THE ELECTRON DENSITY DISTRIBUTIONS IN HYDRIDE MOLECULES

Transcription

1 THE ELECTRON DENSITY DISTRIBUTIONS IN HYDRIDE MOLECULES 111. HYDROGEN FLUORIDE1 R. F. W. BADER~ AND G. A. JONES~ Department of Chenzistry, University of Ottawa, Ottawa, Canada Received April 2, 1963 ABSTRACT An electron density distribution is determined for the hydrogen fluoride molecule in its equilibrium configuration by requiring the distribution to predict the correct dipole moment and to exert forces on the nuclei equal and opposite to the nuclear forces of repulsion. The ability of the density function to give good estimates of the expectation values of other one-electron operators is found to be satisfactory. It is shown that an electron population analysis of the orbital densities, when interpreted in terms of the electronic forces which these densities exert on the nuclei, is of value in the understanding of chemical binding. IKTRODUCTION In two previous papers (1, 2) we have given the electron density distributions for the equilibrium configuration of the water and ammonia n~olecules as determined by the requirement that the densities exert forces on the nuclei which are equal and opposite to the nuclear forces of repulsion. In each case the general characteristics, e.g., hybridization and bond polarity, of the densities so obtained are in good agreement with the results obtained from S.C.F. calculations. In addition, the densities determined by the requirement of zero resultant forces provide expectation values for the one-electron operators which are in as good, or better, agreement with experiment as the values predicted by wave functions obtained from S.C.F. calculations. This points to the main reason for carrying out such calculations. It was felt that a reliable picture of the oneelectron density could be readily obtained by requiring the distribution to reproduce a property which is itself determined directly by the one-electron density, i.e., the forces on the nuclei. The wave function derived from a variational calculation is obtained by minimizing the energy with respect to the second-order density distribution. Therefore, the resulting one-electron density is not necessarily the best possible one obtainable from the given basis set of orbitals. This was elegant]? demonstrated by Mukherji and Karplus in their use of a constrained wave function for hydrogen fluoride (3). In the present paper we determine the electron density distribution for the hydrogen fluoride molecule by requiring that it balance the forces on the hydrogen and fluorine nuclei and, in addition, give the observed dipole moment. It is worth noting that the criteria of zero resultant forces fixes the electric field due to the density in the region of each nucleus while the requirement of reproducing the observed dipole moment governs the overall disposition of the density in the molecule. The present results substantiate the previous ones in that good estimates are obtained by this method for those molecular properties which depend on the density. In addition, however, we wish to illustrate the advantage gained in the understanding and interpretation of the binding in molecules (4). Since the molecular forces are determined by the one-electron density, their discussion is rigorously carried out in terins of classical electrostatics, thus affording an excellent medium in which to interpret the nature of the binding in a molecule. 'Abstracted from the thesis submitted by G. A. Jones in partial fulfillment of the requirements for the degree of Doctor of Philosophy. 2Present adduess: Department of Chemistry, McMasteu LTniversity, Hamilton, Ontario. 3Recipient of a Commonwealth Scholarship. Canadian Journal of Chemistry. Volume 41 (1963)

2 2252 CANADIAN JOURNAL OF CHEMISTRY. VOL The Determination of the Density Distribution The electron density is obtained in terms of the single determinantal L.C.A.O. M.0. approximation. The basis set includes Is, 2s, and 2p atomic orbitals on fluorine (denoted by so, s, and p respectively) and a 1s orbital on hydrogen (denoted by h). The set is later extended to include d orbitals on the fluorine atom. The orbitals are assigned the orbital exponents determined by Ransil (5) in a S.C.F. calculation for the hydrogen fluoride molecule. By using these values (termed the B.L.M.O. set) a meaningful comparison of the S.C.F. and the present results may be made. The molecular orbitals written in the equivalent orbital form are: the fluorine atom inner shell orbital the bond orbital and three doubly occupied lone pair orbitals which we write as one sigma lone pair orbital along the bond axis and two?r orbitals 4, = cos s - sin E, p 4n = pa. These latter three orbitals may of course be combined to give three equivalent lone pair orbitals. The angle the equivalent lone pair orbitals make with the bond axis will be determined by the relative amount of 2s and 29 in the 4, orbital." The symbols h, and s in the above expressions denote orbitals orthogonal to the so orbital of fluorine. The orbitals embody three independent parameters: eb, the bond hybridization parameter, el, the corresponding parameter for the lone pair orbital, and the polarity factor XIp. These three variables are fixed by requiring the density to balance the two nuclear forces and duplicate the dipole moment. The remaining orbital coefficients are uniquely determined by the orthogonality conditions which are listed in the Appendix. The values of the parameters which satisfy the three conditions are given in Table I together with the TABLE I Parameters for the electrostatic and S.C.F. molecular orbitals for HF Dipole Force on-h 1 X/P (a.u.)* Force on F (a.u.) moment (Dl Electrostatic method 112'46' - 6"O' S.C.F. (4) ' 6"41' *A negative force is one in the same direction as the force of nuclear repulsion. values of the same parameters obtained from the transformed S.C.F. calculation of Ransil (5).t These parameters succinctly characterize the density distribution. The value of E, for the electrostatic density indicates that the sigma lone pair orbital is 99% 2s and 1% 2p *For exanzple, when 1 = 0, the equivalent lone pair orbitals are sp2 hybrids in a plane perpendicular to the bond axis through the fluorine nucleus, and when 1 = 30, the orbitals are sp3 hybrids which project out behind the fluorine nucleus symmetrically placed around the bond axis at an angle of 70 Sd' with this axis.?equivalent orbitals are obtained from the 2u and 3u symmetry orbitals of Ransil by an orthogonal transfornzation which yields one orbital with no hydrogen contribution (the lone pair) and another containing all of the hydrogen contribution (the bond orbital). This results in a unique set of orbitals.

