Range-separated density-functional theory applied to the beryllium dimer and trimer

Abstract

The beryllium dimer and trimer are, despite their small number of electrons, excellent systems for assessing electronic-structure computational methods. With reference data provided by multi-reference averaged coupled-pair functional calculations, we assess several variants of range-separated density-functional theory, combining long-range second-order perturbation theory or coupled-cluster theory with a short-range density functional. The results show that (i) long-range second-order perturbation theory is not sufficient, (ii) long-range coupled-cluster theory gives reasonably accurate potential energy curves, but (iii) provided a relatively large value of $\mu =1$ bohr⁻¹ for the range-separation parameter is used.

1 Introduction

Density-functional theory (DFT) has become in the last 20 years a widely used tool in quantum chemistry due to its good performance for structural and thermochemical properties at an advantageous cost even for medium-size molecules and larger entities. Nevertheless, research is still very active to improve the usual DFT approximations, notably for the treatment of long-range dispersion interactions and the treatment of genuine multi-reference cases.

To address the former issue, János Ángyán, to the memory of whom this article is dedicated, worked on the development of range-separated density-functional theory (RS-DFT), in a long-standing and fruitful collaboration with the present authors (see, e.g., Refs. [1,2,3,4,5]). A never-published work that was started with János some years ago was on the application of RS-DFT on the dimer and trimer of Beryllium, which contain both dispersion and multi-reference effects. For this special issue, we have reexamined this work and completed it.

Be₂ is weakly bound and has been the subject to a long-standing discussion between theoreticians and experimentalists [6,7,8,9,10,11,12,13], whereas Be₃ is a fairly stable aggregate, thus of completely different nature in bonding. This has been reviewed by, e.g., Kalemos in a recent publication [14]. The goal of the present paper is to see whether different flavors of RS-DFT can correctly describe these different chemical bonds.

The paper is organized as follows. After a short recall of RS-DFT, we give a brief survey of existing data and construct consistent reference results for both the dimer and the trimer. To these we compare different flavors of RS-DFT and discuss their performance. All further technical details are collected in “Appendix”, and the underlying datasets can be consulted in the Supplementary Material.

2 Range-separated density-functional theory

The common Kohn–Sham procedure [15] minimizes the following energy expression with a single Slater determinant wave function $\Phi$

$$\begin{aligned} E_{\mathrm{exact}} = \min _{\Phi }\left\{ \langle \Phi |\hat{T}+\hat{V}_{\mathrm{ne}}|\Phi \rangle + E_{\mathrm{Hxc}}\left[ n_{\Phi }\right] \right\} , \end{aligned}$$

(1)

where $\hat{T}$ is the kinetic energy operator, $\hat{V}_{\mathrm{ne}}$ is the electron–nucleus attraction operator, and $E_{\mathrm{Hxc}}\left[ n_{\Phi }\right]$ is the Hartree-exchange-correlation functional evaluated at the density $n_\Phi$ produced by $\Phi$. With the exact density functional $E_{\mathrm{Hxc}}[n]$, the exact ground-state energy would be obtained, in the limit of a complete basis set, by virtue of the Hohenberg–Kohn theorem.

The Kohn–Sham method can be considered as a special case of a more general RS-DFT scheme (see, e.g., Ref. [16])

$$\begin{aligned} E_{\mathrm{exact}} = \min _{\Psi }\left\{ \langle \Psi |\hat{T}+\hat{V}_{ \mathrm ne }+\hat{W}_{\mathrm{ee}}^{\mathrm{lr}}|\Psi \rangle + E^{\mathrm{sr }}_{\mathrm{Hxc }}\left[ n_{\Psi }\right] \right\} , \end{aligned}$$

(2)

where $\Psi$ is a general, multi-determinant wave function, $\hat{W}_{\mathrm{ee}}^{\mathrm{lr}} = \sum _{i<j} w_{\text {ee}}^{\mathrm{lr}}(r_{ij})$ is a long-range electron–electron interaction, and $E^{\mathrm{sr}}_{\mathrm{Hxc}}[n]$ the associated short-range complement Hartree-exchange-correlation density functional. The range separation of the electron–electron interaction

$$\begin{aligned} \frac{1}{r_{ij}} = w_{\mathrm{ee}}^{\mathrm{lr}}(r_{ij}) + \left( \frac{1}{r_{ij}}- w_{\mathrm{ee}}^{\mathrm{lr}}(r_{ij}) \right) , \end{aligned}$$

