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Semiempirical quantum mechanical calculation of the electronic structure of DNA. Molecular orbitals correlation and orbital energy shifts in the double hydrogen bonding of the adenine—thymine base pair
Journal of Molecular Structure, 88 (1982) 283-293 THEOCHEM Elsevier Scientific Publishing Company, Amsterdam -Printed
in The Netherlands
SEMIEMPIRICAL QUANTUM MECHANICAL CALCULATION OF THE ELECTRONIC STRUCTURE OF DNA. MOLECULAR ORBITALS CORRELATION AND ORBITAL ENERGY SHIFTS IN THE DOUBLE HYDROGEN BONDING OF THE ADENINE+HYMINE BASE PAIR
JULIO MARANON* Laboratorio Universidad
de Fisica TeBrica, Departamento de Fisica, Facultad de Ciencias Exactas, National de La Plata, C.C. No 67, 1900 La Plata (Argentina)
HORACIO GRINBERG* Departamento de Quimica Orghica, Facultad de Ciencias Exactasy de Buenos Aires, 1428 Buenos Aires (Argentina)
Naturales,
Universidad
(Received 16 September 1981)
ABSTRACT The double hydrogen bonding in the adenine-thymine nucleotide base pair has been investigated in the CNDO/S semiempirical approximation. Correlation of the molecular orbitals for the double proton transfer in the normal and tautomeric configurations shows that the n molecular orbitals are only slightly perturbed, whereas the D molecular orbitals are delocalized on both units of the base pair. Analysis via perturbation theory in order to elucidate the formation of the hydrogen-bonded complex has been performed. The results suggest that an unsymmetrical charge transfer is involved in the double proton transfer process. The first-order contribution to the perturbed orbital energies of the u and a molecular orbitals localized on the same unit of the base pair, is predominantly from exchange repulsion energy, whereas the second-order contributor is mainly polarization energy. In the normal configuration of the base pair, the contribution of the deepest (T molecular orbitals (up to -30 eV) to the energy of formation of the hydrogen-bonded complex shows a stabilizing character, whereas at higher energies the opposite trend predominates. The behaviour of the molecular orbitals of the tautomeric configuration is quite different since only some of the D orbitals (those laying between -40 and -30 eV) localized on thymine help stabilize the complex, whereas the remainder contribute to destabilization of the configuration. Hence the normal configuration of the base pair is more stable than the tautomeric configuration.
INTRODUCTION
A great deal is known about hydrogen bonding due to its well established importance in chemistry and biology. A broad range of instrumental techniques has been applied and theoretical models have been formulated in an attempt to account for this phenomenon [l-6] . It is known that hydrogen *MCIC CONICET, Argentina. OlSS-1280/82/0000-0000/$02.75
0 1982 Elsevier Scientific Publishing Company
284
bonding is an intermediate range intermolecular interaction between an electron-deficient hydrogen atom and a region of high electron density. Although the emphasis was originally on electrostatic forces, it was soon recognized that short-range repulsions played an important role and, further, that charge-transfer and polarization effects also contributed. However, the relative importance of the different contributing factors is still disputed [ 3,7]. Clearly this stems from the fact that, until very recently, calculations on the hydrogen bond were of a very approximate nature: owing to the size of even the smallest hydrogen-bonded system, either the theoretical method had to be oversimplified, or the system studied had to be truncated. Molecular orbital (MO) studies have been of two types, involving either perturbation theory, which attempts to rationalize the various types of energy contributing to both the energy of formation and equilibrium conformations [ 81, or the SCF total complex approach. For long-range weak interactions, where intermolecular overlap is negligible, the Coulomb, polarization and dispersion forces, which result from standard second-order perturbation theory, may be readily calculated from the wavefunctions of the isolated molecules [ 71 . However, in view of the difficulties encountered in precise experimental work, the most reliable results are probably those yielded by accurate SCF calculations [ 91, although part of the dispersion energy is not obtained from such calculations. A further analysis of the hydrogen bond using the method of separated electronic groups as applied to adeninethymine (A-T) and guanine-cytosine systems has also been reported [lo] . Whether hydrogen-bond energies are obtained from experiment or from theoretical calculations, the final results themselves do not reveal the origin of the hydrogen-bond stability (as measured by -AH), nor how characteristic differences between different systems should be interpreted. Interpretations therefore tend to be arbitrary, since they are usually based on non-observable features of the interacting molecules. Nevertheless, they can be useful in organizing the large amount of data available. With these problems in mind and following our general approach [ 11-141 this paper presents the results of semiempirical quantum mechanical calculations of the double hydrogen bonding in the A-T base pair. Certain concepts such as the polarity of the NH.. -0 and 0. -aHN bonds and the hybridization of the lone-pair orbital on 0 and N explicitly enter the calculations, thereby allowing an evaluation of their usefulness for interpretative purposes. In particular, an attempt is made to rationalize the SCF results obtained for the formation of the double hydrogen bonding of this hydrogen-bonded complex by analysing the correlation of the molecular orbitals and orbital energy shifts. RESULTS
AND DISCUSSION
The method adopted here involves treating the hydrogen-bonded complex as a single supermolecular system by the LCAO self-consistent procedure. Wavefunctions of the A-T base pair (in both normal and tautomeric
285
configurations) were generated by the all-valence electrons semiempirical CNDO/S method [ 151. The program was modified to suit local input-output requirements and to facilitate changing dimension statements [ 161. The numbering scheme for the hydrogen-bonded system under consideration is shown in Fig. 1. Bond lengths and angles were obtained from ref. 17 and references cited therein. The interatomic distances for the atoms involved in the hydrogen bonds were taken from Arnott et al. [ 181. One of the problems in these types of studies [19-221 is that, since the exact geometries of the rare tautomeric forms are not known experimentally, the energies are determined using assumed geometries, which retain the ring structure of the normal form for the rare form. Molecular orbital correlation The molecular orbitals correlation was performed on the basis of the conservation of orbital symmetry principle [ 231. Thus, starting with the isolated molecules 30 A apart, the nucleic bases were allowed to approach each other until the equilibrium configuration of the double hydrogen bonding was reached. Both units of the base pair were considered to be planar and since the hydrogen bond has directional character, the only plausible reaction path is that in which both bases move on the same plane in such a direction as to simultaneously allow the formation of the two hydrogen bonds. The u and 71 MOs were identified in both units of the base pair at the equilibrium separation. This identification was straightforward since by virtue of the planarity of the hydrogen-bonded system, its Slater determinant is composed of two well defined blocks [24] corresponding to the u and n MOs.
