- Email: [email protected]
Ab initio basis set and correlation limit interaction energies for He–He, He–H2, and H–H2
4 May 2001
Chemical Physics Letters 339 (2001) 133±139
www.elsevier.nl/locate/cplett
Ab initio basis set and correlation limit interaction energies for He±He, He±H2, and H±H2 Jae Shin Lee * Department of Chemistry, College of Natural Sciences, Ajou University, Suwon 442-380, Republic of Korea Received 4 December 2000; in ®nal form 27 February 2001
Abstract An ab initio search for the exact nonrelativistic interaction energies for He±He, He±H2 , and H±H2 systems was made by estimating the basis set limit CCSD(T) interaction energies from the extrapolation of counterpoise (CP) interaction energies with aug-cc-pV5Z and aug-cc-pV6Z basis sets by 1= X 13 (X 5; 6) and correcting for the dierence between the FCI and CCSD(T) energies. While the result for He2 is in perfect agreement with the exact quantum Monte Carlo result, the result for H3 suggests the barrier height for colinear H H2 ! H2 H reaction is close to 9.60 kcal/ mol, 0.01 kcal/mol lower than the best result reported so far Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The error of an ab initio atomic and molecular electronic computation mainly comes from two sources: de®ciency in electron correlation treatment and incompleteness of one-particle basis set employed in the calculation. Depending on the problems of interest, one source of error could be more dominant than the other or both could be comparable to each other. For example, in many hydrogen bonded complexes, the eect of basis set extension appears more important than the electron correlation eect as the binding could be explained reasonably well at the second-order Mùller±Plesset (MP2) level or even at the SCF level. On the contrary, in very weakly bound systems such as van der Waals molecules or complexes, employment of a high-level electron correlation method and large ¯exible basis set are *
Fax: +82-31-219-1615. E-mail address: [email protected] (J.S. Lee).
both important to understand the binding as the interatomic or intermolecular binding forces are usually dominated by the instantaneous longrange dispersion force. Therefore, in this case, an accurate determination of the binding energy can only be achieved by properly considering the eect of basis set extension and electron correlation on the results of the calculation with incomplete basis set and electron correlation treatment. It would be highly desirable to have a scheme to systematically approach the correlation and basis set limit as the limit is approached. In this respect, the employment of the family of correlation-consistent basis sets (aug-)cc-pVXZ (X cardinal number, D(2), T(3), Q(4), 5, 6) developed by Dunning and coworkers [1±6] in an ab initio electronic computation would give an advantage as the energy and other properties were known to systematically approach the basis set limit as the cardinal number X increases [2±4,7±11]. By extrapolating the computed energies using appropriate functions at dierent correlation levels one could predict the
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 3 1 7 - 7
134
J.S. Lee / Chemical Physics Letters 339 (2001) 133±139
basis set limit without actual computation at the basis set limit, thereby separating the basis set incompleteness error from the error caused by the de®ciency in electron correlation treatment at respective level of theory. Although there have been several suggestions about the most eective extrapolation scheme to estimate the basis set limit from the results with the correlation-consistent basis sets, we recently have shown [12] that the extrapolation of two successive total electronic energies with correlation-consistent basis sets 3 cc-pVXZ (or aug-cc-pVXZ) by X k could provide the highly accurate basis set limit energies as the basis set limit is approached, with k being the parameter depending on the electron correlation level employed. Although this type of functional form was motivated from the previous partial wave expansion studies of correlation energies of small atomic systems [13±15], extrapolation of total energies rather than correlation energies may be more appropriate to exhibit the manner in which correlation-consistent basis sets are built, that is, the basis functions of higher angular momenta are added systematically to saturate the basis function space, both in radial and angular directions, simultaneously. By adjusting the parameter k, one could control the convergence rate with correlation-consistent basis set, depending on the correlation level employed. The recommended values for k were 0 for MP2 and )1 for CCSD and CCSD(T) level. Although the application of this extrapolation scheme to He, H2 , and He2 have been shown to yield very accurate total and interaction energies [12], its general applicability to more complex and larger systems, especially with the energies of the complex and fragments in the counterpoise (CP) scheme (both with the complex basis set), has yet to be tested. In this Letter, on the basis of this limiting behavior of total electronic energies of the complexes and fragments with the correlation-consistent basis set in the CP scheme [16], we pursue the ab initio limit of interaction energies of He±He, H±H2 , and He±H2 for which very accurate interaction energies are available from the previous studies for comparison. These systems could be considered as the model systems for weak (He±He, He±H2 ) and strong (H±H2 ) interaction, respectively. Further-
more, the accurate determination of the barrier height to H H2 ! H2 H reaction has been shown to be critical to understand the dynamic characteristics of this reaction, especially under the resonance condition [17,18]. It will be shown that by carefully measuring the eect of electron correlation and basis set extension toward the full con®guration interaction/complete basis set (FCI/ CBS) limit, one can get a very accurate estimate to the exact result, which may be only compared to the result by exact quantum Monte Carlo (QMC) method or FCI result with a large basis set. In the following section we explain the methodology and computational details to reach the ab initio FCI/ CBS limit. In Section 3, the results and discussions for He2 , H3 , and HeH2 are presented along with the comparison with the previous theoretical and experimental results on these systems. The conclusion is in Section 4. 2. Method and computational approach In this study, the nonrelativistic interaction energy under Born±Oppenheimer approximation for He±He, He±H2 , and H±H2 was obtained in two steps. First, the basis set limit interaction energies at the CCSD(T) level was estimated through the extrapolation of the CP corrected interaction energies with aug-cc-pV5Z and aug-cc-pV6Z basis sets by 1= X 13 (X 5; 6). This comes from the fact that once the basis set limit total energies of the complex (EAB 1) and the fragments (EA 1 and EB 1) can be estimated from the extrapolation of total energies of the complex EAB X and fragments (EA X and EB X ) with aug-cc-pV5Z and aug-cc-pV6Z basis sets by 1= X 13 formula (X 5; 6), then the basis set limit interaction energies DEAB 1 can be estimated from the extrapolation of corresponding interaction energies DEAB X with aug-cc-pV5Z and aug-cc-pV6Z set 3 using the same formula of 1= X 1 as described in the following. Let EAB 1 EAB X
