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Derivation of Correlation Functions to Predict Bond Properties of Phenyl-CH Bonds Based on Vibrational and 1H NMR Spectroscopic Quantities Martin Presselt,†,‡ Christoph Schnedermann,† Michael Mu¨ller,† Michael Schmitt,† and Ju¨rgen Popp*,†,§ Institute of Physical Chemistry, Friedrich-Schiller-UniVersity Jena, Helmholzweg 4, 07743 Jena, Germany, and Institute of Photonic Technology, Albert-Einstein-Str. 9, 07745 Jena, Germany ReceiVed: June 10, 2010; ReVised Manuscript ReceiVed: August 3, 2010
The study of electron density properties significantly contributes to the determination of important chemical relations. The experimental determination of the electron density distribution is limited to single crystals. However, equivalent information is often desired for molecules, which do not crystallize in a sufficient manner. Furthermore, it is of high impact to study changes in the electron density distribution (i.e., related reactivities) upon environmental variations. Consequently, here we investigate methods to derive electron density properties from spectroscopic data. In particular correlation functions are introduced, which are able to predict electron density properties in all five CH-bonds of monosubstituted benzene derivatives at once. The prediction performance for electron densities and the corresponding Laplacians is lower as compared to previously introduced local functions [Presselt et al. J. Phys. Chem. A 2009, 113, 3210], but far less spectroscopic input data are needed. However, for ellipticities a higher prediction performance than this obtained for the previously derived local functions could be obtained despite the fact that less spectroscopic data were used. Thus, ellipticities are best predicted using 1H NMR data for the para position of monosubstituted benzene derivatives. Motivation and Approach It is well-known that experimentally determined electron density distributions F(r) significantly contribute to the elucidation of interesting bonding situations.1-11 However, the experimental determination of F(r) is limited to single crystals. Nonetheless, F(r) properties of substances which do not tend to form single crystals or of substances in various noncrystalline environments are also of huge interest.12,13 Therefore, we recently introduced the concept of deriving correlation functions based on spectroscopic parameters enabling the prediction of electron density properties in bond-critical points (BCPs).14 The spectroscopic parameters we utilized were in particular vibrational wavenumber values and IR and Raman intensities of inherently localized CD stretching vibrations as well as 1H NMR shifts. These quantities which can be easily determined for substances in various physical states were used as variables for linear functions to predict the electron density F, the respective Laplacian ∇2F, or the ellipticity ε in BCPs. These functions were optimized to describe specific CH-bonds in monosubstituted benzene derivatives. To obtain an excellent prediction performance, at least two spectroscopic variables within the linear functions describing F, ∇2F, or ε were required. Hence, to reliably describe F, ∇2F, or ε in the BCPs of each ph-CH bond via linear functions the determination of about 10 (six using mean values) spectroscopic values was necessary. The work presented within this study focuses on a combined description of all five aromatic CH bonds using less than 10 spectroscopic variables, aiming to achieve a comparable prediction performance as that for the independent individual predic* To whom correspondence should be addressed. † Friedrich-Schiller-University Jena. ‡ Present address: Institute of Physics, Ilmenau University of Technology, Weimarer Str. 32, 98693 Ilmenau, Germany. § Institute of Photonic Technology.
tion of the CH bond properties.14 Since plotting the aromatic CH positions versus the bond characterizing electron density values yields approximately symmetric plots for most of the studied monosubstituted benzene derivatives as shown in our preceding article14 (see also Figure 2), we aim in the following to derive functions representing these plots. Hence, the purpose of the present study is to describe electron densities, the respective Laplacian, or the ellipticities in BCPs of all aromatic CH bonds with just one global function dependent on the spectroscopic data. Calculations The density functional theory (DFT) data used within this work were taken from a recent study of ours, where the electron density target values were validated via the MP2 method.14,15 The F-, ∇2F-, and ε-values were extracted from the electron density distribution via QTAIM implemented in AIM2000.16-20 All electron density characteristics used throughout the work, as shown exemplarily in Figure 1 for aniline, and further characteristics of general interest can be found in the Supporting Information. Geometry optimizations and computations of spectroscopic data were performed with GAUSSIAN03 (g03)21 choosing the pure density functional of Becke and Perdew BP8622,23 combined with the triple-ξ TZVP basis set.24,25 Table 1 gives an overview of the investigated benzene derivatives. The left part of Figure 2 displays the electron density target values of the BCPs of the investigated phenyl CH bonds for each benzene derivative plotted against the five different phenyl CH positions. Since the DFT calculations lead to fixed geometries, the electron density values for equivalent ph positions (ortho1 and ortho5, meta2 and meta4) differ for asymmetric substituents. To account for the free rotation around the ph-substituent bond, taking place in solution or gas phase, all properties for equivalent but opposite CH bonds were
10.1021/jp105348d 2010 American Chemical Society Published on Web 08/24/2010
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Figure 1. Summary of electron density characteristics F, ∇2F, and ε of aniline (gray, carbon; red, nitrogen; turquoise, hydrogen). Data sets written in black refer to the bond critical points, whereas the red written data set refers to a ring critical point.
