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The role of electron density and potential at the bond midpoint in homonuclear diatomic binding
Journal of Molecular Structure (Theochem), 209 (1990) 433-437 Elsevier Science Publishers B.V., Amsterdam
433
THE ROLE OF ELECTRON DENSITY AND POTENTIAL AT THE BOND MIDPOINT IN HOMONUCLEAR DIATOMIC BINDING
N.H. MARCH and P.M. KOZLOWSKI Theoretical Chemistry Department, University of Oxford, 5 South Parks Road, Oxford OX1 3UB (Gt. Britain) F. PERROT Centre d%tudes de Lime&Valenton,
94195 Villeneuve, St. Georges Cede+ (France)
(Received 26 February 1990)
ABSTRACT Motivated by a recent study of molecular binding in cold dense plasmas, a density-gradient expansion of the single-particle kinetic energy is utilized to express the chemical potential of homonuclear diatomic molecules at equilibrium separation in terms of electron density and potential at the midpoint of the chemical bond. By utilizing known approximate relations between molecular-energy terms at equilibrium, it is argued that one can characterize D/N2 by the same quantities at the bond midpoint, D being the dissociation energy and N the total number of electrons in the molecule.
INTRODUCTION
It has been known for a long time that the total energy of a homonuclear molecule built from all but the very lightest atoms, at its equilibrium separation, is quite close to the sum of the energies of the separated atoms. Thus, direct calculation of the dissociation energy by calculating the molecular energy and subtracting the total energy of the separated atoms is, at best, a hazardous way to proceed. Mucci and March [l] therefore proposed a quitre different line of attack; namely, to make a merit out of Teller’s theorem [ 2 1, This theorem asserts that molecular binding is not possible in a local density, Thomas-Fermi, theory. Therefore, Mucci and March argued for an intimate connection between electron-density gradients in the molecular charge clond at equilibrium, and molecular dissociation energy D. Their proposal was subsequently refined and extended [ 3,4] and their general conclusion substantiated by quantum-chemical calculations. Briefly, these results [ 3,4] exhibited a definite correlation between D/W and TZ in a series of light molecules, N being the total number 0166-1280/90/$03.50
0 1990 -
Elsevier Science Publishers B.V.
434
of electrons and T2 the lowest order inhomogeneity correction to the ThomasFermi kinetic energy T,,, where To =ck
{p(r)}5’3dr
3h2 3 2’3 Ck=lOm 0 8a
and
fi2 -(@)2dr T -2-72m
Ip
The next-order correction, T4, is also known from the work of Hodges [ 51 and can be added to the ensuing formulae should numerical accuracy eventually necessitate that. We have been motivated to reopen the above programme as a result of recent progress made in treating molecular binding in cold dense plasmas [ 61. Briefly, analytical progress proved possible, within the same density gradient framework involving T,,, T2 and T4, because of the existence of a small parameter; namely the ratio of charge displaced in the plasma at the centre of the homonuclear bond to the mean charge density in the plasma in the absence of the molecule. Unfortunately, no such small parameter exists in free-space homonuclear diatomic molecules. CHEMICAL POTENTIAL AND BOND MIDPOINT PROPERTIES
Notwithstanding the absence of a small parameter our purpose in the present work is to show that, just as in ref. [6] for dense plasmas, the electron density and its low-order gradients, plus the potential, at the midpoint of the chemical bond, play a major role in interpreting molecular formation and in characterizing bond dissociation energies. The argument below proceeds in two stages. First, we give the usual Euler equation of density functional theory explicitly to order To+ T,: as already mentioned the inclusion of T4 is straightforward but lengthy, should it eventually be required for fully quantitative results. Secondly, some known approximate relations between molecular-energy terms at equilibrium are invoked. The Euler equation of density functional theory is essentially a statement of the constancy of the chemical potential p (R) at every point in the molecular charge cloud, with electron density p (r,R) at nuclear separation R. Of course the kinetic contributions coming from To+ T2 + .... and the corresponding potential-energy piece, vary individually from point to point in the electronic distribution. The constancy of the sum of these contributions underlies the condition that the electronic redistribution necessitated by bond formation is complete. Then the Euler equation
435
6E
ST,
’
=----+Vv,(r)
+ V,(r) +L @(r)
ap(r)
(3)