p= X
3
1 ;
EA 1 EA X
q= X
13 ;
EB 1 EB X
r= X
13 :
J.S. Lee / Chemical Physics Letters 339 (2001) 133±139
Then DEAB 1 DEAB X
s= X
3
1 ;
where DEAB 1 EAB 1 DEAB X EAB X sp q r
EA 1 EA X
EB 1; EB X ;
p; q; r and s are fitting parameters: The contracted basis functions for aug-cc-pV5Z and aug-cc-pV6Z set are [6s 5p 4d 3f 2g] and [7s 6p 5d 4f 3g 2h], respectively, which include the diuse functions of each angular momenta type [19]. In our study, we extrapolated the CP corrected interaction energies rather than uncorrected interaction energies as CP correction was shown necessary to make the correlation contribution of interaction energy dominated by the electronic Coulomb cusp without the interference of BSSE [20,21]. Although total (interaction) energies (SCF correlation energy) rather than correlation energies were extrapolated using a single type of function in this study based on previous results for He, H2 and He2 , the validity of CP procedure was expected to be equally applicable to total energies as well as correlation energies as the BSSE in HF energies with aug-cc-pV5Z and aug-cc-pV6Z set would be negligible. Once the basis set limit interaction energies are estimated at the CCSD(T) level, the next step is to estimate the dierence between the interaction energies at the CCSD(T) and FCI level. Since the FCI calculations with a large basis set (including f or higher angular-type functions) were impractical in actual calculations, this was done by performing FCI calculations with aug-cc-pVDZ and aug-ccpVTZ basis set and comparing the dierences between interaction energies at the CCSD(T) and FCI level. After the dierences between two levels with aug-cc-pVDZ and aug-cc-pVTZ basis sets were computed, they were extrapolated using 3 1= X 1 formula (X 2; 3) to estimate the basis set limit dierence between two levels, assuming 3 the same 1= X 1 behavior for the total energies at the FCI level as the CCSD(T) level. Fortunately, the dierence between the CCSD(T) and FCI interaction energies appeared to converge
135
rapidly with basis set, assuring that the dierence with aug-cc-pVTZ basis set is already very close to the basis set limit dierence. For He2 and H3 , the calculation has been carried out at the internuclear distance of 5.6 and 1.707825 a.u. (with linear symmetric con®guration for H±H±H) to compare with the previous QMC results [22,23]. For H3 , this linear con®guration corresponds to the saddle point (transition state) geometry for H H2 ! H2 H reaction. The activation barrier for this reaction is then computed as the dierence between the energies of the complex and fragments (H, H2 ). The equilibrium distance of H2 in the fragments was set to 1.40108 a.u. [24]. For He±H2 complex, to compare with the previous experimental and theoretical results for He±H2 system [25,26], the interaction energy was calculated at the geometry where He atom is distanced 6.5 a.u. from the center of mass of H2 molecule with H±H distance ®xed at the vibrationally averaged bond distance of 1.449 a.u.. The coupled cluster and FCI calculations were performed using GA U S S I A N and GA M E S S program packages, respectively [27,28]. 3. Results and discussion In Table 1, we present the estimated CCSD(T) basis set limit total and interaction energies obtained from extrapolation of corresponding energies in the CP scheme with aug-cc-pV5Z and augcc-pV6Z basis sets for He±He, He±H2 and H±H2 systems. Both CP corrected and uncorrected interaction energies are presented to estimate the possible error bound of the obtained basis set limit. More often than not, the exact basis set limit lies between CP corrected and uncorrected results. For H atom energy at the basis set limit, exact value of )0.5 Eh was employed. Although it is dicult to evaluate the accuracy of extrapolated results due to the absence of the `exact' CCSD(T) basis set limit interaction energies except for He2 system, the excellent agreement of extrapolated total and interaction energies with the known basis set limit at the CCSD(T) level for He, H2 , and He2 system [12], along with the very small dierence between two extrapolated basis set limits of CP corrected and uncorrected interaction energies
136
J.S. Lee / Chemical Physics Letters 339 (2001) 133±139
Table 1 Basis set limit total (E) and interaction DE energies for He±He, He±H2 and H±H2 at the CCSD(T) levela aug-cc-pV5Z
aug-cc-pV6Z
Estimated CBS limitb
CBS limit
EHe2 EHe c DEd
)5.8064335 )2.9032012 )31.1 ()32.5)
)5.8069436 )2.9034556 )32.4 ()32.8)
)5.8074788 )2.9037225 )33.8 ()33.2)
)33.80.3e
H± H2
EH3 EH2 c EH c DEd
)1.6587923 )1.1742645 )0.4999950 15 467 (15 455)
)1.6589165 )1.1743656 )0.4999993 15 448 (15 443)
)1.6590468 )1.1744717 )0.5 15 425 (15 426)
He± H2
EHeH2 EHe c EH2 c DEd
)4.0771051 )2.9032024 )1.1738560 )46.7 ()49.0)
)4.0774636 )2.9034559 )1.1739603 )47.4 ()48.0)
)4.0778397 )2.9037219 )1.1740697 )48.1 ()47.0)
Complexnbasis set He±He
a
Geometries at which total and interaction energies are computed are given in the text (Section 2). Extrapolated results of total and interaction energies by 1= X 13 (X 5; 6). c Energies of the fragments were computed with dimer-centered (complex) basis set. d CP corrected (and uncorrected, in parentheses) interaction energies in units of lEh . Total energies are in units of Eh . e From Ref. [12]. b
