QTAIM Investigation of the Electronic Structure and Large Raman

ARTICLE pubs.acs.org/JPCA

QTAIM Investigation of the Electronic Structure and Large Raman Scattering Intensity of Bicyclo-[1.1.1]-pentane Richard Dawes,*,†,|| Jason R. Dwyer,*,‡,|| Weixing Qu,§,^ and Kathleen M. Gough*,§ †

Missouri University of Science and Technology, Rolla, Missouri 65409, United States University of Rhode Island, Kingston, Rhode Island 02881, United States § University of Manitoba, Winnipeg, Manitoba R3T 2N2 ‡

ABSTRACT: Our previous studies of the variation of Raman scattering intensities in saturated hydrocarbons have identified a number of structural descriptors that correlate with calculated polarizability derivatives for particular bond displacements: ring strain, steric hindrance, and alignment and location of a C
H group within the molecular framework (e.g., endo-/exo-, axial/equatorial, in-plane/out-of-plane). The bridgehead C
H bond intensities in bicyclo-[1.1.1]-pentane appear to be extraordinarily large, given its size and structure. Molecular polarizability and derivatives are analyzed here for bicyclo-[1.1.1]-pentane and propane, with HF, MP2, CCSD, B3LYP, M06, and M062X levels of theory and the Dunning AVTZ basis set. Analyses of calculated electronic charge densities were performed with two implementations of QTAIM, including an origin-dependent method and an implementation with origin-independent atomic moments. Numerically accurate atomic partitioning of mean molecular polarizabilities is achievable with either; however, accurate partitioning of polarizability derivatives places stringent requirements on the numerical integration, more so for this highly strained bicyclic structure. QTAIM reveals that most of the polarizability (∼90%) can be attributed to charge transfer between atomic basins. Calculated Raman intensities are in accord with our experimental data, notably in the prediction of large trace scattering intensities for stretching of the bridgehead CH in bicyclo-[1.1.1]-pentane and for the methyl in-plane C
H in propane. Density difference plots illustrate the effects of bond displacements on the electron densities and the resultant changes in polarizability. Stretching of the bridgehead C
H bond in bicyclo-[1.1.1]-pentane produces electron density changes that are similar to those encountered upon stretching the methyl in-plane C
H of propane.

1. INTRODUCTION The analysis of Raman trace scattering intensities offers powerful insights into chemical bonding that are accessible experimentally through Raman spectroscopy and theoretically through electronic structure calculations and the quantum theory of atoms in molecules (QTAIM).1 Raman trace scattering intensities arise from derivatives of the mean molecular polarizability with respect to the molecular vibration. We have used C
H vibrations in several alkane series to probe the factors that determine the changes that vibrations of individual bonds induce in the molecular polarizability. The redistributions of electronic density realized in the molecular polarizability derivative are correlated to familiar molecular structural features, such as ring strain, steric hindrance, and alignment of a C
H group with the molecular framework (e.g., endo-/exo-, axial/equatorial, in-plane/out-of-plane). These local structural descriptors are essentially a partitioning of a single, global molecular structure into components. We seek more insight into the nature of the interaction between these components of molecular structure change and electron density redistribution. Within the assumptions of the Placzek polarizability theory,2 the trace, or isotropic, Raman scattering is due to the change in r 2011 American Chemical Society

the mean molecular polarizability, ∂R/∂q, where ! ∂αyy ∂R 1 ∂αxx ∂αzz ¼ þ þ ∂q 3 ∂q ∂q ∂q

ð1Þ

αxx, αyy, and αzz are the diagonal elements of the molecular polarizability tensor. These experiments thus reveal the overall changes in the polarizability of the electron density with respect to atomic displacements along some vibrational mode, q. QTAIM analysis can be used to connect the changes in the molecular polarizability to the underlying changes in the electron density, resolved into changes at each atom of the molecule. This atomby-atom decomposition allows exploration of the underlying mechanisms connecting familiar local structural features to the molecular polarizability derivative. In our initial investigations, vibration of the bridgehead C
H in bicyclo-[1.1.1]-pentane (bcp) was predicted to generate an exceptionally high molecular polarizability derivative, the highest Special Issue: A: Richard F. W. Bader Festschrift Received: June 16, 2011 Revised: September 6, 2011 Published: September 26, 2011 13149

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for the set of molecules examined.3,4 Although bcp is a saturated alkane, it nevertheless possesses a rather unusual molecular geometry, including the shortest known nonbonded bridgehead C
C distance.5 Effects due to the unusual electronic structure of bcp have been noted in NMR studies where strong cross-cage electronic interactions produce large couplings.6
8 Given the unusual electronic structure, we followed our initial HartreeFock survey calculations with a combination of experiment and higher-level theory. Measurement of the absolute Raman trace scattering intensity confirmed the large derivative and, in addition, corrected the spectral assignment.5 The supporting theoretical analysis required not only the highly accurate CCSD(T) method and the aug-cc-pVTZ (AVTZ) basis set, but also a quartic force-field analysis to treat anharmonic resonances and ultimately produced near-quantitative agreement with the experimental result. We now seek to use QTAIM to probe the atomic-level origins of this exceptionally large molecular polarizability derivative and to explore the robustness of the existing structural parametrization that arose from earlier QTAIM analysis of much simpler molecules.9
11 To explore the role of electronic effects, we have carried out calculations and QTAIM analyses of α and ∂R/∂q at several levels of theory: HF, MP2, CCSD, CCSD(T), B3LYP, M06, and M062X, all with the AVTZ basis. We present the results and discuss the origins of this unusual molecule’s large polarizability derivative.

