J . Chem. In5 Comput. Sci. 1992, 32, 407410
407
The Numbers of Structural Isomers, Stereoisomers, and Chiral and Achiral Stereoisomers of Fluorochloroalkanes Fangzhi Gu' Computer Department, Inner Mongolia Normal University, Huhehaote, 010022, China Jianji Wang Department of Chemistry, Inner Mongolia Normal University, Huhehaote, 010022, China Received January 2, 1992 Three recurrence formulas are presented for counting structural isomers, stereoisomers, and achiral stereoisomers of "fluorochloroalkyls", CiH2i+l-j-kCljFk (j = 0-2i 1 ; k = 0-2i 1 -j). A generating function for counting chiral stereoisomers of fluorochloroalkyls, CiH2i+l-j-kCl,Fk,and four generating functions for counting structural isomers, stereoisomers, and achiral and chiral stereoisomers of fluorochloroalkanes,CiH2i+2-,-kCljFt (j = 0-2i + 2; k = 0-2i + 2 - j ) are obtained. Some results are tabulated.
+
The enumerationsof structural isomers, stereoisomers, chiral stereoisomers, and achiral stereoisomers of some acyclic compounds have been reported.'-' In particular, Read4 has discussed fully and clearly the enumerations of structural isomers and stereoisomers of acyclic compounds. However, the enumeration of isomers of fluorochloroalkanes has not been reported yet, though they are useful in many aspects. We present three recurrence formulas for counting structural isomers, stereoisomers, and achiral stereoisomers of fluorochloroalkyls and obtain the numbers of structural isomers, stereoisomers, and chiral and achiral stereoisomers of fluorochloroalkanes using P6lya's t h ~ r e m . ~ , ~ 1. DEFINITIONS
1.1. Quartic Tree with Three Different Kinds of Points. Consider a tree Twhich consists of a finite set of three different kinds of points. One kind of point has a degree of 1-4 and is colored black, representing the carbon atom; the second kind of point has only degree 1 and is colored red, representing fluorine atoms; and the third kind of point also has only degree 1 and is colored blue, representing chlorine atoms. Thenumber of points can be an arbitrary natural number. The tree is called a quartic tree with three different kinds of points. Quartic trees with threedifferent kinds of points represent structural formulas for the acyclic saturated hydrocarbons substituted by two kinds of halogen atoms C1 and F (fluorochloroalkanes), in which hydrogen atoms are omitted. In this paper, all fluorochloroalkanesinclude alkanes (j = 0 and k = 0), fluoroalkanes, chloroalkanes, and fluorochloroalkanes; all fluorochloroalkyls include alkyls, fluoroalkyls, chloroalkyls, and fluorochloroalkyls. 1.2. Planted Quartic Tree with Three Different Kinds of Points. A planted quartic tree with three different kinds of points is a quartic tree with three different kinds of points, in which, beside those points, there is a distinguishable point, with a degree 1-3; this point is called a root. There is a direct correspondence between planted quartic trees with three different kinds of points and structural formulas of fluorochloroalkyls. 1.3. Steric Tree with Three Different Kinds of Points. A steric tree with three different kinds of points is a quartic tree in which every carbon point has four neighbors (neighboring points) in a tetrahedral configuration. A steric tree with three different kinds of points represents a stereostructural formula of a fluorochloroalkane. 0095-2338/92/1632-0407$03.00/0
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1.4. A planted steric tree with three different kindsof points is a steric tree which contains a distinguishable root point. There is a direct correspondencebetween the planted steric trees with three different kinds of points and stereoisomersof fluorochloroalkyls.
