A theoretical study of the collinear reaction F+H2→HF+H using multiconfigurational second-order perturbation theory (CASPT2)

Chemical Physics 171 (1993) 107-l 18 North-Holland

A theoretical study of the collinear reaction F + H2 --+HF + H using multiconfigurational second-order perturbation theory (CASPT2 ) Remedios Gonzalez-Luque,

Manuela MerchAn

Departamento de Quimlca Fisica, Vniversitat de Vakncia, Dr. Moliner SO,Burjassot, 46 100 Valencia, Spain

and Bjiirn 0. Roos Department of Theoretical Chemistry, Chemical Centre, P.O.B. 124, S-221 00 Lund, Sweden Received 12 October 1992

The second-order perturbation method (CASPTZ) with a single state multiconfigurational reference function generated in complete active self-consistent field (CASSCF) calculations has been used to compute the collinear barrier height, saddle point geometry, and exothermicity of the reaction F + H Z+HF+H. Companson with full configuration (FCI) calculations with small basis sets shows that the CASPT2 method is capable of reproducing accurately the exact benchmark results correlating seven electrons. Large atomic natural orbital basis sets are used at the seven- and nine-electron level of correlation. With the largest AN0 basis set used, F[ 7s6p5d4f&] /H [ 6s5p4d2f], the computed nine-electron barrier height, 2.47 kcal/mol, is about 0.5 kcal/ mol higher than results obtained using MRCI+Q and CCSD(T) methods employing basis sets of similar quality. The nineelectron exothermicity obtained with this basis set is 3 1.25 kcal/mol, about 0.5 kcal/mol below the experimental value.

1. Introduction The complete active space (US) SCF methods1 introduced in 1980 [2 ] has become a valuable tool for describing the electronic structure of molecular systems in the ground and excited states. The CASSCF method includes the, so-called, static electron correlation effects due to strong configuration mixing (near degeneracy), but does not take into account the dynamical correlation effects. One simple way to treat dynamical correlation effects in molecular systems is to use second-order perturbation theory. A second-order perturbation treatment has recently been developed [ 3,4], where the CASSCF wavefunction is used as the reference function. The approach, called the CASPT2 approximation [ 41, has been successfully applied to compute a number of properties of the ozone molecule [ $6 1. It has also been used in studies of the excited state of several n1 For a review of the CASSCF method see ref. [ 11. 0301-0104/93/$06.00

systems [ 7- 10 1, yielding a surprisingly accurate description of the spectra. The same approach was also recently applied to predict the geometries and binding energies of a large set of test molecules [ 111. The results of the test show that the CASSCF/CASPT2 method is capable of yielding accurate molecular structures, not only for molecules with “normal” chemical bonds, but also for molecules in excited states and for weakly bonded systems. Heats of reactions for isogyric reactions were predicted with an accuracy of If:2 kcal/mol. The performance of CASPTZ theory has, however, not yet been tested for the calculation of potential energy surfaces. The reaction F + HZ+ HF + H is a good candidate for such a test, since extensive and accurate experimental [ 12 ] and theoretical data [ 13- 17 ] are available. Full CI (FCI ) calculations using different basis sets have been performed [ 13,15,17] on the collinear barrier height. With a F[4s3pld]/H[2slp] Dunning-Huzinaga [ 18,19 ] basis set and no correlation of the F 2s electrons, FCI( 7), a barrier height

0 1993 Elsevier Science Publishers B.V. All rights reserved.

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R. Gonzdez-Luque et al. /Chemical Physics 171(1993)

of 4.5 kcal/mol was found [ 131. A better description of molecular correlation effects is obtained with basis sets derived from correlated atomic calculations. Basis sets of atomic natural orbitals (ANOs) provide an excellent description of molecular correlation effects. It has been established, however, that compact sets of primitive Gaussian functions can describe molecular correlation effects equally well as the AN0 sets provided the exponents are optimized in correlated calculations on the atoms [ 201. The most extensive nine-electron FCI calculation, FCI (9)) has been performed by Knowles et al. [ 171 (KSW) with such a basis set. They used a F[4s3pld]/H[3slp] set, which was an extension of the Dunning correlation consistent (CC) valence double-zeta basis set [ 201 and obtained a barrier height of 3.169 + 0.045 kcal/mol. In addition to the FCI calculations a large number of theoretical studies has been performed using different levels of treatment [ 13-l 7,2 1-28 1. Schaefer and co-workers originally estimated that further improvement in ab initio calculations would not lower the barrier height under about 2.4 kcal/mol [ 21-231. However, lower barrier heights have been obtained by other authors in agreement with the experimental estimate ( z l-2 kcal/mol). Truhlar and co-workers, using the scaled external correlation method (SEC) with an F[ 8s5p3dlf] / [251 in conjunction H [ 3slp Id] Gaussian basis set, computed a barrier height of 1.6 kcal/mol. Scuseria [ 161 has recently used the coupled cluster method including all single, double, and perturbative triple excitations, CCSD (T) [29,30], to predict the collinear barrier height. Including correlation of the 2s electrons and using an F [ 7s7p5d4f2g] /H [ 6s5p4d2f] AN0 [ 3 1] basis set, a barrier of 2.05 kcal/mol was found. Bauschlicher et al. (BWLTJ) [14] used an F[5s5p3d2flg]/ H [ 4s3p2d] AN0 [ 3 1 ] basis set and performed a complete geometry optimization at the MRCI level of theory, including the Davidson correction [ 321 ( +Q). Including 2s correlation, a barrier height between 1.35 and 1.86 kcal/mol was estimated. Employing the same type of AN0 basis set, Bauschlicher et al. (BLLT) [ 151 used the averaged coupled-pair function (ACPF) method [ 331 and correlated both seven- and nine-electrons. When 2s correlation was included the size consistent ACPF method predicted an energy barrier of 1.85 kcal/mol. BLLT [ 151 esti-