TABLE 1: Summary of the Investigated Benzene Derivatives
averaged, thus leading to perfectly symmetric graphs as shown in the right part of Figure 2. Fit Methodologies Selected F-, ∇2F-, and ε-value data involving averaged values for the ortho and meta positions are exemplarily shown in Figure 3. Considering arithmetic means of data, referring to the two ortho or to the two meta positions facilitates the comparison with experimental data, since the mean data account for the free rotation of the substituents expected for real liquids. To describe the electron density target values (F, ∇2F, ε) for the different ph-CH positions with just one function, the following three symmetric functions were investigated (see Figure 3):
f1(CH-BCP) ) P1 + (A - B)(P2 · x2 + P3 · e-x ) 2
2
(1)
f2(CH-BCP) ) P1 + (A - B)(P2 · x2 + P3 · ex )
(2)
f3(CH-BCP) ) P1 + (A - B)(P2 · x2 + P3 · x4)
(3)
Here, f illustrates the target quantity (F-, ∇2F-, or ε-values), and x is the position of the CH-BCPs (0 ) para, 1 ) meta, 2 )
ortho). P1, P2, and P3 are parameters of the global function, and (A - B) is a switching factor, included to account for the shape of the function (either “W” or “M” shape as shown in Figure 3). If (A - B) is set to 1, the switch factor is switched off. Consequently, for (A - B) ) 1, parameters P2 and P3 are determining the shape of the functions exclusively. If the switch factor (A - B) is positive, functions f1-f3 possess a ”W” shape; if the switch factor (A - B) is negative, these functions possess a “M” shape. The switch factor is calculated from the vibrational data of a substance (A) using the corresponding benzene data (B) as a reference (denoted benzene). A further possibility to differentiate between “M”- and “W”-shape is to apply an internal switch factor by subtracting a vibrational parameter of the metaposition (B) from the corresponding one for the ortho-position (A) of the same substance (denoted internal). By applying eqs 1-3 to fit the DFT calculated target data shown in the right part of Figure 2, a parameter set Pi (P1, P2, P3) for every benzene derivative j (1-18) is generated, thus leading to the parameter set Pi,j. Subsequently, the three parameter sets P1,j, P2,j, and P3,j were expressed via relations to spectroscopic data, which were chosen to be as simple as possible; that is, the parameter sets Pi,j were fitted by applying a linear relation depending on one or on two spectroscopic variables, as shown in eq 4 where ai, bi, and ci are parameters evaluated within the fitting procedure and Xi,j,1 and Xi,j,2 are the
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Figure 2. Left: electron densities in the individual BCPs of the CH bonds of the phenyl group. Right: electron densities averaged for the corresponding ortho and meta positions.
Figure 3. Progression of F-, ∇2F-, and ε-values in the ph-CH BCPs of benzoyl chloride (circle) and aniline (triangle) derived by means of three example fitting functions (green f1, blue f2, red f3).
spectroscopic data (1H NMR shifts, CD vibrational wavenumber values, and IR and Raman intensities). Exemplarily, a typical fit would consist of fitting the parameter set P1,j by using vibrational wavenumber values (X1,j,1 ) WNj) together with the corresponding IR intensities (X1,j,2 ) IRj) of every benzene derivative j.
Pi,j(Xi,j,1, Xi,j,2) ) ai + bi · Xi,j,1(+ci · Xi,j,2)
(4)
By doing so the major goal is to achieve the highest possible prediction performance of the functions f1, f2, or f3, depending on the spectroscopic variables Xi,j. Therefore, all permutations of parameter functions Pi,j(Xi,j,1,Xi,j,2) were inserted into the global
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functions f(x) to re-evaluate the corresponding global fit performance. The phenyl CH positions are addressed by setting x in the global functions f(x) to 0 (para), 1 (meta), or 2 (ortho). However, the fit performance in the end depends exclusively on the spectroscopic variables Xi,j, thus leading to f(Xi,j). After comparing all permutations of parameter functions P1,j(X1,j,1,X1,j,2), P2,j(X2,j,1,X2,j,2), and P3,j(X3,j,1,X3,j,2), the combination resulting in the lowest χ2 of the global function f(Xi,j) was identified as the best global fit function. All fits were performed by applying “Mathematica”.26 The fit performance to describe the electron density target data depending on the spectroscopic variables of the functions f1, f2, or f3 is quantified by applying the sum of the squared errors χ2 (eq 5) and by the validated sum of squared errors valχ2 (eq 6), where n is the number of substances used within this study, fj are the F-, ∇2F-, or ε-values of each benzene derivative j, and ffit are the corresponding global fit function values. frefit will be explained in the following paragraph.
The resulting total errors ∆Frel are calculated according to eq 8, where ex-χ2 denotes the maximum error, resulting after including the experimental errors and comparing the resulting fj-target values with the original function f. In detail, for each spectroscopic parameter two new data sets were built up, one accounting for experimental errors decreasing the input data and the other one for errors increasing the input data. For each parameter Pi, the data set leading to the maximum error was chosen. Subsequently, the resulting maximum errors were squared and summed, thus yielding the error arising from measurement inaccuracies ex-χ2. This error estimation is rather pessimistic, since the experimental errors assumed in this work are rather large and can be easily minimized using sophisticated experimental setups as already discussed in our previous study.14,30
SDrel )
∆Frel )
val-χ2 n-1
(7)
ex-χ2 n-1
(8)
n
χ2 )
∑ (fj - ffit)2
(5)
j)1
Results
n
val-χ2 )
∑ (fj - frefit)2
(6)
j)1
val-χ2 was calculated to evaluate the fit prediction performance for new substances which were not included in the input data. Therefore, every substance is treated as an unknown one once. The global function was refitted by using a fixed composition of Pi,j(Xi,j,1, Xi,j,2) applying a reduced data set with one substance left out; that is, the parameters ai, bi, and ci of every linear fit were adjusted yielding a different function frefit. The spectroscopic data of the substance left out were inserted into the global refitted function predicting the corresponding electron density data. The electron density data predicted via this approach (frefit) are compared with the corresponding DFT calculated data fj according to eq 6 to calculate the val-χ2 values. This was done to include the possibility of predicting F-, ∇2F-, and ε-values of new substances. In the following we will focus on this valχ2 value as the main statistical value. Furthermore, we tested another input data set containing relative values of all spectroscopic variables. This data set consists of relative wavenumber values (difference to benzene), 1 H NMR shifts (which is already relative to benzene as reference), and IR and Raman intensities (normalized to the benzene signals). The latter were calculated according to Schro¨tter and Klo¨ckner using the respective Raman activities and assuming a temperature of 293.15 K with an excitation wavelength of 752.488 nm.27,28 The reliability of fits utilizing these relative input data sets will be evaluated by comparing the standard deviations of these fits SDrel with the uncertainty of the relative experimental measurements ∆Frel. The standard deviation SDrel was calculated from the val-χ2 values according to eq 7. Since the accuracy of experimental data depends on the applied setup, external conditions, sample properties, and so forth, common errors for each spectroscopic method were estimated: for the experimental determination of 1H NMR shifts a usual error of 0.01 ppm was assumed, while vibrational wavenumbers can be determined with an accuracy of typically 1 cm-1. For the ratios of IR29 as well as Raman bands an usual error of about 0.05 was assumed.