A(o, n) =
oMOs j 0 ______-______+_______________ 0
ADENINE
Fig. 1. Adenine-thymine
n MOs
THYMINE
base pair. Hydrogen bonds are shown as dotted lines.
286
Fig. 2 shows the results of the correlation of the double proton transfer (DPT) for the normal and tautomeric configurations. In both correlations the n MOs are only slightly perturbed, i.e. their relative localization is conserved. The u MOs, however, are delocalized on both units of the base pair. In particular, these MOs are strongly perturbed through the DPT process when both molecules are allowed to approach within 0.78 A of the equilibrium distance of the hydrogen bonding. The conclusion which emerges from these considerations is that the s MOs are not delocalized because the planarity of the hydrogen-bonded complex is not altered by the DPT process.
Normal A A-T
Tautomerlc A* A*-T’
T
T* 3
-8
w--++
-12 II
-13 -14 -
I-I
-15 -16 -
l-I
-17 -* -18 -19 -
Fig. 2. Correlation of selected occupied and virtual molecular orbitals of the normal and tautomeric configurations of the adenine-thymine base pair.
287
Since U~T interaction was found to be negligible, the formation of the hydrogen-bonded complex in both normal and tautomeric configurations requires the bonding u MOs to participate, at the expense of the antibonding u MOs, so that the perturbation of the system occurs on MOs of the same symmetry. When the units of the base pair were 9 A apart, two crossings (among the n occupied MOs) were detected upon generation of the tautomeric configuration; another two crossings (among the 7~virtual MOs) were found to change its relative localization [A(T) -+ T(A)] without modifying the eigenvectors after the crossing (Fig. 3). Interaction of the molecular orbitals It is interesting to determine which orbitals contribute to both the wavefunctions of the double hydrogen bonding and to the splitting of the orbital energy with respect to the non-perturbed orbital. Following the perturbation scheme of Libit and Hoffmann [25] , the perturbed wavefunction can be written as
making explicit the consequence of intra- and inter-group mixing. The former will assume a crucial role in our analysis of the polarization phenomenon. The MOs Gy) denote the unperturbed wavefunctions while cji are the mixing coefficients factored into subsets of orbitals on A and on T. Each mixing coefficient can be broken down into zero-, first- and second-order contributions Cki
= cg + c$ + cg) + . . .
Higher-order contributions are neglected. Wavefunctions to a given order in perturbation theory actually determine the perturbed energies to a higher order. However, while wavefunctions to second order are required in this case, the attendant energies beyond that order are unnecessary. n(A)
n(T)
n(T)
n(A)
Fig. 3. See text.
