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Long-range interactions in model DNA systems
Journal of Molecular Structure (Theochem), 123 (1985) 121-127 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
LONG-RANGE INTERACTIONS IN MODEL DNA SYSTEMS
H. CHOJNACKI and Z. LASKOWSKI Institute of Organic and Physical Wroctaw (Poland)
Chemistry,
Z-4, Wyb. Wyspiahskiego
27, 50-370
(Received 4 October 1984)
ABSTRACT The role of the long-range interactions has been examined in model doubly stranded DNA systems within the semiempirical approach. The essential influence of long-range corrections to the proton transfer potential was found, in general to result in a notably more unsymmetrical potential energy curve. However, in the case of singlet excited (n, x*) electronic states a much more symmetrical potential is predicted. It is concluded that highly polar sugar-phosphate species are of essential importance for interaction energy components (mainly electrostatic terms) and thus for related proton transfer processes. INTRODUCTION
In most papers dealing with intermolecular interactions, only nearestneighbouring molecules or unit cells in the case of periodic systems are taken into account. However, it seems that this is rather far from the satisfactory as the long-range interactions have to be included to reach coherent results, for highly polar interacting subsystems in particular. In that respect Kertesz [l] has carefully investigated the linear metallic-type chain of hydrogen atoms and showed how sensitive the energetic quantities are to the number of the included unit cells. Similarly, in their study of the infinite linear hydrogen fluoride chain, Karpfen and Schuster [2] have corrected the total energy of the reference unit cell for electrostatic interactions by using the multipole expansion. It was also shown [3], that long-range effects play an important role in providing consistent results for predicting conformational stability of polymers. Therefore, it seems justified to suppose that the similar (or greater) effects may be expected in DNA systems or in other biopolymers. The DNA helix, truly three-dimensional, can be seen as built up of a unit cell infinitely repeated along one direction of the physical space exhibiting quasitranslational symmetry. In this highly polar system the long-range interactions are expected to play a fundamental role in determing its conformational stability. These effects also seem to be essential for intercalation processes as well as in the possible proton transfer processes occurring between the hydrogen bonded complementary base pairs.
0166-1280/85/$03.30
o 1985 Elsevier Science Publishers B.V.
122
In a somewhat different context Prohofsky et al. [4] introduced longrange effects using a modified Coulomb potential for ten neighbouring base pairs and for interpretation of the acoustic modes of the double helix [ 51. On the other hand, Claverie [6] calculated stacking interaction energies as a function of the helical movement for ten base pairs to evaluate B-DNA potential parameters. In general, the stability of the structure of nucleic acid helices is very important for an understanding of many unsolved problems of molecular biology. COMPUTATIONAL
METHOD
The electronic structure of isolated unit cell has been calculated at the modified all-valence INDO level [ 71. The method with the two-centre resonance integrals evaluated by applying the Heisenberg equation of motion [8], is capable of reproducing reasonably the ground state properties as well as the low-lying excited state energies within the unified set of parameters. The long-range energy corrections between the reference and the neighbouring unit cells have been evaluated as the sum of electrostatic dispersion and repulsion components. For calculation of the monopole-monopole energy term of interacting subsystems A and B, the Coulomb relationship
(1) was used with k = 1/4ne, and net qi, qj charges on the respective non-bonded atoms. To evaluate the dispersion and repulsion energy terms, the slightly modified empirical relations E,p=
--C
1
1
iEA
jEB
kiki/Z6
(2)
and E rep=D
x
1
iEA
jEB
kikj exp (--Ez)
(3)
proposed by Caillet and Claverie [9] have been utilized, where 2 = 0.5
rij/JR&
and Rip Rj denote the Van der Waals radii of atoms i and j, respectively, whereas ki, kj are the parameters depending on the atoms under consideration [9]. The parameters C, D, and E were kept independent of the atomic species assuming C = 0.895, D = 1.966 - lo’, and E = 13.8 for kJ mol-’ and 8, units.