Prediction Performance of Global Functions. In the following the prediction performance of the global functions f1, f2, and f3 are discussed and compared to the results obtained via functions related to the individual phenyl positions presented in a recent paper of ours.14 The functions f1, f2, and f3 are calculated using either no switch factor, a switch factor related to benzene values as reference (benzene), or an internal switch factor referring to the difference in the target value between ortho and meta positions (internal). For the latter two cases vibrational spectroscopic parameters are applied (see section fit methodologies). To achieve a descriptive illustration of the fitting results the different functions f1, f2, and f3 were compared by focusing on a combination of parameter functions P1,j(X1,j,1,X1,j,2), P2,j(X2,j,1,X2,j,2), and P3,j(X3,j,1,X3,j,2), leading to the best fit performance. In Table 2 the functions f1, f2, and f3 possessing the highest performance in describing the F-values in all phenyl CH-BCPs are compared. Besides the lowest χ2 and val-χ2 values also the corresponding switch factors and the spectroscopic variables used to describe the parameters Pi,j are listed. These parameters are fitted each using one spectroscopic variable (monolinear fits). In the case of f1 the lowest val-χ2 value is obtained for the function applying the wavenumber differences between the ortho and the meta CD-bond stretching vibrations of each benzene derivative as the switch value. In contrast to this, the functions f2 and f3 let to the lowest val-χ2 values without applying any switching factor. Comparing all three functions (f1, f2, f3), the biquadratic function f3 yields the lowest val-χ2 value, while employing only the two spectroscopic variables σH(ortho) and WN(ortho). Thus, only spectroscopic values of the ortho CH bond are needed to describe the electron densities in all phenyl CH positions of any monosubstituted benzene derivative if the parameters Pi,j are fitted with only one spectroscopic parameter, that is, if monolinear fits are applied. Table 3 summarizes the fit results in case the parameters Pi,j are fitted with two spectroscopic parameters (bilinear) according to eq 4 (without brackets). Here, f1 possesses the highest prediction performance for a wavenumber switch factor in relation to benzene (WNbenzene-switch). f2 and f3 achieve the
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TABLE 2: Best Electron Density Fit Functions of f1, f2, and f3 Using Monolinear Descriptions of the Parameters Pi,ja Density: Monolinear Parameter Fit Results According to f1 switch
P1
P2
P3
statistics
var.
ref.-typ
var.
pos
var.
pos
var.
pos
χ2
val-χ2
WN IR RA WN IR RA
benzene benzene benzene internal internal internal
WN RA WN WN WN σH WN
ortho meta ortho ortho para ortho ortho
WN IR IR IR RA RA IR
ortho ortho ortho ortho ortho ortho ortho
WN WN RA RA RA IR WN
meta para ortho ortho ortho ortho meta
2.774 × 10-5 5.930 × 10-5 1.454 × 10-4 1.030 × 10-4 2.717 × 10-5 7.201 × 10-4 5.441 × 10-5
3.511 × 10-5 8.325 × 10-5 1.980 × 10-4 1.425 × 10-4 3.443 × 10-5 8.730 × 10-4 7.875 × 10-5
WN IR RA WN IR RA
benzene benzene benzene internal internal internal
σH σH σH σH σH σH σH
Density: Monolinear Parameter Fit Results According ortho WN ortho WN ortho WN ortho WN ortho IR ortho IR ortho IR ortho IR ortho IR ortho IR meta IR ortho IR ortho IR ortho IR
to f2 ortho ortho ortho ortho ortho ortho ortho
1.342 × 10-5 7.522 × 10-5 3.882 × 10-5 2.271 × 10-5 1.699 × 10-5 1.651 × 10-4 1.609 × 10-5
1.719 × 10-5 1.030 × 10-4 5.476 × 10-5 3.209 × 10-5 2.171 × 10-5 2.127 × 10-4 2.288 × 10-5
WN IR RA WN IR RA
benzene benzene benzene internal internal internal
σH σH σH σH σH σH σH
Density: Monolinear Parameter Fit Results According ortho WN ortho WN ortho WN ortho WN ortho IR ortho IR ortho IR ortho IR ortho IR ortho IR meta IR ortho IR ortho IR ortho IR
to f3 ortho ortho ortho ortho ortho ortho ortho
1.333 × 10-5 7.896 × 10-5 3.734 × 10-5 2.232 × 10-5 1.725 × 10-5 1.572 × 10-4 1.613 × 10-5
1.707 × 10-5 1.079 × 10-4 5.262 × 10-5 3.141 × 10-5 2.200 × 10-5 2.025 × 10-4 2.279 × 10-5
a var denotes the spectroscopic variable describing a particular parameter Pi, and pos denotes the corresponding position at the ph-ring the variable refers to. The ref.-typ refers to the kind of reference of the switch, that is, the given variable in ortho position with respect to the same variable of benzene in ortho-pos. (benzene) or with respect of the meta position of the same molecule (internal). (WN: wavenumber, RA: Raman activity, IR: IR intensity, σH: 1H NMR shift).