107-I 18

mated that the nine-electron ACPF and MRCI + Q barriers in the basis set limit would be about 1.65 kcal/mol, in good agreement with previous estimates by Thrular and co-workers [ 25 1, and slightly larger than the previous lower bound suggested by BWLTJ. It was found in the BLLT study that reference conligurations of the F--type in the MRCI wavefunction were important for a correct description of the barrier height. Inclusion of F-HZ configurations (extending the number of reference configurations to 13) affects the barrier height at the ACPF level of theory, but to a smaller extent at the MRCI+Q level, when nine electrons are correlated. The most extensive CASSCF, MRCI ( + Q ), and ACPF calculations have been reported by KSW [ 171. MRCI calculations were ’ performed using full CASSCF reference spaces. The FCI (9 ) barrier height is very close to the corresponding MRCI+Q result obtained with the same oneelectron set, which gives support to the value of 1.93 kcal/mol obtained with the same MRCI+Q wavefunction, but in an extended orbital basis set [ 171. In order to calibrate the performance of the CASPT2 approach for studies of chemical reactions, two series of test calculations have been carried out on the collinear barrier height and exothermicity for the reaction F+ H2-+HF + H. The purpose of the first series was to compare the CASPT2 results with the FCI results. While FCI basis sets cannot yield results of high accuracy, they provide an excellent test of the ability of other CI schemes to reproduce an “exact” (within a given basis) result. The second series of calculations, using extended basis sets, was performed in order to compare the present approach with corresponding MRCI, ACPF and CCSD(T) results [ 14-171.

2. The CASSCFXASPT2

approach

The computational model used in the present study consists of two major steps: a CASSCF calculation followed by a CASPT2 calculation. The CASSCF approach is today well known and needs no further description [ 11. The CASPT2 method [ 3,4] computes the first-order wavefunction and the second-order energy in the full CI space without any further approximation with a CASSCF wavefunction constituting the reference function. The zeroth-order

R. Gonzdez-Luque

et al. /Chemrcal Physics I71 (1993) 107-I 18

Hamiltonian is defined as a Fock type one-electron operator and is constructed in such a way that a MBller-Plesset type perturbation treatment is obtained in the closed shell single determinant case. Two different formulations of the zeroth-order Hamiltonian are possible: one which uses only the diagonal part of the Fock matrix (called PT2D) and one which includes also the non-diagonal elements (PT2F). The first choice is computationally simpler and leads in most cases to results not very different from PT2F. It should be emphasized, however, that only the nondiagonal approach is invariant to rotation of the molecular orbitals. The full approach must therefore be used in cases, like this, where such an invariance is important. Both approaches have been used here, giving a thorough test of the PT2D alternative, which leads to cheaper calculations. The CASPT2 equations are formulated exclusively in terms of one-, two-, and three-body density matrices and are therefore independent of the actual size of the reference function. The limiting factor is not the number of correlated electrons, but the number of active orbitals, which determines the size of the density matrices. The current implementation of the CASPT2 method allows a maximum of 14 active orbitals. A larger number is rarely needed in any application and could not be handled by the CASSCF program anyway, except in cases with very few active electrons (or holes). For further details of the CASPT2 approach we refer the reader to the original papers [ 3,4 1. The free parameters in a CASSCF/CASPT2 calculation are: the basis set, the number of correlated electrons, and the number and type of active orbitals. Two different kinds of basis sets were used in this work. They are presented in table 1. For comparison of the CASPTZ results with the Bauschlicher and Taylor (BT) [ 131 FCI calculations, the same double-zeta basis set was used, originated from the Dunning-Huzinaga [ 18,191 contraction (a scale factor of 1.2 was used for s( H) ) and augmented with a diffuse p set on F and polarization functions on all the atoms. This basis set is designated BS 1. The 3s component of the fluorine d set was deleted in all the calculations. The 1s and 2s electrons on F were not correlated in this FCI calculation, FCI ( 7 ). To take into account the effect of 2s correlation at the FCI level, FCI (9)) BLLT [ 15 ] used a smaller basis set,

109

BS2. This basis set is obtained from BSl by omitting the polarization functions on the F and H atoms. In order to reduce the error in the correlation energy and to get results closer to the experimental value, basis sets which include functions with high angular moments (d, f, g), and which have been optimized to describe correlation effects in atoms must be used. We have therefore also performed calculations using the recently developed averaged atomic natural orbital (ANO) basis sets [ 341. They have been constructed by averaging the corresponding density matrix over several atomic states, positive and negative ions and the atom in an external electric field. The different density matrices have been obtained from correlated atomic wavefunctions. The starting primitive sets are F[ 14s9p4d3f] and H[8s4p3d]. The largest AN0 basis set used in this work is F[ 7s6p5d4f2g] /H [ 6s5p4d2f]. Starting from the contraction even-temF [ 7s6p4d3f] /H [ 6s4p3d], pered diffuse polarization functions have been added: on F one even-tempered d-(factor 2.86) and one f(factor 2.5), and on H one p-(factor 2.86) and one d-(factor 2.5)type Gaussian function. The exponent of g and f polarization functions on the fluorine and hydrogen atoms, were chosen from the criteria of maximum overlap with the corresponding f-, and dtype functions. Only the pure spherical harmonic components of the d-, f-, and g-type functions were included in the AN0 basis set. To compare with similar quality basis set calculations [ 14-161 the effect of a p-type diffuse function on the fluorine atom with an exponent of 0.0 18684 1 was checked. CASSCF wavefunctions were used as the starting point in the second-order perturbation treatment. To analyze the influence of the active space in this treatment and compare with MRCI and ACPF results three active spaces were used. The minimum active space used is the valence set. It includes the F 2p, and H 1s orbitals. Calculations were performed in CZV symmetry. So, the minimum active space is denoted (31 lo), where integers refer to the number of a,, bi, b2 and a2 orbitals in the active space. The second CASSCF space denoted as (3220) is the valence set plus an additional set of correlating n orbitals. The inclusion of 2p+2p’ excitations in the CASSCF and MRCI reference space is known to significantly reduce the barrier height of the reaction. In the next extension an additional set of correlating o orbitals is

