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A sum rule for harmonic vibrational frequencies valid within a series of pairs of isomeric isotopomers
Volume 149, number 5,6
2 September 1988
CHEMICAL PHYSICS LETTERS
A SUM RULE FOR HARMONIC VIBRATIONAL FREQUENCIES VALID WITHIN A SERIES OF PAIRS OF ISOMERIC ISOTOPOMERS
*,**
Zden6k SLANINA ’ The J. Heyrovskj Institute of Physical Chemistry and Electrochemistry, Czechoslovak Academy of Sciences. Michova 7, 12138 Prague 2 - Vinohrady, Czechoslovakia Received 1 March 1988: in final form 17 May 1988
The difference in the sums of squares of harmonic vibrational frequencies for two isotopomers which are isomeric, differing by the interchange of isotopic labels at two positions, is invariant with respect to the isotopic composition of the rest of the molecule, This has been illustrated for substituted ethylenes and water-dimer isotopomers as examples. It has been shown that the waterdimer isotopomen can be decomposed into four classes.
1. Introduction
Various relations exist between the vibrational frequencies of isotopic modifications of molecular systems which are useful for vibrational analysis, see e.g. refs. [l-6]. It is often appropriate to separate the isomeric isotopomers as a special subclass of all isotopomers of a given species [ 7- 15 1. Quite recently observations have been reported concerning the relative stabilities of various deuterated water dimers in krypton matrices, and the zero-point vibrational energy differences have been estimated for seven isomerizations of these isotopomers [ 161. For a rationalization of the relations between isomeric isotopomers it is useful to analyze the problem at the harmonic level, which is the topic of the present communication.
2. General formulation Let us consider a molecular system and, therein,
two positions, 1 and 2, which can be occupied by two different isotopes of the same atom. Without making the problem less general we shall consider the pair involved to be H,D. The arrangement in which H adopts the 1 position and D the 2 position will be denoted as H/D, and the isomeric configuration formed by the interchange of H and D atoms (i.e. D atom at position 1 and H atom at position 2) will be assigned the symbol D/H. All the other atoms of the system except the mutually interchanging H,D pair will be called the frame. For our purposes the vibrational problem will be treated in terms [ 1] of the force-constant matrix, F’, in 3N mass-weighted Cartesian coordinates. The vibrational frequencies oi satisfy the relationship J+i=4x2c$ with respect to the eigenvalues A, of the matrix F’, IF’-AEI
* Dedicated to Professor E. Bright Wilson on the occasion of his 80th birthday. ** Part XXXVIIl in the series “Multi-molecular Clusters and Their Isomerism”. For part XXXVII, see ref. [241. ’ Reprint-request address: The J. Heyrovsky Institute of Physical Chemistry and Electrochemistry, Czechoslovak Academy of Sciences, DolejSkova 3, 182 23 Prague 8 - Kobylisy, Czechoslovakia.
=O.
(1)
Furthermore, let us also introduce the force-constant matrix in Cartesian coordinates, F: F;=Fij/(mimj)“‘,
(2)
where mk denotes the mass of the kth atom. From matrix algebra it is well known that [ 171
0 009-2614/88/$ 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
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where the summation runs over all 3N degrees of freedom (it makes no difference that in the case of a non-linear molecule six of the eigenvalues li become zero). Assuming the validity of the Born-Oppenheimer approximation, the force fields (F, constants) are identical for both configurations H/D and D/H. As the atomic masses within the frame are fixed, the sum IFi, over the degrees of freedom of the frame for both isomers is constant, hence
2 September
1988
tutions in the frame. Depending on the possibilities of isotopic modifications which are compatible with the given frame we can obtain series of pairs of isomerit isotopomers. In principle, isotopic substitution could be considered for every atom of the frame; nevertheless, the most interesting is gradual substitution of the hydrogen atoms of the frame by deuterium. Finally, let us note that the differences (6) can also be formally taken as the changes belonging to ( isomerization ) D/H-t the reaction H/D.