TABLE 3: Best Electron Density Fit Functions of f1, f2, and f3 Using Bilinear Descriptions of the Parameters Pi,ja Density: Bilinear Parameter Fit Results According to f1 switch var.
ref.-typ
P1 var.
P2
pos
var. H
pos
var. WN WN IR IR IR WN IR
P3
pos
var.
ortho ortho ortho ortho ortho ortho ortho
H
pos
WN WN IR IR WN WN IR
statistics
pos
var.
para para ortho ortho para ortho ortho
H
σ IR σH RA IR IR RA
pos
χ2
val-χ2
para meta meta ortho meta ortho ortho
-6
4.888 × 10 4.051 × 10-5 1.031 × 10-4 9.334 × 10-5 1.946 × 10-5 6.463 × 10-4 4.350 × 10-5
1.452 × 10-4 6.857 × 10-5 2.197 × 10-4 1.498 × 10-4 1.476 × 10-4 8.941 × 10-4 7.552 × 10-5
WN IR RA WN IR RA
benzene benzene benzene internal internal internal
WN WN WN WN σH WN WN
ortho ortho ortho ortho meta ortho ortho
σ RA RA WN σH σH WN
ortho para para meta para ortho meta
WN IR RA WN IR RA
benzene benzene benzene internal internal internal
WN WN WN RA IR RA IR
meta para meta ortho para para ortho
σH IR IR σH σH σH σH
Density: Bilinear Parameter Fit Results ortho IR meta σH meta para WN ortho RA para meta WN meta IR ortho para IR ortho RA meta ortho WN ortho RA para meta WN meta IR ortho ortho IR ortho RA meta
According to f2 WN ortho WN ortho IR ortho IR ortho WN ortho IR ortho IR ortho
σH RA σH RA RA RA RA
para para meta ortho para ortho ortho
3.104 × 10-6 6.181 × 10-5 2.774 × 10-5 1.804 × 10-5 7.372 × 10-6 1.292 × 10-4 9.605 × 10-6
9.509 × 10-5 1.157 × 10-4 9.456 × 10-5 7.174 × 10-5 5.884 × 10-5 3.047 × 10-4 3.818 × 10-5
WN IR RA WN IR RA
benzene benzene benzene internal internal internal
WN WN WN IR IR RA IR
meta para meta ortho para meta ortho
RA IR IR σH σH σH σH
Density: Bilinear Parameter Fit Results para IR meta σH meta para WN ortho RA para meta IR ortho IR meta para IR ortho RA meta ortho WN ortho RA para meta IR ortho RA meta ortho IR ortho RA meta
According to f3 WN ortho WN ortho IR ortho IR ortho WN ortho IR ortho IR ortho
σH RA IR RA RA RA RA
para para para meta para ortho ortho
3.431 × 10-6 6.510 × 10-5 2.812 × 10-5 1.832 × 10-5 7.110 × 10-6 1.219 × 10-4 9.281 × 10-6
8.881 × 10-5 1.216 × 10-4 5.924 × 10-5 8.499 × 10-5 6.307 × 10-5 2.798 × 10-4 3.856 × 10-5
σ RA σH RA σH RA RA
ortho para meta ortho para ortho ortho
var.
a var denotes the spectroscopic variable describing a particular parameter Pi, and pos denotes the corresponding position at the ph-ring the variable refers to. The ref.-typ refers to the kind of reference of the switch, that is, the given variable in ortho position with respect to the same variable of benzene in ortho-pos. (benzene) or with respect of the meta position of the same molecule (internal). (WN: wavenumber, RA: Raman activity, IR: IR intensity, σH: 1H NMR shift).
highest prediction performance for an internal Raman activity (RAint) switch. The bilinear fit functions reveal that f2 has the highest prediction accuracy of all bilinear fits, when using RA(ortho), RA(meta), IR(ortho), and σH(ortho) to describe the Pi,j (P1,j(IRortho,σH,ortho), P2,j(IRortho,RAmeta), P3,j(IRortho,RAortho)). Because of the higher number of degrees of freedom, these
bilinear fits result in an improved fit performance (lower χ2 values) for each kind of global function as compared to the monolinear fits (compare Tables 2 and 3). In contrast to this, the majority of functions exhibit higher prediction performances (lower val-χ2 values) for monolinear fits as compared to bilinear fits. Consequently, the bilinear fit functions cannot improve the
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TABLE 4: Parameters of the Best Global Density Fit Function (Absolute) and Best Global Fit Function Using a Relative Data Set (RelatiWe) As Described Earlier Parameters of Best Fits Using f1 absolute
mono
switch
WN_int.
