The accurate calculation of the binding energy of the beryllium dimer is a challenging theoretical problem. In this study, the binding energy of Be2 is calculated using the diffusion Monte Carlo (DMC) method, using single Slater determinant and multiconfigurational trial functions. DMC calculations using single-determinant trial wave functions of orbitals obtained from density functional theory calculations overestimate the binding energy, while DMC calculations using Hartree-Fock or CAS(4,8), complete active space trial functions significantly underestimate the binding energy. In order to obtain an accurate value of the binding energy of Be2 from DMC calculations, it is necessary to employ trial functions that include excitations outside the valence space. Our best estimate DMC result for the binding energy of Be2, obtained by using configuration interaction trial functions and extrapolating in the threshold for the configurations retained in the trial function, is 908 cm−1, only slightly below the 935 cm−1 value derived from experiment.

The beryllium dimer has been the subject of numerous experimental and theoretical studies.1–34 In 1984, Bondybey and English, using ro-vibrational data from near the bottom of the ground state 1Σg+ potential energy curve of Be2, deduced a value of 790 ± 30 cm−1 for the binding energy (here defined from the potential energy minimum, i.e., neglecting vibrational zero-point energy).1–3 Based on rotational structure in the v = 0 level, Bondybey and English determined a bond length of 2.45 Å. More recently, Merritt and coworkers experimentally observed eleven vibrational levels of Be2, allowing them to obtain a more refined estimate of 929.74 cm−1 for the well depth.4 This was subsequently revised to 934.9 cm−1 upon further analysis of the experimental data.5 Over the past few years, several electronic structure calculations have been reported that obtained well depths close to the recent experimental value.20–29 The keys to the successful calculations are the use of large, flexible basis sets and the recovery of a large portion of the correlation energy including contributions from the 1s core orbitals. To illustrate the difficulty of calculating an accurate binding energy of Be2, we note that a complete basis set limit coupled cluster singles plus doubles with perturbative triples [CCSD(T)] calculation including correlation of the 1s core electrons underestimates the binding energy by 224 cm−1.28 Moreover, basis functions beyond those included in the aug-cc-pCVQZ basis set35,36 contribute 79 cm−1 to the CCSD(T) value of the binding energy.28 

In this study, we apply the diffusion Monte Carlo (DMC) method37–40 to the Be dimer. The DMC method is capable of giving the exact ground state energy under the constraint of the fixed-node approximation,41–46 which is required to maintain the fermionic nature of the wave function. The constraint is imposed by the use of a trial function often taken to be a single Slater determinant of Hartree-Fock (HF) or density functional theory (DFT) orbitals. If the nodal surface of the trial wave function were exact, then the DMC method, if run for a sufficient number of steps and extrapolated to zero time step, would give the exact ground state energy. It is generally assumed42 that for weakly interacting dimers, the errors introduced by the use of single determinant trial functions to impose the fixed nodes largely cancel when the interaction energy is calculated by subtracting the sum of the energies of the two monomers from that of the dimer, and this has been confirmed for systems such as the water dimer and the methane dimer.31 However, it is not clear that this will be the case for weakly interacting species for which static correlation effects are important. The Be dimer is thus a particularly interesting test system, as the ground state wavefunction of Be has considerable 2s2 → 2p2 character. Indeed, all-electron DMC calculations on Be using a CAS(2,4), complete active space, trial function allowing for 2s2 → 2p2 mixing give a significantly lower total energy than do DMC calculations using a single Slater determinant trial function.24,33,44 However, DMC calculations using a CAS(4,8) trial function for the dimer and a CAS(2,4) trial function for the atom considerably underestimate the binding of the dimer.22 Harkless and Irikura24 used a truncated CAS(4,8) space and Anderson and Goddard32 used a generalized valence bond (GVB) trial function and each reported DMC values of the binding energy of Be2 in good agreement with experiment. As will be discussed later in the manuscript, this is likely to be fortuitous. In the present study, we calculate the binding energy of Be2 using the DMC method in conjunction with more flexible multiconfigurational trial functions than were employed in earlier studies.

The experimental value of the equilibrium bond length, 2.453 603 Å,4 was used for all calculations on the beryllium dimer. In the first set of calculations, single determinant trial functions were considered, with the orbitals being obtained from the HF approximation and from several DFT methods including the local density approximation (LDA), the Perdew-Burke-Ernzerhof (PBE)47 and Becke-Lee-Yang-Parr (BLYP)48,49 generalized gradient approximation (GGA) functionals, and the Becke3LYP,49–51 PBE0,52 and Becke half and half exchange plus LYP correlation (BH&HLYP)53 hybrid functionals, which contain 20%, 25%, and 50% exact exchange, respectively. In addition, a trial function comprised of a single Slater determinant of Brueckner orbitals54,55 was considered. The cc-pVQZ-g 5s4p3d2f contracted Gaussian-type orbital basis set35,36 was used to represent the orbitals in the single Slater determinant trial functions. Both cc-pVQZ-fg and cc-pVQZ-g basis sets were used in generating the multiconfigurational trial functions. Here, -fg indicates that both the f and g functions were omitted from the basis set, while -g indicates that only the g functions were omitted.