2. COUNTING 2.1. Let A(x,y,z) be the generating function for counting the planted trees with three different kinds of points, in which the coefficient a(ij,k) of the term d9zk is the number of structural isomers of fluorochloroalkyls containing i carbon atoms, j chlorine atoms, and k fluorine atoms. We establish the recurrence formula
2 2 2
3 3 3
'/6x[A3(x,Y,Z) 3A(xyJ)A(x ,Y ,Z + 2A(x ,Y ,Z 11 (1) where a(O,O,O) = 1, a(0,0,1) = 1, a(0,1,0) = 1, and a ( i j , k ) = o(i,kj) = a(ij,2i 1-j - k) = a(i,k,2i + 1 - j - k ) = a(i,2i + 1 - j - kj) = a(i,2i + 1 - j - k,k). 2.2. Let B(x,y,z) be the generating function for counting the quartic trees with three different kinds of points, in which the coefficient b ( i j , k ) of the term xfpzk is the number of structural isomers of fluorochloroalkanescontaining i carbon atoms, j chlorine atoms and k fluorine atoms. Using the method of Harary et a1.,3p4 we can obtain the function
+
m
~ ( x , y , z= )
2i+2 2i+2-j
7 j=o 1b(ij,k)x'yizk= i=l
k=O
/24x[A4(x~,z) + 6A2(xy,zM(x2Y2,z21+ 3A2(x2y2,z2)+ 8A(x,y,z)A(x3,y 3,z3) 6A(x4y4,z4)]1
+
'/2[A21(x,Y,Z)- A,(x2,y2,z211 ( 2 )
where A,(x,y,z) = A(x,y,z) -Y - z - 1 (3) and b(ij,k) = b(i,kj) = b(ij,2i 2 - j - k ) = b(i,k,2i 2 - j - k ) = b(i,2i + 2 - j - k j ) = b(i,2i + 2 - j - k,k). 2.3. Let C(x,y,z) be the generating function for counting the planted steric trees with three different kinds of points, in which the coefficient c(ij,k) of the term xi'zk is the number of stereoisomers of fluorochloroalkyls containing i carbon
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0 1992 American Chemical Society
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408 J. Chem. Inf. Comput. Sci., Vol. 32, No. 5, 1992
Gu AND WANG
atoms,j chlorine atoms, and k fluorine atoms. We establish the recurrence formula
1
2i+l Zi+l-i
+ y + z + '/3x[c3(x,y,z)+ 2C(x3,y3,z3)](4)
generating function of stereoisomer of fluorochloroalkanes containing one labeled carbon atom. Let P(x,y,z) be the generating function of achiral stereoisomer of fluorochloroalkanes containing one labeled carbon atom and let Dl(x,y,z) be the generating function of chiral stereoisomer of fluorochloroalkanes containing one labeled carbon atom. Then
where c(O,O,O) = 1, c(O,O,l) = 1, c(O,l,O) = 1, and c(ij,k) = c ( i , k j ) = c(ij,2i + 1 - j - k) = c(i,k,2i + 1 - j - k) = c(i,2i + 1 - j - kj) = c(i,2i 1 - j - k,k). The c(ij,k) is the number of stereoisomers of fluorochloroalkyls containing i carbon atoms, j chlorine atoms, and k fluorine atoms. 2.4. Let D(x,y,z) be the generating function for counting steric trees with three different kinds of points. Using the method of Harary et aL3v4we can obtain the followingfunction:
+
+ 3@(x2 ,y2 ,z 2 ) + 8C(x,y,z)C(x3 ,y 3 ,z 3 11 2 2 2 1/2[c:(x,Y,z)- C,(X ,y ,z 11 ( 5 )
'/12X[C4(X,Y,~>
where C,(x,y,z) = C(x,y,z) - y - z
-1
(6) Here, d ( i j , k ) is the number of stereoisomers of fluorochloroalkanes containing i carbon atoms, j chlorine atoms, and k fluorine atoms, and d ( i j , k ) = d ( i , k j ) = d(ij,2i 2 -j-k)=d(i,k,2i+2-j-k) = d ( i , 2 i + 2 - j - k j ) =d(i,2i + 2 - j - k,k). 2.5. Let E(x,y,z) be the generating function for counting achiral planted steric trees with three different kinds of points. Let F(x,y,z) be the generating function for counting chiral planted steric trees with three different kinds of points. We get the functional relations
+
+ z + '/6X[C3(X,Y,Z)+ 2 2 2 3 3 3 3 ~ ( ~ , Y , z ) C,y( x,z 1 + 2C(X JJ,z )I
