The Journal of
Physical Chemistry VOLUME 97, NUMBER 17, APRIL 29,1993
0 Copyright 1993 by the American Chemical Society
LETTERS Hydrogen Bonding: The Role of Hydrogen Delocalization in the AHB Bond P. P. Schmidt Office of Naval Research, Chemistry & Materials Divisions, 800 N . Quincy St., Arlington, Virginia 2221 7-5000 Received: December 29, 1992; In Final Form: March 1 , I993
The behavior of the A-H.-B multiple minimum hydrogen bond can be understood in terms of an analysis similar to the molecular orbital method in molecular electronic quantum theory. Delocalization of hydrogen about two or more wells is described in terms of a linear combination of Gaussian orbitals, each referred to a well minimum. The analysis yields an effective A-B potential mediated by hydrogen. The role of a model threecenter interaction in the potential is found to be important and can account for much of the strength of the hydrogen bond.
Hydrogen bonds are widely distributed in nature; among the best examples are water, upon which both the liquid and solid phases sensitively depend, and the genetic material In spite of years of attention, however, there is still no widely accepted, adequate potential energy function that can duplicate the physically complex behavior observed. Hydrogen bonding frequently involves double well potentials. Even in the case that hydrogen is shared between two identical atoms, for certain configurations of the system, the hydrogen atom can localize near one or the other of the heavier atoms. In the general case in which the two atoms that share the hydrogen are different, hydrogen generally has a stronger association with one than the other. One well is usually deeper and perhaps narrower than the other. In any case, the fact that hydrogen can migrate from one well to the other on a quantal time scale makes it difficult to model the potential with simple functions, such as a single Morse or Lennard-Jones form. There have been efforts to model the hydrogen-bonded interaction between two atoms with these atoms bonded in turn to other atoms in a larger molecular f r a m e ~ o r k .The ~ recent time-dependent treatment of fluctuating double well potentials of Siems et. al.5-6 is similar in intent but different in execution to that used here. The treatment developed below has elements in common with Werthamer’~’-~ quantum crystal method. We assert that it is indeed possible to construct a potential for the hydrogen-mediated A-B bond once one knows more about the
behavior of hydrogen as a bonding agent. To this end, the analysis specificallyuses simple potential forms, in particular, the Morse function, to describe the simultaneous interaction of hydrogen with two atomic fragments A and B within a larger system. In the treatment here, however, we will consider only the symmetric, homonuclear case; similar results (unpublished) have been found for the heteronuclear examples. In addition, a three-center term is considered, in the manner introduced by Halllo a few years ago, to provide for an adjustable barrier on the reaction path. Hydrogen delocalization is handled in exactly the same manner as is done with the electron in, for example, H2+.11 The wave function for hydrogen in its ground state is a linear combination of atomic functions, each referred to an origin at a minimum in the double well system. The energies of the ground and excited states for given values of RABare then derived. Finally, an effective direct interaction between A and B is added in much the same manner as the Coulomb repulsion is added between the two H atoms in H2+. The direct interaction is assumed to be either an exponential repulsion or a Morse potential that operates between the heavy fragments A and A’ (or B in the heteronuclear case). In their work on the melting transition in DNA, Prohovsky and co-workers1*J3and Pitici and Svirschevski14have emphasized the role played by anharmonic relative motions of the DNA bases. Theyahave not, however, directly considered the dynamics of hydrogen in the bonds between the bases. Instead, they have assumed that base pairs interact as described by a Morse potential;
This article not subject to U S . Copyright. Published 1993 by the American Chemical Society
4250 The Journal of Physical Chemistry, Vol. 97, No. 17, 1993
the binding hydrogen is assumed to respond adiabatically to motions of the nucleic acid chains. Their assumption is completely in line with the explicit assumption of this paper that the delocalization of hydrogen in the bond needs to be considered in order to account for the binding. The objective is, in effect, to derive the parameters of the binding potential in terms of the individual parameters of the A-H interactions and the direct A-A' interaction. In line with this goal, Jaroszewski et. al.15 recently carried out an ab initio quantum chemical analysis of proton transfer in the systems [HjN-H-NHj]+ and [HjN-HOH2]+ to see the effects of stretching and squeezing the N-N and 0-N distances on proton transfer. Their interest also was to probe quantum effects on the transfer. As the double Morse potential examined below shows, the ab initio calculations also indicate that the barrier to transfer is sensitive to the distance between heavy atoms. The more fundamental quantum mechanical treatment, based on electronic