123 RESULTS
AND CONCLUSIONS
The systems were considered as quasi-periodic and stacked one-dimensional complexes built of complementary base pairs repeated along one direction. In all cases assumed hypothetical unit cells were kept neutral. The point charge distribution for the respective unit cell, considered as a super-molecule, has been evaluated by the above-mentioned all-valence approach with polarization d orbitals for the phosphorus atom explicitly taken into account. The interaction energy components between the reference and the successive unit cells were evaluated according to the relationships (l)-(3) using the computer program written for generating the assumed double helix structure based on an experimental geometry [lo]. The repulsion long-range energy correction shown in Fig. 1 is significant only for the nearest-neighbouring unit cell but not the electrostatic and dispersion contributions, the former being non-negligible for ten or even more next-neighbours. There was no essential difference in this dependence for A-DNA and B-DNA when unit cells were chosen in some other way. It also seems that the repulsive region occurring at some intermediate distances, see Fig. 1, is a general feature of the electrostatic monopole-monopole interaction energy term of the double helix DNA systems. However, such a dependence was not found in the calculations for doubly-stranded helices built of complementary base pairs alone without sugar-phosphate backbone [ 111. It should be noted that there is an important difference in the total interaction energy, as well as in its components, for the same base pair sequences
Repulsion B-DNA
,X-X
I8
(A.....T)
20 Unit cell no.
Coulomb
Fig. 1. Coulomb monopole-monopole, dispersion and repulsion interaction energy components as a function of the unit cell number for the’quasi-periodic adenine-thymine B-DNA system.
124
of A-DNA and B-DNA, the latter being more stable by ca. 200 kJ mol-’ (see Table 1). Generally, the long-range interactions seem to play a more important role in the A-DNA double helix, where the molecular packing is closer than that of B-DNA. On the other hand, the meaningful importance of the base pair sequences can be visualized in the interaction energies up to several unit cells. This fact may be of great importance for a possible theoretical prediction of the palindromic sequences and modifications of the helix structure. It also seems to be of importance for considerations of possible solitonic excitations of the DNA double helix [ 121. At this point, it is interesting to note that the long-range interstrand interactions lead to some destabilization of the double helix by 36.51 kJ mol-’ and 18.74 kJ mol-’ for A-DNA and B-DNA, respectively. According to the refined X-ray studies [lo], the complementary base pairs are obtused within the heavy atom plane with the hydrogen bond N6-H* - 04 in adenine-thymine, Fig. 2, and N4--Ha *-06 of guaninecytosine, Fig. 3, shorter than is usually assumed in previous calculations. In particular, for the former case, the 2.70 .& N4--H~~~06 hydrogen bond length, normally assumed 2.84 W, may be expected to influence the possibility of interbase proton transfer. According to the submolecular Lowdin hypothesis [13], the proton tunnelling between the double well potential could be responsible for the partial loss of the genetic code and for spontaneous or induced mutations in biological systems. Therefore, it is of fundamental importance to know the potential energy curves for the proton motion within the hydrogen bridged complementary base pairs. It is expected that the unsymmetrical potential for the proton transfer in isolated pairs may be significantly altered when the long-range interactions are taken into account. We have found that for the single proton transfer its potential energy curve becomes only slightly more symmetrical in comparison with the respective isolated base pairs. However, in the case of the double proton transfer, the long-range effects lead to a notably more unsymmetrical curves involving single minimum, both for the ground and the lowest (71, n*) excited states (see Table 2). Thus, merely for the lowest (n, n*) excited states a much more symmetrical potential was obtained. These results allow one to understand the great stability of the genetic code in the light of the above-mentioned LBwdin hypothesis. l
TABLE 1 Total interaction energies (kJ mol-‘) in quasi-periodic A-DNA Unit cell
A-DNA
A-*-T G...C
-228.00 -201.74
8::::
-305.80
B-DNA -459.36 -474.58 -522.51
and B-DNA
systems
125
02 H-h
Fig. 2. Molecular structure of the adenine-thymine base pair according to the refined X-ray studies [lo] with the proton shifts represented by the respective arrows.
