate index for weak interactions on the energy basis, which are the sum of Vb(rc) and Gb(rc) (kinetic energy densities) at BCPs.1,2,12,13 Eqs 1 and 2 show the relations between the functions.

Hb(rc) ) Gb(rc) + Vb(rc)

(1)

(p2 /8m)∇2Fb(rc) ) Hb(rc) - Vb(rc)/2 ) Gb(rc) + Vb(rc)/2 (2) One might imagine that the dynamic nature of interactions plays only an additional role compared with the static nature. However, the former enables us to classify the interactions that cannot be accomplished based on the latter only. For instance, it seems difficult to distinguish hydrogen bonds from van der Waals interactions, since such interactions would extend over the two areas successively. However, they can be discussed * To whom correspondence should be addressed. Telephone: +81 73 457 8252. Fax: +81 73 457 8253, E-mail: [email protected].

Figure 1. Polar (R, θ) coordinate representation of Hb(rc) vs Hb(rc) Vb(rc)/2.

separately based on the dynamic nature (θp < 90° for vdW whereas θp > 90° for HB, see text for θp).10,11 Similarly, the charge-transfer (CT) interactions in the molecular complexes (MC) can be distinguished from the adducts of trigonal bipyramidal structures (TBP) (θp < 180° for CT-MC, whereas θp > 180° for CT-TBP, see also Table 2).10,11 The dynamic nature of interactions must be closely related to the basic properties of interactions; therefore, it is indispensable for the better understanding of molecules and adducts. Perturbed structures are necessary to elucidate the dynamic behaviors. They can be obtained by the partial-optimization method (POM), where the distances (r) or angles (θs) in question are fixed suitably for the perturbed structures (c.f.: eqs 4 and 5).10,11 POM is useful if the structures are well obtained. However, POM may contain some disadvantages, since an interaction will be chosen arbitrarily and the perturbed structure is determined by the partial optimization. A bond in polycyclic compounds such as cubanes and fullerenes, an interaction in multi-interaction systems such as a hydrogen bond in enzymes, peptides, or DNA, and an interaction in the extended hypervalent bonds are the examples that would involve some difficulties in POM. Dynamic behaviors of interactions in the reaction

10.1021/jp104278j  2010 American Chemical Society Published on Web 06/14/2010

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coordinates of transition states (TS) must also be difficult to apply POM to, since perturbed structures around TS with POM would be different from those on intrinsic reaction coordinates (IRC). How can the perturbed structures in question be visualized? A motion around an interaction must include those of others in a molecule. The internal motions must essentially obey Hooke’s law, which are described by harmonic oscillators.14 The internal motions of a molecule become more intensive with a rise in temperature, although they never freeze even at zero Kelvin in the ground state of the molecule, since they have zero-point energies.14 Such internal motions perturb bond lengths and angles in molecules, relative to those of the full-optimized structures. Namely, internal vibrations must be closely related to dynamic behaviors of interactions in molecules. This idea led us to an expectation that the perturbed structures can be constructed using the normal coordinates of internal vibrations (NIV). NIV must contain the high possibility to clear up the above-mentioned difficulties in POM. The idea will also be applicable to weak interactions15 such as those in vdW,16 H-bonded,17 and CT adducts,18 if the internal vibrations are suitably located on the interactions in question. The analysis by the plots of Hb(rc) versus Hb(rc) - Vb(rc)/2 () (p2/8m)32Fb(rc)) is favorable to classify and evaluate weak to strong interactions, since interactions are usually characterized by the signs of Hb(rc) and 32Fb(rc). Interactions in the region of 32Fb(rc) < 0 are called shared-shell (SS) interactions, where Fb(rc) are locally concentrated relative to the average distribution around rc, and they are closed-shell (CS) interactions with the depleted Fb(rc) for 32Fb(rc) > 0. Interactions exhibit covalent nature when Hb(rc) < 0, where electrons at BCPs are stabilized in the region, whereas they exhibit no covalency if Hb(rc) > 0 due to the destabilization of electrons at BCPs under the conditions. Hb(rc) must be negative when 32Fb(rc) < 0 as shown by eq 2 (Vb(rc) < 0 at all rc).10,11 Consequently, 32Fb(rc) < 0 and Hb(rc) < 0 for SS interactions. The CS interactions are especially called “pure” CS interactions for Hb(rc) > 0 and 32Fb(rc) > 0, since electrons at BCPs are depleted and destabilized under the conditions.13a Electrons in the intermediate region between SS and pure CS are locally depleted but stabilized at BCPs, since 32Fb(rc) > 0 but Hb(rc) < 0. Such CS will be called “regular” CS here,11 when it is necessary to distinguish from pure CS. The polar (R, θ) coordinate representation10,11 is applied to the plots of Hb(rc) versus Hb(rc) - Vb(rc)/2, which is more favorable to classify, evaluate, and understand weak to strong interactions in a unified form. Figure 1 explains the treatment, where data at BCPs of full-optimized structures and perturbed structures are employed for the plots. Although R [) (x2 + y2)1/2 where (x, y) ) (Hb(rc) - Vb(rc)/2, Hb(rc))] in (R, θ) corresponds to the energy for an interaction at BCP relative to that without any interactions in the (x, y) plane,19 θ [) 90° - tan-1(y/x)] controls the helical stream of the plot, which is measured from the y-axis. Each plot for an interaction shows a specific curve, as shown in Figure 1. It is expressed by θp and κp: θp [) 90° - tan-1(dy/dx)] corresponds to the tangent line for the plot measured from the y-direction, and κp [) |d2y/dx2|/{1 + (dy/ dx)2}3/2] is the curvature of the plot at BCP of the full-optimized structure. Although (R, θ) correspond to the static nature of interactions, (θp, κp) values clarify the dynamic nature. Here, we propose a method to evaluate the dynamic nature of interactions: Perturbed structures in question are generated employing NIV, which cover the changes in distances and angles. The method must contain the high possibility to apply to a wide range of interactions under various conditions.