288
Ei = I$“’ + El” + EP’
(3)
It is assumed that the MOs are initially localized on each unit of the base pair, these being at such a distance apart that intermolecular overlap is negligible. Perturbation then occurs as a result of moving one molecule towards the other, thus increasing intermolecular overlap. No change of geometry or of basis within a group is allowed, maintaining the original orthogonal group orbitals. It is also assumed that there is no change in intramolecular Hamiltonian matrix elements. A consequence of these restrictions is that, because of the expression [25] (4) c!?) II = 0 since S$) = 0. It also follows that EI’) = 0. As long as the CNDO/S eigenvectors are used, differential overlap is neglected and the MOs become delocalized. Therefore, the particular type of perturbation theory resolved by Imamura [ 261 can be used for the mixing (generated by the first- and second-order interactions) of the basis orbitals; neglecting overlap, the first- and second-order coefficients are given by (5)
cw = ti
c
jfi
(0) ctj
EF’-Ey)
cc r
s
fpc(!)c@)
”
rz
sJ
(6)
where c$ represents the atomic coefficients of the unperturbed MOs, #r), and second-order matrix elements of H have been neglected. These expressions are similar when combined with the corresponding approximations of the CNDO/S scheme [ 151 to those obtained within the perturbational MO formalism in SCF-MO theory. The large number of orbitals and localization problems make it impossible to use the semiqualitative method of Libit and Hoffmann [ 251. However, as long as the expansion coefficients of the normal (A-T) and tautomeric (A*-T*) configurations at both the equilibrium distance of the hydrogen bonding and that in which the units are 30 a apart are known, eqns. (5) and (6) can be used in order to determine what orbit& contribute to the firstand second-order mixing coefficients. The results obtained suggest that the n MOs do not participate in the hydrogen bonding formation since they are perpendicular to the molecular plane. If a u or a n MO is localized on A(T), the first- and second-order mixing coefficients only contain the u or 71MO localized on A(T). This behaviour partially accounts for the orbital polarization in the formation of the double hydrogen bonding, since there occurs an intramolecular charge transfer from an unperturbed orbital, making the overall polarization the algebraic sum of the orbital polarizations. In fact, considering the perturbed A:~) wavefunction
289
and the interaction
of the two groups of orbitals, eqn. (1) can be written
as
(7) where the polarization is achieved by second-order in-group mixing. In this case polarization is due to the fact that the charge transfer is not symmetrical as long as its intensity is inversely proportional to the square of the energy difference. The u MOs are mostly delocalized over both units of the base pair when the double hydrogen bonding is in its equilibrium configuration. From perturbation analysis, it can be inferred that the contribution to the first- and second-order mixing coefficients comes from the atomic orbital coefficients localized in the original molecule. On the other hand, the MOs that are not localized in the original molecule only contribute to the first-order mixing coefficients. These results suggest that, as regards the u MOs delocalized over both units, two polarization mechanisms play major roles. The first is that mentioned above by in-group mixing in a second-order polarization effect; the second is accounted for by intermolecular charge transfer from A(T) to T(A), as can be seen from an examination of the atomic coefficients that correspond to the delocalization of the orbital. That is, a transfer of the orbital charge to the other group is achieved, which causes a polarization in T(A). As stated above, this charge transfer is not symmetrical, since more charge is transferred to the MOs that are closest in energy; i.e., a polarization of the gross atomic population is involved. In this case, the perturbed wavefunction can be expressed in the form .P@)
1
=
&r)(l I
ACT) + cg’) + z C;)a;~) jfi
T(A)
+ 1
+$(A)
(8)
k
that is, ui of A mixes with other orbitals of A at the second-order level but with orbitals of T at the first-order level. For the perturbation felt by the MOs of both molecules upon formation of the base pair (normal and tautomerit configurations), the first- and second-order orbital energy shifts are
(9)
where the overlap S and second-order matrix elements of H have again been neglected. The results obtained from the analysis performed using eqns. (9)
290
and (10) can be summarized by the following: the first- and second-order contributions to the orbital energy shift in both u and 71MOs localized in the original molecule, as well as the contribution to the u MOs delocalized on both molecules of the pair, is due to a self-interaction of the orbital energy. This result coincides with the assumption made above, namely that Ey) (intermolecular) = 0. In order to interpret the results of the orbital energy interactions, the partition technique of the SCF orbital energy [ 1,4,8,28-311 was used. Thus, it is concluded that the first-order energy is due to Coulomb attraction plus electronic exchange energy due to antisymmetrization, E, E(l)= E, + E,
(11)
whereas polarization and charge-transfer contributions to the second-order energy were detected. It is widely recognized that at the equilibrium distance, the most important energetic contribution to hydrogen bonding is Coulomb attraction, second-order effects being less important. The contribution of the exchange repulsion energy to the perturbed orbital energies of the (I and n MOs localized on the same unit of the base pair, was found to be predominant at the first-order level, whereas at the second-order level polarization energy is the dominant factor. The delocalized u MOs contribute to the charge-transfer energy at the second-order level; since these orbitals are delocalized over the entire system they cause a redistribution of the overall electronic population upon formation of the hydrogen-bonded complex. Orbital energy shifts In the Hartree-Fock
scheme, the total electronic
E, = x kiei - c x (2Jij -Kij) i i i
energy can be written
as
(12)
where ei is the orbital energy, ki the occupation number and Jij and K, the molecular Coulomb and exchange integrals. Since the hydrogen-bonded complex is treated as a single super-molecular system in this case, the formation of the base pair is accompanied by a modification of the total energy, AE, , mainly due to the variation in the energy of the occupied orbitals [ 15,321. AE,
~ A
~ i
ki’i = ~ i
ki”i
(13)
Hence, by studying the behaviour of Aei it is possible to characterize the interaction produced upon formation of the double hydrogen bonding in both normal (A-T) and tautomeric (A*-T*) configurations. In order to illustrate this behaviour, a correlation of the orbital energy shift as a function of the u - and n-orbital energies is shown in Fig. 4.
291
Normol
e .v.
Toutomerlc
e.v
Fig. 4. Orbital energy shifts (normal and tautomeric configurations) of occupied molecular orbitals in the adenine-thymine base pair.
