2 study of some model systems

J. theor. Biol. (1978) 71, 49-72

Environmental Effects on H-bond Potentials: A SCRF MO CNDOj2 Study of Some Model Systemst 0. TAPIA:,

F. SLJSSMANQ;AND E. POULNN

Quantum Chemistry Group, Uppsula Universirv, Uppsala, Sweden and Centre de MPcanique Ondulatoire AppliquPe, Paris, France (Received 17 February 1977, and in revisedform 6 July 1977) The self-consistentreaction field (SCRF) theory of solvent effects upon dipoiar specieshas been applied to the study of H-bonded systems.A model to representthe enzyme core medium is discussed.It is shown, albeit heuristically, that this type of environment may be representedwith the combined effect of a reaction field and an inhomogeneousexternal electric field acting over the site system,this latter being provided by the main chain dipolar peptide residues.The generalization of the reaction field concept attained within the SCRF theory allowed for this extention to be operationally implemented. The model systemsconsidered are: a formaldehyde-water, acetonewater, a water dimer and a tetrahedrally arranged water trimer. The interest has beenfocuzed on the study of the dependanceupon the sitesurrounding coupling strength of the intermolecular potentials and interactions related to the H-bond. The orientational and intermolecular distancedependanceof the pair potential and the proton potentials have beenconsidered. The perturbing effect of the enzyme core mediumis manifestedthrough the drastic variation of many (but not all) of the H-bond potentials. The changesshowed by the orientational component of the intermolecular potential together with that of the proton potential shapeillustrate this point. As far as the proton and charge relay mechanismsare concerned, the results reported, while confirming our previous ones, shed a new light upon thesematters. The sensitizationof the H-bridge towards an external electric field is made possibleby the presenceof the reaction field, in absenceof this latter, the external field effect is not significant. t Theabstractof thispaperappearedin the Research Reportsof the Symposium Future of QuantumChemistry,in honourof P. 0. Ltiwdin, heldin Dalseter,Norway, September 1976.

$ Correspondence to: Centrede MkcaniqueOndulatoireAppliquke,23 rue du Maroc, 75019Paris,France. SPermanent address:ChemicalPhysicsDepartment,TheWeizmannInstitute of Science, P.O.B.26, Rehovot,Israel. 49

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1. Introduction

The self-consistent reaction field (SCRF) theory of solvent effects (Tapia & Goscinski, 1975) has been applied to the study of H-bonded systems (Tapia, Poulain 8z Sussman, 1975; Tapia & Poulain, 1977). The numerical results obtained illustrate the modifications produced by a polarizable medium upon some properties related to the H-bond. The SCRF theory can be considered to be a generalization of Onsager’s (1936) reaction field concept. The theory permits calculation of the electronic properties of a given dipolar system surrounded by a polarizable medium. The dipolar system is treated quantum mechanically; the coupling with the medium contains only the electrostatic interaction; the interactions such as solute to medium charge transfer and solute to medium H-bonds are not included. The environments to be described with this approach may be those encountered in molecular crystals, polar and non-polar solvents (liquids). It was, nevertheless, stated that very probably the interior 01 a globular protein may provide polarizable environments to specific groups of atoms and molecules linked through H-bond, which would alter their electronic properties as a function of the folding of the globular protein. It was suggested that the SCRF theory leads to a representation of such environments but no further qualification of the nature of the protein medium was given there (Tapia, er al., 1975, Tapia & Poulain, 1977). Richards (1974) has calculated packing densities of a number of proteins. Packing densities, averaged over a relatively small number of atoms (5-15), appear to vary substantially in different parts of the same protein. Low densities representing packing defects may permit relatively easy motions; surrounding areas of high density may serve as relatively incompressible regions. These calculations illustrate the fact that the interiors of proteins are quite inhomogeneous. Assume now that a given dipolar system is embedded in a region inside a globular protein. It may be an H-bond system or an electron Donor-Acceptor one. A conformational transformation may produce a change of the packing density around this system. It is then reasonable to think that the susceptibility toward polarization would also be concomitantly changed. As we know from our calculations, the H-bond properties may accordingly be changed. Nevertheless, it is not obvious that the SCRF theory, as it stands now, may be directly applied to represent the effect of this medium on polar systems. In section 2 it is shown with simple arguments that the SCRF theory can be used as an approximate method to calculate the electronic wave function of a dipolar system inside a globular protein. It is worth noticing that solvent polarization effect upon H-bonded as well as other dipolar systems have been considered in different manners by a

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number of authors. Onsager’s (1936) reaction field approach has been widely used (see for instance, Barrio1 8c Weisbecker, 1967; Sinanoglu & Abdulnur, 1965; Yomosa, 1967, 1974; Beveridge, Kelly & Randa, 1974); Rinaldi & Rivail (1973) have used this theory together with a self-consistent molecular orbital (MO) theory to study the conformation of an H-bonded water dimer. In a further communication Rivail & Rinaldi (1976) have generalized their approach to include a number of multipole terms in the interaction potential. Hylton, Christoffersen & Hall (1974) and McCreery, Christoffersen & Hall (1976u) have developed a quantum mechanical solvent effect model which was used by Burch, Krishnan, Raghuveer &; Christoffersen (1976) to study an H-bonded formamide-water system. The rationale of the theoretical approach proposed by these authors has been challenged by Rivail & Rinaldi (1976). An instructive discussion on these matters was given by Brown & Coulson (1958). These authors have suggested that it is unappropriate to consider the dielectric as instantaneously polarized by the fields of all the charges in the solute. Instead, an average field acting over the solute molecule is preferred. The SCRF theory is actually based on a static reaction field, which takes account of polarization interactions as well as interactions involving permanent moments, provided the latter type of interactions are not averaged out to zero (Tapia & Goscinski, 1975). This theory presents the formal advantage, over the one based on the continuum approach to the environment of the molecule, that different regions of the protein core may be associated with different values of the reaction field susceptibility. It is obvious that the continuum model, while of value in treating polar liquids or solutions, cannot give an adequate description of such inhomogeneous media as those found inside a globular protein. On the contrary, the method developed by Warshel & Levitt (1976) appears to be quite appropriate in this connection. Many other approaches are currently being used to study environmental effects on molecular structure and properties. The reader is referred to the book edited by Pullman (1976) for further details (see also, McCreery, Christoffersen & Hall, 1976b and references therein). The theoretical scheme proposed in section 2 has been implemented within the same scheme already used by Tapia & Goscinski (1975). An homogeneous electric field I’e has been added in order to include the effect originating in the dipole moments of the amide groups of the folded main chain. In this work we study the response of several H-bonded systems toward different reaction field strengths (RFS) and superposed external fields. They are only intended as an illustration of the kind of variation undergone by the ground state electronic properties. No attempts are made here to accomplish a molecular orbital study of the enzymic reaction mechanism of a particular