Nakanishi and Hayashi Applicability of NIV is discussed, exemplified mainly by CT interactions, as the first step to evaluate the dynamic nature by the method. Methodological Details in Calculations. Molecules and adducts are optimized and the frequency analysis is performed on the optimized structures with the 6-311++G(3df,3pd) basis sets of the Gaussian 03 program,20 unless otherwise noted. The Møller-Plesset second-order energy correlation (MP2) level is applied to the calculations.21,22 The k-th perturbed-structures in question (Skw) will be given by the addition of the normal coordinates of the k-th internal vibration (Nk) to the coordinates of a full-optimized structure (So) in the matrix representation.23 Eq 3 explains the method. The coefficient fkw in eq 3 controls the difference in structures between Skw and So: fkw are determined as shown in eqs 4 and 5 here, where ro and θso show the distances and angles of the full-optimized structures with ao of Bohr radius (0.52918 Å) and bo of one radian (57.296°).24 Eq 6 is employed in place of eq 5 when θso ) 180°, since w ) +0.025 and -0.025 in θs ) θso + wbo will give the same structure when θso ) 180°, so do w ) 0.05 and -0.05. Nk of five digits are employed to predict Skw, although only two digits are usually printed out.25

Skw ) So + fkw • Nk

(3)

r ) ro + wao (w ) (0), (0.05, and (0.1; ao ) 0.52918 Å) (4) θs ) θso + wbo (w ) (0), (0.025, and (0.05; bo ) 57.296o)

(5)

θs ) θso - wbo (w ) (0), 0.0125, 0.025, 0.0375, and 0.05; bo ) 57.296o)

(6)

AIM functions are calculated with the Gaussian 03 program20 and analyzed by the AIM2000 program.26 POM is also applied to obtain the perturbed structures by optimizing partially with the distance (r) of an interaction or angle (θs) around it in question fixed as shown in eqs 4-6.10,11 Results and Discussion Dynamic Behaviors of CT Interactions. The NIV method is applied to clarify the dynamic nature of interactions, together with POM. CT interactions are mainly discussed as the first step to demonstrate the availability of NIV. Figure 2 illustrates some typical internal vibrations of Br2 (D∞h), Br3- (D∞h), H2Se (C2V), H2SesBr2 (MC: Cs),27 and H2Se · Br2 (TBP: C2V).27 Table 1 shows AIM functions of Hb(rc) - Vb(rc)/2 and Hb(rc) and AIM parameters of k () Vb(rc)/Gb(rc)), R, and θ at BCPs for the fulloptimized structures, together with the optimized distances (ro) in question. AIM functions and parameters in Table 1 correspond to the static nature of the interactions. Table 2 collects the θp and κp values calculated by NIV (denoted by θp;NIV and κp;NIV, respectively) with the frequencies (ν) and force constants (kf) for the interactions, together with the θp and κp values by POM (θp;POM and κp;POM, respectively). The values in Table 2 show the dynamic nature. The θp and κp values are also calculated for various CT interactions by NIV and POM. Table 3 summarizes the AIM functions and parameters for the CT interactions, together with the corresponding ν and kf. Structures of the molecules and adducts are given in the Supporting

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Figure 2. Some internal vibrations in Br2 (D∞h) (a), Br3- (D∞h) (b), H2Se (C2V) (c), H2SesBr2 (MC: Cs) (d), and H2Se · Br2 (TBP: C2V) (e).

TABLE 1: AIM Functions and Parameters for Some Interactions Evaluated at Full Optimized Structuresa,b species Br-*-Br (D∞h) 3 Br-*-1Br-*-2Br- (D∞h) H-*-Se-*-H (C2V)h H2Se–*–1Br-*-2Br (Cs)i 1

Br-*-(H2)Se-*-2Br (C2V)j

BCP

roc (Å)

c32Fb(rc)d (au)

Hb(rc) (au)

ke

Rf (au)

θg (°)

Br-*-Br Br-*-Br Se-*-H Se-*-H Se–*–1Br 1 Br-*-2Br Se-*-H Se-*-1Br

2.2806 2.5594 1.4587 1.4598 3.1936 2.3100 1.4587 2.5158

-0.0044 0.0075 -0.0388 -0.0407 0.0059 -0.0023 -0.0555 0.0023

-0.0574 -0.0179 -0.1607 -0.1607 0.0004 -0.0514 -0.1750 -0.0276

-2.179 -1.557 -2.935 -3.029 -0.966 -2.099 -3.733 -1.859

0.0576 0.0203 0.1654 0.1658 0.0059 0.0514 0.1835 0.0277

184.3 158.3 193.6 194.2 86.2 182.6 197.6 175.3

a The 6-311++G(3df,3pd) basis sets being employed at the MP2 level. b Given values corresponding to the static nature of interactions. c The full-optimized structures are given in the Supporting Information. d c32Fb(rc) ) Hb(rc) - Vb(rc)/2 where c ) p2/8m. e k ) Vb(rc)/Gb(rc). f R ) (x2 + y2)1/2, where (x, y) ) (Hb(rc) - Vb(rc)/2, Hb(rc)). g θ ) 90° - tan-1(y/x). h ∠HSeH ) 91.28°. i ∠HSeH ) 91.35°; ∠HSe1Br ) 91.96°; ∠Se1Br2Br ) 179.34°. j ∠HSeH ) 94.22°; ∠HSe1Br ) 86.75°; ∠1BrSe2Br ) 170.45°.