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R. Gonzcilez-Luque et al. /Chemical Physics 171(1993)

107-l

18

Table 1 Basis sets used in the present and previous studies of F+ Hz Basis set

Type

Original primitive

FCI basis sets BSl CGTO BS2

CGTG

BS3

CGTO H(4s)

Extended basis sets BSAl AN0 BSBl BSCl BSDl

AN0 AN0 AN0

BSEl

AN0

BSA2

AN0

BSB2

AN0

BSC2

AN0

BSD2

AN0

BSE2

AN0

BSF

CGTO

F( 14s9p4d3f) H(8s4p3d) BSAl plus BSB 1 plus F( 14s9p4d3f) H(Ss4p3d) F( 14s9p4d3f) H(Ss4p3d) F( 13s8p6d4f) H( 8s6p4d3f) F( 13s8p6d4fLg) H( 8s6p4d3f) F( 13sSp6d4fZg) H(Ss6p4d3f) F( 13sSp6d4f2g) H(8s6p4d3f) F( 13s8p6d4Qg) H(8s6p4d3f) F( 16s7p3d2flg) H(8s3p2d)

Additional

Contraction

Reference.

p(O.O74), d( 1.4)

F[4s3pld]

[13],thiswork

P(0.8)

H[2slpl F[~S~PI H[2sl F[4s3pld]

d(3.01073, 1.25335,0.40089) s(O.l1435), ~(0.08440) s(O.O3), ~(0.727)

s(O.O9), p(O.O6), d(0.20), f(0.38), g(0.84) s(O.3), P(O.l)

included in the active space, corresponding to the active space (4220). However, the 6a, orbital is only weakly occupied and is of the same importance as the 20, orbital in Hz. Therefore a (5220) active space must be used to eliminate the possibility of multiple solutions. This space is flexible enough to correlate both the F 2p orbitals and to provide radial correlation for HZ. Calculations were performed at three collinear F...H...H geometries, corresponding to reactants, transition state, and products. The CASPT2 barrier height is referred to F...H2 at 100.0 au. The exothermicity is computed using FH...H at 100.0 au. The geometry was optimized using a second-order polynomial fit with a grid size of 0.0 1 au. All calculations have been performed on the IBM 9021/500-2VF computer of the University of

1171

H[3slpl F [ 5s4p3d2f] H[4s3p2d]

F: g(0.8) H: f(0.8) g(O.8,0.3) f(O.8,0.3) d(O.O7496),f(O.2048), g(O.8,0.3) p(O.O3455), d(0.11642), g(O.8,0.3) ~(0.059326)

[15],thiswork

F[6s5p4d3f2g] H [ Ss4p3d2f] F[ 7s6pSd4fZg] H [ 6s5p4d2f] F[ 5s5p3d2f] H [ 4s3p2d] F[Ss5p3d2flg] H [ 4s3p2d] F[5s5p3d2flg] H[4s3p2dlf] F[ 6s6p4d3f2g] H [ 5s4p3d2f] F[ 7s7pSd4f2g] H [ 6s5p4d2f] F [ 7s6p4d3Q]

this work this work this work this work this work [ 14-161 [ 14-161

[I61 [I61 1161 [I71

H [ 6s4p2d]

Valencia Computer Centre using the MOLCAS-2 quantum chemistry software [ 35 1, which includes the CASPT2 program as one module.

3. Results and discussion As a calibration test of the ability of the CASPT2 method to predict the barrier height and exothermicity of the reaction F + Hz-) HF + H, CASPT2 calculations have been performed at the seven-electron level (H 1s and F 2p) of correlation using the F[4s3pld]/H[2slp]fullCIbasisset,BSl,attheFCI optimized geometry. In table 2 we compare the CASPT2 results with the FCI (7) results [ 13 1. Several active spaces have been considered for the CASSCF wavefunction, which is the reference func-

R. Gonzctlez-Lugue et al. /Chemical PhysicsI71 (I 993) 107-l I8 Table 2 FCI calibration of the barrier height and exothermicity of the F+Hr-rHF+H reaction using the BSl FCI basis set at the FCI geometry a) Method

Barrier (kcal/mol)

Exothermicity (kcal/mol)

PT2D(3110) PT2F(3110) PTZD( 3220) PT2F( 3220) PTZD( 5220) PTZF( 5220) MRCI(3220)(0.025) b, MRCI(3220)(0.025)+Qb’ MRCI(5220) (0.025) b, MRCI(5220)(0.025)+Qb’ ACPF(3220)(0.025)=’ CISDTQ d, CCSD *’ CCSD( T) d, FCI =’

5.53 5.55 4.60 4.56 4.62 4.54 4.73 4.5 1 4.55 4.32 4.56 4.41 4.91 4.46 4.50

30.70 30.01 30.59 30.83 30.30 29.67 29.17 28.80 29.41 29.31 28.89 27.30 26.52 27.26 28.84

a) Calculations were done correlating Sevenelectrons. Saddle point geometry (au) (2.761, 1.467). Reactant geometry: R(HH)= 1.404 au. Product geometry: R (HF) = 1.740 au. b)Ref. [14]. “‘Ref. [15]. d)Ref. [16]. “‘Ref. [13].

