Frozen fragment reduced variational space analysis of hydrogen bonding interactions. Application to the water dimer

Volume 139, number 1

CHEMICAL PHYSICS LETTERS

FROZEN FRAGMENT REDUCED VARIATIONAL OF HYDROGEN BONDING INTERACTIONS. APPLICATION TO THE WATER DIMER

14 August 1987

SPACE ANALYSIS

Walter J. STEVENS Molecular Spectroscopy Division, National Bureau of Standards, Gaithersburg, MD 20899, USA

and William H. FINK Department of Chemistry, Universrty of Califortua, Davis, CA 95616, USA Received 19 March 1987; in final form 22 May 1987

A reduced variational space method is presented for analyzing hydrogen bonding interactions in terms of Coulomb and exchange, polarizability, and charge-transfer components. The method relies on the use of SCF optimized monomer orbitals in dimer calculations in which the wavefunction of one monomer is held frozen while the other is optimized with a basis set including selected subsets of the unoccupied monomer orbitals. Freezing the monomer wavefunctions allows the polarizability and charge-transfer interactions to be ascribed to specific monomers. Applications are presented for the interaction energy and dipole moment of the water dimer.

1. Introduction

The study of weakly bound molecular complexes using high-resolution spectroscopic methods is currently a very active area of research [ 1,2]. With techniques such as molecular beam electric resonance spectroscopy [ 31 and pulsed-nozzle Fouriertransform microwave spectroscopy [4,5], it is now possible to determine accurate geometric conformations for the ground states of weakly bound molecular complexes in the gas phase. The availability of accurate experimental data has rekindled theoretical interest in both the accurate prediction of structures and the analysis of the forces that hold together such complexes. It is customary to decompose the weak interactions between closed-shell molecules into several energy components including electrostatic, exchange repulsion, polarization, dispersion, and charge transfer [ 61. There has been considerable discussion over the years about the relative magnitudes of these energy components, particularly in hydrogen-bonded

complexes #I. Although electrostatic terms clearly dominate the long-range interaction between two polar molecules [ 8,9], exchange repulsion interactions dominate at short range, and charge-transfer may become important in the region of the potential minimum where the electronic wavefunctions begin to overlap. Charge transfer has been used to explain binding energy trends and spectroscopic intensity behavior in hydrogen-bonded complexes [ lo], and, more recently, experimentally determined structures of several weakly bound systems [ 1l- 161. In these cases it has been stated that electrostatic models are incapable of predicting or rationalizing experimentally determined properties and structures. Very recently, Buckingham and Fowler [ 17,181 and Liu and Dykstra [ 191 have argued that the geometries of weakly bound complexes can be predicted on the basis of electrostatic and polarization interactions alone, if sufficiently accurate fragment electrostatic potentials and polarizabilities are used #’ For a good discussion see ref. [ 71. 15

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in conjunction with some suitable distance constraint such as atomic hard sphere radii. They have shown that the failures of previous electrostatic structure predictions resulted from including too few terms in the multipolar expansions of the fragment potentials and polarizabilities. With the exception of dispersion, interactions between closed-shell molecules are determined quite accurately by ab initio Hartree-Fock single-configuration self-consistent-field (SCF) calculations. Several methods have been proposed for analyzing ab initio calculations in terms of the energy components discussed above [20-261. The method of Kitaura and Morokuma [23,24] is by far the most widely used. In that method, the wavefunction of the dimer complex is constructed from the occupied and unoccupied molecular orbitals of the isolated fragments. The energy components are determined from the change in the total energy when well-defined interaction matrix elements are eliminated from the Fock matrix. Using this method, energy analyses of many weakly bound complexes have indicated that electrostatic interactions are dominant, but that polarization and charge-transfer effects are not negligible at the minimum of the interaction potential surface [24] HZ. For many complexes, the chargetransfer and polarization energies are very nearly cancelled by the exchange repulsion energy, and the electrostatic energy mimics the total interaction energy [27]. Recently, Weinhold and co-workers [ 25,26,28] have proposed a new method for analyzing molecular interactions called “natural bond orbital analysis”. Using a “natural hybrid orbital” procedure [ 29,301, the SCF wavefunction of the complex is transformed into strongly occupied inner-shell, bonding, and lone-pair orbitals, and weakly occupied antibonding and Rydberg orbitals. The interaction energy components are determined by systematically eliminating the weakly occupied orbitals from the variational space and recalculating the energy. Applications of this method to the water dimer [ 261 and several other hydrogen-bonded systems [ 281 have determined that the charge-transfer interaction is dominant. In fact, the analysis predicts that the interaction between two water molecules, at #’ For a recent analysis see ref. [ 271. 16

