15800
J. Phys. Chem. 1996, 100, 15800-15804
Theory of Acyclic Chemical Networks and Enumeration of Polyenoids via Two-Dimensional Chirality Chin-yah Yeh Department of Anesthesiology, The UniVersity of Utah School of Medicine, Salt Lake City, Utah 84132 ReceiVed: May 17, 1996; In Final Form: July 17, 1996X
Procedure for constructing acyclic chemical networks is delineated through the example of polyenoids, CNHN+2. Chirality in two-dimensional space is introduced to enumerate polyenoids. Cis-trans dichotomies, each arising from a 2D chiral center, are stacked to form stereoisomers. Redundancy caused by rotating the molecular plane in 3D space is eliminated to obtain the net count of isomers.
1. Introduction Cyvin et al.’s work1 on the enumeration of polyenoids actuated the present study and provided details that led to the analysis used for enumerating polyenoids in this paper. A polyenoid, CNHN+2, is a hydrocarbon made of fully conjugated π bonds. For even N, polyenoids comprise the majority of polyenes; the rest are radicals. A polyene is an integral multiple of double bonds. For odd N, only radicals exist. Electron delocalization keeps atoms in place by providing three equally spaced bonding sites surrounding each carbon atom. Therefore, the acyclic networks of polyenoids have equilateral triangular building blocks. Each link consists of a σ bond and a delocalized π bond that is shared by every carbon atom. As an appetizer, let us list the isomer counts of the backbones of some common chemicals. For retinals, C20H28O, the backbone consists of 5 conjugated double bonds and forms 733 isomers, when side groups (-CH3 and -CHO) and ring closure are ignored. For carotenes, C40H56, the 11 double bonds as a whole form 1 905 930 975 isomers. This paper is organized as follows. A recurrence relation is constructed, based on the concept that regular triangles are used as building blocks to form acyclic networks. Procedures for enumerating both the constitutional and steric isomers of free polyenoids drawn in two-dimensional space are described. In a digression, the concept of assigning chirality to appropriate nodes is applied to count the stereoisomers of alkanes. Then, the method to eliminate redundant counts caused by the free rotation of planar structures in three-dimensional space is depicted. In this paper, the generic term “acyclic network” is used interchangeably with the term tree, especially when the use of the latter may inadvertently imply a live species. Node and atom are synonyms, so are link and bond (if overall bond between atoms combining σ and π bonds is considered). Valence means number of sites available for linking. 2. Three Steps To Construct Acyclic Networks2,3 The first step of enumerating a family of networks is to find the recurrence relation for growing networks. To do so, one needs to characterize the building block. Is it structureless (the most symmetrical form)? If not, what symmetry has it? How many valences are allowed at each node? Then one constructs a recurrence relation embodying these characteristics. A network is grown by starting from a distinct part of structure which is marked as the root. One finds monosubstituted X
Abstract published in AdVance ACS Abstracts, September 1, 1996.
S0022-3654(96)01433-5 CCC: $12.00
networks before substituent-free networks. Note that a root can be either a node attached by a functional group as shown in the step of constructing monosubstituted networks or an arbitrarily labeled node as depicted in next step. Therefore, one could view monosubstituted trees as being rooted with one less branch at the root, whereas single-node labeled trees are rooted with fully grown root. The second step is to construct single-node labeled trees with fixed valence on each node. In the same way monosubstituted networks are constructed, one partitions the exponent of each term in the expansion series, but now the root has no substituent or functional group attached. Consequently, one more branch is grown from the root. For trivalent networks like polyenoids, two branches are