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enzyme. Instead, simple H-bonded systems related to proton relay structures and salt bridges which may be associated with proton translocation have been considered. The model systems computed are: formaldehyde-water, acetone-water. water dimer and a tetrahedrally arranged water trimer. The calculational method and definitions are included in section 2, while the discussion of the computed examples is to be found in section 3. The dependence of the intermolecular potentials and related effects associated with the H-bond upon the site coupling strength have been emphasized. Attention has been focused on: (i) orientation, (ii) intermolecular distance dependence, and (iii) proton potentials. A general discussion is presented in section 4.

2. Enzyme Core Representation (A)

A HEURISTIC

APPROACH

Globular protcins are rather tightly packed structures (Richards, 1974), in the interior of which solvent is excluded. In many cases there are a varying number of water molecules located at specific siteqdependingupon the specific enzyme, which play a structural or a functional role, but which do not constitute a solvent in the macroscopic sense. For a number of globular proteins it is generally acknowledged that most of the polar side chains are in contact with the solvent, while the non-polar ones tend to cluster in the core of the enzyme. This makes the neutral dipolar peptide units appear as the principal, but not the only, source of electric fields in the protein core. To obtain a model of the enzyme core the protein cage concept will be considered (Johannin & Kellershohn, 1972). This term designates any domain of the protein core which is surrounded and thereby affected by the electrostatic field of the neighbouring peptide groups. For instance, an imidazole ring of a histidine residue may be considered as filling a cage, or the cage may be the whole active site where different groups are assembled to form the active catalyst. A proton relay structure may also be filling such a cage. The magnitude of the electric field effective within these cages can be ascertained from Johannin & Kellershohn’s (1972) model calculations. They found there to be a rather inhomogeneous electric field over the indole ring plane of tryptophan-172 of a-chymotrypsin, with strengths varying between O-1 up to approximately 1.0 V A- ‘. Therefore the peptide dipolar distribution may produce high electrostatic fields. This can result, as pointed out by Johannin & Kellershohn (1972), in a perturbing effect at any point of the protein core, thereby on every molecule

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53

or group of atoms making part of the protein structure (side chains) that are present in the protein core. Since attention is focussed upon the calculation of the electronic wave function of a particular system in the cage, one must also consider the polarization of the medium. This latter is produced in part by the cage dipolar field. The polarization field reacts back on the cage system, this leads to a coupling operator which is a function of the cage dipole moment. The theory outlined below does take these points into account appropriately. (B)

THE EFFECTIVE

HAMILTONIAN

Consider then an n-electron system in a cage endowed with a finite dipole moment M, and zero charge. If the overail system is considered as a supermolecule, the electrostatic interaction potential V between the cage molecule and the protein core medium in a fixed geometrical configuration is known. The construction of an effective Hamiltonian for the fixed cage system imply an average over the internal and configurational co-ordinates of molecules or molecular fragments in the medium. The former being a quantum mechanical average, the latter standing for a classical statistical mechanical average taken over the relevant variables. This implies that a separability of the total wave function between the cage and the surrounding medium is satisfied. Therefore the cage molecule may be treated as an independent composite particle subjected to the static fields produced by the molecular species in the medium. Overlap and exchange interactions between both systems are therefore neglected. Once this program has been accomplished one can write the effective Hamiltonian in atomic units as follows:

where s is the number of nuclei; ri and R, are respectively the position vector operator of the ith electron and the ccth nucleus with charge 2,; H, denotes the electronic Hamiltonian of the isolated molecule in the clamped nuclei approximation. V,, is the effective potential produced by the polarized medium on the cage system. It can be written within the dipole-dipotc approximation (Buckingham, 1967) as follows:

where p, is the total dipoIe moment pc=

operator of the cage system:

i Z,R, - i ri; LX=1 i= 1

(3)

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the dipolar tensor ?Tis defined by: T(cp) = R,‘(3RcpRcpR&i), (4) with R,, = R, - R,; R, and R, are the vector position of a given origin inside respectively the pth amide group and the system inside the cage; i defines a unit tensor. The quantum mechanical average over the medium system allows us to write down M, as the dipole moment of the pth amide group in presence of the cage system, this implies that the polarization effects are included; finally, the statistical average is indicated by ( ). In order to establish the averages, one searches for the relevant variables which define the conformational degrees of freedom. As is well known, under the assumption of peptide unit planarity, the polypeptide main chain is endowed with a high degree of flexibility due to the rotations about the N-C” and C”-C’ single bonds adjacent to the C”- carbons. The corresponding angles are named respectively the @- and Y-angle (Edsall et (II., 1966; see also IUPAC-IUB commission, 1971). The conformation of a pair of peptides units is specified by these angles. For a polypeptide main chain, the set of N-(a, Y)-angles provides a configurational space where partition functions can be built up to describe the folding process. The structural features associated with the folded chain are described with a potential function VcN) (Scheraga, 1963). This potential would include the potential of hindered rotations, that is, the part of the energy of each peptide unit which is independent of the other units. It would also contain terms representing the energy of formation of an or-helix, and so forth. The statistical-mechanical theories (Zimm & Bragg, 1959; Lifson & Roig, 1961; Lifson, 1964; Go, Go & Scheraga, 1968, 1970) of the helix coil transition in polyamino acids are based on the assumption that the phase space is divided into subspaces, each of which is represented by a particular sequence of helices (h) and coils (c), and that the contribution of the corresponding conformation to the partition function Z0 may be expressed as products of statistical weights of alternating coil and helix sequences. The interaction between the cage and the protein main chain dipoles introduces a modification of the preceding distribution function. For the sake of record, let us write it down as follows: . ., W,)}], f(Wl, w,,. . .YW,; z) = exp [-j3{Pc”)+E,(Wl,. (5) where fi-’ = kT, is the thermal energy; Wi = (Oi, ‘Pi); r is the configurational co-ordinate system in the cage (site system) which includes the position and orientation co-ordinates; PcN) is defined by: VN) = VN’( w, , . , W,) +E,,,( w, , . . ) w,, T). where EpO, is the work necessary to polarize the main chain dipoles; EC