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the hydrogen bond equilibrium conformation, would be repulsive without charge transfer. A possible weakness in this approach is that the energies of the monomers change when the weakly occupied orbitals are eliminated from the basis set. The dependence of the monomer energies on the weakly occupied orbitals may not be constant over the whole potential surface, and it may not be possible to uniquely separate intramolecular and intermolecular energy effects. Very recently, Hurst et al. [ 3 1] have employed an intermolecular perturbation theory developed by Hayes and Stone [ 32,331 to analyze the interaction energy of several van der Waals dimers. The method employs non-orthogonal monomer orbitals to evaluate the first-order electrostatic and exchange repulsion contributions to the interaction energy and the second-order polarization and charge-transfer terms as well as the dispersion contribution. The analysis of several dimers has led to the conclusion that the electrostatic interactions dominate and have the same angular dependence as the total interaction energy [ 3 11. Also, while charge transfer in hydrogen-bonded dimers from the proton acceptor molecule to the proton donor molecule is found to be important in determining the absolute binding energy, it is not the dominant term, and, in most cases its magnitude is similar to the polarization term. In this note, we demonstrate a new method for analyzing weak interactions based on frozen-fragment SCF calculations. The method is closely related to the most recent version of the Morokuma and Kitaura method [ 241, but, instead, uses the group function approach to molecular structure put forth by McWeeney [ 34,351 and implemented several years ago by Fink [ 361. The function space is divided so that the orbitals of one fragment may be optimized in the field of the frozen orbitals of the other fragment. In addition, the variational space may be truncated by removing the unoccupied orbitals of either fragment in order to isolate polarization, charge-transfer and basis set superposition effects. The method has been used recently by Bagus et al. [ 371 to analyze charge-transfer and polarization effects in ligand-metal bonding. In the next sections we describe the method, which we call the “reduced variational space” self-consistent-field (RVS SCF)

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method, and present an analysis of the hydrogen bond in the water dimer as an example.

2. Method Frozen fragment reduced variational space calculations were carried out using a series of computer programs that implement a single-determinant SCF version of McWeeney’s group function approach [ 34,351. A matrix implementation of these procedures has been reported in detail [ 361 that enables a rigorous direct transfer of the orbitals from a previous calculation to the composite problem of interest. We have recently installed these procedures into the HONDO series of programs [ 381, and all reported calculations have been carried out with these routines or other related HONDO series programs. For the problem of interest, the orbitals of the monomers may be transferred to the dimer calculation, the occupied orbitals of one monomer frozen, the vacant (virtual) orbitals of each monomer either included or excluded from the variational space, and the remaining orbitals Gramm-Schmidt orthogonalized to the frozen monomer orbitals. Use of the Gramm-Schmidt procedure ensures that the orbitals of the frozen monomer remain exactly as they were for the isolated molecule calculation. The reference point for subsequent calculations has then been defined by solution of the monomer problem and not by any otherwise imposed condition. Optimization of the remaining occupied orbitals of the dimer complex within the reduced variational space enables the isolation of conceptual interaction energies between the monomers of the complex. Table 1 defines the relationships between the molecular orbital basis sets and the energy components in the RVS SCF calculations. The simplest dimer wavefunction is constructed from only the optimum occupied orbitals of the monomers. Exclusion of all vacant monomer orbitals from the basis set results in a dimer wavefunction with no in situ variational flexibility. The components of the interaction energy in such a dimer product include the Coulomb (electrostatic) interaction, exchange repulsion, and orthogonality between the occupied monomer orbitals. The energy of the dimer product wavefunction is independent of the procedure used