generated on each stem in monosubstituted networks, whereas three branches are generated from the root in single node labeled networks. In other words, if f(x) is the counting function of monosubstituted trees, then the trees with an extra link at the root are represented by x + xf(x) and are used to construct trees with fully grown root, namely, singlenode labeled trees. The third step is to construct free trees. The dissimilarity characteristic theorem is used. Various interpretations of this theorem exist but the essential idea is as follows. Each member in the set of single-node labeled trees is reassiged: one labels an extra node next to the root in the direction toward the center and obtains a set of twin-rooted trees. In the set, there are trees of which one cannot label a second root if the root is already at the center; the number of these trees, plus the trees with a twin root at the center, is equal to the number of free trees. On the other hand, by combining two sets of monosubstituted trees at the root and eliminating the two substituents, one obtains pairs of twin-rooted trees with fully grown roots, except that symmetrical ones are not paired. A simple subtraction gives the the number of free trees. 3. Constructing Monosubstituted Networks 3a. Constitutional Isomers. A recurrence relation results in an infinite series expressed in powers of a variable, say x. The Nth coefficient in the series counts all networks of size N. The series designated as f(x) is obtained by iterating the form
f(x) )
∞
∞
N)1
n)1
∑ bNxN ) x∏(1 - xn)-bn
(1)
This is a procedure used by Cayley4 for counting the alkane series and is generalized in the following to encompass networks made of structured nodes, polyenoids in our case. The rightmost © 1996 American Chemical Society
Enumeration of Polyenoids
J. Phys. Chem., Vol. 100, No. 39, 1996 15801
expression of eq 1 is expanded, and terms of the same power are collected and then equated to the corresponding term in the left side, resulting in N-1
bN )
(
b + in - 1
n ∑ ∏ in ∑ ni )N-1 n)1 n
n
)
(2)
Equation 2 counts trees of unlimited branching.5 The summation in eq 2 covers all combinations of powers of x that result in an (N - 1)th power of x. Each combination is uniquely represented by a partition of the integer N - 1, hence the constraint ∑nnin ) N - 1 is imposed on the summation. Meanwhile, the product operation in eq 2 gives a cross-sectional view of the growth process for trees. Each product represents a growth pattern, as shown through building the two monosubstituted networks of polyenoids in this section. The growth process starts at the root and repeats itself. Each node is grown in the same fashion, resulting in a set of networks. The previous paragraph applies to chemical networks as well. But chemical networks differ from trees in two aspects: nodes have limited valency and often are structured. The first difference simplifies the iterating relation in two ways, to be shown in the next paragraph, whereas the second difference is elaborated in section 3c. Firstly, there is an upper limit in the number of terms each product has in eq 2. For polyenoids, it is two, representing the two branches that are allowed to grow from any stem. Secondly, in carrying out the summation in eq 2 a second constraint, ∑nin e 2, is imposed. In other words, the valency not only limits the number of terms included in the summation but also demands that in the first constraint, ∑nnin ) N - 1, the integer N - 1 be broken into at most two parts as k1 + k2 ) N - 1. The three alternatives, k2 ) 0, k1 > k2 > 0 and k1 ) k2 > 0 (for odd N only), lead us to rewrite eq 2 as
bN ) bN(k2)0) +
∑
k1+k2)N-1
bN(k1>k2>0) + bN(k1)k2>0)
bN(k2)0) ) bN-1 bN(k1>k2>0) ) bk1bk2
(3)
b′N )
1 bN(k1)k2>0) ) b(N-1)/2(b(N-1)/2 + 1)(N mod 2) 2
f2(x) )
∑
N)1
[
bN-1 +
1 N-2
∑ bk bN-k -1 +
2 k1)1
1
1
1 1 1 ) x 1 + f2(x) + f22(x) + f2(x2) 2 2
(
]
]
b(N-1)/2 (N mod 2) xN
2
∑
k1+k2)N-1
[b′N(k1>k2)0) + b′N(k1>k2>0)] + b′N(k1)k2)