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is the energy of the cage system at a given configuration (IV,, . . ., W.v, ~5) including the interaction energy between the dipole moment M, (W,, . . . . W,, T) and the electric field G, (W,, . . ., W,, z). The average appearing in (2) could be evaluated with equation (5). Actually there is no need, at this stage, to calculate it. Instead, what we need is the qualitative information hidden therein. One has to notice that for a prescribed number of secondary structures, which characterize the folded state of the protein, it is only necessary to know the corresponding sequence of k and c’s to evaluate the average. Thus, with the cage system kept fixed, the fraction of the phase space involved will be different for the different conformations of the protein main chain. It is now evident that (G,), the average electric field, is a function of the conformation of the globular protein. The point is to work out a tractable form of (G,) to be used in the quantum mechanical theory. To bring about this program, one separates the amide dipole moment M, into a permanent component, M;, and an induced component. Thi\ latter is given by E$,*T(pc) -MC, where EP stands for the electronic polarizahility tensor of the pth amide group. The permanent dipole Mi is the one obtained by a virtual de-charging process whereby the cage dipole is changed reversibly from its full value down to zero, while the medium configuration is kept fixed. The effective Hamiltonian can be written now as follows: Her = K,-PJ(G:)

+($,.M,)j,

(6)

T(cp). h, * T(pc),

(7)

where : N

DI = J,

E

is the high frequency reaction field sysceptibility at a fixed configuration of the medium system. Formally, (Gz) includes the orientational correlations between the site dipole and the N-amide groups. In this sense it is a part of the reaction field. An estimation of (GO,} can be made if one follows a technique similar to the one used by Johannin & Kellershohn (1972). This is due to the fact that the co-ordinates obtained from X-ray diffraction experiments correspond to averages over the accessible micro-conformations compatible with the overall conformation of the protein. Therefore, the second term in equation (6) can be obtained experimentally. The third term in equation (6) is now decoupled. Defining the fluctuation field F = (Jr. M,) - (Gr). (M,), that contribution is written as: -r,.
= -P,.f
(8)

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Bearing in mind the relative rigidity of the protein structure compared to the liquid state, m, maintains its average value through all the thermal motion, therefore the fluctuation field contribution vanishes. It is claimed now that there exist an electronic wave function Y, which describes this cage system under the effect of the field (J,) .m+ (G”,), and that it is the solution of the non-linear Schrbdinger equation (9):

IX,-P. ((W +GI). <&lV)l~

= fU-II^ =
(9

This implies that m is given as the expectation value of p, taken over r, i.e. m = (Ylp,M). Aside from the term (G”,) = TO, this equation is similar to the SCRF equation derived by Tapia 8c Goscinski (1975) for a polar solute in solution. The effective energy (EC) = E,, is the energy of the cage system under the effect of the mean field produced by the environmental molecules. The electrostatic contribution to the internal energy contributed by the cage system is : E efr =

(C)

Eeff

THE MOLECULAR

+(&ml).

ORBITAL

EQUATIONS

Consider a closed-shell electronic system with n = 2 m electrons. We start with a Slater determinant ‘P built over an orthonormal set of spatial orbitals 41, 439.. .T hn, each holding two electrons. The Hamiltonian H, is given by (McWeeny, 1969): HII

=

i&

49

+3

Jj

(10)

llrij,

where h(i) the one-electron core operator; rz;’ is the Coulomb repulsion operator between two electrons in atomic units. The variational principle is applied to get the Hartree-Fock equations for the molecular orbitals. The problem is to determine the functions 4,, . . ., 4, such that: S(YppqY) with the orthonormality

= O,

and the reaction field restriction

i # j = 1,. . . ,2m,

(1.9

(Tapia & Goscinski,

J&m.g.m)

[

(11)

constraints:

&4i/+j)

These conditions,

= 0

1975; Tapia, 1977):

= 0.

(13)

(11) to (13), lead to the Hartree-Fock-like

h+ t j=l

(2jj-Rj)+r.((G).m+r,)

+i 1

=

equations: ci$!)i9

(14)

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57

with :

The effective dipole moment has to be evaluated with the MO’s which are solution of equation (14), as well as the standard Coulomb ji and exchange (rZ,> operators, which are defined by: Jj(l>4(1> = J 432>4(2>rF2’

dr,4(1),

(16)

and Rj(lW(l> = f 4j(2)$(2>rFz’ dr,dj(l), (17) $J~and ek stand for the kth MO and its corresponding energy. Equation (14) is solved in a hnear basis set of valence atomic orbitals centered on the nuclei. The matrix elements have been calculated according to the CND0/2 prescriptions (Pople & Beveridge, 1970 and references therein). Sometimes convergence was not attained with the usual iterative technique. In this case the following procedure is used. Once the first-order density matrix P” and P’, respectively corresponding to the HiickelHamiltonian and to the output of the first diagonalization of equation (14) have been obtained, an approximate density matrix P’ is built up as I,‘2 (P”+P’). The procedure is applied for the following steps. Thus the density matrix for the (it- 1)th cycle is given by: P ri+l = 1/2(P’i+P’i-l), i=l,2,... (18) until convergence is attained. The standard iterative procedure is then reentered until convergence is attained. This procedure has converged for all the situations encountered so far. Correlation effects have been introduced via second order perturbation theory. As is well known they are related to the London-van der Waals dispersive forces. The same procedure used by Tapia, Nogales & Campano (1974) was employed. There are two energy quantities relevant to the discussion of H-bond potentials. The first is the expectation value of the nonlinear operator evaluated with the determinantal wave function built from the lowest energy molecular orbital solutions of equation (14). This corresponds to the energy of the cage molecule under the effect of the combined electric fields PO and the reaction field. EsCRF stands for this energy quantity. It is given by the equation: E SCRF

=

(~SCRFIHOI~SCRF)-MSCRF.

