Type conformations and pseudorotation interconversion path-way between conformations: A tool to...

Journal of Molecular Structure 1027 (2012) 186–199

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Journal of Molecular Structure

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Type conformations and pseudorotation interconversion path-way betweenconformations: A tool to study medium size (5–9 atoms) rings

José Pérez a,⇑, Luis García a, Eduardo Pérez a, José Luis Serrano a, Mathieu Kessler b

a Dpto de Ingeniería Minera, Geológica y Cartográfica. Área de Química Inorgánica. Universidad Politécnica de Cartagena, 30203 Cartagena (Murcia), Spainb Dpto de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Spain

h i g h l i g h t s

" A comprehensive characterization of type conformations by torsion angles." A very simple description of pseudorotation process." A systematic characterization of pseudorotation mechanism by variation in torsion angles.

a r t i c l e i n f o

Article history:Received 8 April 2012Received in revised form 28 May 2012Accepted 30 May 2012Available online 16 June 2012

Keywords:PseudorotationConformational analysisMedium sized rings

0022-2860/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.molstruc.2012.05.074

⇑ Corresponding author.E-mail address: [email protected] (J. Pérez).

a b s t r a c t

Type conformations for the conformational characterization of medium size rings (5–9 atoms) are pre-sented. Pseudorotation interconversion path-way between conformations is described in terms ofintra-ring torsion angles. This allows simple conformational analysis of any ring. Several examples ofapplication are presented.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction and the pseudorotation paths of interconversion between its

The study of conformational preferences in small molecules,and particularly in medium size rings, has attracted continuousinterest in many biologically relevant compounds. Different ap-proaches have been made to this field from a wide variety of disci-plines such as medicinal [1], physical [2] or computationalchemistry [3] among others, which have all devoted renewed ef-forts as more powerful tools have been available. A nice exampleabout conformational pathways of simple six-membered rings thatincludes a historical perspective has been recently reported [4].

From an experimental point of view most conformational studiesmake use of an input of torsion angles data, which can be obtainedeither using NMR extracted from vicinal three-bond coupling con-stants and appropriate software [3] or X-ray crystallography. In thissense, the continuous growth of the Cambridge Structural Database(CSD) has incredibly promoted the research areas that use crystalstructure data [5], including conformational studies. We have alsotaken advantage of the benefits offered by the CSD, as shown inour recent contributions to this field about eight-membered rings

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type-conformations [6–8]. Additional results have been producedrecently concerning five and six rings, as complementary structuralsections included in our more synthetic-oriented papers [9]. Duringthe course of these studies we realized that information on typeconformations (model conformations characterized by intra-ringtorsion angles and point symmetry, usually widely used as refer-ence conformation), somehow at the core of any research to be car-ried out, is fairly scattered and accordingly a comprehensive workgathering it together would be of interest for a wide audience ofresearchers on the areas above mentioned.

On the other hand, in spite of being the most important inter-conversion pathway and hence the most profusely studied, notwidely applicable pseudorotation description can be found in thebibliography. In this paper we present a simple tool of wide and di-rect applicability to study conformational changes in medium-sizerings. It is based on torsion angles data and provides numerical val-ues for each step avoiding the use of sophisticated mathematictreatment. Furthermore, no reference to energy will be madethrough the text because we want to present a geometrical toolapplicable to rings having different kinds of atoms. Usually thepseudorotation occurs along a low energy path. Indeed it has beenargued that the path must be isoenergetic for the motion to be

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-60

060

60

-60

0

0

0 60

60

-60

-60

boat C2v

Fig. 1. Boat conformation in six-membered rings.

Table 1Torsion angles along pseudorotation path in five-membered rings.

Da (�) s0 (�) s1 (�) s2 (�) s3 (�) s4 (�) Symmetry

0 48.1 �38.9 38.9 38.9 �38.9 half-chair (C2)

36 45.7 �45.7 28.3 0 �28.3 envelope (Cs)72 38.9 �48.1 38.9 �14.9 �14.9 half-chair (C2)

108 28.3 �45.7 45.7 �28.3 0 envelope (Cs)144 14.9 �38.9 48.1 �38.9 14.9 half-chair (C2)

180 0 �28.3 45.7 �45.7 28.3 envelope (Cs)216 �14.9 �14.9 38.9 �48.1 38.9 half-chair (C2)

252 �28.3 0 28.3 �45.7 45.7 envelope (Cs)288 �38.9 14.9 14.9 �38.9 48.1 half-chair (C2)

324 �45.7 28.3 0 �28.3 45.7 envelope (Cs)360 �48.1 38.9 �14.9 �14.9 38.9 half-chair (C2)

396 �45.7 45.7 �28.3 0 28.3 envelope (Cs)432 �38.9 48.1 �38.9 14.9 14.9 half-chair (C2)

468 �28.3 45.7 �45.7 28.3 0 envelope (Cs)504 �14.9 38.9 �48.1 38.9 �14.9 half-chair (C2)

J. Pérez et al. / Journal of Molecular Structure 1027 (2012) 186–199 187

called pseudorotation. However, the process is also called pseudo-rotation for molecules for which the path is not isoenergetic. Theauthors have successfully applied the geometrical aspects ofpseudorotation to the conformational study of very varied rings,in some cases including transition metal atoms.

540 0 28.3 �45.7 45.7 �28.3 envelope (Cs)576 14.9 14.9 �38.9 48.1 �38.9 half-chair (C2)

612 28.3 0 �28.3 45.7 �45.7 envelope (Cs)648 38.9 �14.9 �14.9 38.9 �48.1 half-chair (C2)

684 45.7 �28.3 0 28.3 �45.7 envelope (Cs)720 48.1 �38.9 38.9 38.9 �38.9 half-chair (C2)

In Tables 1–17 underlined bold stand out that in the pseudorrotation path an atomsequentially takes up each of the possible ring positions. Bolditalic stand out thatinitial position is reached again.

a D is a conformational parameter, a ‘‘phase angle’’ according to [20], this appliesalso to Eq. (1).

2. Results and discussion

2.1. General aspects

Type conformations characterize the conformation of N-mem-bered rings by the use of the N intra-torsion angles. Some remarksare necessary.

1. Symmetry axes keep torsion angles unchanged while sym-metry planes change the sign maintaining the absolutevalue for a torsion angle. This is illustrated in Fig. 1 for boatconformation but it applies through all conformations inthis paper.

2. Point symmetry in Fig. 1 applies to torsion angles. Throughthe paper point symmetry is indicated because it is veryhelpful in the analysis but it refers exclusively to torsionangles. The introduction of structurally different atomslowers the symmetry but keeps intact the usefulness oftype conformations: only torsion angles but not bond dis-tance and angles are considered in type conformations.Some examples of applications to rings having differentkinds of atoms are shown in this paper.

2.1.1. Puckering coordinatesAn N-membered ring (with no additional restrictions) is com-

pletely described by N bond distances, N-3 bond angles and N-3torsions, that is 3N-6 parameters. The use of type conformationsimplies the use of N parameters and avoids the use of bond dis-tances and angles. For a planar ring, only distances and anglesare necessary, that is 2N-3 parameters. Puckering is a measure ofthe non-planarity, it can be described by (3N-6) � (2N-3) = N-3parameters (alternatively, a plane is defined by three atoms, puck-ering can be described by the distances of the remaining atomsfrom the plane, requiring N-3 parameters).

Cremer and Pople puckering coordinates [10] are a valuable toolin the conformational mapping and classification of rings fragments,especially for medium size rings. They provide a measure of pucker-ing of the ring based on the out-of-plane atomic displacements from

Fig. 2. Type conformations in five-membered rings.

a planar basis conformation (these are not the unique [11] but themost used puckering coordinates). This plane goes through the geo-metrical center of the ring and is uniquely defined, irrespectively ofthe choice of the starting atom or the direction in the numberingprocess, and is moreover common to the ring and its mirror image.Cremer and Pople puckering coordinates characterize the conforma-tion of an N-membered ring by N-3 parameters while type confor-mations use N parameters.

