Weakly bound molecular dimers: Intramolecular vibrational fundamentals, overtones, and tunneling splittings from full-dimensional quantum calculations using compact contracted bases of intramolecular and low-energy rigid-monomer intermolecular eigenstates

Accurate calculation of the vibrational levels of weakly bound, van der Waals and hydrogen-bonded dimers, where both monomers are diatomic or polyatomic molecules, has posed a major challenge to theory for the past couple of decades. The challenges arise from several properties of these complexes. One of them is high dimensionality—dimers of diatomic molecules, e.g., (HF)₂, are six-dimensional (6D), while those composed of triatomic molecules, e.g., (H₂O)₂, are 12D. The other stems from the features of the intermolecular potential energy surfaces (PESs) of the weakly bound dimers,^1,2 and floppy molecular systems, in general:³ shallow, often multiple minima separated by low barriers, allowing for coupled, low-frequency, strongly anharmonic large-amplitude intermolecular vibrations, extensive wave function delocalization, and tunneling. In addition, dimers composed of identical monomers, such as (HF)₂ and (H₂O)₂, exhibit a complex hydrogen-bond network rearrangement dynamics that takes place along low-barrier tunneling pathways connecting equivalent minima, one such pathway for (HF)₂ and (HCl)₂^2,4 and three pathways for the water dimer.^5,6 These tunneling motions manifest in the splitting of each (ro)vibrational level of the dimers into sets of two^2,4 or more^5,6 levels.

In weakly bound dimers, the vibrational modes fall in two distinct classes: intramolecular vibrations of the monomers and intermolecular vibrations of the complex. Typically, the intramolecular frequencies are at least an order of magnitude higher than those of the intermolecular modes. As a result, in the vast majority of cases, the calculations of the vibrational levels of the dimers have relied on an adiabatic, Born-Oppenheimer separation between their high-frequency intramolecular and low-frequency intermolecular degrees of freedom (DOFs). In practice, generally, this has meant that the monomers are taken to be rigid, and only the coupled intermolecular vibrations of the dimer are considered. More sophisticated adiabatic treatments have been developed as well, for (HF)₂^7,8 and (H₂O)₂,^9–11 allowing the inclusion of the monomer flexibility in an approximate manner.

The rigid-monomer and vibrationally adiabatic eigenstate calculations are often highly accurate and yield results in very good agreement with the experiment. Nevertheless, it is important to have the ability to perform rigorous full-dimensional bound-state calculations, for flexible monomers, treating the intra- and intermolecular DOFs of the dimer as fully coupled. Only such calculations can provide a definitive and complete assessment of the accuracy of both the dynamical approximations and the potential energy surface (PES) employed. Moreover, certain spectroscopic properties of the dimers, such as the intramolecular vibrational frequencies and their shifts from the gas-phase values, can be obtained reliably only from the full-dimensional quantum treatment. Of course, going from the rigid- to flexible-monomer treatment increases significantly the dimensionality of the problem, for example, from 4D to 6D for (HF)₂ and from 6D to 12D for (H₂O)₂. This greatly increases the difficulties and the cost of both computing the PESs and the bound-state calculations on them, relative to the rigid-monomer approach.

To date, full-dimensional quantum bound state calculations have been reported for only a few hydrogen-bonded dimers: (HF)₂ and isotopologs,^12–17 (HCl)₂,^18,19 and most recently (H₂O)₂.²⁰ Such calculations are demanding computationally already when the monomers are in the ground vibrational state. But extending these computations to the case when one monomer or both monomers are vibrationally excited poses an entirely new set of difficult challenges. The energies of the intramolecular vibrational fundamentals of the monomers are typically high, ≈3900 cm⁻¹ for (HF)₂ and in the case of (H₂O)₂, ≈3650 cm⁻¹ and ≈3750 cm⁻¹ for the symmetric and asymmetric stretch fundamentals, respectively, and ≈1600 cm⁻¹ for the bending vibration. The energies of the corresponding overtone excitations are almost a factor of two larger. At these energies, the density of the intermolecular vibrational states belonging to the intramolecular ground-state manifold is very high. The intramolecular vibrational fundamentals and overtones of the monomers generally lie above the dissociation limit of the dimer. Consequently, these states are actually quasibound, sharp, long-lived resonances embedded in the continuum. Although the focus of this paper is on molecular dimers, it should be mentioned that vibrational levels of H₂O-atom (halide) dimers, including their intramolecular fundamentals (and water-bend overtone) and intermolecular vibrational eigenstates, have been obtained from full-dimensional quantum calculations.²¹ 