Information. The first and second frequencies of the internal vibrations are also given in the Supporting Information. Whereas Br-*-Br and Se-*-H in Br2 (D∞h), H2Se (C2V), H2SesBr2 (MC: Cs) and/or H2Se · Br2 (TBP: C2V) are SS interactions with negative values of Hb(rc) - Vb(rc)/2 and Hb(rc), Se-*-Br in H2Se · Br2 (TBP) and Br-*-Br in Br3- are (regular) CS interactions since Hb(rc) - Vb(rc)/2 > 0 and Hb(rc) < 0, and the pure CS interaction for Se–*–Br in H2SesBr2 (MC) has positive values for both.11 The Hb(rc) - Vb(rc)/2 and Hb(rc) values for Se-*-H decrease (stronger) in the order of H2Se g H2SesBr2 (MC) > H2Se · Br2 (TBP), although r(Se, H) do not change so much. On the other hand, the values for Br-*-Br become more positive (weaker) in the order of Br2 < H2SesBr2 (MC) < Br3-, where r(Br, Br) increase in this order. Dynamic behaviors of interactions can be evaluated by θp and κp. The values for Br-*-Br, Se-*-H, and Se-*-Br in Br2, Br3-, H2Se, H2SesBr2 (MC), and/or H2Se · Br2 (TBP) are given in Table 2. The results with NIV and POM are exactly the same for the diatomic molecule of Br2: θp and κp are evaluated to be 190.9° and 0.27 au-1, respectively. Although the θp;NIV and κp;NIV values for Br-*-Br in Br3- are 180.1° and 4.66 au-1, respectively, if ν3 of 170.1 cm-1 is employed for the evaluation, then the values with ν4 of 198.9 cm-1 are 186.1° and 7.07 au-1, respectively. The θp;POM and κp;POM values are 184.2° and 10.01 au-1, respectively. The θp;POM and κp;POM values are closer to the corresponding θp;NIV and κp;NIV values evaluated with ν4, relative to those with ν3, respectively. The perturbation on a bond must affect others. We call the perturbed bond in POM the major bond and the influenced bonds

minor ones, so do θp and κp, here. Similar notation will be employed in NIV: Such bond from ν will be major that is perturbed mostly by ν, and those lesser ones will be minor in NIV. Minor values of θp and κp must also play a reasonable role in the dynamic behaviors of interactions. Minor values of θp and κp for 1Br-*-3Br in 2Br-*-1Br-*-3Br- are substantially different from those of the major one for 1Br-*-2Br in POM. The w′/w ratio is -0.357 (when w ) 0.100) in POM-b (Table 2), where w′ in r ) ro + w′ao is for r(1Br-*-3Br) and w in r ) ro + wao for r(1Br-*-2Br). The latter is major and the former is minor in this case. The θp;NIV and κp;NIV values are the same for 1 Br-*-2Br and 1Br-*-3Br, respectively, since the magnitudes in the changes of the bond distances are equally predicted for the internal vibrations of ν3 and ν4, although w′/w ) 1 for ν3 whereas w′/w ) -1 for ν4.28 The θp values for the major bonds must be affected by the behaviors of the minor ones: θp must also be controlled by w′, which are closely related to w. For example, θp are linearly and excellently correlated to w′/w in Br3- (θp ) -3.00w′/w + 183.1: R2 ) 1.000 (square of correlation coefficient)). The intercept of 183.1° corresponds to the averaged value of θp;NIV. Table 3 lists the θp;NIV values evaluated with ν of antisymmetric stretching modes or close to the modes. The antisymmetric stretching modes seem more suitable than the symmetric ones when dynamic behaviors of interactions are discussed by NIV in relation to those by POM. The results may show that POM essentially produces the perturbed structure close to that with the stretching mode of the antisymmetric type. The κp;POM value is also closer to κp;NIV evaluated with ν4 than that with ν3,

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TABLE 2: θp and Kp Values for Some Interactions Evaluated with NIV and POM, Together with Corresponding ν, kf, and w′/wa,b method

BCPc

NIV-ν1

Br-*-Br

ν (cm-1)

kfd (mdyne Å-1)

342.7

5.460

θpf (°)

Br-*-Br (D∞h): 1.000

190.9

0.27

νSG: r(Br-Br)

185.7k 180.1 186.1 196.0k 184.2 191.0

1.19k 4.66 7.07 10.01 7.82

νPIU: ∠BrBrBr νSGG: r(Br-Br) νSGU: r(Br-Br) f: ∠BrBrBr f: r(1Br-2Br)

207.6k 185.0 186.2 209.1k 185.8 132.8

175.01k 4.83 4.24 48.00k 4.56 79.84

νA1: ∠HSeH νA1: r(H-Se-H) νB2: r(H-Se-H) f: ∠HSeH f: Se-1H f: Se-1H

H2Se–*–1Br-*-2Br (Cs): 0.064 1.000n 0.064 -0.055 4.335 1.000o 4.335 -0.460 3.800 1.000p 3.845 1.000q 1.000 -0.069 1.000 -1.662

124.8 201.4 191.2 146.7 186.3 185.7 124.3 198.8 191.9 129.9

259.34 34.46 0.31 503.90 4.62 4.09 261.18 0.35 0.34 328.97

νA′: Ses1Br νA′: Ses1Br νA′: 1Br-2Br νA′: 1Br-2Br νA′: 1H-Se-2H νA′: 1H-Se-2H f: Ses1Br f: Ses1Br f: 1Br-2Br f: 1Br-2Br

1 Br-*-(H2)Se-*-2Br (C2V): 1.542 1.000s 3.094 1.000t 1.000u -0.181

185.9 188.9 187.7 195.7

0.31 0.27 1.28 36.90

3

NIV-ν1 NIV-ν3 NIV-ν4 POM-a POM-b

Br-*-2Brj 1 Br-*-2Brj 1 Br-*-2Brj 1 Br-*-2Brj 1 Br-*-2Br 1 Br-*-3Br

93.9 170.1 198.9

NIV-ν1 NIV-ν2 NIV-ν3 POM-a POM-b

Se-*-1Hm Se-*-1Hm Se-*-1Hm Se-*-1Hm Se-*-1H Se-*-2H

1072.1 2520.7 2533.4

NIV-ν3

Se–*–1Br 1 Br-*-2Br 1 Br-*-2Br Se–*–1Br Se-*-H Se-*-H Se–*–1Br 1 Br-*-2Br 1 Br-*-2Br Se–*–1Br