3. Illustration on some substituted ethylenes ,z F:i(HID) =const. +
r
F,, +F,, +FM mH
t
J-44+F,, +FM mD
3 (4)
3.v ,;,
CAD/H)
=const. t
F,, +Fz +F33 + f’u +F,, +Fc6 mD
9
(5)
mH
where mH and mb are the masses of the H and D atoms, respectively. From the relations (4) and (5) the following expression is obtained for the difference between the sums of squares of vibrational frequencies of the two isomers: 4x2 F o:(H/D) /=I
-4~~ ‘c”wf(D/H) i=l
(6) We now carry out an isotopic substitution within the frame - to obtain a new pair of isomeric isotopomers H/D and D/H with a new value for the common constant (clearly both isomeric pairs, the former and the newer one, generally will not form four mutually isomeric species but only two pairs of isomers). As, however, the value of this constant cancels in eq. (6), the following useful conclusion can be made: For all H/D and D/H isomeric pairs formed in this way the value of the difference of the sum of squares of their harmonic vibrational frequencies is a constant, invariant with respect to isotopic substi498
To illustrate the possible applications of the rule derived in section 2 we shall use the substituted ethylenes H,C=CHX in which X is either a halogen atom (X =F, Cl, Br) or hydrogen. There are three topologically different ways for the choice of centres representing the interchanging H, D pair (fig. 1). Let us start from the arrangement HDC-CHX; here the frame and its isotopic modification are represented by the species C=CHX and C=CDX, respectively. If X = H, then, of course, both members of the pair resulting from the abovementioned frame are identical, i.e. the difference of both sums in eq. (6) is zero. From the application of the summation rule it follows that the difference (6) must be zero also for the frame C=CHD, which in other words means the same values of sums of squares of the harmonic vibrational frequencies for cis- and trans-ethylene-& Analogous considerations can also be carried out for the remaining two choices of positions of the mutually interchanging H, D pair. It follows that a-ethylene-& also has the same sum of squares of harmonic vibrational frequencies as the two abovementioned dideuteroethylenes. As the observed values of normal vibrational frequencies are available for all three species [ 41, the validity of the three relations can be tested for dideuteroethylenes. It can clearly be seen (tables 1 and 2) that the changes in the sums of squares of the frequencies for all three isomerizations considered in the second line of fig. 1 are relatively close to zero. (Note that a deviation from zero of the sum by lo4 cme2 can be caused by an error of about 1.7 cm-’ in the determination of a frequency with a value of 3000 cm-‘.) The fre-
Volume 149, number 56
2 September I988
CHEMICAL PHYSICS LETTERS cis-h2
trans- h2
trans.- h2
a-h2
cis -h2
a-h2
trawd2
cis-d2
a-d2
vans-d2
a-d-2
cis-d2
Fig. 1. Three classes ((i)-(iii) ) of monosubstituted (X) ethylene isotopomer isomerizations classified according to the differences in the sums of squares of their vibrational frequencies; the interchanging H, D pair is shown in each reactant in circles.
Table 1 Sums of squares of vibrational frequencies a) of mono- and dideuterated ethylenes H,D,XC? (n + m = 3) Species
X=H
X=F
X=CI
X=Br
cis-hz tram-h, cis-da tram-d,
43.12 b, 43.72 b’ 38.52 38.53 43.72 b’ 38.52
34.47 34.31 29.06 28.94 34.18 29.19
33.13 33.14 27.85 27.94 32.98 28.05
32.66 32.75 27.52 27.42 32.61 27.12
a-h, a-d,
u) In ( IO3 cm-‘)2; b, Ethylene-d,.
derived from compilation [4].
quencies observed are not strictly harmonic and, in addition, although originating predominantly from gas-phase measurements, they are partially completed [ 4 ] by liquid-state measurements. The relations are less good in the case of monohalogen derivatives of ethylene. Here the application of the sum rule (see fig. 1) leads to the requirement of equality of the change in the sum of squares of harmonic vibrational frequencies for the two isomerizations: cis-HDC=CHX+trans-HDC=CHX and
Table 2 Differences in sums ‘) of squares of vibrational frequencies along the isomerizations of the isotopomers shown in fig. 1 Position ofH,D pair b’
Frame
(i)
C,HX CaDX
X=H
X=F
X=Cl
X=Br
trans-HDC=CDX=cis-HDC=CDX
,
(7)
for trans-HDC=CHX=H,C=CDX and
(ii)
C,HX C,DX
(iii)
CzHX CzDX
0” -13
-103 120
-96
87 91
0 cJ IO
- 195 -255
-158 -104
-84 - 300
0 r,
-298 -135
-153 -200
-204
-3
8’In lo3 cm-l; based on data of table 1. b, Seefig. 1. c1 Autoisomerization of ethylene-d,.
5
3
D, C=CHX = trans-HDC=CDX ,
(8)
and finally for cis-HDC=CHX =H, C=CDX and DZC=CHX%is-HDC=CDX (obviously, reactions (7)-(9)
(9) are not linearly in-
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Volume 149, number 5,6
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PHYSICS LETTERS
dependent: the processes (9) are the sum of reactions ( 7 ) and ( 8 ) ) , Again we can utilize the available data [ 41 for X = F, Cl, Br for numerical illustration (tables 1 and 2). Bearing in mind that the harmonic frequencies are not available, and not all frequencies originate from the gas phase, then the agreement can be regarded as satisfactory. The condition (6) is best fulfilled with the bromo derivative in the reactions ( 7 ). On the other hand, the largest discrepancies are encountered with the fluoro derivative, again in the reactions (7). Let us note, however, that in the last unfavourable case mentioned perfect agreement could be attained by a 37 cm-’ shift in a frequency of around 3000 cm-‘. (For the sake of completeness let us note, however, that for another type of reaction it was shown [ 18-201 that the anharmonicity contribution and other higher-order corrections to the zero-point vibrational energy change are negligible. ) It is obvious that the rule suggested enables a systematical generation of identities which could serve for testing or completing the data observed for the vibrational spectra of some isotopomers.