3 BADER AND JONES: HYDROGEN FLUORIDE 2253 in character. Since E, is negative, the small amount of 2p character present actually projects density towards the hydrogen atom rather than away from it. This gives three equivalent lone pair orbitals of almost sp2 character (1.00 part 2s, 2.03 parts 2p) which niake an angle of 86" with the bond axis, and are directed towards the H atom. Conceivably, the sigma lone pair orbital could possess an orbital character of either 2s or 2p or any intermediate hybrid. As previously pointed out (I) only the limiting 2s form for this orbital will result in a balanced force on either the fluorine or hydrogen nuclei. Any lone pair s-p hybridization on the fluorine atom in the conventional direction results in a large force on the fluorine nucleus in a direction away from the hydrogen. This same s-p polarization removes density from the binding region and places it behind the fluorine nucleus, thus resulting in a greatly decreased shielding with respect to the hydrogen nucleus. Of the two possible limiting forms 2s and 29, the lone pair will become a 2s orbital and the bond orbital will then be primarily 2p in character. This form for the bond orbital exerts a much larger binding force than does the 2s form (1). Similar results were obtained for the water and ammonia molecules, both of which were predicted to possess lone pair orbitals with the maximum possible amount of 2s character. To reach the limiting 2s form for the lone pair orbital orthogonality restrictions require the introduction of negative 2s character into the bond orbital, i.e., eb > 90". In the water and ainmoiiia n~olecules the amount of negative 2s character so introduced was reduced to a minimum by delocalization of the hydrogen 1s orbitals. No such mechanism is possible in the case of hydrogen fluoride and consequently a larger percentage (actually 15%) of negative 2s character must be included in the bond orbital. It is interesting, however, that the nuclear force on hydrogen can still be balanced in spite of the density removed from the binding region by this adverse polarization of the fluorine density. Such is not the case for water and ammonia, where the forces on hydrogen cannot be balanced for large negative values of cos eb. Thus, where delocalization is possible, it is found to be essential, and where not possible, not necessary in order to achieve electrostatic equilibrium. Actually the slight projection of the sigma lone pair orbital towards the hydrogen atom in hydrogen fluoride compensates for the reverse polarization in the bond orbital. A large bond polarity XIp is necessary to meet the observed dipole moment. There is a small positive atomic dipole contribution frorn the reverse hybridization in the bond orbital but the major contribution to the dipole moment is due to removal of charge from hydrogen (small p) to the fluorine (large A). This again is similar to the previous results for water and ammonia where the dipole was found to be due almost entirely to the bond polarity. A density contour map of the electron density for hydrogen fluoride by the electrostatic method is shown in Fig. 1. The general features exhibited by the S.C.F. density distribution are similar to those found from the electrostatic approach. Lone pair hybridization is unimportant (although the small amount of hybridization, el = 6041f, predicted by the S.C.F. function has a very adverse effect on the forces). The bond orbital contains 3y0 negative 2s character. Polarity is significantly smaller, as reflected in the low value of the predicted dipole moment. Duncan and Pople (1.5) determined density distributions for the hydrogen fluoride, water, and ammonia molecules by requiring the density to duplicate only the observed dipole moment. In all three cases their results predicted a large degree of hybridization for the lone pair electrons (approaching overall sp3 hybridization in all three molecules). We have previously shown (1, 2) these results to be untenable for water and ammonia,