Cremer and Pople puckering coordinates are calculated fromcartesian coordinates, so they depend upon bond distances and an-gles. In fact, when Cremer and Pople puckering coordinates areused to characterize the conformation of a ring, it is thus implicitly

Fig. 3. Type conformations for six-membered rings.

-60

060

60

-60

0

33 33

3333

-70-70

Fig. 4. Projection of B (left) and TB (right) conformation.

Fig. 5. Type conformations for seven-membered rings.

Table 3Pseudorotation path-way C-TC in seven-membered rings.

s0 (�) s1 (�) s2 (�) s3 (�) s4 (�) s5 (�) s6 (�) Symmetry

63.8 �83.5 66.1 0 �66.1 83.5 �63.8 chair (Cs)

72.3 �88.1 39.1 39.1 �88.1 72.3 �54.3 twist-chair (C2)83.3 �66.1 0 66.1 �83.5 63.8 �63.8 chair (Cs)

88.1 �39.1 �39.1 88.1 �72.3 54.3 �72.3 twist-chair (C2)66.1 0 �66.1 83.5 �63.8 63.8 �83.5 chair (Cs)

39.1 39.1 �88.1 72.3 �54.3 72.3 �88.1 twist-chair (C2)0 66.1 �83.5 �63.8 �63.8 83.5 �66.1 chair (Cs)

�39.1 88.1 �72.3 54.3 �72.3 88.1 �39.1 twist-chair (C2)�66.1 83.5 �63.8 63.8 �83.5 66.1 0 chair (Cs)

�88.1 72.3 �54.3 72.3 �88.1 39.1 39.1 twist-chair (C )

188 J. Pérez et al. / Journal of Molecular Structure 1027 (2012) 186–199

assumed that all bond distances are equal and that no variation isfound between bond angles.

No simple relation between torsions and puckering coordinatescan be found despite the fact that puckering has lower dimension-ality than torsions (N-3 vs. N). Some effort has been put in describ-ing how Cremer and Pople coordinates can be obtained fromtorsion angles [12,13] but these relations include bond lengthsand angles and only correspond to a local infinitesimalapproximation.

To perform a conformational study, it is very common to beginthe exploration of the dataset under consideration by an analysis ofthe Cremer and Pople coordinates. This is particularly informativefor rings of size 6 or below, since in this case a low-dimensionalgraphical representation of the puckering coordinates is possiblein a 2-dimensional or 3-dimensional device. In the case of six-membered rings for example, a simple transformation of the Cre-mer and Pople coordinates allows to locate each ring on a polarsphere [14]. For seven-membered rings and eight-membered rings,four and five puckering coordinates are respectively required todescribe each fragment. It is of course difficult to picture four-dimensional, or even worse, five-dimensional objects in thethree-dimensional minds eye. Bocian and Strauss [15] use the torito map the conformational equivalents of a given cycloheptanering using puckering coordinates, while Evans and Boeyens [16]suggest to use a three-dimensional representation which consistsof a set of tori on the surface of a sphere. This may explain why,even if the Cremer and Pople coordinates are considered usefulparticularly in a first exploration of the conformational dataset,many of the published conformational studies use torsion anglesas input data.

An additional argument may be given in favor of torsion angles.There is a well known difficulty to be faced in a conformationalanalysis based on experimental crystallographic observation: addi-tionally to the presence of conformational enantiomers, many ofthe chemically interesting fragments present topological symme-try, which implies that fragments that can be deduced from one

Table 2Pseudorotation path-way B-TB in six-membered rings.

s0 (�) s1 (�) s2 (�) s3 (�) s4 (�) s5 (�) Symmetry

60 �60 0 60 �60 0 boat (C2v)

70 �33 �33 70 �33 �33 twist-boat (D2)60 0 �60 60 0 �60 boat (C2v)

33 33 �70 33 33 �70 twist-boat (D2)0 60 �60 0 60 �60 boat (C2v)

�33 70 �33 �33 70 �33 twist-boat (D2)�60 60 0 �60 60 0 boat (C2v)

�70 33 33 �70 33 33 twist-boat (D2)�60 0 60 �60 0 60 boat (C2v)

�33 �33 70 �33 �33 70 twist-boat (D2)0 �60 60 0 �60 60 boat (C2v)

33 �70 33 33 �70 33 twist-boat (D2)

60 �60 0 60 �60 0 boat (C2v)

another by atomic permutations should be considered as equiva-lent. It is rather simple to describe the action of permutationaloperators or coordinate inversion on the set of torsion angles asso-ciated to a chemical fragment, it is however more involved to ob-tain a similar description for their action on the Cremer andPople coordinates.

2.1.2. Pseudorotation mechanismPseudorotation mode (w) may be described as a concerted,

change of dihedral angles such that each ring atom (and each ringbond) sequentially takes up each of the possible ring positions; thissequence of rings positions is called the pseudorotation itinerary

2

�83.5 �63.8 63.8 83.5 �66.1 0 66.1 chair (Cs)

�72.3 54.3 �72.3 88.1 �39.1 �39.1 88.1 twist-chair (C2)

�63.8 63.8 �83.5 66.1 0 �66.1 83.5 chair (Cs)

�54.3 72.3 �88.1 39.1 39.1 �88.1 72.3 twist-chair (C2)�63.8 83.5 �66.1 0 66.1 �83.5 �63.8 chair (Cs)

�72.3 88.1 �39.1 �39.1 88.1 �72.3 54.3 twist-chair (C2)�83.5 66.1 0 �66.1 83.5 �63.8 63.8 chair (Cs)

�88.1 39.1 39.1 �88.1 72.3 �54.3 72.3 twist-chair (C2)�66.1 0 66.1 �83.5 �63.8 �63.8 83.5 chair (Cs)

�39.1 �39.1 88.1 �72.3 54.3 �72.3 88.1 twist-chair (C2)0 �66.1 83.5 �63.8 63.8 �83.5 66.1 chair (Cs)

39.1 �88.1 72.3 �54.3 72.3 �88.1 39.1 twist-chair (C2)66.1 �83.5 �63.8 �63.8 83.5 �66.1 0 chair (Cs)

88.1 �72.3 54.3 �72.3 88.1 �39.1 �39.1 twist-chair (C2)83.5 �63.8 63.8 �83.5 66.1 0 �66.1 chair (Cs)

72.3 �54.3 72.3 �88.1 39.1 39.1 �88.1 twist-chair (C2)�63.8 �63.8 83.5 �66.1 0 66.1 �83.5 chair (Cs)

54.3 �72.3 88.1 �39.1 �39.1 88.1 �72.3 twist-chair (C2)

63.8 �83.5 66.1 0 �66.1 83.5 �63.8 chair (Cs)

Fig. 6. Type conformations for eight-membered rings.


[17]. In one step of the itinerary for a symmetrical ring, this mustpass out of its symmetry through a continuum of asymmetricalforms (or retaining a symmetry axis perpendicular to the ring),to a form of opposite symmetry, and ultimately back to the initialsymmetry form in which each atom now occupies the next ring po-sition on the itinerary and the torsion angle signs have been in-verted but the symmetry remain (this is an important point tomention, though often neglected). After N steps (in even rings) or2N steps (in odd rings) the starting conformation of the itineraryis reached again.