strongly suggests the estimated basis set limit interaction energies are very close to the exact CBS limits, with the possible error bound being the dierence between two extrapolated basis set limits of CP corrected and uncorrected interaction energies. It also has to be noted that though extrapolation of fragment total energies such as He or H2 with the dimer (complex) basis set may yield the less accurate basis set limit total energies than extrapolation of corresponding energies with monomer basis set, interaction energies could be more accurate in the case of former than the latter due to the consistent extrapolated results for the fragments and complex in the CP scheme, which is well manifested in the case of He2 . To compare the extrapolated results with the experimental data or exact QMC results directly, one has to consider the eect of full (connected) triple (in case of H3 ) and quadruple (in case of He2 and HeH2 ) excitations on the interaction energies. This was done by examining the dierence between the CCSD(T) and FCI energies of the complexes as the CCSD energies of the fragments (He and H2 ) are equivalent to FCI energies. In Table 2, we present the estimated dierence between the CCSD(T) and FCI interaction energies at the basis set limit along with the exact QMC results for He±He and H±H2 systems. Since the dierence between the FCI and CCSD(T) interaction energies is equal to the dif-
ference between the total energies of the complex at two theoretical levels, we ®rst computed the dierence between total energies of the complexes at the CCSD(T) and FCI level with aug-cc-pVDZ and aug-cc-pVTZ basis sets, which were then ex3 trapolated by 1= X 1 (X 2; 3) to estimate the dierence at the basis set limit. For HeH2 and H3 , due to a huge number of con®guration state functions at the FCI/aug-cc-pVTZ level, some of the less important diuse functions were excluded from the basis set. Therefore, while d-type diuse functions on the hydrogen atom were omitted from the original aug-cc-pVTZ basis set for HeH2 , all diuse functions on the central hydrogen atom were excluded from the corresponding aug-ccpVTZ basis set for H3 . Exclusion of these basis functions from aug-cc-pVTZ basis sets was expected to have a negligible eect on the dierence between two levels as the increase beyond aug-ccpVDZ set is shown to little change the dierence between two levels in Table 2. Actually it is shown from Table 2 that the extrapolation of dierences 3 with DZ and TZ basis set by 1= X 1 (X 2; 3) yields virtually the same results with aug-cc-pVTZ basis set in all cases. Therefore, one could conclude that further basis set increase beyond aug-ccpVTZ would little aect the magnitude of the dierence between the CCSD(T) and FCI interaction energies. This is consistent with the previous
J.S. Lee / Chemical Physics Letters 339 (2001) 133±139
137
Table 2 Dierence dE between the CCSD(T) and FCI total energies (E) for He±He, H±H2 and He±H2 a aug-cc-pVDZ
aug-cc-pVTZb
E ECCSD T dEd
)5.7791399 )5.7790139 0.9
EFCI ECCSD T dEd EFCI ECCSD T dEd
Complexnbasis set He±He
H± H2
He±H2
FCI
Estimated CBS limitc
FCI/CBS limit
)5.8012278 )5.8012268 1.0
1.0 ()34.5)
34:8 0:3e
)1.6493284 )1.6492040 124.4
)1.6568731 )1.6567486 124.5
125 (15 300)
15 319f
)4.0545239 )4.0545222 1.7
)4.0727743 )4.0727726 1.7
1.7 ()49.8)
±
a
Geometries at which total and interaction energies are computed are given in the text (Section 2). b For HeH2 and H3 , some of diuse functions were excluded from the original aug-cc-pVTZ basis set (see the text). c Extrapolated results of dierences between the CCSD(T) and FCI level by 1= X 13 (X 2:3). Values in parentheses are estimated FCI/CBS limit interaction energies (see the text). d dE ECCSD T EFCI 106 . E and dE are in units of Eh and lEh , respectively. e From Ref. [22]. f From Ref. [23].
result on He2 which showed the dierence of 1:0 lEh between CCSD(T) and FCI energies with a large basis set including bond functions [12,29]. As a result we expect the extrapolated dierences in Table 2 are accurate to exact values within few tenths of lEh . If one adds these dierences to the CCSD(T) basis set limit interaction energies (CP corrected results), one would obtain the estimated FCI/CBS limit interaction energies which are also shown in Table 2. For He±He system, our result is in perfect agreement with the exact QMC result [22], con®rming the reliability of the extrapolation scheme employed in this study. For H±H2 , the discrepancy between the estimated FCI/CBS limit and QMC result could be mainly due to the fact that the QMC barrier height in Table 2 is the barrier height in the potential energy surface (PES) constructed by ®tting the QMC potential points with statistical sampling error of 16 lEh 0.10 kcal/mole). Considering the remarkable accuracy of our extrapolation scheme demonstrated for He, H2 , and He2 , it is highly probable that the extrapolated barrier height of 15 300 lEh (or 9.601 kcal/mole) would be closer to the exact barrier height than the QMC-based result of 15 319 lEh (or 9.613 kcal/mole). This is further supported by the narrow margin between CP corrected and uncorrected interaction energies at the CCSD(T) level (see Table 1). It is known that the resonance
characteristics for H± H2 interaction is very sensitive to the detailed PES near saddle point region and dierence in barrier height of a little more than 0.01 kcal/mole could aect those characteristics in a detectable manner [17,18,32]. Therefore, it now appears that the exact barrier height for colinear H±H2 interaction is lower than any other values predicted from the previously reported PES for this system [18,23,30±32]. For He±H2 interaction, our CCSD(T) interaction energy of )48:1 lEh is in good agreement with the previous MP4 result of )47:2 lEh with a large basis set containing bond functions by Tao [26] and experimentally derived result of )48 lEh by Rodwell and Scoles [25]. However, if one considers the dierence between the CCSD(T) and FCI results and also relaxation eect of He±H2 and H±H distances to search for the global minimum of the complex, it is highly probable that the electronic binding energy between He and H2 could be more than 50 lEh . This value again is larger than any previous results for this system [25,26,33]. Furthermore, comparing He±He and He±H2 interaction, atom±diatom dispersion interaction appears to be about 1.5 times larger than isoelectronic atom±atom dispersion interaction, though other factors such as dipole moment and polarizability of the interacting species would certainly aect the actual binding energy.
138
J.S. Lee / Chemical Physics Letters 339 (2001) 133±139
Therefore, we expect that similar behavior in the dierence of the interaction energies between Ne± Ne and Ne±B2 system (or Ar±Ar and Ar±F2 ) could be observed. In this respect, it is interesting to note that while the interaction energy for Ne±HF is larger than for Ne±Ne by a factor of more than 2.0, the interaction energy for Ne±FH appears to be larger than for Ne±Ne by a factor of less than 1.5 [34,35].
small energy dierence in the PES could aect the dynamical behavior signi®cantly. The application of similar extrapolation scheme to larger and more complex systems is in progress.