axis. The length xj is the Ωth atom’s position projected onto the jth axis. The rows and columns of the atomic polarizability tensor correspond to the direction of the applied electric field and the spatial response in the charge density of the atom, respectively. The atomic dipole contribution reflects the distortion of electron density within a given atomic basin, and the charge transfer contribution reflects the transfer of electron density between atomic basins, due to an external electric field. In the AIMAll implementation of QTAIM, total molecular polarizability is instead calculated from a sum of field-induced changes in origin-independent atomic dipoles, eliminating the formal need for separate charge transfer and origin-dependent dipole terms. The changes in atomic basin charges can still be examined to assess the importance of charge transfer between atomic basins to molecular electronic redistributions arising from structural or field-induced perturbations. In either case, once the molecular polarizability tensor elements have been obtained, we calculate the derivative tensor with respect to a selected bond vibration as the numerical difference between the polarizability tensors of the equilibrium (optimized) and the nonequilibrium (stretched or contracted) geometry of the molecule. These are found by summing over the atomic contributions,

2. THEORETICAL METHODS Wave functions were obtained from electronic structure calculations using the Gaussian 2009 suite of software.12 Calculations were performed in the standard orientation, which for bcp and propane diagonalizes the moment of inertia and polarizability tensors. The principal molecular symmetry axis was aligned with the z-axis. Strict SCF convergence tolerances (10
11) and an ultrafine integration grid were used. In this work, we have used our previous approach based on output from AIM200013,14 and an alternate method available in the AIMAll implementation of QTAIM.15 Both methods have been outlined in detail elsewhere.1,16 Here, we present a short outline of the analytic QTAIM theory and the numerical steps required to carry out the calculations. QTAIM partitions molecular electron density into rigorously defined atomic basins and this permits the electron density redistribution in response to a structural perturbation or to an external electric field to be partitioned into atomic contributions. The molecular polarizability is simply the sum of the polarizabilities of the atoms in the molecule,

where in AIM2000

αij ¼

Na

∑ αij, Ω Ω¼1

ð2Þ

where Na is the number of atoms in the molecule and i,j run through the axes. With the AIM2000 implementation, the atomic polarizability is calculated as the sum of an atomic charge transfer contribution (αCT) and an atomic dipole contribution (αAD): αij, Ω ¼ fαCT gij, Ω þ fαAD gij, Ω   μj, Ω
μ0, Ω Ni, Ω
N0, Ω xj ¼ þ Ei Ei

ð3Þ

where Ni,Ω (or N0,Ω) and μi.Ω (or μ0.Ω) are the Ωth atom’s electron population and induced dipole component along the jth axis without (0), or with (i), the electric field applied along the ith

∂αij ¼ ∂r

Na

∑ Ω¼1

∂αij, Ω ∂r

ΔfαAD gij, Ω ∂αij, Ω ΔfαCT gij, Ω ≈ þ ∂r nΔr nΔr

ð4Þ

ð5Þ

gives the Ωth atom’s polarizability derivative tensor; n is the number of symmetrically equivalent stretched or contracted bonds in the calculation. To connect the computational analysis to a readily measured experimental parameter (the intensity of Raman trace scattering spectra), we calculate the derivative of the mean molecular polarizability, R, with respect to the totally symmetric stretch (or contraction) of the bond (here, C
H) of the molecule, ∂R/∂r, according to eq 1, where r replaces the vibrational mode q. The following steps are carried out: Electronic structure calculations12 a Calculation of the equilibrium molecular polarizability, R(req), at the B3LYP/AVTZ optimized structure. b Displacements of all symmetry-equivalent bonds of a chosen functional group (e.g., bridgehead C
H bonds in bcp) and calculation of the molecular polarizability at these displaced geometries, R(req ( Δr); Δr = 0.01 Å. c Approximation of the molecular polarizability derivative with the second-order accurate central difference formula, ∂R/∂r ≈ (R(req + Δr)
R(req
Δr))/n2Δr, where n is the number of symmetrically equivalent bonds. d Comparison with experimental Raman data to assess the quality of the theoretical model. e Compute wave functions for QTAIM analysis at equilibrium and displaced geometries with zero-field and with fields applied in the x, y, and z directions (field strengths 0.001 a.u.). The numerical integration error in QTAIM ( ∼1% for polarizability) is enough to cause problems with interpretation of derivatives. For the QTAIM calculations, a larger value (0.04 Å) for Δr was chosen in an effort to ensure that the small differences in polarizability would not be 13150

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overwhelmed by numerical integration limitations. This step has been shown to be acceptable.1 QTAIM calculations (AIM 2000, AIMAll) a Integration of electron density over atomic basins using eq 2 to numerically recover the atomic contributions to polarizability. For AIM2000, the relative and absolute accuracy parameters were set to 10
6, and maximum path length to 3  106. Note: the AIM2000 numerical integration algorithm adjusts some of these parameters during the integration. For AIMAll, the promega1 algorithm was used with a dense integration grid (boaq=superhigh, briaq=4). b Comparison with values from electronic structure calculation to assess degree of recoverability by the numerical algorithm. c Analysis of atomic contributions within the context of the molecular framework. Table 1. Optimized Bond Lengths in Bicyclo-[1.1.1]pentane (Å)