In this review we describe pseudorotation as a function of tor-sion angles for three reasons: (i) torsion angles are simple andcommon structural parameters; (ii) for a good approximationalong pseudorotation, rings can be assumed to have constant bondlengths and often constant skeletal bond angles as well [18]. Thepseudorotation motion involves then mainly changes in the tor-sional angles; (iii) we try to present a tool of general applicabilityirrespectively of the kind of atoms in the ring.

2.2. Five-membered rings

2.2.1. Type conformationsIn the case of five-membered rings, two type conformations

have been described [19]: envelope (E) and half-chair (HC). Fig. 2

shows these conformations together with torsion angles andsymmetry.

2.2.2. PseudorotationThe type conformations described for five-membered rings

interconvert by a pseudorotation mechanism.Starting from a half-chair symmetry an envelope conformation

is reached and then a new HC symmetry is obtained where eachatom occupies the next ring position on the itinerary and torsionangles have been sign inverted (this is illustrated in Table 1 bythe torsion angle bold). According to this, as N = 5 is odd, 2N stepsare necessary to reach the starting conformation.

In this case any of the infinity of conformations a five-mem-bered ring can assume in the course of that E M HC path can beapproximated by a simple analytical expression [20]:

sj ¼ smax cosðD=2þ jdÞ ð1Þ

where smax is the maximum attainable torsional angle (48.1�),j = 0,1,2,3,4 and d = 144�. Asymmetrical forms appear between typesymmetrical conformations. When D = 720� the initial conforma-tion is reached again.

Next file contains an animation showing the pseudorotationpath E M HC in five membered rings.

Table 4Pseudorotation path-way B-TB in seven-membered rings.

s0 (�) s1 (�) s2 (�) s3 (�) s4 (�) s5 (�) s6 (�) Symmetry

57.5 30.9 �69.9 0 69.9 �30.9 �57.5 boat (Cs)

17.9 64.4 �45.4 �45.4 64.4 17.9 �74.6 twist-boat (C2)�30.9 69.9 0 �69.9 30.9 57.5 �57.5 boat (Cs)

�64.4 45.4 45.4 �64.4 �17.9 74.6 �17.9 twist-boat (C2)�69.9 0 69.9 �30.9 �57.5 57.5 30.9 boat (Cs)

�45.4 �45.4 64.4 17.9 �74.6 17.9 64.4 twist-boat (C2)0 �69.9 30.9 57.5 �57.5 �30.9 69.9 boat (Cs)

45.4 �64.4 �17.9 74.6 �17.9 �64.4 45.4 twist-boat (C2)69.9 �30.9 �57.5 57.5 30.9 �69.9 0 boat (Cs)

64.4 17.9 �74.6 17.9 64.4 �45.4 �45.4 twist-boat (C2)30.9 57.5 �57.5 �30.9 69.9 0 �69.9 boat (Cs)

�17.9 74.6 �17.9 �64.4 45.4 45.4 �64.4 twist-boat (C2)�57.5 57.5 30.9 �69.9 0 69.9 �30.9 boat (Cs)

�74.6 17.9 64.4 �45.4 �45.4 64.4 17.9 twist-boat (C2)

�57.5 �30.9 69.9 0 �69.9 30.9 57.5 boat (Cs)

�17.9 �64.4 45.4 45.4 �64.4 �17.9 74.6 twist-boat (C2)30.9 �69.9 0 69.9 �30.9 �57.5 57.5 boat (Cs)

64.4 �45.4 �45.4 64.4 17.9 �74.6 17.9 twist-boat (C2)69.9 0 �69.9 30.9 57.5 �57.5 �30.9 boat (Cs)

45.4 45.4 �64.4 �17.9 74.6 �17.9 �64.4 twist-boat (C2)0 69.9 �30.9 �57.5 57.5 30.9 �69.9 boat (Cs)

�45.4 64.4 17.9 �74.6 17.9 64.4 �45.4 twist-boat (C2)�69.9 30.9 57.5 �57.5 �30.9 69.9 0 boat (Cs)

�64.4 �17.9 74.6 �17.9 �64.4 45.4 45.4 twist-boat (C2)�30.9 �57.5 57.5 30.9 �69.9 0 69.9 boat (Cs)

17.9 �74.6 17.9 64.4 �45.4 �45.4 64.4 twist-boat (C2)57.5 �57.5 �30.9 69.9 0 �69.9 30.9 boat (Cs)

74.6 �17.9 �64.4 45.4 45.4 �64.4 �17.9 twist-boat (C2)

57.5 30.9 �69.9 0 69.9 �30.9 �57.5 boat (Cs)


2.3. Six-membered rings

2.3.1. Type conformationsConformations in six-membered rings can be characterized by

using six type conformations [21]: chair (C), boat (B), twist-boat(TB), envelope (E), half-chair (HC) and screw-boat (SB). Sometimesenvelope is also referred as half-boat conformation. Fig. 3 showsthese conformations together with torsion angles and symmetries.

2.3.2. PseudorotationIn the pseudorotation mode a continuous change of dihedral an-

gle takes place. It applies to B M TB transformation.Fig. 4 shows the symmetry elements and torsion angle values in

both conformations, B and TB. As can be seen the change in two an-

Table 5Pseudorotation path-way C-TC in eight-membered rings.

s0 (�) s1 (�) s2 (�) s3 (�) s4 (�)

119.9 �76.2 0 76.2 �119.9109.3 �37.3 �37.3 109.3 �109.3

76.2 0 �76.2 119.9 �76.2

37.3 37.3 �109.3 109.3 �37.30 76.2 �119.9 76.2 0

�37.3 109.3 �109.3 37.3 37.3�76.2 119.9 �76.2 0 76.2

�109.3 109.3 �37.3 �37.3 109.3�119.9 76.2 0 �76.2 119.9�109.3 37.3 37.3 �109.3 109.3�76.2 0 76.2 �119.9 76.2

�37.3 �37.3 109.3 �109.3 37.30 �76.2 119.9 �76.2 0

37.3 �109.3 109.3 �37.3 �37.376.2 �119.9 76.2 0 �76.2

109.3 �109.3 37.3 37.3 �109.3

gles is small (�60� to �70�) and the remaining angles alternatelyincrease and decrease.

Table 2 shows the torsion angles for the conversion B M TBalong the pseudorotational path-way until the starting conforma-tion is reached again. It shows the variation of each torsion anglealong the complete cycle. This is independent from the kind ofatoms in the ring, so it is of general applicability to identify the po-sition of any ring in the pseudorotation path by internal torsion an-gles only.

Any of the infinity of conformations a 6-membered ring can as-sume in the course of that B M TB path in this case can be approx-imated by a simple analytical expression [22]:

sj ¼ smax cosðDþ jdÞ ð2Þ

where smax is the maximum attainable torsional angle (60�),j = 0,1,2,3,4,5 and d = 120�. In this case when D = 360� the initialconformation is reached again. At D = 0�, 60�, 120�, . . .. TB conform-ers are obtained; at D = 30�, 90�, 150�, . . . B conformers are ob-tained. (Note that using Eq. (2) and smax = 60� the maximumtorsion angle obtained in TB conformations is 60� and not 70� as ap-pears in Table 2).

Next file contains an animation showing the pseudorotationpath B M TB in six membered rings.

2.4. Seven-membered rings

2.4.1. Type conformationsConformations in seven-membered rings can be described [23]

using four type conformations: chair (C), boat (B), twist-chair (TC)and twist-boat (TB). Fig. 5 shows these conformations togetherwith torsion angles and symmetry.

2.4.2. Interconversion between type conformationsIn seven-membered rings there are two pseudorotations modes

[18]: C M TC and B M TB.In the C M TC mode, starting from a C symmetry a TC conforma-

tion is reached and then a new C symmetry is obtained when eachatom occupies the next ring position on the itinerary and torsionangles are sign inverted (this is emphasized in Table 3 by the boldtorsion angle). The same applies to B M TB mode. Tables 3 and 4collect the torsion angles variation along pseudorotation pathwaysinterconversion. After 2N steps (odd ring) the starting conforma-tion of the itinerary is reached again.