Acknowledgements This work was supported by Korea Research Foundation Grant (KRF-1999-015-DI0051).
4. Conclusion For He±He, H±H2 , and He±H2 interaction, we have shown that by applying a simple two-point 3 (X 1 X 5; 6) extrapolation scheme to the CP corrected CCSD(T) interaction energies (with aug-cc-pV5Z and aug-cc-pV6Z basis set) to obtain the basis set limit and correcting for the dierence between the CCSD(T) and full CI basis set limit result, one can get a highly accurate interaction energy which may be only comparable to the exact QMC result (or FCI result with a large basis set which may not be feasible computationally at the present time). This was possible due to the systematically convergent behavior of total electronic energy with correlation-consistent basis set in these molecular systems. In the case of H3 , our extrapolated results appear to strongly suggest that the barrier height for linear H H2 ! H2 H reaction is about 0:010 0:012 kcal/mole lower than the values predicted from the best potentials currently available for this system [23,32]. Since the dynamics of this reaction is known to be sensitive to the colinear barrier height [30,32], it would be interesting to examine the eect of this reduced barrier height on the dynamical characteristics such as the location of scattering resonance or thermal rate constant. We also found that the binding energy for He±H2 system would be more than 50 lEh , which suggests atom±diatom (homonuclear) dispersion interaction would be roughly 1.5 times larger than isoelectronic atom± atom dispersion interaction. Therefore, though simple, this kind of extrapolation scheme could be exploited as an eective tool to determine the accurate relative energies of weakly bound complexes and other molecular systems for which
References [1] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007. [2] D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 98 (1993) 1358. [3] D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 100 (1994) 2975. [4] K.A. Peterson, D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 100 (1994) 7410. [5] A. Wilson, T. van Mourik, T.H. Dunning Jr., J. Mol. Struct. (Theochem.) 388 (1997) 339. [6] A. Wilson, D.E. Woon, K.A. Peterson, T.H. Dunning Jr., J. Chem. Phys. 110 (1999) 7667. [7] D. Feller, J. Chem. Phys. 96 (1992) 6104. [8] S.S. Xantheas, T.H. Dunning Jr., J. Phys. Chem. 97 (1993) 18. [9] K.A. Peterson, T.H. Dunning Jr., J. Chem. Phys. 102 (1995) 2032. [10] T. Helgaker, W. Klopper, H. Koch, J. Noga, J. Chem. Phys. 106 (1997) 9639. [11] A. Halkier, T. Helgaker, P. Jorgensen, W. Klopper, J. Olsen, Chem. Phys. Lett. 302 (1999) 437. [12] J.S. Lee, S.Y. Park, J. Chem. Phys. 112 (2000) 10746. [13] C. Schwartz, Phys. Rev. 126 (1962) 1015. [14] R.N. Hill, J. Chem. Phys. 83 (1985) 1173. [15] W. Kutzelnigg, J.D. Morgan III, J. Chem. Phys. 96 (1992) 4484. [16] S.F. Boys, F. Bernardi, Mol. Phys. 19 (1970) 553. [17] A. Kuppermann, Y.S.M. Wu, Chem. Phys. Lett. 241 (1995) 229. [18] A.I. Boothroyd, W.J. Keogh, P.G. Martin, M.R. Peterson, J. Chem. Phys. 95 (1991) 4343. [19] Basis sets were obtained from the Extensible Computational Chemistry Environment Basis Set Database, as developed and distributed by the Molecular Science Computing Facility, Environmental and Molecular Sciences Laboratory which is part of the Paci®c Northwest Laboratory, P.O. Box 999, Richland, WA 99352, USA, and funded by the US Department of Energy. The Paci®c Northwest Laboratory is a multi-program laboratory operated by Battelle Memorial Institue for the US Department of Energy under contract DE-AC06-76RLO
J.S. Lee / Chemical Physics Letters 339 (2001) 133±139
[20] [21] [22] [23] [24] [25] [26] [27]
1830. Contact David Feller or Karen Schuchardt for further information. A. Halkier, W. Klopper, T. Helgaker, P. Jorgensen, P.R. Taylor, J. Chem. Phys. 111 (1999) 9157. A. Wilson, T.H. Dunning Jr., J. Chem. Phys. 106 (1997) 8718. J.B. Anderson, C.A. Traynor, B.M. Boghosian, J. Chem. Phys. 99 (1993) 345. Y.S.M. Wu, A. Kuppermann, J.B. Anderson, Phys. Chem. Chem. Phys. 1 (1999) 929. W. Kolos, L. Wolniewicz, J. Chem. Phys. 49 (1968) 404. W.R. Rodwell, G. Scoles, J. Phys. Chem. 86 (1982) 1053. F.M. Tao, J. Chem. Phys. 100 (1994) 4947. M.J. Frisch, et al., GA U S S I A N 98, Gaussian, Inc., Pittsburgh PA, 1998.
139
[28] M.W. Schmidt, et al., Gamess, J. Comput. Chem. 14 (1993) 1347. [29] T.v. Mourik, J.H.v. Lenthe, J. Chem. Phys. 102 (1995) 7479. [30] D.G. Truhlar, C.J. Horowitz, J. Chem. Phys. 68 (1978) 2466. [31] A.J.C. Varandas, F.C. Brown, C.A. Mead, D.G. Truhlar, N.C. Blais, J. Chem. Phys. 86 (1987) 6258. [32] A.I. Boothroyd, W.J. Keogh, P.G. Martin, M.R. Peterson, J. Chem. Phys. 104 (1996) 7139. [33] W. Meyer, P.C. Hariharan, W. Kutzelnigg, J. Chem. Phys. 73 (1980) 1880. [34] S.V. O'Neil, D.J. Nesbitt, P. Rosmus, H.J. Werner, D.C. Clary, J. Chem. Phys. 91 (1989) 711. [35] D.E. Woon, J. Chem. Phys. 100 (1994) 2838.