The assumption made in all fully coupled bound-state calculations to date of weakly bound dimers with the monomers in vibrationally excited states^{13,16,17,19,20} is that for obtaining accurate energies of intramolecular vibrational fundamentals and overtones, it is essential to have converged highly excited intermolecular vibrational eigenstates in the intramolecular ground-state manifold that are energetically close to the intramolecular excitation(s) considered. This daunting task was approached in two ways. In one, employed in the earlier treatments,^13,16,19 a (contracted) basis was tailored to the fundamental (and overtone) vibrational levels of interest by truncating it from both above and below their energies. This leaves bands of basis functions, each centered on the particular intramolecular fundamentals. More recently, it has become possible, at great computational effort, to compute directly the vibrational levels of the HF dimer from the ground state up, with both monomers in their ground vibrational states and also when one is vibrationally excited.¹⁷ A very large basis set of dimension 3 600 000 had to be used. Clearly, extending these calculations to the HF-stretch overtone manifold would be prohibitively costly. Carrington’s 12D calculations of the vibrational levels of (H₂O)₂ reach the manifold of the excited water bend vibration.²⁰ They also used a large basis, one with the dimension 1 387 000, and did not treat the higher-energy water stretching fundamentals. The high density of eigenstates generated by these calculations^17,20 creates an additional problem of identifying correctly the energy levels corresponding to the fundamental (and overtone) monomer excitations among the many closely spaced quasibound continuum states. Thus, if the full-dimensional calculations of the monomer fundamentals and overtones in molecular dimers must indeed include highly excited intermolecular vibrational states of the ground-state manifold lying below, and all the way up to, these intramolecular excitations, then in the foreseeable future, they will remain prohibitively costly for routine applications, in particular, when one monomer or both monomers are polyatomic.

However, from our recent fully coupled quantum 6D calculations of the vibration-translation-rotation (TR) eigenstates of H₂, HD, and D₂ inside the clathrate hydrate cage,²² it emerged that this widely held view may not be correct, making the task much easier than previously thought. In that study, we showed that a converged first excited (⁠$v$ = 1) vibrational state of the caged H₂ can be obtained without converging the very large number of intermolecular translation-rotation (TR) states belonging to the $v$ = 0 manifold up to the energy of the intramolecular stretch fundamental, ≈4100 cm⁻¹. In fact, it sufficed to have only a modest number of converged TR states in the $v$ = 0 zero manifold up to at most 400–450 cm⁻¹ above the ground state, far below the H₂ stretch fundamental, and absolutely none within several thousand wave numbers of its energy.²² This finding was most surprising and was explained in terms of extremely weak coupling between the intramolecular stretch of the guest H₂ and highly excited $v$ = 0 intermolecular TR states. This conclusion is supported by the remarkable agreement of the quantum 6D results with those from our quantum 5D (rigid H₂) treatment of the frequency shift in this system,²³ which assumes complete decoupling between the H₂ intramolecular vibration and the TR modes.

Weakly bound molecular dimers (and larger clusters) also exhibit very weak coupling between the high-frequency intramolecular modes and the low-frequency intermolecular vibrations. The valuable insight gained in the above study of the hydrate-encapsulated H₂²² suggested to us that it should be possible to calculate efficiently accurate energies of the fundamental, and overtone, excitations of their intramolecular modes by means of full-dimensional, fully coupled quantum bound-state calculations, employing a modest-size contracted basis for the intermolecular DOFs covering only a small portion of the energy range far below the fundamental and overtone excitations of interest.

In this paper, we present such an approach to the calculation of the vibrational states of weakly bound molecular dimers, for the monomers in both the ground and excited intramolecular vibrational states, with the emphasis on the latter. Given the full-dimensional vibrational Hamiltonian, Ĥ, associated with such states, which depends on the collection of intermolecular coordinates, Q, and intramolecular coordinates, q, we effect a separation of that operator into two reduced-dimension Hamiltonians as follows:

This partitioning is done solely for the purpose of generating contracted intra- and intermolecular basis functions, as described below. In Eq. (1), Ĥ_inter(Q; q₀) is a rigid-monomer intermolecular vibrational Hamiltonian corresponding to monomer geometries defined by a set of constant values for the intramolecular coordinates q = q₀. Ĥ_intra(q; Q₀) is a Hamiltonian for all the intramolecular vibrational DOFs when all the intermolecular coordinates are fixed to reference values Q = Q₀. Finally,

We then focus on computing the vibrational eigenstates of Ĥ in Eq. (1) that correspond to the monomers in the ground vibrational state, and also their fundamental and overtone excitations, together with the relatively low energy intermolecular vibrational states in each of these intramolecular state manifolds. In this task, we rely on the supposition that, by a judicious choice of q₀ and Q₀, a basis set very well-suited to that problem might be constructed from a small number of products of the eigenfunctions of Ĥ_inter and Ĥ_intra. The physical reasoning behind this is that the geometries of semirigid molecules do not change very much with the intramolecular vibrational state. Hence, intermolecular interaction potentials and intermolecular-state manifolds should not be markedly dependent on the intramolecular vibrational states in which the monomers reside. The reason why the above contracted basis functions are expected to provide a compact, efficient representation of the problem is that they are already tailored to it and reflect in part the PES and the intermode couplings.

