Journal of Atomic, Molecular, and Optical PhysicsVolume 2012 (2012), Article ID 916510, 11 pagesdoi:10.1155/2012/916510

Research Article

Electronic-Rotational Coupling in Cl–para-H₂ Van der Waals Dimers

Robert J. Hinde

Department of Chemistry, University of Tennessee, Knoxville, TN 37996-1600, USA

Received 31 July 2012; Accepted 12 November 2012

Academic Editor: Hari P. Saha

Copyright © 2012 Robert J. Hinde. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We examine the interaction between an open-shell chlorine atom and a para-H₂ molecule in the region of configuration space that corresponds to a weakly bound Cl–para-H₂ van der Waals dimer. By constructing and diagonalizing the Hamiltonian matrix that represents the coupled Cl atom electronic and H₂ rotational degrees of freedom, we obtain one-dimensional energy curves for the Cl–para-H₂ system in this region of configuration space. We find that the dimer exhibits fairly strong electronic-rotational coupling when the Cl–H₂ distance R is close to ; however, this coupling does not modify substantially the positions and depths of the van der Waals wells in the dimer’s curves. An approximation in which the para-H₂ fragment is treated in the strict limit thus appears to yield an accurate representation of those states of the weakly bound Cl–para-H₂ dimer that correlate with H₂ in the limit.

1. Introduction

Experimental studies [1] of the infrared absorption spectra of solid para-H₂ matrices that contain chlorine atoms as substitutional impurities indicate that Cl-H₂ interactions raise the transition energy associated with the spin-orbit (SO) transitions of the Cl impurities. In these systems, the H₂ molecules in the Cl atom’s first “solvation shell” reside in the van der Waals region of the Cl-H₂ potential energy surface [2]. A detailed analysis of the matrix-induced blue shift of the SO transition for Cl atoms embedded in solid para-H₂ would thus provide insight into the shape of the Cl-H₂ potential energy surface in this region of configuration space. This in turn could help us better understand the dynamics of the Cl + H₂ → HCl + H reaction, which has long been considered a benchmark system in chemical reaction dynamics [3]. For example, theoretical studies [4] of the HCl/DCl product branching ratio of the Cl + HD reaction suggest that the van der Waals region of the potential energy surface plays a key role in controlling the reaction’s dynamics at low collision energies.

In the para-H₂ matrix, the Cl atom’s SO transition is blue shifted by about 60 cm⁻¹ by Cl-H₂ interactions [1], which amounts to a shift of about 5 cm⁻¹ for each of the twelve H₂ molecules in the Cl atom’s first solvation shell. The matrix-induced blue shift of the Cl SO transition can be qualitatively understood as arising from subtle differences in the van der Waals interactions of the ground and excited SO states of the Cl atom with nearby H₂ molecules. For a simulation to reproduce quantitatively the observed blue shift, we might therefore anticipate that the potential energy surfaces used in the simulation should be accurate to better than 1 cm⁻¹. Theoretical studies of the matrix-induced shift would thus represent a fairly stringent test of the Cl-H₂ potential energy surfaces involving both the ground and excited SO states of the Cl atom.

Previous simulations [5, 6] of open-shell atomic impurities in para-H₂ matrices have employed a simplified model for the para-H₂ matrix, a model in which the H₂ molecules are treated as spherical particles. This is equivalent to assuming that the H₂ molecules in the matrix remain in the pure ground rotational state, with their orientational degrees of freedom completely unperturbed by anisotropic interactions with the open-shell impurity. However, a careful study of the B-H₂ and Al-H₂ van der Waals dimers [7] indicated that the binding energies of these dimers increased by 15% to 20% (or several cm⁻¹) when the restriction for the H₂ monomer was relaxed and rotational states were allowed to mix into the H₂ molecule’s rotational wave function. Similar behavior in the Cl-H₂ dimer might raise concerns that the effective Cl-H₂ potential energy functions obtained from a pure approximation for the H₂ molecule would be insufficiently accurate for a quantitative study of the matrix-induced blue shift of the Cl SO transition. In such a case, more sophisticated treatments [8] of the H₂ molecules’ rotational degrees of freedom might be needed.