Chemical Physics Letters 339 (2001) 133±139
www.elsevier.nl/locate/cplett
Ab initio basis set and correlation limit interaction energies for He±He, He±H2, and H±H2 Jae Shin Lee * Department of Chemistry, College of Natural Sciences, Ajou University, Suwon 442-380, Republic of Korea Received 4 December 2000; in ®nal form 27 February 2001
Abstract An ab initio search for the exact nonrelativistic interaction energies for He±He, He±H2 , and H±H2 systems was made by estimating the basis set limit CCSD(T) interaction energies from the extrapolation of counterpoise (CP) interaction energies with aug-cc-pV5Z and aug-cc-pV6Z basis sets by 1= X 13 (X 5; 6) and correcting for the dierence between the FCI and CCSD(T) energies. While the result for He2 is in perfect agreement with the exact quantum Monte Carlo result, the result for H3 suggests the barrier height for colinear H H2 ! H2 H reaction is close to 9.60 kcal/ mol, 0.01 kcal/mol lower than the best result reported so far Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The error of an ab initio atomic and molecular electronic computation mainly comes from two sources: de®ciency in electron correlation treatment and incompleteness of one-particle basis set employed in the calculation. Depending on the problems of interest, one source of error could be more dominant than the other or both could be comparable to each other. For example, in many hydrogen bonded complexes, the eect of basis set extension appears more important than the electron correlation eect as the binding could be explained reasonably well at the second-order Mùller±Plesset (MP2) level or even at the SCF level. On the contrary, in very weakly bound systems such as van der Waals molecules or complexes, employment of a high-level electron correlation method and large ¯exible basis set are *
Fax: +82-31-219-1615. E-mail address: [email protected] (J.S. Lee).
both important to understand the binding as the interatomic or intermolecular binding forces are usually dominated by the instantaneous longrange dispersion force. Therefore, in this case, an accurate determination of the binding energy can only be achieved by properly considering the eect of basis set extension and electron correlation on the results of the calculation with incomplete basis set and electron correlation treatment. It would be highly desirable to have a scheme to systematically approach the correlation and basis set limit as the limit is approached. In this respect, the employment of the family of correlation-consistent basis sets (aug-)cc-pVXZ (X cardinal number, D(2), T(3), Q(4), 5, 6) developed by Dunning and coworkers [1±6] in an ab initio electronic computation would give an advantage as the energy and other properties were known to systematically approach the basis set limit as the cardinal number X increases [2±4,7±11]. By extrapolating the computed energies using appropriate functions at dierent correlation levels one could predict the
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 3 1 7 - 7
134
J.S. Lee / Chemical Physics Letters 339 (2001) 133±139
basis set limit without actual computation at the basis set limit, thereby separating the basis set incompleteness error from the error caused by the de®ciency in electron correlation treatment at respective level of theory. Although there have been several suggestions about the most eective extrapolation scheme to estimate the basis set limit from the results with the correlation-consistent basis sets, we recently have shown [12] that the extrapolation of two successive total electronic energies with correlation-consistent basis sets 3 cc-pVXZ (or aug-cc-pVXZ) by X k could provide the highly accurate basis set limit energies as the basis set limit is approached, with k being the parameter depending on the electron correlation level employed. Although this type of functional form was motivated from the previous partial wave expansion studies of correlation energies of small atomic systems [13±15], extrapolation of total energies rather than correlation energies may be more appropriate to exhibit the manner in which correlation-consistent basis sets are built, that is, the basis functions of higher angular momenta are added systematically to saturate the basis function space, both in radial and angular directions, simultaneously. By adjusting the parameter k, one could control the convergence rate with correlation-consistent basis set, depending on the correlation level employed. The recommended values for k were 0 for MP2 and )1 for CCSD and CCSD(T) level. Although the application of this extrapolation scheme to He, H2 , and He2 have been shown to yield very accurate total and interaction energies [12], its general applicability to more complex and larger systems, especially with the energies of the complex and fragments in the counterpoise (CP) scheme (both with the complex basis set), has yet to be tested. In this Letter, on the basis of this limiting behavior of total electronic energies of the complexes and fragments with the correlation-consistent basis set in the CP scheme [16], we pursue the ab initio limit of interaction energies of He±He, H±H2 , and He±H2 for which very accurate interaction energies are available from the previous studies for comparison. These systems could be considered as the model systems for weak (He±He, He±H2 ) and strong (H±H2 ) interaction, respectively. Further-
more, the accurate determination of the barrier height to H H2 ! H2 H reaction has been shown to be critical to understand the dynamic characteristics of this reaction, especially under the resonance condition [17,18]. It will be shown that by carefully measuring the eect of electron correlation and basis set extension toward the full con®guration interaction/complete basis set (FCI/ CBS) limit, one can get a very accurate estimate to the exact result, which may be only compared to the result by exact quantum Monte Carlo (QMC) method or FCI result with a large basis set. In the following section we explain the methodology and computational details to reach the ab initio FCI/ CBS limit. In Section 3, the results and discussions for He2 , H3 , and HeH2 are presented along with the comparison with the previous theoretical and experimental results on these systems. The conclusion is in Section 4. 2. Method and computational approach In this study, the nonrelativistic interaction energy under Born±Oppenheimer approximation for He±He, He±H2 , and H±H2 was obtained in two steps. First, the basis set limit interaction energies at the CCSD(T) level was estimated through the extrapolation of the CP corrected interaction energies with aug-cc-pV5Z and aug-cc-pV6Z basis sets by 1= X 13 (X 5; 6). This comes from the fact that once the basis set limit total energies of the complex (EAB 1) and the fragments (EA 1 and EB 1) can be estimated from the extrapolation of total energies of the complex EAB X and fragments (EA X and EB X ) with aug-cc-pV5Z and aug-cc-pV6Z basis sets by 1= X 13 formula (X 5; 6), then the basis set limit interaction energies DEAB 1 can be estimated from the extrapolation of corresponding interaction energies DEAB X with aug-cc-pV5Z and aug-cc-pV6Z set 3 using the same formula of 1= X 1 as described in the following. Let EAB 1 EAB X