(3)

is achieved through the use of the error function with an arbitrary parameter $\mu$

$$\begin{aligned} w_{\mathrm{ee}}^{\mathrm{lr}} (r_{ij}) \ = \ \frac{\mathrm{erf}(\mu \, r_{ij})}{r_{ij}}. \end{aligned}$$

(4)

In practice, approximations must be used for the wave function $\Psi$. A first step is to use only a single-determinant wave function $\Phi$, which leads to the range-separated hybrid (RSH) approximation [1]

$$\begin{aligned} E_{\mathrm{RSH}} = \min _{\Phi }\left\{ \langle \Phi |\hat{T}+\hat{V}_{\mathrm{ne}} + \hat{W}_{\mathrm{ee}}^{\mathrm{lr}}|\Phi \rangle + E^{\mathrm{sr}}_{\mathrm{Hxc}}\left[ n_{\Phi }\right] \right\} , \end{aligned}$$

(5)

the expectation value of the long-range interaction, $\langle \Phi |\hat{W}_{\mathrm{ee}}^{\mathrm{lr}}|\Phi \rangle$, giving a long-range Hartree-Fock (HF) contribution to the energy. The minimization in Eq. (5) leads to self-consistent Kohn–Sham-like RSH equations for the orbitals $|\phi _i \rangle$ and orbital energies $\epsilon _i$

$$\begin{aligned} \left( \hat{T} + \hat{V}_{\mathrm{ne}} + \hat{V}_{\mathrm{H}} + \hat{V}_{\mathrm{x,HF}}^{\mathrm{lr}} + \hat{V}_{\mathrm{xc}}^{\mathrm{sr}} \right) \,|\phi _i\rangle \ = \ \epsilon _i\, |\phi _i\rangle , \end{aligned}$$

(6)

where $\hat{V}_{\mathrm{H}}$ is the full-range Hartree potential, $\hat{V}_{\mathrm{x,HF}}^{\mathrm{lr}}$ is the long-range nonlocal HF exchange potential (evaluated with the “erf” part of the electron–electron interaction), and $\hat{V}_{\mathrm{xc}}^{\mathrm{sr}}$ is the short-range exchange-correlation potential (obtained from the functional derivative of $E^{\mathrm{sr}}_{\mathrm{xc}}[n]$).

Equation (5) does not yet include the long-range correlation energy. The exact energy is formally the sum of the RSH energy and the long-range correlation energy:

$$\begin{aligned} E_{\mathrm{exact}} = E_{\mathrm{RSH}} + E^{\mathrm{lr}}_{\mathrm{c}} \end{aligned}$$

(7)

as depicted schematically in Fig. 1. Whereas for instance dispersion interactions are often added to DFT via ad hoc corrections based on the evaluation of atomic polarizabilities [17,18,19], the RS-DFT formalism allows us to include explicitly these important contributions to intermolecular interactions, only based on (i) the single parameter of the range separation $\mu$, (ii) the choice of the short-range exchange-correlation functional, and (iii) the long-range correlation method. The long-range correlation energy can be calculated by second-order Møller–Plesset (MP2) perturbation theory [1], which leads to exactly the same equations as standard MP2 [20], i.e., for closed-shell systems:

$$\begin{aligned} E^{\mathrm{lr}}_{\mathrm{c,MP2}} = \sum _{ijab}\frac{[2(ia|jb)^{\mathrm{lr}}-(ib|ja)^{\mathrm{lr}}](ia|jb)^{\mathrm{lr}}}{\epsilon _i + \epsilon _j-\epsilon _a-\epsilon _b}, \end{aligned}$$

(8)

where i, j and a, b refer to occupied and virtual RSH orbitals, respectively, and $(ia|jb)^{\mathrm{lr}}$ are the long-range two-electron integrals (using the “erf” interaction). Of course, all these quantities (integrals and orbital energies) depend on the range-separation parameter $\mu$. Similarly, the long-range correlation energy can be calculated by coupled-cluster singles, doubles and perturbative triples (CCSD(T)) [3, 21].