3. RESULTS The optimized structural parameters of bcp are displayed in Table 1 for all methods used in this work; the molecular structure, axes, and atomic numbering system are shown in Figure 1. To facilitate QTAIM-inspired comparisons between methods via difference density plots, all subsequent calculations were performed at the optimized B3LYP geometry. Since bond distances vary with method by much less than our 0.04 Å bond stretch and the second derivative of the polarizability is fairly small near equilibrium, the calculated molecular polarizability derivative is not strongly sensitive to the choice of initial structure. The equilibrium polarizability elements, mean molecular polarizability, and polarizability derivatives for bridgehead CH bond stretch are given in Table 2. There is only slight variation in equilibrium polarizability with method, and the molecular polarizability derivatives are fairly close to the experimental result. The calculated polarizability derivatives exceed the experimental value by between about 3% (CCSD, CCSD(T)) and 10% (B3LYP), whereas the

optimized bond distances (Å) a

bridgehead C
H

methylene C
H

C
C

HF MP2

1.0801 1.0881

1.0825 1.0903

1.5444 1.5509

B3LYP

1.0886

1.0901

1.5543

M06

1.0891

1.0911

1.5422

M062X

1.0876

1.0892

1.5460

method

a

Dunning’s AVTZ basis was used for all methods.

Figure 1. Molecular axes and atomic numbering for bicyclo-[1.1.1]pentane and propane.

Figure 2. Density difference plots (method-HF) for equilibrium geometry with no applied field. Isosurfaces are 0.0020 a.u. (reduced charge density = relatively more positive and increased charge density = relatively more negative, H atoms are white spheres). Electron correlation reduces density along bonds while increasing density around carbon atoms and between bridgehead carbons. The numerical value underneath each figure shows the method-HF difference at the cage critical point in a.u.

Figure 3. Density difference plots for propane from (a) MP2-B3LYP and (b) M06-B3LYP and comparable plots (c, d) for bicyclo-[1.1.1]pentane, calculated at B3LYP equilibrium geometries with no applied field. Iso surfaces are 0.0020 a.u.; color scheme as in Figure 2.

Table 2. Calculated Diagonal Elements of the Equilibrium Molecular Polarizibility Tensor, Mean Molecular Polarizability, And Polarizability Derivatives for Stretch of the Bridgehead CH in Bicyclo-[1.1.1]-pentane, for All Methods polarizability αxx (a.u.)

αyy (a.u.)

αzz (a.u.)

R (a.u.)

R (cm2 V
1)

derivative, ∂α̅ /∂r (cm V
1)

HF

55.40

55.40

51.40

54.07

8.91

1.62

MP2

58.12

58.12

54.32

56.86

9.37

1.67

CCSD

56.57

56.57

52.68

55.27

9.11

1.59

CCSD(T)

57.32

57.32

53.44

56.03

9.24

1.59

B3LYP

58.82

58.82

54.64

57.43

9.47

1.70

M06

58.23

58.09

53.98

56.77

9.36

1.60

M062X exptb

57.69

57.62

53.63

56.31

9.28

1.66 1.54

a

method

a

Dunning’s AVTZ basis was used for all methods. b Ref 5. 13151

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experimental measurement was assigned an error of no more than 5%.5 It appears that although the equilibrium density is influenced by electron correlation, the dipole response to an applied field is similar for the various methods. In fact, polarizability derivatives are found to depend much more strongly on basis set than method. For the HF method, the polarizability derivative associated with the bridgehead C
H stretch increases by 17% as the basis is increased from D95** (1.38)3 to AVTZ (1.62, these results).

The effect of electron correlation on the electron density for bcp can be seen in Figure 2, which shows the 0.0020 a.u. isosurface for the density difference between each method and that of the HF method. Figure 3 shows the density differences among the MP2, M06, and B3LYP treatments of electron correlation in propane and bcp. Electron correlation methods were anticipated to affect the electron density inside the bcp cage. Propane calculations were also performed for comparison, since bcp contains what may be thought of as three propane chains sharing common carbons at the bridgeheads. Calculation of the molecular polarizability derivative via QTAIM requires multiple numerical analyses of potentially complex vibrationally and field-perturbed interatomic surfaces and is particularly challenging for an unusual molecule such as bcp. Table 3 shows the sensitivity of this numerical calculation to the method; that is, to the capture of the detailed electron distribution within the molecule. Although the numerical recovery of polarizability is quite good, the roughly 1% errors are on the order of the differences between stretched and equilibrium structures; hence, the QTAIM-recovered polarizability derivatives are generally not accurate enough for quantitative analysis.

