PETER POLITZER
1174
SnCL lie on either side. Also, the reduced density of BiC13, a pyramidal molecule like NH3, lies close to that of NH3. (The data for BiBr3 would lie just below those for BiC4. They were omitted for reasons of clarity.) A linear dependence of I p - pol on (T, - T)"8
was found graphically for both the liquid and vapor down to a temperature about 50" below T,. This is a much greater temperature range than the SO" found for N20.13 (13) D. Cook, Trans. Faraday Soc., 49, 716 (1953).
A Study of the Bonding in the Hydrogen Molecule'
by Peter Politzer Quantum Chemistry Group, Chemislry Department, Indiana Uniuersity, Bloomington, Indiana (Received October 81,1966)
An analysis has been made of a number of hydrogen molecule wave functions in terms of the electrostatic forces operating within the molecule and the electronic density distributions. No correlation was found between the calculated energies and the degrees to which the functions satisfy the requirement that there be a zero resultant force upon each nucleus. Similarly, the energies do not directly reflect the extent to which charge density is concentrated at the midpoint of the bond in each case. Surprisingly, however, it was found from the most accurate wave function studied that the formation of the molecule from the free atoms involves a nearly constant increase in charge density at all points along the axis from one nucleus to the other.
In the hope of obtaining a better understanding of the nature of covalent bonding, a comparative study has been carried out of a number of different wave functions which have been proposed for the hydrogen molecule. These functions have been analyzed in terms of two properties: the attractive force which the electrons are found in each case to exert upon either nucleus and the manner in which the electronic distribution in the molecule differs from the hypothetical situation of two noninteracting hydrogen atoms at the same separation. According to the Hellmann-Feynman theorem,293 the electronic attractive force upon a nucleus can be calculated by classical electrostatics, using, however, the quantum mechanical electronic distribution. In the case of the hydrogen molecule, if the wave function describing the two electrons is $(r,, rz), then the reThe Journal of PhyskaE Chemistry
sulting force component along the molecular axis which acts upon nucleus A is given by
where p = 2,f$(rl, rz)$*(rcl, r2) drz, r A is the radial distance from nucleus A, and BA is measured away from the molecular axis, a t the nucleus. The right side of eq 1 can generally be expanded as a sum of integrals over the particular basis orbitals used in the wave function #(rl, rz); there are formulas available (1) This work was supported by grants from the National Science Foundation and the U. 8.Air Force Office of Scientific Research. (2) R. P. Feynman, Phys. Reu., 56, 340 (1939). (3)An interesting discussion of the Hellmann-Feynman theorem and some of its applications has been given by L. Salem, Ann. Phya. (Paris), 8 , 169 (1963).
BONDING IN
THE
1175
HYDROGEN MOLECULE
Table I : Summary of Force Analyses of Various Ht Wave Functions (Values Given Are in Atomic Units) Function
Wang
+ Weinbaum $‘ = 1 ~ ~ + 1 1s ~~ ~ +1 +
$ = 1 ~ ~ 1 1s ~~ ~ 1 s ~
% imbalance
Energy
Attraotion
Repulsion
Ref
- 1.13905
0.4033
0.5073
20.5
a
-1.14796
0.4246
0.4888
13.1
b
-1.1485
0.5011
0.4987
0.441
C
-1.1501
0.4882
0.4904
0.449
d
-1.13211
0.5326
0.5102
4.39
e
- 1.15661
0.4987
0.5102
2.23
e
-1.13349
0.5151
0.5088
1.24
e
- 1.15919
0.5185
0.5088
1.91
e
-1.13357
0.5127’
0.5102
0.490
f
-1.169785
0.5121
0.5102
0.372
9
-1,17444
0.5098
0,5098‘
0.000
i
s ~
U(~SA~SA ~SB~SB)
Rosen
+ 1SA + X2pOA Floating orbita.1 $‘ = PAPB
PBPA,
PA =
(based on Weinbaum function) LCAO-SCF (all exponents equal) LCAO-SCF-CI (all exponents equal) LCAO-SCF (exponents varied) LCAO-SCF-CI (exponents varied) SCF (nine basis functions of type Amfin exp( - 0.75X)) Hagstrom (four-term truncated natural orbital expansion of 33-term CI function) Experimental
S. C. Wang, Phys. Rev., 31, 579 (1928).
‘S. Weinbaum, J. Chem. Phys., 1, 593 (1933).