4 CANADIAN JOURNAL OF CHEMISTRY. VOL. 41, 1963 Distance from fluorine nucleus in 0.U. FIG. 1. The upper diagram gives lines of constant density in a plane through the hydrogen fluoride molecule, as determined by the electrostatic method. Only the valence orbital contributions are shown. The concentration of the lone pair electrons in p, orbitals and in the 2s orbital lzads to a pileup of density in the plane through fluorine perpendicular to the bond axis and an effective hole" behind the fluorine nucleus. The lone pair density is concentrated toward the hydrogen nucleus, where it can exert a binding force on both nuclei. Below the contour map is a graph showing the variation in electron density along the bond axis. The dotted line is the density distribution in an isolated hydrogen atom. It can be seen that the increased orbital exponent of the h orbital in the molecule tends to counteract the effect of a high bond polarity, so that the peak height at the proton is hardly changed on bond formation. where such lone pair hybridization results in enormous forces of repulsion on all of the atoins in the molecule. The failure of their method in these two cases was due to their relating the degree of hybridization to bond geometry (i.e., not allowing for bent bonds or delocalization). In their treatment of hydrogen fluoride they exclude the possibility arrived at in this paper of three equivalent lone pair orbitals projected in a plane perpendicular to the bond axis through the fluorine nucleus (sp2 hybridization). They do this on the grounds that such hybridization would of necessity lead to negative XIIL values, i.e., correspond to an antibonding bond orbital. This is not a consequence, however, if one allows for negative values of cos cb as in the present treatment or as found in the S.C.F. wave function. (This can be seen from their equation number 4.2.) The distribution favored by Duncan and Pople has three lone pair orbitals of approximately sp3 hybridization projected out behind the fluorine nucleus. This distribution results in large unbalanced forces of repulsion on both the hydrogen and fluorine nuclei. For the sp3 case, the final force on the proton is -1 a.u. and on the fluorine nucleus is -13 a.u. These figures should be compared with those in Table I.

5 BADER AND JONES: HYDROGEN FLUORIDE 2255 The electrostatic density distribution predicted for hydrogen fluoride agrees with the rule previously proposed (I):"in hydride molecules which possess at least one lone pair of electrons the bond orbitals will be primarily of P character and the lone pair orbitals will contain close to the maximum possible amount of s character". The "maximum possible amount of s character" corresponds in each case to the lone pair density being symmetrically placed with respect to a plane passing through the heavy nucleus and perpendicular to the main symmetry axis: sp2 orbitals in HF, sp orbitals in H20, and an s orbital in NH3. These forms for the orbitals do not exert any force on the heavy nucleus, and in addition they provide the maximum possible screening of the heavy nucleus from the protons. Any increase in p character from these lower limits displaces charge from the hydrogen side to the remote side of the F, 0, and N nuclei and results in large imbalanced forces on both the hydrogen and heavy-atom nuclei. It is always found that it is necessary to introduce inner shell polarization to an appreciable extent to balance the nuclear force of repulsion on the heavy atom in the hydride molecules with lone pair electrons. This polarization is brought about by including a 2p contribution in the inner shell orbital. In the present calculation the coefficient determining the amount of 2p so introduced (62) has a value of * This is larger than that predicted by variational calculations but it must be pointed out that the S.C.F. results for HzO, NH3, and HF all predict a resultant force on the heavy atom approximately equal to the full nuclear force of repulsion. Such a large discrepancy denotes a serious lack of flexibility in the basis orbitals chosen for the S.C.F. calculations. Because of the l/r,2 dependence of the forces, any inadequacy of the wave function in regions close to a heavy nucleus is emphasized more than in an energy calculation. The large value of cz found necessary for electrostatic equilibrium could be markedly reduced if an orbital which permitted polarization of the Is density close to the heavy nucleus was included in the basis set. A good choice for such an orbital would be a "Kp" orbital, i.e. one of the form cos 0 e-"'. Since this orbital, unlike the 29, has no pre-exponential radial dependence it peaks at the nucleus and, therefore, provides a much larger polarization and hence larger force than the same amount of 2-p. The present results and those for water and ammonia (1, 2) suggest that the inclusion of such an orbital will be necessary for obtaining wave functions from S.C.F. calculations which satisfy the criterion of electrostatic equilibrium. Another form of polarization not represented in the basis set is that of 2p with 3d. To be most effective these should be "shrunken" 3d orbitals, of the same radius as the 29 orbitals. In order to investigate the effect of 3d polarization by the present method it is necessary to ignore inner shell polarization as we have only three restraining conditions available. It is found that invoking 3d polarization cannot of itself enable the system to attain electrostatic equilibrium. The d, or d, polarization removes density from the outermost regions of the fluorine atom and concentrates it in the region between the nuclei. This displaced density increases the force on the hydrogen nucleus as well as the force on fluorine. Regardless of the values assigned to the other parameters the amount of 3d polarizatioil necessary to balance the force on fluorine always results in too large a force on the proton. On the other hand, inner shell polarization can be introduced to balance the force on fluorine without greatly changing the force on the proton and thus the system can attain electrostatic equilibrium. This is due to the fact that inner shell polarization displaces charge close to the fluorine nucleus rather than moving it into the overlap *We preoioztsly repovted values of cz for the water and aflzmonia molecules about a factor of 10 less than the present value. This is due to our neglect of the overlap between the heavy atom Is orbital and the hydrogen Is orbital. When this overlap is taken into considevation the cp values are comparable to that obtained for HF but the remaining pavanzetevs in the Hz0 and NHa densities remain unchanged.