((~).“SCR~+r~),

(19)

where He is the molecular hamiltonian in the clamped nuclei approximation, and MSC~F is the dipole moment evaluated with the MO’s which are the solutions of (14). The electrostatic contribution to the internal energy of the

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global system (.sSCRF),i.e. the cage system plus the environment, equation : &RF

=

E,cRF+1/2M,cR~.(g").M~c~~,

is given by the (204

where the last term corresponds to the energy expended to polarize the medium. &scRFcan also be written as: &RF

=

(~sCRFINOIICISCRF)-~/~MSCRF.(S).MSCRF-MSCRF~O.

(2Ob)

If the second-order perturbation (sop) energy EEop is added to EsCRF or the corresponding energy quantities will be labeled by E and E %CRF, respectively. They correspond to E,, and E,~ respectively. For the environment provided by the enzyme core it is expected that the RF direction may differ from the direction of the dipole moment. Nevertheless, due to the approximate scheme used, we have taken a scalar RF susceptibility tensor (g) = g . 1, where 1 represents a unit tensor. The g-value gauges the reaction field strength (RFS) acting on the cage. To take a fixed value of g would be equivalent to take a fixed dielectric constant in the Warshel & Levitt (1976) microscopic model. 3. Results and Discussion (A)

l~lternmlecul~r

g-EFFECT

UPON

INTERMOLECULAR

POTENTIALS

distance dependence

The g-dependance of the intermolecular potential has been studied inside a range of 5 O-2 A around the locus of the energy minimum found for the isolated complex (g = 0.0). It is therefore assumed that a fixed g-value may be used to remove a significant portion of the potential energy curve from the computational scheme. The g-values have been chosen to produce RFS’s ranging from O-1 up to 2. VA-‘. The electrostatic contribution to the internal energy as a function of the intermolecular distance at different g-values has been calculated for the acetone-water, formaldehyde-water and the water dimer systems. The results of the acetone-water system are displayed in Table 1; they illustrate the trends found in all cases. Perusal of Table 1 shows that the locus of the energy minimum is shifted from 2.55 A corresponding to g = 0, down to 2.40 A for g = 0.01 (bohrm3), while the RFS rises from zero to l-91 V A- ‘. Thus, in spite of the rather high RFS attained, the effect upon the locus of the minimum might be considered as minor. Incidentally a similar result is obtained for the H-O equilibrium distance in the H-bond; the RF produces a minor effect upon this quantity also.

TOWARDS

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OF TABLE

PROTEIN

CORE

59

EFFECTS

1

Acetone-water H-bonded system. Internal energy: E = E + (l/2) g M’,,,,, as a function of the intermolecular distance and the reaction field susceptibility coupling g. The RF strength is indicated under the energ?, entry. The energy is in Hartrees, and the RFS in VA- 1 = O-01944 a.u. RO?4’ Cl.0 ClaOl oaO5 0.010

2.25 64.116023 0.0 64.119815 0.1446 64.137300 0.8424 64.165768 2G603

2.35 64.128634 0.0 64.132057 0.1375 64.147850 O-8010 64.173353 1.9601

0 (‘4 2.45 64.133122 0.0 64.136270 0.1315 64.150643 0.7648 64.173663 I .8698

2.55 64.i 33404 0.0 64.136315 0.1268 64.149570 0.7343 64.170585 1.7920

2.65 64.131753 0.0 64.134486 0.1228 64.146860 0.7093 64.166298 1.7266

In contrast, the shape of the potential becomes easily perturbed. FOI 0 . . .O’ distances shorter than the locus of the minimum, the slopes decrease in absolute value, while in the outer region the opposite behaviour is found. Therefore, as g increases the shape at the bottom becomes more parabolic. It is also apparent from Table 1 that the RFS increases in a non-lineal manner with g at a given intermolecular distance. Angular dependence

The angular dependence of H-bond potentials between two water molec:ules in two different arrangements has been calculated for a number of g-values. A linear planar [Fig. l(a)] and linear [Fig. l(b)] system have been chosen. The angular variable corresponds to the q-angle formed betwe:n the bisector line of the proton acceptor partner, which is kept fixed in both cases, and the 0’. . . H-O axis. In the linear planar arrangement both molecules are in the same plane, whereas in the linear system the 0’. . . . H-O axis is in a plane perpendicular to and passing aiong the bisector of the proton acceptor partner. A glance at Fig. l(a) and l(b) shows that these angular dependence potentials are extremely sensitive to the effect of the surroundings. The characteristic features of the linear plane arrangement for g = 0 (bohre3), i.e. a slight asymmetry of the angular dependency, an almost parabolic shape and a shallow minimum, are changed into a sharp angular dependency once the coupling with the surrounding is allowed for. The linear system displays simi!ar features. In this case one starts with an almost

60

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-1084*0

-1085.5 s -2% w

-1085.0 ='?,0125 q 0*015

-1Of36.5r I 6040

-60 20 0 -20 -40

I1 ,

60 40 20 0 -73 -GO x0 -40 -80

FIG. 1. Potential energy curves of (a) linear planar water dimer and (b) a linear water dimer, as a function of the relative orientation of the partners and for different reaction field susceptibilities 9. For all figures the proton co-ordinates are measured from the oxygen acceptor atom.

spherical potential shape for g = 0 (bohrW3) to end up with an acute orientational dependency. Differences in details can easily be appreciated in the figures. As expected, there is a connection between the above mentioned behaviour of the potential and the total dipole moment variation as a function of the angular co-ordinates. In both casesthe loci of the energy minima almost coincides with the corresponding loci of the maxima attained by the dipole moment. Further studies on model systemsrelating to the folding processin protein would be of interest The overall results seemto show that the orientational dependent potential of H-bonded systems is very much sensitive to the presence of a polarizable environment.

TOWARDS

A REPRESENTATION (B)

PROTON

OF

PROTEIN

CORE

EFFECTS

61

POTENTIALS

The proton potential of the isolated complexes is simply given, within the Born-Oppenheimer approximation, as a plot of the total electronic energy as a function of the proton co-ordinate in the bridge. In this theory two different energy quantities appear, namely E (or ESCRF)and E (or ascRp). E includes the work done by the system to polarize the surroundings. This may be counterbalanced by other external work sources, which do not necessarily cancel out the effect of the RF upon the MO’s. In this sense, one would say that E (EsCRF) is a quantity describing an intrinsic property of the system, whereas E (ssCRF) corresponds to the description of the whole system. Hence, both quantities will give complementary information when plotted as a function of the proton co-ordinate. They will be considered separately. The effect of, (a) the intermolecular distance 0. . .O’, (b) the coupling parameter strength g, and (c) superposed homogeneous electric Geld F,, on the proton potentials are illustrated.