In this work, we examine electronic-rotational coupling in the Cl-para-H₂ van der Waals dimer, using the potential energy surfaces for the dimer presented in [9] as the foundation for our study. We find some evidence for moderately strong electronic-rotational coupling in the low-energy repulsive region of the Cl-H₂ potential energy surface. This coupling does not, however, substantially change the positions or depths of the van der Waals minima for dimers composed of a Cl atom and a para-H₂ molecule. This suggests that treating the H₂ molecules in Cl-doped solid para-H₂ as pure molecules should be a reasonably good approximation.

2. Spin-Orbit States of an Isolated Cl Atom

We begin our investigation by reviewing the effects of SO coupling in an isolated chlorine atom. We first construct a basis set of antisymmetrized many-electron functions for the atom’s subshell; we will use this basis set to evaluate the matrix elements of the atom’s effective SO operator. For future convenience, we choose these antisymmetrized many-electron functions to be Slater determinants of Cartesian spin-orbitals in a space-fixed coordinate system. We use and , respectively, to represent a one-electron Cartesian orbital paired with either the spin-up or spin-down electron spin function. For specificity, we note that the function is positive along the positive axis. The six many-electron functions that span the subspace of interest are these Slater determinants:

The Cl atom’s SO operator is written as where and are the total orbital and spin angular momentum operators for the five electrons in the subshell and the constant is two-thirds of the energy gap between the lower and upper SO energy levels of the Cl atom; this energy gap is [10]. In a basis set consisting of these six Slater determinants, has the matrix representation where the basis set is ordered from left to right (and top to bottom). In the same basis set, the operator corresponding to the -axis projection of the total angular momentum of the subshell, has the matrix representation

Because and commute with one another, we can simultaneously diagonalize the and matrices. The normalized many-electron functions that do this are The four functions labeled with the subscript are degenerate, with energy . They represent the four components of the Cl atom’s lower SO energy level; the fraction listed in each function’s subscript, when multiplied by , gives that function’s eigenvalue. The two functions labeled with the subscript are degenerate components of the Cl atom’s upper SO energy level and have energy ; these functions have eigenvalues of , as indicated by their respective subscripts. Later, we will use the symbol to represent the subscripts attached to the six many-electron SO wave functions listed above; the eigenvalue for each function is thus .

Next we examine the charge densities associated with the Cl atom’s lower and upper SO energy levels. The charge density associated with one of the six electronic wave functions listed above is obtained by forming the product , integrating over the five electrons’ spin coordinates and the spatial coordinates of four of the five electrons, and multiplying by five. The final multiplication by five accounts for the fact that the many-electron function is normalized so that integrating over all five electrons’ spatial coordinates gives the value one, while integrating the charge density over all space should give the value five.

To emphasize the shapes of these charge densities, we note that the charge density associated with the Slater determinant , which corresponds to a singly occupied orbital aligned with the space-fixed Cartesian axis and doubly occupied orbitals aligned with the other two space-fixed Cartesian axes, can be pictured as a charge density “hole” superimposed on the isotropic charge density function associated with a filled six-electron subshell. We will represent this hole-plus-filled-subshell charge distribution using the symbol .

After computing the charge densities associated with the six SO wave functions listed above and writing them in terms of this symbol, we obtain This shows that the Cl atom’s upper SO energy level has an isotropic charge density distribution, while the charge densities associated with the lower SO energy level are anisotropic. The wave functions have higher levels of electron density along the space-fixed Cartesian -axis and lower levels of electron density in the plane; the situation is reversed for the wave functions.

3. Cl-H₂ without Spin-Orbit Coupling

Now we consider a Cl-H₂ dimer in which the Cl atom is located at the origin of the space-fixed coordinate system and the H₂ center of mass is located on the positive space-fixed axis. We use and to represent the polar and azimuthal angles, respectively, of the H–H covalent bond in the space-fixed coordinate system. The H–H covalent bond length is denoted , and is the Cl-H₂ distance. In this section, we neglect the effect of SO coupling in the Cl atom.

3.1. Characteristics of the Cl-H₂ Potential Energy Surface

First we set the azimuthal angle to , so that the H₂ molecule resides in the plane. If we neglect effects related to SO coupling, then there are three Cl-H₂ adiabatic electronic states, each of which is doubly degenerate because of the spin-1/2 nature of the Cl-H₂ dimer [2, 9, 11]. Two of these states have spatial symmetry with respect to the plane; these are states in which the -orbital “hole” of the Cl atom resides in the plane. The third state has spatial symmetry; in this state, the Cl atom’s -orbital hole is aligned with the -axis.