In seven-membered rings analytical expressions for torsion an-gles along pseudorotation path have been reported by FourierTransform model using two components with two amplitude

s5 (�) s6 (�) s7 (�) Symmetry

76.2 0 �76.2 Chair (C2h)

37.3 37.3 �109.3 Twist-Chair (C2h)0 76.2 �119.9 Chair (C2h)

�37.3 109.3 �109.3 Twist-Chair�76.2 �119.9 �76.2 Chair (C2h)

�109.3 109.3 �37.3 Twist-Chair (C2h)

�119.9 76.2 0 Chair (C2h)

�109.3 37.3 37.3 Twist-Chair (C2h)�76.2 0 76.2 Chair (C2h)�37.3 �37.3 109.3 Twist-Chair (C2h)

0 �76.2 119.9 Chair (C2h)

37.3 �109.3 109.3 Twist-Chair (C2h)76.2 �119.9 76.2 Chair (C2h)

109.3 �109.3 37.3 Twist-Chair (C2h)119.9 �76.2 0 Chair (C2h)

109.3 �37.3 �37.3 Twist-Chair (C2h)

Table 6Pseudorotation path-way BC-TBC in eight-membered rings.

s0 (�) s1 (�) s2 (�) s3 (�) s4 (�) s5 (�) s6 (�) s7 (�) Symmetry

65.0 44.7 �102.2 65.0 �65.0 102.2 �44.7 �65.0 Boat-Chair (Cs)

44.8 51.9 �93.2 88.0 �93.2 51.9 44.8 �115.6 Twist-Boat-Chair (C2)�44.7 102.2 �65.0 65.0 �102.2 44.7 65.0 �65.0 Boat-Chair (Cs)

�51.9 93.2 �88.0 93.2 �51.9 �44.8 115.6 �44.8 Twist-Boat-Chair (C2)�102.2 65.0 �65.0 102.2 �44.7 �65.0 �65.0 44.7 Boat-Chair (Cs)

�93.2 88.0 �93.2 51.9 44.8 �115.6 44.8 51.9 Twist-Boat-Chair (C2)�65.0 65.0 �102.2 44.7 65.0 �65.0 �44.7 102.2 Boat-Chair (Cs)

�88.0 93.2 �51.9 �44.8 115.6 �44.8 �51.9 93.2 Twist-Boat-Chair (C2)�65.0 102.2 �44.7 �65.0 65.0 44.7 �102.2 65.0 Boat-Chair (Cs)

�93.2 51.9 44.8 �115.6 44.8 51.9 �93.2 88.0 Twist-Boat-Chair (C2)�102.2 44.7 65.0 �65.0 �44.7 102.2 �65.0 65.0 Boat-Chair (Cs)

�51.9 �44.8 115.6 �44.8 �51.9 93.2 �88.0 93.2 Twist-Boat-Chair (C2)�44.7 �65.0 65.0 44.7 �102.2 65.0 �65.0 102.2 Boat-Chair (Cs)

44.8 �115.6 44.8 51.9 �93.2 88.0 �93.2 51.9 Twist-Boat-Chair (C2)65.0 �65.0 �44.7 102.2 �65.0 65.0 �102.2 44.7 Boat-Chair (Cs)

115.6 �44.8 �51.9 93.2 �88.0 93.2 �51.9 �44.8 Twist-Boat-Chair (C2)

Table 7Pseudorotation path-way CC-TCC in eight-membered rings.


66.0 �105.2 105.2 �66.0 66.0 �105.2 105.2 �66.0 chair-chair (C2v)

82.4 �114.6 82.4 �56.2 82.4 �114.6 82.4 �56.2 twist-chair-chair (D2)105.2 �105.2 66.0 �66.0 105.2 �105.2 66.0 �66.0 chair-chair (C2v)

114.6 �82.4 56.2 �82.4 114.6 �82.4 56.2 �82.4 twist-chair-chair (D2) (D2)105.2 �66.0 66.0 �105.2 105.2 �66.0 66.0 �105.2 chair-chair (C2v)

82.4 �56.2 82.4 �114.6 82.4 �56.2 82.4 �114.6 twist-chair-chair (D2)66.0 �66.0 105.2 �105.2 66.0 �66.0 105.2 �105.2 chair-chair (C2v)





56.2 �82.4 114.6 �82.4 56.2 �82.4 114.6 �82.4 twist-chair-chair (D2)

66.0 �105.2 105.2 �66.0 66.0 �105.2 105.2 �66.0 chair-chair (C2v)


parameters, obtained from a least-squares fit using the torsion an-gles predicted by force-field calculations [24], and one phase angle[11b,24].

sj ¼ A cosð�ðD=2Þ � ð13p=14Þ þ ð6p=7ÞjÞ þ B cosð�ð3D=2Þ� ð25p=14Þ þ ð4p=7ÞjÞ; j ¼ 0;1; . . . ;6 ð3Þ

2.5. Eight-membered rings

2.5.1. Type conformationsIn the case of eight membered rings, ten type conformations

have been described [25] crown (CR), boat-boat (BB), sofa (S), boat(B), chair (C), twist-chair (TC), twist-chair-chair (TCC), chair-chair(CC), boat-chair (BC) and twist-boat-chair (TBC). Sofa conformationis also called twist-boat (TB). Fig. 6 shows these conformations to-gether with torsion angles and symmetries.

2.5.2. PseudorotationIn eight-membered rings there are four pseudorotations modes

[26]: C M TC; BC M TBC; CC M TCC; B M TB M BB. Tables 5–8 col-lect torsion angles along pseudorotation pathways interconver-sion. After N steps (even ring) the starting conformation of theitinerary is reached again.

In this case, because of the high symmetry of the type conforma-tions only N/2 steps are necessary to reach the startingconformation.

In a similar way to the CC M TCC transformation, in theB M TB M BB case only N/2 steps are necessary to obtain the start-ing conformation because of the high symmetry.

2.6. Nine-membered rings

2.6.1. Type conformationsIn the case of nine-membered rings, sixteen type conformations

have been described [27]. Fig. 7 shows these conformations andTable 9 shows the associated torsion angles and the existingsymmetries.

2.6.2. PseudorotationIn nine-membered rings there are eight pseudorotations

modes: BC M TBC; CC M TCC; C M TC; B M TB; BB M TBB;CB M TCB; BC00 M TBC00; CC00 M TCC00. Tables 10 and 11 illustrateBC M TBC; CC M TCC modes; the remaining modes are includedin Appendix A (Tables 12–17).

In this case, because the high symmetry of the type conforma-tions only N/3 steps are necessary to reach starting conformation.

Table 8Pseudorotation path-way B-TB-BB in eight-membered rings.


