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Potential curves for alkaline-earth dimers by density functional theory with long-range correlation corrections
Chemical Physics Letters 416 (2005) 370–375 www.elsevier.com/locate/cplett
Potential curves for alkaline-earth dimers by density functional theory with long-range correlation corrections ´ ngya´n Iann C. Gerber, Ja´nos G. A
*
Laboratoire de Cristallographie et de Mode´lisation des Mate´riaux Mine´raux et Biologiques, UMR 7036, CNRS – Universite´ Henri Poincare´, B.P. 239, F-54506 Vandœuvre-le`s-Nancy, France Received 12 August 2005 Available online 19 October 2005
Abstract Potential curves of alkaline-earth dimers, Be2, Mg2 and Ca2, bound by London dispersion forces, are determined with the recently developed range separated hybrid method with perturbational long-range correlation corrections (RSH + MP2), and are compared to wave function approaches (MP2), to conventional density functional (LDA, PBE, BLYP, BPW91) and hybrid functional (PBE0, B3LYP) results. In contrast to rare gas dimers, functionals involving BeckeÕs exchange are not repulsive, but yield a relatively profound potential well. The RSH + MP2 approach outperforms the MP2 method considerably and reduces the basis set superposition error. It represents a promising alternative for the study of larger alkaline-earth clusters. Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction One of the major shortcomings of practical density functional theory (DFT) is its failure in describing van der Waals (London dispersion) forces [1]. For instance, the binding in rare gas dimers is strongly overestimated by the local density approximation, while semi-local functionals, using generalized gradient approximations (GGA), behave in a rather erratic manner [2–5]. Recently, a new scheme has been proposed to extend the applicability of DFT methods to London dispersion forces [6]. This approach is based on the idea of a range separation [7], i.e., on the separation of the electron–electron interactions to a short and a long-range component. The short-range interactions, involving the electron–electron cusp, can be efficiently described by appropriately designed local or semi-local functionals, while the long-range exchange and correlation effects are taken into account explicitly by a wave function description. Inclusion of long
*
Corresponding author. Fax: +33 383 406 492. E-mail addresses: [email protected] (I.C. Gerber), ´ ngya´n). [email protected] (J.G. A 0009-2614/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.09.059
range Hartree–Fock exchange leads to a range separated hybrid functional (RSH), which can be corrected for long-range correlation effects, responsible for the dispersion interactions, by a long-range second order Møller– Plesset (MP2) perturbation theory. The resulting RSH + MP2 approach has been proved to be successful for rare gas dimers [6], and it seems to be a promising method for the description of the interactions in dispersionbound alkaline-earth dimers as well. The recent growth in the interest for alkaline-earth dimers is due to their use in cold atoms collisions. The laser cooled and trapped Group II atomic species, like Mg, Ca, Sr, and Ba, can be used in ultra-precise optical clocks or quantum information processing devices; see [8] and references therein. Unfortunately, the determination of the potential energy surfaces of small alkaline-earth dimers and clusters, like Ben, Mgn [9], or Can [10], necessary for a theoretical modelling of such processes, are among the most difficult challenges in quantum chemistry. For small values of n, where the metallic behaviour is not yet manifested, these systems composed of closed shell singlet atoms, are held together by weak dispersion forces as it has been shown by the decomposition of the interaction energy components within the SAPT framework [11].
I.C. Gerber, J.G. A´ngya´n / Chemical Physics Letters 416 (2005) 370–375
Alkaline-earth clusters have been studied by a great variety of approaches. Approximate density functionals [10,12,13] have been reputed to have a relatively good performance for such systems, but a closer look on the results reveals that the predicted potential curves are not reliable. Wave function studies have been done with MP2-R12 [14], MP4(SDQT) [15], coupled cluster (CCSD, CCSD(T)) [9], multireference averaged quadratic cluster (MR-AQCC) [16], multireference CI (MRCI) [17,18], CASSCF/CASPT2 [19], extended geminal [20] and full CI (FCI) [21,22] approaches. Various theoretical methods predict a dissociation energy of Be2 around 800–950 cm1 with an equilibrium dis˚ . The comparison with experiment is tance of about 2.5 A somewhat occulted by the fact that experimental dissociation energy has been determined from an analytical potential curve, fitted to measured vibrational frequencies. For instance, the dissociation energy obtained from a Morse potential fit has been found to be 790 ± 30 cm1 [23], while a Morse + C6/R6 fit has led to 839 ± 10 cm1 [24]. Accurate theoretical predictions and experimental studies are relatively scarce for Mg2 and Ca2 dimers. In the case of Mg2, early experiments gave 424 cm1 for the well-depth ˚ for the equilibrium distance [25]. The groundand 3.89 A state potential of Ca2 has been determined recently by Fourier-transform spectroscopy and numerical reconstruction [26], leading to a binding energy of 1102 cm1 with an equi˚. librium distance of 4.28 A The importance of the inclusion of core states in the correlation treatment in the Be2 system has been emphasized by Martin [22], who estimated the frozen core (FC) full CI (FCI) dissociation limit 875 ± 20 cm1, while the all electron (AE) FCI dissociation limit is by 70 cm1 larger, 945 ± 20 cm1. This is in agreement with earlier estimates of 80–90 cm1 for the FC/AE difference, mainly attributed to core–valence correlation effects [17]. Similar trends can be expected for the Mg2 and Ca2 systems. A new promising density functional approach is the ‘‘ab initio DFT’’ method of Bartlett and co-workers [27], which consists in determining system-specific correlation functional and potential on a many-body basis. The resulting optimized effective potential second order many-body perturbation theory approach, i.e., OEP-MBPT(2), has been recently applied to van der Waals complexes, like He2, Ne2 as well as Be2 [28]. In particular, a significant improvement has been observed with respect to a simple MP2 treatment for the case of Be2: while MP2 practically does not bind, the bound potential given by the OEPMBPT(2) method is only by about 30% too low with respect to the reference curve. A drawback of this calculation is, however, that the basis set superposition error (BSSE) could not be evaluated due to convergence problems. An alternative method, based also on the density functional theory, is the RSH + MP2 approach, which combines long-range Hartree–Fock exchange with an explicit second order perturbational treatment of the correlation. The relatively weak van der Waals bond in rare gas