tion in the second-order perturbation calculation. As can be seen in table 2, the CASPT2 barrier height is in much better agreement with the FCI result when the 2p+2p’ correlation is included in the active space. The reduction in the barrier height with the inclusion of 2p+2p’ correlation arises from the improvement in the description of the electron afftnity of F, as there is a significant degree of H+F- character at the saddle point geometry. It has been shown that similar reference configurations are also needed in an MRCI treatment [ 15 1. The expansion of the active space from ( 3 110) to (3220) yields a PT2F barrier height of 4.56 kcal/mol, only 0.06 kcal/mol apart from the FCI result (4.50 kcal/mol). The inclusion of an additional set of correlating o orbitals on fluorine, and a 20, orbital in Hz, (5220) active space, decreases the CASPT2 barrier by only 0.02 kcal/mol. This can be understood if we look at the weight of the reference functions. With this basis set, when the reference space is expanded from (3110) to (3220), the weight of the reference wavefunctions increases from 97% to 99%. However, when an additional set of correlating o orbitals are added to the (3220) active space, the weight of the

111

reference function does not significantly change. Results obtained with other approximate methods are also included in table 2. At the MRCI level of treatment several reference spaces have been considered. Results obtained with two of them, using a 0.025 configuration threshold selection, are included. The MRCI barrier heights are in much better agreement with the FCI result when correction is made for quadruple excitations. Using the (3220) active space and the multireference Davidson correction ( +Q) BWLTJ [ 141 computed an energy barrier of 4.51 kcal/mol in good agreement with the FCI result. The ACPF( 3220) result is in better agreement with FCI than MRCI (3220), but does not reach the accuracy of the MRCI (3220) +Q approach. The CCSD(T) barrier is only 0.04 kcal/mol smaller than the FCI result. A comparison between CCSD and CCSD (T ) results shows that there is an important connected triples contribution to the barrier height of 0.45 kcal/ mol. The barrier height obtained at the CISDTQ level is smaller than the FCI result. The CISDTQ result was obtained using SCF MOs, which usually exhibit more 2s/2p mixing than CASSCF MOs (specially for HF). Thus, the single reference methods include more 2s correlation than multi-reference methods based on CASSCF wavefunctions also in the seven-electron calculation. The exothermicity of the reaction is also presented in table 2. The CASPT2 results using different active spaces are very similar. The result obtained with the largest active space is 0.83 kcal/mol larger than the FCI value. MRCI predictions for the exothermicity are in slightly better agreement with the FCI result than CASPT2 and single reference methods. Single reference methods underestimate the exothermicity while the CASPTZ approach overestimates it. As was pointed out earlier the 2s/2p mixing effect in methods which use SCF MOs can be the reason behind this difference. Similar conclusions can be drawn from table 3 where the optimized geometry at each level of correlation treatment is used (only the more reliable PT2F results are reported in these tables). By comparing the results in tables 2 and 3, it is clear that optimization of the geometry has only a small effect on the barrier height and exothermicity. The nine-electron results are also presented in table 3. The inclusion of 2s correlation decreases the barrier height using the

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R. Gonzdez-Luque et al. /Chemical Physics I71 (1993) 107-118

Table 3

Comparison of different theoretical methods for the barrier height, exothermicity, and geometries of the F+Hr-rHF+H tained with the BSl FCI basis set

reaction, ob-

Saddle point method

PT2F(3110) PT2F( 3220) PTZF( 5220) SDCI ” SDCI+Qa’ CPF ” MRCI(3110)“’ MRCI(3110)+Q” MRCI ( 3000) b, MRCI(3000)+Qb’ MRCI(3220) (0.025) c, MRCI(3220)(0.025)+Qc’ ACPF( 3220) d, CISDTQ =) FCI ‘)

R(HF)

(au)

R(HH)

barrier (kcal/mol)

(au)

?e-

9e-

le-

9e-

7e-

9e-

2s effect

2.748 2.177 2.175 2.512 2.782 2.801 2.722 2.113 2.740 2.195 2.761 2.155 2.760 2.763 2.761

2.718 2.163 2.770 2.516 2.717 2.805 2.711 2.781

1.472 1.471 1.468 1.488 1.468 1.467 1.473 1.464 1.476 1.467 1.474 1.475 1.47s 1.465 1.467

1.466 1.471 1.468 1.494 1.474 1.468 1.414 1.462

5.56 4.56 4.54 6.76 4.49 4.40 5.15 4.39 5.16 4.42 4.70 4.49 4.56 4.45 4.50

4.16 4.26 4.14 7.93 4.81 4.09 4.65 3.64

0.80 0.30 0.40 -1.17 -0.32 0.31 0.50 0.75

Reactant (Hz) and product (HF) method

PT2F(31 IO) PTZF( 3220) PTZF( 5220) SDCI ” SDCI+Q” CPF” MRC1(3110)*’ MRCI(31lO)+Q” FCI ”

WHH)

(au)

exothermicity (kcal/mol)

R(HF) (au)

7e-

9e-

?e-

9e-

le-

9e-

2s effect

I .409 1.409 1.407 1.399 1.403 1.403 1.407 1.405 1.404

1.409 1.409 1.407 1.398 1.403 1.403 1.407 1.405

1.739 1.736 1.735 1.728 1.I32 1.731 1.761 1.759 1.740

1.740 1.740 1.740 1.I33 1.739 1.738 1.748 1.750

30.01 30.83 29.67 26.43 26.72 26.41 28.68 29.21 28.84

28.90 27.63 26.70 25.21 25.56 25.23 25.58 26.00

1.11 3.20 2.97 1.16 1.16 1.24 3.10 3.21

‘) Ref. [13]. b)Ref. [14]. =) 12 reference configurations chosen from the CASSCF wavefunction (cf. ref. [ 141). d, Includes the additional configuration F-H: found to be important in the ACPF calculation (cf. ref. [ 15 ] ). ‘) Six-component d functions are employed (cf. ref. [ 241).

