Research Article  Open Access
Robert J. Hinde, "ElectronicRotational Coupling in Cl–paraH_{2} Van der Waals Dimers", Journal of Atomic and Molecular Physics, vol. 2012, Article ID 916510, 11 pages, 2012. https://doi.org/10.1155/2012/916510
ElectronicRotational Coupling in Cl–paraH_{2} Van der Waals Dimers
Abstract
We examine the interaction between an openshell chlorine atom and a paraH_{2} molecule in the region of configuration space that corresponds to a weakly bound Cl–paraH_{2} van der Waals dimer. By constructing and diagonalizing the Hamiltonian matrix that represents the coupled Cl atom electronic and H_{2} rotational degrees of freedom, we obtain onedimensional energy curves for the Cl–paraH_{2} system in this region of configuration space. We find that the dimer exhibits fairly strong electronicrotational coupling when the Cl–H_{2} 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 paraH_{2} fragment is treated in the strict limit thus appears to yield an accurate representation of those states of the weakly bound Cl–paraH_{2} dimer that correlate with H_{2} in the limit.
1. Introduction
Experimental studies [1] of the infrared absorption spectra of solid paraH_{2} matrices that contain chlorine atoms as substitutional impurities indicate that ClH_{2} interactions raise the transition energy associated with the spinorbit (SO) transitions of the Cl impurities. In these systems, the H_{2} molecules in the Cl atom’s first “solvation shell” reside in the van der Waals region of the ClH_{2} potential energy surface [2]. A detailed analysis of the matrixinduced blue shift of the SO transition for Cl atoms embedded in solid paraH_{2} would thus provide insight into the shape of the ClH_{2} potential energy surface in this region of configuration space. This in turn could help us better understand the dynamics of the Cl + H_{2} → 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 paraH_{2} matrix, the Cl atom’s SO transition is blue shifted by about 60 cm^{−1} by ClH_{2} interactions [1], which amounts to a shift of about 5 cm^{−1} for each of the twelve H_{2} molecules in the Cl atom’s first solvation shell. The matrixinduced 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 H_{2} 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 matrixinduced shift would thus represent a fairly stringent test of the ClH_{2} potential energy surfaces involving both the ground and excited SO states of the Cl atom.
Previous simulations [5, 6] of openshell atomic impurities in paraH_{2} matrices have employed a simplified model for the paraH_{2} matrix, a model in which the H_{2} molecules are treated as spherical particles. This is equivalent to assuming that the H_{2} molecules in the matrix remain in the pure ground rotational state, with their orientational degrees of freedom completely unperturbed by anisotropic interactions with the openshell impurity. However, a careful study of the BH_{2} and AlH_{2} 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 H_{2} monomer was relaxed and rotational states were allowed to mix into the H_{2} molecule’s rotational wave function. Similar behavior in the ClH_{2} dimer might raise concerns that the effective ClH_{2} potential energy functions obtained from a pure approximation for the H_{2} molecule would be insufficiently accurate for a quantitative study of the matrixinduced blue shift of the Cl SO transition. In such a case, more sophisticated treatments [8] of the H_{2} molecules’ rotational degrees of freedom might be needed.
In this work, we examine electronicrotational coupling in the ClparaH_{2} 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 electronicrotational coupling in the lowenergy repulsive region of the ClH_{2} 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 paraH_{2} molecule. This suggests that treating the H_{2} molecules in Cldoped solid paraH_{2} as pure molecules should be a reasonably good approximation.
2. SpinOrbit 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 manyelectron 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 manyelectron functions to be Slater determinants of Cartesian spinorbitals in a spacefixed coordinate system. We use and , respectively, to represent a oneelectron Cartesian orbital paired with either the spinup or spindown electron spin function. For specificity, we note that the function is positive along the positive axis. The six manyelectron 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 twothirds 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 manyelectron 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 manyelectron 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 manyelectron 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 spacefixed Cartesian axis and doubly occupied orbitals aligned with the other two spacefixed Cartesian axes, can be pictured as a charge density “hole” superimposed on the isotropic charge density function associated with a filled sixelectron subshell. We will represent this holeplusfilledsubshell 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 spacefixed Cartesian axis and lower levels of electron density in the plane; the situation is reversed for the wave functions.
3. ClH_{2} without SpinOrbit Coupling
Now we consider a ClH_{2} dimer in which the Cl atom is located at the origin of the spacefixed coordinate system and the H_{2} center of mass is located on the positive spacefixed axis. We use and to represent the polar and azimuthal angles, respectively, of the H–H covalent bond in the spacefixed coordinate system. The H–H covalent bond length is denoted , and is the ClH_{2} distance. In this section, we neglect the effect of SO coupling in the Cl atom.