One of the central advantages of these methods with respect to standard quantum chemistry approaches like MP2 or coupled-cluster theory is the considerably weaker basis-set dependence as the short-range correlation hole is taken already into account via the density-functional part. As a consequence, basis-set superposition errors (BSSE) should be significantly reduced with respect to standard wave-function-based correlation calculations.

Fig. 1

Schematical composition of the RS-DFT total energy as the RSH energy plus the long-range correlation energy. For $\mu =0$, one obtains standard Kohn–Sham theory without explicit long-range correlation correction, and for $\mu \rightarrow \infty$ RSH reduces to HF (no short-range functional) to which the correlation energy is added

Concerning the short-range exchange-correlation functional, several approximations [16, 21,22,23,24,25,26,27] have been proposed. We employed from the different possibilities the short-range local-density approximation (LDA) of Ref. [25] and the short-range Perdew–Burke–Ernzerhof (PBE) functional of Ref. [22]. We will refer to the range-separated methods as RSHLDA + MP2 and RSHLDA +CCSD(T), and RSHPBE + MP2 and RSHPBE + CCSD(T). Of course, setting the range-separation parameter $\mu$ to infinity, we obtain the standard MP2 or CCSD(T) approach without a density-functional part.

3 Brief review of existing data

In the past, Be clusters have been abundantly studied, and an outstanding object of interest is the dimer (for a recent review, see [28]). Compared to the heavier alkaline-earth dimers, Be₂ already shows a shorter and stronger binding than expected for a purely dispersive interaction with an interatomic equilibrium distance and cohesion energy of about 2.4 Å and 2.5 kcal/mol, respectively.

This binding energy and interatomic distance have been a subject of long debates, experimentally and theoretically (see Refs. [6,7,8,9,10,11,12]). The experimental difficulties are summarized in a recent article [13], and from the theoretical side both the near degeneracy of the occupied 2s orbital with the unoccupied 2p orbitals [29, 30] and the shallowness of the potential well without being a purely dispersion interaction require a careful treatment of electron correlation [28].

It is therefore not surprising that at the HF level, Be₂ is not bound at all. Single-reference configuration-interaction (CI) schemes truncated at the level of single and double excitations (CISD) are not suited for this system [29, 31], even when including all exclusion-principle-violating (EPV) diagrams in a coupled-electron-pair approximation (called full CEPA or self-consistent size-consistent CI) [32]. One has to go indeed to multi-reference or coupled-cluster (including triple excitations [31, 33]) correlation methods for a correct description.

With the work of Røeggen and Veseth [9], Patkowski et al. [12], Schmidt et al. [11], and corresponding experimental data [13], we can consider that the potential well of the Be dimer is precisely known. DFT with usual semilocal approximations overestimates the interaction energy; however, the work of Jones [6] was the first theoretical one to claim the Be dimer to be significantly bound. RS-DFT with long-range MP2 or random-phase-approximation (RPA) approaches was applied to the Be dimer [2, 34, 35], giving significantly underestimated interaction energies. Range separation in combination with multi-reference perturbation theory (NEVPT2) has been published by Fromager et al. [36] with the conclusion that differences to the reference data despite the multi-reference long-range correlation treatment should be imputed to deficiencies of the employed short-range PBE functional.

For Be₃ in D_3h symmetry, HF theory gives no overall binding [37,38,39,40,41]. Nevertheless, a distinct local minimum is produced, contrary to Be₂ where the HF potential curve is purely repulsive. Including electron correlation [37,38,39,40,41] results in a cohesion energy of the triatomic “molecule” between 15 and 30 kcal/mol, with an equilibrium interatomic distance of about 2.2 Å, which is about the same as for the local minimum in HF. The significant difference in the binding of the dimer and the trimer (i.e., the importance of non-additivity or “3-body interactions”) has been discussed by Novaro and Kołos [42] and Daudey et al. [43] already in the 1970s in the framework of HF theory. A comparative study [44] of the dimer, trimer and tetramer of Be and of Mg, based on MP2, CCSD(T), and DFT methods, estimates the utility of different basis-set extrapolation formulae and concludes that an extrapolation from the difference between double-zeta and triple-zeta basis sets is sufficient to estimate converged results for binding energies. As in the case of rare-gas dimers and the Be dimer, the LDA overestimates the binding energy of the trimer [45, 46]. Even gradient-corrected functionals such as BPW91 result in too high binding energies [47], the effect being less important but still present for the B3LYP hybrid functional [48].