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Table 1 Relationships between molecular orbital basis sets and energy components in the reduced variational space SCF calculations. The “occ” subscript refers to occupied fragment SCF orbitals, and “vat” refers to vacant (or virtual) fragment orbitals. Brackets around an orbital subset name indicate that that subset is frozen in the calculation MO basis set

Energy components a)

Aocc Bocc A,, + B,,

64 Et3

[&,I +L+L [&I +&+L,+&,, A,,, + [BowI+ A,,, A,, + 1J&l+ A,,, + L A,, + A,,, + Bv,, B, + Bv,, +A,,,

EA+EB+CEX EA + EB + CEX + POLs

E,+Ee+CEX+POLB + CTa_A + SUPs EA + Ee +CEX + POL, EA + E,+ CEX + POL, +CT*_s+SUP* E,+SUPA EB+SUP,

‘) Energy component notation: E=energy of isolated fragment; CEX = Coulomb and exchange energy between orthogonalized fragments; POL=polarization energy; CT=charge-transfer energy; SUP=energy lowering due to superposition of basis sets.

to orthogonalize the monomer orbitals. Freezing the orbitals of one fragment while variationally optimizing the other fragment in a space that includes subsets of the vacant monomer orbitals allows various interaction energies to be defined and assigned to a specific monomer. For example, a calculation freezing the occupied orbitals of one of the monomers and optimizing the other including all available variational flexibility would conceptually include all electrostatic interactions, all exchange interactions, repolarization of the variational monomer by the frozen one, and charge transfer or bonding interactions from the variational monomer toward the frozen one. Exclusion of the vacant orbitals of the frozen monomer from the available variational space precludes the possibility of charge transfer or bonding interactions between the monomers. Unlike the Morokuma analysis, we do not attempt to isolate the exchange or orthogonality effects, and all wavefunctions are properly antisymmetrized. Care must be taken while using an orbital partitioning scheme to define energy components, because, at finite distances, the orbitals of the monomers are not perfectly spatially separated. Basis set superposition effects, which arise when the wave17

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function of one fragment is improved by variationally incorporating orbital space from the other fragment, can bias the analysis in favor of chargetransfer effects if the monomer basis sets are not complete enough to yield accurate monomer wavefunctions. Usually, the counterpoise method of Boys and Bemardi [ 391 is used to estimate the magnitude of the superposition effect. In that procedure, at each geometry, the energy of each monomer is recalculated using the combined basis sets of both monomers. The RVS SCF method allows the implementation of a modified counterpoise correction, as suggested by Morokuma and Kitaura [ 241, in which the magnitude of the superposition effect is determined by combining the basis set of one monomer with only the vacant orbitals of the other monomer. The occupied orbitals of the second monomer do not contribute to the superposition effect because they are frozen in the dimer calculations. A second type of superposition effect, which has been discussed by Loushin et al. [40], arises when the charge polarization of one monomer is enhanced by incorporating the vacant orbitals of the other monomer into the wavefunction. This “polarization superposition” can also be improperly identified as charge transfer if the individual monomer basis sets are not complete enough. This effect is more difficult to quantify, but should be manifested by a change in the balance of polarization and charge-transfer energies as the monomer basis sets are systematically improved.