We demonstrate the method here in application to the HF dimer. This is one of two paradigmatic hydrogen-bonded dimers, the other being (H₂O)₂, providing the simplest example of the hydrogen-bond rearrangement dynamics ubiquitous in hydrogen-bonded systems of all levels of complexity. The most intensely studied aspect of the dynamics of the HF dimer (and HCl dimer) has been the large-amplitude tunneling motion between the two equivalent equilibrium configurations of (HF)₂. It interchanges the roles of the two monomers as hydrogen-bond (or proton) donor and acceptor (also referred to as the “bound” and “free” monomers, respectively) and, as mentioned earlier, splits the levels of the dimer, both inter- and intramolecular, into closely spaced pairs of states.^2,4,24 Both experiments and calculations have revealed a strong dependence of the tunneling splitting on the excitation of any of the vibrational modes of the dimer,^2,4 evidence of the highly coupled nature of the tunneling dynamics in the HF dimer.

The HF dimer provides an excellent test of the accuracy and efficiency of the approach presented in this work, for two reasons. First, converged vibrational levels, and their tunneling splittings, from full-dimensional (6D) calculations have been reported for the intermolecular states built off the ν₁ + ν₂ ≤ 2 dimer intramolecular states for several PESs of the HF dimer.^12–17 (In denoting the intramolecular vibrational excitations, we employ the usual convention¹³ by which ν₁ and ν₂ refer to the stretching vibrations of the free and bound HF, respectively.) Hence, there are ample literature results to which we can compare the results of the present approach. Second, intramolecular vibrational frequency shifts upon complexation are quite substantial for the HF dimer. Thus, the species represents a stringent test case for our method, given the assumptions on which it is based. What makes this comparison particularly discriminating and insightful is the fact that all full-dimensional bound-state treatments to date for HF-stretch excited states have involved calculations of a large number of highly excited intermolecular vibrational eigenstates below and above the energies of the intramolecular fundamentals^13,16,17 and overtones.¹³ In contrast, in the present approach, the highest-energy intermolecular vibrational states calculated in the ν₁ = ν₂ = 0 manifold lie about 3000 cm⁻¹ below the ν₁ and ν₂ fundamentals.

This paper is organized as follows: In Sec. II, we describe the specifics of the application of the above to the HF dimer. This includes presenting the vibrational Hamiltonian of the species and its partitioning, the procedures for calculating the intermolecular and intramolecular vibrational basis states, and the details associated with diagonalizing the full Hamiltonian. Section III presents the results of the method for eigenstates in the intramolecular ν₁ + ν₂ = 0, 1, and 2 manifolds of (HF)₂. These results, for two dimer PESs, are compared with the ones from the literature. While the latter were obtained by approaches more computationally intensive than ours, agreement is nevertheless very good. We also present in Sec. III metrics that speak to the quality of 6D basis constructed by the present approach. Here, we show that a very small number of such basis states contributes overwhelmingly to each of the low-energy states in the ν₁ + ν₂ ≤ 2 manifolds. Section IV presents the conclusions.

The J = 0 6D vibrational Hamiltonian for the HF dimer can be written in terms of the body-fixed (BF) Jacobi coordinates Q = (R, θ₁, θ₂, ϕ) and q = (r₁, r₂).¹² These coordinates are defined in terms of the vector R, with the magnitude R, which goes from the center of mass of HF # 1 to that of HF # 2, and the vectors r_i (i = 1, 2), with the magnitudes r_i, each of which goes from the F nucleus of the relevant monomer moiety to the H nucleus of that moiety. θ_i is the angle between R and r_i, while ϕ is the dihedral angle between r₁ and r₂ along R. One then has

where μ_D is the reduced mass of the dimer, μ is the reduced mass of each monomer, $j^i$ is the vector operator associated with the angular momentum of monomer i in the BF frame, and $ĵi2$ is the operator associated with the square of that angular momentum. Two variants of V_tot, the full 6D PES for the dimer, were used herein—the SQSBDE PES by Suhm and Quack²⁵ and the recent surface reported by Huang et al.¹⁷ We will henceforth refer to the latter as the HYZX PES.