3.1. Characteristics of the ClH_{2} Potential Energy Surface
First we set the azimuthal angle to , so that the H_{2} molecule resides in the plane. If we neglect effects related to SO coupling, then there are three ClH_{2} adiabatic electronic states, each of which is doubly degenerate because of the spin1/2 nature of the ClH_{2} 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 H_{2} polar angle is either or , then symmetry considerations make the electronic states corresponding to the and Slater determinants good adiabatic electronic states of the ClH_{2} 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 ClH_{2} interaction energy obtained in [9] for these two adiabatic states depends on the ClH_{2} distance , both for geometries with (in which the H–H bond is collinear with the ClH_{2} distance) and for geometries with (in which the H–H bond is perpendicular to the ClH_{2} distance). In these plots, the H–H covalent bond length is held fixed at , its vibrationally averaged value.
(a) Collinear
(b) Tshaped
Note that for collinear approach of the H_{2} 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 ClH_{2} 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 H_{2} molecule; this leads to the most favorable quadrupolequadrupole interaction between the H_{2} 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 H_{2} center of mass.
Now we investigate how the ClH_{2} interaction changes as the H_{2} 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 ClH_{2} 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 H_{2} quadrupole.
(a)
(b)
3.2. ElectronicRotational Coupling in ClH_{2} Dimers
Now we lift the restriction that and investigate how the rotational energy levels of the H_{2} 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 ClH_{2} dimer that is a direct product of functions describing the Cl atom’s electronic state and functions describing the H_{2} 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 H_{2} rotational quantum number .
In this work, we limit the possible values to and ; this leads to a basis set with 36 orthonormal direct product electronicrotational basis functions. In the absence of SO coupling, however, the Hamiltonian matrix splits into two uncoupled blocks that are, respectively, associated with the spinup and spindown Slater determinants. The two blocks have a common set of eigenvalues.
We could exploit the cylindrical symmetry of the ClH_{2} 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 H_{2} rotational angular momentum, is rigorously good in the absence of SO coupling. If we were to express the SOfree 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 SOfree ClH_{2} 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 ClH_{2} wave functions with , 1, 2, or 3 as , or states, respectively.
For a fixed value of the ClH_{2} distance , the elements of the electronicrotational Hamiltonian matrix can be written as where and represent spacefixed Cartesian directions, is the H_{2} rotational constant (here taken to have the value 59.06 cm^{−1}), and describes the ClH_{2} 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 spinup and a spindown 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 H_{2} angles and , the potential energy portion of (24) becomes where Here, we compute the integrals in (26) by numerical quadrature, using the 240point spherical design specified in [13].
To evaluate , we introduce a rotated system of Cartesian axes, which we denote . The and axes rotate in the spacefixed plane as the H_{2} azimuthal angle changes, so that the H_{2} molecule always resides in the plane. The Slater determinants and , defined in the spacefixed 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 spacefixed 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, ClH_{2} electronicrotational basis functions with different values of are coupled only by the trigonometric functions of that are shown explicitly in (30) through (34). and couple H_{2} rotational states with or 2, couples H_{2} rotational states with , and and couple H_{2} rotational states with . has no dependence on and therefore does not couple basis functions with different values.
Constructing and diagonalizing the electronicrotational Hamiltonian matrix at a series of values gives the ClH_{2} 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 H_{2} molecule as , while those in groups B and C correlate with states of the H_{2} molecule in this limit. All of the B curves, at large , correlate with electronicrotational basis functions that involve the Cl atom’s Slater determinant. The C curves, on the other hand, correlate at large with electronicrotational basis functions involving the and Slater determinants of the Cl atom.
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 spacefixed axis, which is the direction of approach of the incoming H_{2} 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 shortrange repulsive overlap interactions between Cl and H_{2} and permits closer approach of the incoming H_{2} 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 spacefixed axis for large . For the curve, the H_{2} rotational state at large corresponds to the spherical harmonic. In this state, the H_{2} molecule can be viewed classically as rotating in the plane. This orientation of the rotating H_{2} molecule both maximizes its attractive electrostatic quadrupolequadrupole interaction with the Cl atom and minimizes shortrange ClH_{2} 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 electronicrotational basis functions that are direct products of the H_{2} spherical harmonics and the Cl atom’s Slater determinant. As , therefore, the character of this curve is associated with the rotational motion of the H_{2} 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 electronicrotational 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.