The two adiabatic electronic states, which we label and in order of increasing energy, can be viewed as linear combinations of diabatic states associated with the Slater determinants and . The mixing angle defines these linear combinations [2, 9, 11]: The mixing angle can be computed by applying an approximate diabatization procedure to the adiabatic electronic wave functions obtained from a conventional ab initio calculation [2, 9, 11, 12].

When the H₂ polar angle is either or , then symmetry considerations make the electronic states corresponding to the and Slater determinants good adiabatic electronic states of the Cl-H₂ system. For these geometries, the mixing angle must be either or . For , the state is lower in energy than the state, while for , the state is lower in energy than the state. In Figure 1, we show how the Cl-H₂ interaction energy obtained in [9] for these two adiabatic states depends on the Cl-H₂ distance , both for geometries with (in which the H–H bond is collinear with the Cl-H₂ distance) and for geometries with (in which the H–H bond is perpendicular to the Cl-H₂ distance). In these plots, the H–H covalent bond length is held fixed at , its vibrationally averaged value.

(a)  Collinear

(b) T-shaped

Figure 1: (a) Adiabatic Cl-H₂ potential energy surfaces, as functions of the Cl-H₂ distance , for (a) collinear and (b) T-shaped geometries. Solid lines indicate the potential energy for the Cl electronic configuration in which the hole is aligned with the space-fixed -axis; dashed lines indicate the potential energy for the Cl electronic configuration in which the hole is aligned with the space-fixed -axis. In both panels, is held fixed at .

Note that for collinear approach of the H₂ molecule, the adiabatic and states undergo a crossing near . At that point, the mixing angle changes discontinuously from at large to at small . We can explain this behavior using a simple picture of the Cl-H₂ interaction based on electrostatic and overlap contributions. At large , it is more favorable for the “hole” in the Cl atom’s subshell to be oriented at right angles to the covalent bond of the incoming H₂ molecule; this leads to the most favorable quadrupole-quadrupole interaction between the H₂ molecule and the Cl atom. At small , however, repulsive overlap interactions become more important; these are minimized if the Cl atom’s hole orients itself along the axis that connects the Cl nucleus and the H₂ center of mass.

Now we investigate how the Cl-H₂ interaction changes as the H₂ molecule rotates from to , with both and held fixed. As before, we neglect SO coupling and we set to its vibrationally averaged value of . Figure 2(a) shows, for the two adiabatic electronic states, how the Cl-H₂ interaction energy depends on at . In Figure 2(b) we plot, as functions of , the contributions that the and diabatic states make to the adiabatic state; as we see from (20), these contributions are simply and . Figure 2(b) demonstrates that for the state, the hole in the Cl atom’s subshell rotates in space as changes so that the Cl atom maintains a favorable electrostatic interaction with the H₂ quadrupole.

(a)

(b)

Figure 2: (a) Adiabatic potential energy surfaces for planar Cl-H₂ geometries shown as functions of the H₂ polar angle for and . (b) Contributions made by the Cl atom’s (dashed line) and (solid line) Slater determinants to the lowest-energy electronic state shown as functions of the H₂ polar angle for and .

3.2. Electronic-Rotational Coupling in Cl-H₂ Dimers

Now we lift the restriction that and investigate how the rotational energy levels of the H₂ molecule are perturbed by its interaction with the Cl atom. We continue to neglect SO coupling in the Cl atom.

We construct a basis set for the coupled rotational and electronic degrees of freedom of the Cl-H₂ dimer that is a direct product of functions describing the Cl atom’s electronic state and functions describing the H₂ molecule’s rotation. For the electronic portion of the direct product basis, we use the six Slater determinants defined in (1) through (6); this will simplify the subsequent incorporation of SO coupling. For the rotational portion of the direct product basis, we use the spherical harmonics with even values of the H₂ rotational quantum number .

In this work, we limit the possible values to and ; this leads to a basis set with 36 orthonormal direct product electronic-rotational basis functions. In the absence of SO coupling, however, the Hamiltonian matrix splits into two uncoupled blocks that are, respectively, associated with the spin-up and spin-down Slater determinants. The two blocks have a common set of eigenvalues.

