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A general coupled cluster study of the N2 molecule
31 August 2001
Chemical Physics Letters 344 (2001) 578±586
www.elsevier.com/locate/cplett
A general coupled cluster study of the N2 molecule Jesper Wisborg Krogh *, Jeppe Olsen Department of Chemistry, University of Aarhus, DK-8000 Aarhus, Denmark Received 28 September 2000
Abstract The equilibrium distance, harmonic frequency and potential curve of the nitrogen molecule are investigated using the cc-pVDZ basis and various single- and multi-reference coupled cluster (CC) methods. Including single and double excitations from all determinants of the smallest active space that ensures correct dissociation, the CC method gives for the equilibrium distance, 1 cm 1 for the frequency, deviations from full con®guration interaction (FCI) of 0.0001 A and a non-parallelity error (NPE) of 0:0006 Eh for the potential curve. Restricting the single and double excitations from the active space to those that are at most quadruple excited compared to the Hartree±Fock determinant, produces results that are very close to those obtained including all excitations up to quadruple excitations. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The coupled cluster (CC) method [1±3] provides geometries and relative energies with an accuracy that often is comparable to or surpasses the accuracy oered by experiment [2,4±6]. Thus, the coupled cluster method has emerged as the standard method for very accurate calculations of equilibrium geometries, atomization energies, and reaction enthalpies for small molecules. The standard formulation of the CC theory uses a single determinant as a reference state, and this has traditionally been considered to imply that the standard CC method is accurate only for wave functions that are dominated by a single con®guration. To accommodate for systems including static correlation where several con®gurations are
important, numerous multi-reference CC methods have been advanced (see [3] for a recent review). It may, however, be argued, that the standard single reference method may be turned into a multi-reference method simply by modifying the excitation manifold [7,8]. Consider for example the excitation manifold ^ sl ^ s0l ; s^1l ; s^2l
1
and a single reference determinant jrefi, so s^0l jrefi are the determinants of a complete active space (CAS), and s^1l jrefi (^ s2l jrefi) are the determinants that can be obtained by single (double) excitations from the CAS determinants. As all the excitation operators commute we may write the corresponding CC wave function as ! X X X T^ 2 2 1 1 0 0 e jrefi exp cl s^l cl s^l cl s^l jrefi l
*
Corresponding author. Fax: +45-8619-6199. E-mail address: [email protected] (J.W. Krogh).
exp
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 8 5 3 - 3
X l
l
c2l s^2l
X l
!
l
c1l s^1l jCASi;
2
J.W. Krogh, J. Olsen / Chemical Physics Letters 344 (2001) 578±586
P
where jCASi exp l c0l s^0l jrefi is a general CAS state that may be considered as the reference state. The distinction between single- and multi-reference CC expansions is therefore not clearcut. Note, however, that the single determinant reference still is present through the choice of the excitation manifold. Irrespective of whether the CC expansion of Eq. (2) should be considered a multior single-reference expansion, it provides a theoretically sound foundation for CC calculations. If the unlinked CC equations for the excitation manifold of Eq. (1) are linearized, one recovers the multi-reference con®guration interaction with single and double excitations (MRCISD) equations, which are independent of the choice of the reference determinant. We will in the following denote CC calculations using the excitation manifold of Eq. (1) as multi-reference CC single and double (MRCCSD) calculations, mainly to distinguish these calculations from standard CC calculations. The use of CC wave functions of the form of Eq. (2) was pioneered by Adamowicz [7±10], and such expansions have subsequently been used among others by Bartlett and coworkers [11], Piecuch and coworkers [12], and Li and Paldus [13]. A drawback of this approach is that selected operators with high excitation levels are required, and these excitation levels have traditionally been dicult to include in CC