We must check that P(x,y,z) does not contain meso forms (R+ - R). Let Q(x,y,z) be the generating function of achiral stereoisomer of fluorochloroalkanes containing a labeled C-C bond, in which meso forms are not included. Though meso forms of fluorochloroalkanes containing one labeled C-C bond can be achiral. So
W , y , z ) + I/2F(X,Y,Z) = 1 + y
(7)
+ '/2[E(x2,y2,z2) - y 2 - z2 - 11 (16)
y - z - 11'
and E(x,y,z) + F(x,y,z) = C(X,Y,Z> From eqs 4, 7, and 8 we get m
~ ( x , y , z=)
(8)
Let S(x,y,z) be the generating function of achiral stereoisomer of R-R and R'OR' forms, in which the meso forms also are not included. So
2i+l 2i+l-j
7,
i=O j=O
S(x,y,z) = [ E ( x2 ,y 2 ,z 2) - y2 - z2 - 11
e ( i j , k ) x y z k=
(17)
j=O
1
+ y + z + xE(x,y,z)C(x2,y~,z2)( 9 )
where e(O,O,O) = 1, e(0,0,1) = 1, e(0,1,0) = 1, and e ( i j , k ) = e(i,kj) = e(ij,2i + 1 - J - k) = e(i,k,2i + 1 - j - k) = e(i,2i + 1 - j - kj) = e(i,2i + 1 - J - k,k). Here, e(ij,k) is the number of achiral stereoisomers of fluorochloroalkyls containing i carbon atoms, j chlorine atoms, and k fluorine atoms. The results are given in Table I. From eq 8 we get
However meso forms are achiral and must be counted. Let M(x,y,z) be the generating function of the meso forms of fluorochloroalkanes. Then M(x,y,z) = I/z[C(x2
2 9.Y
,z 21 - y 2 - z 2 - 11 1
and G(x,y,z) =
-
2
2
2
/2[E(X ,y ,z 1 -Y2 - z2 - 11 (18)
2i+2 2i+2-j
7, 7 i = l j=o
g(ij,k)xiYzk = P(x,y,z) j=o
4
wheref(ij,k) is the number of chiral stereoisomers of fluorochloroalkyls containing i carbon atoms, j chlorine atoms, and k fluorine atoms, andf(ij,k) = f ( i , k J )=flij,2i + 1 j-k)=f(i,k,2i+ l - j - k ) = f ( i , 2 i + l - j - k j ) = f ( i , 2 i + 1 - j - k,k). 2.6. Let G(x,y,z) be the generating function of achiral stereoisomer of fluorochloroalkanes and D'(x,y,z) be the
4
4
Q(x,y,z) + S(x,y,z) + M(x,y,z) = 1 / 2 x C ( x,Y J ) + 1 /*xE2(x,y,z)C(x2,y2,zZ) - 1/2[E(X,y,Z)- y 2 - z2- 112 + 1
/JC(X 2 ,y 2 ,z 21 - y 2 - z2 - 11 (19) where g(ij,k) is the number of achiral stereoisomers of fluorochloroalkanes containing i carbon atoms, j chlorine atoms, and k fluorine atoms. The results are given in Table I.
NUMBERS OF STEREO~SOMERS OF FLUOROCHLOROALKANES Table I. Numbers of Isomers of Fluorochloroalkvls and Fluorochloroalkanesa i j k m f(iJ,k) 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8
1 1 1 2 1 1 1 2 2 1 1 1 1 2 2 2 3 1 1 1 1 1 2 2 2 2 3 3 4 1 1 1 1 1 1 2 2 2 2 2 3 3 3 4 4 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4 4 4 5 1 1 1 1 1 1 1 1 2 2 2 2
1 1 2 2 1 2 3 2 3 1 2 3 4 2 3 4 3 1 2 3 4 5 2 3 4 5 3 4 4 1 2 3 4 5 6 2 3 4 5 6 3 4 5 4 5 1 2 3 4 5 6 7 2 3 4 5 6 7 3 4 5 6 4 5 6 5 1 2 3 4 5 6 7 8 2 3 4 5
1 3 2 1 5 4 3 3 2 7 6 5 4 5 4 3 3 9 8 7 6 5 7 6 5 4 5 4 3 11 10 9 8 7 6 9
a
7 6 5 7 6 5 5 4 13 12 11 10 9 8 7 11 10 9 8 7 6 9 8 7 6 7 6 5 5 15 14 13 12 11 10 9 8 13 12 11 10
1 4 5 5 13 23 27 35 35 42 91 138 159 182 25 1 25 1 30 1 131 338 620 878 979 829 1422 1840 1840 2231 2586 2586 402 1200 2563 4263 5709 6287 3462 7047 11006 13642 13642 13472 19462 21910 25529 25529 1218 4128 10020 19043 29456 38002 41309 13658 31999 58012 84725 101932 101932 71523 122343 166602 184350 195071 244298 244298 276507 3657 13856 37607 80087 139749 205 120 256869 276669 51640 136418 280153 467550
2 8 10 10 32 60 72 100 100 120 300 488 570 700 1020 1020 1276 438 1365 2788 4158 4732 4060 7714 10430 10430 13300 15806 15806 1572 5856 14364 25960 36432 40688 21100 49170 82940 106612 106612 107028 165438 190432 229020 229020 5568 24120 690 16 146208 242784 325804 358800 101920 280584 564560 878808 1089560 1089560 734120 1385376 1992976 2243932 24 16440 3 157104 3 157104 3663520 19532 96440 315156 764904 1463280 2282296 2959632 3223994 467040 1480962 3452960 6283424