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complexes has been successfully described by this approach and it is interesting to test if the RSH + MP2 method is able to describe dimers of the much more polarizable alkaline-earth atoms as well. Another attractive feature of the RSH + MP2 approach, observed in the rare gas dimers studies, is its relative insensitivity with respect to BSSE [6], which makes it particularly advantageous for the study of medium size clusters, where the application of the counterpoise correction would be prohibitively complicated. In the following we study the potential curves of Be2, Mg2 and Ca2 dimers by the RSH + MP2 method and compare our results with available experimental data and accurate ab initio potentials taken from the literature and also with conventional density functional results. The basic principles of the RSH + MP2 method along with the technical details of the calculations are described in the following section, followed by the discussion of the results (Section 3). We conclude by some generals remarks and perspectives. 2. Methods and computational details The RSH + MP2 method consists in determining first the range separated hybrid orbitals, /k(r), by the self-consistent solution of the following Kohn–Sham-like equations 1 sr;l r2 þ vne ðrÞ þ vH ðrÞ þ vxc ðrÞ þ ^vlr;l x;HF /k ðrÞ ¼ ek /k ðrÞ; 2 ð1Þ where vne(r) is the nuclear attraction, vH(r) is the Hartree potential; vsr;l xc ðrÞ is the short-range exchange-correlation (xc) potential, obtained as the functional derivative of the short-range xc functional, Esr;l vlr;l x;HF is the long-range xc ½q, and ^ Hartree–Fock exchange potential. The range separation parameter, l, determines the reach of the short-range component of the electron–electron interactions, defined as sr;l wee ðrij Þ ¼ erfcðlrij Þ=rij . The total energy corresponding to the range separated hybrid solution for a closed-shell system Z X 1 l sr;l ERSH ¼ 2 ek dr vxc ðrÞ þ vH ðrÞ qðrÞ 2 k X 1 þ Esr;l hkljlkilr;l ð2Þ xc ½q þ 2 kl is corrected in a second step for long-range correlation effects by an MP2-like expression,where i, j and a, b are occupied and virtual orbitals, respectively, ElMP2 ¼
2 1 X jhijabilr;l j ; 4 ijab ei þ ej ea eb
ð3Þ
where the antisymmetrized two-electron integrals are calculated over the RSH orbitals with the long-range component of the Coulomb operator, wlr;l ee ðrij Þ ¼ erfðlr ij Þ=r ij . A more detailed description of the RSH + MP2 method can be found in [6].
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The RSH + MP2 scheme has been implemented in a development version of MOLPRO package. All calculations have been done with the LDA variant of the RSH + MP2 method, i.e., the short-range exchangecorrelation terms are treated on the local density approximation level [29], using a value of the range separation parameter, l = 0.5, the same as in our previous study of rare gas dimers. In order to compare our results directly with recently published results for the same system [28], the Be2 calculations have been done in the augmented triple zeta ANO [30] basis set with all the s functions uncontracted. Higher angular momentum components were used in cartesian form. This basis set will be referred to as ANO-TZ in the following. For the Mg2 dimer the correlation consistent polarized Core/Valence Triple Zeta (cc-pCVTZ) basis set [31] was applied, while the Ca2 dimer was calculated with standard cc-pV5Z basis set of Dunning [32]. Basis set superposition error (BSSE) has been removed by the standard counterpoise method, and the grid superposition error was eliminated by enforcing the same numerical quadrature of high accuracy, with a threshold of 1010 for both dimer and monomer calculations. High convergence criterium (1010 a.u.) was used in orbital optimization in the SCF cycle.
Table 1 Parameters of the Be2 potential curves in the ANO-TZ basis set Method
De (cm1)
Re (a.u)
r (a.u)
xe (cm1)
LDA PBE BLYP BPW91 PBE0 B3LYP
4517 3402 2137 2873 2268 1436
4.54 4.59 4.59 4.63 4.73 4.73
3.61 3.72 3.86 3.79 3.91 4.00
324 327 327 327 327 327
MP2 (AE) RSH + MP2 (AE)
369 (435) 747 (9)
5.19 (-0.23) 4.86 (0.00)
4.47 4.19
327 327
MP2 (FC) RSH + MP2 (FC)
340 (28) 737 (1)
5.28 (-0.05) 4.86 (0.00)
4.52 4.20
334 302
EXGEM [20] r12-MRCI [18]
945 903
4.63 4.62
4.02 4.02
318 270
Significant BSSE corrections are given in parentheses.
0
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3. Results and discussion -500 PBE BLYP BPW91 B3LYP PBE0 EXGEM Røeggen (’05)
-1
U [cm ]
3.1. Be2 dimer The difficulty of the Be2 system stems mainly from the fact that one is supposed to handle simultaneously strong static correlations due to the presence of low-lying (2s2p) configuration in the Be atoms and long-range dynamical correlations between the two subsystems of the complex [17]. Most of the standard correlation approaches are deficient to describe one of these effects. We have chosen as ab initio reference potentials the extended geminal (EXGEM) calculations of Røeggen [20] and the (r12)-MRCI results of Gdanitz [18]. The EXGEM potential gives a binding energy of 945 cm1, slightly stronger than the experimental estimate, and yields a harmonic vibrational frequency overestimated by 40 cm1 (317 cm1 against the experimental value of 275 cm1). The (r12)-MRCI potential curve is in a better agreement with experimental values: it predicts a well-depth of around 900 cm1 and a harmonic frequency of 270 cm1. Table 1 summarizes some characteristic features of the potential energy curves obtained by various standard density functionals, (LDA, PBE, BPW91, BLYP, PBE0 and B3LYP), by the AE-MP2 method and by the RSH + MP2 approach. The results are presented in terms of well-depths De, equilibrium distances Re, and hard core radii, r, defined by E(r) = 0. The harmonic vibrational frequency, xe characterizes the curvature of the potential in the minimum. The full potential energy curves are presented in Figs. 1 and 2.