calculations. A very general CC method, the general active space CC (GASCC) method, that allows arbitrary divisions of orbital spaces and general levels of excitations, has recently been introduced by one of us [15]. Several other methods using CI-technology to enable CC calculations with general excitation rank have recently been published [16,17]. We will here apply the GASCC method and code using wave functions of the form of Eq. (2) to study the nitrogen molecule. In addition to the potential curve, the equilibrium distance and harmonic frequency are obtained and compared to the full con®guration interaction (FCI) results of Larsen et al. [18]. Several recent studies have used the CC method to study the equilibrium constants and potential curve of N2 . Li and Paldus [13] used their reduced multi reference CC (RMRCCSD) method to study the potential curve. They did not, however, con-
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sider the equilibrium properties, nor did their study include triple and higher excitations obtained within the CC framework. Using large basis sets, Kucharski et al. [14] have studied the performance of various single-reference methods for the determination of the potential curve of N2 . In the following section, we brie¯y summarize the algorithms currently used for the GASCC method, Section 3 discusses the results, and the conclusions are given in Section 4. 2. The general active space coupled cluster method 2.1. The coupled cluster equations In the CC method [1,2], the wave function is obtained from a reference wave function jrefi by applying an exponential operator as given in Eq. (2). We will in the following assume that the reference function is a single Slater-determinant, allowing a division of the molecular spin-orbitals into occupied and virtual spin-orbitals. The operator T^ is written in the spin-orbital basis as X T^ tl s^l ; 3 l
where s^l is an spin-orbital excitation operator exciting from the occupied spin-orbitals to unoccupied spin-orbitals and tl is the associated amplitude. The energy and amplitudes are obtained from the transformed Schr odinger equation ^ ^ e T H^ eT jrefi Ejrefi
by projecting jrefi; s^l jrefi
with
4 the
Slater-determinants
^ ^ hrefje T H^ eT jrefi E;
5
^ ^ hrefj^ syl e T H^ eT jrefi 0:
6
If all spin-orbital excitation operators, s^l , are included in the CC excitation operator, Eq. (3), the CC equations, Eqs. (5) and (6), constitute an alternative form of FCI. Usually, only a restricted set of excitation operators and amplitudes are included, for example all single and double excitations from the reference state.
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The CC wave function is obtained by an iterative scheme and in each iteration, one evaluates the CC vector function ^ ^ fl t hrefj^ syl e T H^ eT jrefi:
7
Highly ecient algorithms have been developed for evaluating the standard CC vector functions, CC with single and double excitations (CCSD) [19], and with single, double, and triple excitations (CCSDT) [20]. In the present context, a more general approach is taken. In the initial implementation, the con®guration interaction (CI) codes of LUCIA 1 have been modi®ed and extended to allow the evaluation of the CC vector function. We will here brie¯y sketch the approach and refer to [15] for further details. The evaluation of the vector function f proceeds in four steps. First the following CI-expansion is obtained ! X 1 T^ n ^ T jrefi: jai e jrefi 8 n! n0 The above step is realized by evaluating ^ T^jrefi; T^2 jrefi; . . . until all terms in eT jrefi that contribute to the vector function have been obtained. To evaluate the eect of T^ on a CI vector, the action of general n-body operators on a state must be evaluated. In the second step, the action of the Hamiltonian H^ on jai is evaluated using the standard direct CI approach jbi H^ jai:
9
The exponential e ^
jci e T jbi
T^
is then applied yielding ! X 1n n T^ jbi 10 n! n0
and ®nally the CC vector function is obtained as the general transition density matrix fl
hrefj^ syl jci:
11
As discussed in [15], jai must be evaluated in a space that is obtained by allowing single and double excitations out from jrefi; s^l jrefi, 1 LUCIA, a general CI code written by J. Olsen, University of Aarhus with contributions from H. Larsen and M. F ulscher.