In the normal configuration of the base pair, the contribution of the deepest u MOs (up to -30 eV) to the energy of formation of the hydrogenbonded complex shows a stabilizing character, whereas at higher energies the opposite trend predominates. The 71MOs, however, show a destabilizing contribution, The correlation of the energy shifts of either A or T u MOs is similar to that deduced by Imamura [ 261 for second-order perturbation energies, where the relative stabilization (destabilization) depends upon the barrier height of the DPT process. In the correlation of the tautomeric configuration, the behaviour is totally different since only some of the u MOs (those laying between -40 and -30 eV) localized on T help stabilize the complex, whereas the remainder destabilize the configuration. This accounts for the fact that the normal configuration of the base pair is more stable than the tautomeric configuration. From Figs. 2 and 4 it can be deduced that the shift of the u MOs localized on T contributes significantly to the stabilization of both configurations, this
292
contribution being greater than that from A. The qualitative character of these correlations can be partly explained by the perturbative study performed on both the wavefunctions and orbital energies. The extent of stabilization or destabilization is mainly due to the second-order contribution of the polarization energy and to a significant charge-transfer component. The Coulomb attraction energy, however, does not manifest itself through the contributions of the orbital energy shifts to the energy of formation of the hydrogen-bonded complex. These considerations are only valid for the normal form of the base pair since the tautomeric configuration shows a different behaviour. The results presented constitute a reasonable theoretical description of the elementary molecular interactions involved in nucleic acids and nucleic acidnucleic acid associations. Thus, a physical model by which important biological mechanisms can be described is provided. In particular, the role of hydrogen bonds in the hypochromism of double-stranded polynucleotides calls for further studies of model compounds. An investigation along these lines is underway and will be published at a later date. ACKNOWLEDGMENT
We are grateful to CONICET,
Argentina
for support
of this work.
REFERENCES 1 P. A. Kollman and L. C. Allen, Chem. Rev., 72 (1972) 283. 2 R. Rein, in B. Pullman (Ed.), Intermolecular Interactions: From Diatomics to Biopolymers, John Wiley and Sons, 1978, Chap. 3; see also P. Schuster, Chap. 4. 3 S. Bratoz, Adv. Quantum Chem., 3 (1967) 209. 4 L. C. Allen, J. Am. Chem. Sot., 97 (1975) 6921. 5 H. Morita and S. Nagakura, Theor. Chim. Acta, 27 (1972) 325. 6 S. Scheiner, Theor. Chim. Acta, 57 (1980) 71. 7 G. R. Pack, G. H. Loew, S. Yamabe and K. Morokuma, Int. J. Quantum Chem., Symp., 5 (1978) 417. 8 F. B. van Duijneveldt and J. N. Murrell, J. Chem. Phys., 46 (1967) 1759; J. G. C. M. van Duijneveldt-van de Rijdt and F. B. van Duijneveldt, J. Am. Chem. Sot., 93 (1971) 5644. 9 H. J. Gold, J. Am. Chem. Sot., 93 (1971) 6387; H. J. Gold, Chemical and Biochemical Reactivity, The IVth Jerusalem Symposia on Quantum Chemistry and Biochemistry, Jerusalem, 1974. 10 J. A. Fornes, 0. M. Sorarrain, L. M. Boggia and R. Veronessi, Z. Phys. Chem. (Leipzig), 250 (1972) 23. 11 J. Maraiibn and 0. M. Sorarrain, Z. Naturforsch., Teil C, 32 (1977) 647,870. 12 J. MararXn,O. M. Sorarrain, H. Grinberg, S. Lamdan and C. H. Gaozza, J. Theor. Biol., 74 (1978) 11. 13 H. Grinberg, A. L. Capparelli, A. Spina, J. Maraiion and 0. M. Sorarrain, J. Phys. Chem., 85 (1981) 2751. 14 J. Maraiion and H. Grinberg, to be published. 15 J. Del Bene and H. H. Jaffe, J. Chem. Phys., 48 (1968) 4050; 50 (1969) 1126; R. L. Ellis, G. Kuehnlenz and H. H. Jaffe, Theor. Chim. Acta, 26 (1972) 131.