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-1500
4
6
8
10
12
14
R [a0] Fig. 1. Be2 dissociation curves for different approximate functionals with BSSE correction applied in the Roos-TZ basis set.
Good agreement was found with the literature values for LDA and BLYP [12] as well as for BPW91 and B3LYP [13,33] binding energies and geometries. All functionals have a tendency for overbinding by a factor ranging from nearly 5 (LDA) to 1.5 (B3LYP), while the equilibrium geometry remains relatively good (less than 2% error with respect to the reference value of Røeggen). Adding some Hartree–Fock exchange, i.e., switching from PBE to PBE0, or from BLYP to B3LYP, improves substantially the description of the van der Waals minimum by reducing
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250 MP2 corrected RSH+MP2 corrected MP2 non-corrected RSH+MP2 non-corrected r12-MRCI Gdanitz (’99) EXGEM Røeggen (’05)
0
-1
U [cm ]
-250
-500
-750
-1000 4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
R [a0] Fig. 2. Influence of the BSSE correction in the intermolecular interaction energy for the beryllium dimer in the ANO-TZ basis set, and comparison with the accurate results of [18] and [20].
the binding energy as well as by shifting the repulsive wall and making it more repulsive, as reflected by the hard core radii reported in Table 1. The fact that all functionals lead to bound potentials is somewhat surprising in comparison with the trend found for the GGA description of rare gas dimers. In this latter case a clear distinction has been found between functionals using BeckeÕs exchange, leading always to fully repulsive potentials and the PBE exchange, which yields potential curves of binding character. The decisive role of the exchange component of the functional has been emphasized by several authors [3], and the particular behaviour of BeckeÕs exchange has been correlated to the violation of the local Lieb–Oxford bound [34] and to the behaviour of the enhancement factor for large reduced density gradients [35]. The particular behaviour of the Be2 dimer indicates that the above cited empirical rules, established for the rare gas dimers, cannot be generalized for other kind of van der Waals systems. A further surprise comes from the shape of the potential curves in the long-range region, where the functionals with BeckeÕs exchange component display a small bump, i.e., a local maximum, as it can be seen in Fig. 1. A possible explanation might be that the overlap of the more diffuse Be atoms decays much slower than for the dimers of the more compact rare gas atoms. Due to persisting overlap, the overbinding behaviour of the functional is prolonged, because of the small reduced density gradient values, and
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all GGA functionals lead to bound curves. The enhancement factor of the Becke exchange varies too steeply with the reduced density gradient, making the attractive exchange energy too small with respect to the repulsive kinetic energy, giving rise to a predominance of the repulsion for an intermediate distance and leading to the appearance of the bump on the BLYP, BPW91 and B3LYP curves. Fig. 2 compares the RSH + MP2 potential curve with the conventional MP2 results in the AE approach. MP2 is obviously unable to describe the bonding in this case, as it has already been discussed by several authors [18,28]. The situation is complicated by a BSSE which is very large in comparison to the small binding energy. In a sharp contrast to conventional MP2, the RSH + MP2 calculation with the same basis set is not only doing a very good job in describing the potential well, but it is also practically BSSE free. The absence of BSSE in the RSH + MP2 approach can be attributed to the fact that the inner shell correlation is treated by the short-range functional which is much less sensitive to basis set superposition effects than the wave function methods. In effect, as it can be seen from the comparison of the AE and FC MP2 results in Table 1, most of the BSSE comes from the core–core and core–valence correlation in the wave function approach, while the valence– valence correlation, responsible for the van der Waals bonding, has a much smaller BSSE. The RSH + MP2 handles only this latter component of the correlation by wave function method, which explains the insensitivity with respect to the BSSE. This behaviour is, however, not a general feature of the RSH + MP2 formalism, since it depends on the particular value of the range-separation parameter, l. The value of l = 0.5 seems to be favorable for this system. In spite of the considerable improvements brought by the RSH + MP2 approach, a fully quantitative description would require improvement both in the short-range functional part by a gradient correction, and in the wave function description of the long-range correlation, e.g., by including triple or even higher excitations [28]. As far as the asymptotic region is concerned, it is known that the RSH + MP2 approach is not able to provide an exact answer. The leading C6 coefficient of the power series expansion of the potential, given by the Casimir–Polder formula, Z 3h dxaA ðixÞaB ðixÞ; C AB ¼ ð4Þ 6 p where aX(ix) are the exact frequency-dependent mean dipole polarizabilities, would require the knowledge of the exact dipole polarizabilities of the monomers. Within the RSH + MP2 framework, in the asymptotic limit, the aX(ix) is obtained from a uncorrelated response function instead of the exact one. Nevertheless, the asymptotic limits of the potential curves seem to be quite reasonable.