0 0 2 2 0 4 0 8 8 0 8 0 10 20 20 20 0 0 17 0 26 0 52 52 70 70 0 70 70 0 32 0 60 0 72 120 120 200 200 200 0 200 0 280 280 0 64 0 140 0 198 0 280 280 560 560 688 688 0 560 0
688 980 980 980 0 0 120 0
300 0 488 0 570 600 600 1400 1400
2 8 8 8 32 56 72 92 92 120 292 488 560 680 1000 1000 1276 438 1348 2788 4132 4732 4008 7662 10360 10360 13300 15736 15736 1572 5824 14364 25900 36432 406 16 20980 49050 82740 106412 106412 107028 165238 190432 228740 228740 5568 24056 69016 146068 242784 325606 358800 101640 280304 564000 878248 1088872 1088872 734120 1384816 1992976 2243244 2415460 31 56124 3 156124 3663520 19532 96320 315156 764604 1463280 2281808 2959632 3223424 466440 1480362 3451560 6282024
J. Chem. Int Comput. Sci., Vol. 32, No. 5, 1992 409
n 2 4 3 2 6 5 4 4 3 8 7 6 5 6 5 4 4 10 9 8 7 6 8 7 6 5 6 5 4 12 11 10 9 8 7 10 9 8 7 6 8 7 6 6 5 14 13 12 11 10 9 8 12 11 10 9 8 7 10 9 8
7 8 7 6 6 16 15 14 13 12 11 10 9 14 13 12 11
diJ,k)
12 16 18 12 27 44 55 59 86 101 118 31 81 154 232 282 207 372 527 583 632 814 946 80 240 525 912 1295 1539 712 1494 2482 3300 3643 3012 468 1 5787 6748 7565 210 711 1753 3436 5549 7569 8811 2386 5726 10841 16693 21516 23355 13299 23984 34907 41935 40833 55399 61277 69217 555 2094 5741 12515 22568 34564 45594 52266 7871 21159 44899 78027
1 3 4 6 8 16 22 32 36 23 60 106 138 154 244 290 356 69 219 47 1 758 952 69 1 1394 2092 2366 2644 3570 4256 208 786 2000 3814 5742 7006 2944 7182 13010 18216 20380 16708 28 158 36178 43498 49984 636 2784 8186 18056 3 1608 45284 53996 12092 34508 73174 121392 163001 179400 94884 190760 296796 367900 359526 516536 581584 674152 1963 9766 32588 81636 162750 267066 368052 430924 48280 157578 382602 731640
1 1 2 4 2 4 4 8 8 3 8 8 10 22 20 30 20 5 17 17 26 26 43 52 74 70 52 70 106 8 32 32 60 60 72 108 120 220 200 272 120 200 200 390 280 14 64 64 140 140 198 198 204 280 490 560 723 688 280 560 560 688 1050 980 1324 980 23 120 120 300 300 488 488 570 480 600 1300 1400
1 1 1
6 12 18 24 28 20 52 98 128 132 224 260 336 64 202 454 732 926 648 1342 2018 2296 2592 3500 4150 200 754 1968 3754 5682 6934 2836 7062 12790 18016 20 108 16588 27958 35978 43108 49704 622 2720 8122 17916 31468 45086 53798 11888 34228 72684 120832 162278 178712 94604 190200 296236 367212 358476 5 15556 580260 673172 1940 9646 32468 81336 162450 266578 367564 430354 47800 156978 38 1302 730240
Gu AND WANC
410 J. Chem. If. Comput. Sci.. Vol. 32, No. 5, 1992 Table I. (Continued) i
j
k
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
2 2 2
6 7 8
3
3 3 3 3 3 4 4 4 4 5 5 6
1 4 5 6 1 4 5 6 7 5 6 6
m 9 8 7 1 10 9 8 7 9 8 7 6 7 6 5
a(ij,k)
c(ij,k)
e(ij,k)
Aij,k)
650517 764915 764915 347462 682016 1078542 1408556 1537621 1268161 1882161 2281740 2281740 2593063 2879642 2879642
9226336 11134160 11134160 45 13492 10015156 17163440 23457982 25989200 20931624 33392884 41902660 41902660 48912080 55399484 55399484
2040 2040 2040 0 1400 0 2040 0 2940 2940 3680 3680 0 3680 3680
9224296 11132120 11132120 4513492 10013756 17163440 23455942 25989200 20928684 33389944 41898980 41898980 48912080 55395804 55395804
n 10 9 8 12 11 10 9 8 10 9 8 7 8 7 6
b(ij,k)
d(ij,k)
g(iJ,k)
h(ij,k)
114361 142969 154005 55458 113308 188426 261601 307276 221301 348577 454452 495633 515671 624204 692751
1141392 1479816 1612282 499196 1163152 2115724 3105862 3747630 2577656 4412800 6028562 6677400 7030940 8817320 9982942