3. Calculations RVS SCF calculations were carried out for the water dimer at a variety of geometries. The monomer basis sets consisted of the (9~5~14s) Gaussian primitives of Huzinaga [ 4 1 ] contracted to ( 3s2p 12s) [ 421and augmented by a d polarization function on oxygen ( (x = 0.85 ) and a p polarization function on each hydrogen ( cx= 1.OO). For some calculations, this “double-zeta plus polarization” (DZP) basis. set was augmented by an additional d (DZP + d) polarization function on oxygen (0 = 0.20)) which improves the description of the monomer dipole moment and polarizability. In all dimer calculations, the monomer geometries were held fixed at Ro,=0.956 8, 18

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Table 2 Calculated SCF properties of the Hz0 monomer *) Property

DZP basis set

DZP + d basis set

energy (au) P(D)

- 76.046479 2.224 3.030 5.389 7.177

- 76.049966 1.976 6.563 7.559 8.593

.a,, (au1 o,, (au) a:.- (au)

a) The z-axis is the Cz axis, and the x-axis is perpendicular to the Hz0 plane. Hz0 geometry: Ro,=0.956 A, LHOH= 105.2”.

and L HOH= 105.2”, which is near the experimental equilibrium. The monomer properties with the two basis sets are given in table 2. In addition to overestimating the dipole moment, the DZP basis set gives a poor description of the out-of-plane polarizability of water. Both the dipole moment and the polarizability are improved significantly at the DZP + d level. The experimental dipole moment and average polarizability of water are 1.8546 D [ 431 and 9.64 au [44] respectively. With the monomer geometries held fixed, the C,symmetry, open structure for the dimer shown in fig. 1 was optimized at the SCF level using the DZP basis set. Near the minimum energy configuration, the surface is very flat with respect to angle variation, and no attempt was made to find the absolute minimum. The energy gradient for the intermolecular coordinates was minimized to within 0.0005 au, which only determines the angles to within a few degrees. The values of Ro,=3.0085 A, 8=53.84”, and 0 =O" are close to the experimentally determined [45] values of 2.976(4), 57(5), and 6(8),

Fig. 1. The coordinates used to describe the geometry of the water dimer. The values of the coordinates at the reference geometry are Roe = 3.0085 A, @= 53.84”, and @ ~0”. The angles are positive as drawn.

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respectively. A very recent, high-quality theoretical calculation [46] gives 2.911, 56.8, and 4.5.

4. Energy decomposition at the reference geometry At the reference geometry of fig. 1, the interaction energy between the two monomers was analyzed using both the DZP and the DZP+d basis sets. The results of the analysis are shown in table 3. The total interaction energy is given as a function of the orbital subsets included in each RVS SCF calculation, and the energy components are calculated according to the prescription of table 1. The sum of the Coulomb and exchange interactions (CEX), which is defined by the non-variational product wavefunction constructed from the occupied monomer orbitals, is attractive, and accounts for 60 to 70°h of the total interaction energy. The magnitude of CEX is smaller in the DZP+d basis set, which seems reasonable since the monomer dipole moments are reduced with the larger basis set. The energetic contributions of polarization and Table 3 Analysis of the water dimer interaction energy. The reference geometry is given in fig. 1.All energies are in units of kcal/mol (1 kcal/mol= 4.185 kJlmo1) MO basis set

Interaction energy DZP

DZP+d

A,, +A,., + R,,, &,+&,~+A,,

3.44 3.83 3.95 3.65 4.54 0.15 a) 0.04 a’

2.58 2.95 3.11 2.95 3.76 0.09 a’ 0.03 =’

component analysis CEX POL, POLB CT,-, CT.+,

3.44 0.21 0.39 0.14 0.08

2.58 0.37 0.37 0.12 0.13

A,, + I&

lA,,,l+n,,+n,,, [A,,,l+n,,+n,,,+A~., A,+ [r-&I +A,., A,+ [J&l +A,,,+&

total full variational LJ)

(71%) (4%) (8%) (15%) (2%)

4.86 4.89

a1 Energy lowering due to basis set superposition. w Corrected for superposition.