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 H_{2} molecule. The crossover in the description of the curve of group A reflects a strong interaction between these two zeroorder 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 ClH_{2} 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 zeroorder state makes to the curve. Figure 4 indicates that this zeroorder 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 paraH_{2} molecule and a Cl atom described by the Slater determinant.
4. ClH_{2} Including SpinOrbit 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 ClH_{2} van der Waals region, the triatomic system’s SO matrix elements are nearly independent of the ClH_{2} 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 ClH_{2} 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 H_{2} 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 blockdiagonalize the Hamiltonian matrix, we use to classify the symmetry of the ClH_{2} 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 ClH_{2} energy curves.

As Table 1 indicates, the state of the H_{2} molecule, which has as well, can either form a ClH_{2} state with by combining with the state of the Cl atom or form a ClH_{2} state with by combining with the state or the state of the Cl atom. Consequently, there are three distinct ClH_{2} curves that correlate as with the ground rotational state of the H_{2} 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 paraH_{2} molecule.
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 ClH_{2} 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 H_{2} molecule. Consequently, of the three curves shown in Figure 5, this curve allows the H_{2} 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 H_{2} molecule distorts the electronic structure of the Cl atom. To answer this question, we take the ClH_{2} 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 H_{2} fragment’s rotational states. Figure 6(a) shows, as a function of the ClH_{2} distance , the fractional contributions that these two pure atomic SO states make to the corresponding ClH_{2} dimer wave functions. For large , the Cl electronic structure is essentially unperturbed by the H_{2} molecule. As decreases below about , however, the impinging H_{2} 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)
For these two curves, ClH_{2} interactions also perturb the rotational degrees of freedom of the H_{2} fragment at small values. This leads to a mixing of some character into the H_{2} molecule’s rotational wave function. To measure this perturbation, we project the two ClH_{2} wave functions onto the H_{2} 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 H_{2} 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 H_{2} rotational state.
Figures 7 through 9 complete our survey of the ClH_{2} 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 H_{2} 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 shortrange 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.
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 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 H_{2}; the Cl atomic state coupled with the rotational state of H_{2}; and the Cl atomic state coupled with the rotational state of H_{2}. 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 H_{2} 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 ClH_{2} dimer exhibits strong electronicrotational coupling, these regions of configuration space are at moderately high energies on the repulsive wall of the dimer’s potential energy surface.
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 H_{2} molecule; the others correlate with the H_{2} 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 H_{2} molecule.
We close by computing the ClH_{2} interaction curves , including the effects of SO coupling, under the assumption that the H_{2} 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 ClH_{2} dimer wave function; this leads to a matrix representation of the ClH_{2} 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 H_{2} 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 H_{2} fragment. The three curves obtained from the larger Hamiltonian matrix were first shown in Figure 5.
(a)
(b)
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 BH_{2} and AlH_{2} dimers [7], for which the zeropointcorrected binding energies change by 15% and 22%, respectively, when the H_{2} fragment is frozen in its state. It appears that the anisotropy of the underlying diabatic potential energy surfaces is substantially larger for the BH_{2} and AlH_{2} systems than for the ClH_{2} system; this leads to a greater perturbation of the H_{2} molecule’s rotational wave function in the region of the van der Waals minima of the BH_{2} and AlH_{2} systems.
For the two curves shown in Figure 12, we see that the pure approximation reproduces the actual ClH_{2} 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 H_{2} 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 ClH_{2} interaction for values below ; this is the region of configuration space where strong electronicrotational 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 Cldoped solid paraH_{2} are carried out at K, it is unlikely that the Cl atom’s H_{2} neighbors spend a significant portion of time exploring this portion of the potential energy curve.
5. Summary
We have computed a series of onedimensional energy curves for the ClparaH_{2} dimer by diagonalizing the dimer’s combined electronicrotational 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 electronicrotational coupling in the ClH_{2} dimer is strong for ClH_{2} distances near . In the system where SO effects are ignored, this coupling gives rise to an avoided crossing between two ClH_{2} states with symmetry: one that correlates at large with the H_{2} 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 ClH_{2} dimer’s states: one of which correlates at large with the H_{2} rotational level and two of which correlate with the rotational level.
By removing H_{2} rotational states from the direct product basis set used to express the ClH_{2} Hamiltonian, we can assess how the assumption that the paraH_{2} fragment retains its pure identity might affect the shape of the ClparaH_{2} energy curves. We find that the positions and depths of the system’s van der Waals minima are virtually unaffected when the H_{2} fragment is restricted to its pure rotational state. This suggests that the H_{2} molecules in Cldoped solid paraH_{2} 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 CHE0848841.
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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.