E-proton potentials The E-proton potentials of a water dimer at three intermolecular distances, namely, 2.6, 2.8, and 3.OA for different g-values have been plotted in Figs. 2(a), 2(b) and 2(c) respectively. The proton co-ordinate is measured from the proton acceptor oxygen. ESCRF-proton potential curves of a water dimer have already been reported (Tapia, et al. 1975; Tapia & Poulain, 1977). The fundamental role played by the intermolecular distance in shaping the proton potential can be seen when the curves with the same g-values are compared. Consider the set with g = 0.01 (bohrm3). While the RFS at the H-bond minimum (region I) is roughly constant within the R,, .o’ range, ;t double well potential shape is favoured when the partners are stretched outwards from the equilibrium distance of 2.6 A. Moreover, the detailed features of the double well depend upon the intermolecular distance. This can be seen from the set of potentials having g = 0.02 (bohre3). At 2.6 A the proton potential is almost symmetric with a small barrier in between, while for larger intermolecular distances the asymmetry increases quite rapidly. On the other hand, as was already noted by us, the proton potential at distances smaller than the equilibrium one hardly displays a second minimum at all in spite of the rather large RFS attained. Another aspect that deserves att.ention is the RFS variation in region I compared to the zone where the structure H,O OH is present (region II). Consider the curves with g = 0.01 (bohrm3). In region I the RFS slightly decreases, whereas in region 11 the RFS increases rapidly, when R,. . . o. goes up. The same trend is found when

i r . L

TOWARDS

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CORE EFFECTS

63

both regions present a minimum of the potential curve; the set with g = 0.02 (bohre3) illustrates this. Taking now a fixed intermolecular distance, the RFS increases in both regions when g increases, the rate of increment being significantly larger in region II. These results and many others can be qualitatively understood in terms of the simple theory of charge transfer complexes (Bratos, 1967). In theAppendix further details are given in this connection. The carbonyl-water proton potentials behave in a similar way to the preceding example with respect to the intermolecular distance effect. However, at the calculated equilibrium distance the shape of the proton potential does not display the double well no matter how large the g-value might be within the selected range.

L-A.--l

I.0

I.2

1 I.4

I.6

I I.8 2aO

d(i)

FIG. 3. External homogeneous electric field (E) effect upon the proton potential curves of a water trimer. The points have been calculated from a coupled Hartree-Fock or selfconsistent field perturbation theory. The direction of the external field is taken along the x-axis. E+ points along the positive x-direction, the opposite choice for E-.

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External eIectric$eld effect on E (ES,,,)-proton potentials

The effect of an external homogeneous electric field upon Es,,,-proton potential of a tetrahedrally arranged water trimer is displayed in Fig. 3. A water dimer has already been discussed (Tapia & Poulain, 1977). An important result comes out from the calculations. It is related to the sensitization of the proton potential towards an external field produced by the presence of a reaction field. The results are as follows. A given external field of appropriate direction is found to produce sensitive changes upon a proton potential curve only if the system is coupled with the surrounding. Even a small g-value, e.g. g = 0401 (bohM3) case in Fig. 3, may be operational in this connection.

TABLE 2

H-bonded water dimer: external electric field eflect on proton potentials. E- and E-proton potential .for g = 0.020 (bohe3); The interaction energy between the dipolar system and the externaljeld IYXis calculated injrst-order perturbation theory: M, r,. The total energy E (l?,) = E-M, lYx is given as a .function of the proton position in the bridge R 04,

P(A) 1.04 1.14 1.24 1.34 1.44 1.54 1.64 1.74 1.84

- E(eV) 1088.3415 1087.3049 1085%316 1084.9348 1084.7939 1085.1854 1085.7067 1085.9490 1085.4150

- c(eV) 1081.1988 1081.5502 1081.8165 1082.3061 1083.0195 1083.8380 1084.5689 1084.9170 10844405 R+-0,

1.04 1.14 1.24 1.34 144 1.54 164 1.74 1.84 1.94 2.04

1090.8150 1089+3020 1087.9127 1085.6118 1083.9660 1083.5463 1083.9351 1084.6645 1085.3887 1085.7539 1085.2957

= 2.8

-e(l)

.A4, * r,(eV) 2.6731 2.3865 1.9790 1.5442 1.2062 0.9900 0.8613 0.7877 0.7446

1083.8719 1083.9367 1083.7955 1083.8503 1084.2257 1084+3280 1085.4302 1085.7047 1085.1851

3.0669 2.9060 2.6246 2.1721 1.6467 1.2474 1 a066 0.8692 o-791 1 0.7471 0.7220

1084.7250 1084.3148 1083.6397 1082.9582 1082.6756 1082.8998 1083.5643 1084.3836 1085*1419 1085.5215 1085.0703

= 3.0

1081.6581 1081.4088 1081.0151 1080.7861 1081.0289 1081.6524 1082.5577 1083.5144 1084.3508 1084.7744 1084.3483

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EFFECTS

A second result concerns the differential response shown by region I (H-bonded bridge) and region II (H30 HOH OH). It is exemplified by the proton potentials obtained with g = 0405 (bohr-‘) and the fields E* which are directed along the positive and negative x-axis respectively. As expected (see Appendix) region II is much more sensitive to the external field effect. This is due to the large dipole moment contributed to the wave function by the ionic component. It is worth noting that the E+ field completely counterbalances the RF eflect at zone I, while region II retains the feature produced by the RF, namely the relative minimum. e-proton potentials

In Table 2 for the water dimer and Table 3 for the water trimer, the energies E, E, E (r,) as a function of the proton co-ordinate in the bridge are given. E (I’,) is defined by E (I’,) = E-M.I-~ TABLE

3

H-bonded water trimer. Comparison between the H-bonded and ion-pair structures, located respectively around d = 1.95 t% and d = I.05 A. The electrostatic contribution to the internal energy has been calculated to within the SCRF level; the external field effects are calculated to first-order of perturbation theory. The energy is given in Hartrees. 1 Hartree = 21.21 eV. - e(Hartr.)