degrees of freedom, supports the approach taken here. The followinganalysis shows that the effective potential between the two atomic fragments A and B depends sensitively on the presence and size of the three-center term. This particular term has an interesting effect on thecharacter of the hydrogen bonding. Considering a symmetric double well potential for hydrogen taken at a fixed value of and plotted simply as a classical hydrogen atom moving on a line between A and B, one finds that the threecenter term appears mainly to shift the complete potential upward in energy compared to the simple sum of two Morse functions; compare Figures 1 and 2. When used in the quantum mechanical analysis, its presence can spell the difference between a binding or a repulsive, antibinding state depending on the value of the Hall coefficient A (see eq 1). For an asymmetric potential, the Hall repulsion can have an important effect on both the classical and quantum mechanical behavior of hydrogen. If such behavior is not simply an artifact of the model potential, if indeed threecenter terms represent real effects, then it is possible that molecular combinations within chemically complex environments, such as a living cell, can trigger a sudden change from binding to antibinding. Although subtle, the strength of the environmental influence on hydrogen bonding may be essential to account for the physical basis of DNA melting12-14and, perhaps, of mutation.16 The model potential energy function is constructed as the sum of two Morse potentials and a three-center term, A exp[-f(rAH - riH)] exp[-{(rBH) - riH)], introduced by Hall a few years ago:I0
Letters
and these coordinates can be used to obtain the transition-state values of rAHand rBH. For a symmetric transfer, PA,= Z)BH and Xt = I.?. For a nonsymmetric potential, it is nevertheless still the case that 2 3 can ~ equal ~ BBH accidentally. According to eq 4, one Xt = I.?. Now, however, riH # riH,and (5)
The position of the saddle point in real space is not necessarily at the center of the AB line. The meaning of the Hall term is understood best from the fact that, in its absence, the maximum height of the barrier cannot be greater than the larger of the dissociation energies of the two Morse potentials. In actual systems, therefore, it is likely to be the case that the barrier will be larger than a double pair potential approximation alone will give. The additional barrier height needed can be gained, in part, through the use of many-body terms. The Hall term is the simplest of what is in fact a generic collection of many-body model potentials based on exponential (or better) decay of the effect with bond distance and a polynomial functional dependence on bond displacements; see, for example, the extensive discussion in Murre11 et al., cited in ref 17. For the purpose of this treatment, the binding hydrogen is restricted to the line that joins A and B. A general treatment requires account of the possibility that the binding proton lies off the AB axis." For the off-axis treatment, one needs additional spatially oriented terms in the complete potential energy function to ensure the directionality of the bonds involved. Account of noncollinear hydrogen bonding, and angular bonds in general, can be given using many-body potential forms that depend only on bond displacements; a discussion of this is given in ref 17. We consider only the ground state. The fact that hydrogen can occupy either of two energy minima in the double potential well under normal circumstances suggests that one consider an expansion of the state function in terms of elementary basis functions. The obvious choice for the points of expansion are the locations of the well minima. Thus, the ground vibrational state is a linear combination of basis functions:l8
and (A((B)is the overlap integral. The basis functions JA) and IB) are chosen to be Cartesian Gaussian 1s functions:
(A) = exp[-a(r - A)2]
The first two terms are the familiar Morse potential with 23 the dissociation energy associated with the free pair bonds (AH, BH). The role of the three-center energy, the third term in eq 1, is to define a barrier on the AH-.A+ A-HA reaction hypersurface.10 The analysis of the reaction potential energy surface can be mapped from an exponential dependence to a quadratic form:"J let
(7) for the H in the vicinity of the well minimum nearest to atom A. A similar term is used for the second minimum nearest to atom B. The Gaussian exponent a(b) is associated with the function expanded a t the point A(B), where A and B also serve as basis function labels. The Hamiltonian operator is the sum of the kinetic energy operator and the potential, eq 1. Considering normalization, the energies of the ground and first excited state are (in obvious notation)
The matrix elements of the kinetic energy operator are of course well known for Gaussian basis functions. The matrix elements
The Journal of Physical Chemistry, Vol. 97,No. 17, 1993 4251
Letters of the exponential terms in the Morse-Hall potential are equally simple. First of all, one writes the potential as an expansion in terms of Gaussians using the H e h r e S t e ~ a r t - P o p l e lexpression ~ for the 1s Slater function, a simple exponential. Thus,I7 6
V,(r)
= 2, exp(tro)Cdk[exp(tro- 4ukr2)- 2 exp(-ukr2)] k=l
-1.42
/-.