is formally exact in the many-electron problem. Here, T, is the single-particle kinetic energy, which can alternatively be handled using Slater-Kohn-Sham [ 7,8] single-particle equations if desired. The terms VN(r) and V,(r) represent the nuclear potential and the potential created by the charge density p (r,R). Finally, E, [p] is the, as yet unknown, exchange and correlation functional. In the following, T, is replaced by the approximation To + T2, as given in eqns. (1) and (Z), while the exchange and correlation contribution V,,(r) =6&/6p(r) to the one-body potential energy V(r) (4)
V(r) = VN(r) + Ve(r) + V,,(r)
is assumed to have the local density form V,, (p (I:) ). Then eqn. (3) takes the explicit form [ 91
--_--
+V(r)
(5)
the first set of terms omitted from eqn. (5) coming from T4 given in ref. 5. Our work on molecular binding in plasmas [ 61 in the same density-gradient framework now motivates the use of eqn. (5 ) at the midpoint of the bond in homonuclear diatomics to evaluate the chemical potential, p, at the equilibrium separation R = R,. Taking the origin r = 0 at the midpoint of the bond, one evidently has, up to and including order T2:
Re r=O
R. r=O
Since, from eqn. (4)) for molecules of like atoms with atomic number 2,
WW,) =
-F+
Ve(0J-L)+ Vxc(p(O,R,) 1
(7)
e
the sum of the first two terms on the right-hand side being the electrostatic potential at r =O, the local density approximation assumed for V,, has the immediate consequence that only this electrostatic potential plus the density, both evaluated at the midpoint of the chemical bond, are involved in determining the chemical potential p(R,). We need only to add here that the midpoint of the bond is a favourable place for evaluating 6Ts/6p(r) in the formally exact eqn. (3 ) because, by symmetry, the density is flat at the midpoint, along the bond axis, which helps to validate the density gradient expansion invoked in writing eqn. (6).
436 USE OF RELATION BETWEEN MOLECULAR-ENERGY
TERMS
Let us turn at this point to the second stage of the argument referred to above. Various relations between molecular-energy contributions have been proposed, the work of Politzer [lo] being especially prominent. To utilize this body of knowledge, let us next multiply eqn. (5) by p( r ) and integrate over the whole of space. Then invoking again the constancy of p leads to
where U,, is the electron-nuclear potential energy while U,, is the electrostatic self-energy of the molecular charge density. The term in eqn. (5) involving the Laplacian of p integrates to zero through space. In what follows we neglect V,,; obviously in fully quantitative work this term can be incorporated at the local density level as above. Furthermore, let us utilize the fact that, at the order to which we have worked, the total kinetic energy, T, is the equal to the sum To+ Tz and hence eqn. (8) can be rewritten as
Np=fT-;T2
+ U,, +2U,,
At equilibrium, Mucci and March [ 111 and Politzer [lo] of the so-called scaling relations. K?,+2u,,
T
=-- 5 3
have confirmed one
(10)
by direct appeal to numerical self-consistent field calculations on a whole set of light molecules. Thus, utilizing eqn. (10) in eqn. (9) strongly suggests that Np (R,) should correlate with T2(R,), although further work is needed before it can be assumed that ,u(R,) z - (2/3) Tz (R,) /N is sufficiently precise for quantitative work. At least qualitative support for this correlation comes from Fig. 1 of Lee and Ghosh [ 41, where large values of Tz frequently, though not exclusively, involve molecules containing the most electronegative elements 0 and F, when one recalls the connection of chemical potential with electronegativity [ 12,131. DISCUSSION AND SUMMARY
In summary, eqn. (6) points strongly to the fact that, just as we found [6] previously for the homonuclear molecules Na, and Bez in cold dense plasmas, the electron density and potential at the bond midpoint can be used to characterize molecular properties at equilibrium. If we then add to eqn. (6) the
437
more approximate considerations based on combining eqns. (9) and (lo), the further conclusion is that p(R,) should correlate with Z’, (R,). Since the work of refs. [ 31 and [ 41, following that of Mucci and March [ 11, correlates D/N2 with T2 (R,) , it is expected that an intimate relation exists between D and the bond midpoint properties p (O,R,) and V( O,R,) . However, we reiterate the caution given in ref. [ 31 that there is no claim that the properties of an equilibrium molecule, such as T2 (R, ), can lead to an absolute determination of D. ACKNOWLEDGEMENTS
Two of us (N.H.M. and F.P.) acknowledge that the motivation for their contribution to the present study was their attendance at the 1989 Workshop at ITP, UCSB on atoms and ions in strongly coupled plasmas. Financial support from the National Science Foundation under Grant No. PHY82-17853, supplemented by funds from the National Aeronautics and Space Administration, is also acknowledged.