- M.T

- @>

I.0 1.05 1 .lO

59406368 59.418285 59.415232

0.149473 0.146899 0.143472

59.555841 59.565184 59.558704

1.9776 1.9436 1.8982

1.90 1.95 2.0 -

59.674211 59.687159 59.683453

0.031192 0.030665 0.030282

59.705403 59-717823 59.713735

0.4127 0.4057 0.4007

IRI

-M.T

-E(r)

59.554202 59.562595 59.554952

0.155393 0.153952 0.152035

59.709595 59.716547 59.706987

4.1119 4.0737 4.0229

59.680697 59.693393 59.689505

0.032710 0*031980 0.031442

59.713407 59.725373 59.720947

0.8655 0.8462 0.8320

--E

IR!

where r0 is a homogeneous external field of strength 1 V A-‘. It can be seen that the work done by the system on the surroundings produces striking changes on the dimer s-proton potential. Consider the water dimer at &,. . o, = 2.8 A. Th e d ou bl e well E-proton potential is changed into a single well c-proton potential corresponding to the H-bonded complex. However, at &. . . o, = 3.0 A the double well feature remains, though the relative position of the energy minima is inverted. The complex of region II is destablized faster than that corresponding to region I. These results T.B.

3

66

0.

TAPlA

ET

AL.

emphasize the influence of the intermolecular distance upon the potential shape. The a-proton potential of the trimer is given at R, . . . o, = 3.0 8, as a function of g in Table 3. It can be seen that the double well shape is retained when passing from one energy quantity to the other, and is true for all the g-values considered so far. These results show that in absence of any other external work source, the H-bonded minimum (region I) is always energetically favoured with respect to the ion-pair one (region II). Nevertheless, due to the presence of a double minimum, the structure corresponding to region II can be trapped in a metastable situation. This would lead to an equilibrium in a thermodynamical sense. It turns out that an external electric field may profoundly affect the foregoing equilibrium situation. This can easily be appreciated from the entries entitled E (I?,) in both tables. It is interesting to note that the field strength used here, which has the order of magnitude found inside cr-chymotrypsin (Johannin & Kellershohn, 1972), brings the energy difference between regions I and II for the trimer with g = 0.01 (bohM3) from 3.6 eV down to 0.24 eV. The last figure is almost the same order of magnitude as the energy balance for a proton translocation in molecular biology. Similar changes can be produced on the proton potential of an association of two water molecules. The foregoing results confirm and further emphasize the fundamental role that would play a RF susceptibility of a protein environment in sensitizing potential proton relay systems to the presence of external electric fields. The ratio between the ion-pair population, N,r, and the H-bonded association, N,, is given by the Boltzman factor: NIINIr

= exp i - (Q - WW.

and is related to the equilibrium constant K between both species. As our results show, the energy difference is a function of the RF susceptibility as well as the external field strength Fe, i.e. E,- .sII = As(g, Fe). It is remarkable that on the one hand A&(0, Fe) does not correspond to any equilibrium situation for the model systems studied, and on the other hand, the presence of a g # 0 should lead to a conspicuous effect when an external field is present. Therefore as pointed out by Tapia & Poulain (1977) the pK may be changed either by a change of the RF susceptibility, or by the effect of a superposed external field, or both. This behaviour of the model systems may clearly be correlated with the pK changes of liver alcohol dehydrogenase when the NAD+ enters to its binding site (Eklund, 1976). Considered from a more general point of view, the results obtained

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67

with this theory document the contention of Johannin & Kellershohn (197’2). They illustrate the profound changes that sonic cage molecular properties may undergo under the influence of intraproteic electric fields.

4. General Discussion An effective Hamiltonian representing a dipolar system in a protein cage has been proposed. The analysis leading to it follows a pattern similar to Tapia & Goscinski’s (1975) self-consistent reaction field theory of solvent effects. The average field acting within the cage is provided by the dipole moments of the amide groups. An average over the distribution of the peptide internal rotational angles is implied. Due to the inhomogeneous distribution of peptide units in the interior of a globular protein, different regions are likely to be associated with cages of different RF susceptibilities. Thus, for instance, a cage near a p-pleated sheet when compared to a corresponding one in the neighbourhood of a loop section would have a smaller contribution to the orientational part of the RF susceptibility. It seems likely that the supersecondary structures are producing particular types of susceptibilities over the active sites. The point is that these situations may be associated with different g-tensors and thereby the effect on the electronic state of the site system can be calculated. Once the average field (Gz) and the corresponding reaction field susceptibility tensor (g) are known, the quantum chemical scheme implied by equation (14) can, at least in principle, be solved. A representation of (GO,) may be obtained if the co-ordinates of the polypeptide main chain atoms are known. An appropriate assignment of a dipole moment to the amide group will lead to an estimate of the electric field at any point of the protein core. The determination of (y’) would imply a rather delicate operation. For instance, the adjustment of a given (y’) in order to reproduce the shifts of a specific electronic property of a cage system would yield knowledge of it. For the time being (y’) can be taken as a parametrical object. An approximate molecular orbital scheme has been used to implement the general theory. The numerical results obtained permit an assessment of the sort of changes that a polarizable environment may produce on the electronic properties of a given system in the cage. It is believed that, despite the rather crude approximations made, the main electrostatic effects produced by an enzyme core medium have been taken into account to an acceptable extent. The results concerning the proton potentials we have shown so far document the idea that the equilibrium between an acid and a base inside the

68 proton might reaction :

0.

TAPlA

CT

AL.

depend upon the surrounding.