/
\
...
-
3.75A 4.00A
\ \
(9)
and the coefficients dk and uk are given by
I I
___---__
-1.58
,
\
/
- 3.00A _ _ 3.25A MOA
',
-_-1 .so -1 .sa -2.06 -2.1 4
The values of the quantities (Yls,k and 6ls,k are given by Hehre et
-2.22 0.00
01.19
The matrix element of the Morse potential contribution in the s basis functions is
(AI VMI B ) =
0.20
0.10
0.30
0.40
0.50
0.60
0.70
0.80 030
1.00
R, (scaled units)
Figure 1. Potential energy of hydrogen between two identical atoms for fixed values of RAB.The fixed values are indicated above each curve. The distance RAHis scaled in order to fit each curve on the same graph. The scale factor is AH = 0.2Ri, r with 0 Ir I1. The values of the Morse potentials for the identical atoms are Z)AH = 1.4 eV, [ = 1.1 A-I, and r:H = 0.9 A. For these curves, the Hall term 34 = 0.0.
+
The matrix element of the exponential operator, (AIEZC(uk)lB), is given by
- 3.00A -1 3 3
(AIE2'(Uk)lB) =
-1 3 9 -1.45 -1 .5t
+
+
P is defined by P = (UAH ~ B H ) / ( u b) with AH and BH the coordinates of the expansion points for the basis functions. Finally, the overlap matrix (AIIB) is
1,,\
/
- \ , \ \
/
i
3.25A 3.50A 3.75A 4.0OA
1 \-
----------
7
-1.87
For the Hall contribution to the potential,
-193' 000
' 010
'
'
020
'
030
040
0.50
060
070
080
0.90
100
R, (scaled units)
Figure 2. Same as Figure 1 except the Hall term 34 = 1.0 eV.
one uses the following expression for the ss matrix elementZ0
can define an effective potential for AA' in terms of a direct potential between A and B that is either a nonbonding exponential repulsion or a Morse form.
VAA, = D A A ~ ~ ~ P [ - - ~ ( R - R ~ ) I ( ~ ~ P [ ~ ((18) R-R~)I-~~) The parameters used are D A A=~ 0.5 eV, y = 3.0 A-', and Ro = 3.0 A; R is the AA' distance and 6 = (0,l) depending on whether the interaction is nonbonding and repulsive or an attractive Morse potential. The complete effective potential is therefore
The vectors C and H are defined as the distances between the bonding hydrogen and the end atoms A and B, respectively. The vector P is the same as in aq 12, and L(i,, is given by
These matrix elements were used in eq 8 to obtain the energies discussed below. The values of the Gaussian exponents a and b used were determined from the harmonic vibrational force constant derived for the Morse potential: 2121) for r = rO. Optimization of the energies with respect to a and b was not carried out. Except for small separations R A Athere ~ , is very little splitting, i.e., E+ - E0; see Figure 4. Thus, quantum mechanical effects-evidenced by H exchange and splittingappear only for small separations. The hydrogen-mediated interaction between atoms A and A' is given by EAHAT = E- where E- is the ground-state energy. We
=
VAH, = EAHA4R) + VAAdR) (19) This form tacitly assumes (in the senseof the Born-Oppenheimer approximation) that adiabatic kinetic energy contributions are negligible. The values of the dissociation energy and other constants in the VAAi (repulsive)potential were chosen, after some experimentation, to yield a composite potential VAHA, similar to the potential used by Prohovsky and c o - w o r k e r ~ . ~ ~ J ~ The following results apply to the case of identical atoms in the system AHA. Figures 1 and 2 illustrate the potential energy of hydrogen regarded as a point mass moving between identical atoms A for various fixed values of the distance RAA,.In Figure 1, A = 0; Le., there is no Hall contribution, and an overall attractive AHA configuration exists on the reaction hypersurface.I0 For large enough values of R A Aa~double well potential exists. In Figure 2, A = 1 eV, a value that only slightly shifts the potential. We note that low barrier, reasonably symmetric well structures can arise for some sets of parameters in the case of the heteronuclear system AHB; this might correspond, for example, to the interaction of dissimilar DNA bases that,