whereas jbi and jci must be evaluated in the space jrefi; s^l jrefi. The use of an intermediate space that is two excitation levels higher than the chosen CC space, makes the present approach signi®cantly more time-consuming than the conventional CC approach. For a CC calculation containing all excitations up through level k, a conventional CC approach scales like Ok V k2 , where O is the number of occupied orbitals and V is the number of virtual orbitals. In the present implementation, the operation count scales as Ok2 V k2 and this initial implementation is therefore not suitable for production calculations. A new general CC code with the same scaling as conventional CC algorithms is currently being tested. The new method is based on direct manipulations of spin-strings without invoking intermediate CI-expansions. A number of features that are present in the ®nal GASCC codes have been added to the program. The single excitations, T^1 , are excluded from ^ ^ the operators eT and e T , and are instead included by replacing the Hamiltonian H^ with the similarity transformed Hamiltonian 1 H^ T e
T^1
^1 H^ eT :
12
The requirement that the CC excitation operator has singlet spin is used to halve the number of CC coecients that need to be calculated and stored. For example, the coecients for exciting from alpha-orbitals i and j to alpha-orbitals a and b, is equal to the coecient for exciting the corresponding beta-orbitals. For a CCSD calculation, the number of doubles amplitudes that needs to be calculated in the present spin-orbital basis is thereby reduced to 34O2 V 2 which is only 50% larger than the number of coecients in a spin-adapted basis. 2.2. The generalized active space In the original CC methods, the orbital space was divided into occupied and virtual orbitals. This simple division of the orbitals has been very important for the generation of black box methods, but is not general enough to allow compact descriptions of, for example, bond-breakings. To describe the breaking of bonds and other multicon®gurational wave functions, one must use at
J.W. Krogh, J. Olsen / Chemical Physics Letters 344 (2001) 578±586
least three sets of orbitals [21]: inactive orbitals that are doubly occupied in all dominating con®gurations, active orbitals that have variable occupations in the important con®gurations, and secondary orbitals that are unoccupied in all important con®gurations. This threefold division of the orbital space has allowed the generation of multi-reference CI methods that accurately describe potential curves of small molecules. The division of orbitals into three spaces is not sucient for the development of commutative CC multi-reference methods, as these assume the division of the orbitals in hole-orbitals where electrons are excited from and particle orbitals where electrons are excited to. To obtain the hole-particle division, the occupation of the reference determinant is used to divide the active orbitals into a set of active hole orbitals, where electrons are excited from, and a set of active particle orbitals, where electrons are excited to. At least four orbital spaces are thus required to set up a commutative multireference method. In the present approach we use a more general ansatz, the generalized active space (GAS), where the orbital space may be divided into an arbitrary number of subspaces, and arbitrary restrictions may be imposed on the number of electrons that may be annihilated and created in the hole and particle spaces. 3. Results In the following we discuss calculations on N2 using the cc-pVDZ basis of Dunning and coworkers [22]. In all calculations, the 1rg and 1ru orbitals are doubly occupied. Canonical Hartree± Fock orbitals are used in the single reference calculations (CCSD, CCSDT, etc.) whereas the multi-reference calculations use orbitals obtained from a CASSCF calculation. In the CASSCF calculations, the orbitals 1rg ; 1ru ; 2rg ; 2ru are doubly occupied inactive orbitals, 6 electrons are distributed in 6 active orbitals (3rg ; 3ru ; 1pg ; 1pu ), and the remaining orbitals are unoccupied. This active space is the smallest active space that allows correct dissociation of the nitrogen molecule. Note that the doubly occupied orbitals were re-optimized in the CASSCF calculations. The results are
581
compared with the recent FCI potential curve of Larsen et al. [18] obtained using the canonical Hartree±Fock orbitals. As the doubly occupied orbitals were re-optimized in the CASSCF calculations, the FCI calculations employing the CASSCF orbitals dier slightly from the reported FCI calculations. From calculations at the equilibrium distance (2.068 a.u.), it was established that the dierence between the FCI energies using the two orbital sets was of no signi®cance for the following comparisons. The multi-reference CI (MRCISD) calculations included the determinants of the CAS and all determinants obtained by single and double excitations out from these determinants. Several types of multi-reference CC calculations were carried out, all using the ground state determinant as the reference state. Calculations were carried out using the same set of excitations as the multi-reference CI calculations (MRCCSD). The operator manifold in the MRCCSD calculations includes up to eightfold excitations from the reference determinant. In addition, calculations were carried out with the excitation manifold restricted to those excitations that are at most fourfold excited compared to the reference determinant (MRCCSD(4ex)). Finally, we performed coupled cluster calculations, where the triple and quadruple excitations were obtained from MRCISD calculations and frozen in the following CC calculations, that only optimized the single and double excitations. The latter calculations are examples of the use of the reduced multireference method suggested by Li and Paldus [23], and are therefore referred to as the RMRCCSD calculations. Note that the triple and quadruple excitations in the RMRCCSD calculations are obtained from the full MRCISD calculations. 3.1. Calculations at the equilibrium distance To study the convergence of the CC method for the nitrogen molecule at the equilibrium distance, various single reference coupled cluster and CI calculations were carried out. In addition to the SD and SDT expansions, expansions containing up to quadruple excitations (SDTQ), up to quintuple excitations (SDTQ5) and up to hextuple excitations (SDTQ56) were studied. This analysis
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may be considered as a continuation of a previous study of CH2 and H2 O [15]. In Table 1, we give the energies relative to FCI as a function of excitation level for the CI and CC calculations, and in Fig. 1 these data are presented in the form of a singlelogarithmic plot. The convergence of the CC hierarchy has been theoretically analyzed by Kutzelnigg [24] and later by Helgaker et al. [2]. The energy obtained from the CC wave function including up to n-fold excitation is correct through order 3n=2 in the ¯uctuation potential, where x is the largest inteTable 1 The convergence of the energy in the CC and CI hierarchies for the ground state of N2 using the cc-pVDZ basisa
a
Excitation level N
ECC N
0 2 3 4 5 6
0.3219744 0.0134653 0.0016261 0.0001925 0.0000165 0.0000013
EFCI
ECI N
EFCI
0.3219744 0.0342770 0.0243304 0.0020562 0.0007113 0.0000391
The energy is given in Eh as the deviation from the FCI energy. The full CI energy is 109:27652761 Eh .