We could exploit the cylindrical symmetry of the Cl-H₂ system to further factorize the Hamiltonian matrix. This would involve rewriting the electronic portion of the basis set in terms of complex linear combinations of and that are eigenfunctions of the operator, with eigenvalues : Because of the system’s cylindrical symmetry, the quantum number , which, after multiplication by , is the -axis projection of the sum of the electronic orbital angular momentum and the H₂ rotational angular momentum, is rigorously good in the absence of SO coupling. If we were to express the SO-free Hamiltonian matrix in terms of the complex basis functions of (22) and (23), the two blocks of the matrix would therefore factor into uncoupled subblocks corresponding to the allowed values of . Although we do not employ this approach here, we will use to classify the overall symmetry of the SO-free Cl-H₂ wave functions. For the basis set employed here, ranges from to in steps of one. In analogy with the nomenclature for diatomic molecules, we will denote Cl-H₂ wave functions with , 1, 2, or 3 as , or states, respectively.

For a fixed value of the Cl-H₂ distance , the elements of the electronic-rotational Hamiltonian matrix can be written as where and represent space-fixed Cartesian directions, is the H₂ rotational constant (here taken to have the value 59.06 cm⁻¹), and describes the Cl-H₂ interaction. The analogous matrix element in which the electronic portions of the basis functions are represented by and has the same value as the matrix element in (24). Matrix elements that involve a spin-up and a spin-down Slater determinant are zero in the absence of SO coupling.

Because the Slater determinants and are orthogonal unless , the kinetic energy portion of (24) can be evaluated quite easily; it is given by After writing out explicitly the integration over the H₂ angles and , the potential energy portion of (24) becomes where Here, we compute the integrals in (26) by numerical quadrature, using the 240-point spherical -design specified in [13].

To evaluate , we introduce a rotated system of Cartesian axes, which we denote . The and axes rotate in the space-fixed plane as the H₂ azimuthal angle changes, so that the H₂ molecule always resides in the plane. The Slater determinants and , defined in the space-fixed coordinate system, are related to their analogues and , defined in the rotated coordinate system, through Equipped with this relationship, we can now evaluate , where and are directions in the space-fixed coordinate system, in terms of integrals over , and . Doing this, we obtain where, for the sake of brevity, we have suppressed the dependence of . Note that and are both zero by reasons of symmetry.

The quantities , and are the same as those in through of [9]. They depend on and but are independent of . As a consequence, Cl-H₂ electronic-rotational basis functions with different values of are coupled only by the trigonometric functions of that are shown explicitly in (30) through (34). and couple H₂ rotational states with or 2, couples H₂ rotational states with , and and couple H₂ rotational states with . has no dependence on and therefore does not couple basis functions with different values.

Constructing and diagonalizing the electronic-rotational Hamiltonian matrix at a series of values gives the Cl-H₂ curves shown in Figure 3, which we now discuss in some detail. The curves fall into three distinct groups, labeled as A, B, and C. The two curves in group A correlate smoothly with the rotational level of the H₂ molecule as , while those in groups B and C correlate with states of the H₂ molecule in this limit. All of the B curves, at large , correlate with electronic-rotational basis functions that involve the Cl atom’s Slater determinant. The C curves, on the other hand, correlate at large with electronic-rotational basis functions involving the and Slater determinants of the Cl atom.

Figure 3: Coupled electronic-rotational Cl-H₂ energy curves that correlate with the and states of H₂ shown as functions of the Cl-H₂ distance . Spin-orbit effects are neglected, and the H₂ bond length is held fixed at .

We begin by considering the two curves (one and one ) in group A. Both curves are attractive at long range, but the minimum of the curve is deeper and occurs at a smaller value. The minimum for the curve occurs at , where meV. The minimum for the curve is at , where meV. The curve correlates at large with the Slater determinant of the Cl atom’s subshell; for this Slater determinant, the hole of the Cl atom is oriented along the space-fixed axis, which is the direction of approach of the incoming H₂ molecule. The curve, by contrast, correlates at large values with the and Slater determinants, for which the Cl atom’s hole is oriented in the plane. The curve thus minimizes short-range repulsive overlap interactions between Cl and H₂ and permits closer approach of the incoming H₂ molecule.

