Molecular structure and pseudorotation of 1,1-dichlorocyclopentane as determined by gas-phase electron diffraction and ab initio calculations: A large amplitude treatment

Article abstract of DOI:10.1021/jp031172w

The molecular structure of 1,1-dichlorocyclopentane (DCICP) has been investigated by means of gas-phase electron diffraction and ab initio calculations. Although the electron diffraction data could be fairly well reproduced by a C_s (envelope) model we found it more pertinent to apply a pseudorotation model to account for the dynamic and large amplitude motion in DClCP. On the basis of this model we analyzed the dependency of the ring geometric parameters and vibrational mean amplitudes on the phase angle φ. For a better elucidation of this distinct dependency we developed particular equations which describe the dependency of the distribution of the delocalized net charges throughout the ring on the phase angle φ. The joint electron diffraction and ab initio study has led to the following r_a structural parameters of DClCP (C_s conformer): r(C-Cl)_ax = 1.787(3) A, r(C-Cl)_eq = 1.769(3) A, average r(C-C) _ring = 1.535(1) A, r(C-H)_av = 1.114(5) A, ∠(C5-C1-C2) = 103.0(9)°, ∠(Cl-C-Cl) = 108.6(3)°, and ∠(H-C-H) = 104.6(26)°. The puckering amplitude for the five-membered ring was determined to be q = 0.400(11) A. The quantum mechanical calculations were carried out by utilizing the Hartree-Fock, density functional B3PW91, and perturbation MP2 methods and applying the following basis sets: cc-pVDZ, 6-31G(d,p), 6-311G(df,pd), 6-311+G(d,p), 6-311++G(d,p), 6-311+G(2df,2pd), and 6-311++G(2df,2pd). For the purpose of comparison and systematic study, we optimized the geometries and calculated the charge distributions using the natural population analysis (NPA) and Mulliken population analysis (MPA) of 1,1-difluorocyclopentane, 1,1-dibromocyclopentane, and their corresponding monohalogenated derivatives.

Full text of DOI:10.1021/jp031172w

4658
J. Phys. Chem. A 2004, 108, 4658-4673
Molecular Structure and Pseudorotation of 1,1-Dichlorocyclopentane As Determined by
Gas-Phase Electron Diffraction and ab Initio Calculations: A Large Amplitude Treatment
Marwan Dakkouri,*,† Volker Typke,*,‡ and Tilman Schauwecker†
Department of Electrochemistry and Communication and Information Center, UniVersity of Ulm,
D-89069 Ulm, Germany
ReceiVed: October 16, 2003; In Final Form: March 17, 2004
The molecular structure of 1,1-dichlorocyclopentane (DClCP) has been investigated by means of gas-phase
electron diffraction and ab initio calculations. Although the electron diffraction data could be fairly well
reproduced by a Cs (envelope) model we found it more pertinent to apply a pseudorotation model to account
for the dynamic and large amplitude motion in DClCP. On the basis of this model we analyzed the dependency
of the ring geometric parameters and vibrational mean amplitudes on the phase angle φ. For a better elucidation
of this distinct dependency we developed particular equations which describe the dependency of the distribution
of the delocalized net charges throughout the ring on the phase angle φ. The joint electron diffraction and ab
initio study has led to the following ra structural parameters of DClCP (Cs conformer): r(C-Cl)ax ) 1.787(3)
Å, r(C-Cl)eq ) 1.769(3) Å, average r(C-C)ring ) 1.535(1) Å, r(C-H)av ) 1.114(5) Å, (C5-C1-C2) )
103.0(9)°, (Cl-C-Cl) ) 108.6(3)°, and (H-C-H) ) 104.6(26)°. The puckering amplitude for the five-
membered ring was determined to be q ) 0.400(11) Å. The quantum mechanical calculations were carried
out by utilizing the Hartree-Fock, density functional B3PW91, and perturbation MP2 methods and applying
the following basis sets: cc-pVDZ, 6-31G(d,p), 6-311G(df,pd), 6-311+G(d,p), 6-311++G(d,p), 6-311+
G(2df,2pd), and 6-311++G(2df,2pd). For the purpose of comparison and systematic study, we optimized the
geometries and calculated the charge distributions using the natural population analysis (NPA) and Mulliken
population analysis (MPA) of 1,1-difluorocyclopentane, 1,1-dibromocyclopentane, and their corresponding
monohalogenated derivatives.
Introduction
is some certain correlation between the nature of the geminal
substituents and these parameters.
In a very recent study1 we analyzed the structure of
1,1-dicyanoycyclopentane and investigated the hindered pseu-
dorotational motion in this five-membered-ring molecule. Vari-
ous intriguing features emerged from this study and among them
was the noticeable dependency of the geometry parameters,
vibrational amplitudes, and charge distribution on the phase
angle φ.
These interesting results have prompted us to study one
further geminally substituted cyclopentane with substituents
possessing different electronic properties than the cyano groups.
Accordingly, we investigated the structure of 1,1-dichlorocy-
clopentane (DClCP) by means of electron diffraction and ab
initio calculations. Static one-conformer (Cs and C1 forms) and
two-conformer (Cs plus C2) models as well as a model
considering the pseudorotational motion have been used to fit
the experimental data.
The main perspectives for conducting the present work are
as follows: (1) to obtain additional relevant results which may
contribute to a more complete and perhaps detailed understand-
ing of the hindered pseudorotational motion in five-membered
rings; (2) to explore the influence of the geminal chlorine atoms
on the pseudorotational parameters of the five-membered ring,
the phase angle φ, and the puckering amplitude q; and (3) from
the comparison of the structural and pseudorotational parameters
in 1,1-dicyanocyclopentane and DClCP to deduce whether there
Synthesis and Experimental Data
Synthesis. The sample of 1,1-dichlorocyclopentane was
prepared by the reaction of phosphorus pentachloride with
cyclopentanone following a modified route of Domnin et al.2
We used benzene as a solvent instead of ligroin and performed
the reaction at 10 °C instead of 0 °C. Furthermore, to prevent
the formation of monochlorocyclopentene the cyclopentanon-
benzene solution was added very slowly (one drop per 5 s) to
the PCl5/benzene slurry and the entire reaction was carried out
under anhydrous conditions by applying a stream of dry
nitrogen. The subsequent decomposition of POCl3 was under-
taken very carefully at about 5 °C and under vigerous stirring.
The crude product was purified by using a 100 cm spinning
band column. The main fraction was obtained at 51 °C/30 Torr
25
(nD ) 1.4685). Further purification was carried out by using
gas chromatography separation. To ascertain the purity of the
sample, IR and mass spectra were taken.
The gas-phase electron diffraction intensities were recorded
on Kodak Electron Image plates with a Balzers KD-G2
diffraction unit at the University of Tu¨bingen. The accelerating
voltage was 60 keV, and the recording temperature was about
30 °C. Data were collected at camera distances of 50 and 25
cm (nominal) yielding molecular intensity values at intervals
(∆s) of 0.2 Å-1 ranging from s ) 1.6 to 17.6 Å-1 and from 9.0
to 33.8 Å-1. The electron wavelength was calibrated with ZnO
diffraction patterns giving λ500 ) 0.049243 ( 0.000058 Å and
† Department of Electrochemistry.
‡ Communication and Information Center.
10.1021/jp031172w CCC: $27.50 © 2004 American Chemical Society
Published on Web 05/04/2004