WN A -1
P1 P2 P3 relatiVe
mono
switch
WN_int. P1 P2 P3
absolute switch
IR
value
pos
σH
RA
value
pos
value
pos
value
pos
1.23 × 10 1.46 × 10-4 3.35 × 10-4
6.66 × 10 -
-5
para -
-
-
-4.44 × 10-6 -1.00 × 10-5
ortho ortho
-
-
2.76 × 10-1 1.46 × 10-4 3.32 × 10-4
6.66 × 10-5 -
para -
-
-
-2.24 × 10-4 -5.01 × 10-4
ortho ortho
-
-
3.59 × 10-1 -6.44 × 10-3 2.94 × 10-2
-3.27 × 10-5 2.55 × 10-6 -1.27 × 10-5
ortho ortho para
-3.69 × 10-5
meta
-1.38 × 10-4 1.11 × 10-5 -
para para -
-
-
2.84 × 10-1 -6.03 × 10-4 3.72 × 10-4
-3.32 × 10-5 2.59 × 10-6 -1.27 × 10-5
ortho ortho para
-2.50 × 10-4
meta
-6.94 × 10-3 5.57 × 10-4 -
para para -
-
-
χ2
val-χ2 -5
2.72 × 10
3.44 × 10-5
χ2 2.59 × 10-5 SD 1.39 × 10-3
val-χ2 3.27 × 10-5 ∆F 3.38 × 10-1
χ2 4.05 × 10-5
val-χ2 6.86 × 10-5
χ2 4.14 × 10-5
val-χ2 8.70 × 10-5
χ2
val-χ2
bi WN_benz. P1 P2 P3
relatiVe
bi
switch
WN_benz. P1 P2 P3
Parameters of Best Fits Using f2 absolute
mono
switch
P1 P2 P3
relatiVe
WN A
value
IR pos
value
σH
RA pos
value
-1
2.76 × 10 1.70 × 10-2 -4.94 × 10-3
-7.43 × 10-6 -4.94 × 10-3
ortho ortho
-
-
-
2.76 × 10-1 -6.94 × 10-6 -4.57 × 10-6
-7.42 × 10-6 2.15 × 10-6
ortho ortho
-
-
2.77 × 10-1 2.54 × 10-5 3.81 × 10-6
-
-
-8.68 × 10-5 3.58 × 10-6 -6.23 × 10-7
2.77 × 10-1 1.19 × 10-3 1.96 × 10-4
-
-
-5.87 × 10-4 1.29 × 10-3 -2.17 × 10-4
pos
value
pos
-
1.06 × 10 -
-3
ortho -
-
-
1.06 × 10-3 -
ortho -
ortho ortho ortho
-6.07 × 10-7 -5.93 × 10-8
meta ortho
7.84 × 10-4 -
ortho -
ortho ortho ortho
-1.49 × 10-3 -1.55 × 10-4
meta ortho
7.84 × 10-4 -
ortho -
-5
1.34 × 10
1.72 × 10-5
χ2 1.34 × 10-5 SD 1.01 × 10-3
val-χ2 1.72 × 10-5 ∆F 2.32 × 10-4
χ2 9.60 × 10-6
val-χ2 3.82 × 10-5
χ2 1.02 × 10-5
val-χ2 3.93 × 10-5
mono
switch
P1 P2 P3
absolute
bi
switch
RA_int. P1 P2 P3
relatiVe
bi
switch
RA_int. P1 P2 P3
Parameters of Best Fits Using f3 absolute
mono
switch
P1 P2 P3
relatiVe
WN
IR
σH
RA
A
value
pos
value
pos
value
pos
value
pos
χ2
val-χ2
2.76 × 10-1 2.78 × 10-2 -1.92 × 10-2
-1.21 × 10-5 8.38 × 10-6
ortho ortho
-
-
-
-
1.13 × 10-3 -
ortho -
1.33 × 10-5
1.71 × 10-5
2.76 × 10-1 3.01 × 10-6 -1.78 × 10-5
-1.21 × 10-5 8.37 × 10-6
ortho ortho
-
-
-
-
1.13 × 10-3 -
ortho -
χ2 1.33 × 10-5 SD 1.00 × 10-3
val-χ2 1.71 × 10-5 ∆F 2.86 × 10-4
2.77 × 10-1 2.10 × 10-5 1.48 × 10-5
-
-
-9.26 × 10-5 5.22 × 10-6 -2.43 × 10-6
ortho ortho ortho
-6.18 × 10-7 -2.31 × 10-7
meta ortho
8.38 × 10-4 -
ortho -
χ2 9.28 × 10-6
val-χ2 3.86 × 10-5
2.77 × 10-1 9.57 × 10-4 7.62 × 10-4
-
-
-6.25 × 10-4 1.86 × 10-3 -8.43 × 10-4
ortho ortho ortho
-1.52 × 10-3 -6.02 × 10-4
meta ortho
8.38 × 10-4 -
ortho -
χ2 9.86 × 10-6
val-χ2 3.97 × 10-5
mono
switch
P1 P2 P3
absolute
bi
switch
RA_int. P1 P2 P3
relatiVe
bi
switch
Ra_int. P1 P2 P3
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TABLE 5: Best Fit Results for the Respective Laplacian of the Electron Density Using f2 and Relative Values Laplacian Parameters for f2 absolute
mono
switch
-
WN
IR
value
P1 P2 P3 relatiVe
A
pos
σH
RA
value
pos
value
-1
2.33 × 10 5.12 × 10-2 -1.11 × 10-2
-2.23 × 10-5 4.83 × 10-6
ortho ortho
-
-
-
2.33 × 10-1 1.28 × 10-4 -1.96 × 10-5
-2.23 × 10-5 4.83 × 10-6
ortho ortho
-
-
-3.30 × 10-1 5.76 × 10-3 -1.02 × 10-3
2.45 × 10-4 -2.04 × 10-6 3.93 × 10-7
meta ortho ortho
-
2.32 × 10-1 1.07 × 10-3 -1.21 × 10-4
2.45 × 10-4 -2.12 × 10-6 4.02 × 10-7
meta ortho ortho
-
pos
value
χ2
pos
-
2.55 × 10 -
-3
ortho -
-
-
2.55 × 10-3 -
ortho -
-
-2.04 × 10-5 2.29 × 10-6
para para
1.17 × 10-3 -
ortho -
-
-1.02 × 10-3 1.14 × 10-4
para para
1.17 × 10-3 -
ortho -
val-χ2 -5
8.00 × 10-5
χ2 6.33 × 10-5 SD 2.17 × 10-3
val-χ2 8.00 × 10-5 ∆F 5.59 × 10-4
χ2 2.96 × 10-5
val-χ2 7.73 × 10-5
χ2 2.90 × 10-5
val-χ2 3.66 × 10-4
6.33 × 10
mono
switch
P1 P2 P3
absolute
bi
switch
WN_int. P1 P2 P3
relatiVe
bi
switch
WN_int. P1 P2 P3
TABLE 6: Best Ellipticity Fit Results Including Relative Values for All No-Switch Functions Ellipticity Best Mono Fit Results via f1 absolute
mono
switch
P1 P2 P3
relatiVe
WN A
value
IR pos
value
σH
RA pos
value
-2
1.64 × 10 -4.36 × 10-4 -2.45 × 10-3
-
-
-
-
-
1.64 × 10-2 -4.36 × 10-4 -2.45 × 10-3
-
-
-
-
-
pos
value
pos
-
-2
2.33 × 10 -9.51 × 10-3 -3.63 × 10-2
para para para
-
2.33 × 10-2 -9.51 × 10-3 -3.63 × 10-2
χ2
val-χ2 -5
3.46 × 10-5
para para para
χ2 2.52 × 10-5 SD 1.43 × 10-3
val-χ2 3.46 × 10-5 ∆F 1.28 × 10-3
2.52 × 10
mono
switch
P1 P2 P3
Ellipticity Best Mono Fit Results via f2 absolute
mono
switch
P1 P2 P3
relatiVe
WN
IR
σH
RA
A
value
pos
value
pos
value
pos
value
pos
χ2
val-χ2
1.40 × 10-2 1.25 × 10-3 -8.10 × 10-5
-
-
-
-
-
-
-1.18 × 10-2 1.55 × 10-2 -1.20 × 10-3
para para para
2.52 × 10-5