4. The water-dimer isotopomers In contrast to the observed data it is possible that the condition (6) will be perfectly fulfilled by the vibrational frequencies of isotopomers derived computationally from a single common force field. This approach will be applied to the water-dimer isotopomers, and our description of the force field will start from the MCY water interaction potential [ 2 1 ] in its MCY-B modification [ 22 ] which was recently shown [23] to lead to the best agreement with the thermodynamic data observed for water-dimer formation in the gas phase. A number of studies (see e.g. refs. [ 2 l-23 ] ) have shown that the water dimer exhibits a C, structure (see fig. 2), i.e. a structure with one symmetry plane which superimposes the hydrogen atoms of the acceptor water molecule in the dimer. If no such symmetry plane existed, there would be altogether 24 various (H,O), isotopomers (the oxygen isotopes are not considered). The presence of the symmetry element reduces the number of isotopomers to 12. Of these, however, two are irrelevant ( (H,0)2 and 500
2 September
I988
( D20) *) from the point of view of isomerization, hence ten structures are left for further consideration. Among these structures three are monodeuterated, four are dideuterated, and three are trideuterated. From the mono-, di-, and tri-deuterated dimers it is possible to compose (:), (Z), and (: ) isomerizations, respectively, i.e. altogether 12 isomerizations. Fig. 2 presents a systematic way of generating these isomerizations that clearly shows the correspondence to the classes created by the sum rule. There are three topologically different choices of the mutually interchanging H,D pair, e.g. the first column of fig. 2 analyzes the choice which locates both atoms on the donor molecule. In this case the frame is formed by the OOHz grouping which allows two possible hydrogen isotopic modifications - OOHD and OODz. Therefore, for this choice of the H,D pair there exist three isomerizations between the water-dimer isotopomers which exhibit the same value for the change in the sum of squares of harmonic vibrational frequencies. Similarly, fig. 2 shows how to generate two further classes corresponding to the remaining two possible choices of the position of mutually interchanging H,D pairs. As can be seen from fig. 2, altogether eleven isomerizations are obtained in these three classes. The final, twelfth one cannot be generated by the interchange of only two atoms. This twelfth isomerization involves a process in which all four light atoms take part simultaneously: DOD*OH* =HOH.OD,
.
(10)
Thus without any calculations we arrived at the interesting conclusion that the group of all twelve isomerizations considered can be decomposed into four classes, each having its own value of the constant term in eq. (6) common to all members of a particular class. However, it must be noted that only eight out of these twelve isomerizations are such that they involve cleavage of no chemical (intramolecular) bond but only intermolecular hydrogen bond. Table 3 illustrates these conclusions numerically, showing that the differences between the values of the term (6) are considerable between the four individual classes. The term (6), of course, is not accessible by observation. Experiment [ 16 ] determines the zero-point vibration energy changes which would necessitate a transition from the second to the first
Volume 149, number 5,6
O-@OH2=
CHEMICAL PHYSICS LETTERS 0-H.C@H’
0-D.OH2
0-GOHO=
dD
0-@OD2= dD
P
-D.OHD
0-H.C@D=
O-DOD2 Id
-H.OD2
O+OgH=
0-H.OHD o/
p+@=
O-H.002 Id
H
K
6
0-H.OHD I4
o/
H
6D
Id
/
H
Is
0-_D.cw:
0-D.OHD
O-@@H=
0-H.OHD
Q
d
6D
2 September 1988
&-D.@=(-0.002
Fig. 2. Three (of four possible) classes of water-dimer isotopomer isomerizations classified according to the differences in the sums of squares of their vibrational frequencies (each class is assembled into one column); the interchanging H,D pair is shown in each reactant in circles.
powers of frequencies. Consequently the strict decomposition into four classes (table 2) disappears. Nevertheless it can be seen that a certain degree of correspondence is maintained - those isomerizations corresponding to the same class in terms of eq.