DMC calculations were also carried out using multiconfigurational trial functions generated from CAS and configuration interaction (CI) calculations. For the beryllium dimer, both CAS(4,8) and CAS(4,16) trial functions were considered. The CAS(4,8) wave function allows all arrangements of the four valence electrons in the space of the molecular orbitals (MOs) derived predominantly from the 2s and 2p atomic orbitals (AOs). The CAS(4,16) wavefunction expands the active space to include the πg, πu, σg, σu molecular orbitals derived from the 3s and 3p atomic orbitals and has 816 configuration state functions (CSFs). The DMC calculations were carried out retaining all CSFs with coefficients greater than 0.001, 0.0025, 0.005, and 0.01 in magnitude, and these results were used to extrapolate the energies to the value for the full configuration space. The extrapolation is shown in Figure 1. With the 0.001 coefficient threshold, 341 CSFs are retained from the CAS(4,16) space. Truncations of the configuration space were carried out after the CAS (or CI) calculations using GAMESS but prior to the optimizations in the variational Monte Carlo (VMC) calculations.

FIG. 1.

Extrapolation of the DMC energies of the beryllium dimer in the calculations using the CAS(4,16)/cc-pVQZ-g trial function, as described in the text. The dashed red line is a linear fit to the DMC energies (blue squares).

FIG. 1.

Extrapolation of the DMC energies of the beryllium dimer in the calculations using the CAS(4,16)/cc-pVQZ-g trial function, as described in the text. The dashed red line is a linear fit to the DMC energies (blue squares).

Close modal

CI trial functions were generated by carrying out configuration interaction calculations, allowing for up to four electron excitations from the valence space into the full virtual space and employing CAS(4,8) orbitals. Natural orbitals were then generated and used to carry out subsequent CI calculations allowing up to quadruple excitations in the space of all natural orbitals with occupations greater than 0.0001 in the first CI calculation (again keeping the 1σg and 1σu orbitals frozen). Trial functions with reduced configuration spaces were then generated by discarding configurations with coefficients above particular thresholds (0.01, 0.005, 0.0025, and 0.001). For the smallest threshold (0.001) 484 of 4500 CSFs were retained. For calculating the binding energy, a single plus double excitation CI (SDCI) calculation was carried out on the atom using CAS(2,8) orbitals and followed by a subsequent SDCI calculation using natural orbitals with occupations greater than 0.0001.

Each of the trial functions was combined with a Jastrow factor56 with electron-electron, electron-nucleus, and electron-electron-nucleus terms. VMC calculations were used to optimize the Jastrow factors via energy minimization. For the multiconfigurational trial functions, the coefficients of the CSFs were optimized simultaneously with the parameters in the Jastrow function. The resulting trial functions, including the Jastrow factors, were then used to carry out DMC simulations using 40 000-50 000 walkers at a single time step of 0.001 a.u. The correction scheme of Ma et al.57 was used to account for the electron-nuclear cusps. For estimating statistical errors, the blocking procedure of Flyvbjerg and Petersen was used.58 For one set of DMC calculations using the CAS(4,16) trial function, time steps of 0.0005, 0.003, and 0.005 a.u. were also used, allowing extrapolation of the energies to the zero time step limit. This extrapolation is shown in Figure 2.

FIG. 2.

Extrapolation to zero time step of the DMC energy of the Be dimer at the equilibrium bond length of 2.453 603 Å. The calculations were based on the CAS(4,16)/cc-pVQZ-g trial function and used a 0.001 threshold on the CI coefficients.

FIG. 2.

Extrapolation to zero time step of the DMC energy of the Be dimer at the equilibrium bond length of 2.453 603 Å. The calculations were based on the CAS(4,16)/cc-pVQZ-g trial function and used a 0.001 threshold on the CI coefficients.