56.3 56.3 309.0 309.0 2515.4 2528.6

Se-*-1Brr Se-*-1Brr Se-*-1Br Se-*-2Br

182.8 257.2

1

1

NIV-ν8 NIV-ν9 POM-a POM-b

NIV-ν3 NIV-ν4 POM-a

2

0.692 3.816 3.860

νchrcth or fi

-

Br-*- Br-*- Br (D∞h): 0.410 -0.030 1.345 1.000 1.840 -1.000 -0.011 1.000 -0.357 1

NIV-ν6

κpg (au-1)

w′/we

H-*-Se-*-2H (C2V) -0.027 1.000 -1.000 -0.036 1.000 0.007

l

νA1: 1Br-Se-2Br νB2: 1Br-Se-2Br f: Se-1Br f: Se-1Br

a The 6-311+G(3df,3pd) basis sets being employed at the MP2 level. b Values are given in plane for the major interactions and in italic for the minor ones: See text about the major and minor interactions. c BCP (*) on the interaction in question of which AIM data being given. d Force constant corresponding to the frequency. e Ratio of w′/w where w′ in r ) ro + w′ao or θs ) θso + w′bo is for the minor interaction whereas w in r ) ro + wao or θs ) θso + wbo for the major one in POM: Most perturbed interaction by ν is called major interaction whereas others are minor ones in NIV. f θp ) 90° - tan-1(dy/dx) where (x, y) ) (Hb(rc) - Vb(rc)/2, Hb(rc)). g κp ) |d2y/dx2|/[1 + (dy/dx)2]3/2. h Main character of frequency in NIV. i Fixed interaction which corresponds to the major one in POM. j The same values being predicted for 1Br-*-3Br. k Some error might contain due to small w′/w. l Error must be very large due to very small w′/w (w′ ) 0.0006). m The same value(s) being predicted for Se-*-2H. n w′/w ) -0.0008 for Se-*-H. o w′/w ) 0.0021 for Se-*-H. p w′/w ) -0.0008 for Se–*–1Br and -0.0002 for 1Br-*-2Br. q Substantially not detected for Se–*–1Br and 1Br-*-2Br. r The same values are given for Se-*-2Br. s w′/w ) 0.012 for Se-*-H and -0.071 for ∠HSeH. t w′/w ) 0.009 for Se-*-H and -0.007 for ∠HSeH. u w′/w ) 0.005 for Se-*-H and -0.028 for ∠HSeH.

although κp;POM seems somewhat larger than κp;NIV. The perturbed structures by POM remind us of those encountered in the thermal processes. Changes in r(Br-Br) are small when NIV is applied to ∠BrBrBr with ν1 of 93.9 cm-1 (w′/w ) -0.030; w′ ) 0.0015 when w ) -0.050) and POM is applied to ∠BrBrBr (w′/w ) -0.011; w′ ) 0.00057 when w ) -0.050). Calculated θp and κp values would not be so reliable due to the small w′/w ratios with the small w′ values when NIV and POM are applied to angles. Consequently, the two motions can be analyzed separately as the first step to evaluate the dynamic behaviors of interactions by the AIM dual functional analysis. The results are summarized in Figure 3. The θp;NIV and κp;NIV values for Se-*-H in SeH2 are evaluated to be 185.0-186.2° and 4.24-4.83 au-1, respectively, with ν2 of 2521 cm-1 (A1) and ν3 of 2533 cm-1 (B2). The θp;POM and κp;POM values are 185.8° and 4.56 au-1, respectively (see POM-b in Table 2), which are very close to the averaged values of the former, respectively. The change in the Se-*-2H distance is less than 1% of that in Se-*-1H if POM is applied to Se-*-1H. Changes in r(Se-H) were small when NIV and POM were

applied to ∠HSeH: w′/w ) -0.027 in NIV (w′ ) -0.0013 with w ) 0.050) and -0.036 in POM (w′ ) -0.0018 with w ) 0.050). The coupled motion between r(Se-H) and ∠HSeH seems very small in SeH2. Consequently, the stretching motion can be analyzed separately from the bending in H2Se, as the first order of the AIM dual functional analysis. The θp;NIV and κp;NIV values for Se-*-H in H2SesBr2 (MC) are very close to those in H2Se, respectively, if evaluated employing ν8 of 2515 cm-1 and ν9 of 2529 cm-1. The coupled motion of r(Se-H) with r(SesBr) and r(Br-Br) is negligible for H2SesBr2 (MC); therefore, they can also be discussed separately. The θp;NIV and κp;NIV values for Se–*–Br evaluated with ν3 of 56 cm-1 are 124.8° and 259.3 au-1, respectively, which are very close to θp;POM of 124.3° and κp;POM of 261.2 au-1, respectively. The values for Br-*-Br by NIV with ν6 of 309 cm-1 are 191.2° and 0.31 au-1, respectively, which are also very close to θp;POM of 191.9° and κp;POM of 0.34 au-1, respectively. The minor values for Se–*–Br and Br-*-Br in H2SesBr2 (MC) are substantially different from those of the major ones, which is discussed later again.