Group B includes three curves: one , one , and one . These curves all correlate smoothly with the Slater determinant as , so the Cl atom’s hole is aligned with the space-fixed axis for large . For the curve, the H₂ rotational state at large corresponds to the spherical harmonic. In this state, the H₂ molecule can be viewed classically as rotating in the plane. This orientation of the rotating H₂ molecule both maximizes its attractive electrostatic quadrupole-quadrupole interaction with the Cl atom and minimizes short-range Cl-H₂ repulsive interactions at small values. Consequently, of the three group B curves, the curve has the deepest minimum.

The curve from group B correlates at large with electronic-rotational basis functions that are direct products of the H₂   spherical harmonics and the Cl atom’s Slater determinant. As , therefore, the character of this curve is associated with the rotational motion of the H₂ molecule. Figure 3 shows, however, that this curve undergoes an avoided crossing with the group A curve near .

To understand the nature of this avoided crossing in more detail, we project the wave function for the group A curve onto two pairs of electronic-rotational direct product basis functions: the two functions and and the two functions and . The first pair of functions describes the group A curve at large values, while the second pair of functions describes the group B curve at large values. Figure 4 shows the dependence of the contributions that these two pairs of basis functions make to the curve of group A.

Figure 4: Solid line: the contribution made by the basis function to the lowest Cl-H₂   curve shown in Figure 3. Dashed lines: contributions made by the pair of basis functions and the pair of basis functions to the lowest Cl-H₂   curve shown in Figure 3.

It is clear that, for the entire range of values shown in Figure 4, the makeup of the group A curve is dominated by these two pairs of basis functions; summed together, their contributions represent over 99% of the curve’s character. At large values, the curve can be described entirely by the and direct product functions, demonstrating that the character of the state arises from the Cl atom’s subshell. At small values, however, the curve is described entirely by the and direct product functions; here, the character of the state arises from the rotational motion of the H₂ molecule. The crossover in the description of the curve of group A reflects a strong interaction between these two zero-order states, an interaction that arises from the and functions of (33) and (34). The result is to strongly couple the electronic and rotational degrees of freedom of the Cl-H₂ dimer near .

We have also taken the group A curve and projected it onto the direct product basis function to assess the contribution that this zero-order state makes to the curve. Figure 4 indicates that this zero-order state provides a very good description of the group A curve, accounting for more than 90% of its character over the entire range of values shown there. This demonstrates that the group A curve can be closely approximated by the interaction between a pure   para-H₂ molecule and a Cl atom described by the Slater determinant.

4. Cl-H₂ Including Spin-Orbit Coupling

Now we ask how the results summarized in Figures 3 and 4 change when we include the effects of SO coupling. To do this, we simply add the appropriate matrix elements of the SO operator (see (8)) to the Hamiltonian matrix expressed in the direct product basis set, and then compute the eigenvalues of the new matrix at a series of values. The calculations of [9] indicate that, in the Cl-H₂ van der Waals region, the triatomic system’s SO matrix elements are nearly independent of the Cl-H₂ geometry and have values nearly identical to those for an isolated Cl atom. This justifies the approximation implicit in our calculations, namely that the matrix for the free Cl atom is a good approximation of the effects of SO coupling in the Cl-H₂ dimer. For a given value of , and in the absence of accidental degeneracies, the new Hamiltonian matrix has 18 distinct doubly degenerate eigenvalues. Kramers [14] explained that a system with a single unpaired electron would always, in the absence of an external magnetic field, have doubly degenerate energy levels; the two states that make up a single doubly degenerate energy level are sometimes called a Kramers pair.

As before, the system’s cylindrical symmetry introduces a rigorously good quantum number that, when multiplied by , gives the -axis projection of the system’s total (electronic spin plus electronic orbital plus H₂ rotational) angular momentum. Here we call this quantum number ; its value is given by , and for the basis set employed here, the possible values of range from to in steps of one. Although we have not exploited the system’s cylindrical symmetry to block-diagonalize the Hamiltonian matrix, we use to classify the symmetry of the Cl-H₂ wave functions and to gain some physical insight into the shapes of the system’s energy curves .