4.17 4.21

(62%) (9%) (9%) (17%) (3%)

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charge transfer are comparable to each other. In the DZP calculation, the polarization of the “A” fragment is a little small due to the poor description of the out-of-plane component of the polarizability. In the case of charge transfer, the two basis sets give very similar results. Charge transfer from the “A” fragment (proton acceptor) to the “B” fragment (proton donor) is the dominant CT component. This agrees qualitatively with the analysis of Weinhold et al. [ 251, but the magnitude is much less. In our analysis, charge transfer does not dominate the bonding. Our results are similar to the recent Morokumatype analyses of Singh and Kollman [47] and Morokuma and Kitaura [ 241, which were obtained with a DZP basis set at a slightly different geometry. The results of Sir@ and Kollman are CEX=2.68 kcal/mol, POLA + POL, = 0.90 kcal/mol, and + CTB_* = 1.28 kcal/mol. No breakdown was CTA-B given of the contributions from each fragment independently. The results of Morokuma and Kitaura are CEX = 3.2 kcal/mol, POL, + POLB = 0.5 kcal/ mol, CT*-a= 1.7 kcal/mol, and CTB_A=O. 15 kcal/ mol. Although our DZP analysis gives a similar total interaction energy, the components are somewhat different. In particular, the sum of the CT components is quite a bit smaller (0.82 kcal/mol). The total interaction energy obtained by adding the RVS SCF components agrees very well with the fully variational result after correcting for basis set superposition. This demonstrates that second-order effects and coupling among the various components are minimal in the RVS SCF analysis. The superposition errors are not large, and are reduced by about one third with the DZP+d basis set.

5. Geometry dependence of the interaction energy components RVS SCF calculations using the DZP basis set were carried out for several values of Roe,8,and @ near the reference geometry. Each coordinate was varied with the others held fixed. The results of the energy component analysis are shown in table 4. We have also added an estimate of the correlation contribution to the interaction, A& 2), which was calculated using second-order Moller-Plesset perturbation theory [ 481. Frisch et al. [ 461 have shown that this is

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Table 4 Geometry dependence of the components of the water dimer interaction energy using the DZP basis set. See fig. 1 for definition of coordinates. As each coordinate was varied, all other coordinates were fixed at the reference geometry. All energies are in units of kcall mol (1 kcal/mol= 4. I85 kJ/mol) Coordinate

CEX

POL,

POLa

CT A-8

CT B-A

W2)