4 Results and discussion

In order to dispose of a coherent reference dataset for the two systems, we perform MP2, MP4, CCSD(T), MRCI, MR-ACPF [49], and MR-AQCC [50,51,52] calculations with the MOLPRO program package [53]. For the multi-reference calculations, two electrons and four orbitals of each Be atom are included in the complete active space (CAS), forming the reference space to which single and double excitations in the size-consistency-corrected ACPF or AQCC correlation formalism are added.

We employ the aug-cc-pVXZ basis sets (X = D, T, Q — see “Appendix”), and for the reference calculations we extrapolate the correlation energy via the inverse-cubic scheme $E_{\text {c}}(X)=E_{\text {c}}(\infty )+B/X^3$ to an estimate of the complete (valence) basis-set limit. For RS-DFT calculations, however, it has been shown that an exponential form $E_{\text {c}}(X)=E_{\text {c}}(\infty )+B\exp (-\beta X)$ is more adequate [54]—we will therefore employ this scheme for the RS-DFT results. All data are corrected for the BSSE by the counterpoise scheme of Boys and Bernardi [55], even though this correction remains always smaller than 0.2 kcal/mol. The HF energy is considered to be converged with the quadruple-zeta basis set.

For the wave-function-based correlation calculations, the core orbitals were kept frozen, i.e., only the valence shells are explicitly correlated. Indeed, this choice is consistent with the fact that the aug-cc-pVXZ basis sets do not contain functions to correlate core electrons. In the short-range functional, however, core correlation is automatically included even when long-range correlation is treated by a frozen-core wave-function-based method.

In order to study core-valence correlation explicitly, we did some exploratory calculations with the core-polarization potentials (CPPs) of the Stuttgart group [56] added to the frozen-core calculations, as well as correlated all-electron calculations in an aug-cc-pwCVTZ basis set with core functions [57] added to the standard aug-cc-pVTZ set.

4.1 Wave-function-based reference calculations

For the dimer, the extrapolated MR-ACPF and MR-AQCC interaction energy curves are close to the reference potential of Refs. [9] or [12] (see Table 1, left column for bond lengths and potential depths), while the CCSD(T) curve has a similar shape but underestimates the interaction energy significantly (see Supplementary material for data and graph). As discussed above, CCSD is not at all adequate here, and perturbation theory yields as well quite different interaction potentials, MP2 underestimating at the equilibrium distance and MP4 overestimating at larger distances.

Fig. 2

Basis-set extrapolated interaction energy curves for the Be trimer, as obtained by wave-function-based methods. The right panel is a zoom into the minimum-energy region of the left panel

For the trimer (see Fig. 2), we do not dispose of an accurate reference curve from the literature, but we see that CCSD(T) and multi-reference methods give potentials of similar shape, and again second-order perturbation theory yields significantly different results, but in the opposite way with respect to Be₂. MP4 overestimates the interaction energy as much as CCSD(T) underestimates it (Table 1, right column). CCSD again largely underbinds.

Table 1 Equilibrium bond lengths (Å) and interaction energies (kcal/mol) from inverse-cubic basis-set extrapolated potential curves for the Be dimer and trimer

By taking into account core and core-valence correlations, the interaction energy is lowered (in a triple-zeta basis set) by about 0.1 kcal/mol for the all-electron calculations, and by about 0.4 kcal/mol using the CPPs, for the dimer. For the Be trimer, the effect is about 1 kcal/mol for the inclusion of core correlations and about 4 kcal/mol for the use of the CPPs. We thus use the frozen-core data, and, due to these estimations, we keep in mind that the true interaction potential may be slightly lower than our actual multi-reference data.