Close modal

The single determinant trial functions were generated using Gaussian0951 and the multiconfigurational trial functions were generated using GAMESS.59 The quantum Monte Carlo calculations were carried out using the CASINO60 and QMCPACK61 codes for the single determinant and multideterminant trial functions, respectively. QMCPACK was used for the latter calculations due to its implementation of an efficient algorithm for handling multideterminant trial functions.62 

The results of the DMC calculations at the 0.001 a.u. time step are reported in Table I. With the HF trial function, the DMC calculations give a binding energy of 724 cm−1, significantly smaller than the experimental value of 935 cm−1. On the other hand, the DMC calculations using trial functions employing LDA or GGA orbitals considerably overestimate the binding energy of Be2. Significantly, improved agreement with experiment is obtained when using orbitals from hybrid functionals containing a component of exact exchange or from Brueckner CCSD calculations. Specifically, the DMC calculations using PBE0, BH&HLYP, and Brueckner orbitals give binding energies of 992, 966, and 955 cm−1, respectively. Toulouse and Umrigar30 obtained a binding energy of 1008 cm−1 from DMC calculations using single determinant trial functions but optimizing the orbitals and basis functions in the VMC step. For both Be and Be2, regardless of the orbitals used, the DMC calculations using single determinant trial functions give energies considerably above the exact energies of these species, suggesting that the good agreement with experiment of these calculated binding energies is fortuitous. Support for this conjecture is provided by Fig. 3 from which it is seen that the calculations that give binding energies closest to experiment do so because they give a higher energy for the dimer. It should be noted that all single determinant trial functions should give the same DMC energy of the Be atom.63 The spread in the DMC energies of the Be atom calculated using different single determinant trial functions is only about 27 cm−1 with part of that being statistical and part being due to finite time step errors (i.e., using a time step of 0.001 a.u.).

TABLE I.

Total energies of Be and Be2 and the Be2 dissociation energy computed with DMC using various trial functions.

Total energy (a.u.)
Trial functiona Beb Be2 De (cm−1)
HF/QZ-g  −14.657 30(4)  −29.317 89(6)  724(21) 
LDA/QZ-g  −14.657 21(4)  −29.319 77(7)  1174(25) 
PBE/QZ-g  −14.657 31(5)  −29.319 60(8)  1094(26) 
BLYP/QZ-g  −14.657 25(4)  −29.319 56(8)  1113(26) 
B3LYP/QZ-g  −14.657 27(3)  −29.319 46(8)  1079(23) 
PBE0/QZ-g  −14.657 28(3)  −29.319 07(8)  992(21) 
BH&HLYP/QZ-g  −14.657 26(5)  −29.318 91(7)  966(26) 
BD/QZ-g  −14.657 18(4)  −29.318 72(7)  955(24) 
CAS(4,8)/QZ-fgc  −14.667 23(1)  −29.337 07(3)  573(8) 
CAS(4,16)/QZ-fgc  −14.667 30(1)  −29.338 32(3)  819(8) 
Ext. CAS(4,16)/QZ-fg  −14.667 30(1)  −29.338 41(2)  838(7) 
CAS(4,16)/QZ-gc  −14.667 27(2)  −29.338 38(3)  845(8) 
Ext. CAS(4,16)/QZ-g  −14.667 27(2)  −29.338 45(2)  857(9) 
CI/QZ-gc  −14.667 25(1)  −29.338 48(2)  873(6) 
Ext. CI/QZ-g  −14.667 25(1)  −29.338 64(2)  908(6) 
Experimentald  −14.667 356  −29.338 97  934.9(4) 
Total energy (a.u.)
Trial functiona Beb Be2 De (cm−1)
HF/QZ-g  −14.657 30(4)  −29.317 89(6)  724(21) 
LDA/QZ-g  −14.657 21(4)  −29.319 77(7)  1174(25) 
PBE/QZ-g  −14.657 31(5)  −29.319 60(8)  1094(26) 
BLYP/QZ-g  −14.657 25(4)  −29.319 56(8)  1113(26) 
B3LYP/QZ-g  −14.657 27(3)  −29.319 46(8)  1079(23) 
PBE0/QZ-g  −14.657 28(3)  −29.319 07(8)  992(21) 
BH&HLYP/QZ-g  −14.657 26(5)  −29.318 91(7)  966(26) 
BD/QZ-g  −14.657 18(4)  −29.318 72(7)  955(24) 
CAS(4,8)/QZ-fgc  −14.667 23(1)  −29.337 07(3)  573(8) 
CAS(4,16)/QZ-fgc  −14.667 30(1)  −29.338 32(3)  819(8) 
Ext. CAS(4,16)/QZ-fg  −14.667 30(1)  −29.338 41(2)  838(7) 
CAS(4,16)/QZ-gc  −14.667 27(2)  −29.338 38(3)  845(8) 
Ext. CAS(4,16)/QZ-g  −14.667 27(2)  −29.338 45(2)  857(9) 
CI/QZ-gc  −14.667 25(1)  −29.338 48(2)  873(6) 
Ext. CI/QZ-g  −14.667 25(1)  −29.338 64(2)  908(6) 
Experimentald  −14.667 356  −29.338 97  934.9(4) 
a

QZ refers to the cc-pVQZ basis set. The “-g” and “-fg” indicate, respectively, that the g functions and f and g functions were omitted from the basis sets. Ext. refers to CAS and CI results extrapolated to the full configuration space for the active orbital list as described in the text.

b

The DMC energies of the Be atom calculated using various single determinant trial functions should agree. The spread of the energies in the table is the result of statistical errors and the use of a finite (0.001 a.u.) time step.

c

0.001 threshold on CI coefficients for retained configurations.

d

The experimental De value for Be2 is from Ref. 5. The non-relativistic energy of the Be atom is from Ref. 60.