4252 The Journal of Physical Chemistry, Vol. 97, No. 17, 1993
Letters
0.3
450
r
0.2 0.1 0.0
-0.1 -0.2 -0.3 -0.4 -0.5 -0.6
’
-0.7 3.09
3.27
3.45
3.63
3.81
3.99
4.17
4.35
4.53
4.71
4.69
3.09
3.1 5
3.1 2
3.18
3.21
3.24
3.27
3.30
3.33
3.36
3.39
RAN
RAN
Figure 3. Composition of the effective potential between A and A‘ mediated by the hydrogen bond: A = OeV. V~~operatesdirectly between A and A’. VAHAis due to the presence of the mobile hydrogen atom. The sum is the total effective potential: VAA, EAHA.The dotted curves indicate the deuterium-mediated interaction, which is small. The atoms A and A‘ are identical. The potential parameters are as in Figure 1.
+
nevertheless, possess similar sets of chemical and physical characteristics. The behavior of hydrogen as a classical particle moving between A and A‘ under the influence of a Morse interaction with each A species only suggests behavior as a binding agent in the hydrogen bond. Moreover, the addition of the Hall repulsion shows little effect; the main contribution of the Hall interaction is to raise the overall energy for H moving between fixed centers A and B as is seen by comparing Figures 1 and 2. The role of hydrogen as a binding agent is only seen in the quantum mechanical analysis and the contribution of that analysis to the total energy of the system. As numerical experimentation with the potential shows (see Figures 1 and 2), for sufficiently small AA’separations, the barrier vanishes. Of course, the total energy increases due to the strong interparticle repulsion. Nevertheless, under these conditions the trapped hydrogen ceases to tunnel but vibrates in a single well that becomes increasingly harmonic as the AA‘distance decreases. In order to account accurately for all possibilities, and especially in this kind of system, one can consider using a single center analysis with a large basis set. Such a calculation was tried with the Gaussian expansion of the Morse potential. It was found, however, that for small AA’ distances the normally insignificant numerical inaccuracies in the HehreSteward-Pople e x p a n ~ i o n ~ ~ are sufficiently amplified to render the quantum calculation useless. On the other hand, for most systems of interest, the AA’ separation will be sufficiently great to ensure the intervening hydrogen moves in a double well system. Thus, the two-center analysis outlined above is appropriate. The role of hydrogen in bonding is illustrated in Figure 3. In this figure, as well as in Figure 5 illustrating a nearly nonbonding case, the direct short-ranged repulsion potential is plotted as V,A,. The AHA bonding contribution is normalized to the A H A dissociation limit. For Figure 3 also, A = 0; i.e., there is no Hall contribution. The hydrogen-mediated potential EAHA~is attractive for the AA’separationsshown. This part ofthecompleteeffective potential combined with the direct AA’interaction yields an energy minimum indicating the existence of a stable hydrogen bond. Because of the fact that the double well system collapses into a single well for small separations, it is not possible to map the entire potential with the two-center approach used here. The figures show that the AA’ bond energy due to hydrogen bonding decreases as a function of decreasing AA‘ distance in the absence of the Hall repulsion. In the presence of the Hall repulsion, the energy increases. Part of the quantum mechanical aspect of the hydrogen bonding is indicated in Figure 4 which shows the ground-state splitting of the A H hydrogen levels as a function of the AA’ distance.
+
Figure 4. Plot of the splitting of the ground-state energy level for the AH bond due to tunneling as a function of the AA’ separation. The parameters used are the same as in Figure 3.
-
-
? 2 2.
0.1 I
f
0.14F‘-
W
0.09
----_
-----------
1
0.05
‘
\
-.