6 2256 CANADIAN JOURXAL OF CHEMISTRY. VOL region. In a fuller treatment using this extended basis set, the 3d orbitals may provide part of the shielding obtained by negative lone pair hybridization in the present treatment. Sternheimer (6) has calculated the electric field at various nuclei in atoms and ions when they are subject to an external perturbing field. He concluded that inner shell polarization was very important for the alkali atoms (7), but in the negative fluoride ion (8) he found that the largest contribution to the electric field came from a p-d type perturbation. In the latter treatment the inner shell was approximated by a Slater 1s orbital as opposed to the Hartree-Fock orbitals employed for the valence electrons. Sternheimer estimated an error of ~20% in the inner shell contribution as a result (8), but since this contribution was small compared to those of the s-9, p-s, or p-d perturbations, the final result was not affected. It seems therefore that 3d polarization is much more important in the diffuse negative ion than in the neutral fluorine atom taking part in bond formation. A Force Analysis of the Wave Fz~nctions for HF An investigation of the forces exerted by an electron density distribution provides insight into the nature of molecular binding (4). When the L.C.A.O. approach is used in the construction of molecular orbitals the resulting density expression will consist of a sum of products of atomic orbitals of the form c,c,4,,4,p where c, and c, are the orbital coefficients and 4, and 4, denote the type of orbital (Is, 2s, etc.) situated on nucleus a or P. \Ye thus obtain the three basic electron populations, the atomic population terms or 4z8~jP and the overlap population $~,,+~p. Each of these three electron populations gives rise to its own distinct contribution to the force acting on a particular nucleus. Atomic Force The atomic force is the force exerted on nucleus a by the density +,,+,,. If 4,,+,, possesses a center of symmetry, then this density will not exert a force on nucleus a. Holyever, the introduction of any asymmetry, due to hybridization or polarization, gives rise to a force on nucleus a. Consequently the atomic force term is sensitive to the degree of charge polarization on a nucleus. Penetration Force The penetration force is the force exerted on nucleus a by the atomic charge density situated entirely on another nucleus, 41843P. It is a measure of the electronic shielding of nucleus from nucleus a and the amount of this shielding is determined by the extent to which nucleus a penetrates the density around P. When 4,p+,p possesses spherical symmetry the penetration force has a particularly simple physical interpretation. It can be shown from a well-known theorem of electrostatics that the total force exerted on a by a spherical charge density on 6 is simply the force that would be exerted by that density contained within a sphere of radius d (the bond length) if all this charge was situated at nucleus 0. Thus the penetration force due to spherical orbitals is numerically equal to the number of electrons contained in a sphere of radius equal to the bond length, all divided by d2. As d decreases, a smaller fraction of the electrons is contained in the smaller sphere and nucleus P is less shielded from nucleus a. The density of tightly held inner shell electrons lies completely within the sphere defined by the bond length and thus they effectively screen an equal number of nuclear charges. The more diffuse valence shell electrons do not. The electron density in a p, orbital on P, since it places charge along the bond axis, exerts a shielding force greater than the same amount of density spherically disposed. Electrons in p, orbitals, on the other hand, are less effective than s electrons at shielding since their density is principally directed perpendicular to the bond axis. As would be