p= X
3
1 ;
EA 1 EA X
q= X
13 ;
EB 1 EB X
r= X
13 :
J.S. Lee / Chemical Physics Letters 339 (2001) 133±139
Then DEAB 1 DEAB X
s= X
3
1 ;
where DEAB 1 EAB 1 DEAB X EAB X sp q r
EA 1 EA X
EB 1; EB X ;
p; q; r and s are fitting parameters: The contracted basis functions for aug-cc-pV5Z and aug-cc-pV6Z set are [6s 5p 4d 3f 2g] and [7s 6p 5d 4f 3g 2h], respectively, which include the diuse functions of each angular momenta type [19]. In our study, we extrapolated the CP corrected interaction energies rather than uncorrected interaction energies as CP correction was shown necessary to make the correlation contribution of interaction energy dominated by the electronic Coulomb cusp without the interference of BSSE [20,21]. Although total (interaction) energies (SCF correlation energy) rather than correlation energies were extrapolated using a single type of function in this study based on previous results for He, H2 and He2 , the validity of CP procedure was expected to be equally applicable to total energies as well as correlation energies as the BSSE in HF energies with aug-cc-pV5Z and aug-cc-pV6Z set would be negligible. Once the basis set limit interaction energies are estimated at the CCSD(T) level, the next step is to estimate the dierence between the interaction energies at the CCSD(T) and FCI level. Since the FCI calculations with a large basis set (including f or higher angular-type functions) were impractical in actual calculations, this was done by performing FCI calculations with aug-cc-pVDZ and aug-ccpVTZ basis set and comparing the dierences between interaction energies at the CCSD(T) and FCI level. After the dierences between two levels with aug-cc-pVDZ and aug-cc-pVTZ basis sets were computed, they were extrapolated using 3 1= X 1 formula (X 2; 3) to estimate the basis set limit dierence between two levels, assuming 3 the same 1= X 1 behavior for the total energies at the FCI level as the CCSD(T) level. Fortunately, the dierence between the CCSD(T) and FCI interaction energies appeared to converge
135
rapidly with basis set, assuring that the dierence with aug-cc-pVTZ basis set is already very close to the basis set limit dierence. For He2 and H3 , the calculation has been carried out at the internuclear distance of 5.6 and 1.707825 a.u. (with linear symmetric con®guration for H±H±H) to compare with the previous QMC results [22,23]. For H3 , this linear con®guration corresponds to the saddle point (transition state) geometry for H H2 ! H2 H reaction. The activation barrier for this reaction is then computed as the dierence between the energies of the complex and fragments (H, H2 ). The equilibrium distance of H2 in the fragments was set to 1.40108 a.u. [24]. For He±H2 complex, to compare with the previous experimental and theoretical results for He±H2 system [25,26], the interaction energy was calculated at the geometry where He atom is distanced 6.5 a.u. from the center of mass of H2 molecule with H±H distance ®xed at the vibrationally averaged bond distance of 1.449 a.u.. The coupled cluster and FCI calculations were performed using GA U S S I A N and GA M E S S program packages, respectively [27,28]. 3. Results and discussion In Table 1, we present the estimated CCSD(T) basis set limit total and interaction energies obtained from extrapolation of corresponding energies in the CP scheme with aug-cc-pV5Z and augcc-pV6Z basis sets for He±He, He±H2 and H±H2 systems. Both CP corrected and uncorrected interaction energies are presented to estimate the possible error bound of the obtained basis set limit. More often than not, the exact basis set limit lies between CP corrected and uncorrected results. For H atom energy at the basis set limit, exact value of )0.5 Eh was employed. Although it is dicult to evaluate the accuracy of extrapolated results due to the absence of the `exact' CCSD(T) basis set limit interaction energies except for He2 system, the excellent agreement of extrapolated total and interaction energies with the known basis set limit at the CCSD(T) level for He, H2 , and He2 system [12], along with the very small dierence between two extrapolated basis set limits of CP corrected and uncorrected interaction energies
136
J.S. Lee / Chemical Physics Letters 339 (2001) 133±139
Table 1 Basis set limit total (E) and interaction DE energies for He±He, He±H2 and H±H2 at the CCSD(T) levela aug-cc-pV5Z
aug-cc-pV6Z
Estimated CBS limitb
CBS limit
EHe2 EHe c DEd
)5.8064335 )2.9032012 )31.1 ()32.5)
)5.8069436 )2.9034556 )32.4 ()32.8)
)5.8074788 )2.9037225 )33.8 ()33.2)
)33.80.3e
H± H2
EH3 EH2 c EH c DEd
)1.6587923 )1.1742645 )0.4999950 15 467 (15 455)
)1.6589165 )1.1743656 )0.4999993 15 448 (15 443)
)1.6590468 )1.1744717 )0.5 15 425 (15 426)
He± H2
EHeH2 EHe c EH2 c DEd
)4.0771051 )2.9032024 )1.1738560 )46.7 ()49.0)
)4.0774636 )2.9034559 )1.1739603 )47.4 ()48.0)
)4.0778397 )2.9037219 )1.1740697 )48.1 ()47.0)
Complexnbasis set He±He
a
Geometries at which total and interaction energies are computed are given in the text (Section 2). Extrapolated results of total and interaction energies by 1= X 13 (X 5; 6). c Energies of the fragments were computed with dimer-centered (complex) basis set. d CP corrected (and uncorrected, in parentheses) interaction energies in units of lEh . Total energies are in units of Eh . e From Ref. [12]. b