Molecular Structure of 1,1-Dichlorocyclopentane
J. Phys. Chem. A, Vol. 108, No. 21, 2004 4659
TABLE 1: Optimized Geometrical Parameters of 1,1-Dichlorocyclopentane (DClCP) and 1,1-Difluorocyclopentane (DFCP) As
Predicted by Different Levels of Theory
DClCP
DFCP
Cs symmetry
C2 symmetry
Cs symmetry
C2 symmetry
B3PW91
/6-311
MP2
/6-311
B3PW91
/6-311
MP2
/6-311
B3PW91
/6-311
MP2
/6-311
B3PW91
/6-311
MP2
/6-311
parameter
+G(df,pd)
+G(2df,2pd)
+G(df,pd)
+G(2df,2pd)
+G(df,pd)
+G(2df,2pd)
+G(df,pd)
+G(2df,2pd)
bond distances (Å)
C1-C2
1.522
1.536
1.553
0.395
1.811
1.792
1.091
1.519
1.535
1.554
0.419
1.794
1.776
1.088
1.542
1.529
1.529
0.421
1.806
1.806
1.092
1.541
1.527
1.528
0.450
1.790
1.790
1.089
1.516
1.536
1.556
0.370
1.375
1.363
1.091
1.509
1.534
1.557
0.400
1.373
1.360
1.087
1.530
1.530
1.532
0.403
1.370
1.370
1.092
1.525
1.528
1.531
0.432
1.368
1.368
1.089
C2-C3
C3-C4
qa
C1-Cl(F)6
C1-Cl(F)7
b
C-H
bond angles (deg)
C1-C2-C3
103.8
105.9
103.9
108.4
108.0
107.1
122.7
128.8
39.8
103.2
105.7
103.5
109.0
108.5
107.4
122.1
129.0
42.3
104.7
102.8
106.4
107.6
107.8
107.6
126.2
126.2
104.1
102.0
106.3
108.1
108.3
108.1
126.0
126.0
103.7
106.2
105.6
105.8
108.2
106.8
123.8
130.4
37.7
102.8
105.9
105.4
106.4
108.8
107.2
122.8
130.9
40.1
104.7
103.1
107.4
105.2
108.1
107.4
127.4
127.4
104.2
102.4
107.5
105.7
108.6
107.9
127.2
127.2
C2-C3-C4
C5-C1-C2
Cl(F)-C-Cl(F)
H-C2-H
H-C3-H
c
â6
d
â7
Re
dihedral angles (deg)
C1-C2-C3-C4
C4-C5-C1-C2
C2-C3-C4-C5
23.9
39.3
0.0
25.3
41.7
0.0
-35.1
13.4
43.6
-37.3
14.3
46.5
22.4
37.2
0.0
24.1
40.4
0.0
-33.5
12.8
41.6
-35.7
13.7
44.4
µf
2.73
2.76
2.84
2.88
2.68
2.83
2.75
2.92
a Puckering amplitude for the ring, calculated with the program RING.12,13 b Average value. c Angle between the bond Cl(F)6-C1 and the plane
C5C1C2. d Angle between the bond Cl(F)7-C1 and the plane C5C1C2. e Angle between the C5C1C2 plane and C2C3C4C5 plane (flap angle). f Dipole
moment.
λ250 ) 0.049376 ( 0.000046 Å. Optical densities from three
photographic plates for both distances were recorded on our
computer-controlled and -modified microdensitometer ELSCAN
E-2500 (Optronics International, Chelmsford, MA). Our usual
data reduction and refinement procedures were used,3,4 and the
atomic scattering amplitudes and phases of Haase5 were applied.
The intensity data used for the determination of the molecular
structure are provided in the Supporting Information.
of the results, all calculations were conducted with the same
theoretical methods and basis sets that have been applied in
the case of DClCP. For brevity reasons the results obtained for
DBrCP and MBrCP have not been listed in the tables incor-
porated in this paper.
Tables 1 and S1 (S indicates Supporting Information) display
the most prominent structural parameters for DClCP as predicted
by the quantum mechanical methods which have been applied.
For atomic numbering see Figure 1.
Ab initio Calculations
Pseudorotation
The quantum mechanical computations were carried out on
various levels of theory, using the programs Gaussian98,6
Spartan,7 and MOLPRO.8 The following computational proce-
dures were applied: HF/cc-pVDZ, HF/6-31G(d,p), HF/6-
311+G(d,p), density functional (DFT) hybrid B3PW91/6-
311+G(df,pd), and Møller-Plesset perturbation method MP2
in combination with the basis sets 6-311G(d,p), 6-311+G(d,p),
6-311++G(d,p), 6-311+G(2df,2pd), and 6-311++G(2df,2pd).
It should be pointed out, however, that the augmentation of these
basis sets by adding diffuse functions and higher polarization
functions on both heavy atoms and hydrogen does not signifi-
cantly affect the structural results. It remains to note that all
these methods, particularly the perturbation methods, provided
results, which are in good agreement with the experiment.
Saturated five-membered rings are known to undergo a special
form of vibrational motion, the pseudorotation.9-11 A close
examination of the variation of nonbonded distances during one
cycle of pseudorotation shows that in DClCP the intramolecular
distances for several pairs of atoms change by up to 2 Å. This
indicates that the analysis of electron diffraction patterns with
a large amplitude treatment of the pseudorotation should be more
appropriate than a static model allowing only for Cs symmetry.
To apply this concept various problems have to be solved. Most
importantly the dependency of the geometry on the phase angle
φ must be analyzed because it is required to establish the ring
geometry for any phase angle φ. Inspection of Tables 1 and S1
shows that the ring C-C bond distances as obtained from all
ab initio methods for the Cs and C2 conformers depend strongly
on the phase angle φ: the C1-C2 distance increases by about
0.021 Å in going from φ ) 0 to π/2 and thereby the C3-C4
distance decreases by 0.024 Å. These changes in atomic
distances are easily observable in the analysis of electron
diffraction patterns.
In the course of our systematic study of the effect of
substituents and to examine whether certain correlations exist
between the kind of substituent and structural and/or confor-
mational changes we also performed ab initio calculations on
the following related molecules: monofluorocyclopentane
(MFCP), monochlorocyclopentane (MClCP), monobromocy-
clopentane (MBrCP), 1,1-difluorocyclopentane (DFCP), and 1,1-
dibromocyclopentane (DBrCP). For more proper comparison
In 1,1-disubstituted five-membered rings the pseudorotational
motion is not a virtually free motion, but is characterized by a

4660 J. Phys. Chem. A, Vol. 108, No. 21, 2004
Dakkouri et al.
Figure 1. Atomic numbering of 1,1-dichlorocyclopentane (DClCP) and monochlorocyclopentane (MClCP).
hindering potential function of the form (neglecting higher
terms)
not constant during one cycle of pseudorotation. Subsequently
the calculated values for the energy were analyzed to derive
the coefficients of the potential function V(φ). We found that
the coefficient V6 is required to adequately describe the
calculated energy values. The same analysis was also performed
for the puckering amplitude with a formula similar to eq 1:
V(φ) ) 1/2V2(1 - cos2φ) + 1/2V4(1 - cos4φ) +
1/2V6(1 - cos6φ) (1)
where φ is the phase of the pseudorotation. This gives stable
configurations of Cs symmetry at φ ) 0, (π, (2π, ..., and
configurations of C2 symmetry at φ ) (π/2, (3π/2, .... It
depends on the actual values of the coefficients whether the
configurations of C2 symmetry are stable or not. The energy
difference between configurations of Cs and C2 symmetry is
given by V2 + V6.
q(φ) ) q0 + q2 cos2φ + q4 cos4φ + q6 cos6φ
(3)
In Table 2 are collected the minimum values of the puckering
amplitude qmin(φ) and the relative energy Emin(φ) ) E(φ*180°)
- E(φ)180°) at distinct values of φ as well as the derived
coefficients of eqs 1 and 3. Figure 2 shows the potential function
for 1,1-dichlorocyclopentane as obtained with the HF/cc-pVDZ
method and the calculated energy values. The points marked
“×” are obtained assuming a constant puckering amplitude q
for all phase angles φ.
From the calculations described above it is obvious that the
molecular structure is available for the complete cycle of
pseudorotation. This offers the possibility to analyze the
deformation of the ring due to the pseudorotation. Following
the approach we presented in a previous paper1 we applied a
model which assumes that the density distribution of the
delocalized net charges, which represents the overall distribution
of charges throughout the ring, depends markedly on the phase
angle φ. Assuming that this dependency can be described by a
function, d, of the form shown in Figure 4 we write for the
density distribution
To get reasonable estimates for the potential parameters V2,
V4, and V6, additional calculations with the program MOLPRO
and the HF/cc-pVDZ method have been performed. Basically,
the potential function V(φ) can be calculated by optimizing 3N
- 7 ) 38 geometrical parameters for a number of configurations
with a fixed value of the phase angle φ. Unfortunately the
geometrical parameters used in the available ab initio programs
for setting up the Z matrix do not allow for an immediate
description of the pseudorotation. Therefore we tried to fix one
of the dihedral angles which were used to locate the ring C
atoms at values in a suitable range while the remaining
parameters were optimized. Subsequently, we applied the
program RING12,13 to evaluate the corresponding puckering
amplitude q and phase angle φ. It turns out that no such dihedral
angle can be used to cover the full range in going from Cs
symmetry to C2 symmetry. Also, the resulting structures do not
necessarily represent the configuration of lowest energy at given
values of φ and q. Therefore we calculated the energy hyper-
surface in the following way: the program MOLPRO provides
a mode to optimize the geometry on the basis of Cartesian
coordinates and thereby fixing selected coordinates to some
deliberately chosen value. We decided to fix the x-coordinate
of atom C1 to zero, and to fix the z-coordinates of all ring C
atoms to a value calculated from the formula given by Kilpatrick
et al.14
d(R, φ) ) d1 cosR + d2 cos2R + dc2c cosR cos2φ +
d
s2s sinR sin2φ + dc4c cosR cos4φ + ds4s sinR sin4φ +
d
2c2c cos2R cos2φ + d2s2s sin2R sin2φ +
d2c4c cos2R cos4φ + d2s4s sin2R sin4φ (4)
where R designates the angle between the axis from the center
of the ring to atom C1 and every position along the ring, both
projected to the mean plane as defined by Cremer and Pople.13
The coefficients d1, d2, etc. are adjustable parameters. Compared
to the relatively simple formula given previously,1 eq 4 includes
more terms to cover the values of all ring C-C bond distances
during one cycle of pseudorotation. With this density function
we now write for the contribution of the delocalized net charges
to the bond distance Ci-Cj
x
zi ) 2/5 q cos(φ + 4π(i - 1)/5)
(i ) 1, ..., 5) (2)
This leaves 3N - 6 Cartesians to be adjusted in the geometry
optimization. We varied the puckering amplitude q in a
reasonable range with a step width of 0.01 Å, and φ with a step
width of 10° in the range 90° to 270°. The results of these
calculations are shown in Figure 3. The line indicating the
bottom of the energy valley corresponds to the minimum energy
at a given value of the phase angle φ, and it also gives the
optimal puckering amplitude q at this φ. It turns out that q is
Rj
rij ) rCC
+
d(R,φ) dR
(5)
∫
Ri
Assuming a regular pentagon in the mean plane of the five-
membered ring the integral can be easily evaluated. If we adjust
the parameters rCC, d1, d2, etc. to the ab initio distances by