7 BADER AND JONES: HYDROGEN FLUORIDE 2257 anticipated a closed shell in which all the orbitals have the same orbital exponent, i.e. a n ~~np,~np,~, acts as a spherical distribution and the force is simply eight times that of a single electron in an ns atomic orbital. Overlap Force The overlap force is the force on nucleus a due to the density +i,+jp resulting from the overlap of the two atomic orbitals. This force term is a sensitive measure of the binding power of a ~nolecular orbital. The binding is not simply proportional to the extent of overlap, of equal importance is where this density is placed. Thus a 2sls overlap is larger than is a 2pls overlap, but the latter is much more effective at binding since more of the overlap charge density is placed along the bond axis, where it exerts the maximum force. IVe shall illustrate the advantages of a force analysis by applying the method to Ransil's S.C.F. results for hydrogen fluoride. We consider the forces exerted on the hydrogen atom first. The forces due to the atomic and overlap populations for each of the symmetry orbitals are given in Table 11. Since the atomic density on hydrogen possesses spherical TABLE I1 Forces on the proton in HF (S.C.F. density) in a.u. of (l/d2) Overlap Orbital Penetration force PE force OH Electronic force Nuclear re~ulsion Net force Total symmetry (1s orbital) there are no atomic force contributions in this case. Great simplification is achieved by omitting the factor l/d2 from the numerical values of the forces. The nuclear repulsion is then simply -Z,Z, = -9. The sum of the electronic forces must equal minus this same number to attain electrostatic equilibrium. In the case of the separated atoms where there is no penetration each electron shields one unit of nuclear charge and the net force is zero." We therefore break up the total nuclear repulsion of -9 into contributions for each orbital as determined by the shielding each orbital would exert in the case of the separated atoms. Consider the lu orbital. This orbital is made up almost entirely from the 1s orbital on fluorine. The pair of electrons in this orbital is very tightly held by the fluorine nucleus as is evidenced by the fact that the penetration force equals simply Thus these two electrons effectively shield 2 units of positive charge on the fluorine nucleus from the hydrogen nucleus. The overlap force term in this case is insignificant and the net force due to this orbital and its share of the nuclear charge is zero. That is, in the separated atoms or in the molecule the inner shell orbital on fluorine simply acts to shield two units of nuclear charge, exerting no resultant force on the hydrogen nucleus. This is termed a nonbinding orbital in accordance with the definition previously proposed (4). In the limit of separated atoms the 2u orbital becomes the doubly occupied 2s orbital on fluorine and so is again assigned two units of the nuclear repulsion. The penetration of this charge density by the proton is much greater than in the lu case as evidenced by would quadrupole force operative due to the fact that does not hane a closed-shell

8 2258 CANADIAN JOURNAL OF CHEMISTRY. VOL. 41, 1963 the value for P,(2u) being less than 2. However, this loss is offset by the gain in force due to the density transferred to the overlap region. The net force is +O.l5, which thus makes the 2u orbital a net-binding orbital. The fluorine contribution to the 3u orbital in the limit of separated atoms becomes a singly occupied 2p orbital. Thus it is assigned a corresponding nuclear repulsion of - 1 unit. The molecular orbital of course contains two electrons, but of these one is gained from the hydrogen in bond formation. Thus the penetration for the 3a orbital is to be measured against -1 and due to the large p character of this orbital it is seen to approach this limiting value quite closely. In addition, there is a large overlap force which points to this orbital as being the main source of the binding of the proton in the hydrogen fluoride molecule. An analysis of the 2u and 3u orbitals in terms of Mulliken's method of overlap populations (9) leads to the opposite conclusion. In such an analysis the bonding character of an orbital is related to the magnitude of the overlap population irrespective of where this overlap density is placed. The 2s orbital of fluorine, which is the main constituent of 2u, has a larger numerical overlap with hydrogen than does the 2p, orbital found primarily in the 3u orbital. Thus the 2u orbital is found to have the highest overlap population and is thus rated as being the most strongly bonding orbital. As was pointed out in the discussion of the overlap forces, the 29, overlap, while numerically less than the 2s overlap, concentrates the overlap density along the bond axis, where it exerts a much greater force on the proton than does the 2s overlap density. Thus the force analysis which relates the strength of binding not to the density itself but to the actual force exerted by the density leads to the conclusion that the 3u orbital is the orbital principally responsible for binding the proton in the hydrogen fluoride molecule. The 3u orbital is again a net-binding orbital, as the resultant force is one which draws the two nuclei together. The degenerate 9, orbitals have no overlap with the hydrogen and are thus restricted to a penetration force. Out of the four units of nuclear charge they shield in the separated atoms, only approximately three of these are shielded at the equilibriu~n bond length. These orbitals thus give a net force of -1, which tends to push the twp nuclei apart. The p, orbitals are, therefore, net antibinding. Since Ransil's wave function does not predict a zero resultant force for the hydrogen nucleus the total electronic force does not equal the total nuclear force in this case as can be seen from the totals given in the last row of the table. The totals of the separate force contributions are of sonle interest. The density in the overlap population exerts 11% of the total electronic force which binds the proton in HF. The other 89% of the binding is due entirely to the density on the fluorine nucleus. It should be recalled that at infinite separation P, = 9.00 and OH = Thus in forming the molecule the proton penetrates the fluorine atom density to the extent that effectively only 7.8 of the nuclear charges are screened but there is an accompanying transfer of density to the binding region which gives rise to the overlap forces and thus to the formation of a stable molecule. Table I11 gives the corresponding breakdown for the forces on hydrogen obtained in terms of equivalent orbitals by the electrostatic method. If Ransil's orbitals are transformed into equivalent orbitals, the 2u and 3u orbitals become the 41 and +b orbitals. The 4, and p, orbitals give almost identical results in the two cases (the 9,'s exactly so). Thus to compare the densities we may compare the sum of the contributions of 4, and c$~ with those from 2u and 3f~. The electrostatic density is seen to predict a greater shielding of the fluorine nucleus, actually giving a result of 3.21, which is greater than the 3 units of nuclear charge associated with these two orbitals in the separated atoms. This is obtained by a large XIp value, removing charge from hydrogen and placing it on the fluorine atom in the bonding orbital, and by