coupled pair functional (CPF) method [ 36 1. However, the 2s correlation shows the opposite trend at the SDCI( +Q) level. When 2s electrons are correlated the CASPT2 results follow the same trends as MRCI. It may be noted that the effect of 2s correlation is smaller at the CASPT2 level when 2p-+2p’ correlation is included in the active space. Inclusion of the 2s electrons in the correlation treatment also diminishes the computed exothennicity in the

CASPT2 and MRCI calculation. The 2s effect is considerably smaller in the single-reference case. The saddle point, reactant, and product optimized geometries are also reported in table 3. The CASPT2 results can be compared to FCI at the 7e- treatment. The CASPT2 saddle point geometry is similar to FCl( 7) when the 2p-+2p’ excitations are included in the active space. PT2F theory then gives an HF distance slightly longer than FCI( 7), while if these ex-

R. Gonzcilez-Luque et al. /Chemical Physics I71 (1993) 107-I 18 Table 4 Study of the 2s effect on the barrier height (kcal/mol) of the F+H,+HF+H reaction using the [4s2p]/ [2s] Dunning-Huzinaga basis set a) Method

(7e-)

(9e-)

2s effect

PT2F(3110) PT2F( 3220) PTZF( 5220) MRCI(3220) b, MRCI(3220)+Qb’ ACPF( 3220) b, CISDTQ b, CCSD(T) c, FCI b’

9.33 8.85 8.88 8.78 8.74 8.75 8.82 8.90 8.77

7.52 7.3 1 7.07 6.68 6.50 6.53 6.82 6.84 6.64

1.81 1.54 1.81 2.10 2.24 2.22 2.00 2.06 2.13

‘) Saddle point geometries (au): seven electrons (2.470, 1.583); nine electrons(2.550, 1.550) (cf. ref. [ 151). b)Rev. [IS]. “Ref. [16].

citations are handled perturbationally, the opposite trend is found. Comparing the CASPT2 results with results obtained with other correlation treatments, the CASPT2 theory gives a saddle point geometry of similar quality as the multi-reference MRCI and ACPF methods. On the other hand, CASPT2 method gives a product HF distance closer to the FCI value than MRCI (3 1 lo), with and without correction for quadruple excitations. When the 2s electrons are correlated, the CASPT2 HF distances increases, while the MRCI result shows the opposite trend. The effect of the 2s correlation effect on the barrier height has been considered at the FCI level by BLLT [ 15 1, using the double-zeta Dunning and Huzinaga, BS2, basis set. This basis set is formed from BSl by omitting the d- and p-type polarization functions on F and H, respectively. We have performed CASPTZ calculations with this basis set at the geometry used in the work of BLLT [ 15 1. Our results are compared with the FCI results in table 4. As was previously pointed out, the FCI result cannot be obtained using only the valence orbitals in the CASSCF reference function. Correlating seven electrons leads to a barrier height only 0.08 kcal/mol larger than the FCI( 7) result is at the PTZF( 3220) level. When the 2s electrons are also correlated, the barrier height decreases as the reference space increases. PT2F( 5220) gives a barrier height 0.43 kcal/ mol larger than the FCI (9) result. For comparison energy barriers obtained with other correlation methods, and the 2s correlation effect, are also in-

113

Table 5 Comparison of the CASPTZ results with FCI(9) and MRCI results using a [4s3pld] / [ 3slp] basis set ‘) Method

Barrier (kcal/mol)

Exothermicity (kcal/mol)

PTZD( 3220) PTZF( 3220) MRCI( 3220) b, MRCI(3220)+Qb’ ACPF( 3220) b, FCI b,

3.53 3.50 3.48 3.17 3.31 3.169f0.045

26.93 28.63 28.66 27.76 27.92 27.9lkO.02

a) CASPTZ results were obtained using the averaged AN0 basis set [ 341. FCI, MRCI, and ACPF calculations were performed using an extension of the valence double-zeta basis set reported by Dunning [ 201 (see text). The saddle point geometry was optimized at the 9e- MRCI( 3220) (0.025) + Q level using the BSB2 AN0 basis set (cf. table 6). Experimental geometries were used for reactants and products [ 391. b, Ref. [ 171.

eluded in table 4. At the nine-electron level of correlation these methods give a barrier height lower than the CASPT2 approach. Thus the effect of 2s correlation seems to be somewhat underestimated at the CASPT2 level of theory. It is worth noting that the multi-reference MRCI + Q and ACPF methods overestimated the effect of 2s correlation giving a nineelectron barrier lower than the FCI (9 ) result. Without the multi-reference Davidson correction the MRCI method gives a barrier height and 2s effect similar to the FCI result. The effect of correlating the 2s electrons in the CCSD (T) calculation is close to the FCI result. However, in total the CCSD(T) method gives energy barriers higher than FCI. In order to get a better quantification of the 2s correlation effect, FCI calculations at the 7e- and 9e- level should be performed with a more flexible basis set. KSW [ 171 have recently computed the FCI barrier height and exothermicity of the reaction F + H2 at the 9e- level with a valence triple-zeta plus polarization AN0 basis set (BS3 ). It comprises the valence double-zeta s and p sets for F and H [ 201 augmented with s, p, and d functions (cf. table 1 ), where the exponents were optimized at the MRCI level for F-. It has been shown [ 201 that the performance of such a basis set is comparable to corresponding AN0 sets [ 3 11, even though the latter contains more primitive functions. We have computed the barrier height and exothermicity at the geometry used by KSW [ 171