Each component in bN represents a distinct growth pattern. Let f2(x) be an altered form of f(x), denoting the generating function for constitutional isomers of polyenoids, with the subscript 2 added to indicate that there can be two branches growing from each stem. The same symbols, bN’s, are kept, now giving the constitutional isomer counts of polyenoids. Summing over N in eq 3, one obtains f2(x) as ∞
stituted polyenoids. If instead of structureless nodes, regular triangles are preferred to be thought as building blocks for constructing constitutional isomers of polyenoids, then each triangle is able to rotate freely in the three-dimensional space. The count assumes no discrimination from cis-trans or cisoidtransoid isomerism. Or equivalently, only all-trans configurations are counted. 3b. Two-Dimensional Chirality. Two elements are needed to incorporate the geometric isomerism of polyenoids into the enumeration scheme: nodes are structured as space-filling equilateral triangles and the network is viewed to be grown on a plane. Both demand that chirality come into play. Much as in 3D space, an object not superimposable with its mirror image is said to have chirality. But in 2D space the mirror is not through a planar surfacesrather it is through a straight line. Just as a chiral carbon atom in the three-dimensional space has four distinct substituent groups, a two-dimensional chiral center in a polyenoid has three distinct groups attached to it. The 2D chirality is annihilated by free rotations in 3D space but the isomerism caused by 2D chirality is only reduced in 3D sapce. This point is dealt with further in section 7. For trees representing the chemical structures of monosubstituted polyenoids, the two branches growing from each stem are not interchangeable if not equivalent. Moreover, both branches differ from the stem since the stem contains the root, which is defined as the distinct part in an otherwise uniformly structured molecule. Thus, 2D chirality results in cis-trans isomerism around double bonds and cisoid-transoid isomerism around single bonds, giving geometric isomers as shown in the next subsection. 3c. Geometric Isomers. One imposes chirality while collecting terms of the same power in eq 1. Therefore, all counts in eq 3 are doubled except for the configurations with two identical branches, which yield achiral nodes. The configurations corresponding to the situation k1 ) k2 > 0 have two branches of the same size. But only bk1 counts of these configurations have identical branches and need to be waived from being doubled. Notations for the counting function and its expansion coefficients in eq 1 are modified for the geometric isomers of polyenoids as f2c(x) ) ∑N)1∼∞b′NxN. Complying with the above, one modifies eq 3 to include 2D chirality as
(4)
where b0 ) 1 is assumed, albeit b0 does not exist in eq 1. This recurrence relation dictates that nodes are structureless and the three binding sites on each node are fully permutational. Hence, the relation gives the constitutional isomer count of monosub-
b′N(k1>k2)0) ) 2b′N-1 b′N(k1>k2>0) ) 2b′k1b′k2
[(
b′N(k1)k2) ) 2
)
(5)
]
b′(N-1)/2 + 1 - b′(N-1)/2 (N mod 2) 2
) (b′(N-1)/2) (N mod 2) 2
Again b′0 ) 1 is used. Summing up eq 5 over N, one finds the recurrence relation
f2c(x) ) x[1 + 2f2c(x) + f2c2(x)]
(6)
from which a simple analytical solution results
f2c(x) )
1 1 - 1 - (1 - 4x)1/2 2x 2x
(7)
4. Constructing Single-Node Labeled Networks 4a. Constitutional Isomers. Carrying out the expansion of eq 1 again, by using the expansion coefficients found in
15802 J. Phys. Chem., Vol. 100, No. 39, 1996
Yeh
f2(x) as exponents, but with the constraint changed to ∑nin e 3, there results another function g3(x) ) ∑N)1∼∞g3NxN, which represents all trivalent acyclic rooted networks. A root can be a carbon atom labeled in a certain way (isotopically, for example). Likewise, the exponent is partitioned into N ) 1 + k1 + k2 + k3, not all of the components being unlike. Growth patterns with k3 ) 0 are already counted in f2(x) and are hence separated as in
g3cN(k1)k2>k3) ) [b′k1(b′k1 + 1) - b′k1]b′k3 ) b′k1b′k2b′k3 (12) g3cN(k1>k2)k3) ) b′k1b′k2b′k3
(
g3cN(k1)k2)k3) ) 2
∞
g3(x) ) f2(x) +