0.00 3.09
3.27
3.45
.- _ 3.63
3.81
,
3.99
4.17
4.35
4.53
4.71
4.89
RAN
Figure 5. Same as Figure 3 with the Hall term A = 0.25 eV.
Because of the fact that the two-center analysis does not carry through to the single center limit-a difficulty reminiscent of the separated atom/united atom problem of molecular electronic quantum mechanics-it is not possible to trace the coalescence of the levels back to a single level as the double well collapses on AA’ compression. Nevertheless, the figure shows an apparent tendency to bend over at the low end of the AA’ separation. Halllo noted that, for values of A < a,stable AHA molecular associations exist on the reactive AH.-A to A.-HA hypersurface. We have found, and show in Figure 5 , that for a small value of A = 0.25 eV, EAHA,is nearly antibinding. The parameters used in the direct interaction, V A A , , indicate a short-ranged repulsion between the A species. A bond exists in this case because of hydrogen delocalization, but the hydrogen-mediating part of the whole bond is not strongly stabilizing against the AA’ repulsion. It is necessary to note that it is possible to find combinations of parameters for eq 1 that yield a reasonable effective AA’hydrogenmediated binding energy with small, but finite, values of A. Changes in A can then lead to nonbonding states in the same manner as shown above for A = (0,0.25 eV). An interesting feature of these results is the fact that, for a relatively small change in the three-center Hall term, there can be a large change in the binding, all other parameters remaining the same. This effect suggests the that a small change in the chemical composition in the vicinity of a hydrogen bond may lead to a large scale change in the binding, a kind of chemically induced predissociation. If, for example, through an inductive effect, the character of the interaction between hydrogen and the sources to which it is attached changes, as manifested in a change in the Hall repulsion, one might expect the sudden emergence of an antibinding state to result in the break of the hydrogen bond. Such a postulated effect seems reasonable. Hoffmann and Ladik2I postulated a similar effect based on their molecular electronic analysis of the binding of DNA bases using a tight binding approximation. Enzymatic action, for example, to cleave
Letters
The Journal of Physical Chemistry. Vol, 97, No. 17, 1993 4253
the strands of DNA, should not necessarily involve reactive chemical change of DNA, its bases, or the enzyme other than the alteration of the hydrogen bonds in DNA. Watson and Crick15 in their initial announcement of the structure of DNA suggested that cleavageof the hydrogen bonds was central to the mechanism of reproduction. The active site of the enzyme could, if this picture is correct, simply be a physical catalytic point. Once cleaved, the bases in the separate strands could then associate with base complements to generate two emergent, new double helices, as occurs. The results of the calculations carried out indicate that it is indeed possible to determine values of the parameters of the AH Morse potentials and the direct AA' potential to duplicate the values oftheeffective Morse potentialused by Prohovsky et a1.12-14 This suggests, further, that it should be possible to determine values of the parameters through fitting both to experimental and theoretical data.