Total

R,=2.8085 2.9085 3.0085 3.1085 3.2085 3.3085

1.80 2.88 3.44 3.66 3.68 3.58

0.37 0.28 0.21 0.17 0.13 0.10

0.70 0.52 0.39 0.30 0.24 0.18

1.28 0.96 0.74 0.60 0.50 0.42

0.15 0.12 0.08 0.07 0.04 0.03

1.49 1.26 1.07 0.91 0.78 0.67

5.79 6.02 5.93 5.71 5.37 4.98

8=33.84 43.84 53.84 63.84

3.56 3.54 3.44 3.23

0.23 0.22 0.21 0.21

0.43 0.41 0.39 0.37

0.59 0.65 0.74 0.81

0.09 0.09 0.08 0.07

0.94 1.00 1.07 1.14

5.84 5.91 5.93 5.83

@=-15.0 0.0 15.0

2.95 3.44 3.41

0.17 0.21 0.18

0.33 0.39 0.35

0.69 0.74 0.60

0.07 0.08 0.07

1.05 1.07 0.93

5.26 5.93 5.54

a good approximation for hydrogen-bonded systems. In the water dimer system, correlation contributes about 20% of the binding energy. In systems containing second-row and heavier atoms, the correlation contribution is known to be much larger [ 491. As expected, all of the components in table 4 increase as Roe decreases, with the exception of CEX, which goes through a minimum because the exchange repulsion effects become dominant at small separations. If charge transfer and correlation are omitted, the sum of the Coulomb, exchange and polarization energies gives a radial minimum that is too large by more than 0.1 A. The 8 dependence of the interaction potential is very flat. The polarization components are not very strongly dependent on this coordinate, and the minimum energy is determined by a balance of the CEX, POL,,,, and AE(2) terms. CEX favors smaller angles, while charge transfer and correlation favor larger angles. These results are in agreement with the Morokuma-type analysis recently published by Rendell et al. [28]. The dependence of the energy on the deviation of the hydrogen bond from linearity (the @ coordinate) is stronger than the 8 dependence. All components favor near-linear configurations. The CEX term is reduced rather sharply for angles which put the proton of the donor molecule on the same side of the O-O axis as the protons of the acceptor molecule. Motion in this direction takes the partially 20

unshielded donor proton away from the negative electrostatic potential of the oxygen lone-pair electrons. All other components are not as strongly dependent on this angle. Although the CEX energy component is always dominant in this analysis of the water dimer, the radial and angular dependence of the interaction energy cannot be reproduced by this term alone. The surface is so flat, that the weak dependence of the other components on angular orientation help to determine the absolute minimum.

6. Dipole moment analysis The reduced variational space procedure can also be used to analyze the dipole moment of the water dimer complex. SCF calculations using the DZP and DZP + d basis sets at the reference geometry of fig. 1 produce dimer dipole moments of -3.118 and - 2.873 D, respectively. The smaller moment for the DZP + d basis set is in reasonable agreement with the experimental value of 2.643 D [ 451, and reflects the smaller moment of the monomer as seen in table 1. It is interesting to determine whether polarization or charge transfer dominates the dipole moment enhancement upon dimer formation. An early calculation by Yamabe and Morokuma [ 501 indicated that polarization and charge-transfer contributions are nearly the same, but they used a small 4-31G basis

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Table 5 Analysis of the water dimer dipole moment.

The reference

MO basis set

PHYSICS

geometry

+ Rx,

[A,,1 +L,+& [A,,1 +B,+L+& &,+ [&,I +A,,, A,, + I &,I + A,,,+ B,,,

-2.564 -2.757 -2.752 -2.690 -2.790

component CEX POL, POLa CT,-, CTe-A

-0.018 -0.126 -0.193 -0.100 0.005

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is given in fig. 1. All moments

DZP 1.x

A,

LETTERS

1987

are in units of D

DZP+d PI,

,Kr

PC,

0.787 0.849 0.847 0.764 0.796

-2.282 -2.471 - 2.460 - 2.489 -2.550

0.701 0.769 0.766 0.715 0.729

0.005 -0.023 0.062 0.032 - 0.002

-0.021 -0.207 -0.189 -0.061 0.011

0.007 0.014 0.068 0.014 -0.003

analysis

enhancement K4+/k

-0.432 -2.546

0.074 0.782

-0.467 -2.261

0.100 0.694

total full variational

-2.978 - 2.999

0.856 0.854

-2.728 -2.761

0.794 0.796

set, which undoubtedly led to an overestimate the charge-transfer contribution. Table 5 shows the dipole moment results from the various RVS SCF calculations at the reference geometry. Also shown are the components of the moment change calculated from the appropriate differences as given in table 1. Clearly, the changes due to polarization are more dominant than the changes ascribed to charge transfer. All of the components, with the exception of the weak CT,,, term, contribute to the enhancement of the dimer dipole moment relative to the sum of the monomer moments. The total enhancement, and each of the components, lie mostly along the O-O axis, which agrees with experimental Stark effect measurements [ 451. Charge transfer from A to B, while an order of magnitude more important than B to A transfer, is still small when compared to the total effect of polarization. This situation is even more pronounced at larger intermolecular separations where the overlap is small and the chargetransfer effect is reduced relative to polarization. At the DZP level of calculation, the dipole moment change due to polarization of fragment B is larger than for polarization of A. This is an artifact of the small out-of-plane polarizability of the monomer obtained with the DZP basis set (see table 1). The results from the DZP + d calculation reveal compa-

rable polarization effects for the two fragments. Adding the monomer dipole moments to the sum of the components in table 5 gives total dipole moments of 3.119 and 2.9 18 D for the complex using the DZP and DZP+d basis sets respectively. These values are close to the full variational values, which again demonstrates the consistency of the analysis. Superposition effects on the dipole moment of each monomer were found to be small, and the effect on the dimer moment is negligible because the two monomer superposition errors tend to cancel.