4.2 RS-DFT calculations

From the previous section, we conclude that today’s “gold standard” CCSD(T) with basis-set extrapolation gives for the trimer too low a cohesion energy, by about 2.5 kcal/mol or 10%. The calculations need a non-negligible effort, which may be avoided by resorting to DFT-based calculations while resulting in not worse an error. In Fig. 3, we display the evolution of the RS-DFT interaction energies with the range-separation parameter $\mu$, in comparison with our MR-ACPF reference data, and the “region of confidence” around the reference energy as shaded area, which we chose to be the deviation of basis-set extrapolated CCSD(T) around the MR-ACPF reference. In order to remain inside this region, we may take a value for the range-separation parameter $\mu$ between 0.7 and 1.7 a.u., and either short-range functional combined with long-range CCSD(T). In the figure, we see as well the reduction in the influence of the extrapolation procedure when going from large $\mu$ to $\mu =0$.

For simplicity, we will use for further comparisons $\mu =1.0$ a.u. at the lower end of the interval of confidence, however significantly larger than the commonly employed value of 0.5 a.u. (see, e.g., Refs. [5, 22, 58]). For this value of $\mu =1.0$ a.u., the contribution from the short-range functional with its advantages (low computational cost, small basis-set superposition error, and weak basis-set dependence) is still present. Without loosing too much in accuracy, we still may carry out the calculations in the aug-cc-pVTZ basis set.

Fig. 3

Effect of the basis-set extrapolation for the equilibrium cohesion energies and bond lengths as a function of the range-separation parameter $\mu$, for Be₃. Full lines are for the aug-cc-pVTZ basis set, and dashed lines represent the corresponding extrapolated values

Fig. 4

Interaction energy curves of the RS-DFT methods for the Be dimer, compared to the reference data. Basis set is the aug-cc-pVTZ one, and the range-separation parameter $\mu$ is taken equal to 1 a.u.. Of the corresponding curves, the RSHPBE ones lie always above the RSHLDA ones

Fig. 5

Interaction energy curves of the RS-DFT methods for the Be trimer, compared to our own reference data. Same parameters as for Fig. 4. RSHLDA interaction energies are always smaller than the RSHPBE ones

Table 2 Equilibrium bond lengths (Å) and interaction energies (kcal/mol) for different methods and basis sets for the Be dimer and trimer

Figures 4 and 5 show the RSHLDA + MP2, RSHPBE + MP2, RSHLDA + CCSD(T) and RSHPBE + CCSD(T) interaction energies with aug-cc-pVTZ basis for $\mu =1\,$ a.u., for the dimer and for the trimer, in comparison with the available reference data. The equilibrium bond lengths and interaction energies obtained from RS-DFT calculations with a series of basis sets are reported in Table 2, and compared to those obtained from standard DFT, MP2, and CCSD(T) calculations.

Calculations with standard density-functional approximations overestimate the interaction energies considerably. For instance in the case of the dimer with a factor of 5 for LDA, and still a factor of 1.5 for B3LYP. RSHLDA + CCSD(T) and RSHPBE + CCSD(T) produce interaction energies in good agreement with the reference data. In contrast, RSHLDA + MP2 and RSHPBE + MP2 significantly underbind the dimer and overbind the trimer, as we have seen already from Figs. 4 and 5. Note that for Be₃, RSH + CCSD(T) (Fig. 5) and CCSD(T) (Fig. 2) give the same deviation from the reference data for large interatomic distances.

For the Be₂ case, the energy differences due to different basis sets are small, and we may look first at the Be₃ results. We see that if we take the difference in binding energies between the DZ and TZ as one, the difference between TZ and QZ is about one quarter, consistently for all methods. In absolute values, the differences due to the basis sets are divided by a factor of two between MP2/CCSD(T) and the RS-DFT interaction energies. However, using RS-DFT this dependence is twice as large as for the pure density-functional calculations, employing the standard functionals LDA, PBE, B3LYP or the short-range functionals only. The same factor of 4 between the DZ–TZ and the TZ–QZ differences is found in the case of the Be₂ in the RS-DFT calculations.