REFERENCES 1
2 3 4 5 6 I 8 9 10 11 12 13
J.F. Mucci and N.H. March, J. Chem. Phys., 78 (1983) 6187. E. Teller, Rev. Mod. Phys., 34 (1962) 627. N.L. Allen, C.G. West, D.L. Cooper, P.J. Grout and N.H. March, J. Chem. Phys., 83 (1985) 4562. C. Lee and S.K. Ghosh, Phys. Rev., Ser. A, 33 (1986) 3506. C.H. Hodges, Can. J. Phys., 51 (1973) 1428. F. Perrot and N.H. March, Phys. Rev. A., 1990, in press. J.C. Slater, Phys. Rev., 81 (1951) 385. W. Kohn and L.J. Sham, Phys. Rev., Ser. A, 140 (1965) 1133. See, for example, N.H. March, J. Phys. Chem., 86 (1982) 2262. P. Politzer, J. Chem. Phys., 64 (1976) 4239. J.F. Mucci and N.H. March, J. Chem. Phys., 71 (1979) 5270. R.T. Sanderson, Science, 121 (1955) 207; Chemical Bonds and Bond Energy, Academic Press, New York, 1978. R.G. Parr, R.A. Donnelly, M. Levy and W.E. Palke, J. Chem. Phys., 68 (1978) 3801.
433
THE ROLE OF ELECTRON DENSITY AND POTENTIAL AT THE BOND MIDPOINT IN HOMONUCLEAR DIATOMIC BINDING
N.H. MARCH and P.M. KOZLOWSKI Theoretical Chemistry Department, University of Oxford, 5 South Parks Road, Oxford OX1 3UB (Gt. Britain) F. PERROT Centre d%tudes de Lime&Valenton,
94195 Villeneuve, St. Georges Cede+ (France)
(Received 26 February 1990)
ABSTRACT Motivated by a recent study of molecular binding in cold dense plasmas, a density-gradient expansion of the single-particle kinetic energy is utilized to express the chemical potential of homonuclear diatomic molecules at equilibrium separation in terms of electron density and potential at the midpoint of the chemical bond. By utilizing known approximate relations between molecular-energy terms at equilibrium, it is argued that one can characterize D/N2 by the same quantities at the bond midpoint, D being the dissociation energy and N the total number of electrons in the molecule.
INTRODUCTION
It has been known for a long time that the total energy of a homonuclear molecule built from all but the very lightest atoms, at its equilibrium separation, is quite close to the sum of the energies of the separated atoms. Thus, direct calculation of the dissociation energy by calculating the molecular energy and subtracting the total energy of the separated atoms is, at best, a hazardous way to proceed. Mucci and March [l] therefore proposed a quitre different line of attack; namely, to make a merit out of Teller’s theorem [ 2 1, This theorem asserts that molecular binding is not possible in a local density, Thomas-Fermi, theory. Therefore, Mucci and March argued for an intimate connection between electron-density gradients in the molecular charge clond at equilibrium, and molecular dissociation energy D. Their proposal was subsequently refined and extended [ 3,4] and their general conclusion substantiated by quantum-chemical calculations. Briefly, these results [ 3,4] exhibited a definite correlation between D/W and TZ in a series of light molecules, N being the total number 0166-1280/90/$03.50
0 1990 -
Elsevier Science Publishers B.V.