g3cN(k1>k2>k3) ) 2b′k1b′k2b′k3
∑ ∑ N)1
g3N(k1gk2gk3>0)xN
(8)
3 k )N-1 ∑i)1 i
The expansion coefficients, g3N’s in eq 8, are partitioned according to the partition of N - 1 and those not counted in f2(x) are listed in the following
2
1 g3N(k1>k2)k3>0) ) (bk1b2k2 + bk1bk2) 2
(
(9)
)
g3N(k1)k2)k3) ) bk1 + 2 3
One sums over all terms listed in eq 9 to find the final form of g3(x) as
x g3(x) ) f2(x) + [f23(x) + 3f2(x)f2(x2) + 2f2(x3)] (10) 6 which can be verified by substituting the equalities ∞
f23(x) )
[(N-1)/2]
∑{
+3
3 bN/3
N)1 (N mod 3)0)
′ bk2bN-2k + ∑ k)1 6
∑ ∑ k )N
bk1bk2bk3}xN
i i
∞
N-1
∑{
3 bN/3
+
N)1 (N mod 3)0)
∑ ′bkb[(N-k)/2][(N - k -
∞
∑ N)1
bN/3
xN
(N mod 3)0)
In eq 11, the primed summation, ∑′, means to exclude the term of k ) N/3; a bracketed number as in [z] is defined as the largest integer not larger than z. 4b. Geometric Isomers. Counting single-node labeled geometric isomers of polyenoids is similar, except that each count from the partition of the exponent may yield two isomeric configurations or one, depending on whether the node is chiral in 2D space or not. The counting function for singly labeled trivalent networks, g3c(x), has the components ∞
g3c(x) )
- (b′k1)2
(
b′k1 + 1 2
)
- b′k1 ) (b′k1)2
and are waived from doubling. Note that f2c(x) is not neatly separated in g3c(x), as is f2(x) in g3(x). That is, for polyenoids, counts in monosubstituted isomers are not fully transmigrated into single-node labeled isomers. This is because the situation k1 > k2 ) k3 ) 0 generates achiral isomers in singly labeled networks but the k1 > k2 ) 0 situation in f2c(x) gives chiral monosubstituted polyenoids in 2D space. The use of equalities, eq 11, on eq 12 results in
1 2 g3c(x) ) x 1 + f2c(x) + f2c2(x) + f2c3(x) + f2c(x3) 3 3
[
]
(13)
Both eqs 10 and 13 comply with Po´lya’s theorem.6,7 Singly labeled acyclic networks can be derived from the cycle index with the appropriate symmetry of the building block. Functions 1 + f2(x) and 1 + f2c(x) are used in cycle indices based on the symmetries D3 and C3 to generate g3(x) and g3c(x), respectively.
Here a free network means being substituent-free and labelfree. One has to delabel the single-node labeled trees, which are counted by g3(x) and g3c(x), for constitutional and geometrical isomers of polyenoids, respectively, in order to yield the isomer counts in 2D space, F3(x) and F3c(x). For both, one uses the dissimilarity characteristic theorem8,9
k)1
1) mod 2]}xN (11) f2(x3) )
)
5. Constructing Free Networks
k1>k2>k3>0
f2(x)f2(x2) )
3
The remarks on obtaining the geometric isomers of monosubstituted polyenoids in eq 5 are identically applied. Besides, in the situation k1 ) k2 ) k3, the count as expressed in the last row of eq 9 for constitutional isomers is doubled except for the structures having reflective symmetry in 2D space, which amount to
g3N(k1>k2>k3>0) ) bk1bk2bk3 1 g3N(k1)k2>k3>0) ) (b2k bk3 + bk1bk3) 2
b′k1 + 2
∑ ∑ N)1
3 k )N-1 ∑i)1 i
g3cN(k1gk2gk3)xN
1 F3(x) ) g3(x) - [f22(x) - f2(x2)] 2
(14a)
1 F3c(x) ) g3c(x) - [f2c2(x) - f2c(x2)] 2
(14b)
6. Digression: Stereoisomers of Alkanes Just as stereoisomers of polyenoids are counted as acyclic networks built from trivalent nodes, alkanes are built from tetravalent nodes. There are four valences in each carbon atom; those not linked are satisfied with hydrogen atoms. Starting from the carbon where the root is attached, three branches can be grown. Let f3c(x) be the counting function of the stereoisomers of alkanes, with the subscript 3 indicating the three branches growing from each stem. The three branches can be rotated but not pairwise interchanged without generating a new isomer. The expansion coefficients in f3c(x) ) ∑N)1∼∞bN′′xN are in parallel with those of g3c(x) in eq 12 since both functions involve the growth patterns of three branches
Enumeration of Polyenoids
b′′N )
J. Phys. Chem., Vol. 100, No. 39, 1996 15803
∑
b′′N(k1gk2gk3)
3 k )N-1 ∑i)1 i
b′′N(k1)k2>k3), b′′N(k1>k2)k3) ) b′′k1b′′k2b′′k3 b′′N(k1>k2>k3) ) 2b′′k1b′′k2b′′k3 b′′N(k1)k2)k3) )
(
b′′k1 + 2 3
Figure 1. Three types of achiral polyenoids in 2D space.