I.C. Gerber, J.G. A´ngya´n / Chemical Physics Letters 416 (2005) 370–375
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3.2. Mg2 dimer Characteristic features of the Mg2 potential curves are reported in Table 2, while full shape of the RSH + MP2 and MP2 potentials are illustrated in Fig. 3. Standard density functionals present again a rather erratic behaviour. Like for Be2, all functionals yield bound potentials, but now BLYP and B3LYP produce very shallow potential wells. LDA, PBE, BPW91 and PBE0 overbind by a factor of 4–2. The equilibrium distances and hard core radii, pre-
Table 2 Reduced parameters of the Mg2 potential curves in the cc-pCVTZ basis set, reference values taken from [25]: Dref = 424 cm1, Rref = 7.35 a.u., xref = 51.1 cm1, and rref = 0.83 Method
De/Dref
Re/Rref
r
xe/xref
LDA PBE BLYP BPW91 PBE0 B3LYP
3.96 2.58 0.11 1.83 2.05 0.19
0.88 0.90 1.00 0.91 0.93 1.02
0.70 0.75 0.93 0.78 0.78 0.92
0.93 0.97 0.97 0.97 0.98 0.97
MP2 (AE) RSH + MP2 (AE)
0.69 (0.18) 0.74 (0.07)
1.06 (0.03) 1.02 (0.01)
0.89 0.87
0.96 0.92
MP2 (FC) RSH + MP2 (FC)
0.64 (0.03) 0.69 (0.06)
1.07 (0.01) 1.03 (0.01)
0.90 0.87
0.98 0.93
Significant BSSE corrections are given in parentheses.
200 MP2 RSH+MP2 FCI Partridge (’90) MP2-R12 Klopper (’93) CCSD(T) Stoll (’01) Exp. Balfour (’70)
100
-100
-1
U [cm ]
0
dicted by conventional functionals, behave much less systematically than for the Be2 system. On the other hand, one obtains quite reasonable values for the harmonic frequencies which have less than 7% of error. Again, the RSH + MP2 approach brings an improvement for practically all of the parameters, not only in comparison to conventional density functionals, but also with respect to MP2 calculations. Again, the reduction of the BSSE is significant as compared to the MP2 all-electron calculations and the total BSSE correction of the equilibrium distance is only one third of that of a usual MP2 calculation. Note however that the improvement is less spectacular for this system. 3.3. Ca2 dimer Finally, we report an attempt to describe the Ca2 system, which has not yet been extensively studied neither by DFT [10] nor by accurate ab initio methods [15]. Compilation of reduced potential curve parameters obtained by various functionals is given in Table 3. Two distinct groups of functionals can be distinguished: two of them, BLYP and B3LYP, simply underestimate the binding energies by more than 20% and 39%, respectively, which is much less problematic than in the magnesium dimer for instance. The rest of the functionals overestimates the exchange energy contribution dramatically, producing too deep minima. Nevertheless, one should note the probably fortuitous agreement of the equilibrium distances obtained by all the functionals with respect to the experimental value, with errors less than 4%. In contrast to the rare gas dimers [3], it seems that these weakly bound systems are more sensitive to the influence of correlation. As a matter of fact, switching from the LYP correlation to the PW91 one, keeping the same exchange functional (B88), leads to a sizeable change of the potential curve with a severe overestimation of the binding energy as well as a significant decrease of the hard-core radius. The same trends can be observed in the case of Be2 and Mg2, which means that PW91 correlation functional is able, in a certain way, to take into account the strong static correlation in the beryllium atom, even if this effect is lar-
-200 Table 3 Reduced parameters of the Ca2 potential curves in the cc-pV5Z basis set in frozen core approximation, reference values taken from [26]: Dref = 1102 cm1, Rref = 8.09 a.u., xref = 63.7 cm1, and rref = 0.85
-300
-400
-500
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
R [a0] Fig. 3. Dissociation curves for MP2 and RSH + MP2 for the Mg2 system in the cc-pCVTZ basis set, after BSSE correction.
Method
De/Dref
Re/Rref
r
xe/xref
LDA PBE BLYP BPW91 PBE0 B3LYP
2.62 2.01 0.79 1.74 1.55 0.61
1.00 0.96 0.98 0.96 0.98 1.01
0.76 0.79 0.86 0.80 0.82 0.88
0.91 0.77 0.76 0.77 0.78 0.56
MP2 RSH + MP2
0.78 (0.01) 0.84 (0.01)
1.06 (0.00) 1.02 (0.00)
0.90 0.87
1.00 0.95
Significant BSSE corrections are given in parentheses.
I.C. Gerber, J.G. A´ngya´n / Chemical Physics Letters 416 (2005) 370–375
suffers less from the BSSE than any other ab initio correlated wave function methods and allows for studies of larger van der Waals complexes in a reasonable computational cost.
500 MP2 RSH+MP2 MP4(SDTQ) Kaplan (’00) Exp. Allard (’02)
250
375
Acknowledgements The authors are indebted to Dr. A. Savin (Paris) for useful discussions, and to Prof. R.J. Bartlett and V. Lotrich (University of Florida) for providing us details of their Be2 calculations.
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-1
U [cm ]
0
-500
References
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7
8
9
10
11
12
13
14
R [a0] Fig. 4. Dissociation curves for MP2 and RSH + MP2 for the Ca2 system in the cc-pV5Z basis set, after BSSE correction.