9 BADER 4XD JOSES: HYDROGES FLUORIDE TABLE I11 Forces on the proton in HF (electrostatic density) Electronic Nuclear xet Orbital PH OH force force force d,, Total ' having a negative el for the lone pair which removes charge from behind the fluorine riucleus and places it on the side facing the proton. The total overlap force is actually less in the electrostatic case, but the total electronic force is now large enough to achieve electrostatic equilibrium. In Table I\.' we give the breakdown of the forces operating on the fluorine nucleus for TABLE IV Forces on the fluorine nucleus in HF (in units of 9/d2 a.u.) Electronic Orbital A R OF Ps force Electrostatic density $ h 4 I Total S.C.F. density la h Total both the electrostatic and S.C.F. densities. As previously stated, the S.C.F. calculation falls far short of predicting a zero resultant force on the fluorine nucleus. The principal reason for this is the large negative atomic force due to the sp hybridization in the sigma lone pair orbital. In the electrostatic calculation this atomic force term is of positive sign (due to the negative value for el). These figures illustrate the great sensitivity of the force on a heavy nucleus to atomic polarization of the charge density. Any hybridization must be proposed with care as it results in exceedingly large forces acting on the nucleus in question. The Calcatlntion of the ;Ifolec~~lar Properties Determined by the One-electron Density Rlany- of the illo~nents of hydrogen fluoride which can be compared to observed properties of the moleci~le have been calculated by Karplus and Lipsco~nb and co-workers (10-13). They found that wave functions which gave reasonable estimates of energies did not necessarily yield good estimates of these properties. A method of "constrained" variation calculations was suggested by Karplus and Mukherji (3). In this approach it was required that integrals of certain operators agreed with the known values for the corresponding properties. It was hoped that the resultant increase in energy would be more than