N-H

0 /

Thus, for the equilibrium

-

N...H-A

dw

H-N

0 ,@, .-I

N-H

A@,

the position of the equilibrium will be displaced according to the changes that may be produced in g. Thus, an increment of g would lead to a displacement to the right. The effect of external fields will also be particularly enhanced by the presence of the polarizable surrounding. Recent calculations of the imidazol-methylic alcohol system actually display this trend (Sanhueza & Tapia, 1978). Hydration effects upon a H-bonded model system have been studied by a cluster approach to the environment (Schuster, Jakubetz, Beier, Meyer & Rode, 1974). They found that the appearance of a double well in the proton potential depends critically upon the intermolecular distance as well as the very presence of the environment. These results agree with the trends found in this work. It follows that the intermolecular distance does play an essential role in setting up efficient proton relay systems. Moreover, distances shorter and sometimes equal to the H-bond minimum do not favour the appearance of a double well. The sensitivity of the orientation-dependent potential to the environmental coupling is appealing. In general, a proton relay structure is the result of assembling composite groups. One can imagine that the polarizing surroundings inside the globular protein would be efficient in assembling these groups with very definite orientations. The system would lose some of the accessible conformations which are otherwise available to the isolated system (intrinsic conformations), or there is relative destabilization by the effect of the surrounding (environmental forces). In this connection the quantum chemical studies of the conformational properties of the polar heads of the phospholipids are relevant (Pullman, Berthod & Gresh, 1975). The numerical results show an intrinsic preference toward highly folded structures. When a hydration shell is allowed for, the results show that more extended conformations are favoured as a result of environmental forces. It is interesting to note that the extended configuration should have a higher dipole moment than the folded configuration. The RF effect, which is implicitly included in the supermolecule approach, may be one of the factors accounting for these results. The statistical mechanical rationale behind the consideration of solvent effects on conformational stability has been discussed by Beveridge & Schnuelle (1974). Since our effective energy differs from the one used by these authors, there is reason to be cautious in applying directly the partition functions stated by them.

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69

The corresponding partition function in our treatment is: zj= J dw,... c,(,Nj dW,exp {-jk’(N)} exP {-fl@cfE&} CJWl) where Ci(Wi) denotes the region of the conformational phase space accessible to Wi corresponding to the jth folded state of the protein. By the cumulant theorem (Kubo, 1962) one gets: 2’ z Zf, exp {-NE, +E,,l))o} = Zb exp ( - htf}, where we have neglected the cumulants of higher order in 8, e.g.

+P”{(tEc+E,,J% - (t-J%+&,,,)%)

02)

and so forth. To within the constant fact of Zb the equation (22) is the analogue of the conformational partition function in the continuum model discussed by Beveridge & Schnuelle (1974). If one can approximate ((E,+E,,,)> by ((E~+E,,,t))c, then seff is given by sscRF. Reminding the equation (5) of these authors, one should notice that the energy of the solute in a given conformation in our treatment contains the deformation effect produced by the reaction field, i.e. &,,Utc (5) = (TIHJY), where Y is the wave function of the solute in the medium. warshe & Levitt (1976) have introduced the electrostatic polarization of the enzyme atoms and the orientations of the solvent water molecules through a microscopic dielectric model. Their conclusion, obtained from the theoretical study of the stabilization of the carbonium ion in the reaction of lysozyme, is that the solvation energy resulting from the medium polarization is considerable and that without it, acidic groups can never become ionized. Similar conclusions can be drawn from our calculations. In general, the properties associated with a variation of the total dipole moment are bound to be affected by changes operating in the environments. The calculations of Burch et al. (1976) on the formamide-water system also substantiate this statement. A number of interesting approaches exists to treat the environmental effects on biomolecules from the quantum mechanical side that will not be discussed here (Port & Pullman, 1970; Colin Baird, 1974). Also, the work of Christoffersen & Maggiora (see for instance: Davis, Christoffersen & Maggiora, 1975; Oie, Maggiora & Christoffersen, 1976) is relevent in this respect. Of course the relative merits of different approaches have to be assessed, nevertheless, at the level where we have implemented our theory, e.g. CND0/2, INDO or MIND0/3 schemes, we believe that it is too early to compare the results coming from all these approaches. An ab initio

70

0.

TAPIA

ET

AL.

version of our program is currently being worked out and it will be used to study some enzymic reaction mechanism as well as environmental effects on biomolecules. Undoubtly the theoretical representation of an enzyme core medium is a complex task, nevertheless, it is believed that the theory outlined here is a small step towards the attainment of such a goal, In fact, it provides a rationale to the microscopic scheme put forward by Warshel & Levitt. This can be seenif, instead of using the dipole-dipole approximation in (1) the average potential produced by the amide groups (or the atoms of the enzyme) is considered to build up the model Hamiltonian. The multipole approximation must then be used at the level where the evaluation of the matrix elements of equation (14), in the LCAO basis, is made. These matters will be considered elsewhere. To sum up, the results reported here show to what extent the combined effect of a reaction field and an external electric field may change some properties associated to H-bonded model systems. In general, the reaction field enhance the susceptibility of the H-bond to the external field effect. This is done in such a way that the ion-pair structure, associatedto the proton transposition along the bridge, may be stabilized, and therefore the pK becomes sensitive to environmental changes, e.g. the presence of charged speciesnear the proton relay system. It is also expected that the reactivity as well as reaction rates will be found to be profoundly affected by the combined effect of (G”,) and y”.M,. We are grateful to ProfessorP. 0. Lowdin and Dr 0. Goscinski from Uppsaia for encouragementsand friendly hospitality. The financial support of the Swedish Institute is gratefully acknowledged.One of us (O.T.) is indebted to Professors C-I Brand&r and B. Plapp, and Drs E. Zeppezauer and H. Eklund for many interestingdiscussionson protein structure and enzymic mechanisms.The interest in this work and the hospitality given to him by Professor R. Daudel and Dr M. Allavena from Paris is acknowledged.We wish to thank the refereesof this Journal for their criticism and suggestions. REFERENCES BARRIOL, J. & WEISBECKER, A. (1967).Compt. Rend. AC. Sci. (Paris) C265, 1372. BEVERIDGE, D. L., KELLY, M. M. & RADNA, R. J. (1974). J. Am. Chem. Sot. 96, 3769. BEVERIDGE, D. L. & SCHNUELLE, G. W. (1974). J. phys. Chem. 78.2064. BRATOS, S..(1967). Adv. Quant. &hem. 3,.209. BROWN. R. D. & COULSON. C. A. (1958). Coil. It& CNRS 82. 311. BUCKIN~HAM, A. D. (1967): Adv. &em. khy.s. 11, 107. ’ BURCH, J. L., RAGHUVEER, K. S. & CHRISTOFFERSEN, R. E. (1976). In Environmental Effects

on Molecular

Structure and Properties, (B. Pullman,ed.), p. 17.Dordrecht: Reidel.