In Table 1, we list the combinations of the and quantum numbers that can generate each of the four positive allowed values. The corresponding negative values can be obtained by reversing the sign of both and . In the following discussion, we consider only the states that have positive values, as the states with negative values have identical energies. We will refer to this table and to the charge densities of the two SO levels of an isolated Cl atom ((17) through (19)), as we discuss the role that SO coupling has on the Cl-H₂ energy curves.

Table 1: Combinations () of azimuthal angular momentum quantum numbers that can couple together to produce a given positive value.

(+3/2, −1)
(+3/2, 0)
(+3/2, +1)
(+3/2, +2)

(+1/2, 0)
(+1/2, +1)
(+1/2, +2)

(−1/2, +1)
(−1/2, +2)

(−3/2, +2)

As Table 1 indicates, the state of the H₂ molecule, which has as well, can either form a Cl-H₂ state with by combining with the state of the Cl atom or form a Cl-H₂ state with by combining with the state or the state of the Cl atom. Consequently, there are three distinct Cl-H₂ curves that correlate as with the ground rotational state of the H₂ molecule. Two of the three curves are associated with the Cl atom’s lower SO energy level, and one is associated with the atom’s upper SO energy level. These curves, shown in Figure 5, are the most important ones for understanding the interaction between Cl and an approaching para-H₂ molecule.

Figure 5: Coupled electronic-rotational Cl-H₂ energy curves that correlate with the rotational state of H₂ shown as functions of the Cl-H₂ distance . Spin-orbit effects are included; the horizontal dashed lines indicate the asymptotic energies of the Cl-H₂ dimer as for the lower and upper spin-orbit levels of the Cl atom.

All three curves exhibit relatively shallow van der Waals minima. The state that correlates with the Cl atom’s lower SO energy level has the deepest minimum, which occurs at . At this value, the depth of the Cl-H₂ curve, measured with respect to the asymptote, is 10.0 meV. The other two states have shallower minima, which occur at larger values: the minimum for the state that correlates with the upper SO energy level is at , while that for the state is at . The depths of these minima, measured with respect to the asymptotes, are, respectively, 7.3 meV and 6.5 meV. The relative locations and depths of the three minima can be understood by examining the charge densities of the Cl electronic states that are associated with each curve.

For the curve that correlates with the lower SO energy level, the Cl atom is described at large by the electronic wave function. This Cl atomic state has a depletion of electron density along the axis (see (18)), which is the direction of approach of the incoming H₂ molecule. Consequently, of the three curves shown in Figure 5, this curve allows the H₂ molecule to approach the Cl atom most closely. For the curve, on the other hand, the Cl atom is described at large by the electronic wave function; as (17) shows, the electron density for this state of the Cl atom is depleted in the plane and built up along the axis. Of the three curves shown in Figure 5, this curve has its minimum at the largest value. Finally, for the curve that correlates with the upper SO energy level, the Cl atom’s large- character is that of the state. In this state, the atom’s “hole” is completely orientationally delocalized (see (19)). This leads to a minimum that occurs at an value between the minima for the two other curves.

We now ask, for the two curves depicted in Figure 5, how the approach of the incoming H₂ molecule distorts the electronic structure of the Cl atom. To answer this question, we take the Cl-H₂ wave functions for the two curves and project them onto either the (for the lower curve) or the (for the upper curve) Cl atomic wave function. In computing these projections, we sum over all of the H₂ fragment’s rotational states. Figure 6(a) shows, as a function of the Cl-H₂ distance , the fractional contributions that these two pure atomic SO states make to the corresponding Cl-H₂ dimer wave functions. For large , the Cl electronic structure is essentially unperturbed by the H₂ molecule. As decreases below about , however, the impinging H₂ molecule begins to mix some excited SO character into the wave function of the lower curve, and some ground SO character into the wave function of the upper curve.

(a)

(b)

Figure 6: (a) The contribution made by the Cl atom’s respective or spin-orbit electronic state to the two curves shown in Figure 5. (b) The contribution made by the H₂   rotational state to the Cl-H₂ wave function for the two curves shown in Figure 5.

For these two curves, Cl-H₂ interactions also perturb the rotational degrees of freedom of the H₂ fragment at small values. This leads to a mixing of some character into the H₂ molecule’s rotational wave function. To measure this perturbation, we project the two Cl-H₂   wave functions onto the   H₂ molecular rotational wave function. In computing these projections, we sum over all of the Cl fragment’s SO states. Figure 6(b) shows how the contribution that the H₂   state makes to the overall dimer wave function changes with . For the range of values shown here, the character of both curves is dominated by the H₂ rotational state.