H
Fig. 3. Molecular structure of the guaninecytosine base pair according to the refined X-ray studies [lo] with the proton shifts represented by the respective arrows. The N4--He 06 hydrogen bond length amounts to 2.70 A instead of 2.84 A usually assumed. l
l
The significant role of ionic species in the proton transfer processes seems also to be confirmed in our calculations. The strong perturbing effect of a metal ion [ 141 leads to drastic changes in the potential energy curve, either that of the ground state or the low-lying excited electronic states resulting,
126 TABLE 2 Potential barrier parameters h, and h, (kJ mol-I) for the double proton transfer in the isolated base pairs and in quasi-periodic DNA systems with long-range interaction corrections. h, is the energy difference between the lower minimum and the maximum, whereas h, denotes the energy difference between both minima on the potential energy curve. The results are given for the ground and the lowest singlet excited states S,(n, n*) and S,(R, x*) Proton transfer in base pair
Electronic state
Isolated base pair h,
h,
A-DNA
B-DNA
h,
h,
h,
h,
A*..T
Ground S,(x, x*) S,(n, n*)
217.1 194.9 258.6
167.9 117.7 128.3
252.8 190.1 243.1
239.3 108.1 94.5
228.7 214.2 240.2
191.0 155.3 87.8
G*.*C
Ground S,(n, n*) S,(n, m*)
220.0 237.3 292.3
146.6 166.0 165.0
254.7 255.7 283.7
215.4 202.6 147.6
231.6 231.6 285.6
168.8 160.2 150.5
in some cases, in a symmetrical potential. Our results are also consistent with the earlier studies [15] concerning the role of the probe charge on the potential barrier for the double proton transfer in the formic acid dimer. The separate problem is posed by the approach used in this calculation and the potential function in particular. Based on the results of Caillet and Claverie [9] and a number of others, we feel that the polarization interactions, not explicitly considered here, might be of significant magnitude, but approximately constant and unlikely to effect our conclusions in a major way. According to FGmer et al. [ 161, a similar argument may be put forward concerning the charge-transfer terms. On the other hand, our procedure has been tested for smaller systems like hydrogen fluoride and imidazole chains. In the former case, the long-range interaction energy was qualitatively similar to that of Karpfen and Schuster [2]. However, the method can be incorrect and misleading when the electrostatic potentials are given for the total system without providing the means to estimate the error due to the local additivity assumption [17]. It should also be noted that the point charge model does not represent charge cloud penetration properly. Anyway, it must be stressed that in the light of recent papers [ 18, 191 showing that electrostatic interactions predict structures of Van der Waals molecules very well, our calculation method may be justified, at least to some extent. We acknowledge the tentative nature of our results, which must be considered as a first step towards the elucidation of the long-range interaction effects in the DNA systems. In order to come to a better understanding of DNA structure, we must look at it as the complete system with its ion and solvent environment. All those facts distinctly point out that more detailed studies are unavoidable in this field.
127 ACKNOWLEDGEMENTS
This work has been sponsored, in part, by the Polish Academy of Sciences and by the Senate and Institute of Organic and Physical Chemistry Fund of the Wroclaw Technical University. The authors are also indebted to dot. L. Piela for valuable discussions. REFERENCES 1 M. Kertesz, Acta Phys. Acad. Sci. Hung., 41 (1976) 107. 2 A. Karpfen and P. Schuster, Chem. Phys. Lett., 44 (1976) 453. 3 J.-L. Bredas, J.-M. Andre and J. Delhalle, Chem. Phys., 45 (1980) 109. 4 B. F. Putman, E. W. Prohofsky and L. L. Van Zandt, Biopolymers, 21 (1982) 885. 5 E. W. Prohofsky, Jerusalem Symp. Quantum Chem. Biochem., 15 (1982) 573. 6P. Claverie in B. Pullman (Ed.), Molecular Associations in Biology, Academic Press, New York, 1968. 7 J. Lipinski, A. Nowek and H. Chojnacki, Acta Phys. Pol. A, 53 (1978) 229. 8 J. Linderberg and L. Seamans, Int. J. Quantum. Chem., 8 (1974) 925. 9 J. Caillet and P. Claverie, Acta Crystallogr. A, 31 (1975) 448. 10 S. Arnott and D. W. L. Hukins, Biochem. Res. Commun., 47 (1972) 1504. 11 H. Chojnacki and Z. Laskowski, to be published. 12 S. W. Englander, N. R. Kallenbach, A. J. Heeger, J. A. Krumhansl and S. Litwin, Proc. Nat. Acad. Sci. U.S.A., 77 (1980) 7222. 13 P. 0. Lowdin, Rev. Mod. Phys., 35 (1963) 724. 14 J. Lipinski and E. Gorzkowska, Chem. Phys Lett., 93 (1983) 479. 15H. Chojnacki, J. Lipiiiski and W. A. Sokslski, Int. J. Quantum Chem., 19 (1981) 330. 16 W. Fiimer, P. Otto and J. Ladik, Chem. Phys., 86 (1984) 49. 17 E. Clementi and G. Corongiu, J. Chem. Phys., 72 (1980) 3979. 18 A. D. Buckingham and P. W. Fowler, J. Chem. Phys., 79 (1983) 6426. 19 F. A. Baiocchi, W. Reiher and W. Klemperer, J. Chem. Phys., 79 (1983) 6428.