strongly suggests the estimated basis set limit interaction energies are very close to the exact CBS limits, with the possible error bound being the dierence between two extrapolated basis set limits of CP corrected and uncorrected interaction energies. It also has to be noted that though extrapolation of fragment total energies such as He or H2 with the dimer (complex) basis set may yield the less accurate basis set limit total energies than extrapolation of corresponding energies with monomer basis set, interaction energies could be more accurate in the case of former than the latter due to the consistent extrapolated results for the fragments and complex in the CP scheme, which is well manifested in the case of He2 . To compare the extrapolated results with the experimental data or exact QMC results directly, one has to consider the eect of full (connected) triple (in case of H3 ) and quadruple (in case of He2 and HeH2 ) excitations on the interaction energies. This was done by examining the dierence between the CCSD(T) and FCI energies of the complexes as the CCSD energies of the fragments (He and H2 ) are equivalent to FCI energies. In Table 2, we present the estimated dierence between the CCSD(T) and FCI interaction energies at the basis set limit along with the exact QMC results for He±He and H±H2 systems. Since the dierence between the FCI and CCSD(T) interaction energies is equal to the dif-
ference between the total energies of the complex at two theoretical levels, we ®rst computed the dierence between total energies of the complexes at the CCSD(T) and FCI level with aug-cc-pVDZ and aug-cc-pVTZ basis sets, which were then ex3 trapolated by 1= X 1 (X 2; 3) to estimate the dierence at the basis set limit. For HeH2 and H3 , due to a huge number of con®guration state functions at the FCI/aug-cc-pVTZ level, some of the less important diuse functions were excluded from the basis set. Therefore, while d-type diuse functions on the hydrogen atom were omitted from the original aug-cc-pVTZ basis set for HeH2 , all diuse functions on the central hydrogen atom were excluded from the corresponding aug-ccpVTZ basis set for H3 . Exclusion of these basis functions from aug-cc-pVTZ basis sets was expected to have a negligible eect on the dierence between two levels as the increase beyond aug-ccpVDZ set is shown to little change the dierence between two levels in Table 2. Actually it is shown from Table 2 that the extrapolation of dierences 3 with DZ and TZ basis set by 1= X 1 (X 2; 3) yields virtually the same results with aug-cc-pVTZ basis set in all cases. Therefore, one could conclude that further basis set increase beyond aug-ccpVTZ would little aect the magnitude of the dierence between the CCSD(T) and FCI interaction energies. This is consistent with the previous
J.S. Lee / Chemical Physics Letters 339 (2001) 133±139
137
Table 2 Dierence dE between the CCSD(T) and FCI total energies (E) for He±He, H±H2 and He±H2 a aug-cc-pVDZ
aug-cc-pVTZb
E ECCSD T dEd
)5.7791399 )5.7790139 0.9
EFCI ECCSD T dEd EFCI ECCSD T dEd
Complexnbasis set He±He
H± H2
He±H2
FCI
Estimated CBS limitc
FCI/CBS limit
)5.8012278 )5.8012268 1.0
1.0 ()34.5)
34:8 0:3e
)1.6493284 )1.6492040 124.4
)1.6568731 )1.6567486 124.5
125 (15 300)
15 319f
)4.0545239 )4.0545222 1.7
)4.0727743 )4.0727726 1.7
1.7 ()49.8)
±
a
Geometries at which total and interaction energies are computed are given in the text (Section 2). b For HeH2 and H3 , some of diuse functions were excluded from the original aug-cc-pVTZ basis set (see the text). c Extrapolated results of dierences between the CCSD(T) and FCI level by 1= X 13 (X 2:3). Values in parentheses are estimated FCI/CBS limit interaction energies (see the text). d dE ECCSD T EFCI 106 . E and dE are in units of Eh and lEh , respectively. e From Ref. [22]. f From Ref. [23].
result on He2 which showed the dierence of 1:0 lEh between CCSD(T) and FCI energies with a large basis set including bond functions [12,29]. As a result we expect the extrapolated dierences in Table 2 are accurate to exact values within few tenths of lEh . If one adds these dierences to the CCSD(T) basis set limit interaction energies (CP corrected results), one would obtain the estimated FCI/CBS limit interaction energies which are also shown in Table 2. For He±He system, our result is in perfect agreement with the exact QMC result [22], con®rming the reliability of the extrapolation scheme employed in this study. For H±H2 , the discrepancy between the estimated FCI/CBS limit and QMC result could be mainly due to the fact that the QMC barrier height in Table 2 is the barrier height in the potential energy surface (PES) constructed by ®tting the QMC potential points with statistical sampling error of 16 lEh 0.10 kcal/mole). Considering the remarkable accuracy of our extrapolation scheme demonstrated for He, H2 , and He2 , it is highly probable that the extrapolated barrier height of 15 300 lEh (or 9.601 kcal/mole) would be closer to the exact barrier height than the QMC-based result of 15 319 lEh (or 9.613 kcal/mole). This is further supported by the narrow margin between CP corrected and uncorrected interaction energies at the CCSD(T) level (see Table 1). It is known that the resonance
characteristics for H± H2 interaction is very sensitive to the detailed PES near saddle point region and dierence in barrier height of a little more than 0.01 kcal/mole could aect those characteristics in a detectable manner [17,18,32]. Therefore, it now appears that the exact barrier height for colinear H±H2 interaction is lower than any other values predicted from the previously reported PES for this system [18,23,30±32]. For He±H2 interaction, our CCSD(T) interaction energy of )48:1 lEh is in good agreement with the previous MP4 result of )47:2 lEh with a large basis set containing bond functions by Tao [26] and experimentally derived result of )48 lEh by Rodwell and Scoles [25]. However, if one considers the dierence between the CCSD(T) and FCI results and also relaxation eect of He±H2 and H±H distances to search for the global minimum of the complex, it is highly probable that the electronic binding energy between He and H2 could be more than 50 lEh . This value again is larger than any previous results for this system [25,26,33]. Furthermore, comparing He±He and He±H2 interaction, atom±diatom dispersion interaction appears to be about 1.5 times larger than isoelectronic atom±atom dispersion interaction, though other factors such as dipole moment and polarizability of the interacting species would certainly aect the actual binding energy.
138
J.S. Lee / Chemical Physics Letters 339 (2001) 133±139
Therefore, we expect that similar behavior in the dierence of the interaction energies between Ne± Ne and Ne±B2 system (or Ar±Ar and Ar±F2 ) could be observed. In this respect, it is interesting to note that while the interaction energy for Ne±HF is larger than for Ne±Ne by a factor of more than 2.0, the interaction energy for Ne±FH appears to be larger than for Ne±Ne by a factor of less than 1.5 [34,35].
small energy dierence in the PES could aect the dynamical behavior signi®cantly. The application of similar extrapolation scheme to larger and more complex systems is in progress.