Molecular Structure of 1,1-Dichlorocyclopentane
J. Phys. Chem. A, Vol. 108, No. 21, 2004 4661
Figure 2. Potential function of pseudorotation for 1,1-dichlorocyclopentane as obtained from calculations with the method HF/cc-pVDZ. Points
”+” are obtained as the minimum of the calculated energies at constant φ, points “×” are obtained with the assumption of constant q in the
theoretical calculations.
TABLE 2: Minimum Value of the Relative Energy (cal/mol)
and Puckering Amplitude q (Å) at Various Values of Phase
Angle O
TABLE 4: Fourier Coefficients for the Ring C-C Bond
Distances (Eq 6) and the C-C-C Bond Angles (Eq 7)
parameter
r12,r15
r23,r45
r34
phase
qmin
Emin
δrij
c2,ij
c4,ij
s2,ij
s4,ij
-0.0024
-0.0102
0.0010
-0.0075
0.0030
0.0004
0.0041
-0.0017
-0.0104
-0.0004
0.0040
0.0123
0.0014
0.0
90
95
0.423
0.422
0.421
0.417
0.410
0.402
0.396
0.393
0.393
0.394
0.395
1955.6
1935.9
1879.1
1681.3
1428.1
1161.5
883.5
590.1
304.6
84.5
100
110
120
130
140
150
160
170
180
0.0
parameter
R
123,R451
R
234,R345
R512
R0,ijk
c2,ijk
c4,ijk
c6,ijk
s2,ijk
s4,ijk
s6,ijk
104.24
-0.29
0.03
-0.06
-1.27
-0.06
-0.02
104.57
1.54
-0.18
0.01
-1.26
0.18
-0.01
105.29
-1.27
-0.18
0.00
0.0
0.0
0.0
0.0
V
q
V2 ) 1860.6
V4 ) 44.7
V6 ) 93.3
q0 ) 0.4038
q2 ) -0.0151
q4 ) 0.0051
q6 ) 0.0012
The coefficients of this equation have been included in Table
4, and Figure 5 shows the variation of the ring C-C bond
distances with the phase angle φ and the correlation with the
density of the delocalized net charge within these bonds.
The deformation of the ring is further characterized by the
changes in the C-C-C bond angles. It is interesting to note
that the changes of these angles reflect the variation in the
density of the net charge at the apex atom as calculated from
eq 4 in the same way as the C-C bond distances do. In Figure
6 we show the correlation between the variation of the bond
angles C-C-C and of the fluctuation of the delocalized charge
density at the apex atom as a function of the phase angle φ. A
Fourier analysis of the calculated C-C-C bond angles accord-
ing to the equation
TABLE 3: Fitted Constants from Eq 4 for
1,1-Dichlorocyclopentane
parameter
value
parameter
value
rCC
d1
dc2c
ds2s
dc4c
ds4s
1.5369
0.0027
0.0107
0.0097
-0.0004
-0.0009
d2
0.0008
-0.0002
-0.0009
0.0020
d2c2c
d2s2s
d2c4c
d2s4s
0.0026
applying a least-squares procedure we obtain the results shown
in Table 3. It can be easily verified from eq 5 that rCC is the
mean value of the ring C-C bond distances. Combining the
values of the integrals and the constants from Table 3 we find
that the ring C-C bond distances vary as
Rijk ) R0,ijk + c2,ijk cos2φ + c4,ijk cos4φ + c6,ijk cos6φ (
s2,ijk sin2φ ( s4,ijk sin4φ ( s6,ijk sin6φ (7)
yields the coefficients collected in Table 4.
It is also interesting to analyze the variation of geometrical
parameters with the phase angle φ involving ligands attached
rij ) rCC + δrij + c2,ij cos(2φ) + c4,ij cos(4φ) (
s2,ij sin(2φ) ( s4,ij sin(4φ) (6)

4662 J. Phys. Chem. A, Vol. 108, No. 21, 2004
Dakkouri et al.
Figure 3. Calculated relative energies (cal/mol) of 1,1-dichlorocyclopentane at different values of puckering amplitude q and phase angle φ (angles
in deg). Parameters at the curves are the corresponding values of the phase angle φ.
Figure 4. Density of delocalized net charge distribution eq 4 for φ ) 0°, 45°, 70°, and 90°.
to the ring C atoms. There are two types of parameters to be
studied: alterations of the bond distances and bond angles. With
regard to the angle deformations one has to keep in mind that
there are redundancies relating the angles at atoms with four
bonds, and also the ring C-C-C angles are not all independent.
We choose to analyze the rocking (F), wagging (ω), and twisting
(τ) angles as defined by Skancke et al.15 in addition to the ring
C-C-C angles and the Cl-C-Cl and H-C-H angles.
We start by analyzing the variation of the parameters
involving the Cl atoms. We found that the C-Cl bond
lengths are well reproduced by the equation
positions of the C-Cl bond: the upper sign corresponds to the
C-Cl6 bond, the lower sign to the C-Cl7 bond (cf. Figure 1).
In Table 5 the coefficients R0 through R4 are listed, and in Figure
7 we show the correlation between the C-Cl bond distances
and the charge density at atom C1.
Next we investigate the dependency of the angle â ) Cl-
C-Cl on the phase angle φ. It turns out that this angle varies
by only 0.7°. However, it is more interesting to note that the
rocking, wagging, and twisting angles change more drastically.
The Fourier analysis of the calculated values of all these
geometrical parameters results in the following functions:
rCCl ) R0 ( R1 cosφ + R2 cos2φ ( R3 cos3φ + R4 cos4φ
â(φ) ) â0 + â2 cos2φ + â4 cos4φ + â6 cos6φ
F(φ) ) F1 cosφ + F3 cos3φ + F5 cos5φ
(8)
This formula is valid for both the axial and the equatorial

Molecular Structure of 1,1-Dichlorocyclopentane
ω(φ) ) ω2 sin2φ + ω4 sin4φ + ω6 sin6φ
τ(φ) ) τ1 sinφ + τ3 sin3φ + τ5 sin5φ
J. Phys. Chem. A, Vol. 108, No. 21, 2004 4663
TABLE 5: Deformation of the CCl2 Group with Phase
Angle O: The Fourier Coefficients of the Geometrical
Parameters (Eq and 9)
(9)
order
R
â
F
ω
t
0
1
2
3
4
5
6
1.8030
-0.0111
-0.0020
0.0022
-0.0004
0.0
107.66
0.0
0.34
0.0
0.10
0.0
0.02
0.0
7.23
0.0
-1.59
0.0
0.40
0.0
0.0
0.0
1.26
0.0
-0.71
0.0
0.0
0.10
0.0
-0.47
0.0
0.18
0.0
Table 5 also includes the resulting coefficients â0 through τ5
and the corresponding functions are shown in Figure 7.
Finally we examined the deformations of the CH2 groups
during half a cycle of the pseudorotation. It is interesting to
note that all C-H distances can be represented by two
formulas: one for the distances at the C2 and C5 atoms and the
other for the distances at the C3 and C4 atoms. Only the signs
of the coefficients are different for each C-H bond. Similarly
the H-C-H bond angles and the rocking, wagging, and twisting
angles are represented by the following formulas
0.0
0.08
initio results it became evident that most of these assumptions
were quite poor. In particular it turned out to be important to
take into account the differences of the ring C-C bond
distances, the rocking of the CH2 groups, and the difference of
the C-Cl bond distances in the case of the Cs symmetry.
Improving the structural models in this way it soon became
apparent that in analogy to 1,1-dicyanocyclopentane1 the
experimental intensities for DClCP are best reproduced by using
the envelope form (Cs symmetry), but not the C2 conformation,
when the static model was employed. The predominance of the
Cs form is consistent with the results predicted by all ab initio
methods that have been applied for this study (Tables 1 and
S1).
rC H ) r2,0 ( r2,c1 cosφ ( r2,c2 cos2φ ( r2,c3 cos3φ (
2,5
r2,c4 cos4φ ( r2,s1 sinφ ( r2,s2 sin2φ ( r2,s3 sin3φ (
r2,s4 sin4φ
rC H ) r3,0 ( r3,c1 cosφ ( r3,c2 cos2φ ( r3,c3 cos3φ (
3,4
r3,c4 cos4φ ( r3,s1 sinφ ( r3,s2 sin2φ ( r3,s3 sin3φ (
r3,s4 sin4φ
Since the energy values which were obtained from the ab
initio calculations at various levels of theory have shown (Table
7) that some contribution of the C2 conformer to the diffraction
intensities cannot be excluded, we also applied a two-conformer
model and fitted the ratio of the two conformers. In the latter
calculations it proved to be important to additionally take into
account the changes of the geometrical parameters and vibra-
tional amplitudes in going from the Cs to the C2 symmetry. To
limit the number of geometrical parameters the calculations were
finally performed by using the dependency of the geometrical
parameters on φ, as described in the previous chapter, with φ
restricted to 180° (Cs) and 90° (C2).
âHC H ) â2,0 + â2,c2 cos2φ + â2,c4 cos4φ ( â2,s2 sin2φ (
2,5
â2,s4 sin4φ
âHC H ) â3,0 + â3,c2 cos2φ + â3,c4 cos4φ + â3,c6 cos6φ (
3,4
â
3,s2 sin2φ ( â3,s4 sin4φ ( â3,s6 sin6φ
FC H ) F2,c1 cosφ + F2,c3 cos3φ + F2,c5 cos5φ (
2,5
F
2,s1 sinφ ( F2,s3 sin3φ ( F2,s5 sin5φ
FC H ) F3,c1 cosφ + F3,c3 cos3φ + F3,c5 cos5φ (
3,4
To obtain reliable starting values for the vibrational ampli-
tudes lij we carried out a normal coordinate analysis. All
calculations were based on the results of the ab initio calcula-
tions, using the HF/6-31G** method for the Cs and C2
conformers. The procedure we applied for the determination of
these amplitudes has been described previously.16 It is worth-
while to note that initially the values for several calculated
vibrational amplitudes of nonbonded distances were unreason-
ably large. The subsequent elimination of the contributions
originating from the lowest vibrational frequency, however, has
led to generally plausible values.
F
3,s1 sinφ ( F3,s3 sin3φ ( F3,s5 sin5φ
ωC H ) ( ω2,0 ( ω2,c2 cos2φ ( ω2,c4 cos4φ (
2,5
ω
2,c6 cos6φ + ω2,s2 sin2φ + ω2,s4 sin4φ + ω2,s6 sin6φ
ωC3,4H ) ( ω3,0 ( ω3,c2 cos2φ ( ω3,c4 cos4φ (
ω
3,c6 cos6φ + ω3,s2 sin2φ + ω3,s4 sin4φ + ω3,s6 sin6φ
τC H ) ( τ2,c1 cosφ ( τ2,c3 cos3φ ( τ2,c5 cos5φ +
2,5
τ
2,s1 sinφ + τ2,s3 sin3φ + τ2,s5 sin5φ
The final results of the structural analysis gained from the
refinement of the one or two conformer static models are shown
in Table 8.
τC H ) ( τ3,c1 cosφ ( τ3,c3 cos3φ ( τ3,c5 cos5φ +
3,4
τ
3,s1 sinφ + τ3,s3 sin3φ + τ3,s5 sin5φ (10)
However, all calculations described above gave results which
were not fully satisfying, particularly in the region between 3
and 5 Å of the radial distribution function (cf. Figures10 and
11). As can be seen from Figure 11 this is the region with the
strongest dependency of the radial distribution function on the
phase angle φ. This prompted us to start a large-amplitude
treatment of the geometry of DClCP. Such a treatment can be
done by fitting a dynamic model similar to what we described
previously1 for 1,1-dicyanocyclopentane. A detailed description
of an extended treatment is reported in the next section. Another
method of treatment is to fit an “effective” structure by using
a static model of C1 symmetry. Fitting such a model can be
achieved by adding the phase angle φ as a free parameter. This
model improves the agreement between the experimental and
theoretical radial distribution function within the range between
The coefficients in these formulas are displayed in Table 6
and the changes in the geometry are shown in Figure 8 for the
groups H8-C2-H9 and H10-C3-H11; the corresponding rep-
resentations for the remaining CH2 groups are obtained simply
by rotating the figures by 180°.
Static Models
We started the analysis of the electron diffraction data by
applying static models of Cs symmetry and C2 symmetry. Since
not all independent structural parameters of DClCP can be
determined from the electron diffraction intensities, initially a
number of simplifying assumptions had to be introduced for
the data refinement. However, with the availability of the ab