ger not exceeding x. The deviation of the energy from the FCI value should therefore show larger decreases when an even excitation order is added, than when an odd order is added. This behavior has been observed for the water molecule [15]. For the nitrogen molecule we notice marked dierences from the theoretical predictions, with the reductions in the energy errors not depending on whether the added excitation level is even or odd. This behavior re¯ects the complicated electronic structure of N2 making an analysis based on perturbation arguments invalid. It should furthermore be noted that the CC hierarchy converges signi®cantly slower for N2 than for H2 O. Thus, whereas the CCSDTQ energy diers from the FCI energy by 0.00001 Eh [15] for H2 O using the cc-pVDZ basis, the CCSDTQ energy for N2 diers by 0.00019 Eh from the FCI energy. 3.2. Equilibrium distance and harmonic frequency For the various single- and multi-reference calculations, the equilibrium distance and harmonic frequency were determined by calculating
Fig. 1. The deviations of CI and CC energies from the FCI energy as a function of excitation level for N2 using the cc-pVDZ basis set.
J.W. Krogh, J. Olsen / Chemical Physics Letters 344 (2001) 578±586
the energy at ®ve points around the experimental distance and performing quartic interpolation. The resulting equilibrium geometries and harmonic frequencies, and the corresponding FCI results from [18] are given in Table 2. The table also contains the results from [18] obtained using the CCSD(T) method, which augments CCSD with a perturbative estimate of triples contributions [25]. The equilibrium distance is accurately obtained using all the multi-reference methods, deviating at from the FCI distance. Neither the most 0.0002 A use of con®guration interaction instead of CC nor the use of restricted multi-reference CC expansions eects the accuracy of the equilibrium distance. For expansions using smaller active spaces, one may expect larger dierences between the results obtained using CI and CC expansions. For the single-reference calculations, the geometries are similar to those obtained by Kucharski et al. [14], where calculations using CCSD, CCSDT, CCSDTQ in the cc-pVDZ basis were reported. The distance obtained with the CCSD method deviates signi®cantly from the FCI values, and also the full CCSDT methods predict a distance that is inferior to that obtained by the multi-reference methods. It is only with the inclusion of all triple and quadruple excitations in CCSDTQ that the accuracy of the distance becomes comparable to that obtained by the multi-reference methods. The CCSD(T) method predicts a distance very close to the CCSDT distance with a deviation from This deviation is in line the FCI result of 0.0014 A. with the high accuracy generally reported for the Table 2 The equilibrium distance Req and harmonic frequency xe for N2 in the cc-pVDZ basis obtained using FCI and various correlation methods Model
Req (A)
xe (cm 1 )
FCI MRCISD MRCCSD MRCCSD(4ex) RMRCCSD CCSD CCSD(T) CCSDT CCSDTQ
1.1201 1.1200 1.1202 1.1199 1.1201 1.1128 1.1187 1.1185 1.1198
2323 2324 2324 2328 2325 2408 2349 2346 2328
583
CCSD(T) method [2], but is about an order of magnitude larger than the presented deviations of the multi-reference methods. For the harmonic frequency, the MRCISD, MRCCSD, and RMRCCSD methods all produce values within 2 cm 1 of the FCI value, whereas the MRCCSD(4ex) expansion gives a deviation of 5 cm 1 . For the single-reference calculations, the CCSDT method gives a deviation of more than 20 cm 1 which is twenty times larger then the best multi-reference methods. Even the full CCSDTQ calculation gives a frequency that is about 5 cm 1 too high. It is interesting to note that the calculations including all triple and quadruple excitations (CCSDTQ) and the multi-reference calculation including only selected triple and quadruple excitations (MRCCSD(4ex)) give nearly identical results for the equilibrium distance and frequency. We will return to this in the discussion of the potential curve. 