(6) have similar zero-point vibration energy changes as well. An interesting, more general comment should be added concerning another physical meaning of eq. (6). The difference in the sums of squares (6) can be interpreted as a measure of how different posi-
Table 3 Changes in the sum of the squares of vibrational frequencies (A&uf) and zero-point vibrational energies (ACfo,) tions of the water-dimer isotopomers shown in fig. 2 A&f
Isomerization
DOH.OH,=HOD.OH, b, DOH.OHD=HOD.OHD b’ DOH*OD,GHOD.OD, b, DOH,OH,=HOH.OHD DOD.OH,=HOD.OHD DOH,OHD=HOH.OD, DOD,OHD=HOD.OD,
b,
(10’ cm-*)
-248.6
-85.6
b,
HOD,OH,=HOH.OHD b, DODaOH,=DOH*OHD HOD.OHD=HOH.OD, DOD.OHD=DOH.OD, b,
162.9
77.3
alongthe isomeriza-
AX@, (cm-‘) MCY-B
observed ‘)
-18.4 -78.7 -78.9
- ( 110-69) - (70-60) - (68-50)
-4.9 - 12.7 -0.9 -8.5
-6-4
13.5 66.0 77.8 70.4
40-60
65.1
30-50
45-72
a) Ref. [ 161. b, Reconstruction of the intermolecular hydrogen bond only.
501
CHEMICALPHYSICSLETTERS
Volume 149,number 5,6
tions 1 and 2 are in terms of the potential energy (i.e. how different are their sums of diagonal Cartesian force constants ). This is well illustrated by the water dimer. Of the three classes of isomerizations shown in fig. 2, the change in the sums of the squares of the vibrational frequencies (6) (more precisely, its absolute value) is smallest when the interchanging H, D atoms are not involved in the intermolecular bond - see table 3. This agrees with the physical insight that perturbations of the free-monomer force constants are smaller for atoms not involved in the intermolecular bond (although, for a more rigorous discussion, Cartesian coordinate transformations should be properly considered). 5. Concluding remarks The existence of classes of isotopomers formed by pairs of isomers exhibiting the same value for the difference in the sum of squares of their harmonic vibrational frequencies has been shown. This sum rule allows observed vibrational data to be tested or completed. On the basis of this sum rule we can give a rationalization of the (computational) finding of only four values of this difference for the twelve different isomerizations of the water-dimer isotopomers. Acknowledgement The constructive
comments
of the referee
are
highly appreciated. References [ I] E.B. Wilson Jr., J.C. Decius and P.C. Cross, Molecular vibrations (McGraw-Hill, New York, 1955).
2 September 1988
[2] R.P. Bell, The proton in chemistry (Methuen, London, 1959). [ 31 M. Tasumi and G. Zerbi, J. Chem. Phys. 48 ( 1968) 3813. [4] L.M. Sverdlov, M.A. Kovner and E.P. Krainov, Vibrational spectra of polyatomic molecules (Nauka, Moscow, 1970) [in Russian 1. [ 51M.V. Volkenshtein, L.A.Gribov, M.A. Elyashevichand B.I. Stepanov,Vibrations of molecules (Nauka, Moscow, 1972) [in Russian]. [ 61 L. Melander and W.H. Saunders Jr., Reaction rates of isotopic molecules (Wiley, New York, 1980). [ 7 ] W. von E. Doering, V.G. Toscano and G.H. Beasley, Tetrahedron 27 (1971) 5299. [ 81 M.J. Goldstein and MS. Benzon, J. Am. Chem. Sot. 94 (1972) 7147. [ 91 L. Fredin, B. Nelander and G. Ribbegard, Chem. Phys. Letters 36 (1975) 375. [ 10] L. Fredin, B. Nelander and G. Ribbegrd, J. Chem. Phys. 66 (1977) 4065. [ 111L. Fredin, B. Nelander and G. RibbegBrd, J. Chem. Phys. 66 (1977) 4073. [ 121 L.J. Schaad, B.A. Hess Jr. and C.S. Ewig,J. Org. Chem. 47 ( 1982) 2904. [ 13 ] Z. Slanina, Contemporary theory of chemical isomerism (Academia/Reidel, Prague/Dordrecht, 1986). [ 141 Z. Slanina, Thermochim. Acta 112 (1987) 379. [ 15] Z. Slanina, J. Radioanal. Nucl. Chem. 111 (1987) 157. [ 161 A. Engdahl and B. Nelander, J. Chem. Phys. 86 ( 1987) 1819. [ 171 C.E. Pearson, Handbook of applied mathematics (Van Nostrand Reinhold, New York, 1974) p, 1014. [ 181 J.R. Hulston, J. Chem. Phys. 50 (1969) 1483. [ 191 M. Wolfsberg, J. Chem. Phys. 50 (1969) 1484. [20] J. Bron and M. Wolfsberg, J. Chem. Phys. 57 ( 1972) 2682. [21] 0. Matsuoka, E. Clementi and M. Yoshimine, J. Chem. Phys. 64 (1976) 1351. [22] D.G. Bounds, Chem. Phys. Letters 96 ( 1983) 604. [23] Z. Slanina, Chem. Phys. Letters 127 (1986) 67. [24] Z. Slanina, J. Phys. Chem., to be published.