434
of electrons and T2 the lowest order inhomogeneity correction to the ThomasFermi kinetic energy T,,, where To =ck
{p(r)}5’3dr
3h2 3 2’3 Ck=lOm 0 8a
and
fi2 -(@)2dr T -2-72m
Ip
The next-order correction, T4, is also known from the work of Hodges [ 51 and can be added to the ensuing formulae should numerical accuracy eventually necessitate that. We have been motivated to reopen the above programme as a result of recent progress made in treating molecular binding in cold dense plasmas [ 61. Briefly, analytical progress proved possible, within the same density gradient framework involving T,,, T2 and T4, because of the existence of a small parameter; namely the ratio of charge displaced in the plasma at the centre of the homonuclear bond to the mean charge density in the plasma in the absence of the molecule. Unfortunately, no such small parameter exists in free-space homonuclear diatomic molecules. CHEMICAL POTENTIAL AND BOND MIDPOINT PROPERTIES
Notwithstanding the absence of a small parameter our purpose in the present work is to show that, just as in ref. [6] for dense plasmas, the electron density and its low-order gradients, plus the potential, at the midpoint of the chemical bond, play a major role in interpreting molecular formation and in characterizing bond dissociation energies. The argument below proceeds in two stages. First, we give the usual Euler equation of density functional theory explicitly to order To+ T,: as already mentioned the inclusion of T4 is straightforward but lengthy, should it eventually be required for fully quantitative results. Secondly, some known approximate relations between molecular-energy terms at equilibrium are invoked. The Euler equation of density functional theory is essentially a statement of the constancy of the chemical potential p (R) at every point in the molecular charge cloud, with electron density p (r,R) at nuclear separation R. Of course the kinetic contributions coming from To+ T2 + .... and the corresponding potential-energy piece, vary individually from point to point in the electronic distribution. The constancy of the sum of these contributions underlies the condition that the electronic redistribution necessitated by bond formation is complete. Then the Euler equation
435
6E
ST,
’
=----+Vv,(r)
+ V,(r) +L @(r)
ap(r)
(3)
is formally exact in the many-electron problem. Here, T, is the single-particle kinetic energy, which can alternatively be handled using Slater-Kohn-Sham [ 7,8] single-particle equations if desired. The terms VN(r) and V,(r) represent the nuclear potential and the potential created by the charge density p (r,R). Finally, E, [p] is the, as yet unknown, exchange and correlation functional. In the following, T, is replaced by the approximation To + T2, as given in eqns. (1) and (Z), while the exchange and correlation contribution V,,(r) =6&/6p(r) to the one-body potential energy V(r) (4)
V(r) = VN(r) + Ve(r) + V,,(r)
is assumed to have the local density form V,, (p (I:) ). Then eqn. (3) takes the explicit form [ 91
--_--
+V(r)
(5)
the first set of terms omitted from eqn. (5) coming from T4 given in ref. 5. Our work on molecular binding in plasmas [ 61 in the same density-gradient framework now motivates the use of eqn. (5 ) at the midpoint of the bond in homonuclear diatomics to evaluate the chemical potential, p, at the equilibrium separation R = R,. Taking the origin r = 0 at the midpoint of the bond, one evidently has, up to and including order T2:
Re r=O
R. r=O
Since, from eqn. (4)) for molecules of like atoms with atomic number 2,
WW,) =
-F+
Ve(0J-L)+ Vxc(p(O,R,) 1
(7)
e
the sum of the first two terms on the right-hand side being the electrostatic potential at r =O, the local density approximation assumed for V,, has the immediate consequence that only this electrostatic potential plus the density, both evaluated at the midpoint of the chemical bond, are involved in determining the chemical potential p(R,). We need only to add here that the midpoint of the bond is a favourable place for evaluating 6Ts/6p(r) in the formally exact eqn. (3 ) because, by symmetry, the density is flat at the midpoint, along the bond axis, which helps to validate the density gradient expansion invoked in writing eqn. (6).