3.46 × 10-5
1.40 × 10-2 1.25 × 10-3 -8.10 × 10-5
-
-
-
-
-
-
-1.18 × 10-2 1.55 × 10-2 -1.20 × 10-3
para para para
χ2 2.52 × 10-5 SD 1.43 × 10-3
val-χ2 3.46 × 10-5 ∆F 2.08 × 10-3
mono
switch
P1 P2 P3
Ellipticity Best Mono Fit Results via f3 absolute
mono
switch
P1 P2 P3
relatiVe
WN
IR
σH
RA
A
value
pos
value
pos
value
pos
value
pos
χ2
val-χ2
1.39 × 10-2 1.43 × 10-3 -3.15 × 10-4
-
-
-
-
-
-
-1.30 × 10-2 1.81 × 10-2 -4.68 × 10-3
para para para
2.52 × 10-5
3.46 × 10-5
1.39 × 10-2 1.43 × 10-3 -3.15 × 10-4
-
-
-
-
-
-
-1.30 × 10-2 1.81 × 10-2 -4.68 × 10-3
para para para
χ2 2.52 × 10-5 SD 1.43 × 10-3
val-χ2 3.46 × 10-5 ∆F 2.40 × 10-3
mono
switch
P1 P2 P3
performance obtained for all three chosen monolinear functions in predicting unknown electron density values for every phenyl CH-BCP. As shown in Tables 2 and 3, the best function to predict the electron density values in all positions of a monosubstituted benzene molecule, allowing free rotation of the substituent around the phenyl-substituent bond, is the monolinear biqua-
dratic function f3 with no switch factor. The actual parameters for this function f3 as well as the parameters of the functions f1 and f2, for the mono- and bilinear best fit functions, are summarized in Table 4. To facilitate the application of experimentally determined spectroscopic data with an increased accuracy due to the utilization of relative values, the fit procedures were also
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J. Phys. Chem. A, Vol. 114, No. 37, 2010
performed using relative data sets introduced in a recently published study of ours.14 The parameters obtained by applying these relative values are also included in Table 4. It can be easily seen that relative and absolute input data sets generally do not differ significantly in their main statistical values, χ2 and val-χ2 (except for the bilinear fit of f1). However, these results are very promising with respect to the application of accurate experimental spectroscopic data. Furthermore, it can be seen that the monolinear functions tend to preserve the val-χ2 when changing from the absolute to the relative data set values better than the bilinear functions which all lead to less accurate predictions. Since the monolinear functions turned out to have a higher prediction performance than the bilinear functions, only the standard deviation (SDrel, eq 7) of the monolinear fits are compared to experimental errors ∆Frel, calculated according to eq 8. For the prediction of F-values, the standard deviation of the f1-fit is significantly lower than ∆Frel. In contrast to this, the SDrel for the f2- and f3-fits, which are using no switch factor, is roughly one magnitude higher than ∆Frel. Consequently, the latter two functions can be reliably applied when using real experimental spectroscopic data. The same procedure as just described has been also applied to determine the best fit functions for the respective ∇2F- and ε-values. At this point, we focus the discussion on the best functions obtained. The detailed data for the ∇2F- and ε-values comparable to those for the electron density values as summarized in Tables 2 and 3 can be found in the Supporting Information. The bilinear function f2 using an internal wavenumber switch and the four variables, WN (ortho and meta), RA (para), and σH (ortho), achieve the best fit performance to predict the ∇2Fvalues. The corresponding set of parameter values is listed in Table 5. However, the application of the relative input data set within this function leads to a loss of almost 50% in its prediction ability. The best performance for applying the relative data set is achieved by the monolinear function f2 using σH(ortho) to describe P1 and WN(ortho) to describe P2 and P3. Hence, this monolinear function f2(WNortho,σH,ortho) should be applied when working with experimental data, because it has a comparable error as the bilinear function but uses less variables. Furthermore, the deviations expected to occur due to experimental measurement errors (∆Frel) are about one magnitude lower than the standard deviation for the prediction performance for f2(WNortho,σH,ortho). Functions possessing no switch factors are by far best suited to describe and to predict ε-data within the phenyl CH-BCPs. These fit functions lead to χ2 and val-χ2 values about one magnitude lower than the competing functions discussed in detail for the corresponding F-values in the former sections. Surprisingly, all fit functions (f1, f2, f3) with no switching factor end up with