The data used in this work were taken from improved unpublished calculations by Shull and Lowdin; see H. Shull, J. A m . Chem. Soc., 82, 1287 (1960). N. Rosen, Phys. H. Shull and D. D. Ebbing, J . Chem. Phys., 28, 866 (1958). S. Fraga and B. J. Ransil, ibid., 35, 1967 Rev., 38, 2099 (1931). (1961). J. Goodisman, ibid., 39,2397 (1963). S. Hagstrom and H. Shull, Rev. Mod. Phys., 35,624 (1963). * This value is based G. Herzberg, “Spectra of Diatomic Moleon R = 1.4006 au, given by G. Herzberg and L. L. Howe, Can. J. Phys., 37,036 (1959). cules,” D. Van Nostrand Co., Inc., Princeton, N. J., 1950.
’
for the evaluation of such integral^.^" All computations in this work were carried out on a CDC 3600 computer. The results of these calculations for a number of different Hzwave functions are given in Table I; there is also included one result which was taken from previously published work (see ref f in Table I).’ At equilibrium, the net attractive force upon a nucleus due t,o the electrons should be exactly balanced by the repulsion due to the other nuclei; the extent to which this requirement is satisfied is indicated in each case. If the exact Hz wave function were used, there would of course be a perfect balance of the forces on each nucleus. However, there are also certain approximate wave functions which have the property of showing a balance at the calculated bond among these are the “floating” functions, in which the atomic orbitals have been detached from their nuclei, and Hartree-Fock functions. These expectations are confirmed by the results obtained for the floating orbital func-
’
tion, for which the forces are seen to balance to within the precision of the functional parameters, and the SCF (self-consistent field) functions, these being close approximations to the Hartree-Fock.lS It must (4) R. F. W.Bader and G. A. Jones, Can. J. Chem., 39, 1253 (1961). (5) S. Ehrenson and P. E. Phillipson, J . Chem. Phys., 34, 1224 (1961). (6) M.J. Stephen, Proc. Cambridge Phil. SOC.,57, 348 (1961). (7) Some of these forces were also computed by T. P. Das and R. Bersohn [Phya. Reu., 115, 897 (1959)],who examined a number of Ha wave functiom in terms of various electric and magnetic properties. It should be noted that they used a uniform bond length of 1.40 au. (8) A. C. Hurley, Proc. Roy. Sac. (London), A226, 170, 179, 193 (1964). (9) G.G.Hall, Phil. Mag., 6,249 (1961). (10) J. 0. Hirschfelder and C . A. Coulson, J . Chem. Phya., 36, 941 (1962). (11) C.A. Coulson and A. C. Hurley, ibid., 37, 448 (1962). (12) C.W.Kern and M. Karplus, ibid., 40, 1374 (1964). (13) Functions which include incomplete configuration interaction will not necessarily show a balance of forces, even though they may be based upon SCF solutions. This is discussed in ref f of Table I.
Volume 70, Number 4 April 1088
PETERPOLITZER
1176
be noted, however, that this predicted balance of forces in the case of certain approximate functions is at the calculated bond length; there may be significant imbalance when the repulsion is given its actual value a t the true bond length. It is evident that no general correlation exists between the accuracies of the forces and the energies corresponding to the various wave functions. This is not surprising; the force depends upon the oneelectron density, p, and this is not directly related to the energy. For instance, Hartree-Fock functions give good one-electron d e n s i t i e ~ ~despite ~ ' ~ their relatively poor energies. Another striking example of this lack of correlation is the Rosen function, which has only a fair energy but achieves a nearly exact balance of forces. This function gives an unexpectedly good charge distribution, as shall be discussed later in this paper. The effect of bond formation upon the electronic distribution in a system of atoms can be studied by means of the function 6(r), which was introduced by Roux and co-~orkersl~-'~ and has been discussed at length by Rosenfeld.I* It is defined by
6(r) = p(r> -
pF(r>
0.14 I-
0.12
-.