Table 3. QTAIM Recovery of R (a.u.) and DR/Dr from All Methods method

EQ

% recovery

str

% recovery

Δ%

∂R/∂r QTAIM

HF

53.856

99.61

54.914

100.04

0.43

2.181

MP2 CCSD

57.622 55.630

101.35 100.65

58.279 55.683

101.01 99.30


0.34
1.35

1.355 0.108

B3LYP 56.995

99.25

57.350

98.40


0.85

0.732

M06

56.179

98.97

56.708

98.50


0.46

1.090

M062X 56.750

100.78

57.068

99.88


0.90

0.655

Table 4. Origin-Independent Atomic Moments Calculated with AIMAll at MP2 Level of Theory for Equilibrium (EQ) and Stretched (Str) Bridgehead C
H Geometries, without and with Applied Electric Field QTAIM atomic dipoles (a.u.) no applied field EQ

x

y

x direction field z

z direction field

x

y

z

x

y

z

C1

0.0001


0.1739

0.0000


0.0050


0.1742


0.0003

0.0005


0.1740


0.0049

C2 C3

0.1505
0.1507

0.0873 0.0867

0.0002 0.0001

0.1414
0.1603

0.0843 0.0896


0.0004
0.0003

0.1503
0.1509

0.0876 0.0863


0.0047
0.0049

C4

0.0001

0.0000


0.2216


0.0062


0.0001


0.2220

0.0001

0.0000


0.2314

C5

0.0001

0.0000

0.2216


0.0061


0.0001

0.2205

0.0001

0.0000

0.2105

H6

0.0000

0.0000

0.1480


0.0015

0.0000

0.1481

0.0000

0.0000

0.1423

H7

0.0000

0.0000


0.1480


0.0015

0.0000


0.1481

0.0000

0.0000


0.1538

H8


0.0254

0.1621

0.0000


0.0270

0.1637

0.0000


0.0254

0.1621


0.0014

H9


0.1531


0.0591

0.0000


0.1583


0.0603

0.0000


0.1531


0.0591


0.0014

H10 H11

0.1277
0.1277


0.1031
0.1031

0.0000 0.0000

0.1246
0.1309


0.1014
0.1048

0.0000 0.0000

0.1277
0.1277


0.1031
0.1031


0.0014
0.0014

H12

0.0254

0.1621

0.0000

0.0238

0.1608

0.0000

0.0254

0.1621


0.0014

H13

0.1531


0.0591

0.0000

0.1481


0.0579

0.0000

0.1531


0.0591


0.0014

total

0.0001

0.0000

0.0004


0.0588


0.0005


0.0025

0.0000

0.0000


0.0553

C1

0.0002


0.1688

0.0003


0.0050


0.1689


0.0003

0.0002


0.1687


0.0048

C2

0.1460

0.0848

0.0005

0.1369

0.0816


0.0003

0.1459

0.0847


0.0048

C3


0.1463

0.0840

0.0003


0.1557

0.0871


0.0002


0.1462

0.0840


0.0049

C4 C5

0.0000 0.0001

0.0000
0.0001


0.2486 0.2473


0.0063
0.0061

0.0000
0.0001


0.2489 0.2480

0.0000 0.0001


0.0001 0.0000


0.2589 0.2371

H6

0.0000

0.0000

0.1342


0.0015

0.0000

0.1343

0.0000

0.0000

0.1281

H7

0.0000

0.0000


0.1342


0.0015

0.0000


0.1343

0.0000

0.0000


0.1404

H8


0.0253

0.1613

0.0000


0.0269

0.1628

0.0000


0.0253

H9


0.1523


0.0587

0.0000


0.1575


0.0600

0.0000


0.152

Str

0.1613
0.059


0.0014
0.001

H10

0.1270


0.1025

0.0000

0.1240


0.1009

0.0000

0.1270


0.1025


0.0014

H11


0.1270


0.1025

0.0000


0.1302


0.1042

0.0000


0.1270


0.1025


0.0014

H12 H13

0.0253 0.1523

0.1613
0.0587

0.0000 0.0000

0.0237 0.1474

0.1599
0.0576

0.0000 0.0000

0.0253 0.1523

0.1613
0.0587


0.0014
0.0014

total

0.0001


0.0001


0.0003


0.0589


0.0003


0.0018

0.0000


0.0001


0.0570

13152

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Table 5. Charge (a.u.) Transferred into or out of Atomic Basins after Stretch of Bridgehead C
H Bond and Application of z Direction Field (bcp), or Stretch of In-Plane C
H and Application of y Direction Field (propane)a atom bcp

a

method HF

MP2

CCSD

B3LYP

M06

M062X

propane

B3LYP

C1

0.004

0.002

0.002

0.003

0.002

0.002

C1


0.019

C2

0.003

0.002

0.002

0.002

0.002

0.002

C2

0.002

C3 C4bh

0.003
0.029

0.002
0.022

0.002
0.022

0.003
0.020

0.002
0.022

0.002
0.023

H3ip H4

0.008 0.002

C5bh


0.025


0.018


0.019


0.018


0.019


0.020

H5

0.002

H6bh

0.024

0.019

0.019

0.019

0.020

0.022

H6

0.001

H7bh

0.012

0.008

0.008

0.008

0.010

0.010

H7

0.001

H8

0.001

0.001

0.001

0.001

0.001

0.001

C8


0.019

H9

0.001

0.001

0.001

0.001

0.001

0.001

H9ip

0.016

H10

0.001

0.001

0.001

0.001

0.001

0.001

H10

0.003

H11 H12

0.001 0.001

0.001 0.001

0.001 0.001

0.001 0.001

0.001 0.001

0.001 0.001

H11

0.003

Total


0.0002

H13

0.001

0.001

0.001

0.001

0.001

0.001

Total


0.0001


0.0010

0.0005

0.0004

0.0002

0.0004

Field strengths 0.001 a.u.