2208 2040 2610 600 1400 1400 2040 2040 3220 2940 4840 3680 2940 3680 6158
1139184 1477776 1609672 498596 1161752 2114324 3103822 3745590 2574436 4409860 6023722 6673720 7028000 8813640 9976784
* i is the number of carbon atoms. j is the number of chlorine atoms. k is the number of fluorine atoms. m is the number of hydrogen atoms of fluorochloroalkyl. n is the number of hydrogen atoms of fluorochloroalkane. o(ij,k) is the number of structural isomers of fluorochloroalkyls. c(ij,k) is the number of stereoisomers of fluorochloroalkyls. e ( i j , k ) is the number of achiral stereoisomers of fluorochloroalkyls. f ( i j , k ) is the number of chiral stereoisomers of fluorochloroalkyls. b ( i j , k ) is the number of structural isomers of fluorochloroalkancs. d ( i j , k ) is the number of stereoisomers of fluorochloroalkanes. g ( i j , k ) is the number of achiral stereoisomers of fluorochloroalkanes. h ( i j , k ) is the number of chiral stereoisomers of fluorochloroalkanes. -CHCICC1F2 -CHFCClZF -CClFCHClF -CClzCHF2 -CFzCHClz Figure 1. Structural isomers of fluorochloroalkyls having a general formula -C2HC12F2; the number is a(2,2,2). CHzClCClFz CHClFCHClF
F-f:
:lfi
chiral
chiral
l ; J -F
chiral
chiral
F c 1f i l
;$Cl
chiral
;-$;
c1
1
chiral
Let H(x,y,z) be the generating function of chiral stereoisomers of fluorochloroalkanes, we get 2i+2 21+2-j
7,y, j=O
:fz
F achiral
(3.3 F H achiral
b l ;$F c1 H H chiral chiral Figure 4. Stereoisomers of fluorochloroalkyls having a general formula C2H2C12F2; the number is d(2,2,2).
cF1
Some results are given in Table I. As an example, all the structural isomers and stereoisomers (Fischer projections) of the fluorochloroalkyls having the general formula CzHClzF2 and the fluorochloroalkaneshaving the general formula C2H2ClzF2 are given in Figures 1-4.
REFERENCES AND NOTES
achiral Figure 3. Stereoisomers of fluorochloroalkyls having a general formula -C2HCI2F2; the number is c(2,2,2).
i=O
H achiral
The results for some large molecules may not be perfectly satisfactory from a chemicalstandpoint because possiblesteric hindrance is not considered.8
achiral
H(xy,z) =
;$El
chiral
F $- ;
chiral
H
E $-!l:
01
c1 achiral
CHClzCHFz CH,FCCl,F
Figure 2. Structural isomers of fluorochloroalkyls having a general formula C2H2C12F2; the number is b(2,2,2).
H F$E1
H Ff-;
j=O
h(iJ,k)x’y’zk = D(x,y,z)- G(x,y,z)
(20)
where h(iJ,k) is the number of chiral stereoisomers of fluorochloroalkanes containing icarbon atoms,j chlorine atoms, and k fluorine atoms.
(1) Henze, H. R.; Blair, C. M.The Number of Structurally Isomeric Alcohols of the Methanol Series. J . Am. Chem. Soc. 1931,53, 3042. (2) Henze, H. R.; Blair, C. M. The Number of Structurally Isomeric Hydrocarbons of the Ethylene Series. J. Am. Chem. Soc. 1931, 53, 3077. (3) Robinson, R. W.; Harary, F.;Balaban, A. T. The Numbers of Chiral and Achiral Alkanes and Monosubstituted Alkanes. Tetrahedron 1976, 32, 355. (4) Read, R. C. In Chemical Applications of Graph Theory; Balaban, A. T., Ed.; Academic Press: London, 1976; Chapter 4, p 25. (5) Wang, J.; Wang, Q.The Numbers of Constitutions, Configurations, Chiral and Achiral Configurations of Polyhaloalkanes CtHa+2-,Xp Tetrohedron 1991, 47, 2969. (6) Pdya, G.; Kombinatorische Anzahlbestimmungen fur Gruppen, Graphen und Chemische Verbindungen. Acto Math. 1937, 68, 145. (7) . . Wana. J.; Gu. F. Enumeration of Isomers of Polyethers. J . Chem. If. Comput. Sci. 1991, 31, 552. (8) Klein, D. J. Rigorous Reaults for Branched Polymer Models with Excluded Volume. J . Chem. Phys. 1981, 75, 5186.