We proceed by expressing the Ĥ from Eq. (3) in the form of Eq. (1). The rigid-molecule 4D intermolecular vibrational Hamiltonian, Ĥ_inter, is defined as

where r is a constant chosen as a representative HF bond distance and

The 2D intramolecular vibrational Hamiltonian, Ĥ_intra, is defined as

where V_intra(r₁, r₂) is an intramolecular potential energy surface corresponding to a particular set of intermolecular coordinates (for more details, refer to Subsection II D). Finally, the remainder, ΔĤ, is given by

where

To calculate the low-energy 4D eigenvalues and eigenvectors of Ĥ_inter [Eq. (4)], we use the Chebyshev version²⁶ of filter diagonalization.²⁷ That method requires the repeated application of Ĥ_inter on a random initial state vector. We express that vector in the primitive basis

where the |R_s⟩ (s = 1, …, N_R) constitute a Morse discrete variable representation (DVR)²⁸ and the $Yjm$ (j = 0, …, j_max, |m| ≤ m_max ≤ j) are spherical harmonics. In this basis, the matrix elements of the kinetic-energy portion of Ĥ_inter (i.e., $T^inter$⁠) are given by

where

The $⟨Rs′|−12μD∂2∂R2|Rs⟩$ can be readily computed numerically. Hence, operation with $T^inter$ on a state vector is a straightforward matrix-on-vector multiplication.

To operate with V_inter, we first transform the state vector from the basis representation to the 4D grid representation (R_s, θ_1,k, θ_2,l, ϕ_n), where the θ_1,k and θ_2,l (k, l = 1, …, N_θ) are Gauss-Legendre quadrature points and the ϕ_n (n = 1, …, N_ϕ) are Fourier grid points. We then multiply the transformed state function by V_tot(R_s, θ_1,k, θ_2,l, ϕ_n, r, r) and, finally, transform the result back to the original basis representation.

The values of parameters relevant to the diagonalization of Ĥ_inter are given in Table I. For r, we use the average of the expectation values of the monomer’s internuclear distance for the $v$ = 0 and $v$ = 1 vibrational states (SQSBDE potential). To generate the intermolecular Morse DVR, |R_s⟩, we used the relevant value of μ_D from the masses quoted in Table I and the Morse potential

with the values of the parameters D_e, α, and R_e given in Table I. In addition to limiting the 4D basis size by the finite values for N_R, j_max, and m_max, we also used a rotational-energy cutoff to eliminate basis functions with E_rot values greater than $Erotmax$⁠. Here,

where B ≡ 1/(2μr²). The B value used is given in Table I. The total primitive intermolecular basis-set size for the quoted parameters is 7230. The size of the 4D grid is 23 520.

We denote the eigenstate results of the intermolecular calculation generically as |κ⟩ (κ = 1, …, N_inter) and the associated eigenvalues as $Eκinter$⁠. Table II presents the results relevant to the 10 lowest-energy |κ⟩ computed for both the SQSBDE and HYZX PESs. The assignments given in the second column—(ν₃ν₄ν₅ν₆)—refer to the quantum numbers associated with the intermolecular modes in each state:¹² ν₃ denotes that of the in-plane, antigeared (or cis) bend, ν₄ denotes that of the intermolecular stretch, ν₅ denotes that of the in-plane geared (or trans) bend, and ν₆ denotes that of the torsion. The third column gives the G₄ irreducible representation to which each state belongs—we use the A⁺, B⁻, etc., notation of Huang et al.¹⁷ The ΔE values in Table II are reasonably close to analogous values from full 6D calculations, though differences between the two often amount to several wavenumbers.

In constructing 6D bases, we employed seven different intermolecular basis-set sizes, namely, N_inter = 2ⁿ, where n = 1, 2, …, 7. These bases included all intermolecular vibrational states with ΔE values less than or equal to approximately 1, 129/129, 248/251, 374/410, 532/564, 694/732, and 908/945 cm⁻¹, respectively (the numbers are for the SQSBDE/HYZX PESs). Thus, none of the bases contained intermolecular vibrational states with energies anywhere close to those of the HF stretching fundamentals near 3900 cm⁻¹.

To compute the 2D eigenfunctions and eigenvalues of Ĥ_intra in Eq. (6), one first needs to specify V_intra(r₁, r₂). One straightforward choice is to take V_intra as the sum of the free-monomer 1D potential-energy curves (i.e., V_tot for R = ∞). A better choice, perhaps, is to construct a dimer-adapted, effective intramolecular potential by including some information on how the intermolecular interactions in the dimer affect the intramolecular ones.¹⁶ We adopt the latter approach. Specifically, we obtain V_intra(r₁, r₂) from V_tot by fixing the intermolecular coordinates to their values at one of the global minima of V_tot and then by symmetrizing the resulting 2D function with respect to the interchange of r₁ and r₂.

To diagonalize Ĥ_intra, we work with a primitive basis composed of products of 1D Morse DVRs²⁸ for r₁ and r₂: $β1β2≡r1,β1r2,β2$⁠, β₁, β₂ = 1, …, N_r. These DVRs were generated from the 1D vibrational eigenstates corresponding to the reduced mass of HF and a Morse potential generated from the values of the parameters $Deintra$⁠, α^intra, and $reintra$ given in Table I. The matrix elements of Ĥ_intra in this basis are easily obtained numerically. Direct diagonalization of the matrix yields the intramolecular eigenvectors, |γ⟩, and associated eigenvalues, $Eγintra$⁠, γ = 1, 2….