Acknowledgements This work was supported by Korea Research Foundation Grant (KRF-1999-015-DI0051).
4. Conclusion For He±He, H±H2 , and He±H2 interaction, we have shown that by applying a simple two-point 3 (X 1 X 5; 6) extrapolation scheme to the CP corrected CCSD(T) interaction energies (with aug-cc-pV5Z and aug-cc-pV6Z basis set) to obtain the basis set limit and correcting for the dierence between the CCSD(T) and full CI basis set limit result, one can get a highly accurate interaction energy which may be only comparable to the exact QMC result (or FCI result with a large basis set which may not be feasible computationally at the present time). This was possible due to the systematically convergent behavior of total electronic energy with correlation-consistent basis set in these molecular systems. In the case of H3 , our extrapolated results appear to strongly suggest that the barrier height for linear H H2 ! H2 H reaction is about 0:010 0:012 kcal/mole lower than the values predicted from the best potentials currently available for this system [23,32]. Since the dynamics of this reaction is known to be sensitive to the colinear barrier height [30,32], it would be interesting to examine the eect of this reduced barrier height on the dynamical characteristics such as the location of scattering resonance or thermal rate constant. We also found that the binding energy for He±H2 system would be more than 50 lEh , which suggests atom±diatom (homonuclear) dispersion interaction would be roughly 1.5 times larger than isoelectronic atom± atom dispersion interaction. Therefore, though simple, this kind of extrapolation scheme could be exploited as an eective tool to determine the accurate relative energies of weakly bound complexes and other molecular systems for which
References [1] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007. [2] D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 98 (1993) 1358. [3] D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 100 (1994) 2975. [4] K.A. Peterson, D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 100 (1994) 7410. [5] A. Wilson, T. van Mourik, T.H. Dunning Jr., J. Mol. Struct. (Theochem.) 388 (1997) 339. [6] A. Wilson, D.E. Woon, K.A. Peterson, T.H. Dunning Jr., J. Chem. Phys. 110 (1999) 7667. [7] D. Feller, J. Chem. Phys. 96 (1992) 6104. [8] S.S. Xantheas, T.H. Dunning Jr., J. Phys. Chem. 97 (1993) 18. [9] K.A. Peterson, T.H. Dunning Jr., J. Chem. Phys. 102 (1995) 2032. [10] T. Helgaker, W. Klopper, H. Koch, J. Noga, J. Chem. Phys. 106 (1997) 9639. [11] A. Halkier, T. Helgaker, P. Jorgensen, W. Klopper, J. Olsen, Chem. Phys. Lett. 302 (1999) 437. [12] J.S. Lee, S.Y. Park, J. Chem. Phys. 112 (2000) 10746. [13] C. Schwartz, Phys. Rev. 126 (1962) 1015. [14] R.N. Hill, J. Chem. Phys. 83 (1985) 1173. [15] W. Kutzelnigg, J.D. Morgan III, J. Chem. Phys. 96 (1992) 4484. [16] S.F. Boys, F. Bernardi, Mol. Phys. 19 (1970) 553. [17] A. Kuppermann, Y.S.M. Wu, Chem. Phys. Lett. 241 (1995) 229. [18] A.I. Boothroyd, W.J. Keogh, P.G. Martin, M.R. Peterson, J. Chem. Phys. 95 (1991) 4343. [19] Basis sets were obtained from the Extensible Computational Chemistry Environment Basis Set Database, as developed and distributed by the Molecular Science Computing Facility, Environmental and Molecular Sciences Laboratory which is part of the Paci®c Northwest Laboratory, P.O. Box 999, Richland, WA 99352, USA, and funded by the US Department of Energy. The Paci®c Northwest Laboratory is a multi-program laboratory operated by Battelle Memorial Institue for the US Department of Energy under contract DE-AC06-76RLO
J.S. Lee / Chemical Physics Letters 339 (2001) 133±139
[20] [21] [22] [23] [24] [25] [26] [27]
1830. Contact David Feller or Karen Schuchardt for further information. A. Halkier, W. Klopper, T. Helgaker, P. Jorgensen, P.R. Taylor, J. Chem. Phys. 111 (1999) 9157. A. Wilson, T.H. Dunning Jr., J. Chem. Phys. 106 (1997) 8718. J.B. Anderson, C.A. Traynor, B.M. Boghosian, J. Chem. Phys. 99 (1993) 345. Y.S.M. Wu, A. Kuppermann, J.B. Anderson, Phys. Chem. Chem. Phys. 1 (1999) 929. W. Kolos, L. Wolniewicz, J. Chem. Phys. 49 (1968) 404. W.R. Rodwell, G. Scoles, J. Phys. Chem. 86 (1982) 1053. F.M. Tao, J. Chem. Phys. 100 (1994) 4947. M.J. Frisch, et al., GA U S S I A N 98, Gaussian, Inc., Pittsburgh PA, 1998.
139
[28] M.W. Schmidt, et al., Gamess, J. Comput. Chem. 14 (1993) 1347. [29] T.v. Mourik, J.H.v. Lenthe, J. Chem. Phys. 102 (1995) 7479. [30] D.G. Truhlar, C.J. Horowitz, J. Chem. Phys. 68 (1978) 2466. [31] A.J.C. Varandas, F.C. Brown, C.A. Mead, D.G. Truhlar, N.C. Blais, J. Chem. Phys. 86 (1987) 6258. [32] A.I. Boothroyd, W.J. Keogh, P.G. Martin, M.R. Peterson, J. Chem. Phys. 104 (1996) 7139. [33] W. Meyer, P.C. Hariharan, W. Kutzelnigg, J. Chem. Phys. 73 (1980) 1880. [34] S.V. O'Neil, D.J. Nesbitt, P. Rosmus, H.J. Werner, D.C. Clary, J. Chem. Phys. 91 (1989) 711. [35] D.E. Woon, J. Chem. Phys. 100 (1994) 2838.