10 2260 CAXADIAN JOURNAL OF CHEMISTRY. VOL compensated for by an increase in the accuracy with which other one-electron operator functions could be calculated. The method was applied to hydrogen fluoride, by modifying the 'best limited' wave function obtained by Ransil. Karplus and Mukherji chose as constraining functions the dipole moment and the electric field gradient at the deuteron in DF. They found that the diamagnetic and paramagnetic contributions to the susceptibility were in better agreement with experiment; the calculated diamagnetic proton shielding constant decreased, but no accurate measurement is available for comparison; the force on the hydrogen nucleus approached closer to zero, but the'force on fluorine increased. The significance of their results is clear: for the first time it has been demonstrated that, starting with a given set of basis orbitals, that wave function which gives the best energy does not necessarily predict the best values of all other properties. It was concluded from the anomalous value predicted for the force on the fluorine nucleus that "considerably greater distortion of the fluorine atom wave function than that permitted by limitations in the basis set appears to be required". However, the inner shell orbital was left unchanged in the constrained treatment on the grounds that a variation of its coefficients would result in a large increase in energy and only a small change in the dipole moment and field gradient. The present calculations indicate that the force on fluorine would greatly improve if such a variation was included. It is possible to compare three wave functions using the same basis set of orbitals but derived under the following criteria: (i) the minimum energy criterion of the variation method (Ransil); (ii) the same, with the additional requirements that (P2/rD3) and (Pa) give the correct values (Karplus and Mukherji) ; (iii) the criterion of electrostatic equilibrium, which requires (Pl/rH2) and (Pl/rF2) to be correct, with the dipole moment criterion of (Plr). The properties calculated from these functions are listed in Table V. Two densities are given for the electrostatic method to illustrate how the properties change with lone pair hybridization. On transforming the constrained function it is found that E, = 8.5" compared to 0' and -6" for the best electrostatic densities. These latter three functions all have the correct dipole moment and therefore the same polarity of -1.6, compared to a XIp of 1.38 for the Ransil function. Forces and Deuteron Quadrupole Coupling Constant The forces for the electrostatic case are equal to or close to zero since they were used to fix the densities. The difference between the forces predicted by the other two functions can be explained in terms of the parameters describing them. Since more lone pair hybridization is predicted by the constrained function than by Ransil's, the force on fluorine increases. However, the force on hydrogen is more sensitive to the change of polarity than to this small increase in el and it consequently improves. The reason for this unfavorable change in the lone pair hybridization in the constrained function can be found in the criterion of meeting the deuteron quadrupole coupling constant. Kolker and Karplus (12) have discussed the contribution to this constant from different atomic orbitals on fluorine in relation to the corresponding nuclear contributions. They found that the 1s electrons exactly shield their share of charge while the 2p, electrons only shield about two-thirds of the corresponding nuclear charge. Thus both the field and the field gradient at the nuclei, as measured by the forces and the quadrupole coupling constants respectively, are sensitive to the same parameters. The same factors tend to increase or decrease the electronic contribution to both the field and the field gradient

11

12 2262 CANADIAN JOURNAL OF CHEMISTRY. VOL. 41, 1963 at the proton.,as the lone pair hybridization decreases (other factors being equal) both these quantities decrease. At el = 8.5' the calculated coupling constant reproduces the experimental value, but the force on hydrogen is too large; at el = -6' the force on hydrogen is correct, but the calculated coupling constant is lower than the experimental value. The chosen value of the deuteron quadrupole coupling constant is subject to experimental error and the calculated value is subject to an uncertainty in Q, (12). Thus the deuteron quadrupole coupling constant would seern to be a less reliable criterion of the density distribution than that of electrostatic equilibrium. Diamagnetic Sz~sceptibility The diamagnetic contribution to the susceptibility of hydrogen fluoride (Xd) is determined by (rf2) and therefore by the density in the region of the proton, where r~ is large. Changes in hybridization of the fluorine atoll1 would have little effect on this property. The factors which determine the density at the hydrogen nucleus are the polarity and the orbital exponent of the ls(h) orbital. A low polarity and a small orbital exponent both increase this density and, as a result, xd. The first effect is illustrated by the results in Table V. Both the constrained function and the electrostatic functions have the correct dipole moment and give very good estimates of xd. The S.C.F. density has a low polarity, as shown by the dipole moment, and predicts too large a value for x,. It can be concluded from these susceptibility calculations that the electrostatic function with el = 0 gives the best representation of the density at hydrogen. However, the lack of an error estimate for the experimental determination detracts from the significance of the agreement with _this particular point. Proton Shielding Constant No accurate measurement of the proton chemical shift of gaseous hydrogen fluoride is available, due to experinleiltal difficulties (12). The shielding constant of gaseous methane is 3.01 X10p5 and that of unassociated hydrogen fluoride is within 2 p.p.m. of this value. The results given in Table V indicate that the chemical shift is very small relative to methane. Perhaps this is the reason for the failure of Schneider and co-workers (14) to observe a signal due to gaseous HF relative to methane. Again the constrained density and electrostatic densities give very similar results while the S.C.F. density predicts a larger value. The latter can be attributed to the low polarity predicted by the straight energy calculation. Since the shielding constant measures the amount of density right at the nucleus, while the susceptibility is determined by that density in the general region of the proton, it is less sensitive to polarity than it is to choice of orbital exponent. Dipole,140ment The importance of the dipole moment in determining molecular properties is apparent from these results, particularly if they are compared to those calculated from the simple "Slater atom" and "best atom" orbitals (11, 12) which predict very low dipole mon~ents. A wave function predicting the correct dipole moment for which the hydrogen orbital exponent has been optimized might be expected to always yield good estimates of the diamagnetic susceptibilitj~ and proton-shielding constant. The field and field gradient are more sensitive than these properties to the details of the electron distribution on each nucleus, but still need an accurate estimate of the polarity to yield the correct results. The calculations of the molecular properties support our thesis that the forces and dipole moment provide a good test of the overall density distribution. A density satisfying these criteria will also yield good estimates of the other one-electron properties discussed above.