2 September 1988
CHEMICAL PHYSICS LETTERS
A SUM RULE FOR HARMONIC VIBRATIONAL FREQUENCIES VALID WITHIN A SERIES OF PAIRS OF ISOMERIC ISOTOPOMERS
*,**
Zden6k SLANINA ’ The J. Heyrovskj Institute of Physical Chemistry and Electrochemistry, Czechoslovak Academy of Sciences. Michova 7, 12138 Prague 2 - Vinohrady, Czechoslovakia Received 1 March 1988: in final form 17 May 1988
The difference in the sums of squares of harmonic vibrational frequencies for two isotopomers which are isomeric, differing by the interchange of isotopic labels at two positions, is invariant with respect to the isotopic composition of the rest of the molecule, This has been illustrated for substituted ethylenes and water-dimer isotopomers as examples. It has been shown that the waterdimer isotopomen can be decomposed into four classes.
1. Introduction
Various relations exist between the vibrational frequencies of isotopic modifications of molecular systems which are useful for vibrational analysis, see e.g. refs. [l-6]. It is often appropriate to separate the isomeric isotopomers as a special subclass of all isotopomers of a given species [ 7- 15 1. Quite recently observations have been reported concerning the relative stabilities of various deuterated water dimers in krypton matrices, and the zero-point vibrational energy differences have been estimated for seven isomerizations of these isotopomers [ 161. For a rationalization of the relations between isomeric isotopomers it is useful to analyze the problem at the harmonic level, which is the topic of the present communication.
2. General formulation Let us consider a molecular system and, therein,
two positions, 1 and 2, which can be occupied by two different isotopes of the same atom. Without making the problem less general we shall consider the pair involved to be H,D. The arrangement in which H adopts the 1 position and D the 2 position will be denoted as H/D, and the isomeric configuration formed by the interchange of H and D atoms (i.e. D atom at position 1 and H atom at position 2) will be assigned the symbol D/H. All the other atoms of the system except the mutually interchanging H,D pair will be called the frame. For our purposes the vibrational problem will be treated in terms [ 1] of the force-constant matrix, F’, in 3N mass-weighted Cartesian coordinates. The vibrational frequencies oi satisfy the relationship J+i=4x2c$ with respect to the eigenvalues A, of the matrix F’, IF’-AEI
* Dedicated to Professor E. Bright Wilson on the occasion of his 80th birthday. ** Part XXXVIIl in the series “Multi-molecular Clusters and Their Isomerism”. For part XXXVII, see ref. [241. ’ Reprint-request address: The J. Heyrovsky Institute of Physical Chemistry and Electrochemistry, Czechoslovak Academy of Sciences, DolejSkova 3, 182 23 Prague 8 - Kobylisy, Czechoslovakia.
=O.
(1)
Furthermore, let us also introduce the force-constant matrix in Cartesian coordinates, F: F;=Fij/(mimj)“‘,
(2)
where mk denotes the mass of the kth atom. From matrix algebra it is well known that [ 171
0 009-2614/88/$ 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
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where the summation runs over all 3N degrees of freedom (it makes no difference that in the case of a non-linear molecule six of the eigenvalues li become zero). Assuming the validity of the Born-Oppenheimer approximation, the force fields (F, constants) are identical for both configurations H/D and D/H. As the atomic masses within the frame are fixed, the sum IFi, over the degrees of freedom of the frame for both isomers is constant, hence
2 September
1988
tutions in the frame. Depending on the possibilities of isotopic modifications which are compatible with the given frame we can obtain series of pairs of isomerit isotopomers. In principle, isotopic substitution could be considered for every atom of the frame; nevertheless, the most interesting is gradual substitution of the hydrogen atoms of the frame by deuterium. Finally, let us note that the differences (6) can also be formally taken as the changes belonging to ( isomerization ) D/H-t the reaction H/D.