Table III shows the results pertaining to the six lowest-energy |γ⟩ computed per the above for both potentials. Each state is given an assignment in the form |$v$⁠, $v$′⟩^±. Here, $v$ is the number of HF stretch quanta that one monomer has and $v$′ is the number of stretch quanta that the other one has. Since there is interchange symmetry, the states are of the form $v,v′±=1/2[v,v′±v′,v]$ (when $v$ ≠ $v$′), where in the kets on the right the first quantum number refers to monomer #1 and the second to monomer #2. Clearly, when $v$ = $v$′, the states are simply |$v$⁠, $v$⟩. In constructing 6D bases, we included various numbers, N_intra, of the |γ⟩ intramolecular states. Specifically, bases in which all |γ⟩ corresponding to $v$ + $v$′ ≤ 2, 3, 4, and 5 were used. For these, N_intra = 6, 10, 15, and 21, respectively.

With the 4D intermolecular and 2D intramolecular states computed, we construct the 6D product basis

and use that basis to compute the eigenvectors and eigenvalues of the full 6D vibrational Hamiltonian Ĥ in Eq. (3) by direct diagonalization. Of course, in this basis, the Ĥ_inter + Ĥ_intra portion of Ĥ is diagonal with the matrix elements given by

Thus, the main task is to compute the matrix elements ⟨κ′γ′|ΔH|κγ⟩. Those involving the kinetic-energy part of ΔH [see Eq. (7)] are given by

and $⟨κ′|ĵi2|κ⟩$ and $⟨γ′|ri−2|γ⟩$ are readily obtained by expressing the |κ⟩ and |γ⟩ in terms of the primitive basis functions |R_sj₁j₂m⟩ and |β₁β₂⟩, respectively. To compute each of the matrix elements ⟨κ′γ′|ΔV |κγ⟩, we transform |κγ⟩ to the 6D grid representation $Rs,θ1,k,θ2,l,ϕn,r1,β1,r2,β2$⁠, multiply the result by the value of ΔV at each grid point, transform that result back to the 6D basis representation, and evaluate the overlap of that with ⟨κ′γ′|. The 6D grid size was 2 845 920, the product of the intermolecular and intramolecular grid sizes defined by the parameters listed in Table I. As noted above, a number of different basis set sizes, as determined by the values of (N_inter, N_intra), were employed. These were as small as 12 functions (2, 6) and as large as 2688 (128, 21).

Our focus in this work is primarily on the efficient calculation of the energies of intramolecular vibrational excitations in molecular dimers. As such, the rates at which such computed energies converge with respect to basis-set size are especially relevant. Table IV presents the results that pertain to how convergence depends on the size of the intermolecular portion of the 6D basis (i.e., N_inter) with the intramolecular portion remaining fixed (N_intra = 21). In Table IV, which refers to the calculations performed for the SQSBDE potential, each of the values quoted for N_inter = 2, 4, … , 64 in a given row represents the difference between a computed quantity for that N_inter value and the analogous quantity calculated for N_inter = 128 (the “reference basis”). The latter quantity is given as the right-most entry in the row. Furthermore, ΔE₀ refers to the difference between the ground-state energy computed for a given N_inter and that computed for the reference basis. $Δνi±$ and $Δ(2νi)±(i=1,2)$ values refer to analogous differences in the energies of the HF-stretch fundamentals and overtones, respectively. The right superscripts on the ν_i denote the symmetry behavior of the relevant excited vibrational state with respect to the HF interchange, with “+” meaning symmetric and “−” meaning antisymmetric with respect to such interchange.

One notes four points from the results in Table IV. First, except for the overtone excitation energies involving ν₂, all the other quantities are converged to within several hundredths of a wavenumber at N_inter = 64 (total basis-set size equal to 1344). Second, for the vibrational fundamentals, all but the two smallest bases produce results that differ by only several tenths of a wavenumber from the reference-basis value. Third, the excitation energies associated with the two components of an interchange-tunneling doublet (i.e., $νi±$ or $(2νi)±$⁠) behave very similarly with respect to N_inter. This means that tunneling splittings are relatively insensitive to that parameter, varying only on the order of about 0.01 cm⁻¹ over the entire parameter range.