7. Summary and conclusions The reduced variational space SCF method provides a convenient procedure for decomposing the interaction energy of weakly bound complexes into various conceptual components. Charge-transfer and polarization effects can be identified with a particular monomer because the occupied orbitals of the other monomer are frozen in their isolated molecule state. The method also can be used to estimate monomer energy changes due to the superposition of the vacant orbitals of the other monomer. Other properties, such as the dipole moment, can be analyzed in a fashion analogous to the energy analysis. 21

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For the water dimer, the RVS SCF analysis finds the combined Coulomb and exchange interaction to be dominant, but the correct angular dependence of the interaction energy is obtained only if chargetransfer and correlation components are taken into account. All of the components exhibit a strong dependence on the Roe radial coordinate. The Coulomb and exchange term exhibits a radial minimum, but the distance is too large when compared to the minimum energy distance for the sum of all components. In the interaction energy, the polarization and charge-transfer effects have comparable magnitudes. In the dipole moment, however, the polarization of the monomers dominates the enhancement upon dimer formation.

References [ I] A. Weber, ed., Weakly bound molecular complexes (Reidel, Dordrecht, 1987). [2] P. Schuster, ed., Current topics in chemistry, Vol. 120. Hydrogen bonding (Springer, Berlin, 1984). [ 31 T.R. Dyke, B.J. Howard and W. Klemperer, J. Chem. Phys. 56 (1972) 2442. [ 41 T.J. Balle, E.J. Campbell, M.R. Keenan and W.H. Flygare, J. Chem. Phys. 7 1 (1979) 2723; 72 (1980) 922. [5] A.C. Legon, Ann. Rev. Phys. Chem. 34 (1983) 275. [ 6 ] C.A. Coulson, in: Hydrogen bonding, ed. D. Hadzi (Pergamon Press, Oxford, 1959). [7] P.A. Kollman, in: Modern theoretical chemistry, Vol. 4. Applications of electronic structure theory, ed. H.F. Schaefer III (Plenum Press, New York, 1977). [ 81H. Margenau, Rev. Mod. Phys. 11 (1939) 1. [ 91 A.D. Buckingham, Advan. Chem. Phys. 12 (1967) 107. [lo] G.C. Pimentel and A.L. McClellan, The hydrogen bond (Freeman, San Francisco, 1960). [ 111 K.C. Janda, J.M. Steed, SE. Novick and W. Klemperer, J. Chem. Phys. 67 (1977) 5162. [ 121 J.M. Steed, T.A. Dixon and W. Klemperer, J. Chem. Phys. 70 (1979) 4095. [ 131 F.A. Baiocchi and W. Klemperer, J. Chem. Phys. 78 (1983) 3509. [ 141K.R. Leopold, G.T. Fraser and W. Klemperer, J. Chem. Phys. 80 (1984) 1039. [ 151 K.I. Peterson and W. Klemperer, J. Chem. Phys. 80 (1984) 2439. [ 161 G.T. Fraser, K.R. Leopold and W. Klemperer, J. Chem. Phys. 8 1 (1984) 2577. [ 171 A.D. Buckingham and P.W. Fowler, J. Chem. Phys. 79 (1983) 6426. [ 181 A.D. Buckingham and P.W. Fowler, Can. J. Chem. 63 (1985) 2018. [ 191 S.-Y. Liu and C.E. Dykstra, Chem. Phys. 107 (1986) 343.

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