gely overestimated in this approximation and is almost totally missed by LYP functional. In Fig. 4, complete interaction curves obtained by standard MP2 and RSH + MP2 are shown, in comparison to single point result of Kaplan et al. [15], and a potential curve based on experimental spectroscopic data. By giving an equilibrium distance closer to the experimental value of the minimum, and a binding energy deeper than MP2 results, our RSH + MP2 functional ameliorates the description of the dissociation curve of Ca2. Improvement of the repulsive part of the curve can be solely attributed to the correct description made by the short-range DFT approximation, and addition of the long-range perturbational term gives qualitative behaviour similar to the exact one, which is dominated by a standard C6/R6 term. 4. Conclusions and perspectives We have seen that the RSH + MP2 method is able to take into account dispersion effects in weakly bound dimers and offers a dramatic improvement over conventional DFT approaches. This accuracy is due to a combination of short-range DFT with explicit second order perturbational correction of long-range correlation effects. Even if it is clear that a full, quantitative description of Be2, Mg2 and Ca2 systems would require to take into account the neardegeneracy occurring in these systems, our scheme seems to be a reasonable alternative at a low computational cost. Furthermore, it has been demonstrated that our scheme
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Potential curves for alkaline-earth dimers by density functional theory with long-range correlation corrections ´ ngya´n Iann C. Gerber, Ja´nos G. A
*
Laboratoire de Cristallographie et de Mode´lisation des Mate´riaux Mine´raux et Biologiques, UMR 7036, CNRS – Universite´ Henri Poincare´, B.P. 239, F-54506 Vandœuvre-le`s-Nancy, France Received 12 August 2005 Available online 19 October 2005
Abstract Potential curves of alkaline-earth dimers, Be2, Mg2 and Ca2, bound by London dispersion forces, are determined with the recently developed range separated hybrid method with perturbational long-range correlation corrections (RSH + MP2), and are compared to wave function approaches (MP2), to conventional density functional (LDA, PBE, BLYP, BPW91) and hybrid functional (PBE0, B3LYP) results. In contrast to rare gas dimers, functionals involving BeckeÕs exchange are not repulsive, but yield a relatively profound potential well. The RSH + MP2 approach outperforms the MP2 method considerably and reduces the basis set superposition error. It represents a promising alternative for the study of larger alkaline-earth clusters. Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction One of the major shortcomings of practical density functional theory (DFT) is its failure in describing van der Waals (London dispersion) forces [1]. For instance, the binding in rare gas dimers is strongly overestimated by the local density approximation, while semi-local functionals, using generalized gradient approximations (GGA), behave in a rather erratic manner [2–5]. Recently, a new scheme has been proposed to extend the applicability of DFT methods to London dispersion forces [6]. This approach is based on the idea of a range separation [7], i.e., on the separation of the electron–electron interactions to a short and a long-range component. The short-range interactions, involving the electron–electron cusp, can be efficiently described by appropriately designed local or semi-local functionals, while the long-range exchange and correlation effects are taken into account explicitly by a wave function description. Inclusion of long
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Corresponding author. Fax: +33 383 406 492. E-mail addresses: [email protected] (I.C. Gerber), ´ ngya´n). [email protected] (J.G. A 0009-2614/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.09.059
range Hartree–Fock exchange leads to a range separated hybrid functional (RSH), which can be corrected for long-range correlation effects, responsible for the dispersion interactions, by a long-range second order Møller– Plesset (MP2) perturbation theory. The resulting RSH + MP2 approach has been proved to be successful for rare gas dimers [6], and it seems to be a promising method for the description of the interactions in dispersionbound alkaline-earth dimers as well. The recent growth in the interest for alkaline-earth dimers is due to their use in cold atoms collisions. The laser cooled and trapped Group II atomic species, like Mg, Ca, Sr, and Ba, can be used in ultra-precise optical clocks or quantum information processing devices; see [8] and references therein. Unfortunately, the determination of the potential energy surfaces of small alkaline-earth dimers and clusters, like Ben, Mgn [9], or Can [10], necessary for a theoretical modelling of such processes, are among the most difficult challenges in quantum chemistry. For small values of n, where the metallic behaviour is not yet manifested, these systems composed of closed shell singlet atoms, are held together by weak dispersion forces as it has been shown by the decomposition of the interaction energy components within the SAPT framework [11].
I.C. Gerber, J.G. A´ngya´n / Chemical Physics Letters 416 (2005) 370–375
Alkaline-earth clusters have been studied by a great variety of approaches. Approximate density functionals [10,12,13] have been reputed to have a relatively good performance for such systems, but a closer look on the results reveals that the predicted potential curves are not reliable. Wave function studies have been done with MP2-R12 [14], MP4(SDQT) [15], coupled cluster (CCSD, CCSD(T)) [9], multireference averaged quadratic cluster (MR-AQCC) [16], multireference CI (MRCI) [17,18], CASSCF/CASPT2 [19], extended geminal [20] and full CI (FCI) [21,22] approaches. Various theoretical methods predict a dissociation energy of Be2 around 800–950 cm1 with an equilibrium dis˚ . The comparison with experiment is tance of about 2.5 A somewhat occulted by the fact that experimental dissociation energy has been determined from an analytical potential curve, fitted to measured vibrational frequencies. For instance, the dissociation energy obtained from a Morse potential fit has been found to be 790 ± 30 cm1 [23], while a Morse + C6/R6 fit has led to 839 ± 10 cm1 [24]. Accurate theoretical predictions and experimental studies are relatively scarce for Mg2 and Ca2 dimers. In the case of Mg2, early experiments gave 424 cm1 for the well-depth ˚ for the equilibrium distance [25]. The groundand 3.89 A state potential of Ca2 has been determined recently by Fourier-transform spectroscopy and numerical reconstruction [26], leading to a binding energy of 1102 cm1 with an equi˚. librium distance of 4.28 A The importance of the inclusion of core states in the correlation treatment in the Be2 system has been emphasized by Martin [22], who estimated the frozen core (FC) full CI (FCI) dissociation limit 875 ± 20 cm1, while the all electron (AE) FCI dissociation limit is by 70 cm1 larger, 945 ± 20 cm1. This is in agreement with earlier estimates of 80–90 cm1 for the FC/AE difference, mainly