0.10
--
0.08
-n
0'0°
I
(2) Nucleus
R/2
Distance along axis (unite of R ) . Here p(r) is the electronic density of the molecule at a point r, as defined above, and pF(r) is the density which Figure 1. Variation of 6 functions with distance along molecular would result if the atoms could be brought together axis. The distance is measured in units of the respective bond lengths. Ha wave functions represented are: A, Hagstrom; with no perturbing effect upon each other. Since B, Kolos-Roothaan; C, L C A O S C F (variable exponents); 6(r) is a relatively small difference between two larger D, Rosen; E, floating orbital; F, Coulson; G, Weinbaum; quantities, the values assigned to the latter are obH, Wang; and I, Heitler-London. Only the Kolos-Roothaan viously critical; in the present work, pF(r) could and the Heitler-London 6 functions are shown fortunately be stated exactly, as the sum of the den& to the left of the nucleus; the others have essentially the same form as the Kolos-Roothaan. ties of two hydrogen atom 1s orbitals with effective 2 equal to 1.00. Figure 1 shows the 6 function along the molecular six functions are still fairly uncomplicated in form and axis for a group of Hz wave functions. The free atom can readily be given physical interpretations, but they orbitals used in obtaining pF were in each case taken have orbital exponents greater than 1. This can be to be separated by the particular calculated bond length. The Hagstrom and the K ~ l o ~ - R o o t h a a n ~viewed ~ ~ ~ ~ ZM reflecting "promotion" of the atom prior functions which are represented here are the seventerm natural orbital expansions, which have energies (14) M.Cohen and A. Dslgarno, Proc. Phys. SOC.(London), 7 7 , 748 of -1.1713 and -1.1717 au, respectively. These (1961). values differ only slightly from the experimental energy (15) M. Row, 9. Besnainou, and R. Daudel, J. Chim. Phya., 218, of - 1.1744 au, and the Kolos-Roothaan function shall 939 (1956). therefore be taken to be a very close representation (16) M.Row, ibid., 754 (1958). (17) M. Row, M. Cornille, and L. Burnelle, J. Chem. Phys., 37, of the true wave function. 933 (1962). A certain pattern can be discerned in Figure 1. (IS),,J. L. J. Rosenfeld, "Electron Distribution and Chemical BindThe Heitler-London function, in which the orbital ing, Technical Report No. 13, Institute of Theoretical Physics, University of Stockholm, April 3963. exponents (effective nuclear charges) retain the 1.00 (19) W. Kolos and C. C. J. Roothaan, Rev. Mod. Phys., 32, 219 values of the free atoms, presents the usual simple (1960). qualitative description of the chemical bond-a pile(20) E. R. Davidson and L. L. Jones, J. Chem. Phys., 37, 2966 up of charge in the internuclear region. The next (1962). ~~~
The Journal of Physical Chembtry
~
BONDING IN
THE
HYDROGEN MOLECULE
1177
to bond formation.21 Accordingly, these functions show a great increase in electronic density a t the nuclei and a lesser increase in the region between them. Finally, with the very accurate Hagstrom and KolosRoothaan functions, in which nearly all electronic correlation has been taken into account, the internuclear density has been brought up to essentially the same level as at the nuclei.22 This is a surprising and significant result. The usual view of chemical bonding as involving a shift of electronic density into the region between the nuclei is clearly only part of the story-at least in the hydrogen molecule. There is a roughly equal buildup right a t each nucleus, so that the total result is a nearly constant increase in charge density at all points along the axis from one nucleus to the other. I n Table I1 are presented the dissociation energies corresponding to the various H2 wave functions and the respective values of 6(r) at the midpoint of the bond. It is seen that there is no direct relation between the two. This conclusion is contrary to a previous suggestionI8 (which was based upon an investigation of only three Hz wave functions), but it seems quite reasonable once it is realized that the changes occurring near the nuclei during bond formation are as significant as those taking place in the internuclear region. Table I1 : Comparison of Bond Properties for Various Hz Wave Functions De
t
6
Function
ev
(midpoint)
Heitler-L~ndon"~ Coulsonb (MO, variable exponent) L C A O S C F (exponents varied) Wang Weinbaum Rosen Floating orbital Hagstrom Kolos-Roothaan" Experimental
-3.156 -3.488
0.0181 0.0883
-3.632
0.1107
-3.784 -4.026 -4.041 -4.084 -4.662 -4.672 -4.747
0.0687 0.0802 0.1035 0.0953 0.1164 0.1154
W. Heitler and F. London, Z. Physik, 44, 455 (1927); Y. C. A. Coulson, Trans. Faraday Sugiura, ibid., 45, 484 (1927). Soc., 33, 1479 (1937). 'See ref 19 and 20. a
Figure 2 shows a contour diagram of the 6 function in two dimensions for the Kolos-Roothaan four-term wave function. This provides a very accurate picture of the change in electronic distribution which accompanies the formation of the hydrogen molecule.
Nucleus
Midpoint
Figure 2. Two-dimensional 6 function for Kolos-Roothaan Hz wave function. Because of the symmetry of the molecule, the 6 function a t any point in space can be determined simply by rotating the above contours about the internuclear axis.