293
16 Calculations were performed on an IBM 360/50. The closed-shell CNDO/S program was obtained from the Quantum Chemistry Program Exchange, Indiana University, Bloomington, IN 47401, U.S.A. 17 L. E. Sutton, Chem. Sot., Spec. Publ., 11 (1958); 18 (1965). 18 S. Arnott, S. D. Dover and A. J. Wonacott, Acta Crystallogr., Sect. B, 25 (1969) 2192. 19 J. S. Kwiatkowski and B. Pullman, Adv. Heterocycl. Chem., 18 (1975) 199. 20 R. Rein, Int. J. Quantum Chem., Symp., 4 (1971) 341. 21 R. Rein and R. Garduno, in J. Linderberg et al. (Eds.), Quantum Sciences: Methods and Structure, Plenum Press, New York, 1976, p. 549. 22 H. Fujita, A. Imamura and C. Nagata, Bull. Chem. Sot. Jpn., 42 (1969) 1467. 23 R. B. Woodward and R. Hoffmann, The Conservation of Orbital Symmetry, Verlag Chemie GmbH, Academic Press, 1970, p. 10. 24 Because the hydrogens of the methyl group of thymine lie out of the plane of the remainder of the molecule, the MOs of T (and A-T) cannot be strictly identified as pure CJor I MOs. However, such a classification is possible for most of the relevant MOs as they are of predominantly 0 or n type. 25 L. Libit and R. Hoffmann, J. Am. Chem. Sot., 96 (1974) 1370. 26 A. Imamura, Mol. Phys., 15 (1968) 225. 27 M. Whangbo, H. B. Schlegel and S. Wolfe, J. Am. Chem. Sot., 99 (1977) 1296. 28 H. Fujimoto and K. Fukui, in G. Klopman (Ed.), Chemical Reactivity and Reaction Paths, John Wiley and Sons, 1974, Chap. 3. 29 P. A. Kollman and L. C. Allen, Theor. Chim. Acta, 18 (1970) 399. 30 M. Dreyfus and A. Pullman, Theor. Chim. Acta, 19 (1970) 20. 31 K. Morokuma, J. Chem. Phys., 55 (1971) 1236. 32 P. Politzer, J. Chem. Phys., 64 (1976) 4239; K. Ruedenberg, J. Chem. Phys., 66 (1977) 375; A. Anno and Y. Sakai, J. Chem. Phys., 67 (1977) 4771; M. A. Whitehead, J. Chem. Phys., 69 (1978) 497;T. Anno, J. Chem. Phys., 69 (1978) 5213; E. A. Castro, Int. J. Quantum Chem., 15 (1978) 355.
in The Netherlands
SEMIEMPIRICAL QUANTUM MECHANICAL CALCULATION OF THE ELECTRONIC STRUCTURE OF DNA. MOLECULAR ORBITALS CORRELATION AND ORBITAL ENERGY SHIFTS IN THE DOUBLE HYDROGEN BONDING OF THE ADENINE+HYMINE BASE PAIR
JULIO MARANON* Laboratorio Universidad
de Fisica TeBrica, Departamento de Fisica, Facultad de Ciencias Exactas, National de La Plata, C.C. No 67, 1900 La Plata (Argentina)
HORACIO GRINBERG* Departamento de Quimica Orghica, Facultad de Ciencias Exactasy de Buenos Aires, 1428 Buenos Aires (Argentina)
Naturales,
Universidad
(Received 16 September 1981)
ABSTRACT The double hydrogen bonding in the adenine-thymine nucleotide base pair has been investigated in the CNDO/S semiempirical approximation. Correlation of the molecular orbitals for the double proton transfer in the normal and tautomeric configurations shows that the n molecular orbitals are only slightly perturbed, whereas the D molecular orbitals are delocalized on both units of the base pair. Analysis via perturbation theory in order to elucidate the formation of the hydrogen-bonded complex has been performed. The results suggest that an unsymmetrical charge transfer is involved in the double proton transfer process. The first-order contribution to the perturbed orbital energies of the u and a molecular orbitals localized on the same unit of the base pair, is predominantly from exchange repulsion energy, whereas the second-order contributor is mainly polarization energy. In the normal configuration of the base pair, the contribution of the deepest (T molecular orbitals (up to -30 eV) to the energy of formation of the hydrogen-bonded complex shows a stabilizing character, whereas at higher energies the opposite trend predominates. The behaviour of the molecular orbitals of the tautomeric configuration is quite different since only some of the D orbitals (those laying between -40 and -30 eV) localized on thymine help stabilize the complex, whereas the remainder contribute to destabilization of the configuration. Hence the normal configuration of the base pair is more stable than the tautomeric configuration.
INTRODUCTION
A great deal is known about hydrogen bonding due to its well established importance in chemistry and biology. A broad range of instrumental techniques has been applied and theoretical models have been formulated in an attempt to account for this phenomenon [l-6] . It is known that hydrogen *MCIC CONICET, Argentina. OlSS-1280/82/0000-0000/$02.75
0 1982 Elsevier Scientific Publishing Company
284
bonding is an intermediate range intermolecular interaction between an electron-deficient hydrogen atom and a region of high electron density. Although the emphasis was originally on electrostatic forces, it was soon recognized that short-range repulsions played an important role and, further, that charge-transfer and polarization effects also contributed. However, the relative importance of the different contributing factors is still disputed [ 3,7]. Clearly this stems from the fact that, until very recently, calculations on the hydrogen bond were of a very approximate nature: owing to the size of even the smallest hydrogen-bonded system, either the theoretical method had to be oversimplified, or the system studied had to be truncated. Molecular orbital (MO) studies have been of two types, involving either perturbation theory, which attempts to rationalize the various types of energy contributing to both the energy of formation and equilibrium conformations [ 81, or the SCF total complex approach. For long-range weak interactions, where intermolecular overlap is negligible, the Coulomb, polarization and dispersion forces, which result from standard second-order perturbation theory, may be readily calculated from the wavefunctions of the isolated molecules [ 71 . However, in view of the difficulties encountered in precise experimental work, the most reliable results are probably those yielded by accurate SCF calculations [ 91, although part of the dispersion energy is not obtained from such calculations. A further analysis of the hydrogen bond using the method of