4664 J. Phys. Chem. A, Vol. 108, No. 21, 2004
Dakkouri et al.
Figure 5. Variation of the ring C-C bond distances and density of delocalized net charge (-‚-) with the phase angle φ. The density function was
shifted upward by 1.53 to fit into the figure.
Figure 6. Variation of the ring C-C-C bond angles and density of delocalized net charge (-‚-) at the apex atoms with the phase angle φ. The
density function was scaled by 57.3 and shifted upward by 104° or 105° to fit into the figure.

Molecular Structure of 1,1-Dichlorocyclopentane
J. Phys. Chem. A, Vol. 108, No. 21, 2004 4665
TABLE 6: Deformation of the CH2 Groups with Phase Angle O: Fourier Coefficients of the Geometrical Parameters (Eq 10)
sign of coefficient for the bond
name
coefficient
C2-H8
C2-H9
C5-H14
C5-H15
r2,0
r2,c1
r2,c2
r2,c3
r2,c4
r2,s1
r2,s2
r2,s3
r2,s4
1.0887
0.0021
0.0009
0.0003
0.0000
0.0010
0.0014
0.0000
0.0002
+
+
+
-
+
-
+
+
-
-
+
-
+
+
+
-
-
-
-
-
+
+
-
+
+
-
+
-
-
+
+
+
-
+
+
-
sign of coefficient for the bond
name
coefficient
C3-H10
C3-H11
C4-H12
C4-H13
r3,0
1.0908
0.0004
0.0011
0.0000
0.0002
0.0011
0.0001
0.0004
0.0001
+
+
-
-
+
-
-
+
-
+
-
-
+
+
+
-
+
-
-
+
+
-
+
+
+
+
+
-
r3,c1
r3,c2
r3,c3
r3,c4
r3,s1
r3,s2
r3,s3
r3,s4
-
+
+
+
-
-
-
+
order
âHC
FHC
ωC
H
2,5
τC
âHC
FHC
ωC
τC
H
2,5
H
H
H
3,4
H
3,4
H
H
3,4
2,5
2,5
3,4
0
107.95
0.0
0.06
0.0
0.03
0.0
0.0
0.0
0.44
0.0
0.01
0.0
0.0
0.0
-9.58
0.0
0.44
0.0
0.05
0.0
-6.03
0.0
1.90
0.0
-0.20
0.0
-4.99
0.0
-0.78
0.0
0.19
0.0
0.0
0.0
-0.18
0.0
0.21
0.0
-0.05
0.0
-0.49
0.0
-1.18
0.0
0.03
0.0
0.68
0.0
-0.74
0.0
107.32
0.0
-0.29
0.0
0.02
0.0
0.01
0.0
0.27
0.0
0.0
0.77
0.0
1.19
0.0
0.14
0.0
5.69
0.0
0.40
0.0
-0.10
0.0
-1.00
0.0
-0.18
0.0
-0.18
0.0
-0.03
0.0
-0.34
0.0
-0.03
0.0
0.0
-0.83
0.0
-0.89
0.0
0.10
0.0
-0.88
0.0
1.19
0.0
-0.03
0.0
C1
C2
C3
C4
C5
C6
S1
S2
S3
S4
S5
S6
-0.01
0.0
0.01
0.16
0.0
0.01
3 and 5 Å. The results of the calculations from applying a static
model of C1 symmetry have been included in Table 8.
of eqs 3, 6, and 7. However, not all coefficients in these
equations can be refined to fit the electron diffraction patterns.
We decided to fix the higher order Fourier coefficients at the
values displayed in Tables 2 and 4. This leaves the parameters
rCC, q0, R0,512, and R0,123 as adjustable parameters of the ring
geometry. In the fitting process it turned out that the parameter
Large-Amplitude Treatment
To account for large-amplitude motions in the analysis of
gas-phase electron diffraction data the assumption is made that
the large-amplitude motion is a slow motion, and the reduced
total molecular intensity s‚M(s) can be written as17,18
R
0,123 could not be adjusted reasonably. Therefore, this parameter
was maintained at a value that enforced the Cs symmetry at φ
) 180°.
2π w(φ) s‚M(s,R(φ)) dφ/ 2π w(φ) dφ
(11)
Similarly the dependency of the geometry of the CCl2 and
CH2 groups on the phase angle φ was taken into account by
applying eqs 8-10. Again the zeroth order parameters (R0, â0)
and one first-order parameter (F1) were taken as adjustable
parameters for the CCl2 group, and for the CH2 groups the mean
values (r2,0 + r3,0)/2 and (â2,0 + â3,0)/2 were used. The higher
order coefficients were taken from Tables 5 and 6.
A second problem to be solved is the correlation between
the root-mean-square amplitudes lij and the phase angle φ. Using
the scaled quadratic ab initio force fields for both conformations
Cs and C2 we have calculated the vibrational amplitudes within
the range π/2 e φ e 3π/2 with a step width of δφ ) 9°.
Hilderbrandt et al.19,20 tried to establish a dependency of the
vibrational amplitudes on φ of the form
s‚M(s)total
)
∫
∫
0
0
where R(φ) designates the geometry of the molecule at the phase
angle φ. The weight function w(φ) at temperature T with
normalizing factor N is given by
w(φ) ) Ne-V(φ)/RT
(12)
In principle nine independent structural parameters are
required to calculate the positions of the atoms in a five-
membered ring with no symmetry. We chose the 5 C-C bond
distances, the bond angles C5-C1-C2 and C1-C2-C3, the
puckering amplitude q, and the phase angle φ from the
pseudorotational model. Utilizing eq 2 and setting x1 ) 0, the
Cartesian coordinates of all ring atoms can be calculated.
Though this involves the solution of three quadratic equations
there is normally no difficulty in selecting the solutions which
suit the above parameters. To set up the molecular geometry at
a selected value of the phase angle φ additional use was made
lij ) lij0 + a(1 - cosφ)
where a is a constant. However, a detailed analysis of some
selected vibrational amplitudes shows that the dependency of,