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TABLE 3: AIM Functions and Parameters Evaluated for the Charge-transfer (CT) Interactions in Molecular Complexes (CT-MC), Trihalide Ions (X3-), and Chalcogenide Dihalides of Trigonal Bipyramidal Structures (CT-TBP), Calculated at the MP2 Levela species (X-*-Y) g

Me2O-*-Cl2 Me2O-*-Br2g Me2S-*-Cl2g Me2S-*-Br2g Me2Se-*-Cl2g H2Se-*-Br2g Me2Se-*-Br2g [Cl-*-Cl2]-h [Br-*-Br2]-h [Cl-*-BrCl]-h [Br-*-ClBr]-h Me2ClS-*-Clg Me2BrS-*-Brg Me2ClSe-*-Clg H2BrSe-*-Brg Me2BrSe-*-Brg species (X-*-Y) g

Me2O-*-Cl2 Me2O-*-Br2g Me2S-*-Cl2g Me2S-*-Br2g Me2Se-*-Cl2g H2Se-*-Br2g Me2Se-*-Br2g [Cl-*-Cl2]-h [Br-*-Br2]-h [Cl-*-BrCl]-l [Br-*-ClBr]-h Me2ClS-*-Clg Me2BrS-*-Brg Me2ClSe-*-Clg H2BrSe-*-Brg Me2BrSe-*-Brg

ro(X, Y)b (Å)

c32Fb(rc)c (au)

Hb(rc) (au)

νd (cm-1)

kfe (mdyne Å-1)

kf

2.5513 2.5913 2.6331 2.6923 2.5700 3.1936 2.7288 2.2956 2.5594 2.4022 2.4392 2.2649 2.4388 2.3544 2.5158 2.5194

0.0127 0.0122 0.0109 0.0093 0.0091 0.0059 0.0076 0.0133 0.0075 0.0097 0.0103 0.0045 0.0046 0.0047 0.0023 0.0029

0.0007 -0.0004 -0.0056 -0.0078 -0.0128 0.0004 -0.0105 -0.0220 -0.0189 -0.0227 -0.0184 -0.0367 -0.0262 -0.0342 -0.0276 -0.0269

118.0 100.5 104.4 114.9 123.5 56.3 108.8 292.5 198.9 248.3 271.5 334.6 358.5 307.8 257.2 233.4

0.070 0.037 0.044 0.059 0.058 0.064 0.078 1.763 1.840 1.721 1.689 0.389 0.294 0.366 3.094 0.946

-0.972 -1.018 -1.206 -1.296 -1.413 -0.966 -1.408 -1.454 -1.557 -1.538 -1.472 -1.802 -1.741 -1.784 -1.859 -1.820

Ri (au)

θj (°)

θp:POMk (°)

κp:POMl (au-1)

θp:NIVk (°)

κp:NIVl (au-1)

comment

0.0128 0.0122 0.0122 0.0122 0.0157 0.0059 0.0129 0.0257 0.0203 0.0248 0.0211 0.0370 0.0266 0.0346 0.0277 0.0271

86.8 92.1 117.4 130.1 144.6 86.2 144.0 149.0 158.3 156.8 150.8 172.9 170.1 172.2 175.3 173.7

96.4 106.6 163.4 169.9 181.5 124.3 180.1 181.6 183.8 182.9 181.3 191.7 188.5 184.0 187.7 186.6

30.91 49.64 52.73 40.53 9.13 261.18 13.49 10.96 7.38 6.21 11.98 5.31 8.31 0.66 1.28 1.74

98.6 111.7 162.8 171.5 182.3 124.8 181.0 183.1 186.5 185.2 183.4 192.8 188.9 186.3 188.9 189.0

35.54 59.42 51.24 33.00 14.46 259.34 13.82 9.83 6.33 5.58 10.74 4.84 2.89 0.57 0.27 1.95

CT-MC CT-MC CT-MC CT-MC CT-MC CT-MC CT-MC X3X3X3X3CT-TBP CT-TBP CT-TBP CT-TBP CT-TBP

a The 6-311++G(3df,3pd) basis sets being employed at the MP2 level, unless otherwise noted. b Bond distances in the full-optimized structures. c c32Fb(rc) ) Hb(rc) - Vb(rc)/2, where c ) p2/8m. d Internal vibration frequency corresponding to the interaction. e Force constant correspond to the frequency. f k ) Vb(rc)/Gb(rc). g The 6-311+G(3d,2p) basis sets being employed for H and C of the Me group. h See also ref 10. i R ) (x2 + y2)1/2 where (x, y) ) (Hb(rc) - Vb(rc)/2, Hb(rc)). j θ ) 90° - tan-1(y/x). k θp ) 90° - tan-1 (dy/dx). l κp ) |d2y/dx2|/[1 + (dy/dx)2]3/2.

Figure 4 summarizes the results of the analysis for H2SesBr2 (MC). The ratio of w′/w ) -0.069 when POM is applied to r(Se–*–1Br) of H2SesBr2 (MC) where w is for r(Se–*–1Br) and w′ for r(1Br-*-2Br) (see POM-a in Table 2). The former is major and the latter is minor in this case. The w′/w value of -0.069 is close to that for NIV with ν3 (-0.055). The AIM data for Se–*–Br by NIV with ν3 are so close to those by POM (major) that they appear almost overlapped in Figure 4b. Similarly, plots appear almost overlapped in Figure 4c for Br-*-Br from NIV (ν3) and POM (minor), since the AIM data are so close to each other. It is worthwhile to comment that the w′/w ratio could be larger than unity in POM: w′/w ) 1.662 for minor of r(SesBr) when r(Br–Br) is major in H2SesBr2 (MC). The result would be the reflection of the fact that SesBr is much easier to be influenced by the surroundings relative to Br–Br in H2SesBr2 (MC). In the case of H2Se · Br2 (TBP), θp and κp for Se-*-H are also very close to those in H2Se and H2SesBr2 (MC), although not shown in Table 2. The θp;NIV and κp;NIV values for Se-*-1Br in H2Se · Br2 (TBP) are 185.9-188.9° and 0.27-0.31 au-1, respectively, evaluated with ν3 of 182.8 cm-1 and ν4 of 257.2 cm-1. Characters of ν3 and ν4 are symmetric and antisymmetric modes, respectively. The θp;POM and κp;POM values are 187.7° and 1.28 au-1, respectively. The θp;POM value is very close to

Figure 3. Plots of Hb(rc) vs Hb(rc) - Vb(rc)/2 for Br3- (D∞h), where the perturbed structures are generated by NIV and POM. (a) The whole picture of the plot and (b) the partial picture magnified around the point for the full-optimized structure.