separated electronic groups as applied to adeninethymine (A-T) and guanine-cytosine systems has also been reported [lo] . Whether hydrogen-bond energies are obtained from experiment or from theoretical calculations, the final results themselves do not reveal the origin of the hydrogen-bond stability (as measured by -AH), nor how characteristic differences between different systems should be interpreted. Interpretations therefore tend to be arbitrary, since they are usually based on non-observable features of the interacting molecules. Nevertheless, they can be useful in organizing the large amount of data available. With these problems in mind and following our general approach [ 11-141 this paper presents the results of semiempirical quantum mechanical calculations of the double hydrogen bonding in the A-T base pair. Certain concepts such as the polarity of the NH.. -0 and 0. -aHN bonds and the hybridization of the lone-pair orbital on 0 and N explicitly enter the calculations, thereby allowing an evaluation of their usefulness for interpretative purposes. In particular, an attempt is made to rationalize the SCF results obtained for the formation of the double hydrogen bonding of this hydrogen-bonded complex by analysing the correlation of the molecular orbitals and orbital energy shifts. RESULTS
AND DISCUSSION
The method adopted here involves treating the hydrogen-bonded complex as a single supermolecular system by the LCAO self-consistent procedure. Wavefunctions of the A-T base pair (in both normal and tautomeric
285
configurations) were generated by the all-valence electrons semiempirical CNDO/S method [ 151. The program was modified to suit local input-output requirements and to facilitate changing dimension statements [ 161. The numbering scheme for the hydrogen-bonded system under consideration is shown in Fig. 1. Bond lengths and angles were obtained from ref. 17 and references cited therein. The interatomic distances for the atoms involved in the hydrogen bonds were taken from Arnott et al. [ 181. One of the problems in these types of studies [19-221 is that, since the exact geometries of the rare tautomeric forms are not known experimentally, the energies are determined using assumed geometries, which retain the ring structure of the normal form for the rare form. Molecular orbital correlation The molecular orbitals correlation was performed on the basis of the conservation of orbital symmetry principle [ 231. Thus, starting with the isolated molecules 30 A apart, the nucleic bases were allowed to approach each other until the equilibrium configuration of the double hydrogen bonding was reached. Both units of the base pair were considered to be planar and since the hydrogen bond has directional character, the only plausible reaction path is that in which both bases move on the same plane in such a direction as to simultaneously allow the formation of the two hydrogen bonds. The u and 71 MOs were identified in both units of the base pair at the equilibrium separation. This identification was straightforward since by virtue of the planarity of the hydrogen-bonded system, its Slater determinant is composed of two well defined blocks [24] corresponding to the u and n MOs.
A(o, n) =
oMOs j 0 ______-______+_______________ 0
ADENINE
Fig. 1. Adenine-thymine
n MOs
THYMINE
base pair. Hydrogen bonds are shown as dotted lines.
286
Fig. 2 shows the results of the correlation of the double proton transfer (DPT) for the normal and tautomeric configurations. In both correlations the n MOs are only slightly perturbed, i.e. their relative localization is conserved. The u MOs, however, are delocalized on both units of the base pair. In particular, these MOs are strongly perturbed through the DPT process when both molecules are allowed to approach within 0.78 A of the equilibrium distance of the hydrogen bonding. The conclusion which emerges from these considerations is that the s MOs are not delocalized because the planarity of the hydrogen-bonded complex is not altered by the DPT process.
Normal A A-T
Tautomerlc A* A*-T’
T
T* 3
-8
w--++
-12 II
-13 -14 -
I-I
-15 -16 -
l-I
-17 -* -18 -19 -
Fig. 2. Correlation of selected occupied and virtual molecular orbitals of the normal and tautomeric configurations of the adenine-thymine base pair.
287
Since U~T interaction was found to be negligible, the formation of the hydrogen-bonded complex in both normal and tautomeric configurations requires the bonding u MOs to participate, at the expense of the antibonding u MOs, so that the perturbation of the system occurs on MOs of the same symmetry. When the units of the base pair were 9 A apart, two crossings (among the n occupied MOs) were detected upon generation of the tautomeric configuration; another two crossings (among the 7~virtual MOs) were found to change its relative localization [A(T) -+ T(A)] without modifying the eigenvectors after the crossing (Fig. 3). Interaction of the molecular orbitals It is interesting to determine which orbitals contribute to both the wavefunctions of the double hydrogen bonding and to the splitting of the orbital energy with respect to the non-perturbed orbital. Following the perturbation scheme of Libit and Hoffmann [25] , the perturbed wavefunction can be written as
making explicit the consequence of intra- and inter-group mixing. The former will assume a crucial role in our analysis of the polarization phenomenon. The MOs Gy) denote the unperturbed wavefunctions while cji are the mixing coefficients factored into subsets of orbitals on A and on T. Each mixing coefficient can be broken down into zero-, first- and second-order contributions Cki
= cg + c$ + cg) + . . .
Higher-order contributions are neglected. Wavefunctions to a given order in perturbation theory actually determine the perturbed energies to a higher order. However, while wavefunctions to second order are required in this case, the attendant energies beyond that order are unnecessary. n(A)
n(T)
n(T)
n(A)
Fig. 3. See text.