Finally, there is a marked difference in convergence vs N_inter for excitations involving ν₁ and ν₂ modes, the latter converging more slowly. This difference can be understood in terms of the different degrees of coupling between the free (ν₁) vs the bound (ν₂) HF stretch and the intermolecular DOFs. The ν₂ mode is intimately associated with the hydrogen-bonding interaction, and its level of excitation should have a greater effect on intermolecular excitations than that of ν₁. It is precisely the magnitude of this effect that determines how well a finite set (i.e., N_inter) of rigid-monomer intermolecular states can overlap with the intermolecular states associated with different degrees of intramolecular excitation. Notably, even for the fundamental stretching excitations of a hydrogen-bond-donating moiety and N_inter in just the double digits, such overlap is still good enough to yield excitation energies within tenths of a wavenumber of convergence.

Table V pertains to the effect of intramolecular basis-set size on convergence. The format is analogous to that of Table V, except that N_intra varies across a given row and N_inter is fixed to 128. As before, the reference basis is (N_inter, N_intra) = (128, 21). The results in Table V show that convergence to within a few tenths of a wavenumber of the reference basis holds for both the vibrational ground state and the intramolecular fundamentals when the intramolecular basis contains only $v1$ + $v2$ ≤ 2 (N_intra = 6) dimer-adapted states. On the other hand, the intramolecular first-overtone excitations require a basis of N_intra = 15 states (⁠$v1$ + $v2$ ≤ 4) to be comparably converged.

Note also that the ν₁ and ν₂ excitation energies converge at fairly similar rates with respect to N_intra, in contrast to the situation with respect to N_inter convergence. This behavior is probably due in part to our use of a dimer-adapted intramolecular vibrational basis. In any case, in looking ahead to the calculation of intramolecular fundamental frequencies in other dimers, the results here suggest that intramolecular bases containing states with $v$_1,i + $v$_2,i ≤ 3 for each monomer mode i will likely produce values within about a tenth of a wavenumber of the fully converged fundamental frequencies.

The convergence data of Subsection III A suggest that the product basis |κγ⟩ constructed as per Sec. II projects very effectively onto those dimer vibrational eigenstates that have low levels of intermolecular excitation. This point can be made more emphatically by examining directly the basis-set composition of several of the 6D eigenstates. Table VI presents such results computed with the (64, 21) basis for (a) the lowest-energy interchange-tunneling-doublet states in the intramolecular ground-state manifold (denoted $ν0±$⁠), (b) the two pairs of states associated with the interchange-tunneling doublets of the bound and free fundamental HF-stretching excitations (⁠$ν2±$ and $ν1±$⁠, respectively), and (c) the analogous two pairs of states in the first-overtone manifold [$(2ν2)±$ and $(2ν1)±$⁠, respectively]. In Table VI, we denote the intermolecular portion of a basis state as |κ⟩ = |ν₃ν₄ν₅ν₆⟩ as described in Sec. II C and the intramolecular portion as |γ⟩ = |$v$⁠, $v$′⟩^±, as described in Sec. II D. The values appearing for each state, |ψ⟩, correspond to |⟨κγ|ψ⟩|². The “Total” column at table bottom contains the sum of the numbers appearing to the left in the same row. While the results in Table VI are for the SQSBDE potential, similar results are obtained for the HYZX potential.

From Table VI, one sees that the contribution of a single basis function (out of 1388) overwhelmingly dominates for each of the two lowest-energy states in the ground-state manifold. For each of the four lowest-energy eigenstates in the ν₁ + ν₂ = 1 manifold, just two basis states account for over 99% of that state’s norm. Finally, just two basis states dominate, to almost the same extent, in contributing to each of the low-energy states in the first-overtone manifold. Quite clearly, the quality of the basis with respect to the low-energy eigenstates within each of these intramolecular manifolds is very high.

To assess the accuracy and utility of the methodology reported here, it is clearly important to compare our results with those from previous 6D calculations of (HF)₂ vibrational states. For the SQSBDE PES, such results have been reported by Zhang et al.¹² and Vissers et al.¹⁶ for the ν₁ + ν₂ = 0 ground-state manifold. For the ν₁ + ν₂ = 1 fundamental manifold, both Wu et al.¹³ and Vissers et al.¹⁶ have reported results. Finally, Wu et al.¹³ have also reported results for the ν₁ + ν₂ = 2 overtone manifold. Huang et al.¹⁷ have reported results for the ν₁ + ν₂ = 0 and ν₁ + ν₂ = 1 manifolds on the HYZX PES.¹³ In Tables VII–IX, we compare our results with the relevant literature values. In each of these tables, we present two sets of values from this work. These correspond, respectively, to (N_inter, N_intra) = (128, 21), the largest 6D basis used (dimension equal to 128 × 21 = 2688), and a much smaller 6D basis with (N_inter, N_intra) = (32, 21), with dimension of 672. The two bases differ in the number of the 4D intermolecular eigenstates included, 128 and 32, respectively, while the number of the 2D intramolecular eigenstates, 21, is the same in both bases. Our purpose in presenting both sets is to give the reader a sense of the degree to which accuracy is affected by the use of the smaller number of 4D intermolecular eigenstates in the full 6D basis.

