We present a six-dimensional potential energy surface for the H2-H2 dimer based on ab initio electronic structure calculations. The surface is intended to describe accurately the bound and quasibound states of the dimers H2-H2, D2-D2, and H2-D2 that correlate with H2 or D2 monomers in the rovibrational levels (v, j) = (0, 0), (0, 2), (1, 0), and (1, 2). We use four experimentally measured transition energies for these dimers to make two empirical adjustments to the ab initio surface; the adjusted surface gives computed transition energies for 56 experimentally observed transitions that agree with experiment to within 0.036 cm^{-1}. For 29 of the 56 transitions, the agreement between the computed and measured transition energies is within the quoted experimental uncertainty. We use our potential energy surface to predict the energies of another 34 not-yet-observed infrared and Raman transitions for the three dimers.
The (H 2 ) 2 dimer has long been viewed as a prototypical bimolecular van der Waals dimer.
Because the (H 2 ) 2 dimer is electronically simple, it has been the focus of a number of ab the analogous isotopomer spectra, provide information about the vibrational dependence of the H 2 -H 2 interaction. Complementary studies 28 of the far-IR absorption spectrum of the dimer provide information about the anisotropy of the potential energy surface in the region of the van der Waals well.
Recently, the Raman spectrum of the (H 2 ) 2 dimer in the H 2 fundamental region has also been observed. 29 This spectrum provides information about the vibrational dependence of the H 2 -H 2 interaction that is complementary to that provided by the high-resolution IR studies. Specifically, the vibrationally excited state of (H 2 ) 2 that is probed by the IR studies is one in which the vibrational excitation is delocalized across the two H 2 monomers in an antisymmetric fashion, while in the Raman studies, the excited (H 2 ) 2 state is one in which the vibrational excitation is delocalized symmetrically across the two monomers. A comparison of the IR and Raman spectra thus provides insight into the coupling between the two H 2 vibrational modes in the (H 2 ) 2 complex and into the dependence of the H 2 -H 2 potential energy surface on the two monomers’ bond lengths.
Equipped with this new information, we attempt here the construction of a sixdimensional H 2 -H 2 potential energy surface that accurately describes the dimer’s van der Waals well. We begin by computing ab initio H 2 -H 2 interaction energies that are nearly converged with respect to both the one-electron and many-electron basis sets, and then construct a smooth potential energy surface from these computed interaction energies. We then make two small empirical adjustments to the surface; these adjustments soften slightly the surface’s short-range repulsive wall, and increase slightly the strength of the surface’s anisotropic term that couples the rotational degrees of freedom of the two monomers. The empirically adjusted surface gives IR and Raman transition energies for the (para-H 2 ) 2 , (ortho-D 2 ) 2 , and para-H 2 -ortho-D 2 dimers in good agreement with available experimental data. 26,28,29
We consider a space-fixed coordinate system (x, y, z) in which one H 2 molecule (denoted molecule 1) has its center of mass at the origin and the other H 2 molecule (denoted molecule 2) has its center of mass on the positive z axis. The orientation of molecule i is specified by its spherical polar and azimuthal angles (θ i , φ i ). We let R represent the distance between the molecules’ centers of mass, and let r i represent the bond length of molecule i. The H 2 -H 2 potential energy surface can then be expanded in terms of coupled spherical harmonics: 30
where φ = φ 2 -φ 1 , the summation indices l 1 , l 2 , and L are non-negative integers that must satisfy
and the homonuclear symmetry of the two H 2 monomers dictates that l 1 and l 2 are also both even. The angular functions G l 1 ,l 2 ,L have the form
where C is a Clebsch-Gordan coefficient and Y l,m is a spherical harmonic normalized so that Y l,m (0, 0) = δ m,0 (2l + 1)/4π. (We use the Condon-Shortley phase convention for Y l,m .)
The appearance of the Clebsch-Gordan coefficient C in Eq. (3) means that l 1 , l 2 , and L must satisfy the angular momentum triangle rule.
The functions G l 1 ,l 2 ,L constitute a complete, orthogonal basis set for functions of the three angular coordinates (θ 1 , θ 2 , φ). For fixed R, r 1 , and r 2 , the coefficient A l 1 ,l 2 ,L (R, r 1 , r 2 ) can therefore be computed as
where dS i = sin θ i dθ i dφ i .
Earlier studies of the four-dimensional rigid-rotor H 2 -H 2 potential energy surface 14,15,18 show that the surface is dominated by four terms, with (l 1 , l 2 , L) = (0, 0, 0), (0, 2, 2), (2, 0, 2), and (2, 2, 4). In this work, we use numerical quadrature to compute the right-hand side of Eq. (4) for these four (l 1 , l 2 , L) triples. Specifically, at fixed values of R, r 1 , and r 2 , we use the 18-point spherical quadrature rule numbered 25.4.64 in Ref. 31 to evaluate the integrals over both dS 1 and dS 2 in Eq. ( 4). This requires us to compute the H 2 -H 2 interaction energy V (R, r 1 , r 2 , θ 1 , θ 2 , φ), using ab initio quantum chemical methods that we describe in the next subsection, at 12 sets of angles (θ 1 , θ 2 , φ) when r 1 = r 2 and at 19 sets of angles when r 1 = r 2 . Symmetry relationships allow the rest of the 18 2 = 324 interaction energies at fixed (R, r 1 , r 2 ) to be determined from these ab initio calculations.
The accuracy of the A l 1 ,l 2 ,L coefficients computed in this fashion is limited by the fact that the quadrature rule we use fails to reproduce the orthogonality conditions
when l 1 + l ′ 1 ≥ 6 or l 2 + l ′ 2 ≥ 6. This means that the value of A 0,0,0 obtained via quadrature also includes some contamination from A 6,0,6 and A 0,6,6 (if these
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