288
Ei = I$“’ + El” + EP’
(3)
It is assumed that the MOs are initially localized on each unit of the base pair, these being at such a distance apart that intermolecular overlap is negligible. Perturbation then occurs as a result of moving one molecule towards the other, thus increasing intermolecular overlap. No change of geometry or of basis within a group is allowed, maintaining the original orthogonal group orbitals. It is also assumed that there is no change in intramolecular Hamiltonian matrix elements. A consequence of these restrictions is that, because of the expression [25] (4) c!?) II = 0 since S$) = 0. It also follows that EI’) = 0. As long as the CNDO/S eigenvectors are used, differential overlap is neglected and the MOs become delocalized. Therefore, the particular type of perturbation theory resolved by Imamura [ 261 can be used for the mixing (generated by the first- and second-order interactions) of the basis orbitals; neglecting overlap, the first- and second-order coefficients are given by (5)
cw = ti
c
jfi
(0) ctj
EF’-Ey)
cc r
s
fpc(!)c@)
”
rz
sJ
(6)
where c$ represents the atomic coefficients of the unperturbed MOs, #r), and second-order matrix elements of H have been neglected. These expressions are similar when combined with the corresponding approximations of the CNDO/S scheme [ 151 to those obtained within the perturbational MO formalism in SCF-MO theory. The large number of orbitals and localization problems make it impossible to use the semiqualitative method of Libit and Hoffmann [ 251. However, as long as the expansion coefficients of the normal (A-T) and tautomeric (A*-T*) configurations at both the equilibrium distance of the hydrogen bonding and that in which the units are 30 a apart are known, eqns. (5) and (6) can be used in order to determine what orbit& contribute to the firstand second-order mixing coefficients. The results obtained suggest that the n MOs do not participate in the hydrogen bonding formation since they are perpendicular to the molecular plane. If a u or a n MO is localized on A(T), the first- and second-order mixing coefficients only contain the u or 71MO localized on A(T). This behaviour partially accounts for the orbital polarization in the formation of the double hydrogen bonding, since there occurs an intramolecular charge transfer from an unperturbed orbital, making the overall polarization the algebraic sum of the orbital polarizations. In fact, considering the perturbed A:~) wavefunction
289
and the interaction
of the two groups of orbitals, eqn. (1) can be written
as
(7) where the polarization is achieved by second-order in-group mixing. In this case polarization is due to the fact that the charge transfer is not symmetrical as long as its intensity is inversely proportional to the square of the energy difference. The u MOs are mostly delocalized over both units of the base pair when the double hydrogen bonding is in its equilibrium configuration. From perturbation analysis, it can be inferred that the contribution to the first- and second-order mixing coefficients comes from the atomic orbital coefficients localized in the original molecule. On the other hand, the MOs that are not localized in the original molecule only contribute to the first-order mixing coefficients. These results suggest that, as regards the u MOs delocalized over both units, two polarization mechanisms play major roles. The first is that mentioned above by in-group mixing in a second-order polarization effect; the second is accounted for by intermolecular charge transfer from A(T) to T(A), as can be seen from an examination of the atomic coefficients that correspond to the delocalization of the orbital. That is, a transfer of the orbital charge to the other group is achieved, which causes a polarization in T(A). As stated above, this charge transfer is not symmetrical, since more charge is transferred to the MOs that are closest in energy; i.e., a polarization of the gross atomic population is involved. In this case, the perturbed wavefunction can be expressed in the form .P@)
1
=
&r)(l I
ACT) + cg’) + z C;)a;~) jfi
T(A)
+ 1
+$(A)
(8)
k
that is, ui of A mixes with other orbitals of A at the second-order level but with orbitals of T at the first-order level. For the perturbation felt by the MOs of both molecules upon formation of the base pair (normal and tautomerit configurations), the first- and second-order orbital energy shifts are
(9)
where the overlap S and second-order matrix elements of H have again been neglected. The results obtained from the analysis performed using eqns. (9)
290
and (10) can be summarized by the following: the first- and second-order contributions to the orbital energy shift in both u and 71MOs localized in the original molecule, as well as the contribution to the u MOs delocalized on both molecules of the pair, is due to a self-interaction of the orbital energy. This result coincides with the assumption made above, namely that Ey) (intermolecular) = 0. In order to interpret the results of the orbital energy interactions, the partition technique of the SCF orbital energy [ 1,4,8,28-311 was used. Thus, it is concluded that the first-order energy is due to Coulomb attraction plus electronic exchange energy due to antisymmetrization, E, E(l)= E, + E,
(11)
whereas polarization and charge-transfer contributions to the second-order energy were detected. It is widely recognized that at the equilibrium distance, the most important energetic contribution to hydrogen bonding is Coulomb attraction, second-order effects being less important. The contribution of the exchange repulsion energy to the perturbed orbital energies of the (I and n MOs localized on the same unit of the base pair, was found to be predominant at the first-order level, whereas at the second-order level polarization energy is the dominant factor. The delocalized u MOs contribute to the charge-transfer energy at the second-order level; since these orbitals are delocalized over the entire system they cause a redistribution of the overall electronic population upon formation of the hydrogen-bonded complex. Orbital energy shifts In the Hartree-Fock
scheme, the total electronic
E, = x kiei - c x (2Jij -Kij) i i i
energy can be written
as
(12)
where ei is the orbital energy, ki the occupation number and Jij and K, the molecular Coulomb and exchange integrals. Since the hydrogen-bonded complex is treated as a single super-molecular system in this case, the formation of the base pair is accompanied by a modification of the total energy, AE, , mainly due to the variation in the energy of the occupied orbitals [ 15,321. AE,
~ A
~ i
ki’i = ~ i
ki”i
(13)
Hence, by studying the behaviour of Aei it is possible to characterize the interaction produced upon formation of the double hydrogen bonding in both normal (A-T) and tautomeric (A*-T*) configurations. In order to illustrate this behaviour, a correlation of the orbital energy shift as a function of the u - and n-orbital energies is shown in Fig. 4.
