Short‐range intermolecular forces in the H2–H2 system have been approached by a rigorous quantum‐mechanical calculation using a wavefunction consisting of one configuration of nonorthogonal group functions constructed from a minimal basis set of 1s Slater orbitals. The need of avoiding, with restricted wavefunctions, the forced orthogonalization of the atomic‐orbital basis, which amounts to the relaxation of the strong orthogonality condition in a group‐function approach, appears to be essential in order to give the correct angular dependence of the intermolecular energy. About 98% of the potential `barrier' resulting from the minimal‐basis‐set calculation with complete configuration interaction is accounted for by nonorthogonal group functions which allow only for intramolecular correlation. The even simpler one‐determinant wavefunction constructed from SCF bond orbitals gives numerical results within 5% to 10% of the reference value for the intersection and within 3% to 5% for the energy difference between different relative orientations. Both inter‐ and intramolecular correlations seem therefore to be relatively unimportant in determining the relative orientation of the two molecules in the short‐range region. The partitioning of the short‐range interaction energy into a Coulomb component (slightly attractive and slowly varying with the dihedral angle) and into a correction or penetration term, which arises from electron interchange between the electron groups when they begin to overlap, shows that the largest part of the interaction and its orientation dependence arise from the repulsion due to negative overlap between the closed‐shell charge clouds.

1.
(a)
V.
 
Magnasco
 and
G. F.
 
Musso
,
J. Chem. Phys.
 
46
,
4015
(
1967
);
Google Scholar
Crossref
Search ADS
 
(b)
V.
Magnasco
 and
G. F.
Musso
,
47
,
1723
(
1967
). (These papers are referred to as Parts I and II, respectively.),
J. Chem. Phys.
Google Scholar
Crossref
Search ADS
 
2.
R.
McWeeny
,
Proc. Roy. Soc. (London)
A253
,
242
(
1959
).
Google Scholar
 
3.
(a)
R.
 
McWeeny
,
Rev. Mod. Phys.
 
32
,
335
(
1960
).
Google Scholar
Crossref
Search ADS
 
(b) Apart from the “Fermi correlation” between electrons of the same spin in the region of overlap of the two groups.
4.
R. McWeeny, Uppsala Univ. Quantum Chem. Group, Tech. Note No. 59, January 1959 (unpublished).
5.
R.
McWeeny
 and
B. T.
Sutcliffe
,
Proc. Roy. Soc. (London)
A273
,
103
(
1963
).
Google Scholar
 
6.
Calculations using symmetrically orthogonalized atomic orbitals as a basis for a Weinbaum group function of the form (2.5) but satisfying the “strong‐orthogonality” condition, ∫ΦAa(1,xi)ΦBb(1,xj)dx1 = 0 for A≠B or a≠b, have been carried out in our Laboratory (V.M.) and will be published elsewhere. The results are very close to those obtained in the restricted valence‐bond calculation of Part I [Ref. 1(a)].
7.
We adopt the functional notation FR(S) to denote that matrix FR has a functional dependence on the coefficients of group S.
8.
(a)We note that the term “distortion” includes all density changes due to perturbation of each system by the other; it therefore refers to both long‐range effects (polarization and dispersion attractions) and to short‐range effects due to to penetration, these being the more important in the present discussion.
(b) Explicit formulas for the n‐interchange contributions to the density matrices have been given in Ref. 5, Eqs. (4.11) to (4.23).
9.
Here Tr means “Trace” in the sense of a continuous representation, the summation over the diagonal matrix elements being replaced by integration:
Trρ1(n)(1;1′) = ∫ρ1(n)(1;1)dx1
,
Trρ2(n)(12;1′2′) = ∫∫ρ2(n)(12;12)dx1dx2
.
10.
Atomic units are used throughout the paper. Energy atomic units (hartrees) = 27.210 eV, length atomic units (bohrs) = 0.52917 Å, charge atomic units (e) = 4.77×10−10 esu.
11.
The partitioning of the electrostatic part of the interaction energy into Coulomb components as given in (3.26) is strictly correct for unpolarized group functions, namely when the interaction is obtained as a first‐order term in the usual sense of perturbation theory (Ref. 3, Sec. I.3). With self‐consistent group functions the one‐configuration energy embodies part of the second‐order correction, viz., the “polarization energy” arising from single excitations on each molecule. The electronic components of the interaction energy describe now electrostatic and penetration effects between polarized charge clouds. The correct expression for the Coulomb interaction is obtained in this case by supplementing ΔEcb with the difference between EA (self‐energy of the polarized molecule A) and E0A (self‐energy of A when B is at infinity) plus a similar term for B∶ΔEcb′ = ΔEcb+(EA−E0A)+(EB−E0B).
12.
F. O.
Ellison
,
J. Chem. Phys.
34
,
2100
(
1961
).
Google Scholar
Crossref
Search ADS
 