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Figure 5. Plots of Hb(rc) vs Hb(rc) - Vb(rc)/2 for H2Se · Br2 (TBP): Perturbed structures are generated by NIV and POM.

Figure 4. Plots of Hb(rc) vs Hb(rc) - Vb(rc)/2 for H2SesBr2 (MC) where the perturbed structures are generated by NIV and POM. (a) The whole picture of the plots, (b) the partial picture around the Se–*–Br region (data from ν3 being almost overlapped to those from POMmajor in the plots), and (c) for the Br-*-Br region (data from ν3 being almost overlapped to those from POM-minor in the plots).

the averaged value of θp;NIV (187.4°), whereas κp;POM seems somewhat larger than κp;NIV, similarly to the case of Br3-. The θp;NIV and κp:NIV values are predicted to be the same for Se-*1 Br and Se-*-2Br, respectively. However, the minor values of

θp;POM and κp:POM are 195.7° and 38.9 au-1 for Se-*-2Br, respectively, which are substantially larger than the corresponding major values for Se-*-1Br, respectively. This must be the results from w′/w ) -0.181 for r(Se-*-2Br) relative to r(Se-*-1Br) (w′/w ) 1.000) in H2Se · Br2 (TBP) in POM. The obtained w′/w value may show that the Se-*-Br interactions in H2Se · Br2 (TBP) are also affected by the change of surroundings. The results are summarized in Figure 5, which are close to those for Br3-. The differences between the minor and major values in θp are around 6-8° for the most of bromine adducts in Table 2: The difference is 6.8° for Br-*-Br in Br3-, 6.9° for Br-*-Br in H2SesBr2 (MC), and 8.0° for Se-*-Br in H2Se · Br2 (TBP); however, it is 22° for Se–*–Br in H2SesBr2 (MC). The differences would correlate to the susceptibility of the bonds by the surroundings. The comparison in κp seems more complex than that in θp. The θp:NIV and κp;NIV values are very close to the corresponding θp:POM and κp;POM values, respectively, for the species discussed above. How are the dynamic behaviors of the interactions in various molecules and adducts? Those of various CT interactions are discussed, next. Dynamic Behaviors of Interactions. The θp and κp values evaluated by NIV and POM for various CT interactions are collected in Table 3. The CT interactions exhibit the covalent nature except for Me2OsCl2 (MC: Hb(rc) ) 0.0007 au) and H2SesBr2 (MC: Hb(rc) ) 0.0004 au). As shown in Table 3, θp;NIV are very close to θp;POM: The differences in θp (∆θp ) θp;NIV - θp;POM) are -0.6 to 2.7°, except for Me2OsBr2 (MC) of 5.1°. However, the deviation should not be particular for the MC adducts, since the differences are small for Me2OsCl2 (MC: ∆θp ) 2.2°), Me2SsCl2 (MC: -0.6°), Me2SsBr2 (MC: 1.6°), Me2SesCl2 (MC: 0.8°), and Me2SesBr2 (MC: 0.9°). The difference must be small if the internal vibration is well located on the interaction in question. The internal frequency employed for NIV is not located well on the OsBr interaction in Me2OsBr2 (MC). The κp;NIV values are also very close to κp;POM. The differences (∆κp ) κp;NIV - κp;POM) are within (2 au-1 for the usual CT interactions. The ∆κp values are beyond the limit for Me2OsCl2

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(MC: ∆κp ) 4.6 au-1), Me2OsBr2 (MC: 9.8 au-1), Me2SsBr2 (MC: -7.5 au-1), and Me2SesCl2 (MC: 5.3 au-1), although that for Me2SesBr2 (MC: 0.3 au-1) is very small. The deviations in Me2OsBr2 (MC: ∆κp ) 9.8 au-1 with κp;NIV ) 59.4 au-1) and Me2SsBr2 (MC: ∆κp ) -7.5 au-1 with κp;NIV ) 33.0 au-1) are 16 and -23% of the corresponding κp;NIV values, respectively; the deviations in κp usually occur for the adducts of which κp;NIV are large or relatively large. The θp and κp values are similarly evaluated for various interactions in vdW, HB, CT-MC, X3- (trihalide ions), CT-TBP, and covalent bonds of weak (Cov-w) and strong ones (Cov-s), although only those for CT-MC, X3-, and CT-TBP are given in Table 3. The θp;NIV values are plotted versus θp;POM, so are κp;NIV versus κp;POM (38 species). The plots give excellent correlations, which are shown in eqs 7 and 8, respectively.

θp;NIV ) 1.004θp;POM - 0.18 (R2 ) 0.998) κp;NIV ) 0.995κp:POM + 0.06 (R2 ) 0.999)

structures around TS with POM would be different from those on IRC (intrinsic reaction coordinates). The applicability of NIV to the wide range of interactions under various conditions will also be established. Acknowledgment. This work was partially supported by a Grant-in-Aid for Scientific Research (Nos. 19550041 and 20550042) from the Ministry of Education, Culture, Sports, Science, and Technology, Japan. The support of the Wakayama University Original Research Support Project Grant and the Wakayama University Graduate School Project Research Grant is also acknowledged. Supporting Information Available: The full-optimized structures given by Cartesian coordinates for examined molecules and adducts. This material is available free of charge via the Internet at http://pubs.acs.org.