The QTAIM results can nevertheless guide a more detailed investigation of bcp using the results of our earlier QTAIM studies in conjunction with the present molecular electron distribution plots. The molecular polarizabilities were also calculated using AIMAll, which produces origin-independent atomic dipoles that sum to give the molecular polarizability. Table 4 shows these terms for all methods, at equilibrium and stretched bridgehead C
H geometries, with and without applied electric fields. In the AIM2000 implementation, atomic moments are calculated differently, and atomic charge transfer terms are one component of the change in a molecule’s polarizability after a bond stretch (eq 3). The magnitudes of such charge transfer terms can be useful for gaining some insight into the origin of polarizability derivatives, in particular, in conjunction with our earlier analyses. They are shown in Table 5 for all methods, calculated as the difference between the value for the stretched bridgehead structure in the presence of a field applied in the z direction and the corresponding value at equilibrium with zero applied field. Calculated absolute atomic charges are small, as expected for a neutral saturated hydrocarbon. Values for equilibrium and stretched molecular structures (results not shown) showed some method dependence, including changes of the sign of the charge of certain atoms. However, the net changes in atomic charges due to bond stretches and the charge transfer due to applied fields show minimal method dependence, and all exhibit the same trends. As seen in Table 5, for bcp, there is an asymmetry of 0.010
0.012 a.u. (for the various methods) in the charges of the two bridgehead H-atoms (in the field direction), revealing the large endto-end charge transfer in response to the field. The charges on the bridgehead carbon atoms have a smaller asymmetry (∼0.004 a.u.) in the same direction, also contributing to the induced molecular dipole. In contrast, analogous results for propane (Table 5) show less net transfer between the in-plane terminal H atoms (0.08 a.u.) and no asymmetry in charges of the two methyl carbon atoms. The MP2/AVTZ method is chosen for more detailed semiquantitative and visual analysis because it produced a reasonable polarizability derivative and the QTAIM analyses also yielded good results, with a ∂α/∂r recovery error of 15%, sufficient to

Table 6. Polarizability Tensor Elements for Bcp at Equilibrium and C
H Stretched Geometries (MP2) polarizability (a.u.) αxx

αyy

αzz (a.u.)

R

EQ

58.12

58.12

54.32

56.86

str bha

58.31

58.31

56.48

57.70

str Mea

59.03

59.03

54.48

57.51

geometry

a

Two bridgehead C
H bonds stretched +0.04 Å; six methylene C
H bonds stretched +0.013 333 Å.

compare trends. Despite incomplete recovery, the atomic contributions to polarizability, both at equilibrium and in the stretched geometry, can provide insight into the origins of the unusually strong intensity for the stretch of the bridgehead C
H. The individual components of the calculated polarizability tensor, with MP2 method, are shown in Table 6 for equilibrium and C
H stretched geometries of bcp. The bond displacements were chosen to produce equal changes in mean polarizability if the two derivatives were equal. In Figure 4, density differences are plotted for electric fields applied to bcp in the z and x directions and to propane in the x and y directions, obtained for the MP2 method at equilibrium, by subtracting zero-field density from the applied field density. The in-plane methyl and the methylene C
H stretches in propane best match the bridgehead and methylene C
H, respectively, in bcp. Interesting parallels, even in the absence of an external field, are shown in Figure 5, where density difference plots are shown for the effect of stretching each bond type by subtracting equilibrium geometry density from C
H stretched geometry. Finally, in Figure 6, a difference of differences is plotted for the bridgehead stretch and applied z fields (the only case that this is appreciable). This is achieved by taking the z field and subtracting the zero-field to show the polarization, then repeating for the bridgehead stretched geometry. The subtraction of one density difference from the other is then plotted to illustrate origins of the polarizability derivative. 13153

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Figure 4. (a) z direction and (b) x direction polarizability of bcp: density difference plot (0.0001 a.u. isosurfaces) for MP2 method at equilibrium geometry, subtracting zero-field density from applied z or x direction field density. The perspective in b shows a top view down the molecular axis. The density is semiopaque to show the locations of the atoms. (c, d) Analogous plots for propane showing y and x direction polarizability, respectively. Color scheme as for Figure 2.

Figure 6. Origin of increased z direction polarizability upon bridgehead C
H stretch. Density difference plot (0.000 02 isosurfaces) for MP2 method, created by subtracting two density differences from each other (see text).

Figure 5. Density difference plot (0.0001 a.u. isosurfaces) for MP2 method with zero field, subtracting equilibrium geometry density from the C
H stretched density for (a) methyl in-plane CH and (b) methylene in propane and (c) bridgehead C
H and (d) methylene in bicyclo-[1.1.1]pentane.

4. DISCUSSION Measurement of absolute intensities in Raman spectra and interpretation within the context of QTAIM analysis provides an exceptional window on the detailed chemical setting of the functional group within the molecule and an exquisite view of charge rearrangement within a vibrating molecular framework in an external field. Our earlier computational studies of Raman intensities for saturated alkane test sets predicted the bridgehead C
H stretching mode in bicyclo-[1.1.1]-pentane (bcp) to exhibit the highest C
H stretch polarizability derivative of any molecule considered. In contrast, intensities related to the methylene groups of bcp were predicted to be more typical. These polarizability derivatives and the corresponding Raman intensities were initially considered anomalous but are consistent with the trends that are now understood through QTAIM (e.g., alignment, strain, etc.). Many of the survey calculations, performed at the HF/ D95(d,p) level to balance accuracy and throughput, produced readily interpretable trends in the polarizability derivatives. The structural parameters (Table 1) are very similar for B3LYP, MP2, M06, and M062X. As expected, the bridgehead CH is always shorter than the methylene CH. The Hartree
Fock bond lengths are a little shorter than the rest, but even the largest differences are well below our step size for the derivatives. Some CASSCF calculations with various active spaces (data not shown) found the HF configuration to be extremely dominant; hence, it was determined that there was no need for multireference calculations. This is supported by a coupled-cluster T1-diagnostic of only 0.0088; values < 0.02 indicate applicability of single reference methods.