4666 J. Phys. Chem. A, Vol. 108, No. 21, 2004
Dakkouri et al.
Figure 7. Deformation of the CCl2 group dependent on the phase angle φ.
e.g., the vibrational amplitude l(Cl6...H10) on φ is of the form
propagated error σ2 and the total error σtot has been described
elsewhere.16,21 Figure 9 shows the experimental and theoretical
reduced molecular intensity s‚M(s) and Figure 10 the experi-
mental and theoretical radial distribution function from the
pseudorotational model.
It should be noted that the values in Table 8 correspond to
the ra representation of the molecular structure. The data required
to convert the ra representation into the rR representation are
available as Supporting Information (Table S2). For details about
ra, rR, and other representations see for instance Kuchitsu and
Cyvin.22
Table 9 shows a comparison of the bond distances, bond
angles, and dihedral angles which have been obtained from the
electron diffraction study and by using eqs 3 and 6-10 with φ
) 90° (C2 symmetry) and φ ) 180° (Cs symmetry) with those
obtained from the ab initio calculations at the MP2/6-311+G-
(2df,2pd) level.
l6,10 ) l0 + c1(1 - cosφ) + c2(1 - cos2φ) + c3(1 -
cos3φ) + c4(1 - cos4φ) + s1(1 - sinφ) + s2(1 - sin2φ) +
s3(1 - sin3φ) + s4(1 - sin4φ)
Therefore, as described in a previous paper,1 we preferred to
use the differences lij(φ*180°) - lij(φ)180°) of the calculated
vibrational amplitudes: these were introduced into the analyzing
programs as constants to be added to the vibrational amplitudes
of the most stable conformer (symmetry Cs at φ ) 180°). This
limits the possibility of adjusting the vibrational amplitudes in
the least-squares procedure to cases where the fluctuation of
the calculated amplitudes is sufficiently small. Values at
intermediate φ were obtained by interpolation.
To calculate s‚M(s)total from eq 11 we used the usual
assumption that the integral can be appoximated sufficiently
well by finite sums. Therefore, the geometry of the molecule
was set up at various values of the phase angle φ, and the
corresponding M(s,R(φ)) was calculated. For symmetry reasons
it is sufficient to use an interval width of π. Actually, we used
a step width of δφ ) 5° in the interval 90° e φ e 270°, giving
a total of 37 intermediate conformers.
No attempts have been undertaken to extract the potential
constants V2, V4, and V6 from the experimental gas electron
diffraction data. Instead these constants were fixed at the values
given in Table 2.
The fitted structural parameters and most important vibrational
amplitudes resulting from the large-amplitude treatment are
included in Table 8. In this table the different σ values have
the following meaning: σ1 is the single standard error of the
fitting procedure, σ2 is the propagated error resulting from the
estimated uncertainties of the fixed parameters, and σtot is the
total error estimate. The method we used to calculate the
Discussion
The most interesting result of the theoretical calculations for
DClCP is the differences in the ring C-C bond distances and
their dependency on the actual configuration. As can be seen
in Tables 1 and S1 the C1-C2 bond distance increases and the
C3-C4 bond distance decreases on the order of 0.02 Å in going
from the Cs to the C2 conformation. In contrast, the decrease of
the C2-C3 bond length is only on the order of 0.008 Å. These
variations of the ring C-C bond distances are consistent not
only for all computational methods which have been applied in
this work, but also for all other 1,1-disubstituted derivatives of
cyclopentane investigated thus far (Tables 1 and 10, and ref 1).
To rationalize this observation we assume that the variation of
the geometry is due to the variation of the overall density
distribution of charges throughout the ring with the phase angle
φ of the pseudorotational motion. Therefore the constants of

Molecular Structure of 1,1-Dichlorocyclopentane
J. Phys. Chem. A, Vol. 108, No. 21, 2004 4667
Figure 8. Dependency of geometrical parameters in the CH2 groups on the phase angle φ: upper row, C2-H and C3-H distances; middle row,
bond angles H8-C2-H9 and H10-C3-H11; lowest row, rocking, wagging, and twisting angles of the H8-C2-H9 and H10-C3-H11 groups.
TABLE 7: Energy Difference ∆E ) Eeq - Eax (kcal/mol) for MFCP, MClCP, MBrCP, Monocyanocyclopentane (MCCP), and
Monoethynylcyclopentane (MECP) and ∆E ) EC - EC for DFCP, DClCP, and DBrCP from ab Initio Calculations on Various
2
s
Levels of Theory
method
MFCP
MClCP
MBrCP
MCCP
MECP
DFCP
DClCP
DBrCP
HF/cc-pVDZ
HF/6-31G(d,p)
B3LYP/6-311+G(df,pd)
MP2/6-311G(d,p)
MP2/6-311+G(d,p)
MP2/6-311++G(d,p)
MP2/6-311+G(df,pd)
MP2/6-311++G(df,pd)
MP2/6-311+G(2df,2pd)
MP2/6-311++G(2df,2pd)
1.96
2.10
1.72
2.75
2.77
2.72
2.88
2.84
2.38
2.35
1.17
1.55
0.48
0.82
0.28
0.75
-0.47
-0.73
0.58
0.39
-0.46
-0.19
-0.15
-0.20
-0.08
-0.11
-0.30
-0.32
1.82
3.14
0.41
0.85
0.77
0.84
0.90
0.53
0.37
3.05
3.13
3.10
1.45
1.44
1.62
1.62
1.04
1.20
1.08
1.18
0.51
0.50
the function d(R,φ) (eq 4) introduced in the section Pseudoro-
tation were adjusted such that the ring C-C bond distances
obtained at various values of φ and qmin are optimally
reproduced. In Figure 4 the integral of the curve for φ ) 0° (Cs
symmetry, solid line) between atom 3 and atom 4 is clearly
positive: this indicates a reduction in the density of the net
charge between these atoms resulting in the long C3-C4 bond
distance of 1.554 Å (Table S1, HF/cc-pVDZ method). On the
other hand, the respective integral between atom 1 and atom 2
is negative resulting in the short C1-C2 bond distance of 1.525

4668 J. Phys. Chem. A, Vol. 108, No. 21, 2004
Dakkouri et al.
TABLE 8: Final Model Parameters (ra) and Errors for 1,1-Dichlorocyclopentane As Obtained from the Fit of Static Models
and the Pseudorotation Modela
static models
pseudorotation model
m
n
o
Cs
Cs + C2
C1
value
σ1
σ2
σtot
geometrical parameters (distances in Å, angles in deg)
b
c
rC
q0
1.534(2)
0.397(12)
1.781(2)
1.115(8)
1.535(2)
0.404(13)
1.781(2)
1.114(6)
1.535(2)
0.405(14)
1.781(2)
1.114(7)
1.535
0.409
1.780
1.114
0.0005
0.0042
0.0007
0.0021
0.0004
0.0020
0.0002
0.0001
0.001
0.011
0.002
0.005
d
R0
e
rCH
f
R0,512
105.7(11)
180.0(fix)
107.6(4)
103.5(30)
8.1(20)
104.9(11)
90.0/180.0 (fix)
107.9(4)
104.1(31)
8.1(19)
105.4(9)
161.3(11)
107.8(4)
103.1(33)
10.1(33)
104.4
0.33
0.13
0.8
φg
h
â0
108.1
104.6
10.0
0.12
1.03
0.67
0.04
2.22
0.31
0.3
2.6
1.7
i
âHCH
j
F1
fitted vibrational amplitudes (Å)
l(C-C)
0.053(2)
0.056(3)
0.090(7)
0.059(5)
0.101(10)
0.078(19)
0.173(16)
0.084(7)
0.078(6)
0.053(2)
0.056(2)
0.091(6)
0.060(5)
0.103(9)
0.077(5)
0.142(15)
0.079(8)
0.077(6)
94.2(52)
4.55
0.052(2)
0.056(2)
0.089(6)
0.058(5)
0.102(9)
0.074(24)
0.130(54)
0.084(7)
0.077(6)
0.053
0.0006
0.0008
0.0020
0.0015
0.0030
0.0021
0.0046
0.0022
0.0019
0.0001
0.0002
0.0000
0.0017
0.0005
0.0009
0.0039
0.0006
0.0008
0.002
0.002
0.005
0.005
0.007
0.005
0.013
0.006
0.005
l(C-Cl)
l(C-H)
l(C‚‚‚C)
l(C‚‚‚H1)
l(C‚‚‚Cl1)
l(C‚‚‚Cl2)
l(C‚‚‚.Cl3)
l(Cl‚‚‚Cl)
% Cs
0.056
0.090
0.060
0.103
0.075
0.112
0.078
0.076
k
Rlong
5.01
7.78
0.0419
4.61
7.55
4.44
7.15
k
Rshort
7.22
0.0352
l
∑
0.0385
0.0345
a Errors in parentheses (static models, in units of last digit) are three times the standard errors of the fit; for errors of the pseudorotational model
see text. The fitted parameters are the zeroth and first-order coefficients defined in eqs 3 and 6-10. The higher order coefficients given in Tables
2-6 were used. b Constant term of the ring C-C bond distances (eq 6). c Constant term of the puckering amplitude (eq 3). d Constant term of the
bond distances C-Cl (eq 8). e Constant term of the C-H bond distances (eq 10): rCH ) (r2,0 + r3,0)/2. f Constant term of the angle C5-C1-C2 (eq
7). g Phase angle of pseudorotation. h Constant term of the angle Cl-C-Cl (eq 9). i Constant term of the H-C-H angles (eq 10): âHCH ) (â2,0
+
â
3,0)/2. j First Fourier coefficient of the rocking angle of the Cl-C-Cl group at Cs symmetry (eq 9). k R ) [∑wi∆i2/∑(wis2Mi2(obs))]1/2, where ∆i
) sMi(obs) - sMi(calc). l ∑ is the weighted sum of squared residuals. m σ1 is the single standard error of the fit. n σ2 is the propagated error (see
o
2
2
refs 16 and 21). σtot ) [6σ1 + 3σ2 ]1/2
Figure 9. Experimental (×) and theoretical (-) reduced molecular
intensity s‚M(s) for DClCP.
Figure 10. Experimental (×) and theoretical (-) radial distribution
function from the pseudorotational model for DClCP.
Å. In the same way the curve for φ ) 90° (C2 symmetry, dotted
line) shows that the distances C2-C3 and C3-C4 are very similar
while the bond distance C1-C2 is long. Similar results were
also obtained for DCCP.1 We also note that the bond angles
C-C-C are correlated with the variation of the charge density
at the apex atom: this is demonstrated in Figure 6.
One further interesting result of the investigation of the
pseudorotation is the curvature of the valley Emin(φ) and the
corresponding puckering amplitude qmin(φ) as shown in Figure
3. A Fourier analysis shows that terms up to 6th order are
required to satisfactorily reproduce the dependency on φ.
As can be seen from Figure 7, the deformation of the CCl2
group shows an unexpected form. Clearly the local minimum
of the axial C-Cl bond distance at φ ) 180°, the opening of
the Cl-C-Cl bond angle, and the almost constant value of the
rocking angle over the range of about 100° are correlated with
the variation of the charge density at atom 1.
The following remarks emerge from a closer inspection of