436 USE OF RELATION BETWEEN MOLECULAR-ENERGY
TERMS
Let us turn at this point to the second stage of the argument referred to above. Various relations between molecular-energy contributions have been proposed, the work of Politzer [lo] being especially prominent. To utilize this body of knowledge, let us next multiply eqn. (5) by p( r ) and integrate over the whole of space. Then invoking again the constancy of p leads to
where U,, is the electron-nuclear potential energy while U,, is the electrostatic self-energy of the molecular charge density. The term in eqn. (5) involving the Laplacian of p integrates to zero through space. In what follows we neglect V,,; obviously in fully quantitative work this term can be incorporated at the local density level as above. Furthermore, let us utilize the fact that, at the order to which we have worked, the total kinetic energy, T, is the equal to the sum To+ Tz and hence eqn. (8) can be rewritten as
Np=fT-;T2
+ U,, +2U,,
At equilibrium, Mucci and March [ 111 and Politzer [lo] of the so-called scaling relations. K?,+2u,,
T
=-- 5 3
have confirmed one
(10)
by direct appeal to numerical self-consistent field calculations on a whole set of light molecules. Thus, utilizing eqn. (10) in eqn. (9) strongly suggests that Np (R,) should correlate with T2(R,), although further work is needed before it can be assumed that ,u(R,) z - (2/3) Tz (R,) /N is sufficiently precise for quantitative work. At least qualitative support for this correlation comes from Fig. 1 of Lee and Ghosh [ 41, where large values of Tz frequently, though not exclusively, involve molecules containing the most electronegative elements 0 and F, when one recalls the connection of chemical potential with electronegativity [ 12,131. DISCUSSION AND SUMMARY
In summary, eqn. (6) points strongly to the fact that, just as we found [6] previously for the homonuclear molecules Na, and Bez in cold dense plasmas, the electron density and potential at the bond midpoint can be used to characterize molecular properties at equilibrium. If we then add to eqn. (6) the
437
more approximate considerations based on combining eqns. (9) and (lo), the further conclusion is that p(R,) should correlate with Z’, (R,). Since the work of refs. [ 31 and [ 41, following that of Mucci and March [ 11, correlates D/N2 with T2 (R,) , it is expected that an intimate relation exists between D and the bond midpoint properties p (O,R,) and V( O,R,) . However, we reiterate the caution given in ref. [ 31 that there is no claim that the properties of an equilibrium molecule, such as T2 (R, ), can lead to an absolute determination of D. ACKNOWLEDGEMENTS
Two of us (N.H.M. and F.P.) acknowledge that the motivation for their contribution to the present study was their attendance at the 1989 Workshop at ITP, UCSB on atoms and ions in strongly coupled plasmas. Financial support from the National Science Foundation under Grant No. PHY82-17853, supplemented by funds from the National Aeronautics and Space Administration, is also acknowledged.
REFERENCES 1
2 3 4 5 6 I 8 9 10 11 12 13
J.F. Mucci and N.H. March, J. Chem. Phys., 78 (1983) 6187. E. Teller, Rev. Mod. Phys., 34 (1962) 627. N.L. Allen, C.G. West, D.L. Cooper, P.J. Grout and N.H. March, J. Chem. Phys., 83 (1985) 4562. C. Lee and S.K. Ghosh, Phys. Rev., Ser. A, 33 (1986) 3506. C.H. Hodges, Can. J. Phys., 51 (1973) 1428. F. Perrot and N.H. March, Phys. Rev. A., 1990, in press. J.C. Slater, Phys. Rev., 81 (1951) 385. W. Kohn and L.J. Sham, Phys. Rev., Ser. A, 140 (1965) 1133. See, for example, N.H. March, J. Phys. Chem., 86 (1982) 2262. P. Politzer, J. Chem. Phys., 64 (1976) 4239. J.F. Mucci and N.H. March, J. Chem. Phys., 71 (1979) 5270. R.T. Sanderson, Science, 121 (1955) 207; Chemical Bonds and Bond Energy, Academic Press, New York, 1978. R.G. Parr, R.A. Donnelly, M. Levy and W.E. Palke, J. Chem. Phys., 68 (1978) 3801.