The curves plotted in Figure 1, as well as the force analyses discussed earlier, suggest that the LCAOSCF and the Rosen functions give particularly good electronic density distribution^.^^ This supposition was confirmed by comparing two-dimensional contour diagrams for the various 6 functions. It seems, therefore, that the simple picture of the hydrogen molecule as being made up of two atoms slightly polarized toward each other is a reasonably valid one. This model is deficient of course in not taking proper account of the repulsive interactions between the electrons. It is not difficult to see one source of this deficiency, a t least in the case of the Rosen function.2' This function polarizes the atomic charge distributions directly toward each other along the molecular axis, whereas it is to be expected that the repulsion between the electrons would cause these polarized portions to shift somewhat away from each other and hence from the axis. This expectation is confirmed by the very significant effect which is observed when terms of ?r symmetry are added to the Hagstrom and Kolos-Roothaan natural orbital expansions; the energy is improved by nearly 0.3 ev (about 0.0109 tu).^^ Acknowledgments. The author is grateful to Pro(21) K. Ruedenberg, Rev. Mod. Phys., 34, 326 (1962). See also 6(r) for the five-term James-Coolidge function in ref
(22)
15.
(23) This is not surprising in the case of the SCF function, since it is known that Hartree-Fock solutions give good results for one-electron properties. (See ref 9 and 14.) (24) This deficiency is inherent in the SCF function, which, by its very nature, cannot be energetically improved once it has reached the Hartree-Fock limit. (25) J. 0. Hirschfelder and J. W. Linnett [J.Chem. Phys., 18, 130 (1950)l proposed a modified version of the Rosen function which would take account of this character," but there is some question about the accuracy of their numerical results (see ref d in Table I), ' I T
Volume 70,Number 4 April 1968
1178
J. W. WARDAND H. W. HABGOOD
fessor Harrison Shull for stimulating discussions and for his kind support. He also wishes to thank Dr.
Darrell D. Ebbing and Dr. Norman T. Huff for their very useful comments regarding this work.
The Infrared Spectra of Carbon Dioxide Adsorbed on Zeolite X
by J. W. Ward' and H. W. Habgood Contribution No. 323from the Research C o u d l of Alberta, Edmonton,Alberta, Canada (Received October 21, 1966)
Spectra of carbon dioxide adsorbed on the Ca, Sr, and Ba forms of zeolite X showed no evidence of bent, or carbonate-like, structures such as were previously found for carbon dioxide adsorbed on LiX, NaX, and KX. This difference is believed to be due to the absence, in zeolites containing divalent cations, of any cations in the highly exposed typeI11 sites. Carbon dioxide on MgX did show some carbonate-like bands in the 17501300-cm-l region but there was evidence that only partial replacement of monovalent cations had been achieved. The spectra of carbon dioxide on both group I-A and group II-A zeolites showed a strong band between 2375 and 2350 em-', the frequency being higher the stronger the electrical field of the cation, which is believed to be due to a linear species held by an ion-dipole interaction. This band was accompanied (in most cases) by two pairs of weaker side bands symmetrically spaced about the central band.
A previous communication from this laboratory2 described the infrared spectra of carbon dioxide adsorbed at low coverages on the lithium, sodium, and potassium forms of zeolite X. These spectra each included a band near 2350 cm-l attributed to carbon dioxide adsorbed in a linear or near-linear configuration onto the exchangeable cations and at least two pairs of bands in the region 1750-1250 cm-l ascribed to carbon dioxide chemisorbed in a bent configuration onto a surface oxygen to give a carbonate-like structure. The bands near 2350 cm-l showed a slight dependence of frequency upon the nature of the cation while those ascribed to the bent forms varied widely from one cation to another. These observations have now been extended to the alkaline earth ion-exchanged forms of the zeolite and also to higher carbon dioxide coverages, and a number of unexpected results have been found. The Journal of Physical Chemistry
Experimental Section The procedures were similar to those reported previously. The various zeolites were prepared by ion exchange of the sodium form (Linde Lot No. 13916) using 5% solutions of the appropriate chlorides. LiX, KX, CaX, SrX, and BaX were essentially 100% exchanged while in the magnesium zeolite only 70% of the sodium was exchanged and the amount of magnesium taken up corresponded to 35% of the magnesium being added as MgOH+ rather than as Mg2+. A sample of RbX was also prepared but because of the small amount of material available, the degree of exchange was not determined. The pellets were outgassed at 450-500", and doses of carbon dioxide were added at room temperature up (1) Research Council of Alberta Postdoctoral Fellow, 1962-1963. (2) L. Bertsch and H. W. Habgood, J. Phys. C h m . , 67, 1621 (1963).