Electron correlation seems to be less important for calculating polarizability than one might expect, given the unusual bonding in bcp (Table 2). Values are fairly similar for HF, CCSD, and CCSD(T). Polarizability increases slightly, but the derivative does not change between CCSD and CCSD(T). The results for molecular polarizability using the HF method results are the lowest. The QTAIM recovery of polarizability is quite good; however, the recovery of polarizability derivatives for bcp is poor (Table 3). Those calculations are challenging for the present numerical implementations of the analytical QTAIM theory. Integrations are performed over atomic basins with highly unusual shapes before and after small perturbations that arise from both finite applied fields and small geometric (stretching) distortions. The internal checks for consistency that we have employed remain essential. Using the origin-independent QTAIM method, the atomic dipoles for bcp correctly sum to give a molecular dipole moment of 0.000 a.u. For propane, they yield a value of 0.033 a.u., in agreement with both the calculated and literature values.17 The absolute change in polarizability is ∼1.5% of the total mean molecular polarizability at any level of theory; this establishes a desirable benchmark for QTAIM recovery necessary to achieve a meaningful analysis of the charge displacement in the molecule. The difficulty is one of numerical implementation rather than the theory itself; QTAIM has been successfully applied to a number of studies yielding excellent agreement with experimental results and has been key in the analysis of properties from X-ray analysis.1,18 The decision as to which method provides the best results for QTAIM analysis of ∂α/∂r becomes less obvious when considering the origin of these values. The MP2 value appears to be the best, but only because QTAIM over-recovers the molecular polarizability at the equilibrium and stretched geometries to the same extent, that is, ∼101%. HF actually looks better because both recoveries are closer to 100% and are next in line in terms of quality. Importantly, although each of the levels of theory has yielded slightly different values for α and ∂α/∂r, the absolute changes in the atomic moments and atomic charges are the same to within ∼5%. To facilitate comparisons among the methods, images were created of differences in molecular charge densities at equilibrium 13154

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The Journal of Physical Chemistry A and stretched geometries of propane and bcp, with and without applied fields, and for different combinations of HF, MP2, B3LYP, and M06/AVTZ. Since MP2 is built upon the HF reference in a well-defined way, we show a density difference plot between MP2 and HF/AVTZ at the equilibrium geometry, in the absence of an external field (Figure 2). All DFT or post-HF methods employed here produce similar changes relative to the HF calculation: increased electron density near carbon atoms and a reduction of electron density around the hydrogen atoms. The inclusion of electron correlation in the calculations reduces density along bonds while increasing density between bridgehead carbons (inside the cage of bcp). The charge density at the cage critical point is 0.0036 a.u. greater (more negative) in the electron-correlated MP2 calculation than in the HF calculation. Given that the cross-ring interaction is expected to play a role in bcp’s unique properties, we have compared the MP2, M06, and B3LYP electron correlation methods. B3LYP is a standard, widely employed DFT method, whereas the new M06 functional is proposed to be a better alternative for cases when noncovalent interactions are important.19 Differences are more easily appreciated with a visual comparison of the electron density differences, as shown in Figure 3. For both propane and bcp, B3LYP calculations localize more electron density near the nuclei than MP2 calculations. Compared with M06, B3LYP calculations create higher electron density along the bonds of both propane and bcp. MP2 and M06 localize slightly more electron density inside the cage compared with B3LYP; however, the changes within the bonds of the molecular structures show identical trends for propane and bcp. The differences between propane and bcp are then principally the presence of a cage containing appreciable electron density and, of course, the existence of three propane-like chains instead of just one. In the end, the differences in electron density distribution with method do not appreciably affect the computed molecular polarizability. Although the AIMAll method produces a single, originindependent atomic moment for each atomic basin, it is useful to consider the amount of charge transferred between atomic basins as the molecule undergoes different perturbations (stretch, application of external field). The optimized equilibrium geometry from the B3LYP/AVTZ method was used as the equilibrium structure for each method to permit visual comparisons of differences between calculations by method. The magnitudes (and even the sign) of the small static atomic charges (data not shown) vary between methods and reflect the particular electron distribution wrought by the details of incorporation of electron correlation in the method. However, the changes in atomic charges after stretching the bridgehead C
H bond in the presence of an electric field are comparable, regardless of method (Table 5). The QTAIM values for the origin-independent atomic dipoles, calculated from the MP2/AVTZ level of theory (Table 4), are typical of all the methods considered here. The MP2/AVTZ calculations produced a reasonable polarizability derivative with a recovery error of 15% from the electronic structure calculation, sufficient to compare trends. Importantly, although each of the levels of theory has yielded slightly different values for α and ∂α/∂r, the absolute changes in the atomic moments and atomic charges are the same to within about 5%. This table demonstrates that although atomic populations are method-dependent, the changes in atomic populations are not. Despite the residual error in the absolute recovery of the polarizability derivative, the QTAIM analysis can be used to