�73.5 0 73.5 0 �73.5 0 73.5 0 boat (D2d)

�64.9 �37.6 64.9 37.6 �64.9 �37.6 64.9 37.6 twist-boat (S4)�52.5 �52.5 52.5 52.5 �52.5 �52.5 52.5 52.5 boat-boat (D2d)�37.6 �64.9 37.6 64.9 �37.6 �64.9 37.6 64.9 twist-boat (S4)

0 �73.5 0 73.5 0 �73.5 0 73.5 boat (D2d)

37.6 �64.9 �37.6 64.9 37.6 �64.9 �37.6 64.9 twist-boat (S4)52.5 �52.5 �52.5 52.5 52.5 �52.5 �52.5 52.5 boat-boat (D2d)64.9 �37.6 �64.9 37.6 64.9 �37.6 �64.9 37.6 twist-boat (S4)73.5 0 �73.5 0 73.5 0 �73.5 0 boat (D2d)

64.9 37.6 �64.9 �37.6 64.9 37.6 �64.9 �37.6 twist-boat (S4)52.5 52.5 �52.5 �52.5 52.5 52.5 �52.5 �52.5 boat-boat (D2d)37.6 64.9 �37.6 �64.9 37.6 64.9 �37.6 �64.9 twist-boat (S4)

0 73.5 0 �73.5 0 73.5 0 �73.5 boat (D2d)

�37.6 64.9 37.6 �64.9 �37.6 64.9 37.6 �64.9 twist-boat (S4)�52.5 52.5 52.5 �52.5 �52.5 52.5 52.5 �52.5 boat-boat (D2d)�64.9 37.6 64.9 �37.6 �64.9 37.6 64.9 �37.6 twist-boat (S4)�73.5 0 73.5 0 �73.5 0 73.5 0 boat (D2d)

�64.9 �37.6 64.9 37.6 �64.9 �37.6 64.9 37.6 twist-boat (S4)�52.5 �52.5 52.5 52.5 �52.5 �52.5 52.5 52.5 boat-boat (D2d)�37.6 �64.9 37.6 64.9 �37.6 �64.9 37.6 64.9 twist-boat (S4)

0 �73.5 0 73.5 0 �73.5 0 73.5 boat (D2d)

37.6 �64.9 �37.6 64.9 37.6 �64.9 �37.6 64.9 twist-boat (S4)52.5 �52.5 �52.5 52.5 52.5 �52.5 �52.5 52.5 boat-boat (D2d)64.9 �37.6 �64.9 37.6 64.9 �37.6 �64.9 37.6 twist-boat (S4)73.5 0 �73.5 0 73.5 0 �73.5 0 boat (D2d)

64.9 37.6 �64.9 �37.6 64.9 37.6 �64.9 �37.6 twist-boat (S4)52.5 52.5 �52.5 �52.5 52.5 52.5 �52.5 �52.5 boat-boat (D2d)37.6 64.9 �37.6 �64.9 37.6 64.9 �37.6 �64.9 twist-boat (S4)

0 73.5 0 �73.5 0 73.5 0 �73.5 boat (D2d)

�37.6 64.9 37.6 �64.9 �37.6 64.9 37.6 �64.9 twist-boat (S4)�52.5 52.5 52.5 �52.5 �52.5 52.5 52.5 �52.5 boat-boat (D2d)�64.9 37.6 64.9 �37.6 �64.9 37.6 64.9 �37.6 twist-boat (S4)

�73.5 0 73.5 0 �73.5 0 73.5 0 boat (D2d)

Fig. 7. Type conformations for nine-membered rings.


Table 9Symmetry and torsión angles for type conformations in nine-membered rings.

Symmetry s0 (�) s1 (�) s2 (�) s3 (�) s4 (�) s5 (�) s6 (�) s7 (�) s8 (�)

BC C3v �114 0 114 �114 0 114 �114 0 114TBC D2 �57 130 �57 �57 130 �57 �57 130 �57CC Cs �69 108 �139 83 0 �83 139 �108 69TCC C2 �54 126 �115 80 �77 80 �115 126 �54C Cs 121 �30 �80 117 0 �117 80 30 �121TC C2 �70 100 0 �88 125 �88 0 100 �70B Cs �121 39 �80 117 0 �117 80 �39 121TB C2 �70 108 �43 72 �143 72 �43 108 �70BB Cs 67 48 �10 �83 0 83 10 �48 �67TBB C2 80 �70 �10 �34 155 �34 �10 �70 80CB Cs 80 �108 0 90 0 �90 0 108 �80TCB C2 �72 79 44 �105 93 �105 44 79 �72BC’’ Cs 65 51 �140 82 0 �82 140 �51 �65TBC’’ C2 �43 124 �88 �28 117 �28 �88 124 �43CC’’ Cs �98 72 �134 117 0 �117 134 �72 98TCC’’ C2 �62 120 �84 91 �124 91 �84 120 �62

Table 10Pseudorotation path-way BC-TBC in nine-membered rings.

s0 (�) s1 (�) s2 (�) s3 (�) s4 (�) s5 (�) s6 (�) s7 (�) s8 (�) Symmetry

�114 0 114 �114 0 114 �114 0 114 boat-chair (C3v)

�130 57 57 �130 57 57 �130 57 57 twist-boat-chair (D3)�114 114 0 �114 114 0 �114 114 0 boat-chair (C3v)

�57 130 �57 �57 130 �57 �57 130 �57 twist-boat-chair (D3)0 114 �114 0 114 �114 0 114 �114 boat-chair (C3v)

57 57 �130 57 57 �130 57 57 �130 twist-boat-chair (D3)114 0 �114 114 0 �114 114 0 �114 boat-chair (C3v)

130 �57 �57 130 �57 �57 130 �57 �57 twist-boat-chair (D3)114 �114 0 114 �114 0 114 �114 0 boat-chair (C3v)

57 �130 57 57 �130 57 57 �130 57 twist-boat-chair (D3)0 �114 114 0 �114 114 0 �114 114 boat-chair (C3v)

�57 �57 130 �57 �57 130 �57 �57 130 twist-boat-chair (D3)�114 0 114 �114 0 114 �114 0 114 boat-chair (C3v)



57 57 �130 57 57 �130 57 57 �130 twist-boat-chair (D3)

114 0 �114 114 0 �114 114 0 �114 boat-chair (C3v)



�57 �57 130 �57 �57 130 �57 �57 130 twist-boat-chair (D3)�114 0 114 �114 0 114 �114 0 114 boat-chair (C3v)



57 57 �130 57 57 �130 57 57 �130 twist-boat-chair (D3)114 0 �114 114 0 �114 114 0 �114 boat-chair (C3v)



�57 �57 130 �57 �57 130 �57 �57 130 twist-boat-chair (D3)

�114 0 114 �114 0 114 �114 0 114 boat-chair (C3v)


2.7. Other interconversion mechanisms

Pseudorotation is the most widely studied mechanism. Severalothers interconversion mechanisms between conformations havebeen described. Only a few comments about this.

In the symmetrical mode a plane or axis in the ring remainsunchanged [26]. In six-membered rings it applies to C M E M B inter-

conversion and to C M HC M SB M TB. In the first case a plane remainsunchanged, in the second one there is an axis that kept along the inter-conversion. In seven membered symmetrical transformations [26]have been described for C M B conversion (retaining the symmetryplane) and TC M TB conversion (retaining the axis). In eight memberedrings symmetrical mode [26] applies to CR M CC M BC (retaining aplane), CR M TCC (retaining an axis) and BC M BB (retaining a plane).