Molecular Structure of 1,1-Dichlorocyclopentane
J. Phys. Chem. A, Vol. 108, No. 21, 2004 4669
ring C-C bonds in the axial conformer of C1-C2 ) 1.544(8)
Å, C2-C3 ) 1.551(13) Å, and C3-C4 ) 1.525(22) Å. From
Table 10 it is apparent that the average of the predicted values
by the MP2/6-311+G(df,pd) method for the C-C bond lengths
is by 0.017 and 0.006 Å smaller than was obtained by LMH24
and HS,19 respectively. Moreover, the disagreement between
the different C-C bond distances which have been presented
by GLD25 and those shown in Table 10 is significantly large. It
is perhaps worth noting that the experimentally observed average
C-C bond length is 0.018 Å shorter in DClCP (Table 8) than
in MClCP.24 This difference is most likely due to the overes-
timation of the ring C-C bond distances in MClCP and not
the result of the disubstitution in DClCP. Furthermore, the
observed C-Cl bond length in MClCP of 1.810 Å19 agrees quite
well with the predicted value of 1.800 Å (Table 10).
There is a notably good agreement between the theoretically
predicted and experimentally determined values for the bond
angles in MClCP. In the MW studies24,25 it was found that the
angles between the C-Cl bond and the C2-C1-C5 plane, the
C2-C3-C4 angle, and the puckering angle are 123.5°, 105.8°,
and 40.6°, and 124.8°, 106.7°, and 41.6°, respectively. From
Table 10 it is apparent that the predicted values for these angles
are 122.8°, 105.6°, and 41.0°.
Figure 11. Theoretical total radial distribution function and differences
between the contributions from the distinct conformations (φ ) 90°
through 270°) and the total radial distribution function for DClCP.
the results which were obtained from the electron diffraction
analysis and are shown in Table 8:
It is perhaps of interest that we observe a direct relationship
between the puckering amplitude q and the flap angle R for the
envelope Cs conformation of the form:
First, we note that a one- or two-conformer statical model
describes the experimental data surprisingly well. This obser-
avtion is clearly indicated by the value of ∑, the sum of weighted
squared residuals of the fit. Obviously the quality of the fit can
be slightly improved by releasing the restrictions imposed by
enforcing Cs symmetry in favor of C1 symmetry. A rather better
description is achieved by allowing for a two-conformer model
with Cs and C2 symmetry. Therefore it is reasonable that the
pseudorotational model, which in our case allows for 37
conformers, leads to a further improvement of the fit.
Second, the results for the one-conformer model of C1
symmetry can well be taken as an “effective“ structure. The
intramolecular distances obtained for this model compare very
well with the average of the distances obtained from all
configurations with the phase angle φ taken from the interval
π/2 e φ e π, and using the weight function eq 12. A similar
“effective“ structure has been defined in the treatment of the
internal rotation of methyl groups.23
R/q ) σs
where σs is a constant. We checked the validity of this simple
formula for a series of geminally and monosubstituted (axial
and equatorial) cyclopentanes and found that σs ≈ 100 Å-1 for
all those cyclopentanes which we studied in this work by means
of quantum mechanical methods of the type cp-X (where X )
HF, HCl, HBr, F2, Cl2, and Br2). Only the σs value of 102 Å-1
for DFCP deviates slightly from this magnitude. Another
exception is provided by 1,1-dicyanocyclopentane (DCCP) and
monocyanocyclopentane (MCCP) which we studied previously.1
Both compounds possess a σs value of 98 Å-1. On the basis of
these results it is conceivable that a certain series of compounds
possesses its own specific value of σs. It remains to note that
the σs value resulting from the experimental data for DClCP
(Table 9) is also 100(4) Å-1. Assuming a general validity of
this relationship it can be postulated with some reservation that
applying this equation would allow for the prediction of either
the puckering amplitude or the flap angle for a wide range of
substituted cyclopentanes.
Third, consideration of the results in Table 9 reveals that there
is good agreement between the values of the parameters obtained
from the various types of experimental analysis. Equally good
is the consistency between the theoretically predicted values
and those which have been determined from the experiment.
Perhaps the C-Cl and C-H bond distances and the angles Cl-
C-Cl and H-C-H represent some exceptions.
It also should be added that a similar relationship has been
observed for the C2 (half-chair) conformer:
Some years ago Loyd, Mathur, and Harmony (LMH)24 and
Groner, Lee, and Durig (GLD)25 investigated the microwave
spectrum of monochlorocyclopentane (MClCP). In 1982 Hilder-
brandt and Shen (HS)19 also studied the conformation and
structure of MClCP by applying the electron diffraction
technique. All these investigations revealed that MClCP exists
in the envelope Cs form with the chlorine atom either in the
axial or in the equatorial position. Consistently, it also has been
found that the axial conformer is slightly more stable than the
equatorial form. As can be seen from Table 7, this finding is
clearly supported by all ab initio procedures we employed in
this work. While both the microwave24 and the electron
diffraction19 studies provided only an average C-C bond length
of 1.553(5) and 1.542(1) Å, respectively, the microwave
investigation25 presented three different bond distances for the
τ/q ) σ2
where τ is the twist angle C2-C3-C4-C5 and σ2 is a constant.
For those compounds we investigated so far by means of
computational methods, it turned out that σ2 ) 103 Å-1. Should
such a relationship prove to be applicable to many other
substituted cyclopentanes this would permit the prediction of
either q or τ.
In all mono- and disubstituted cyclopentane molecules we
studied in the present work it has been observed that the axial
carbon-halogen bond is always longer than the equatorial one.
The MP2/6-311+G(2df,2pd) calculations for instance suggest
for DClCP and MClCP (axial and equatorial) a bond length
difference of 0.017 Å, and for DFCP and MFCP 0.012 and 0.013
Å (Tables 1 and 10), respectively. Loyd et al.24 tried more or