Figures 7 through 9 complete our survey of the Cl-H₂ energy curves associated with the lower SO energy level. In these figures, the curves whose large- asymptotic values are roughly 7 meV all correlate, as , with the H₂ molecule’s rotational energy level. For the , 5/2, and 7/2 states (Figures 7 and 9), these curves fall into two groups distinguished mainly by the position of their short-range repulsive walls. The curves whose repulsive walls occur at smaller values are associated with the Cl atom’s states, which have a depletion of electron density along the axis. The curves whose repulsive walls occur at larger values are associated with the Cl atom’s states, which do not exhibit depleted electron density along the axis.

Figure 7: All of the Cl-H₂ dimer’s coupled electronic-rotational states that correlate with the lower spin-orbit level of the Cl atom are shown as functions of the Cl-H₂ distance . The horizontal dashed line indicates the dimer’s asymptotic energy, as , for the lower spin-orbit level of the Cl atom and the state of H₂.

The curves depicted in Figure 8 exhibit more complicated behavior. The uppermost curve shown there is associated with the Cl atom’s state, and thus its repulsive wall begins to manifest itself at larger values. The other three curves, however, participate in a pair of avoided crossings near , close to where the avoided crossing observed in Figure 3 occurred. The unusual “kink” seen in Figure 5 is associated with this pair of avoided crossings.

Figure 8: All of the Cl-H₂ dimer’s coupled electronic-rotational states that correlate with the lower spin-orbit level of the Cl atom are shown as functions of the Cl-H₂ distance . The horizontal dashed line indicates the dimer’s asymptotic energy, as , for the lower spin-orbit level of the Cl atom and the state of H₂.

Figure 9: All of the Cl-H₂ dimer’s and coupled electronic-rotational states that correlate with the lower spin-orbit level of the Cl atom are shown as functions of the Cl-H₂ distance . The horizontal dashed line indicates the dimer’s asymptotic energy, as , for the lower spin-orbit level of the Cl atom and the state of H₂.

Figure 10 shows this pair of avoided crossings at higher magnification. The three curves, in order of increasing energy, correlate in the large limit with the Cl atomic state coupled with the rotational state of H₂; the Cl atomic state coupled with the rotational state of H₂; and the Cl atomic state coupled with the rotational state of H₂. The pair of avoided crossings shown in Figure 10 is analogous to the avoided crossing between the group A and group B curves shown in Figure 3. The additional complexity observed in Figure 10 arises from the fact that, as Table 1 shows, three different values of the H₂ molecule’s quantum number can combine with an appropriate SO level of the Cl atom to form a dimer state with . Although the avoided crossings shown in Figure 10, like that shown in Figure 3, demonstrate that there are regions of configuration space where the Cl-H₂ dimer exhibits strong electronic-rotational coupling, these regions of configuration space are at moderately high energies on the repulsive wall of the dimer’s potential energy surface.

Figure 10: A magnified view of three curves shown in Figure 8. This figure shows the pair of avoided crossings near that affect the coupled electronic-rotational states that correlate with the lower Cl spin-orbit level.

Finally, in Figure 11, we show all of the curves that correlate at large with the Cl atom’s upper SO level. Only one of these curves, with , correlates with the state of the H₂ molecule; the others correlate with the H₂ molecule’s excited rotational state. All six curves shown in this figure have very similar shapes; this is because the upper SO level of the Cl atom, with its orientationally delocalized hole, appears isotropic to the incoming H₂ molecule.

Figure 11: All of the Cl-H₂ dimer’s coupled electronic-rotational states that correlate with the upper spin-orbit level of the Cl atom are shown as functions of the Cl-H₂ distance . The horizontal dashed line indicates the dimer’s asymptotic energy, as , for the upper spin-orbit level of the Cl atom and the state of H₂.