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R. Gonzalez-Luque et al. /Chemical Physics 171 (I 993) 107-l 18

with an AN0 basis set of similar quality: F( 4s3p Id) / H (3~1~) [ 341. Comparison of the CASPT2 results with the FCI (9 ) results reported by KSW is not precise due to the use of a slightly different basis set. Anyway, such a comparison is meaningful in order to elucidate the qualitative trends of the CASPT2 results with improvement of the basis set, since the basis set used in the FCI (9) calculation is very similar. It may be noted that the smallest active space used by KSW, denoted 52212, is an expansion in the orbitals 3o-50, and lrr-2~. This active space is the same as the active space denoted (3220) here, which was used also in the previous multi-reference MRCI studies [ 14-l 5 1. It is of interest to note that in contrast to the calculation made in refs. [ 14,15 1, the MRCI and ACPF calculations of KSW were based on complete CASSCF reference spaces, with no conliguration selection. A sequence of internally contracted MRCI [ 371 and ACPF [ 331 calculations have also been performed by KSW, in order to study the convergence of the barrier height and exothermicity with an increased reference space. In table 5 we report the CASPT2 barrier together with the FCI and multi-reference result of KSW. CASPT2 theory gives a barrier height 0.33 kcal/mol larger than FCI. This barrier height is similar to the MRCI result without the multi-reference Davidson correction. A single point calculation at the PT2F(3220) level with the F[4s3pld]/H[2slp] AN0 [ 341 basis set gives a barrier height of 3.6 1 kcal/ mol. Comparing this result with the PT2F( 3220) result (4.26 kcal/mol) obtained using the F [ 4s3p 1d] / H [ 2s 1p ] contracted Gaussian Dunning-Huzinaga basis set (cf. table 3 ) , shows that the barrier height is improved when a basis set is used, which has been optimized to describe correlation effects. From the previous comparison of CASPT2 with FCI results it can be concluded that although the CASPT2 approach gives very good results when seven electrons are correlated, it underestimates the 2s correlation giving a barrier height larger than FCI ( 9 ) . However, as can be inferred from the results given above, the 2s effect is better estimated as the basis set quality is improved: changing the basis set from double-zeta to AN0 triple-zeta decreases the difference between the PT2F(3220) and the FCI result from 0.67 to 0.33 kcal/mol. The PT2F( 3220) exothermicity com-

puted at the 9e- level of correlation is only 0.72 kcal/ mol larger than the FCI result. CASPT2 calculations using more flexible basis sets have been performed with both seven and nine electrons correlated. The smallest of these extended basis sets is the F[ 5s4p3d2f] /H [ 4s3p2d] AN0 basis set (BSAl ). With this basis (3110) and (3220) active spaces were used. Table 6 shows the barrier height at the optimized geometry. As was found out in the previous calibration calculations, the barrier height decreases when 2p-+2p’ excitations are included in the active space. The effect is larger when only seven electrons are correlated. Adding a set of g polarization functions on the fluorine atom (BSB 1) lowers the barrier height 0.12 and 0.15 kcal/mol at the 7eand 9e- level, respectively. This is in agreement with the results obtained by BWLTL [ 141 and Scuseria [ 161 using the externally contracted CI method of a Davidson correction Siegbahn [ 381 with (CC1 + Q), and the CCSD (T) methods, respectively. In table 6 we compare the CASPT2 results with those obtained in other studies with basis sets of similar quality. The F [ 5s4p3d2f] /H [ 4s3p2d] AN0 basis sets [ 3 I 1, augmented with an additional eventempered diffuse 2p function (exp.=0.059326) to better describe F- (BSA2), and additional higher angular momentum functions (BSB2 and BSC2) were used in those studies [ 14- 17 1. Our basis sets BSA 1, BSB 1, and BSC 1 should be of similar quality. We have checked the effect on the barrier height and exothermicity of adding an even-tempered p diffuse function on the fluorine atom (exp. = 0.0 186841) to the BSAl, BSBl, and BSCl basis sets. Single point calculations were performed, using for the saddle point the geometry optimized at the CCSD (T) level [ 161 with the BSC2 AN0 basis set. For reactants and products the experimental geometry was used [ 391. The set of p-type diffuse functions changes the barrier height and exothermicity by less than 0.03 and 0.06 kcal/mol, respectively, for all three basis sets. The conclusion is, that the original basis sets are well suited to describe the F- character. Comparing (cf. table 6) the (7e- ) CASPT2 results with other methods shows that the PT2F( 3220) method gives results of similar quality as the MRCI (3220) (0.025 ) method. The inclusion of quadruple excitations decreases the MRCI barrier to a value 0.63 kcal/mol below that obtained with the

115

R. Gonzdez-Luque et al. /Chemical Physics 171(1993) 107-I 18 Table 6 Comparison of different theoretical methods for the barrier height, exothermicity and geometries of the F+Hs+HF+H extended AN0 basis sets

reaction using

Saddle point basis

BSAl BSAl BSAl BSAl BSA2 BSA2 BSBl BSBl BSB2 BSB2 BSB2 BSB2 BSB2 BSB2 BSB2 BSB2 BSB2 BSB2 BSB2 BSB2

method

PT2D(3110)“’ PT2F(3110)a’ PT2D( 3220) PTZF( 3220) CCI( 3220) ‘) CCI(3220)+Q b, PT2D(3220) PT2F(3220) MRCI( 3000) ‘) MRCI(3000)+Q”’ MRCI(3220)“’ MRCI(3220)+Q” ACPF(3220) d, ACPF( 3220) =) CCSD” CCSD(T) ‘) SDCI+Q b, CPF b’ CCI(3220)s’ CCI(3220)+Qb’

NHF)

(au)

R(HH)

barrier (kcal/mol)