4670 J. Phys. Chem. A, Vol. 108, No. 21, 2004
Dakkouri et al.
TABLE 9: Comparison of ra Distances, Bond Angles, and Dihedral Angles of 1,1-Dichlorocyclopentane (DClCP) at O ) 90° and
180° As Obtained from Electron Diffraction and ab Initio Calculations (MP2/6-311+G(2df,2pd))a
Cs + C2
pseudorot. model
ab initio
Cs
C2
Cs
C2
Cs
C2
bond distances (Å)
C1-C2
1.523(2)
1.537(2)
1.552(3)
0.396(11)
1.788(3)
1.770(3)
1.114(5)
1.543(2)
1.529(2)
1.528(3)
0.423(11)
1.783(2)
1.783(2)
1.114(5)
1.523(2)
1.537(2)
1.552(3)
0.400(11)
1.787(3)
1.769(3)
1.114(5)
1.544(2)
1.529(2)
1.528(3)
0.428(11)
1.782(2)
1.782(2)
1.114(5)
1.519
1.535
1.554
0.419
1.794
1.776
1.088
1.541
1.527
1.528
0.450
1.790
1.790
1.089
C2-C3
C3-C4
qb
C1-Cl6
C1-Cl7
c
C-H
bond angles (deg)
C1-C2-C3
C2-C3-C4
C5-C1-C2
Cl-C-Cl
104.2(5)
105.8(3)
103.4(9)
108.3(3)
104.1(26)
104.1(26)
122.3(7)
129.3(8)
39.5(9)
104.9(6)
102.7(5)
106.0(9)
107.6(3)
104.1(26)
104.1(26)
126.2(2)
126.2(2)
104.3(5)
105.7(3)
103.0(9)
108.6(3)
104.6(26)
104.6(26)
121.3(8)
130.1(9)
39.9(9)
105.0(6)
102.7(5)
105.5(9)
107.9(3)
104.6(26)
104.6(26)
126.1(1)
126.1(1)
103.2
105.7
103.5
109.0
108.5
107.4
122.1
128.9
42.2
104.1
102.1
106.3
108.1
108.3
108.1
125.9
125.9
H-C2-H
H-C3-H
d
â6
e
â7
Rf
dihedral angles (deg)
C1-C2-C3-C4
C4-C5-C1-C2
C2-C3-C4-C5
-24.0(7)
-39.2(10)
0.0(-)
35.3(9)
-13.5(4)
-43.7(9)
-24.4(7)
-39.6(10)
0.0(-)
35.8(9)
-13.7(4)
-44.1(9)
-25.3
-41.7
0.0
37.3
-14.3
-46.5
a Errors in parentheses include the contributions from the estimated uncertainties of the fixed parameters. b Puckering amplitude for the ring.
c Average value. d Angle between the bond Cl6-C1 and the plane C5C1C2. e Angle between the bond Cl7-C1 and the plane C5C1C2. f Angle between
the C5C1C2 plane and C2C3C4C5 plane (flap angle).
TABLE 10: Optimized Geometrical Parameters of Monochlorocyclopentane (MClCP) and Monofluorocyclopentane (MFCP) As
Predicted by Different Levels of Theory
MClCP
MFCP
axial
equatorial
axial
equatorial
B3PW91
/6-311
MP2
/6-311
B3PW91
/6-311
MP2
/6-311
B3PW91
/6-311
MP2
/6-311
B3PW91
/6-311
MP2
/6-311
parameter
+G(df,pd)
+G(2df,2pd)
+G(df,pd)
+G(2df,2pd)
+G(df,pd)
+G(2df,2pd)
+G(df,pd)
+G(2df,2pd)
bond distances (Å)
C1-C2
1.521
1.538
1.549
0.378
1.089
1.822
1.092
1.520
1.537
1.550
0.411
1.086
1.801
1.089
1.521
1.537
1.553
0.414
1.092
1.802
1.092
1.519
1.536
1.554
0.441
1.090
1.784
1.089
1.518
1.539
1.550
0.375
1.094
1.402
1.092
1.513
1.538
1.551
0.405
1.089
1.400
1.088
1.520
1.536
1.555
0.401
1.097
1.390
1.093
1.516
1.534
1.555
0.430
1.093
1.388
1.089
C2-C3
C3-C4
qa
C1-H6
C1-Cl(F)7
b
C-H
bond angles (deg)
C1-C2-C3
C2-C3-C4
C5-C1-C2
Cl(F)-C-H
H-C2-H
104.8
105.8
103.5
104.6
107.4
106.9
131.3
124.0
37.8
104.0
105.6
102.9
105.9
108.0
107.3
131.4
122.8
41.0
103.1
105.9
103.8
104.9
107.7
106.7
125.3
129.8
41.7
102.6
105.6
103.1
106.1
108.1
107.0
124.4
129.5
44.5
104.7
105.9
104.0
106.4
107.5
106.8
131.6
122.0
37.6
103.7
105.6
103.7
107.0
108.1
107.2
132.5
120.6
40.9
103.6
105.9
103.9
106.4
107.7
106.5
125.8
127.9
40.4
102.9
105.5
103.4
107.0
108.2
106.9
123.3
127.7
43.5
H-C3-H
c
â6
â7
d
Re
dihedral angles (deg)
C1-C2-C3-C4
C4-C5-C1-C2
23.1
37.6
25.0
40.7
24.9
41.1
26.5
43.7
22.8
37.4
24.6
40.5
24.2
40.4
25.9
42.8
µf
2.33
2.36
2.44
2.53
1.99
2.10
2.14
2.30
a Puckering amplitude for the ring, calculated with the program RING.12,13 b Average value. c Angle between the bond H6-C1 and the plane
C5C1C2. d Angle between the bond Cl(F)7-C1 and the plane C5C1C2. e Angle between the C5C1C2 plane and the C2C3C4C5 plane (flap angle).
f Dipole moment.
less speculatively to rationalize the higher stability of the axial
conformer in comparison to the equatorial one in MClCP by
invoking dipole-dipole interaction between the C-Cl bond
dipole and the ring dipole. It is conceivable that this kind of
dipolar interaction might be responsible for the difference
between the axial and equatorial C-X (X ) F, Cl, Br) bond
lengths in the investigated molecules. It may also be of interest
to indicate that the C-Cl bond is shortened by about 0.007 Å
on going from MClCP to DClCP, and the C-F bond in DFCP
decreases by about 0.027 Å in comparison to MFCP. This bond
contraction with increasing halogen substitution has commonly
been attributed to the negative hyperconjugation effect. How-
ever, we would like rather to ascribe this bond shortening simply
to increasing bond ionicity with increasing halogen substitution
and subsequent Coulombic attractive interaction between the
negatively charged halogen atoms and the increasingly more

Molecular Structure of 1,1-Dichlorocyclopentane
J. Phys. Chem. A, Vol. 108, No. 21, 2004 4671
TABLE 11: Atomic Charges As Obtained from Natural Population Analysis (NPA) and Mullikan Population Analysis (MPA)
for 1,1-Dichlorocyclopentane (DClCP)
NPA
MPA
Cs symmetry
C2 symmetry
Cs symmetry
C2 symmetry
B3PW91
/6-311
MP2
/6-311
B3PW91
/6-311
MP2
/6-311
B3PW91
/6-311
MP2
/6-311
B3PW91
/6-311
MP2
/6-311
atom
+G(df,pd)
+G(2df,2pd)
+G(df,pd)
+G(2df,2pd)
+G(df,pd)
+G(2df,2pd)
+G(df,pd)
+G(2df,2pd)
C1
-0.063
-0.434
-0.403
-0.028
-0.002
0.233
0.223
0.215
0.213
-0.016
-0.365
-0.339
-0.052
-0.024
0.200
0.186
0.184
0.180
-0.064
-0.433
-0.398
-0.019
-0.019
0.227
0.235
0.216
0.204
-0.018
-0.364
-0.335
-0.042
-0.042
0.192
0.202
0.184
0.171
0.724
-0.647
-0.497
-0.240
0.171
0.208
0.210
1.347
-0.242
-0.029
-0.528
-0.208
-0.011
-0.007
0.000
1.562
-1.049
-0.406
-0.134
-0.134
0.216
0.209
0.184
0.199
1.812
-0.424
0.001
-0.442
-0.442
0.002
-0.007
-0.024
-0.013
C2,5
C3,4
Cl6
Cl7
H8
H9
H10
H11
0.214
0.185
-0.016
positively charged carbon atom. The smaller effect on the C-Cl
bond is due to the smaller difference in the electronegativity
between chlorine and carbon. This kind of rationalization is fully
supported by the atomic charges we obtained from the natural
population analysis (NPA) as shown in Tables 11, S5, S6, S8,
and S10.
We also conducted natural population analysis (NPA)26-29
and Mulliken population analysis (MPA)30 for DFCP, DClCP,
DBrCP, and their monohalogenated derivatives. In addition, we
performed charge distribution analysis on monoethynylcyclo-
pentane (MECP) in its axial and equatorial form. For brevity
only the NPA and MPA atom charges for DClCP are shown in
Table 11. All other NPA and MPA result, which we obtained
for a variety of monosubstituted cyclopentanes, are available
as Supporting Information (Tables S5-S11).
On many occasions it has been demonstrated27,31,32 that the
NPA method has substantial advantages over the MPA proce-
dure and avoids various deficiencies incorporated in the latter
population analysis. The most aggravating disadvantage of the
MPA, however, is its sensitivity to variation of the basis
set.29,32-35 A comparison between the NPA and MPA atom
charges for DClCP (Table 11) and DFCP (Tables S6 and S7)
clearly visualizes the dependency of the MPA method on the
level of theory and the size of the basis set. As is shown in
Table 11 (and Tables S7, S9, and S11), besides the evident
fluctuation of the atom charges with the variation of the basis
set there are instances where the atom charges at some atoms
even reverse their signs.
that all other C-C bond lengths as well as all endocyclic valence
angles remain almost unaffected by the variation of the
substituents (in the mono- and disubstituted series alike). The
only exception is found in DFCP where the C5-C1-C2 angle
widens by approximately 1.8° in comparison to all other
compounds. The pronounced variation of the C1-C2 bond length
and the small change of the remaining C-C bonds with the
alternation of the electronegativity of the substituent are mirrored
in the NPA charge distribution shown in Tables 11, S6, S8,
and S10 and in those values for MBrCP and DBrCP, which are
not shown in this work. The most dramatic changes of the charge
density occur at the anchor atom C1 while the charges at the
other carbon atoms vary only slightly.
Scrutinizing the behavior of the puckering angle R reveals
that consistently all ab initio methods we used demonstrate that
the puckering angle increases upon moving from the axial form
of MClCP to DClCP (Tables 1 and 11). The variation of this
angle in the case of the fluorine counterparts is, however,
negligible (less than 1°). In contrast, this angle widens by about
2° to 3° in the equatorial form of MClCP in comparison to
DClCP. This variation of the puckering angle R parallels the
behavior of the puckering amplitude q. The correlation between
these two magnitudes has been discussed above.
The following general conclusion emerges from the data in
Table 7: It is discernible that in all cases displayed in this table
there is an apparent jump in the ∆E values on going from the
DFT method to the MP2 method and in some cases the
conformational preferability is even reversed. For instance, the
comparison of the ∆E values obtained from B3LYP/6-311+G-
(df,pd) and MP2/6-311+G(df,pd) for the mono- as well as for
the disubstituted cyclopentanes (particularly for DClCP and
DBrCP) distinctly confirms this observation. A closer consid-
eration of the energy differences in Table 7 also reveals the
following: On moving from MFCP to MClCP, MBrCP, MCCP,
and MECP the predominant axial conformer becomes constantly
less favorable in comparison with the equatorial conformer. The
decrease of the stability of the axial conformer from MFCP to
MClCP and MBrCP parallels the decrease of the electronega-
tivity of the halogen atoms F ) 4.00,36 Cl ) 3.07,36 and Br )
2.8136 and thus demonstrates a correlation between the confor-
mational stability within this series and the electronegativity of
the substituent. However, such a relationship does not hold for
MCCP and MECP since in these cases the percentage of axial
form continues to decrease although the electronegativities for
CtCH and CtN are 2.66, and 2.69,37 respectively. This,
however, is not particularly surprising since these electron-rich
moieties exhibit additional electronic interaction since they are
regarded to be strong π- and σ-electron acceptors and weak
It is interesting to note that the dipole moment does not
change noticeably upon going from DClCP to DFCP and DBrCP
(Cs and C2 symmetry) and it decreases only slightly by about
0.2 D on moving from MBrCP to MClCP and MFCP (2.52,
2.36, and 2.12 D, respectively).
Aside from the dependency of the ring C-C bond distances
on the phase angle φ it is perhaps crucial to lend some attention
to the role of the number and kind of substituents by the
determination of the ring geometry. From the results shown in
Tables 1 and 10 and the results for MBrCP and DBrCP, which
are not listed in this paper, it can be generally concluded that
the ring parameters are little affected by the variation of the
substituent and by the successive halogenation except in the
case of the fluoro derivatives. For instance, the C1-C2 bond
length in the axial form of MFCP, MClCP, and MBrCP is 1.516,
1.521, and 1.522 Å, respectively, as predicted by the MP2/6-
311++G(df,pd) procedure (not shown in this work). Using the
same ab initio method we obtained for the Cs conformer of
DCCP, DFCP, DClCP, and DBrCP a C1-C2 bond length of
1.546, 1.513, 1.523, and 1.523 Å, respectively. It is noteworthy