attributed to core–valence correlation effects [17]. Similar trends can be expected for the Mg2 and Ca2 systems. A new promising density functional approach is the ‘‘ab initio DFT’’ method of Bartlett and co-workers [27], which consists in determining system-specific correlation functional and potential on a many-body basis. The resulting optimized effective potential second order many-body perturbation theory approach, i.e., OEP-MBPT(2), has been recently applied to van der Waals complexes, like He2, Ne2 as well as Be2 [28]. In particular, a significant improvement has been observed with respect to a simple MP2 treatment for the case of Be2: while MP2 practically does not bind, the bound potential given by the OEPMBPT(2) method is only by about 30% too low with respect to the reference curve. A drawback of this calculation is, however, that the basis set superposition error (BSSE) could not be evaluated due to convergence problems. An alternative method, based also on the density functional theory, is the RSH + MP2 approach, which combines long-range Hartree–Fock exchange with an explicit second order perturbational treatment of the correlation. The relatively weak van der Waals bond in rare gas
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complexes has been successfully described by this approach and it is interesting to test if the RSH + MP2 method is able to describe dimers of the much more polarizable alkaline-earth atoms as well. Another attractive feature of the RSH + MP2 approach, observed in the rare gas dimers studies, is its relative insensitivity with respect to BSSE [6], which makes it particularly advantageous for the study of medium size clusters, where the application of the counterpoise correction would be prohibitively complicated. In the following we study the potential curves of Be2, Mg2 and Ca2 dimers by the RSH + MP2 method and compare our results with available experimental data and accurate ab initio potentials taken from the literature and also with conventional density functional results. The basic principles of the RSH + MP2 method along with the technical details of the calculations are described in the following section, followed by the discussion of the results (Section 3). We conclude by some generals remarks and perspectives. 2. Methods and computational details The RSH + MP2 method consists in determining first the range separated hybrid orbitals, /k(r), by the self-consistent solution of the following Kohn–Sham-like equations 1 sr;l r2 þ vne ðrÞ þ vH ðrÞ þ vxc ðrÞ þ ^vlr;l x;HF /k ðrÞ ¼ ek /k ðrÞ; 2 ð1Þ where vne(r) is the nuclear attraction, vH(r) is the Hartree potential; vsr;l xc ðrÞ is the short-range exchange-correlation (xc) potential, obtained as the functional derivative of the short-range xc functional, Esr;l vlr;l x;HF is the long-range xc ½q, and ^ Hartree–Fock exchange potential. The range separation parameter, l, determines the reach of the short-range component of the electron–electron interactions, defined as sr;l wee ðrij Þ ¼ erfcðlrij Þ=rij . The total energy corresponding to the range separated hybrid solution for a closed-shell system Z X 1 l sr;l ERSH ¼ 2 ek dr vxc ðrÞ þ vH ðrÞ qðrÞ 2 k X 1 þ Esr;l hkljlkilr;l ð2Þ xc ½q þ 2 kl is corrected in a second step for long-range correlation effects by an MP2-like expression,where i, j and a, b are occupied and virtual orbitals, respectively, ElMP2 ¼
2 1 X jhijabilr;l j ; 4 ijab ei þ ej ea eb
ð3Þ
where the antisymmetrized two-electron integrals are calculated over the RSH orbitals with the long-range component of the Coulomb operator, wlr;l ee ðrij Þ ¼ erfðlr ij Þ=r ij . A more detailed description of the RSH + MP2 method can be found in [6].
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The RSH + MP2 scheme has been implemented in a development version of MOLPRO package. All calculations have been done with the LDA variant of the RSH + MP2 method, i.e., the short-range exchangecorrelation terms are treated on the local density approximation level [29], using a value of the range separation parameter, l = 0.5, the same as in our previous study of rare gas dimers. In order to compare our results directly with recently published results for the same system [28], the Be2 calculations have been done in the augmented triple zeta ANO [30] basis set with all the s functions uncontracted. Higher angular momentum components were used in cartesian form. This basis set will be referred to as ANO-TZ in the following. For the Mg2 dimer the correlation consistent polarized Core/Valence Triple Zeta (cc-pCVTZ) basis set [31] was applied, while the Ca2 dimer was calculated with standard cc-pV5Z basis set of Dunning [32]. Basis set superposition error (BSSE) has been removed by the standard counterpoise method, and the grid superposition error was eliminated by enforcing the same numerical quadrature of high accuracy, with a threshold of 1010 for both dimer and monomer calculations. High convergence criterium (1010 a.u.) was used in orbital optimization in the SCF cycle.
Table 1 Parameters of the Be2 potential curves in the ANO-TZ basis set Method
De (cm1)
Re (a.u)
r (a.u)
xe (cm1)
LDA PBE BLYP BPW91 PBE0 B3LYP
4517 3402 2137 2873 2268 1436
4.54 4.59 4.59 4.63 4.73 4.73
3.61 3.72 3.86 3.79 3.91 4.00
324 327 327 327 327 327
MP2 (AE) RSH + MP2 (AE)
369 (435) 747 (9)
5.19 (-0.23) 4.86 (0.00)
4.47 4.19
327 327
MP2 (FC) RSH + MP2 (FC)
340 (28) 737 (1)
5.28 (-0.05) 4.86 (0.00)
4.52 4.20
334 302
EXGEM [20] r12-MRCI [18]
945 903
4.63 4.62
4.02 4.02
318 270
Significant BSSE corrections are given in parentheses.
0
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3. Results and discussion -500 PBE BLYP BPW91 B3LYP PBE0 EXGEM Røeggen (’05)
-1
U [cm ]
3.1. Be2 dimer The difficulty of the Be2 system stems mainly from the fact that one is supposed to handle simultaneously strong static correlations due to the presence of low-lying (2s2p) configuration in the Be atoms and long-range dynamical correlations between the two subsystems of the complex [17]. Most of the standard correlation approaches are deficient to describe one of these effects. We have chosen as ab initio reference potentials the extended geminal (EXGEM) calculations of Røeggen [20] and the (r12)-MRCI results of Gdanitz [18]. The EXGEM potential gives a binding energy of 945 cm1, slightly stronger than the experimental estimate, and yields a harmonic vibrational frequency overestimated by 40 cm1 (317 cm1 against the experimental value of 275 cm1). The (r12)-MRCI potential curve is in a better agreement with experimental values: it predicts a well-depth of around 900 cm1 and a harmonic frequency of 270 cm1. Table 1 summarizes some characteristic features of the potential energy curves obtained by various standard density functionals, (LDA, PBE, BPW91, BLYP, PBE0 and B3LYP), by the AE-MP2 method and by the RSH + MP2 approach. The results are presented in terms of well-depths De, equilibrium distances Re, and hard core radii, r, defined by E(r) = 0. The harmonic vibrational frequency, xe characterizes the curvature of the potential in the minimum. The full potential energy curves are presented in Figs. 1 and 2.