Table 11Pseudorotation path-way CC-TCC in nine-membered rings.

s0 (�) s1 (�) s2 (�) s3 (�) s4 (�) s5 (�) s6 (�) s7 (�) s8 (�) Symmetry

�69 108 �139 83 0 �83 139 �108 69 chair-chair (Cs)

�77 80 �115 126 �54 �54 126 �115 80 twist-chair-chair (C2)�69 69 �108 139 �83 0 83 �139 108 chair-chair (Cs)

�80 77 �80 �115 �126 54 54 �126 115 twist-chair-chair (C2)�108 69 �69 108 �139 83 0 �83 139 chair-chair (Cs)

�115 80 �77 80 �115 126 �54 �54 126 twist-chair-chair (C2)�139 108 �69 69 �108 139 �83 0 83 chair-chair (Cs)

�126 115 �80 77 �80 115 �126 54 54 twist-chair-chair (C2)�83 139 �108 69 �69 108 �139 83 0 chair-chair (Cs)

�54 126 �115 80 �77 80 �115 126 �54 twist-chair-chair (C2)0 83 �139 108 �69 69 �108 139 �83 chair-chair (Cs)

54 54 �126 115 �80 77 �80 115 �126 twist-chair-chair (C2)83 0 �83 139 �108 69 �69 108 �139 chair-chair (Cs)

126 �54 �54 126 �115 80 �77 80 �115 twist-chair-chair (C2)139 �83 0 83 �139 108 �69 69 �108 chair-chair (Cs)

115 �126 54 54 �126 115 �80 77 �80 twist-chair-chair (C2)108 �139 83 0 �83 139 �108 69 �69 chair-chair (Cs)

80 �115 126 �54 �54 126 �115 80 �77 twist-chair-chair (C2)

69 �108 139 �83 0 83 �139 108 �69 chair-chair (Cs)

77 �80 115 �126 54 54 �126 115 �80 twist-chair-chair (C2)69 �69 108 �139 83 0 �83 139 �108 chair-chair (Cs)

80 �77 80 �115 126 �54 �54 126 �115 twist-chair-chair (C2)108 �69 69 �108 139 �83 0 83 �139 chair-chair (Cs)

115 �80 77 �80 115 �126 54 54 �126 twist-chair-chair (C2)139 �108 69 �69 108 �139 83 0 �83 chair-chair (Cs)

126 �115 80 �77 80 �115 126 �54 �54 twist-chair-chair (C2)83 �139 108 �69 69 �108 139 �83 0 chair-chair (Cs)

54 �126 115 �80 77 �80 115 �126 54 twist-chair-chair (C2)0 �83 139 �108 69 �69 108 �139 83 chair-chair (Cs)

�54 �54 126 �115 80 �77 80 �115 126 twist-chair-chair (C2)�83 0 83 �139 108 �69 69 �108 139 chair-chair (Cs)

�126 54 54 �126 115 �80 77 �80 115 twist-chair-chair (C2)�139 83 0 �83 139 �108 69 �69 108 chair-chair (Cs)

�115 126 �54 �54 126 �115 80 �77 80 twist-chair-chair (C2)�108 139 �83 0 83 �139 108 �69 69 chair-chair (Cs)

�80 115 �126 54 54 �126 115 �80 77 twist-chair-chair (C2)

�69 108 �139 83 0 �83 139 �108 69 chair-chair (Cs)


An asymmetric mode has been described in which only one sideof a symmetrical ring is allowed to change [26] It applies to theinterconversion BC M TC in 8-membered rings, as example.

3. Examples of application

Type conformations and pseudorotation path-way have beenused in the next cases that exemplify the wide applicability ofthe described tool.

1. The five-, and six-membered ring type conformations have beenused to characterize conformations in metalacycles (Pd, Ni, Ru)[9].

2. The eight-membered ring type conformations have been usedto test new probabilistic methods to quantify the proximity ofa experimental conformation to the ideal ones [28].

3. The eight-membered ring type conformations have been usedto characterize conformations in metalacycles (Pd, Ni), amongothers [29].

4. The conformation of copper complexes double bridged by phos-phate and related ligands have been studied [6] by the toolsshown in this review. The ring MOPOMOPO in those complexeshas been studied using the eight-membered ring conforma-tions. In this case, new type conformations appear because ofthe lower topological symmetry: the sequence of torsion angles

may remain unchanged but the associated sequence of atoms isdifferent, a total of 21 possible conformations appear when theMOPOMOPO core is studied referring to the ten canonical con-formations in Fig. 6. A chair and a boat conformations were themost frequent conformations found in the CSD. Energeticallyaccessible interconversion pathways between conformationswere identified by MM including the pseudorotation mecha-nism described above.

5. The conformational preferences in the {M(l-XPX)}2 core(X = O,S) in transition metal complexes have been established[7,8] by statistical methods using torsion angles. Different chairconformations were found in both cases. When X = O, phospho-rus and oxygen atoms are in the ‘‘seat of the chair’’; when X = S,sulfur and metal atoms are in the ‘‘seat of the chair’’. The originof this difference can be justified on the basis of ‘‘non-covalent’’interactions within the ring. Conformation in new complexescontaining the {Mn(l-OPO)}2 core have been studied also [30].

6. The solid state conformational preferences of the macrocycle2,4,4-trimethyl-1,5,9-triazacyclododec-1-ene and related intransition metal complexes have been established using thetype conformations in six-membered rings [31].

Appendix A

See Tables 12–17.

Table 12Pseudorotation path-way C-TC in nine-membered rings.

s0 s1 s2 s3 s4 s5 s6 s7 s8 Symmetry

121 �30 �80 117 0 �117 80 30 �121 chair (Cs)

125 �88 0 100 �70 �70 100 0 �88 twist-chair (C2)121 �121 30 80 �117 0 117 �80 �30 chair (Cs)

88 �125 88 0 �100 70 70 �100 0 twist-chair (C2)30 �121 121 �30 �80 117 0 �117 80 chair (Cs)

0 �88 125 �88 0 100 �70 �70 100 twist-chair (C2)�80 �30 121 �121 30 80 �117 0 117 chair (Cs)

�100 0 88 �125 88 0 �100 70 70 twist-chair (C2)�117 80 30 �121 121 �30 �80 117 0 chair (Cs)

�70 100 0 �88 125 �88 0 100 �70 twist-chair (C2)0 117 �80 �30 121 �121 30 80 �117 chair (Cs)

70 70 �100 0 88 �125 88 0 �100 twist-chair (C2)117 0 �117 80 30 �121 121 �30 �80 chair (Cs)

100 �70 �70 100 0 �88 125 �88 0 twist-chair (C2)80 �117 0 117 �80 �30 121 �121 30 chair (Cs)

0 �100 70 70 �100 0 88 �125 88 twist-chair (C2)�30 �80 117 0 �117 80 30 �121 121 chair (Cs)

�88 0 100 �70 �70 100 0 �88 125 twist-chair (C2)

�121 30 80 �117 0 117 �80 �30 121 chair (Cs)

�125 88 0 �100 70 70 �100 0 88 twist-chair (C2)�121 121 �30 �80 117 0 �117 80 30 chair (Cs)

�88 125 �88 0 100 �70 �70 100 0 twist-chair (C2)�30 121 �121 30 80 �117 0 117 �80 chair (Cs)

0 88 �125 88 0 �100 70 70 �100 twist-chair (C2)80 30 �121 121 �30 �80 117 0 �117 chair (Cs)

100 0 �88 125 �88 0 100 �70 �70 twist-chair (C2)117 �80 �30 121 121 30 80 �117 0 chair (Cs)

70 �100 0 88 �125 88 0 �100 70 twist-chair (C2)0 �117 80 30 �121 121 �30 �80 117 chair (Cs)

�70 �70 100 0 �88 125 �88 0 100 twist-chair (C2)�117 0 117 �80 �30 121 �121 30 80 chair (Cs)

�100 70 70 �100 0 88 �125 88 0 twist-chair (C2)�80 117 0 �117 80 30 �121 121 �30 chair (Cs)

0 100 �70 �70 100 0 �88 125 �88 twist-chair (C2)30 80 �117 0 117 �80 �30 121 �121 chair (Cs)

88 0 �100 70 70 �100 0 88 �125 twist-chair (C2)

121 �30 �80 117 0 �117 80 30 �121 chair (Cs)

Table 13Pseudorotation path-way B-TB in nine-membered rings.