Table VII presents the results relevant to the ν₁ + ν₂ = 0 ground-state manifold of the dimer. The assignments given in the first two columns are in the same format as in Table II. The third column at the top reproduces the full 6D results of Huang et al.¹⁷ and the adjacent fourth and fifth columns show the results obtained by the method reported here. Similarly, the third and fourth columns in the bottom-half of the table reproduce literature results from Zhang et al.¹² and Vissers et al.,¹⁶ respectively, and the fifth and sixth columns show the results that we have obtained.

Tables VIII and IX present analogous 6D results for the ν₁ + ν₂ = 1 fundamental and ν₁ + ν₂ = 2 overtone manifolds, respectively. In these tables, the assignments are given in terms of all six vibrational quantum numbers ν₁ to ν₆.

From Tables VII and VIII, one notes that there is generally excellent agreement between our results and those obtained by others for states in the intramolecular-ground-state and fundamental manifolds. In the former, the differences in comparable ΔE values are generally well within ∼ 0.1 cm⁻¹ for both PESs and for both of our basis sets. More important, our results for both 6D bases, with the respective dimensions of 672 and 2688, pertaining to the states in the low-energy part of the ν₁ + ν₂ = 1 fundamental manifold, differ from prior results on both PESs by, at most, several tenths of a wavenumber. This means, for example, that for the HYZX PES, our approach even when using the smaller of the two 6D bases having the dimension of only 672 gives results in excellent agreement with those of Wu et al.,¹³ employing the basis of dimension 3 600 000. Thus, despite the considerable complexation-induced shifts in the HF intramolecular stretching frequencies (∼−20 to −90 cm⁻¹, depending on the PES and whether the HF is free or bound), the compact-basis approach that we have outlined here is capable of computing accurate energies for intramolecular fundamental excitations.

As to first-overtone excitation energies, ones sees from Table IX that the discrepancies between our results and those from Wu et al.¹³ are about an order of magnitude larger (several wavenumbers) than those that characterize the results for the ground-state and intramolecular-fundamental manifolds. This raises the question as to which set of overtone results is likely to be more accurate. In this regard, we note that the intramolecular portion of the basis employed by Wu et al.¹³ (a) is not dimer-adapted (they used a monomer-stretch basis obtained from the free-HF potential) and (b) includes only states for which $v1$ + $v2$ ≤ 2, where $v1$ and $v2$ are the number of quanta for each of the HF monomers. The intramolecular portion of both our bases is dimer-adapted and includes all states for which $v1$ + $v2$ ≤ 5. These differences in basis, along with the fact that the overtone states are computed by us to be at lower energies relative to the corresponding energies of Wu et al.,¹³ suggest that the former results are better converged than the latter. Further support for this interpretation is evident from the overtone results in Table V, which show that for our method the $v1$ + $v2$ ≤ 2 intramolecular basis leads to results that are several wavenumbers away from those obtained with the $v1$ + $v2$ ≤ 5 basis.

In addition to vibrational excitation energies, the tunneling splittings associated with pairs of states differing only with respect to their symmetry upon monomer interchange are experimentally accessible, sensitive to PES details, and, thus, important targets for computational studies to obtain accurately. Table X presents a comparison between selected tunneling splittings computed here and those reported by others. [In reporting these results, we employ the convention in which the interchange-tunneling splitting, δ(nν_i), of the nν_i vibrational excitation is defined as δ(nν_i) ≡ E(nν_i + ν₅) − E(nν_i).] The results show excellent agreement, at the level of hundredths of wavenumbers, between the present method (both basis sets) and the prior studies with respect to the tunneling splittings of low-energy states in the intramolecular ground-state, fundamental, and first-overtone manifolds.

It is evident from the above that the results obtained by our approach using rather small 6D basis sets (e.g., 672 and 2688 basis functions), regarding the low-energy parts of the intramolecular vibrational fundamental and overtone manifolds, including tunneling splittings, are in excellent agreement with the results in the literature obtained by methods that are considerably more intensive computationally. But what, in our opinion, is most striking conceptually is the fact that highly accurate values of the HF-stretch fundamentals and overtones (and their tunneling splitting) are obtained without computing any of the highly excited intermolecular vibrational states in the ground-state manifold that are energetically close to the ν₁ and ν₂ fundamentals. In fact, for the SQSBDE PES, the 4D intermolecular bases with 32 and 128 intermolecular eigenstates span the excitation energy range up to (only) 532 and 908 cm⁻¹, respectively, relative to the ground-state, far below the HF-stretch fundamentals at ≈3900 cm⁻¹. The situation is very similar for the HYZX PES. Thus, in our calculations, there is a very large energy gap, of several thousand wavenumbers, between the highest-lying computed intermolecular eigenstate of (HF)₂ in the intramolecular ground-state manifold and the energies of the ν₁ and ν₂ fundamentals. Yet, these results are in complete accord with those from (all other) calculations for (HF)₂ in the literature, where the intramolecular fundamentals and overtones are embedded in a dense set of high-lying intermolecular vibrational states.