FIG. 3.

DMC energies of twice the beryllium atom and the dimer for several single-determinant trial wave functions.

FIG. 3.

DMC energies of twice the beryllium atom and the dimer for several single-determinant trial wave functions.

Close modal

As expected, based on earlier studies,22,24 DMC calculations using valence-space CAS trial functions give significantly lower energies for the Be atom and dimer than do the DMC calculations using the trial functions based on single Slater determinants. However, the DMC calculations using the CAS(4,8) trial function for the dimer and CAS(2,4) for the atom give a binding energy of only 573 cm−1, which is even smaller than that obtained using HF trial functions. This indicates that use of valence space CAS trial functions does not result in a balanced treatment of the nodal surfaces of the atom and molecule. Most of the error is due to the inadequacy of the CAS(4,8) space in describing the nodal surfaces of the dimer since the DMC calculations on the atom using the CAS(2,4) trial function give an energy close to the current best estimate64 (−14.667228 vs. −14.667356 a.u.). Expanding the CAS space to include also the MOs derived from the 3s and 3p AOs, giving CAS(2,8) for the atom and CAS(4,16) for the dimer, lowers the DMC energies of the atom and dimer, by 10 and 300 cm−1, respectively, and results in a dimer binding energy of 845 cm−1, at the 0.001 coefficient threshold and using the cc-pVQZ-g basis set. The corresponding binding energy obtained using the cc-pVQZ-fg basis set is 819 cm−1, indicating that the nodal surface of Be2 is slightly improved by including f functions in the basis set. Extrapolating these results along the sequence of coefficient cutoffs gives binding energies of 838 and 857 cm−1 for trial functions expanded in terms of the cc-pVQZ-fg and cc-pVQZ-g basis sets, respectively. The extrapolation to zero time step of the DMC/CAS(4,16) energies obtained with the 0.001 coefficient threshold when using the cc-pVQZ-g basis set gives a DMC binding energy of 849 cm−1 (see Fig. 3) vs. the 845 cm−1 value obtained with the 0.001 a.u. time step. Thus, we conclude that the error due to the use of the finite time step is inconsequential for the calculation of the binding energy of the dimer.

The DMC calculations using CI trial functions with 0.001 coefficient cutoffs and the cc-pVQZ-g basis set yielded a dimer binding energy of 873 cm−1, while the corresponding result obtained by extrapolation to the full configuration space is 908 cm−1, which is only 27 cm−1 smaller than the experimental value of the binding energy. These results demonstrate that correlation effects involving configurations outside the CAS(8,16) space are important for describing the nodal surface of Be2.

It should be noted that the SDTQ CI calculations using the cc-pVQZ-g basis set and freezing the 1s orbitals give a binding energy of only 601 cm−1, which is 334 cm−1 lower than the experimental value. About 70 cm−1 of the error in this result is due to the neglect of the correlation effects involving the core 1s orbitals,29 while the remaining error is due to correlation effects that are not captured due to the basis set truncation. This underscores one of the major advantages of the DMC method, namely, that it achieves convergence with much smaller basis sets (for the trial functions) than required for traditional quantum chemistry methods.

In conclusion, the binding energy of the beryllium dimer has been calculated using the diffusion Monte Carlo method in conjunction with a wide variety of trial wave functions. Even DMC calculations with a trial wave function as flexible as CAS(4,16) considerably underestimate the binding energy of the beryllium dimer. CI trial functions allowing excitations from the valence space into the entire virtual space give a binding energy within 27 cm−1 of the experimental value. It is possible that this small remaining discrepancy from experiment is due to the neglect of excitations from the 1s orbitals in the trial functions used for the DMC calculations. Although DMC calculations using small configurational spaces that give binding energies close to experiment have been reported for Be2, they also give energies for the atom and dimer that are appreciably higher than those obtained using the CI trial functions employed here. Thus, the good agreement of the binding energy of Be2 with the experimental value obtained with such small multiconfigurational trial function spaces is likely fortuitous. We believe that our findings are relevant for a wide range of other dimers, e.g., the benzene dimer, where there is appreciable configuration mixing in the wave functions of the monomers. In particular, achieving well converged binding energies for such systems is likely to require the use of multiconfigurational trial functions allowing for high-order excitations as well as excitations outside the valence space.

This research was supported by Grant No. CHE136234 from the National Science Foundation. The calculations were done on computers in the University of Pittsburgh’s Center for Simulation and Modeling. We acknowledge valuable discussions with M. Morales.

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