�121 39 �80 117 0 �117 80 �39 121 boat (Cs)

�143 72 �43 108 �70 �70 108 �43 72 twist-boat (C2)�121 121 �39 80 �117 0 117 �80 39 boat (Cs)

�72 143 �72 43 �108 70 70 �108 43 twist-boat (C2)�39 121 �121 39 �80 117 0 �117 80 boat (Cs)

�43 72 �143 72 �43 108 �70 �70 108 twist-boat (C2)�80 39 �121 121 �39 80 �117 0 117 boat (Cs)

�108 43 �72 143 �72 43 �108 70 70 twist-boat (C2)�117 80 �39 121 �121 39 �80 117 0 boat (Cs)

�70 108 �43 72 �143 72 �43 108 �70 twist-boat (C2)0 117 �80 39 �121 121 �39 80 �117 boat (Cs)

70 70 �108 43 �72 143 �72 43 �108 twist-boat (C2)117 0 �117 80 �39 121 �121 39 �80 boat (Cs)

108 �70 �70 108 �43 72 �143 72 �43 twist-boat (C2)80 �117 0 117 �80 39 �121 121 �39 boat (Cs)

43 �108 70 70 �108 43 �72 143 �72 twist-boat (C2)39 �80 117 0 �117 80 �39 121 �121 boat (Cs)

72 �43 108 �70 �70 108 �43 72 �143 twist-boat (C2)

121 �39 80 �117 0 117 �80 39 �121 boat (Cs)

143 �72 43 �108 70 70 �108 43 �72 twist-boat (C2)121 �121 39 �80 117 0 �117 80 �39 boat (Cs)

72 �143 72 �43 108 �70 �70 108 �43 twist-boat (C2)39 121 121 �39 80 �117 0 117 �80 boat (Cs)

43 �72 143 �72 43 �108 70 70 �108 twist-boat (C2)

(continued on next page)


Table 13 (continued)


80 �39 121 �121 39 �80 117 0 �117 boat (Cs)

108 �43 72 �143 72 �43 108 �70 �70 twist-boat (C2)117 �80 39 �121 121 �39 80 �117 0 boat (Cs)

70 �108 43 �72 143 �72 43 �108 70 twist-boat (C2)0 �117 80 �39 121 �121 39 �80 117 boat (Cs)

�70 �70 108 �43 72 �143 72 �43 108 twist-boat (C2)�117 0 117 �80 39 �121 121 �39 80 boat (Cs)

�108 70 70 �108 43 �72 143 �72 43 twist-boat (C2)�80 117 0 �117 80 �39 121 �121 39 boat (Cs)

�43 108 �70 �70 108 �43 72 �143 72 twist-boat (C2)�39 80 �117 0 117 �80 39 �121 121 boat (Cs)

�72 43 �108 70 70 �108 43 �72 143 twist-boat (C2)

�121 39 �80 117 0 �117 80 �39 121 boat (Cs)

Table 14Pseudorotation path-way BB-TBB in nine-membered rings.


67 48 �10 �83 0 83 10 �48 �67 boat-boat (Cs)

155 �34 �10 �70 80 80 �70 �10 �34 twist-boat-boat (C2)67 �67 �48 10 83 0 �83 �10 48 boat-boat (Cs)

34 �155 34 10 70 �80 �80 70 10 twist-boat-boat (C2)�48 �67 67 48 �10 �83 0 83 10 boat-boat (Cs)

�10 �34 155 �34 �10 �70 80 80 �70 twist-boat-boat (C2)�10 48 67 �67 �48 10 83 0 �83 boat-boat (Cs)

70 10 34 �155 34 10 70 �80 �80 twist-boat-boat (C2)83 10 �48 �67 67 48 �10 �83 0 boat-boat (Cs)

80 �70 �10 �34 155 �34 �10 �70 80 twist-boat-boat (C2)0 �83 �10 48 67 �67 �48 10 83 boat-boat (Cs)

�80 �80 70 10 34 �155 34 10 70 twist-boat-boat (C2)�83 0 83 10 �48 �67 67 48 �10 boat-boat (Cs)

�70 80 80 �70 �10 �34 155 �34 �10 twist-boat-boat (C2)10 83 0 �83 �10 48 67 �67 �48 boat-boat (Cs)

10 70 �80 �80 70 10 34 �155 34 twist-boat-boat (C2)48 �10 �83 0 83 10 �48 �67 67 boat-boat (Cs)

�34 �10 �70 80 80 �70 �10 �34 155 twist-boat-boat (C2)

�67 �48 10 83 0 �83 �10 48 67 boat-boat (Cs)

�155 34 10 70 �80 �80 70 10 34 twist-boat-boat (C2)�67 �67 48 �10 �83 0 83 10 �48 boat-boat (Cs)

�34 155 �34 �10 �70 80 80 �70 �10 twist-boat-boat (C2)48 67 �67 �48 10 83 0 �83 �10 boat-boat (Cs)

10 34 �155 34 10 70 �80 �80 70 twist-boat-boat (C2)10 �48 �67 �67 48 �10 �83 0 83 boat-boat (Cs)

�70 �10 �34 155 �34 �10 �70 80 80 twist-boat-boat (C2)�83 �10 48 67 �67 �48 10 83 0 boat-boat (Cs)

�80 70 10 34 �155 34 10 70 �80 twist-boat-boat (C2)0 83 10 �48 �67 �67 48 �10 �83 boat-boat (Cs)

80 80 �70 �10 �34 155 �34 �10 �70 twist-boat-boat (C2)83 0 �83 �10 48 67 �67 �48 10 boat-boat (Cs)

70 �80 �80 70 10 34 �155 34 10 twist-boat-boat (C2)�10 �83 0 83 10 �48 �67 67 48 boat-boat (Cs)

�10 �70 80 80 �70 �10 �34 155 �34 twist-boat-boat (C2)�48 10 83 0 �83 �10 48 67 �67 boat-boat (Cs)

34 10 70 �80 �80 70 10 34 �155 twist-boat-boat (C2)

67 48 �10 �83 0 83 10 �48 �67 boat-boat (Cs)

Table 15Pseudorotation path-way CB-TCB in nine-membered rings.


80 �108 0 90 0 �90 0 108 �80 chair-boat (Cs)

93 �105 44 79 �72 �72 79 44 �105 twist-chair-boat (C2)80 �80 108 0 �90 0 90 0 �108 chair-boat (Cs)

105 �93 105 �44 �79 72 72 �79 �44 twist-chair-boat (C2)108 �80 80 �108 0 90 0 �90 0 chair-boat (Cs)




44 �105 93 �105 44 79 �72 �72 79 twist-chair-boat (C2)0 �108 80 �80 108 0 �90 0 90 chair-boat (Cs)

�79 �44 105 �93 105 �44 �79 72 72 twist-chair-boat (C2)�90 0 108 �80 80 �108 0 90 0 chair-boat (Cs)

�72 79 44 �105 93 �105 44 79 �72 twist-chair-boat (C2)0 90 0 �108 80 �80 108 0 �90 chair-boat (Cs)

72 72 �79 �44 105 �93 105 �44 �79 twist-chair-boat (C2)90 0 �90 0 108 �80 80 �108 0 chair-boat (Cs)

79 �72 �72 79 44 �105 93 �105 44 twist-chair-boat (C2)0 �90 0 90 0 �108 80 80 108 chair-boat (Cs)

�44 �79 72 72 �79 �44 105 �93 105 twist-chair-boat (C2)�108 0 90 0 �90 0 108 �80 80 chair-boat (Cs)

�105 44 79 �72 �72 79 44 �105 93 twist-chair-boat (C2)

�80 108 0 �90 0 90 0 �108 80 chair-boat (Cs)

�93 105 �44 �79 72 72 �79 �44 105 twist-chair-boat (C2)�80 80 �108 0 90 0 �90 0 108 chair-boat (Cs)

�105 93 �105 44 79 �72 �72 79 44 twist-chair-boat (C2)�108 80 �80 108 0 �90 0 90 0 chair-boat (Cs)