3. Illustration on some substituted ethylenes ,z F:i(HID) =const. +
r
F,, +F,, +FM mH
t
J-44+F,, +FM mD
3 (4)
3.v ,;,
CAD/H)
=const. t
F,, +Fz +F33 + f’u +F,, +Fc6 mD
9
(5)
mH
where mH and mb are the masses of the H and D atoms, respectively. From the relations (4) and (5) the following expression is obtained for the difference between the sums of squares of vibrational frequencies of the two isomers: 4x2 F o:(H/D) /=I
-4~~ ‘c”wf(D/H) i=l
(6) We now carry out an isotopic substitution within the frame - to obtain a new pair of isomeric isotopomers H/D and D/H with a new value for the common constant (clearly both isomeric pairs, the former and the newer one, generally will not form four mutually isomeric species but only two pairs of isomers). As, however, the value of this constant cancels in eq. (6), the following useful conclusion can be made: For all H/D and D/H isomeric pairs formed in this way the value of the difference of the sum of squares of their harmonic vibrational frequencies is a constant, invariant with respect to isotopic substi498
To illustrate the possible applications of the rule derived in section 2 we shall use the substituted ethylenes H,C=CHX in which X is either a halogen atom (X =F, Cl, Br) or hydrogen. There are three topologically different ways for the choice of centres representing the interchanging H, D pair (fig. 1). Let us start from the arrangement HDC-CHX; here the frame and its isotopic modification are represented by the species C=CHX and C=CDX, respectively. If X = H, then, of course, both members of the pair resulting from the abovementioned frame are identical, i.e. the difference of both sums in eq. (6) is zero. From the application of the summation rule it follows that the difference (6) must be zero also for the frame C=CHD, which in other words means the same values of sums of squares of the harmonic vibrational frequencies for cis- and trans-ethylene-& Analogous considerations can also be carried out for the remaining two choices of positions of the mutually interchanging H, D pair. It follows that a-ethylene-& also has the same sum of squares of harmonic vibrational frequencies as the two abovementioned dideuteroethylenes. As the observed values of normal vibrational frequencies are available for all three species [ 41, the validity of the three relations can be tested for dideuteroethylenes. It can clearly be seen (tables 1 and 2) that the changes in the sums of squares of the frequencies for all three isomerizations considered in the second line of fig. 1 are relatively close to zero. (Note that a deviation from zero of the sum by lo4 cme2 can be caused by an error of about 1.7 cm-’ in the determination of a frequency with a value of 3000 cm-‘.) The fre-
Volume 149, number 56
2 September I988
CHEMICAL PHYSICS LETTERS cis-h2
trans- h2
trans.- h2
a-h2
cis -h2
a-h2
trawd2
cis-d2
a-d2
vans-d2
a-d-2
cis-d2
Fig. 1. Three classes ((i)-(iii) ) of monosubstituted (X) ethylene isotopomer isomerizations classified according to the differences in the sums of squares of their vibrational frequencies; the interchanging H, D pair is shown in each reactant in circles.
Table 1 Sums of squares of vibrational frequencies a) of mono- and dideuterated ethylenes H,D,XC? (n + m = 3) Species
X=H
X=F
X=CI
X=Br
cis-hz tram-h, cis-da tram-d,
43.12 b, 43.72 b’ 38.52 38.53 43.72 b’ 38.52
34.47 34.31 29.06 28.94 34.18 29.19
33.13 33.14 27.85 27.94 32.98 28.05
32.66 32.75 27.52 27.42 32.61 27.12
a-h, a-d,
u) In ( IO3 cm-‘)2; b, Ethylene-d,.
derived from compilation [4].
quencies observed are not strictly harmonic and, in addition, although originating predominantly from gas-phase measurements, they are partially completed [ 4 ] by liquid-state measurements. The relations are less good in the case of monohalogen derivatives of ethylene. Here the application of the sum rule (see fig. 1) leads to the requirement of equality of the change in the sum of squares of harmonic vibrational frequencies for the two isomerizations: cis-HDC=CHX+trans-HDC=CHX and
Table 2 Differences in sums ‘) of squares of vibrational frequencies along the isomerizations of the isotopomers shown in fig. 1 Position ofH,D pair b’
Frame
(i)
C,HX CaDX
X=H
X=F
X=Cl
X=Br
trans-HDC=CDX=cis-HDC=CDX
,
(7)
for trans-HDC=CHX=H,C=CDX and
(ii)
C,HX C,DX
(iii)
CzHX CzDX
0” -13
-103 120
-96
87 91
0 cJ IO
- 195 -255
-158 -104
-84 - 300
0 r,
-298 -135
-153 -200
-204
-3
8’In lo3 cm-l; based on data of table 1. b, Seefig. 1. c1 Autoisomerization of ethylene-d,.
5
3
D, C=CHX = trans-HDC=CDX ,
(8)
and finally for cis-HDC=CHX =H, C=CDX and DZC=CHX%is-HDC=CDX (obviously, reactions (7)-(9)
(9) are not linearly in-
499
Volume 149, number 5,6
CHEMICAL
PHYSICS LETTERS
dependent: the processes (9) are the sum of reactions ( 7 ) and ( 8 ) ) , Again we can utilize the available data [ 41 for X = F, Cl, Br for numerical illustration (tables 1 and 2). Bearing in mind that the harmonic frequencies are not available, and not all frequencies originate from the gas phase, then the agreement can be regarded as satisfactory. The condition (6) is best fulfilled with the bromo derivative in the reactions ( 7 ). On the other hand, the largest discrepancies are encountered with the fluoro derivative, again in the reactions (7). Let us note, however, that in the last unfavourable case mentioned perfect agreement could be attained by a 37 cm-’ shift in a frequency of around 3000 cm-‘. (For the sake of completeness let us note, however, that for another type of reaction it was shown [ 18-201 that the anharmonicity contribution and other higher-order corrections to the zero-point vibrational energy change are negligible. ) It is obvious that the rule suggested enables a systematical generation of identities which could serve for testing or completing the data observed for the vibrational spectra of some isotopomers.