LONG-RANGE INTERACTIONS IN MODEL DNA SYSTEMS
H. CHOJNACKI and Z. LASKOWSKI Institute of Organic and Physical Wroctaw (Poland)
Chemistry,
Z-4, Wyb. Wyspiahskiego
27, 50-370
(Received 4 October 1984)
ABSTRACT The role of the long-range interactions has been examined in model doubly stranded DNA systems within the semiempirical approach. The essential influence of long-range corrections to the proton transfer potential was found, in general to result in a notably more unsymmetrical potential energy curve. However, in the case of singlet excited (n, x*) electronic states a much more symmetrical potential is predicted. It is concluded that highly polar sugar-phosphate species are of essential importance for interaction energy components (mainly electrostatic terms) and thus for related proton transfer processes. INTRODUCTION
In most papers dealing with intermolecular interactions, only nearestneighbouring molecules or unit cells in the case of periodic systems are taken into account. However, it seems that this is rather far from the satisfactory as the long-range interactions have to be included to reach coherent results, for highly polar interacting subsystems in particular. In that respect Kertesz [l] has carefully investigated the linear metallic-type chain of hydrogen atoms and showed how sensitive the energetic quantities are to the number of the included unit cells. Similarly, in their study of the infinite linear hydrogen fluoride chain, Karpfen and Schuster [2] have corrected the total energy of the reference unit cell for electrostatic interactions by using the multipole expansion. It was also shown [3], that long-range effects play an important role in providing consistent results for predicting conformational stability of polymers. Therefore, it seems justified to suppose that the similar (or greater) effects may be expected in DNA systems or in other biopolymers. The DNA helix, truly three-dimensional, can be seen as built up of a unit cell infinitely repeated along one direction of the physical space exhibiting quasitranslational symmetry. In this highly polar system the long-range interactions are expected to play a fundamental role in determing its conformational stability. These effects also seem to be essential for intercalation processes as well as in the possible proton transfer processes occurring between the hydrogen bonded complementary base pairs.
0166-1280/85/$03.30
o 1985 Elsevier Science Publishers B.V.
122
In a somewhat different context Prohofsky et al. [4] introduced longrange effects using a modified Coulomb potential for ten neighbouring base pairs and for interpretation of the acoustic modes of the double helix [ 51. On the other hand, Claverie [6] calculated stacking interaction energies as a function of the helical movement for ten base pairs to evaluate B-DNA potential parameters. In general, the stability of the structure of nucleic acid helices is very important for an understanding of many unsolved problems of molecular biology. COMPUTATIONAL
METHOD
The electronic structure of isolated unit cell has been calculated at the modified all-valence INDO level [ 71. The method with the two-centre resonance integrals evaluated by applying the Heisenberg equation of motion [8], is capable of reproducing reasonably the ground state properties as well as the low-lying excited state energies within the unified set of parameters. The long-range energy corrections between the reference and the neighbouring unit cells have been evaluated as the sum of electrostatic dispersion and repulsion components. For calculation of the monopole-monopole energy term of interacting subsystems A and B, the Coulomb relationship
(1) was used with k = 1/4ne, and net qi, qj charges on the respective non-bonded atoms. To evaluate the dispersion and repulsion energy terms, the slightly modified empirical relations E,p=
--C
1
1
iEA
jEB
kiki/Z6
(2)
and E rep=D
x
1
iEA
jEB
kikj exp (--Ez)
(3)
proposed by Caillet and Claverie [9] have been utilized, where 2 = 0.5
rij/JR&
and Rip Rj denote the Van der Waals radii of atoms i and j, respectively, whereas ki, kj are the parameters depending on the atoms under consideration [9]. The parameters C, D, and E were kept independent of the atomic species assuming C = 0.895, D = 1.966 - lo’, and E = 13.8 for kJ mol-’ and 8, units.