equivalent χ2 and val-χ2 values and depend all on only one particular spectroscopic variable, namely, σH(para) (see Table 6). For all three functions displayed in Table 6, the standard deviation for the prediction performance SDrel is of equivalent magnitude as the deviations arising upon experimental measurement errors ∆Frel. The simple form of the function describing the ellipticity (ε) in all phenyl CH-bonds and the arbitrary assignment of the phpositions (x) to the values (2, ( 1, and 0 allow for a graphical illustration of the fit function (see Figure 4). The function illustrated by the orange surface is determined by function f3 using monolinear descriptions of the parameters Pi according to eq 9 (related to absolute input data, parameters listed in Table 6).
Presselt et al.
Figure 4. Illustration of the function possessing the best ellipticityprediction performance, specified in eq 9, by the orange surface. The blue dots are DFT calculated ε-values in ph-CH bonds (ortho, meta, and para are assigned to (2-, ( 1-, and 0 values at the “ph-pos” axis) of various benzene derivatives (assigned via the σH(para) values (gray mesh-lines)).
f3 ) 0.0139 - 0.0130σH(para) + (0.0014 + 0.0181σH(para))x2 + (-0.0003 - 0.0047σH(para))x4 (9) Equation 9 highlights that for a particular substance the corresponding ε(ph-CH BCP) value (ellipticity), regardless of the position at the phenyl ring, is only related to the σH-value in the para ph-position. This correlation is illustrated by the gray mesh-lines in Figure 4. Comparison with Local Functions. It is of special interest to calculate val-χ2 values for the single positions ortho, meta, and para, to achieve a basis for a comparison of the results obtained in the present study with the ones of the previous study of ours.14 The local val-χ2 values were calculated by inserting the position value into the global function frefit and summing up over all equivalent positions of all substances j. The total valχ2 values are calculated by summation of the local values accounting for the 2-fold existence of meta and ortho positions. The obtained errors are shown in Table 7 for the best predicting functions and are compared with the ones of the best local functions of our recent study using an averaged data input.14 Furthermore, the numbers of required spectroscopic input variables are compared. From Table 7 it can be deduced that in general the global functions presented in this work lead to a worse prediction accuracy (higher val-χ2 values) than the local ones presented in our recent study.14 The prediction of F- or the respective ∇2Fvalues do not possess a higher prediction accuracy as compared to the previous work, neither regarding all nor single positions. However, the approach presented here needs a significantly lower number of variables to describe the properties of the phenyl CH-BCPs in all positions, while the fit performance of the functions predicting the F- and the ∇2F-values in the phenyl CH-BCPs is still reliable. Thus, the functions presented here can be used to obtain an overview over the situation present in a molecule. If the F- or ∇2F-values in all single positions are predicted individually, roughly four times more variables are needed (global: 2, individual: 1-2 for each of the five ph-positions) to increase the accuracy by about a factor of 5 (see Table 7).
Bond Properties of Phenyl-CH Bonds
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TABLE 7: Comparison of val-χ2 Values between Previous Studies14 and the Functions Possessing the Highest Prediction Performance Derived in the Current Study with Respect to the Single Positions as well as the Numbers of Variables Used To Describe the Relationsa F-values current ortho meta para total ∑ var.
∇2F-values previous
-6
4.63 × 10 2.86 × 10-6 2.08 × 10-6 1.71 × 10-5 2
-6
1.60 × 10 3.80 × 10-8 2.90 × 10-7 3.58 × 10-6 8
current
b
ε-values previous
-5
2.09 × 10 1.51 × 10-5 8.17 × 10-6 8.00 × 10-5 2
-6
6.19 × 10 2.06 × 10-7 1.59 × 10-6 1.44 × 10-5 9
current
previous -5
1.04 × 10 1.56 × 10-6 1.07 × 10-5 3.46 × 10-5 1
2.19 × 10-5 1.32 × 10-6 7.47 × 10-6 5.38 × 10-5 8
a
The val-χ2 values of the current study are referring to mono-linear fitted parameters Pi,j. The summation resulting in the total val-χ2 values accounts for the twofold presence of ortho and meta positions. b To be consistent, values referring to the monolinear function f2 are listed instead of the ones referring to the bilinear function, which possesses a higher prediction performance in this particular case.