4. The water-dimer isotopomers In contrast to the observed data it is possible that the condition (6) will be perfectly fulfilled by the vibrational frequencies of isotopomers derived computationally from a single common force field. This approach will be applied to the water-dimer isotopomers, and our description of the force field will start from the MCY water interaction potential [ 2 1 ] in its MCY-B modification [ 22 ] which was recently shown [23] to lead to the best agreement with the thermodynamic data observed for water-dimer formation in the gas phase. A number of studies (see e.g. refs. [ 2 l-23 ] ) have shown that the water dimer exhibits a C, structure (see fig. 2), i.e. a structure with one symmetry plane which superimposes the hydrogen atoms of the acceptor water molecule in the dimer. If no such symmetry plane existed, there would be altogether 24 various (H,O), isotopomers (the oxygen isotopes are not considered). The presence of the symmetry element reduces the number of isotopomers to 12. Of these, however, two are irrelevant ( (H,0)2 and 500
2 September
I988
( D20) *) from the point of view of isomerization, hence ten structures are left for further consideration. Among these structures three are monodeuterated, four are dideuterated, and three are trideuterated. From the mono-, di-, and tri-deuterated dimers it is possible to compose (:), (Z), and (: ) isomerizations, respectively, i.e. altogether 12 isomerizations. Fig. 2 presents a systematic way of generating these isomerizations that clearly shows the correspondence to the classes created by the sum rule. There are three topologically different choices of the mutually interchanging H,D pair, e.g. the first column of fig. 2 analyzes the choice which locates both atoms on the donor molecule. In this case the frame is formed by the OOHz grouping which allows two possible hydrogen isotopic modifications - OOHD and OODz. Therefore, for this choice of the H,D pair there exist three isomerizations between the water-dimer isotopomers which exhibit the same value for the change in the sum of squares of harmonic vibrational frequencies. Similarly, fig. 2 shows how to generate two further classes corresponding to the remaining two possible choices of the position of mutually interchanging H,D pairs. As can be seen from fig. 2, altogether eleven isomerizations are obtained in these three classes. The final, twelfth one cannot be generated by the interchange of only two atoms. This twelfth isomerization involves a process in which all four light atoms take part simultaneously: DOD*OH* =HOH.OD,
.
(10)
Thus without any calculations we arrived at the interesting conclusion that the group of all twelve isomerizations considered can be decomposed into four classes, each having its own value of the constant term in eq. (6) common to all members of a particular class. However, it must be noted that only eight out of these twelve isomerizations are such that they involve cleavage of no chemical (intramolecular) bond but only intermolecular hydrogen bond. Table 3 illustrates these conclusions numerically, showing that the differences between the values of the term (6) are considerable between the four individual classes. The term (6), of course, is not accessible by observation. Experiment [ 16 ] determines the zero-point vibration energy changes which would necessitate a transition from the second to the first
Volume 149, number 5,6
O-@OH2=
CHEMICAL PHYSICS LETTERS 0-H.C@H’
0-D.OH2
0-GOHO=
dD
0-@OD2= dD
P
-D.OHD
0-H.C@D=
O-DOD2 Id
-H.OD2
O+OgH=
0-H.OHD o/
p+@=
O-H.002 Id
H
K
6
0-H.OHD I4
o/
H
6D
Id
/
H
Is
0-_D.cw:
0-D.OHD
O-@@H=
0-H.OHD
Q
d
6D
2 September 1988
&-D.@=(-0.002
Fig. 2. Three (of four possible) classes of water-dimer isotopomer isomerizations classified according to the differences in the sums of squares of their vibrational frequencies (each class is assembled into one column); the interchanging H,D pair is shown in each reactant in circles.
powers of frequencies. Consequently the strict decomposition into four classes (table 2) disappears. Nevertheless it can be seen that a certain degree of correspondence is maintained - those isomerizations corresponding to the same class in terms of eq.