123 RESULTS
AND CONCLUSIONS
The systems were considered as quasi-periodic and stacked one-dimensional complexes built of complementary base pairs repeated along one direction. In all cases assumed hypothetical unit cells were kept neutral. The point charge distribution for the respective unit cell, considered as a super-molecule, has been evaluated by the above-mentioned all-valence approach with polarization d orbitals for the phosphorus atom explicitly taken into account. The interaction energy components between the reference and the successive unit cells were evaluated according to the relationships (l)-(3) using the computer program written for generating the assumed double helix structure based on an experimental geometry [lo]. The repulsion long-range energy correction shown in Fig. 1 is significant only for the nearest-neighbouring unit cell but not the electrostatic and dispersion contributions, the former being non-negligible for ten or even more next-neighbours. There was no essential difference in this dependence for A-DNA and B-DNA when unit cells were chosen in some other way. It also seems that the repulsive region occurring at some intermediate distances, see Fig. 1, is a general feature of the electrostatic monopole-monopole interaction energy term of the double helix DNA systems. However, such a dependence was not found in the calculations for doubly-stranded helices built of complementary base pairs alone without sugar-phosphate backbone [ 111. It should be noted that there is an important difference in the total interaction energy, as well as in its components, for the same base pair sequences
Repulsion B-DNA
,X-X
I8
(A.....T)
20 Unit cell no.
Coulomb
Fig. 1. Coulomb monopole-monopole, dispersion and repulsion interaction energy components as a function of the unit cell number for the’quasi-periodic adenine-thymine B-DNA system.
124
of A-DNA and B-DNA, the latter being more stable by ca. 200 kJ mol-’ (see Table 1). Generally, the long-range interactions seem to play a more important role in the A-DNA double helix, where the molecular packing is closer than that of B-DNA. On the other hand, the meaningful importance of the base pair sequences can be visualized in the interaction energies up to several unit cells. This fact may be of great importance for a possible theoretical prediction of the palindromic sequences and modifications of the helix structure. It also seems to be of importance for considerations of possible solitonic excitations of the DNA double helix [ 121. At this point, it is interesting to note that the long-range interstrand interactions lead to some destabilization of the double helix by 36.51 kJ mol-’ and 18.74 kJ mol-’ for A-DNA and B-DNA, respectively. According to the refined X-ray studies [lo], the complementary base pairs are obtused within the heavy atom plane with the hydrogen bond N6-H* - 04 in adenine-thymine, Fig. 2, and N4--Ha *-06 of guaninecytosine, Fig. 3, shorter than is usually assumed in previous calculations. In particular, for the former case, the 2.70 .& N4--H~~~06 hydrogen bond length, normally assumed 2.84 W, may be expected to influence the possibility of interbase proton transfer. According to the submolecular Lowdin hypothesis [13], the proton tunnelling between the double well potential could be responsible for the partial loss of the genetic code and for spontaneous or induced mutations in biological systems. Therefore, it is of fundamental importance to know the potential energy curves for the proton motion within the hydrogen bridged complementary base pairs. It is expected that the unsymmetrical potential for the proton transfer in isolated pairs may be significantly altered when the long-range interactions are taken into account. We have found that for the single proton transfer its potential energy curve becomes only slightly more symmetrical in comparison with the respective isolated base pairs. However, in the case of the double proton transfer, the long-range effects lead to a notably more unsymmetrical curves involving single minimum, both for the ground and the lowest (71, n*) excited states (see Table 2). Thus, merely for the lowest (n, n*) excited states a much more symmetrical potential was obtained. These results allow one to understand the great stability of the genetic code in the light of the above-mentioned LBwdin hypothesis. l
TABLE 1 Total interaction energies (kJ mol-‘) in quasi-periodic A-DNA Unit cell
A-DNA
A-*-T G...C
-228.00 -201.74
8::::
-305.80
B-DNA -459.36 -474.58 -522.51
and B-DNA
systems
125
02 H-h
Fig. 2. Molecular structure of the adenine-thymine base pair according to the refined X-ray studies [lo] with the proton shifts represented by the respective arrows.