) ( ) +
b′′k1
(15)
3
For easy formulation, b′′0 ) 1 is assumed. Summing over N in eq 15 gives the recurrence relation
f3c(x) ) x[1 + f3c(x) + f3c2(x) + 1/3f3c3(x) + 2/3f3c(x3)] (16)
All chiral structures in 2D space become achiral in 3D space and therefore are doubly counted. To eliminate redundancy, one finds all achiral structures, whose counting function is designated as F3s(x). The final count of geometric isomers of polyenoids, F3d(x), is the count from chiral structures, 1/ F (x), plus that from achiral structures, which appears in both 2 3c F3s(x) and F3c(x). Therefore,
F3d(x) ) 1/2[F3s(x) + F3c(x)]
The counting function and its components for single carbon labeled alkanes are ∞
g4c(x) )
∑ ∑
N
g4cN(k1gk2gk3gk4)x
4 k )N-1 N)1 ∑i)1 i
g4cN(k1>k2>k3>k4) ) 2b′′k1b′′k2b′′k3b′′k4 g4cN(k1)k2>k3>k4), g4cN(k1>k2)k3>k4), g4cN(k1>k2>k3)k4) ) b′′k1b′′k2b′′k3b′′k4 g4cN(k1)k2)k3>k4) )
(
b′′k1 + 2 3
(
g4cN(k1>k2)k3)k4) ) b′′k4
)
( ) ) ( )
b′′k4 +
b′′k2 + 2 3
b′′k1
b′′k4
3
+ b′′k1
b′′k2 3
g4cN(k1)k2>k3)k4) ) /2b′′k1b′′k3(b′′k1b′′k3 + 1) 1
g4cN(k1)k2)k3)k4) )
(
b′′k1 + 3 4
) ( ) +
b′′k1 4
(17)
To find F3s(x), one looks for all polyenoids having reflective symmetry in 2D space, which boil down to three types as shown in Figure 1. Types A1 and A2 together correspond to all symmetrical trivalent networks of even size N and are exactly the set of all two-centered trees,10 although the two-node center of type A2 structures may not be at the center with respect to the two branches R1 and R2. To count two-centered trees in general, one first combines two sets of the monosubstituted trees at the roots to form twin-rooted trees. Among twin-rooted trees are two-centered trees with two equal moieties; they are counted by the functional obtained by substituting the variable of the counting function of monosubstituted trees with its square: f2c(x2). Note that these are also the steps to obtain free trees, as was used in section 5. Type B corresponds to achiral polyenoids in 2D space with odd size N; it is equivalent to a monosubstituted polyenoid with its mirror image geminally grown, and hence is counted by x + xf2c(x2), where the function of x2 corresponds to the size of two identical stems and the multiplier x means the node at the center, which raises the exponent of each term by one. Therefore,
F3s(x) ) [x + xf2c(x2)] + f2c(x2)
which are tedious but straightforward to justify. Hence,
g4c(x) )
x {[1 + f3c(x)]4 + 8[1 + f3c(x)][1 + f3c(x3)] + 12 3[1 + f3c(x2)]2} (18)
Counts in f3c(x) are not truthfully transferred into those in g4c(x) because the k1 > k2 > k3 ) 0 situation leads to chiral configurations in f3c(x) but achiral ones in g4c(x). Therefore f3c(x) is not singled out in g4c(x). Free alkanes are counted by
F4c(x) ) g4c(x) - /2[f3c (x) - f3c(x )] 1
2
2
(20)
(19)
(21)
From eqs 7, 13, 14, 20, and 21, one finds the explicit expression of F3d(x) as
F3d(x) )
(
)
(
1 3 1 1 1 + -1+ (1 - 4x)1/2 + 2 2 2x 6x 2x 24x 8x2 1 1 (1 - 4x2)1/2 - 2(1 - 4x3)1/2 (22) 4x 6x
)
a result already found by Cyvin’s group.1 Equation 22 is expanded in the infinite series F3d(x) ) ∑N)1∼∞F3dNxN and the coefficients are
( )
(
)
Results agree with ref 9. Chiral isomers form pairs but not all diastereoisomers are chiral. A structure consisting of two constitutionally equal but chiral moieties, designated as R and R*, has four combinations, resulting in three isomers: RR, RR* ) R*R, and R*R*, where * means the mirror image. RR* and R*R are equivalent and represent a meso isomer but are distinctly labeled in both the monosubstituted and single-node labeled networks. The distinction is eliminated in free networks.
where the combinatorics of fractional quantities are understood to be zero. F3dN is the final count of polyenoids CNHN+2.
7. Redundancy in 3D Space
8. Results and Discussion
So far, polyenoids are counted in 2D space. When a third dimension is added, extra rotational freedom causes some isomers to lose their identity and results in a redundant count.