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explain the origin of the seemingly unusual bridgehead derivative. In Figure 4, we have plotted density difference plots showing directional polarization for the MP2 method. The net charge transfer (analogous to the charge transfer terms calculated in the AIM2000 implementation) is opposed by alternating regions of increased/decreased density. These regions are analogous to the alternating internal dipoles that we calculated with AIM2000. We have observed before that end-to-end charge transfer is dominant in the equilibrium polarizability, as in a dielectric medium, and, in fact, will be the significant factor in the derivative also;1 thus, the calculations show that bcp is not so very different from other alkanes studied, except in the combination of influences. This becomes clearer from the images in Figure 5, showing the density difference plots for the equilibrium and stretched structures of propane and bcp, in the absence of an external field. Stretching the bridgehead C
H (bcp) or terminal in-plane C
H (propane) reduces bond density, but increases density on the outside of terminal in-plane H-atoms (propane) or on the outside of bridgehead H-atoms (bcp). This is in accord with the increased absolute magnitude of the molecular polarizability in the stretched molecule. However, it also signifies where and how the molecules become more polarizable. The charge density exterior to the bond structure is less tightly held and represents more polarizable regions. We note that there is also an increase in charge density at the cage critical point. Chemically, this makes sense, since complete dissociation of the bridgehead C
H would lead to the formation of propellane (with a bond between the bridgehead carbon atoms). Finally, in Figure 6, we show a difference of differences for the bridgehead stretch and applied z fields (the only case that this is appreciable). By subtracting one density difference from the other, we can see the origin of the increased polarizability. The density difference created by subtracting the zero-field density from the z field density is subtracted from the analogous density difference created at the bridgehead stretched geometry. This indicates that the additional z direction polarizability associated with bridgehead stretching arises from net end-to-end charge transfer, with opposing differences along the C
H bonds. Stretching methylene C
H bonds increases the z direction polarizability only slightly (see Table 6). Stretching methylene C
H bonds also reduces bond densities and increases density on the outside of H atoms. The more significant increase in the x and y direction polarizability is less dramatic than the z direction increase caused by bridgehead C
H stretching because it is smaller in magnitude, it is distributed across both the x and y components, and it is also more delocalized, hence, not appreciable on density difference plots. The numerical trends observed in the QTAIM results are underscored by plots of the electron density. The existence of net end-to-end charge transfer and internal opposing dipoles supports QTAIM-based conclusions. Although the electron density in these plots is not partitioned into atomic basins, the differences between applied field and zero-field densities clearly illustrate the mechanism of polarizability. In bcp, we see that there is a net charge transfer from one end of the molecule to the other in the field direction that is opposed by alternating dipoles. Stretching any C
H bond is seen to increase density outside the H-atom. The magnitude of the change in polarizability that is realized due to C
H stretching then depends on how easily and far that density can be displaced by an applied field. Location and alignment of the C
H are therefore clearly important factors. The response of propane is the same in the sense of end-to-end 13155

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5. CONCLUSION Polarizability and its derivatives were calculated for bicyclo-[1.1.1]pentane (bcp) and propane, using a range of theoretical methods from Hartree
Fock (HF) to post-Hartree
Fock (MP2, CCSD, CCSD(T)) as well as several density functional theory methods (B3LYP, M06, and M062X) and compared with experiment. Only limited variations in calculated values were recorded, with all methods producing reasonable agreement with experiment. The calculated polarizability derivatives exceed the experimental value by between ∼3% (CCSD, CCSD(T)) and 10% (B3LYP). Results of our previous studies using smaller basis sets (the Dunning AVTZ basis was used here) contribute to the conclusion that calculated polarizability in saturated hydrocarbons is more sensitive to the basis set size than to the inclusion of electron correlation (varying by 17% for the HF method between the D95** and AVTZ bases). Density difference plots provide an illustration of how the electronic structure varies with method as well as insight into the mechanisms of polarizability and the variation of its derivative in these systems. Electron correlation results in decreased density along bonds but increased density near heavy atoms and in the nonbonded cage center of bcp. The DFT methods all produce densities qualitatively similar to MP2. Notably, inclusion of electron

correlation was not seen to have a dramatic effect on either the molecular polarizability as a whole or on the local patterns of redistributed density in response to applied fields. QTAIM was used to explore the atomic origins of polarizability and polarizability derivative variations. Numerically partitioning the molecular polarizability into atomic contributions was achievable using either the AIM2000 or AIMAll software to within about 1% error. Despite dense integration grids, the errors in polarizability are on the order of changes due to moderately sized bond stretches, making quantitative comparisons of the atomic contributions to the derivative between methods difficult. In bcp, there is cross-cage charge transfer, but the seemingly anomalous intensity can be classified within our structural features of alignment, steric hindrance, and ring strain. We find as before that there is an end-to-end charge transfer driven by the field, with internal induced dipoles opposing the applied field. QTAIM analysis shows that the basic mechanism, even in these small molecular units, can be understood as being analogous to a dielectric medium responding to an applied field.