H
Fig. 3. Molecular structure of the guaninecytosine base pair according to the refined X-ray studies [lo] with the proton shifts represented by the respective arrows. The N4--He 06 hydrogen bond length amounts to 2.70 A instead of 2.84 A usually assumed. l
l
The significant role of ionic species in the proton transfer processes seems also to be confirmed in our calculations. The strong perturbing effect of a metal ion [ 141 leads to drastic changes in the potential energy curve, either that of the ground state or the low-lying excited electronic states resulting,
126 TABLE 2 Potential barrier parameters h, and h, (kJ mol-I) for the double proton transfer in the isolated base pairs and in quasi-periodic DNA systems with long-range interaction corrections. h, is the energy difference between the lower minimum and the maximum, whereas h, denotes the energy difference between both minima on the potential energy curve. The results are given for the ground and the lowest singlet excited states S,(n, n*) and S,(R, x*) Proton transfer in base pair
Electronic state
Isolated base pair h,
h,
A-DNA
B-DNA
h,
h,
h,
h,
A*..T
Ground S,(x, x*) S,(n, n*)
217.1 194.9 258.6
167.9 117.7 128.3
252.8 190.1 243.1
239.3 108.1 94.5
228.7 214.2 240.2
191.0 155.3 87.8
G*.*C
Ground S,(n, n*) S,(n, m*)
220.0 237.3 292.3
146.6 166.0 165.0
254.7 255.7 283.7
215.4 202.6 147.6
231.6 231.6 285.6
168.8 160.2 150.5
in some cases, in a symmetrical potential. Our results are also consistent with the earlier studies [15] concerning the role of the probe charge on the potential barrier for the double proton transfer in the formic acid dimer. The separate problem is posed by the approach used in this calculation and the potential function in particular. Based on the results of Caillet and Claverie [9] and a number of others, we feel that the polarization interactions, not explicitly considered here, might be of significant magnitude, but approximately constant and unlikely to effect our conclusions in a major way. According to FGmer et al. [ 161, a similar argument may be put forward concerning the charge-transfer terms. On the other hand, our procedure has been tested for smaller systems like hydrogen fluoride and imidazole chains. In the former case, the long-range interaction energy was qualitatively similar to that of Karpfen and Schuster [2]. However, the method can be incorrect and misleading when the electrostatic potentials are given for the total system without providing the means to estimate the error due to the local additivity assumption [17]. It should also be noted that the point charge model does not represent charge cloud penetration properly. Anyway, it must be stressed that in the light of recent papers [ 18, 191 showing that electrostatic interactions predict structures of Van der Waals molecules very well, our calculation method may be justified, at least to some extent. We acknowledge the tentative nature of our results, which must be considered as a first step towards the elucidation of the long-range interaction effects in the DNA systems. In order to come to a better understanding of DNA structure, we must look at it as the complete system with its ion and solvent environment. All those facts distinctly point out that more detailed studies are unavoidable in this field.
127 ACKNOWLEDGEMENTS
This work has been sponsored, in part, by the Polish Academy of Sciences and by the Senate and Institute of Organic and Physical Chemistry Fund of the Wroclaw Technical University. The authors are also indebted to dot. L. Piela for valuable discussions. REFERENCES 1 M. Kertesz, Acta Phys. Acad. Sci. Hung., 41 (1976) 107. 2 A. Karpfen and P. Schuster, Chem. Phys. Lett., 44 (1976) 453. 3 J.-L. Bredas, J.-M. Andre and J. Delhalle, Chem. Phys., 45 (1980) 109. 4 B. F. Putman, E. W. Prohofsky and L. L. Van Zandt, Biopolymers, 21 (1982) 885. 5 E. W. Prohofsky, Jerusalem Symp. Quantum Chem. Biochem., 15 (1982) 573. 6P. Claverie in B. Pullman (Ed.), Molecular Associations in Biology, Academic Press, New York, 1968. 7 J. Lipinski, A. Nowek and H. Chojnacki, Acta Phys. Pol. A, 53 (1978) 229. 8 J. Linderberg and L. Seamans, Int. J. Quantum. Chem., 8 (1974) 925. 9 J. Caillet and P. Claverie, Acta Crystallogr. A, 31 (1975) 448. 10 S. Arnott and D. W. L. Hukins, Biochem. Res. Commun., 47 (1972) 1504. 11 H. Chojnacki and Z. Laskowski, to be published. 12 S. W. Englander, N. R. Kallenbach, A. J. Heeger, J. A. Krumhansl and S. Litwin, Proc. Nat. Acad. Sci. U.S.A., 77 (1980) 7222. 13 P. 0. Lowdin, Rev. Mod. Phys., 35 (1963) 724. 14 J. Lipinski and E. Gorzkowska, Chem. Phys Lett., 93 (1983) 479. 15H. Chojnacki, J. Lipiiiski and W. A. Sokslski, Int. J. Quantum Chem., 19 (1981) 330. 16 W. Fiimer, P. Otto and J. Ladik, Chem. Phys., 86 (1984) 49. 17 E. Clementi and G. Corongiu, J. Chem. Phys., 72 (1980) 3979. 18 A. D. Buckingham and P. W. Fowler, J. Chem. Phys., 79 (1983) 6426. 19 F. A. Baiocchi, W. Reiher and W. Klemperer, J. Chem. Phys., 79 (1983) 6428.