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-1500
4
6
8
10
12
14
R [a0] Fig. 1. Be2 dissociation curves for different approximate functionals with BSSE correction applied in the Roos-TZ basis set.
Good agreement was found with the literature values for LDA and BLYP [12] as well as for BPW91 and B3LYP [13,33] binding energies and geometries. All functionals have a tendency for overbinding by a factor ranging from nearly 5 (LDA) to 1.5 (B3LYP), while the equilibrium geometry remains relatively good (less than 2% error with respect to the reference value of Røeggen). Adding some Hartree–Fock exchange, i.e., switching from PBE to PBE0, or from BLYP to B3LYP, improves substantially the description of the van der Waals minimum by reducing
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250 MP2 corrected RSH+MP2 corrected MP2 non-corrected RSH+MP2 non-corrected r12-MRCI Gdanitz (’99) EXGEM Røeggen (’05)
0
-1
U [cm ]
-250
-500
-750
-1000 4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
R [a0] Fig. 2. Influence of the BSSE correction in the intermolecular interaction energy for the beryllium dimer in the ANO-TZ basis set, and comparison with the accurate results of [18] and [20].
the binding energy as well as by shifting the repulsive wall and making it more repulsive, as reflected by the hard core radii reported in Table 1. The fact that all functionals lead to bound potentials is somewhat surprising in comparison with the trend found for the GGA description of rare gas dimers. In this latter case a clear distinction has been found between functionals using BeckeÕs exchange, leading always to fully repulsive potentials and the PBE exchange, which yields potential curves of binding character. The decisive role of the exchange component of the functional has been emphasized by several authors [3], and the particular behaviour of BeckeÕs exchange has been correlated to the violation of the local Lieb–Oxford bound [34] and to the behaviour of the enhancement factor for large reduced density gradients [35]. The particular behaviour of the Be2 dimer indicates that the above cited empirical rules, established for the rare gas dimers, cannot be generalized for other kind of van der Waals systems. A further surprise comes from the shape of the potential curves in the long-range region, where the functionals with BeckeÕs exchange component display a small bump, i.e., a local maximum, as it can be seen in Fig. 1. A possible explanation might be that the overlap of the more diffuse Be atoms decays much slower than for the dimers of the more compact rare gas atoms. Due to persisting overlap, the overbinding behaviour of the functional is prolonged, because of the small reduced density gradient values, and
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all GGA functionals lead to bound curves. The enhancement factor of the Becke exchange varies too steeply with the reduced density gradient, making the attractive exchange energy too small with respect to the repulsive kinetic energy, giving rise to a predominance of the repulsion for an intermediate distance and leading to the appearance of the bump on the BLYP, BPW91 and B3LYP curves. Fig. 2 compares the RSH + MP2 potential curve with the conventional MP2 results in the AE approach. MP2 is obviously unable to describe the bonding in this case, as it has already been discussed by several authors [18,28]. The situation is complicated by a BSSE which is very large in comparison to the small binding energy. In a sharp contrast to conventional MP2, the RSH + MP2 calculation with the same basis set is not only doing a very good job in describing the potential well, but it is also practically BSSE free. The absence of BSSE in the RSH + MP2 approach can be attributed to the fact that the inner shell correlation is treated by the short-range functional which is much less sensitive to basis set superposition effects than the wave function methods. In effect, as it can be seen from the comparison of the AE and FC MP2 results in Table 1, most of the BSSE comes from the core–core and core–valence correlation in the wave function approach, while the valence– valence correlation, responsible for the van der Waals bonding, has a much smaller BSSE. The RSH + MP2 handles only this latter component of the correlation by wave function method, which explains the insensitivity with respect to the BSSE. This behaviour is, however, not a general feature of the RSH + MP2 formalism, since it depends on the particular value of the range-separation parameter, l. The value of l = 0.5 seems to be favorable for this system. In spite of the considerable improvements brought by the RSH + MP2 approach, a fully quantitative description would require improvement both in the short-range functional part by a gradient correction, and in the wave function description of the long-range correlation, e.g., by including triple or even higher excitations [28]. As far as the asymptotic region is concerned, it is known that the RSH + MP2 approach is not able to provide an exact answer. The leading C6 coefficient of the power series expansion of the potential, given by the Casimir–Polder formula, Z 3h dxaA ðixÞaB ðixÞ; C AB ¼ ð4Þ 6 p where aX(ix) are the exact frequency-dependent mean dipole polarizabilities, would require the knowledge of the exact dipole polarizabilities of the monomers. Within the RSH + MP2 framework, in the asymptotic limit, the aX(ix) is obtained from a uncorrelated response function instead of the exact one. Nevertheless, the asymptotic limits of the potential curves seem to be quite reasonable.