(6) have similar zero-point vibration energy changes as well. An interesting, more general comment should be added concerning another physical meaning of eq. (6). The difference in the sums of squares (6) can be interpreted as a measure of how different posi-
Table 3 Changes in the sum of the squares of vibrational frequencies (A&uf) and zero-point vibrational energies (ACfo,) tions of the water-dimer isotopomers shown in fig. 2 A&f
Isomerization
DOH.OH,=HOD.OH, b, DOH.OHD=HOD.OHD b’ DOH*OD,GHOD.OD, b, DOH,OH,=HOH.OHD DOD.OH,=HOD.OHD DOH,OHD=HOH.OD, DOD,OHD=HOD.OD,
b,
(10’ cm-*)
-248.6
-85.6
b,
HOD,OH,=HOH.OHD b, DODaOH,=DOH*OHD HOD.OHD=HOH.OD, DOD.OHD=DOH.OD, b,
162.9
77.3
alongthe isomeriza-
AX@, (cm-‘) MCY-B
observed ‘)
-18.4 -78.7 -78.9
- ( 110-69) - (70-60) - (68-50)
-4.9 - 12.7 -0.9 -8.5
-6-4
13.5 66.0 77.8 70.4
40-60
65.1
30-50
45-72
a) Ref. [ 161. b, Reconstruction of the intermolecular hydrogen bond only.
501
CHEMICALPHYSICSLETTERS
Volume 149,number 5,6
tions 1 and 2 are in terms of the potential energy (i.e. how different are their sums of diagonal Cartesian force constants ). This is well illustrated by the water dimer. Of the three classes of isomerizations shown in fig. 2, the change in the sums of the squares of the vibrational frequencies (6) (more precisely, its absolute value) is smallest when the interchanging H, D atoms are not involved in the intermolecular bond - see table 3. This agrees with the physical insight that perturbations of the free-monomer force constants are smaller for atoms not involved in the intermolecular bond (although, for a more rigorous discussion, Cartesian coordinate transformations should be properly considered). 5. Concluding remarks The existence of classes of isotopomers formed by pairs of isomers exhibiting the same value for the difference in the sum of squares of their harmonic vibrational frequencies has been shown. This sum rule allows observed vibrational data to be tested or completed. On the basis of this sum rule we can give a rationalization of the (computational) finding of only four values of this difference for the twelve different isomerizations of the water-dimer isotopomers. Acknowledgement The constructive
comments
of the referee
are
highly appreciated. References [ I] E.B. Wilson Jr., J.C. Decius and P.C. Cross, Molecular vibrations (McGraw-Hill, New York, 1955).
2 September 1988
[2] R.P. Bell, The proton in chemistry (Methuen, London, 1959). [ 31 M. Tasumi and G. Zerbi, J. Chem. Phys. 48 ( 1968) 3813. [4] L.M. Sverdlov, M.A. Kovner and E.P. Krainov, Vibrational spectra of polyatomic molecules (Nauka, Moscow, 1970) [in Russian 1. [ 51M.V. Volkenshtein, L.A.Gribov, M.A. Elyashevichand B.I. Stepanov,Vibrations of molecules (Nauka, Moscow, 1972) [in Russian]. [ 61 L. Melander and W.H. Saunders Jr., Reaction rates of isotopic molecules (Wiley, New York, 1980). [ 7 ] W. von E. Doering, V.G. Toscano and G.H. Beasley, Tetrahedron 27 (1971) 5299. [ 81 M.J. Goldstein and MS. Benzon, J. Am. Chem. Sot. 94 (1972) 7147. [ 91 L. Fredin, B. Nelander and G. Ribbegard, Chem. Phys. Letters 36 (1975) 375. [ 10] L. Fredin, B. Nelander and G. Ribbegrd, J. Chem. Phys. 66 (1977) 4065. [ 111L. Fredin, B. Nelander and G. RibbegBrd, J. Chem. Phys. 66 (1977) 4073. [ 121 L.J. Schaad, B.A. Hess Jr. and C.S. Ewig,J. Org. Chem. 47 ( 1982) 2904. [ 13 ] Z. Slanina, Contemporary theory of chemical isomerism (Academia/Reidel, Prague/Dordrecht, 1986). [ 141 Z. Slanina, Thermochim. Acta 112 (1987) 379. [ 15] Z. Slanina, J. Radioanal. Nucl. Chem. 111 (1987) 157. [ 161 A. Engdahl and B. Nelander, J. Chem. Phys. 86 ( 1987) 1819. [ 171 C.E. Pearson, Handbook of applied mathematics (Van Nostrand Reinhold, New York, 1974) p, 1014. [ 181 J.R. Hulston, J. Chem. Phys. 50 (1969) 1483. [ 191 M. Wolfsberg, J. Chem. Phys. 50 (1969) 1484. [20] J. Bron and M. Wolfsberg, J. Chem. Phys. 57 ( 1972) 2682. [21] 0. Matsuoka, E. Clementi and M. Yoshimine, J. Chem. Phys. 64 (1976) 1351. [22] D.G. Bounds, Chem. Phys. Letters 96 ( 1983) 604. [23] Z. Slanina, Chem. Phys. Letters 127 (1986) 67. [24] Z. Slanina, J. Phys. Chem., to be published.