�44 105 �93 105 �44 �79 72 72 �79 twist-chair-boat (C2)0 108 �80 80 �108 0 90 0 �90 chair-boat (Cs)

79 44 �105 93 �105 44 79 �72 �72 twist-chair-boat (C2)90 0 �108 80 �80 108 0 �90 0 chair-boat (Cs)

72 �79 �44 105 �93 105 �44 �79 72 twist-chair-boat (C2)0 �90 0 108 �80 80 �108 0 90 chair-boat (Cs)

�72 �72 79 44 �105 93 �105 44 79 twist-chair-boat (C2)�90 0 90 0 �108 80 �80 108 0 chair-boat (Cs)

�79 72 72 �79 �44 105 �93 105 �44 twist-chair-boat (C2)0 90 0 �90 0 108 �80 80 �108 chair-boat (Cs)

44 79 �72 �72 79 44 �105 93 �105 twist-chair-boat (C2)108 0 �90 0 90 0 �108 80 �80 chair-boat (Cs)

105 �44 �79 72 72 �79 �44 105 �93 twist-chair-boat (C2)

80 �108 0 90 0 �90 0 108 �80 chair-boat (Cs)

Table 16Pseudorotation path-way BC00-TBC00 in nine-membered rings.


65 51 �140 82 0 �82 140 �51 �65 boat-chair00 (Cs)

117 �28 �88 124 �43 �43 124 �88 �28 twist-boat-chair00 (C2)65 �65 �51 140 �82 0 82 �140 51 boat-chair00 (Cs)

28 �117 28 88 �124 43 43 �124 88 twist-boat-chair00 (C2)�51 �65 65 51 �140 82 0 �82 140 boat-chair00 (Cs)

�88 �28 117 �28 �88 124 �43 �43 124 twist-boat-chair00 (C2)�140 51 65 �65 �51 140 �82 0 82 boat-chair00 (Cs)

�124 88 28 �117 28 88 �124 43 43 twist-boat-chair00 (C2)�82 140 �51 �65 65 51 �140 82 0 boat-chair00 (Cs)

�43 124 �88 �28 117 �28 �88 124 �43 twist-boat-chair00 (C2)0 82 �140 51 65 �65 �51 140 �82 boat-chair00 (Cs)

43 43 �124 88 28 �117 28 88 �124 twist-boat-chair00 (C2)82 0 �82 140 �51 �65 65 51 �140 boat-chair00 (Cs)

124 �43 �43 124 �88 �28 117 �28 �88 twist-boat-chair00 (C2)140 �82 0 82 �140 51 65 �65 �51 boat-chair00 (Cs)

88 �124 43 43 �124 88 28 �117 28 twist-boat-chair00 (C2)51 �140 82 0 �82 140 �51 �65 65 boat-chair00 (Cs)

�28 �28 124 �43 �43 124 �88 �28 117 twist-boat-chair00 (C2)

�65 �51 140 �82 0 82 �140 51 65 boat-chair00 (Cs)

�117 28 88 �124 43 43 �124 88 28 twist-boat-chair00 (C2)�65 65 51 �140 82 0 �82 140 �51 boat-chair00 (Cs)

�28 117 �28 �88 124 �43 �43 124 �88 twist-boat-chair00 (C2)51 65 �65 �51 140 �82 0 82 �140 boat-chair00 (Cs)

88 28 �117 28 88 �124 43 43 �124 twist-boat-chair00 (C2)140 �51 �65 65 51 �140 82 0 �82 boat-chair00 (Cs)

124 �88 �28 117 �28 �88 124 �43 �43 twist-boat-chair00 (C2)82 �140 51 65 �65 �51 140 �82 0 boat-chair00 (Cs)

43 �124 88 28 �117 28 88 �124 43 twist-boat-chair00 (C2)0 �82 140 �51 �65 65 51 �140 82 boat-chair00 (Cs)

(continued on next page)




�43 �43 124 �88 �28 117 �28 �88 124 twist-boat-chair00 (C2)�82 0 82 �140 51 65 �65 �51 140 boat-chair00 (Cs)

�124 43 43 �124 88 28 �117 28 88 twist-boat-chair00 (C2)�140 82 0 �82 140 �51 �65 65 51 boat-chair00 (Cs)

�88 124 �43 �43 124 �88 �28 117 �28 twist-boat-chair00 (C2)�51 140 �82 0 82 �140 51 65 �65 boat-chair00 (Cs)

Table 17Pseudorotation path-way CC00-TCC00 in nine-membered rings.


�98 72 �134 117 0 �117 134 �72 98 chair-chair00 (Cs)

�124 91 �84 120 �62 �62 120 �84 91 twist-chair-chair00 (C2)�98 98 �72 134 �117 0 117 �134 72 chair-chair00 (Cs)

�91 124 �91 84 �120 62 62 �120 84 twist-chair-chair00 (C2)�72 98 �98 72 �134 117 0 �117 134 chair-chair00 (Cs)

�84 91 �124 91 �84 120 �62 �62 120 twist-chair-chair00 (C2)�134 72 �98 98 �72 134 �117 0 117 chair-chair00 (Cs)

�120 84 �91 124 �91 84 �120 62 62 twist-chair-chair00 (C2)�117 134 �72 98 �98 72 �134 117 0 chair-chair00 (Cs)

�62 120 �84 91 �124 91 �84 120 �62 twist-chair-chair00 (C2)0 117 �134 72 �98 98 �72 134 �117 chair-chair00 (Cs)

62 62 �120 84 �91 124 �91 84 �120 twist-chair-chair00 (C2)117 0 �117 134 �72 98 �98 72 �134 chair-chair00 (Cs)

120 �62 �62 120 �84 91 �124 91 �84 twist-chair-chair00 (C2)134 �117 0 117 �134 72 �98 98 �72 chair-chair00 (Cs)

84 �120 62 62 �120 84 �91 124 �91 twist-chair-chair00 (C2)72 �134 117 0 �117 134 �72 98 �98 chair-chair00 (Cs)

91 �84 120 �62 �62 120 �84 91 �124 twist-chair-chair00 (C2)

98 �72 134 �117 0 117 �134 72 �98 chair-chair00 (Cs)

124 �91 84 �120 62 62 �120 84 �91 twist-chair-chair00 (C2)98 �98 72 �134 117 0 �117 134 �72 chair-chair00 (Cs)

91 �124 91 �84 120 �62 �62 120 �84 twist-chair-chair00 (C2)72 �98 98 �72 134 �117 0 117 �134 chair-chair00 (Cs)

84 �91 124 �91 84 �120 62 62 �120 twist-chair-chair00 (C2)134 �72 98 �98 72 �134 117 0 �117 chair-chair00 (Cs)

120 �84 91 �124 91 �84 120 �62 �62 twist-chair-chair00 (C2)117 �134 72 �98 98 �72 134 �117 0 chair-chair00 (Cs)

62 �120 84 �91 124 �91 84 �120 62 twist-chair-chair00 (C2)0 �117 134 �72 98 �98 72 �134 117 chair-chair00 (Cs)

�62 �62 120 �84 91 �124 91 �84 120 twist-chair-chair00 (C2)�117 0 117 �134 72 �98 98 �72 134 chair-chair00 (Cs)

�120 62 62 �120 84 �91 124 �91 84 twist-chair-chair00 (C2)�134 117 0 �117 134 �72 98 �98 72 chair-chair00 (Cs)

�84 120 �62 �62 120 �84 91 �124 91 twist-chair-chair00 (C2)�72 134 �117 0 117 �134 72 �98 98 chair-chair00 (Cs)

�91 84 �120 62 62 �120 84 �91 124 twist-chair-chair00 (C2)

�98 72 �134 117 0 �117 134 �72 98 chair-chair00 (Cs)


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