(7)

(8)

The results show that θp;NIV and θp;POM are substantially the same for the weak to strong interactions as a whole, so are κp;NIV and κp;POM. Some deviations occur if the internal vibrations are not well located on the interactions in question. Differences in the w′/w ratios in NIV and POM would be partly responsible for the deviations. Details of the dynamic behaviors of the interactions containing those in vdW and HB will be discussed elsewhere. Conclusion The dynamic behaviors of interactions must be very important, as well as the static behaviors, for the better understanding of molecules and adducts. Perturbed structures from a fulloptimized structure are necessary to evaluate the dynamic nature. Normal coordinates of internal vibrations (NIV) are proposed to generate the perturbed structures. AIM parameters of (θp, κp) are employed to evaluate the dynamic behaviors of interactions in the polar (R, θ) representation of Hb(rc) versus Hb(rc) - Vb(rc)/2 at BCPs: θp corresponds to the tangent line measured from the y-direction and κp is the curvature of the plot at BCP of the full-optimized structure. POM (partial optimization method) is also suitably applied in usual cases if the structures are well obtained. POM contains some disadvantages: NIV must contain the high possibility to clear up the difficulties in POM. The θp;NIV and κp;NIV values are substantially equal to θp;POM and κp;POM, respectively. The differences in θp;NIV and θp;POM are within around (2°, and those in κp;NIV and κp;POM are within around (2 au-1, respectively, for usual interactions. Deviations would occur when the internal vibrations are not located suitably on the interactions in question. The θp;NIV and κp;NIV values must originate from the character of ν employed for NIV in the molecules or adducts. The high utility of NIV is well demonstrated to evaluate dynamic behaviors of interactions for various interactions. Applications of NIV to the CT interactions are discussed in detail in this work. Similar results are obtained for the interactions in vdW and HB adducts. The details of which will be discussed elsewhere. Applications of NIV to those containing difficulties with POM are in progress. Each interaction in polycyclic compounds, multi hydrogen-bonded systems, and extended hypervalent bonds is the target of the proposed method. The analysis of TS must also be of great interest, since perturbed

Note Added after ASAP Publication. This article posted ASAP on June 14, 2010. Tables 1, 2, and 3 have been revised. The correct version posted on June 17, 2010. References and Notes (1) (a) Atoms in Molecules: A Quantum Theory: Bader, R. F. W., Ed.; Oxford University Press: Oxford, UK, 1990. (b) Matta, C. F.; Boyd, R. J., An Introduction to the Quantum Theory of Atoms in Molecules. In The Quantum Theory of Atoms in Molecules: From Solid State to DNA and Drug Design; Matta, C. F.; Boyd, R. J., Eds.;Wiley-VCH: Weinheim, Germany, 2007; Ch. 1. (2) (a) Bader, R. F. W.; Slee, T. S.; Cremer, D.; Kraka, E. J. Am. Chem. Soc. 1983, 105, 5061–5068. (b) Bader, R. F. W. Chem. Res. 1991, 91, 893– 926. (c) Bader, R. F. W. J. Phys. Chem. A 1998, 102, 7314–7323. (d) Biegler-Ko¨nig, F.; Bader, R. F. W.; Tang, T. H. J. Comput. Chem. 1982, 3, 317–328. (e) Bader, R. F. W. Acc. Chem. Res. 1985, 18, 9–15. (f) Tang, T. H.; Bader, R. F. W.; MacDougall, P. Inorg. Chem. 1985, 24, 2047– 2053. (g) Biegler-Ko¨nig, F.; Scho¨nbohm, J.; Bayles, D. J. Comput. Chem. 2001, 22, 545–559. (h) Biegler-Ko¨nig, F.; Scho¨nbohm, J. J. Comput. Chem. 2002, 23, 1489–1494. (3) Molina, J.; Dobado, J. A. Theor. Chem. Acc. 2001, 105, 328–337. (4) Dobado, J. A.; Martinez-Garcia, H.; Molina, J.; Sundberg, M. R. J. Am. Chem. Soc. 2000, 122, 1144–1149. (5) Ignatov, S. K.; Rees, N. H.; Tyrrell, B. R.; Dubberley, S. R.; Razuvaev, A. G.; Mountford, P.; Nikonov, G. I. Chem.sEur. J. 2004, 10, 4991–4999. (6) Tripathi, S. K.; Patel, U.; Roy, D.; Sunoj, R. B.; Singh, H. B.; Wolmersha¨user, G.; Butcher, R. J. J. Org. Chem. 2005, 70, 9237–9247. (7) Boyd, R. J.; Choi, S. C. Chem. Phys. Lett. 1986, 129, 62–65. (8) Carroll, M. T.; Bader, R. F. W. Mol. Phys. 1988, 65, 695–722. (9) The bond order (BO), which corresponds to the strength of a chemical bond, is correlated to Fb(rc) by the form shown below, where A and B are constants that depend on the nature of the bonded atoms.2b BO ) exp[AFb(rc)- B]. (10) (a) Nakanishi, W.; Hayashi, S.; Narahara, K. J. Phys. Chem. A 2009, 113, 10050–10057. (b) See also the original report: Nakanishi, W.; Hayashi, S.; Narahara, K. J. Phys. Chem. A. 2008, 112, 13593–13599. (11) Nakanishi, W.; Hayashi, S. Curent. Org. Chem. 2010, 14, 181– 197. (12) Grabowski, S. J. J. Phys. Chem. A 2001, 105, 10739–10746. (13) (a) Espinosa, E.; Alkorta, I.; Elguero, J.; Molins, E. J. Chem. Phys. 2002, 117, 5529–5542. (b) Rozas, I.; Alkorta, I.; Elguero, J. J. Am. Chem. Soc. 2000, 122, 11154–11161. (c) Espinosa, E.; Molins, E.; Lecomte, C. Chem. Phys. Lett. 1998, 285, 170–173. (14) (a) Molecular Quantum Mechanics, 4th ed.; Atkins, P. W.; Friedman, R. S., Eds.; Oxford University Press: Oxford, UK, 2005; Ch. 10. (b) Menard, K. P. Dynamic Mechanical Analysis: A Practical Introduction; 1st ed.; CRC Press; Boca Raton, Florida, USA, 1999. (15) (a) Molecular Interactions. From Van der Waals to Strongly Bound Complexes; Scheiner, S., Eds.; Wiley: New York, 1997. (b) For example, Avoird, A.; van der Wormer, P. E. S.; Moszynski, R. Theory and Computation of Vibration, Rotation and Tunneling Motions of Van der Waals Complexes and their Spectra, InCh. 4; Del Bene, J. E.; Shavitt, I., The Quest for Reliability in Calculated Properties of Hydrogen-bonded Complexes, in Ch. 5; and Ford, T. A. Ab Initio Predictions of the Vibrational Spectra of Some Molecular Complexes: Comparison with Experiment, in Ch. 6. See also other chapters.