What our results imply is that while highly excited intermolecular vibrational states of (HF)₂ may couple to the intramolecular vibrational excitations, this coupling is extremely weak and has at most a very minor effect on the energies of the intramolecular fundamentals and overtones.

We have introduced an efficient method for the calculation of intramolecular vibrational fundamentals and overtones of weakly bound molecular dimers by means of full-dimensional and fully coupled quantum bound-state calculations. The calculations also yield accurate relatively low-energy intermolecular vibrational levels within each intramolecular vibrational manifold, as well as the tunneling splittings. In this approach, the vibrational Hamiltonian of the dimer is partitioned, for the purpose of basis set generation, into two reduced-dimension Hamiltonians—a rigid-monomer intermolecular vibrational Hamiltonian with the intramolecular coordinates set to fixed values and the intramolecular vibrational Hamiltonian with the monomer geometries defined by setting the intermolecular coordinates to some reference geometry, e.g., those of the global minimum of the PES. The eigenvectors of these two intermediate Hamiltonians serve as contracted basis functions for the inter- and intramolecular degrees of freedom of the dimer. A certain number (small, as it turns out) of the lowest-energy inter- and intramolecular contracted basis functions is included in the final direct-product basis in which the desired eigenstates of the full-dimensional vibrational Hamiltonian are computed. The key to the efficiency of this approach is the expectation, stemming from our previous recent work,²² that obtaining accurate intramolecular vibrational excitations and tunneling splitting will require the inclusion of only a relatively small number of (rigid-monomer) intermolecular vibrational eigenstates in the 6D basis, spanning a range of energies much below those of the intramolecular vibrational fundamentals and overtones considered.

This expectation has been fully borne out by the vibrational-level calculations for (HF)₂ in this work. The system was chosen for two reasons: (1) It is an archetype of hydrogen-bonded dimers, and of hydrogen-bond rearrangement dynamics in general. (2) It provides an excellent test of the accuracy and efficiency of our approach, since several rigorous full-dimensional calculations of its vibrational levels have been reported for the HF monomers in their ground and excited intramolecular vibrational states. Detailed comparison is made between the results in the literature for two 6D PESs and those obtained in this work, for two modest-size 6D direct-product contracted bases, with dimensions of 672 and 2688, respectively. Our results, including the tunneling splittings, are in excellent agreement with those obtained by others, to within several tenths of a wavenumber for the ν₁ + ν₂ = 1 fundamental manifold and well within ∼0.1 cm⁻¹ for the ν₁ + ν₂ = 0 ground-state manifold, for both 6D PESs and both basis sets employed.

What stands out the most about this high level of agreement is the fact that it is achieved without computing the highly excited intermolecular vibrational states in the ground-state manifold that are energetically close to the ν₁ and ν₂ fundamentals. The 4D intermolecular bases used in our calculations with 32 and 128 eigenstates extend to (only) ∼500 and ∼900 cm⁻¹, respectively, above the ground-state, that are much less than the HF-stretch fundamentals at ≈3900 cm⁻¹. As a result, the highest-energy computed 6D intermolecular vibrational eigenstate of (HF)₂ in the intramolecular ground-state manifold lies several thousand wavenumbers below the energies of the ν₁ and ν₂ fundamentals.

This is in sharp contrast with all other full-dimensional vibrational-level calculations for (HF)₂ in the literature, which have invariably involved computation of a large number of highly excited intermolecular vibrational eigenstates below, and above, the energies of the intramolecular vibrational fundamentals and overtones, making such calculations significantly more demanding and costly than those in the present work. It has been long presumed that the quantitative description of the intermolecular vibrational level structure in the immediate vicinity of intramolecular excitations is essential for obtaining accurate energies of the latter, in this and other weakly bound dimers. Our results show that this assumption is not universally valid in the realm of weakly bound molecular complexes.

This crucial insight, together with the methodology presented in this paper, should allow extending full-dimensional, and fully coupled, quantum calculations of intramolecular vibrational fundamentals and overtones to weakly bound dimers with more vibrational degrees of freedom. One obvious target for such calculations is the water dimer, for which full-dimensional calculations have been performed, for the ground vibrational state of the monomers and the excited water bend, but not the excited water stretching modes.²⁰ 

Z.B. is grateful to the National Science Foundation for its partial support of this research through Grant No. CHE-1566085. P.F. is grateful to Professor Daniel Neuhauser for the generous sharing of his computer resources.