I.C. Gerber, J.G. A´ngya´n / Chemical Physics Letters 416 (2005) 370–375
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3.2. Mg2 dimer Characteristic features of the Mg2 potential curves are reported in Table 2, while full shape of the RSH + MP2 and MP2 potentials are illustrated in Fig. 3. Standard density functionals present again a rather erratic behaviour. Like for Be2, all functionals yield bound potentials, but now BLYP and B3LYP produce very shallow potential wells. LDA, PBE, BPW91 and PBE0 overbind by a factor of 4–2. The equilibrium distances and hard core radii, pre-
Table 2 Reduced parameters of the Mg2 potential curves in the cc-pCVTZ basis set, reference values taken from [25]: Dref = 424 cm1, Rref = 7.35 a.u., xref = 51.1 cm1, and rref = 0.83 Method
De/Dref
Re/Rref
r
xe/xref
LDA PBE BLYP BPW91 PBE0 B3LYP
3.96 2.58 0.11 1.83 2.05 0.19
0.88 0.90 1.00 0.91 0.93 1.02
0.70 0.75 0.93 0.78 0.78 0.92
0.93 0.97 0.97 0.97 0.98 0.97
MP2 (AE) RSH + MP2 (AE)
0.69 (0.18) 0.74 (0.07)
1.06 (0.03) 1.02 (0.01)
0.89 0.87
0.96 0.92
MP2 (FC) RSH + MP2 (FC)
0.64 (0.03) 0.69 (0.06)
1.07 (0.01) 1.03 (0.01)
0.90 0.87
0.98 0.93
Significant BSSE corrections are given in parentheses.
200 MP2 RSH+MP2 FCI Partridge (’90) MP2-R12 Klopper (’93) CCSD(T) Stoll (’01) Exp. Balfour (’70)
100
-100
-1
U [cm ]
0
dicted by conventional functionals, behave much less systematically than for the Be2 system. On the other hand, one obtains quite reasonable values for the harmonic frequencies which have less than 7% of error. Again, the RSH + MP2 approach brings an improvement for practically all of the parameters, not only in comparison to conventional density functionals, but also with respect to MP2 calculations. Again, the reduction of the BSSE is significant as compared to the MP2 all-electron calculations and the total BSSE correction of the equilibrium distance is only one third of that of a usual MP2 calculation. Note however that the improvement is less spectacular for this system. 3.3. Ca2 dimer Finally, we report an attempt to describe the Ca2 system, which has not yet been extensively studied neither by DFT [10] nor by accurate ab initio methods [15]. Compilation of reduced potential curve parameters obtained by various functionals is given in Table 3. Two distinct groups of functionals can be distinguished: two of them, BLYP and B3LYP, simply underestimate the binding energies by more than 20% and 39%, respectively, which is much less problematic than in the magnesium dimer for instance. The rest of the functionals overestimates the exchange energy contribution dramatically, producing too deep minima. Nevertheless, one should note the probably fortuitous agreement of the equilibrium distances obtained by all the functionals with respect to the experimental value, with errors less than 4%. In contrast to the rare gas dimers [3], it seems that these weakly bound systems are more sensitive to the influence of correlation. As a matter of fact, switching from the LYP correlation to the PW91 one, keeping the same exchange functional (B88), leads to a sizeable change of the potential curve with a severe overestimation of the binding energy as well as a significant decrease of the hard-core radius. The same trends can be observed in the case of Be2 and Mg2, which means that PW91 correlation functional is able, in a certain way, to take into account the strong static correlation in the beryllium atom, even if this effect is lar-
-200 Table 3 Reduced parameters of the Ca2 potential curves in the cc-pV5Z basis set in frozen core approximation, reference values taken from [26]: Dref = 1102 cm1, Rref = 8.09 a.u., xref = 63.7 cm1, and rref = 0.85
-300
-400
-500
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
R [a0] Fig. 3. Dissociation curves for MP2 and RSH + MP2 for the Mg2 system in the cc-pCVTZ basis set, after BSSE correction.
Method
De/Dref
Re/Rref
r
xe/xref
LDA PBE BLYP BPW91 PBE0 B3LYP
2.62 2.01 0.79 1.74 1.55 0.61
1.00 0.96 0.98 0.96 0.98 1.01
0.76 0.79 0.86 0.80 0.82 0.88
0.91 0.77 0.76 0.77 0.78 0.56
MP2 RSH + MP2
0.78 (0.01) 0.84 (0.01)
1.06 (0.00) 1.02 (0.00)
0.90 0.87
1.00 0.95
Significant BSSE corrections are given in parentheses.
I.C. Gerber, J.G. A´ngya´n / Chemical Physics Letters 416 (2005) 370–375
suffers less from the BSSE than any other ab initio correlated wave function methods and allows for studies of larger van der Waals complexes in a reasonable computational cost.
500 MP2 RSH+MP2 MP4(SDTQ) Kaplan (’00) Exp. Allard (’02)
250
375
Acknowledgements The authors are indebted to Dr. A. Savin (Paris) for useful discussions, and to Prof. R.J. Bartlett and V. Lotrich (University of Florida) for providing us details of their Be2 calculations.
-250
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U [cm ]
0
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References
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7
8
9
10
11
12
13
14
R [a0] Fig. 4. Dissociation curves for MP2 and RSH + MP2 for the Ca2 system in the cc-pV5Z basis set, after BSSE correction.
gely overestimated in this approximation and is almost totally missed by LYP functional. In Fig. 4, complete interaction curves obtained by standard MP2 and RSH + MP2 are shown, in comparison to single point result of Kaplan et al. [15], and a potential curve based on experimental spectroscopic data. By giving an equilibrium distance closer to the experimental value of the minimum, and a binding energy deeper than MP2 results, our RSH + MP2 functional ameliorates the description of the dissociation curve of Ca2. Improvement of the repulsive part of the curve can be solely attributed to the correct description made by the short-range DFT approximation, and addition of the long-range perturbational term gives qualitative behaviour similar to the exact one, which is dominated by a standard C6/R6 term. 4. Conclusions and perspectives We have seen that the RSH + MP2 method is able to take into account dispersion effects in weakly bound dimers and offers a dramatic improvement over conventional DFT approaches. This accuracy is due to a combination of short-range DFT with explicit second order perturbational correction of long-range correlation effects. Even if it is clear that a full, quantitative description of Be2, Mg2 and Ca2 systems would require to take into account the neardegeneracy occurring in these systems, our scheme seems to be a reasonable alternative at a low computational cost. Furthermore, it has been demonstrated that our scheme
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