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(16) For van der Waals interactions, see: (a) Dessent, C. E. H.; Mu¨llerDethlefs, K. Chem. ReV. 2000, 100, 3999–4021. (b) Wormer, P. E. S.; van der Avoird, A. Chem. ReV. 2000, 100, 4109–4143. (17) (a) Hydrogen Bonding s A Theoretical PerspectiVe; Scheiner, S., Ed.; Oxford University Press: New York, 1997. (b) The Weak Hydrogen Bond in Structural Chemistry and Biology; International Union of Crystallography Monographs on Crystallography; Desiraju, G. R.; Steiner, T., Eds.; Oxford University Press: New York, 1999. (c) Nishio, M. Cryst. Eng. Commun. 2004, 6, 130–158. (d) Espinosa, E.; Molins, E.; Lecomte, C. Chem. Phys. Lett. 1998, 285, 170–173. (e) Espinosa, E.; Souhassou, M.; Lachekar, H.; Lecomte, C. Acta Crystallogr. B 1999, 55, 563–563. (f) Espinosa, E.; Lecomte, C.; Molins, E. Chem. Phys. Lett. 1999, 300, 745–748. (g) Espinosa, E.; Alkorta, I.; Rozas, I.; Elguero, J.; Molins, E. Chem. Phys. Lett. 2001, 336, 457–461. (h) Gatti, C.; Bertini, L. Acta Crystallogr A 2004, 60, 438–449. (18) (a) Slifkin, M. A. Charge Transfer Interactions of Biomolecules; Academic Press: London and New York, 1971. (b) Kuznetsov, A. M. Charge Transfer in Physics, Chemistry and Biology: Physical Mechanisms of Elementary Processes and an Introduction to the Theory; Gordon and Breach Science Publishers: Berkshire, 1995. (c) Lippolis, V.; Isaia, F. Charge-Transfer (C.-T.), Adducts and Related Compounds in Handbook of Chalcogen Chemistry: New PerspectiVes in Sulfur, Selenium and Tellurium; Devillanova, F. A., Ed.; Royal Society of Chemistry: Cambridge, 2006, Ch. 8.2, pp 477-499. (19) The origin of the plot corresponds to those without any interactions at BCPs, where Vb(rc) ) Gb(rc) ) 0. See refs 10 and 11. (20) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery Jr., J. A.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C. Pople, J. A., Gaussian 03, ReVision D.02; Gaussian, Inc.: Wallingford, CT, 2004.

Nakanishi and Hayashi (21) Møller, C.; Plesset, M. S. Phys. ReV. 1934, 46, 618–622. Gauss, J. J. Chem. Phys. 1993, 99, 3629–3643. Gauss, J. Ber. Bunsenges, Phys. Chem. 1995, 99, 1001–1008. (22) Calculations can also be performed at the density functional theory (DFT) level of the Becke three parameter hybrid functionals with the LeeYang-Parr correlation functional (B3LYP), if the level is more suitable.29 (23) For the m × n matrix representation, m corresponds to the number of atoms and n () 3) to the x, y, and z components of the space. (24) The values of w ) (0), (0.1, and (0.2 in r ) ro + wao were employed for the perturbed structures in POM in refs 10 and 11, since the bond orders becomes 2/3 and 3/2 times larger at w ) +0.2 and-0.2 relative to the original values at w ) 0, respectively. However, it seems better to employ the perturbed structures as close as possible to the full-optimized ones in NIV. The perturbed structures closer to the full-optimized one will reduce the errors in the AIM functions with the perturbed structures generated by NIV and/or POM. Therefore, w ) (0), (0.05, and (0.1 for r ) ro + wao are employed for the analysis in this paper. Similarly, w ) (0), (0.025, and (0.05 for θs ) θso + wbo are applied to the perturbed structures, since (0.1bo ((5.73°) would be too large as the perturbations for angles. (25) It is achieved by changing the corresponding parameters in Gaussian 03 from the default values to print out the normal coordinates of five digits for the purpose. (26) The AIM2000 program (Version 2.0) is employed to analyze and visualize atoms-in-molecules: (a) Biegler-Ko¨nig, F. J. Comput. Chem. 2000, 21, 1040–1048, see also ref 3g. (27) (a) Chemistry of HyperValent Compounds: Akiba, K.-y., Ed.; WileyVCH: New York, 1999. (b) Nakanishi, W., HyperValent Chalcogen Compounds In Handbook of Chalcogen Chemistry: New PerspectiVes in Sulfur, Selenium and Tellurium: Devillanova, F. A., Ed.; Royal Society of Chemistry: Cambridge, 2006, Ch. 10.3, pp 644-668. (28) The θp and κp for the major bonds are substantially controlled by the characters of the local bonds in question: The influence from the behaviors of the minor bonds would not be so severe for usual cases. (29) Lee, C.; Yang, W.; Parr, R. G. Phys. ReV. B 1988, 37, 785–789. Miehlich, B.; Savin, A.; Stoll, H.; Preuss, H. Chem. Phys. Lett. 1989, 157, 200–2006. Becke, A. D. Phys. ReV. A 1988, 38, 3098–3100. Becke, A. D. J. Chem. Phys. 1993, 98, 5648–5652.

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