4672 J. Phys. Chem. A, Vol. 108, No. 21, 2004
Dakkouri et al.
TABLE 12: Ring Puckering Amplitude q (Å) for MBrCP, Monoethynylcyclopentane (MECP), and DBrCP As Obtained from
ab Initio Calculations on Various Levels of Theory
MBrCP
MECP
DBrCP
cyclopentane
method
ax
eq
ax
eq
Cs
C2
C1
C2
Cs
B3LYP/6-311+G(df,pd)
MP2/6-311G(d,p)
MP2/6-311+G(d,p)
0.368
0.405
0.411
0.437
0.403
0.441
0.439
0.409
0.435
0.434
0.385
0.416
0.416
0.451
MP2/6-311++G(d,p)
MP2/6-311+G(df,pd)
MP2/6-311++G(df,pd)
MP2/6-311++G(2df,2pd)
0.403
0.405
0.406
0.406
0.438
0.435
0.436
0.442
0.415
0.416
0.416
0.451
0.449
0.449
0.436
0.437
0.432
0.432
0.424
0.430
0.424
0.430
0.424
0.430
electron donors as well. Another possible rationalization for the
steadily increasing occurrence of the axial conformer within the
mono derivatives upon going from the bromo to chloro and
fluoro derivatives (Table 7) is the aforementioned dipole-dipole
interaction between the C-X bond dipole and the ring dipole.
From Table 7 it is furthermore apparent that the C2 conformer
of DFCP is slightly more stable than the Cs conformer whereas
in DCCP, DClCP, and DBrCP the Cs form is significantly more
stable than the C2 form. This interesting conformational inter-
conversion is most likely due to the declining electronegativity
of the substituents CtN, Cl, and Br in comparison to the F
atom.
In addition to the puckering amplitudes displayed in Tables
1 and 10 for DFCP, DClCP, and their monosubstituted deriva-
tives we calculated the ring puckering amplitude, q, for axial
and equatorial conformers of MBrCP, MECP, and MCCP, as
well as for DCCP and DBrCP (C2 and Cs forms) (Table 12).
Furthermore, we computed the puckering amplitudes for the
probable C1, C2, and Cs conformations (obviously q ) 0 for
the D5h form) of the parent molecule cyclopentane (Table 12).
From all these values it is evident that there is a general
noticeable increase of the q values upon going from the DFT
method to the perturbation method. This trend parallels the
tendency we observed by the consideration of the ∆E values
listed in Table 7. Inspection of the q values in Tables 1, 10,
and 12 reveals that while the equatorial form is significantly
more puckered than the axial form, the difference between the
q values for the equatorial and axial forms in MECP, however,
is almost negligible. In contrast, there is a noticeable increase
of the q values in going from the mono- to the disubstituted
cyclopentanes.
Pople.13 These authors suggested that the addition of polarization
functions would influence the degree of puckering. Interestingly,
while the value of 0.424 Å is lower than that which was
determined many years ago by means of electron diffraction of
0.438 Å,38 the value of 0.430 Å, however, agrees nicely with
the experimental one. From the comparison of both q values
we obtained for cyclopentane of 0.424 and 0.430 Å with those
for the substituted derivatives shown in Tables 1, 10, and 12 it
can be decisively noticed that these values lie between the values
for puckering amplitudes of the axial and equatorial conformers
and between the values for the Cs and C2 symmetry in the case
of the geminally substituted compounds. It remains to note that
the experimentally determined puckering amplitude for DClCP
(Table 9) of 0.402 Å is considerably smaller than that for
cyclopentane of 0.438 Å.
Finally, it is interesting to note that the experimental values
of the flap angles R of 39.9° (Table 9) and 41.6° 1 for DClCP
and DCCP, respectively, are different. This result is also
supported by all computational methods we applied.
Summary and Conclusions
The following principal conclusions can be drawn from this
study:
(1) The experimental electron diffraction intensities could be
reasonably reproduced by using a two-conformer static model
with Cs and C2 symmetry. Also a proper fit could be achieved
by applying a C1 static model.
(2) The best agreement with the electron diffraction data was
achieved by applying a large-amplitude treatment and a potential
function, which properly describes the hindered pseudorotational
motion in DClCP of the form
Of interest is to compare the puckering parameter, q, for the
substituted cyclopentanes we discussed so far and for the freely
pseudorotating parent molecule cyclopentane. Before carrying
out this comparison, however, we would like to point out that
the C1, C2, and Cs forms of cyclopentane exhibit the same
puckering displacement of 0.424 Å as predicted by the MP2/
6-311+G(df,pd) method. More or less conspicuously, the values
for the total energy for all three nonplanar conformations are
almost equal. It should also be mentioned that almost three
decades ago Cremer and Pople13 published the q values for
various five-membered ring systems, among them cyclopentane.
Using the STO-3G and HF/4-31 methods they obtained for
cyclopentane a puckering amplitude of q ) 0.37 and 0.39 Å,
respectively. These values are evidently smaller than is sug-
gested by the level of theory we applied. This deviation becomes
rather larger when we compare these values with the magnitude
of 0.430 Å, which has been provided by the MP2/6-311++G-
(2df,2pd) method. From this increase in the value for the
puckering amplitude it can be concluded that the augmentation
of the basis set by incorporating polarization and diffusion
functions leads to a larger value of the puckering displacement.
This finding is in accord with the prediction by Cremer and
V(φ) )
1/2V2(1 - cos2φ) + 1/2V4(1 - cos4φ) + 1/2V6(1 - cos6φ)
(3) We could show that there is an inherent correlation
between the fluctuation of the charge density distribution
throughout the ring in DClCP and thus its geometric parameters
and the phase angle φ during the pseudorotation. This correlation
has been described by evaluating an adequate density distribu-
tion function.
(4) Appropriate equations have been developed to characterize
the dependency of the geometry parameters on the pseudoro-
tational parameter φ.
(5) Simple relationships have been found between the phase
angle φ and the flap angle R for the envelope Cs conformation
on one hand and the twist angle τ for the C2 (half-chair)
conformer on the other hand.
(6) The ab initio results obtained from various methods and
different basis sets for a variety of mono and geminally
halogenated cyclopentanes have shown that the ring parameters

Molecular Structure of 1,1-Dichlorocyclopentane
J. Phys. Chem. A, Vol. 108, No. 21, 2004 4673
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remain almost unaffected except the fluoro derivatives exhibit
slight variations.
(7) Upon going from MFCP to MClCP and MBrCP the
predominant axial conformer becomes constantly less favorable
in comparison with the equatorial conformer. This behavior
parallels the decrease of the electronegativity of the halogen
atoms. This demonstrates a correlation between the conforma-
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the substituent. Due to the particular electronic interaction of
the CtCH and the CtN moieties as strong π- and σ-electron
acceptors and weak electron donors MCCP and MECP do not
obey this correlation.
(8) The C2 conformer of DFCP is slightly more stable than
the Cs conformer. In contrast, the Cs form is significantly more
stable than the C2 form for DClCP and DBrCP. This interesting
conformational interconversion may be attributed to the declin-
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to the F atom.
(9) A significant increase of the ring puckering amplitude,
q, has been observed on going from the axial to the equatorial
form within the monosubstituted cyclopentanes MFCP, MClCP,
MBrCP, MCCP, and MECP. A similar trend for the puckering
amplitude is apparent upon proceeding from the Cs to the C2
form along the corresponding geminally substituted series. The
computed ring puckering amplitude for the parent molecule
cyclopentane of 0.424 Å lies between the puckering amplitudes
for the axial and equatorial forms within the monosubstituted
derivatives and between the puckering amplitudes for the Cs
and the C2 form along the geminally substituted homologues.
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Acknowledgment. M.D. gratefully acknowledges the finan-
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Supporting Information Available: Intramolecular atomic
distances and calculated corrections (ra - rR) for 1,1-dichloro-
cyclopentane at phase angle φ ) 180°; additional results of
theoretical calculations on molecular geometry and atomic
charges of mono- and 1,1-dihalogenated cyclopentanes; and
scattering intensities from the diffraction experiment. This
material is available free of charge via the Internet at http://
pubs.acs.org.
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