(au)

le-

9e-

le-

9e-

le-

9e-

2s effect

(2.893) (2.893) 2.891 2.893

2.871 2.844 2.913 2.887 2.879 2.909 2.934 2.907 2.193 2.921 2.914 2.950 (2.950) 2.961 (2.913) (2.913) 2.638 2.939 (2.921) (2.921)

(1.450) (1.450) 1.452 1.450

1.452 1.453 1.450 1.450 1.447 1.445 1.441 1.448 1.465 1.450 1.451 1.450 (1.450) 1.447 (1.445) (1.455) 1.470 1.451 (1.450) (1.450)

4.06 3.95 3.09 3.05

3.54 3.15 2.63 2.58 2.89 2.14 2.49 2.43 3.47 2.29 2.63 1.66 1.17 1.85 3.21 2.21 3.87 2.44 2.79 2.02

0.52 0.80 0.46 0.41

2.906 2.907

2.899 2.910 (2.950) 2.914 (2.917) 2.917

1.450 1.448

1.455 1.456 (1.450) 1.453 (1.443) 1.443

2.97 2.93

2.99 2.42 2.45 2.61 3.24 2.41

0.48 0.50

0.36 0.76 1.28 0.76

Reactant (Hz) and product (HF) basis

BSAl BSAl BSAl BSAl BSBl BSBl BSB2 BSB2 BSB2 BSB2 BSB2 BSB2 BSB2 BSB2 BSB2 BSB2 exp. *)

method

PT2D(3110) PT2F(3110) PT2D(3220) PT2F(3220) PT’ZD( 3220) PTZF( 3220) MRCI( 3000) b, MRCI(3000)+Qb’ MRCI( 3220) =) MRCI(3220)+Qb’ ACPF( 3220) =) CCSDl) CCSD(T) ‘) SDCI+Q b, CPF b, CCI(3220) b, CCI(3220)+Q”’

R(HH)

(au) i,

R(HF)

7e-

9e-

1.406 1.406

1.406 1.406 1.406 1.406

(au)‘)

exothermicity (kcal/mol)

le-

9e-

le-

9e-

2s effect

1.726 1.726

1.743 1.733 1.737 1.733

35.19 34.15 34.06 34.90 34.34 35.18

32.38 32.15 29.28 31.18 29.54 31.47 30.35 31.00 31.61 30.47 30.58 28.87 30.90 29.40 28.85 31.8 30.7 31.73

2.81 1.40 4.78 3.12 4.80 3.11

33.96 33.42 33.54 31.11 33.31

1.401

1.733

2.35 2.95 2.96 2.24 2.23

*) Geometry optimized at the PT2F(3220) level using the BSAl basis set. b, Ref. [ 141. =)With 12 reference configurations chosen from the CASSCF wavefunction using a 0.025 configuration threshold selection (cf. ref.

[151). d,With 12 reference configurations chosen from the CASSCF wavefunction using a 0.025 configuration threshold selection. The MRCI( 3220) (0.025) +Q nine electrons optimal geometry is used (cf. ref. .{ 151). =) Includes additional configuration important at the ACPF level (cf. ref. [ 15 ] ). f, At the (7e-) level, the BSB2 CCSD(T) (7e-) optimal geometry is used. At the (9e-) eeometrv is used (cf. ref. I 16 1)

level, the BSC2 CCSD(T)

(9e-)

optimal

116

R. Gonzdez-Luque

et al. / Chemtcal Physics 171(1993)

PT2F approach. The ACPF and CCSD (T) methods yield barriers about 0.4 kcal/mol smaller than the CASPT2 method. The barrier is decreased when also the 2s electrons are correlated. The CASPT2 result is now below the MRCI (3220) (0.025 ) value and 0.16 kcal/mol above the CCSD(T) result. The lowbarrier est height is obtained at the MRCI(3220)(0.025)+Q level. This is due to the large effect of quadruple excitations. The 2s correlation effect on the barrier is about 0.5 eV with the CASPT2 approach. The effect is somewhat smaller with the MRCI and CCSD(T) methods but larger when the effect of quadruple excitations are included (MRCI+Q and ACPF). As was previously mentioned, when small basis sets are used, the MRCI + Q method overestimates the 2s correlation effect while CASPT2 underestimates it. The exothermicity of reaction F+ H2 has been calculated with the same extended basis sets. To arrive at the experimental result [ 391 2s correlation must be included. If this is not done, the PT2F( 3220) approach gives an exothermicity about 3.0 kcal/mol larger than the experimental value and about 1 kcal/ mol larger than those obtained with the MRCI + Q, ACPF, and CCSD (T) methods. The same tendency was noticed with the BS 1 FCI basis (cf. table 2 ). The effect of correlating the 2s electrons varies somewhat for the different methods (cf. table 6). The largest effect is obtained with the CASPT2 method. The exothermicity obtained in the (9e-) PT2F( 3110) calculation is slightly more than one kcal/mol larger than the experimental value [ 391. The inclusion of 2~42~’ correlation in the active space leads to a 50% decrease in the error. A further improvement is obtained by adding a g-type function to the basis set. The PT2F value now differs from the experimental value [ 391 by only 0.26 kcal/mol. This result is actually closer to experiment than those obtained with the MRCI + Q, ACPF, and CCSD (T) methods using basis sets of similar quality. The transition state geometry has been determined using these extended basis sets at both the (7e-) and (9e-) level of correlation. Correlating the 2s electrons has only a small effect on the interatomic distances, except at the PT2D level of approximation. This approximation is, however, not orbital invariant and therefore not recommended for calculations of geometries and other properties of an energy sur-