Fig. 6

Equilibrium cohesion energies and bond lengths as a function of the range-separation parameter $\mu$, for Be₂ and the aug-cc-pVTZ basis set. We include our basis-set extrapolated MR-ACPF/AQCC reference point

Fig. 7

Equilibrium cohesion energies and bond lengths as a function of the range-separation parameter $\mu$, for Be₃ and the aug-cc-pVTZ basis set

Let us look in more detail into the RS-DFT calculations (Figs. 6, 7). Varying the range-separation parameter, we determine the minima of the cohesion-energy curves, which all lie in a narrow range of distances, but not on energies. Again, we note that MP2 and CCSD(T) long-range correlations have not the same effect on the results. Indeed, when aiming at reproducing the reference calculations, electron correlation of order higher than second-order perturbation theory is needed. This is more pronounced in Be₂ than in Be₃, where RSH + MP2 or RSH + CCSD(T) follow similar trends with the range-separation parameter $\mu$. The interpolation between $\mu =0$ (pure Kohn–Sham DFT) and $\mu \rightarrow \infty$ (no contribution of a functional) appears to be not at all linear, neither for Be₂ nor for Be₃, and the optimal range-separation parameter should be chosen significantly larger than 0.5 a.u., as found before. Of course, for large $\mu$ the difference between the two functionals employed becomes reasonably small.

Before concluding, we might look at the performances of other recent DFT approaches as provided by the double-hybrid functionals including both HF exchange and a MP2 correlation part. Thus, results should lie somewhere between standard MP2 and our RSH + MP2. Using the B2PLYP [59], the XYG3 [60], and the $\omega$B97X-2(TQZ) [61] functionals, we find indeed (data in the Supplementary Material) for the trimer that all functionals overshoot as did MP2 and RSH + MP2 already. For the dimer, the situation is a little different as MP2 and RSH + MP2 resulted in a too weak interaction, but the double hybrids overbind, and the three variants lead to significantly different shapes of the potential.

5 Conclusion

We investigated the challenging systems Be₂ and Be₃ with different wave-function and DFT methods, comparing these to RS-DFT approaches, which are capable of describing explicitly long-range correlations, absent in standard Kohn–Sham theory with semilocal approximations. The interaction energy in small Be clusters is certainly not only due to dispersion as the interatomic distances are far smaller than typical van der Waals interactions, and interaction energies are much higher.

We observe that for this particular type of bonding—no chemical bond properly speaking, but no dispersion-only binding either (as for rare-gas complexes)—the RS-DFT approach, when used with the usual range-separation parameter of $\mu =0.5$ a.u., produces too high binding energies, similar to the commonly employed B3LYP functional. Using a value of $\mu$ around 1 a.u. in our RS-DFT scheme with a CCSD(T) long-range part permits to reproduce well the MR–ACPF or MR–AQCC reference energies of the trimer.

Long-range single-reference CCSD(T) thus seems adequate for obtaining a reliable binding energy—even if perhaps due to fortuitous error cancelation. On the other hand, adding only second-order diagrams for expanding the long-range correlation energy is not sufficient, for any value of the range-separation parameter. This last point is in particular important for the modern development of double hybrids, leaving room for further developments.

Appendix: Technical details

All calculations were performed on 3.3 GHz Intel PCs running CentOS Linux, using Molpro [53] in its version 2008.2 (standard DFT and MCSCF/MRCI calculations) and a local development version, 2008.2, for the RS-DFT calculations. The employed aug-cc-pVXZ basis sets originate from the Gaussian basis set exchange form at Pacific National Laboratories. Only for the calculations on double hybrids, we employed the QChem code, version 5.0 [62].

For the single- and multi-reference methods and the long-range correlation part of the RS-DFT calculations, only the valence electrons of the Be atom were correlated. Energy thresholds were 10$^{-7}\,$a.u. for convergence and 10$^{-9}\,$a.u. as target accuracy for establishing the grid for the numerical density-functional integration.

The multi-reference calculations started from a complete-active-space (CAS) wave function with the two valence electrons in four orbitals for each atom. The subsequent ACPF or AQCC calculations used this CAS wave function as reference space (60 configuration state functions (CSFs) for Be₂, and about 4000 CSFs for Be₃).

Minima in the 1D potential curves were determined through a spline fit with points spaced at 0.1 Å, starting at 1.9 Å and including some large distances to check size consistency.