13.
In the case of systems in nonsinglet states, with which we are not concerned in this paper, other possibilities arise, depending on the spin coupling involved. A simple example is afforded by the interaction of two systems, each in a doublet state with one unpaired electron (e.g., two H atoms), where ΔE1 turns out to be negative (leading to a singlet bound state) or positive (leading to a triplet repulsive state) according to the antiparallel or parallel spin‐coupling scheme of the atomic wavefunctions.
14.
S.
Weinbaum
,
J. Chem. Phys.
1
,
593
(
1933
).
Google Scholar
Crossref
Search ADS
 
15.
These values correspond to the same set of intermolecular separations and relative orientations of the two molecules already used in I. The orbitals that in I and II were labeled (a b c d) are denoted here by (a1a2b1b2) in order to stress that the first two orbitals belong to Molecule A and the last two to molecule B.
16.
All the three‐ and four‐center molecular integrals were evaluated to six significant figures using a three‐dimensional numerical integration in confocal ellipsoidal coordinates. Details and further references for the integration technique are given in I,1a where the values of the distinct multicenter integrals occurring in the calculation have been also reported (Table VII in Appendix C).
17.
This is not unexpected because the “polarization energy” is a second‐order correction and the minimal basis set of atomic orbitals used in the calculation does not seem well enough suited for a thorough description of the effective polarization existing between the two molecules. On the other hand, the individual H2 molecules are nonpolar and have low polarizability.
18.
C. A.
Coulson
,
Trans. Faraday Soc.
33
,
1479
(
1937
).
Google Scholar
Crossref
Search ADS
 
19.
R. M.
Pitzer
 and
W. N.
Lipscomb
,
J. Chem. Phys.
39
,
1995
(
1963
).
Google Scholar
Crossref
Search ADS
 
20.
E.
Clementi
 and
D. D.
Davis
,
J. Chem. Phys.
45
,
2593
(
1966
).
Google Scholar
Crossref
Search ADS
 
21.
Md. A.
Ali
 and
D. B.
Cook
,
J. Chem. Phys.
45
,
3873
(
1966
).
Google Scholar
Crossref
Search ADS
 
22.
R.
McWeeny
 and
K.
Ohno
,
Proc. Roy. Soc. (London)
A255
,
367
(
1960
).
Google Scholar
 
23.
A.
Tsuchida
 and
K.
Ohno
,
J. Chem. Phys.
39
,
600
(
1963
).
Google Scholar
Crossref
Search ADS
 
24.
M.
Klessinger
 and
R.
McWeeny
,
J. Chem. Phys.
42
,
3343
(
1965
).
Google Scholar
Crossref
Search ADS
 
25.
M.
Klessinger
,
J. Chem. Phys.
43
,
S117
(
1965
).
Google Scholar
Crossref
Search ADS
 
26.
Here the “barrier” is the difference in intermolecular energy between the two relative orientations of the two molecules defined by θ = θ = 0° and 180°.
27.
V.
Magnasco
,
Atti Accad. Ligure Sci. Lettere
22
,
367
(
1965
).
Google Scholar
 
28.
V.
Magnasco
 and
D. R.
Ferro
,
Atti Accad. Ligure Sci. Lettere
22
,
375
(
1965
).
Google Scholar
 
This content is only available via PDF.
© 1968 American Institute of Physics.
1968
American Institute of Physics
You do not currently have access to this content.