COLIN BAIRD, N. (1974). Int. J. Quant. Chem.:Quant. Biol. Symp. 1, 49. EDSALL, J. T., FLORY, P. J., KENDREW, J. C., LIQUOR!, A. M., NEMETHY, CHANDRAN, G. N. & SCHERAGA, H. A. (1966). J. mol. Biol. 15, 399. EKLUND, H. (1976).Thesis,Royal AgriculturalCollege,Uppsala.

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Gb, M. & SCHERAGA, H. A. (1968). Proc. nufn. Acad. Sci. U.S.A. 59, 1030. GO, M., Gb, N. & SCHERAGA, H. A. (1970). J. them. Phys. 52,206O. HYLTON, J., CHRISTOFFERSEN, R. E. & HALL, G. G. (1974). Chem. Phys. Lett. 24, 501. IUPAC-IUB Commission (1971). Biochem. J. 121, 577. JOHANNIN, F. & KELLERSHOHN, N. (1972). Biochem. Biophys. Res. Comm. 49, 321. Kuao, R. (1962). J. Phys. Sot. Jap. 17, 1100. LIFSON, S. & ROIG, A. (1961). J. them. Phys. 34, 1963. LIFSON, S. (1964). J. them. Phys. 40,3705. MCCREERY, J. H., CHRISTOFFERSEN, R. E. & HALL, G. G. (1976~). J. Am. them. Sot. 98, 7191. M&REERY, J. H., CHRISTOFFERSEN, R. E. & HALL, G. G. (19766). J. Am. c/rem. Sot. 98, 7198. MCWEENY, R. (1969). Methods of Molecular Quantum Mechanics. London, New York: Academic Press. OIE, R., MAGGIORA, G. M. & CHRISTOFFERSEN, R. E. (1976). Int.J. Quunt. Chem.: Quant. Biol. Symp. 3, 119. ONSAGER, L. (1936). J. Am. Chem. Sot. 58, 1486. POPLE, J. A. & BEVERIDGE, D. L. (1970). Approximate Molecular Orbital Theory. New York : McGraw-Hill. POIRT, G. N. J. & PULLMAN, A. (1974). Int. J. quant. Chem.: Quant. Biol. Symp. 1, 21. PULLMAN, B., BERTHOD, H. & GRESH, N. (1975). FEBS Lett. 53, 199. PULLMAN, B. (ed). (1976). Environmental Effects on Molecular Structure and Properties. Dordrecht : Reidel. RICHARDS, F. M. (1974). J. mol. Biol. 82, 1. RINALDI, D. & RIVAIL, J. L. (1973). Theor. Chim. Acta 32, 57. RI~AIL, J. L. & RINALDI, D. (1976). Chem. Phys. 18, 233. SANHUEZA, E. & TAPIA, 0. (1978), Biochem. Biophys. Res. Comm. in press. SCHERAGA, H. A. (1963). In The Proteins, (H. Neurath, ed.), Vol. 1. p. 477. New York, London : Academic Press. SCIIUSTER, P., JAKUBETZ, W., BEIER, G., MEYER, W. & RODE, B. P. (1974). The Jerusalem Symposia, (E. D. Bergmann & B. Pullman, eds), Vol. 6, p. 257. Dordrecht: Reidel. SINANOGLU, 0. & ABDULNUR, S. (1965). Fed. Proc. 24, S-12. TAPIA, O., NOGALES, A. & CAMPANO, P. (1974). Chem. Phys. Letf. 24, 401. TAPIA, 0. & GOSCINSKI, 0. (1975). Mol. Phys. 29, 1653.

GO, N.,

TAPIA, O., POULAIN, E. & SUSSMAN, F. (1975). Chem. Phys. Lett. TAPIA, 0. & POULAIN, E. (1977). Znt. J. Quunt. Chem. 11, 473. TAPIA, 0. (1978), Theoret. Chim. Actu (Berl.) in press. YOMOSA, S. (1967). Prog. Theor. Phys. (Jap.) Supp. 40, 249. YOMOSA, S. (1974). J. Phys. Sot. Jap. 36, 1655. WARSHEL, A. & LEVITT, M. (1976). J. mol. Biol. 103, 227. ZIMM, B. H. & BRAGG, J. K. (1959). J. Chem. Phys. 31, 526.

33, 65.

APPENDIX

Many of the numerical results concerning environmental effects upon proton potentials and properties associated with H-bonded systemscan be rationalized in terms of the simple theory of charge transfer associations (Bratos, 1967). We give here a brief outline of the theory with an example.

72

0.

TAPIA

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(1) The relevant basic functions arc : $(H,O. . . H-OH) and IC/(H,O+ . . . H-HO-) region I (AI) $(HsO+ . . . OH-) and $(H,O. . . OH) region II WI (2) The total energy associated with each region is g-dependent, as is the energy differences between the minima. Comparisons are made within each region only. One has the following inequalities: E(H,O.. .H-OH) 4 E(H,O+ . . .H-OH-) (A3) and E(HsO+ . . .OH-) < E(H,O.. .OH) (A4) (3) Within the perturbative approach described by Bratos (1967) the wave functions can be written as follows: +I = N, $(H,O.. {

.H-OH)

+

/L$(g)$(H,O+ . . . H-OH-) + (IH~o-~,~o)+CH~O+H~O-gW.(Mn,o+~,oand for region II:

.H-OH)>

(A5)

PrrW(H,O . . . OH) + ~(IH~o-AoH)-G~o+oH- -~MII.(MH~ooH-MH~o+oH-)>

(‘46)

tiII = N,, $(H,O+

. . .OH-)

-MH~o..

+ +

where N is a normalization constant, Z and A are the ionization potential and electron affinity of the corresponding species indicated as subscripts, C the Coulomb interaction energy between the corresponding ions and ~7is the resonance integral. Consider now the dependence of the RFS upon the intermolecular distance. Bearing in mind that the order of magnitude of the g-contribution to the denominators of equation (A5) and (A6) is of the order of a Kcal mol-’ while the others can attain the order of an electronvolt, this term can be neglected. Now when &. . . o, increases the Coulomb term decreases in absolute value. This makes the denominator increase in equation (A5). The result is a decrement of the ionic component of the total wave function, and therefore a decrement of the RFS is expected. It is easy to check that the opposite will happen in region II. These are precisely the results obtained by us by actual computation. A charge transfer formalism has also been used by Yomosa (1974) to study donor-acceptor complexes in solution.