291
Normol
e .v.
Toutomerlc
e.v
Fig. 4. Orbital energy shifts (normal and tautomeric configurations) of occupied molecular orbitals in the adenine-thymine base pair.
In the normal configuration of the base pair, the contribution of the deepest u MOs (up to -30 eV) to the energy of formation of the hydrogenbonded complex shows a stabilizing character, whereas at higher energies the opposite trend predominates. The 71MOs, however, show a destabilizing contribution, The correlation of the energy shifts of either A or T u MOs is similar to that deduced by Imamura [ 261 for second-order perturbation energies, where the relative stabilization (destabilization) depends upon the barrier height of the DPT process. In the correlation of the tautomeric configuration, the behaviour is totally different since only some of the u MOs (those laying between -40 and -30 eV) localized on T help stabilize the complex, whereas the remainder destabilize the configuration. This accounts for the fact that the normal configuration of the base pair is more stable than the tautomeric configuration. From Figs. 2 and 4 it can be deduced that the shift of the u MOs localized on T contributes significantly to the stabilization of both configurations, this
292
contribution being greater than that from A. The qualitative character of these correlations can be partly explained by the perturbative study performed on both the wavefunctions and orbital energies. The extent of stabilization or destabilization is mainly due to the second-order contribution of the polarization energy and to a significant charge-transfer component. The Coulomb attraction energy, however, does not manifest itself through the contributions of the orbital energy shifts to the energy of formation of the hydrogen-bonded complex. These considerations are only valid for the normal form of the base pair since the tautomeric configuration shows a different behaviour. The results presented constitute a reasonable theoretical description of the elementary molecular interactions involved in nucleic acids and nucleic acidnucleic acid associations. Thus, a physical model by which important biological mechanisms can be described is provided. In particular, the role of hydrogen bonds in the hypochromism of double-stranded polynucleotides calls for further studies of model compounds. An investigation along these lines is underway and will be published at a later date. ACKNOWLEDGMENT
We are grateful to CONICET,
Argentina
for support
of this work.
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293
16 Calculations were performed on an IBM 360/50. The closed-shell CNDO/S program was obtained from the Quantum Chemistry Program Exchange, Indiana University, Bloomington, IN 47401, U.S.A. 17 L. E. Sutton, Chem. Sot., Spec. Publ., 11 (1958); 18 (1965). 18 S. Arnott, S. D. Dover and A. J. Wonacott, Acta Crystallogr., Sect. B, 25 (1969) 2192. 19 J. S. Kwiatkowski and B. Pullman, Adv. Heterocycl. Chem., 18 (1975) 199. 20 R. Rein, Int. J. Quantum Chem., Symp., 4 (1971) 341. 21 R. Rein and R. Garduno, in J. Linderberg et al. (Eds.), Quantum Sciences: Methods and Structure, Plenum Press, New York, 1976, p. 549. 22 H. Fujita, A. Imamura and C. Nagata, Bull. Chem. Sot. Jpn., 42 (1969) 1467. 23 R. B. Woodward and R. Hoffmann, The Conservation of Orbital Symmetry, Verlag Chemie GmbH, Academic Press, 1970, p. 10. 24 Because the hydrogens of the methyl group of thymine lie out of the plane of the remainder of the molecule, the MOs of T (and A-T) cannot be strictly identified as pure CJor I MOs. However, such a classification is possible for most of the relevant MOs as they are of predominantly 0 or n type. 25 L. Libit and R. Hoffmann, J. Am. Chem. Sot., 96 (1974) 1370. 26 A. Imamura, Mol. Phys., 15 (1968) 225. 27 M. Whangbo, H. B. Schlegel and S. Wolfe, J. Am. Chem. Sot., 99 (1977) 1296. 28 H. Fujimoto and K. Fukui, in G. Klopman (Ed.), Chemical Reactivity and Reaction Paths, John Wiley and Sons, 1974, Chap. 3. 29 P. A. Kollman and L. C. Allen, Theor. Chim. Acta, 18 (1970) 399. 30 M. Dreyfus and A. Pullman, Theor. Chim. Acta, 19 (1970) 20. 31 K. Morokuma, J. Chem. Phys., 55 (1971) 1236. 32 P. Politzer, J. Chem. Phys., 64 (1976) 4239; K. Ruedenberg, J. Chem. Phys., 66 (1977) 375; A. Anno and Y. Sakai, J. Chem. Phys., 67 (1977) 4771; M. A. Whitehead, J. Chem. Phys., 69 (1978) 497;T. Anno, J. Chem. Phys., 69 (1978) 5213; E. A. Castro, Int. J. Quantum Chem., 15 (1978) 355.