13 BADER AND JOKES: HYDROGEN FLUORIDE 2263 ACKNOIVLEDGMENTS We wish to thank the Computing Centre at the University of Ottawa for their assistance and machine time on the 1.B.M. 650 used in these calculations. m7e are also grateful to nlr. Peter Sefton for his mathematical assistance and to Professor Martin Karplus for the integrals used in the calculation of the deuteron quadrupole coupling constant. APPESDIX The requirement of orthogonality between the orbitals gives rise to three orthogonality conditions. In these, S 2 and S 3 are the overlaps of h, with the s and p orbitals on fluorine. Bond - lone pair condition: Bond - inner orbital condition: tan el = cos eb + S 2 [? X[cl cos EL, + c2 sin eb + coc3] +~[cis~+c~s~] = 0. Lone pair - inner orbital condition: [3I cl cos e, - cz sin e, = 0. Normalizations of the inner orbital and bond orbital yield the additional equations and 151 X2[1 + ~ 3 ~ +p2+2xp[cos ] eb S2 + sin cb Ss] = 1. The limiting case of no inner shell polarization occurs when cz = 0. It follows from equations [2], [3], and [4] that = = 0 and c, = 1. This set of orbitals was used by Duncan and Pople (15) in their calculations on hydrogen fluoride, with the difference that h was not orthogonal to the inner shell so orbital. As a result the overlap integrals were defined differently in the expression they gave for the bond orthogonality with an equivalent lone pair orbital. Machine computation was employed to calculate the parameters and the resulting forces and dipole moment. The three variable parameters chosen for input data were el, XIp, and ca; then the above equations sufficed to find the remaining parameters. In order to solve for eb, the sin and cos terms in equation [2] were expressed in terms of tan (,/a). Integrals All of the force integrals were obtained from analytical expressions listed in ref. 16. They are given in units of l/d2 and therefore have to be divided by to obtain the absolute values. The symbols sf and h refer to Slater 2s and 1s orbitals on fluorine and on hydrogen respectively. In the overlap force integrals O(&,C$~) refers to the force on nucleus a. Overlap integrals S(so,sf) = S(h,so) = S(h,sf) = S(h,p,) =

14 2264 CAXADIAN JOURNAL OF CHEMISTRY. VOL. 41, 1963 Force integrals Orbital Coeficients For the electrostatic density with el = -6' the orbital coefficients are Go = ~~ sf p Gb = ~~ s' ~ h = ~, ~'+0.104j28p. REFERENCES I. R. F. \V. BADER and G. A. JOKES. Can. J. Chem. 41, 586 (1963). 2. R. F. W. BADER and G. A. JOKES. J. Chem. Phys. 38, 2791 (1963). 3. A. MUKHERJI and M. KARPLUS. J. Chem. Phys. 38, 44 (1963). 4. R. F. IT. BADER and G. A. JOKES. Can. J. Chem. 39, 1253 (1961). 5. B. J. RANSIL. Rev. Mod. Phys. 32, 245 (1960). 6. R. R.1. STERXHEIMER. Phys. Rev. 96, 951 (1954). 7. R. M. STER~HEI~IER. Phys. Rev. 127, 1220 (1962). 8. R. M. STERNHEI~IER. Phys. Rev. 115, 1198 (1959). 9. R. S. MULLIKEN. J. Chem. Phys. 36, 3428 (1962). 10. T. P. D-AS and M. KAKPLUS. J. Chem. Phys. 36, 2275 (1962). 11. R. P. HCRST. M. KARPLCS. and T. P. DAS. 1. Chem. Phvs (1962). ~, 12. H. T. KOLKER and RI. KARPLUS. T. Chem. fivs '( C. %I. KERN and UT. N. LIPS COMB.^ J. Chem. Phys. 37, 260 (1962). 14. \I7. G. SCH~EIDER, H. J. BERNSTEIN, and J. A. POPLF. J. Chem. Phys. 28, 601 (1958). 15. A. B. F. DUNCAN and J. A. POPLE. Trans. Faraday Soc. 49, 217 (1953). 16. C. A. COCLEON. Proc. Cambridge Phil. Soc. 38, 210 (1942).