’ AUTHOR INFORMATION Corresponding Authors

*E-mail: [email protected], [email protected], kmgough@ cc.umanitoba.ca. Author Contributions

)

charge transfer being opposed by induced dipoles internal to the chain. Previous calculations on a series of longer, linear, saturated hydrocarbons (butane, pentane, hexane, etc.) showed that the derivative for stretching the terminal C
H bond does increase with chain length, but only gradually. Although the distance for charge transfer increases in longer straight-chain hydrocarbons, the effect is damped by the alternating opposing dipoles along the bonded framework. In the QTAIM results for propane (Table 5), with an applied field, charge is transferred between terminal H-atoms, but charges are equal on the two methyl carbon atoms, indicating efficient damping along the carbon framework. Whereas bcp similarly shows charge transfer between bridgehead H-atoms, the charges on the bridgehead carbons have some asymmetry, reflecting that the interplay among methylene damping, carbon chain alignment, and cross-cage transfer results in a qualitatively different response to the field perturbation that reflects bcp’s unusual molecular nature. In fact, on the basis of the QTAIM results in Table 4, it appears that most charge transfer occurs directly across the cage, since much less polarization of the methylene atoms is observed due to the z direction field. This gives us new insight into how to think about the polarizability derivatives in bcp, bringing the seemingly unusual behavior into the framework that we had established with simpler molecules. Stretching a bond builds up some density on the outside of the atom. How much extra polarizability that allows depends on how easily and how far that density can be moved with the applied field: the induced dipole is fundamentally the product of charge and distance. For propane, stretching the terminal bonds and applying a y field transfers that density the entire length of the molecule and induces a large dipole. Stretching the methylenes places the extra charge at the center of the molecule, where it may distort, but does not have as far to travel. In analogy to propane, the stretch of the methylenes in bcp produce a more typical change in the polarizability, whereas the bridgehead C
H, with their alignment along the z axis allows polarization to occur across the length of the molecule, producing a large induced dipole.

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Authors contributed equally to this work.

Present Addresses ^

Manitoba Water Stewardship, 200 Saulteaux Cres, Winnipeg, MB R3J 3W3

’ ACKNOWLEDGMENT This work was supported by funding from a Natural Sciences and Engineering Research Council of Canada Discovery Grant (K.M.G.), from the University of Rhode Island (J.R.D.) and from the University of Missouri Research Board (R.D.). ’ REFERENCES (1) Gough, K. M.; Dawes, R.; Dwyer, J. R.; Welshman, T. in The Quantum Theory of Atoms in Molecules. From Solid State to DNA and Drug Design; Matta, C., Boyd, R.Eds.; Wiley-VCH: New York, 2007; Chapter 4. (2) Placzek, G.; U.S. Atomic Energy Commission, UCRL-Trans-524(L); 1962. Translated from Handbuch der Radiologie, 2nd ed.; Marx, E., Ed.; Akademisch: Leipzig, 1934; Vol. 6, Part II, pp 205
374. (3) Gough, K. M.; Dwyer, J. R. J. Phys. Chem. A 1998, 102, 2723–2731. (4) Gough, K. M.; Dwyer, J. R.; Dawes, R. Can. J. Chem. 2000, 78, 1035–1043. (5) Dawes, R.; Gough, K. M. J. Chem. Phys. 2004, 121, 1273–1284. (6) Adcock, W.; Blokhin, A. V.; Elsey, G. M.; Head, N. H.; Krstic, A. R.; Levin, M. D.; Michl, J.; Munton, J.; Pinkhassik, E.; Robert, M.; Sav
eant, J.-M.; Shtarev, A.; Stibor, I. J. Org. Chem. 1999, 64, 2618–2625. (7) Giribet, C. G.; Ruiz de Az
ua, M. C.; G
omez, S. B.; Botek, E. L.; Contreras, R. H.; Adcock, W.; Della, E. W.; Krstic, A. R.; Lochert, I. J. J. Comput. Chem. 1998, 19, 181–188. (8) Della, E. W.; Lochert, I. J.; Peralta, J. E.; Contreras, R. H. Magn. Reson. Chem. 2000, 38, 395–402. (9) Gough, K. M.; Yacowar, M. M.; Cleve, R. H.; Dwyer, J. R. Can. J. Chem. 1996, 74, 1139–1144. (10) Gough, K. M.; Srivastava, H. K.; Belohorcova, K. J. Chem. Phys. 1993, 98, 9669–9677. (11) Bader, R. F. W.; Keith, T. A.; Gough, K. M.; Laidig, K. E. Mol. Phys. 1992, 75, 1167–1189. 13156

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(12) Frisch, M. J.Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, N. J.; Klene, M.; Knox, J. E.; 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.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; € Foresman, Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, O.; J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian 09, Revision B.1; Gaussian, Inc.: Wallingford CT, 2010. (13) Biegler-K€onig, F.; Sch€onbohm, J. J. Comput. Chem. 2002, 23 (15), 1489
1494. (14) Gatti, C.; MacDougall, P. J.; Bader, R. F. W. J. Chem. Phys. A 1988, 88, 3792–3804. (15) Keith, T. A. AIMAll (Version 11.04.03), 2011; aim.tkgristmill.com. Accessed April 27, 2011. (16) Keith, T. A. In The Quantum Theory of Atoms in Molecules. From Solid State to DNA and Drug Design; Matta, C., Boyd, R., Eds.; WileyVCH: New York, 2007; Chapter 3. (17) Lide, D. R. J. Chem. Phys. 1960, 33, 1514–1518. (18) Coppens, P. Angew. Chem., Int. Ed. 2005, 44, 6810–6811. (19) Zhao, Y.; Truhlar, D. G. Acc. Chem. Res. 2008, 41, 157–167.

’ NOTE ADDED AFTER ASAP PUBLICATION This article posted ASAP on September 26, 2011. Jason R. Dwyer was added as an additional corresponding author. The correct version posted on October 24, 2011.

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