107-l 18

face. The (7e-) PT2F results are very similar to those obtained using multi-reference CI methods. However, when 2s correlation is included MRCI + Q and ACPF methods give an R (HF) distance about 0.05 au longer than the PT2F approach. The reactant and product geometries obtained at the PT2F level fit the experimental value [ 391. The influence on the barrier height and exothermicity of saturating the basis set at the CASPT2 level was analysed at the seven- and nine-electron level of correlation. The results of this study are presented in tables 7 and 8. CCSD(T) results obtained using similar quality basis sets are also included for comparison. The geometry used for the saddle point in these single point calculations was the one optimized at the CCSD( T) level with the BSC2 AN0 basis set [ 16 1. The smallest extended basis set used is the F [ 5s4p3d2f] /H [ 4s3p2d] basis (BSA 1). The inclusion of a set of g and f polarization functions on F and H atoms, respectively, lowers the CASPT2 barrier height (BSCl ). This effect is slightly larger when also the 2s electrons are correlated. Extending the basis sets beyond this level does not lead to any further decrease of the barrier in the ( 7e- ) calculation, but increases the barrier somewhat when all nine electrons are correlated. On the other hand, employing larger AN0 basis set diminishes the effect of 2s correlation slightly at the CASPT2 level of theory: 0.48 and 0.39 kcal/mol with BSAl and BSEl, respectively. The barrier height obtained at the (9e- ) level of correlation with the largest basis set used in this study (BSEl ) is 2.47 kcal/mol. Comparing with CCSD(T) [ 161, and contracted MRCI+Q calculations [ 171, using similar quality basis sets (BSE2, and BSF), the CASPT2 approach gives a barrier height around 0.5 kcal/mol larger than these methods. A parallel analysis on the exothermicity of the reaction has been performed (cf. table 8). When 2s electrons are correlated, the addition of the first set of g( F) and f(H) polarization functions slightly increases the exothermicity. Further enlargement of the basis set gives an opposite trend. The exothennicity obtained at the (9e-) level using the largest basis set, BSEl, differs with 0.48 kcal/mol from the experimental value [ 39 ] and is 0.35 kcal/mol smaller than the corresponding CCSD (T) results [ 161.

R. Gonzdez-Luqueet al. /Chemical Physrcs 171(1993) 107-118 Table 7 (7e- ) and (9e- ) barrier results Basis ‘)

(kcal/mol) obtained with different extended AN0 basis sets p)

Seven electrons

X=A X=B x=c X=D X=E

117

Nine electrons

PTZD

PTZF

CCSD(T)

PTZD

PTZF

CCSD(T)

3.09 2.97 2.91 2.90 2.90

3.05 2.93 2.87 2.87 2.86

2.58 2.47 2.42 2.27 2.25

2.63 2.49 2.38 2.52 2.53

2.57 2.43 2.32 2.45 2.47

2.4 1 2.27 2.19 2.09 2.05

a) Geometries (au) taken from the CCSD(T) optimization using the BSC2 basrs set. Seven electrons: (2.902, 1.449); nine electrons: (2.913, 1.445) [16]. b, CASPTZ calculations were performed using averaged AN0 basis sets [ 341 (BSX 1) CCSD( T) calculations were performed using the Almlijf and Taylor AN0 basis [ 3 1 ] (BSXZ) (cf. table 1).

Table 8 (7e-) and (9e-)

exothermicities

Basis b,

X=A X=B x=c X=D X=E experimental

(kcal/mol)

obtained with different extended AN0 basis sets’)

Seven electrons

Nine electrons

PTZD

PT2F

CCSD(T)

PT2D

PTZF

34.05 34.34 34.32 34.28 34.30

34.89 35.18 35.16 35.12 35.13

33.1 33.2 33.4 33.7

29.28 29.54 29.62 29.38 29.33

31.18 31.47 31.53 31.31 31.25

[ 39 ]

CCSD(T)

30.9 31.1 31.3 31.6 31.73

‘r The experimental geometry [ 39 ] has been used. ‘) CASPTZ calculations were performed using averaged AN0 basis sets [ 341 (BSX 1) . CCSD( T ) calculations were performed using the Almliif and Taylor AN0 basis [ 3 1 ] (BSXZ ) (cf. table 1) .

4. Summary The F+H2-+FI-I+H reaction has been used as a first test of the behaviour of a multi-configuration second-order perturbation approach (CASSCF/ CASPT2) in calculations of energy barriers for chemical reactions. The calibration of the performance has been made to FCI, using small and medium size A0 basis sets and also to conventional approaches like MRCI, CPF, CCSD (T), etc., here with extended basis sets. The small basis set tests show that the CASPT2 approach is capable of reproducing the barrier heights, saddle point geometries and the heat of reaction with good accuracy. The same requirements for the reference function applies in the perturbation calculation as in corresponding MRCI calculations: the second R orbital shell on F has to be included in order to reproduce the FCI results. Such

similarities between CASPTZ and MRCI have also been found in earlier applications [ 41. The extended basis set calculations show that CASPTZ gives results of very similar quality to those obtained with more advanced methods. The barrier computed with the largest basis set is 0.42 kcal/mol larger than the corresponding CCSD (T) result and the exothermicity is 0.3 kcal/mol smaller. An earlier extended test of heats of reactions for isogyric reactions showed that they are reproduced (at least for systems build from the first row atoms and hydrogen) with an accuracy of f 2 kcal/mol or better, when standard AN0 basis sets areused [ll]. The present results show that the CASPT2 approach may be an interesting alternative for the calculation of energy surfaces for chemical reactions. The method has the virtue of being very general. It can thus be applied not only to ground state surfaces but

118

R. Gonzdez-Luque et al. /Chemical Physws 171(1993)

also to excited states and forbidden reactions, where large configurational mixing occurs in the region around the transition state.

Acknowledgement The research reported in this communication has been supported by a grant from the Swedish Natural Science Research Council (NFR), by IBM Sweden under a joint study contract, and by the Cooperation Cientifica y Ttcnica (Ministerio de Asuntos Exteriores) of Spain.

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