References and Notes ( I ) Bell, R. P. The Protonin Chemistry;Cornell University Press: Ithaca, 1959. Bell, R. P. The TunnelEffectin Chemistry;Chapmanand Hall: London, 1980. (2) See, for example: Barnes, P. In Progress in Liquid Physics; Croxton, C . A., Ed.; John Wiley & Sons: New York, 1976; Chapter 9, p 391. BenNaim, A. Zbid.; Chapter 10, p 429. (3) A large amount of information about all aspects of DNA can be found in: Stryer, L. Biochemistry, 3rd ed.; W. H. Freeman: New York, 1988. (4) Hydrogen bonding is of course very important in biological systems. As a consequence, a number of the molecular mechanics routines bring some form of representationof the hydrogen bonding into thecalculation. Discussion of the model potential energy functions used in the routines can be found in the following references: Brooks, B. R.; Bruccoleri, R. E.; Olafson, B. D.; States, D. T.; Karplus, M. J . Comput. Chem. 1983, 4, 187. Mayo, S. L.; Olafson, B. D.; Goddard, 111, W. A. J . Phys. Chem. 1990, 94, 8897. In addition, hydrogen bonding has been studied in connection with the so-called heavy-light-heavy reactive transitions, of which the hydrogen exchange in malonaldehyde is perhaps the best known. A difference between these treatments and that given in this paper is the reduction of the H bond to a one-dimensional calculation. This work was pioneered by W. H. Miller and his colleagues with recent additions by Barbara and Almlof. See for example: Miller, W. H.; Handy, N. C.; Adams, J. E. J . Chem. Phys. 1980,72,99. Gray,
S.; Miller, W. H.; Yamaguchi, Y.; Schaefer 111, H. F. J. Chem. Phys. 1980, 73, 2733. Miller, W. H. In Tunneling, Jortner, J., Pullman, B., Eds.; D. Reidel Publishing Co.: Amsterdam, 1986; pp91-101. Carrington, T.; Miller, W. H. J . Chem. Phys. 1984,81,3942. Carrington, T.; Miller, W. H. J . Chem. Phys. 1986,84,4364. Shida, N.; Barbara, P. F.;Almlbf, J. E. J . Chem. Phys. 1989,91,4061. Shida, N.; Almlbf, J. E.; Barbara, P. F. Theor. Chim. Acta 1989, 76, 7. Shida, N.; Barbara, P. F.; Almlbf, J. E. J . Chem. Phys. 1991, 94, 3633. The approach we take, in contrast, is carried out in a threedimensional Cartesian space. Although it may not be immediately obvious, it appears to be simpler to work in three dimensions for the reason that, with a reasonable basisset,it may not benecessary tospecify thereactioncoordinate or surface as accurately as done by Miller et al. (loc. cit.) and Shida et al. (loc. cit.). The locations of the points of expansion of the basis functions are adequately taken to be positions of equilibrium for H in the initial and final states. It is possible that for geometrically nonlinear configurations of the transition state, knowledge of the location of the saddle point in coordinate space will be required. This issue is currently under investigation. ( 5 ) Becker, A.; Siems, R. Ferroelectrics 1991, 124, 17. (6) Blaurock, H.; Siems, R. Ferroelectrics 1991, 124, 23. (7) Fredkin, D. R.; Werthamer, N. R. Phys. Reo. 1965, A138, 1527. (8) Gillis, N. S.; Werthamer, N. R. Phys. Reo. 1968, 167, 607. (9) Werthamer, N. R. Am. J. Phys. 1969, 37, 763. (10) Hall, G . G . Theor. Chim. Acta 1985,67,439. ( 1 1) Pilar, F. L. Elementary Quantum Chemistry; McGraw-Hill Book Co.: New York, 1968. (12) Gao, Y.; Devi-Prasad. K. V.: Prohovskv. E. W. J . Chem. Phvs. 1984. 80, '629 1. (13) Kim, Y.; Devi-Prasad, K. V.; Prohovsky, E. W. Phys. Reo. 1985, 832, 5185. (14) Pitici, F.; Svirschevski, S. Phys. Rev. 1991, A44, 8348. ( 1 5) Jaroszewski, L.; Lesyng, B.; Tanner, J. J.; McCammon, J. A. Chem. Phys. Lett. 1990, 175, 282. (16) Watson, J. D.; Crick, F. H. C. Nature 1953, 171,964. (17) Murrell, J. N.; Carter, S.; Farantos, S. C.; Huxley, P.; Varandas, A. J. C. Molecular Potential Energy Functions; John Wiley & Sons: New York, 1984. See also, for example: Jeffrey, G. A.; Saenger, W. Hydrogen Bonding in Biological Structures; Springer-Verlag: Berlin, 1991; Part 1B. (18) Schmidt, P. P. Chem. Phys. Lett. 1989, 159, 511. (19) Hehre, W. J.; Stewart, R. F.; Pople, J. A. J . Chem. Phys. 1969, 51, 2657. (20) Schmidt,P. P. unpublishedresults. Thematrixelementsareinessence many-center overlap integrals that are evaluated by standard methods. See, for example: Obara, S.; Saika, A. J . Chem. Phys. 1986, 84, 3963. Shavitt, I. Methods in Computational Physics; Alder, B., Fernbach, S., Rotenberg, M., Eds.; Academic Press: New York, 1963; Vol. 2. (21) Hoffmann, T. A.; Ladik, J. Adv. Chem. Phys. 1964, 7, 84. Ladik, J. J. Ado. Quantum Chem. 1973, 7, 397.