We close by computing the Cl-H₂ interaction curves , including the effects of SO coupling, under the assumption that the H₂ molecule retains pure character at all values of . We do this simply by removing all of the direct product basis functions with from the basis set used to express the Cl-H₂ dimer wave function; this leads to a matrix representation of the Cl-H₂ Hamiltonian that has three doubly degenerate eigenvalues, or three Kramers pairs. These three eigenvalues represent the interaction curves obtained using the pure approximation for the H₂ fragment. In Figure 12, we show these three curves and compare them with the analogous curves computed using the larger Hamiltonian matrix that includes states of the H₂ fragment. The three curves obtained from the larger Hamiltonian matrix were first shown in Figure 5.

(a)

(b)

Figure 12: A comparison of the three Cl-H₂ curves that correlate with H₂ molecules when the H₂ molecule is either frozen in its state (dashed lines) or allowed to mix some character into its rotational wave function (solid lines). (a) shows, as functions of the Cl-H₂ distance , curves for the two states that correlate with the lower spin-orbit state of the Cl atom; (b) shows curves for the state that correlates with the upper spin-orbit state of the Cl atom. The horizontal dashed lines show the respective asymptotic dimer energies as . The two curves shown in (b) are nearly indistinguishable from one another.

We see from Figure 12 that the depths and positions of the van der Waals minima for these three curves change very little when we impose the pure approximation for the H fragment. For example, the depth of the curve that correlates with the Cl atom’s lower SO state (shown in Figure 12(a)) changes by about 0.5% when we impose this approximation. This is in sharp contrast to the case for the B-H₂ and Al-H₂ dimers [7], for which the zero-point-corrected binding energies change by 15% and 22%, respectively, when the H₂ fragment is frozen in its state. It appears that the anisotropy of the underlying diabatic potential energy surfaces is substantially larger for the B-H₂ and Al-H₂ systems than for the Cl-H₂ system; this leads to a greater perturbation of the H₂ molecule’s rotational wave function in the region of the van der Waals minima of the B-H₂ and Al-H₂ systems.

For the two curves shown in Figure 12, we see that the pure approximation reproduces the actual Cl-H₂ interaction curves very well; the main difference seems to be that the curve that correlates with the lower SO level of the Cl atom has a slightly stiffer repulsive wall if the H₂ molecule is held fixed in its rotational state. For the curve that correlates with the lower SO level of the Cl atom, the pure approximation is of course completely incapable of reproducing the true Cl-H₂ interaction for values below ; this is the region of configuration space where strong electronic-rotational coupling leads to the avoided crossings shown in Figure 10, and the approximation is simply inapplicable there. However, this region of configuration space is relatively high on the repulsive wall of the curve; because the experimental studies of Cl-doped solid para-H₂ are carried out at K, it is unlikely that the Cl atom’s H₂ neighbors spend a significant portion of time exploring this portion of the potential energy curve.

5. Summary

We have computed a series of one-dimensional energy curves for the Cl-para-H₂ dimer by diagonalizing the dimer’s combined electronic-rotational Hamiltonian matrix. We express the Hamiltonian matrix in terms of a direct product basis set that facilitates the inclusion of SO coupling effects. The gross features of the dimer’s curves, such as the depths and positions of the curves’ van der Waals minima, can be rationalized by considering the charge densities associated with the lower and upper SO levels of the Cl atom.

We find that electronic-rotational coupling in the Cl-H₂ dimer is strong for Cl-H₂ distances near . In the system where SO effects are ignored, this coupling gives rise to an avoided crossing between two Cl-H₂ states with symmetry: one that correlates at large with the H₂ rotational level and one that correlates with the rotational level. Once SO effects are included, this avoided crossing becomes a pair of avoided crossings associated with three of the Cl-H₂ dimer’s states: one of which correlates at large with the H₂ rotational level and two of which correlate with the rotational level.

By removing H₂ rotational states from the direct product basis set used to express the Cl-H₂ Hamiltonian, we can assess how the assumption that the para-H₂ fragment retains its pure identity might affect the shape of the Cl-para-H₂ energy curves. We find that the positions and depths of the system’s van der Waals minima are virtually unaffected when the H₂ fragment is restricted to its pure rotational state. This suggests that the H₂ molecules in Cl-doped solid para-H₂ can probably be treated as pure objects without a significant loss of accuracy.

Acknowledgments

R. J. Hinde thanks D. T. Anderson and K. R. Brown for helpful discussions. This work was supported by the US National Science Foundation through Grant CHE-0848841.

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