In contrast to this, the global functions describing the ε-values exhibit a better performance than the local ones (see Table 7). A closer inspection of the single positions reveals that the global ellipticity-function only predicts the ortho position better than the local fits used in our previous study.14 Hence, if ε-values of a phenyl CH bond in a single position are of particular interest, the global function should be applied for the ortho position, while the local functions of the previous work achieve a slightly higher prediction performance for meta and para positions. However, for predicting ε-values in phenyl CH-BCPs the herein introduced global functions are an excellent choice, since the one possessing the highest performance exceeds the total prediction performance of the local functions and is furthermore only dependent on the 1H NMR shift (σH) of the hydrogen atom in the para-ph position. Conclusion In this study we presented a statistical approach to predict the electron density properties of CH-bonds based on DFT calculated spectroscopic data. This approach is of special interest to determine structure-reactivity relations in solution or the gas phase, since the direct experimental determination of the electron density distribution is only possible in single crystals. As target quantities, we have chosen the electron density F, the respective Laplacian ∇2F, and the ellipticity ε within CH BCPs of a series of monosubstituted benzene derivatives. These target quantities were correlated with spectroscopic data, namely, wavenumber shifts, Raman activities, IR intensities of localized CD-stretching vibrations, and 1H NMR shifts (benzene as reference) via one fit-function describing all positions of the ring-CH BCPs at once while assuming a free rotation of the substituent around the ph-substituent bond. It could be shown that the global fit functions are capable of describing the electron density, the respective Laplacians, and the ellipticity in all phenyl CH-BCPs with less variables as compared to our previous work where we used local fit functions.14 We could show that the global functions are a good generalization of the former work due to the fact that prediction performances of the same magnitude are realized, but involving far less spectroscopic input variables as compared to the local functions. In conclusion, these global functions offer an further possibility for the determination of target quantities of unknown monosubstituted benzene substances demanding lower experimental effort as compared to our previously introduced method.14 The presented results prove that particular electron density features even of noncrystalline substances can be obtained via spectroscopy. Furthermore, changes in the electron density distribution during environmental changes or chemical reactions
might be monitored via the spectroscopic techniques Raman, IR, or NMR, yielding the input quantities needed for the correlation functions. To extend this approach beyond utilizing naturally localized vibrations, partitioning schemes might be implemented to express the properties of normal modes in terms of internal coordinates. However, the applicability beyond particular classes of substances has to be investigated. Nonetheless, we have shown here that CD-stretching vibrations in combination with 1 H NMR signals are marker for crucial CH-bond properties. Moreover, it is possible to reliably describe the properties of all phenyl-CH BCPs using one function, thus enabling predictions in phenyl positions which were not individually experimentally investigated. Acknowledgment. The authors gratefully acknowledge the support by the Deutsche Forschungsgemeinschaft (Schwerpunktprogramm 1178) and the Friedrich Schiller University Jena. We also thank Dr. Burkhard Jahn and Dr. Wilhelm Eger for their kind computational efforts. Supporting Information Available: This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Flaig, R.; Koritsanszky, T.; Soyka, R.; Ha¨ming, L.; Luger, P. Angew. Chem., Int. Ed. 2001, 40, 355. (2) Grabowsky, S.; Pfeuffer, T.; Checinska, L.; Weber, M.; Morgenroth, W.; Luger, P.; Schirmeister, T. Eur. J. Org. Chem. 2007, 2759. (3) Luger, P. Org. Biomol. Chem. 2007, 5, 2529. (4) Messerschmidt, M.; Scheins, S.; Grubert, L.; Pa¨tzel, M.; Szeimies, G.; Paulmann, C.; Luger, P. Angew. Chem., Int. Ed. 2005, 44, 3925. (5) Messerschmidt, M.; Wagner, A.; Wong, M. W.; Luger, P. J. Am. Chem. Soc. 2002, 124, 732. (6) Henn, J.; Leusser, D.; Stalke, D. J. Comput. Chem. 2007, 28, 2317. (7) Schirmeister, T.; Breuning, A.; Murso, A.; Stalke, D.; Mladenovic, M.; Engels, B.; Szeghalmi, A. V.; Schmitt, M.; Kiefer, W.; Popp, J. J. Phys. Chem. A 2004, 108, 11398. (8) Hebben, N.; Himmel, H.-J.; Eickerling, G.; Herrmann, C.; Reiher, M.; Herz, V.; Presnitz, M.; Scherer, W. Chem.sEur. J. 2007, 13, 10078. (9) McGrady, G. S.; Sirsch, P.; Chatterton, N. P.; Ostermann, A.; Gatti, C.; Altmannshofer, S.; Herz, V.; Eickerling, G.; Scherer, W. arXiV, e-Print ArchiVe, Physics; Cornell University: Ithaca, NY, 2008; http://arxiv.org/. (10) O’Brien, S. E.; Popelier, P. L. A. Can. J. Chem. 1999, 77, 28. (11) Popelier, P. L. A. J. Phys. Chem. A 1999, 103, 2883. (12) Mladenovic, M.; Arnone, M.; Fink, R. F.; Engels, B. J. Phys. Chem. B 2009, 113, 5072. (13) Chaudry, U. A.; Popelier, P. L. A. J. Phys. Chem. A 2003, 107, 4578. (14) Presselt, M.; Schnedermann, C.; Schmitt, M.; Popp, J. J. Phys. Chem. A 2009, 113, 3210. (15) Moller, C.; Plesset, M. S. Phys. ReV. 1934, 46, 618. (16) Bader, R. F. W. Atoms in Molecules: A Quantum Theory; Oxford University Press: New York, 1990. (17) Bader, R. F. W.; Chang, C. J. Phys. Chem. 1989, 93, 2946.
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