The stereoisomers of polyenoids are explicitly counted through eq 23, whereas the constitutional isomers have to be algorithmically calculated by eq 14a. The code written in Turbo
2N 2N + 2 1 1 + 3(N + 1) N 12(N + 2) N + 1 N 3 1 N-1 1 2(N - 1)/3 + + N + 1 (N - 1)/2 N + 2 (N - 1)/3 2(N + 2) N/2 (23)
F3dN )
( )
(
)
(
)
15804 J. Phys. Chem., Vol. 100, No. 39, 1996 TABLE 1: Numbers of Constitutional and Geometric Isomers of Polyenoids CNHN+2 N
constitutional isomersa
geometric isomersb
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
1 1 1 2 2 4 6 11 18 37 66 135 265 552 1132 2410 5098 11020 23846 52233 114796 254371 565734 1265579 2841632 6408674 14502229 32935002 75021750 171404424
1 1 1 3 4 12 27 82 228 733 2282 7528 24834 83898 285357 983244 3412420 11944614 42080170 149197152 531883768 1905930975 6861221666 24806004996 90036148954 327989004892 1198854697588 4395801203290 16165198379984 59609171366326
a Numbers for N ) 2-21 are listed in ref 12. b Numbers for N ) 1-16 are listed in ref 1.
Pascal is concise and runs efficiently. Variables are compiled in extended double precision with 64-bit mantissa. For such a precision, 19-20 significant digits can be accommodated. The truncation or rounding error begins to show up at N ∼ 40. Both the constitutional and steric isomer counts for N e 30 are listed in Table 1. Equation 14a uses the recurrence relation, eq 4, the solution of which has been paved by Cayley11 in 1875. On the basis of
Yeh eq 4, Cyvin et al.12 have devised a scheme to count the constitutional isomers of polyenoids and tabulated the result up to N e 21. This paper extends the result with a handy algorithm.2 A common drawback in graph-theoretical models of the isomer enumeration,13 including the present one, is that steric strain, or worse, atom overlapping, is not excluded. Steric strain needs to be treated numerically by embedding chemical structures on lattices, such as a hexagonal one for polyenoids14 or a tetrahedral one for alkanes.15,16 Mathematical treatment of the steric strain for counting linear polyenes is partially successful and will appear shortly. In summary, the procedure used by Cayley4,11 for counting chemical series made of repetitive units is generalized to cover networks made of structured nodes. Enumeration of trivalent and tetravalent networks (polyenoids and alkanes) are illustrated. Acknowledgment. Professor S. Cyvin is generous to draw a key reference to the author’s attention. References and Notes (1) Cyvin, S. J.; Brunvoll, J.; Brendsdal, E.; Cyvin, B. N.; Lloyd, E. K. J. Chem. Inf. Comput Sci. 1995, 35, 743. (2) Yeh, C.-Y. J. Chem. Inf. Comput. Sci. 1995, 35, 912. (3) Yeh, C.-Y. J. Chem. Inf. Comput. Sci. 1996, 36, 854. (4) Cayley, A. Am. J. Math. 1881, 4, 266. (5) Cayley, A. Philos. Mag. 1857, 13, 172. (6) Po´lya, G. Acta Math. 1938, 68, 145. (7) Read, R. C. In Chemical Applications of Graph Theory; Balaban, A. T., Ed.; Academic Press: New York, 1976; Chapter 4. (8) Harary, F.; Norman, R. Proc. Am. Math. Soc. 1960, 11, 332. (9) Robinson, R. W.; Harary, F.; Balaban, A. T. Tetrahedron 1976, 32, 355. (10) Harary, F. Graph Theory; Addison-Wesley: Reading, MA, 1969; Chapter 4. (11) Cayley, A. Rep. Br. Assoc. AdV. Sci. 1875, 257. (12) Cyvin, S. J.; Brunvoll, J.; Cyvin, B. N. J. Mol. Struct. 1996, 357, 255. (13) Klein, D. J. J. Chem. Phys. 1981, 75, 5186. (14) Kirby, E. C. J. Math. Chem. 1992, 11, 187. (15) Thorpe, M. F.; Schroll, W. K. J. Chem. Phys. 1981, 75, 5143. (16) Wall, F. T.; Hiller, Jr., L. A.; Wheeler, D. J. J. Chem. Phys. 1954, 22, 1036.
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