3.3. The potential curve In Fig. 2 we have plotted the deviations from the FCI energy as a function of the inter-nuclear distance for the single- and multi-reference methods. The curves are based on the calculations reported in Table 3. In Table 3 we also report the various non-parallelity errors (NPE) de®ned as the dierence between the largest and smallest deviations from FCI. All the multi-reference methods have deviation curves with nearly zero curvature at the equilibrium distance which explains the accurate frequencies obtained with these methods. The energies obtained with the multi-reference CI method have deviations from the FCI values that are 3±4 times larger than the deviations obtained with the multi-reference coupled cluster method. However, the NPE obtained with the MRCISD method (0.0010 Eh ) is lower than obtained with the RMRCCSD(0.0018 Eh ) and MRCCSD(4ex) (0.0031 Eh ) methods, and is only surpassed by the NPE of the MRCCSD method (0.0006 Eh ). The full MRCCSD method gives the smallest energy deviations with absolute errors of less than 0.002 Eh and the smallest NPE of 0.0006 Eh . The MRCCSD(4ex) deviations increase signi®cantly
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J.W. Krogh, J. Olsen / Chemical Physics Letters 344 (2001) 578±586
Fig. 2. The potential curves for N2 using various correlation methods and the cc-PVDZ basis. The energies are in hartrees and the inter-nuclear distances are given in atomic units.
when the inter-nuclear distance is increased with errors of more than 0.004 Eh for distances larger than 2.5 a.u. The RMRCCSD curve is closer than the MRCCSD(4ex) curve to the MRCCSD curves, but starts to show signi®cant deviations for distances larger than 3.0 a.u. It is thus only the MRCISD and MRCCSD methods that are able to provide energies of high accuracy over the complete potential curve. It may seem surprising that the RMRCCSD curve is more accurate than the MRCCSD(4ex) curve, as these calculations employ the same set of CC excitations. However, as discussed above, the RMRCCSD calculation uses triple and quadruple excitations obtained from a complete MRCISD calculation, and should therefore be considered as an approximation to the full MRCCSD calculation and not as an approximation to the MRCCSD(4ex) calculation. Not surprisingly, the single reference methods show deviations from FCI that are very geometry dependent, and they have therefore signi®cant NPEs. Thus for the shorter geometries, the CCSD(T), CCSDT, and CCSDTQ methods have
smaller deviations from the FCI energy than the multi-reference methods, but the deviations increase as the bond is stretched and have a maximum around 2.75 a.u. Furthermore, the curvature of the CCSD(T) and CCSDT deviation curves are signi®cant already at the equilibrium distance, which explains why these methods give harmonic frequencies that dier by more than 20 cm 1 from the FCI value. It is interesting to note that the curves for CCSDTQ and MRCCSD(4ex) are nearly parallel. The triple and quadruple excitations included in the MRCCSD(4ex) calculations are thus those that are most dependent on the geometry. On the other hand, the ®ve-fold and higher excitations excluded in the CCSDTQ and the MRCCSD(4ex) calculations are necessary to obtain an NPE of less than 0.001 Eh . 4. Conclusion The equilibrium distance, harmonic frequency, and potential curve for N2 have been investigated
0.0026 0.0099 0.0070 0.0662 0.0018 0.0031 Energies and deviations are in hartrees.
± NPE
0.0010
0.0006 a
CCSDTQ CCSDT
0.000400 0.000844 0.001627 0.001759 0.001839 0.002080 0.002351 0.003631 0.006834 0.01031 0.0081 0.000638 0.001076 0.001565 0.001807 0.001879 0.002180 0.002475 0.003087 0.005182 0.00770 0.0076
CCSD(T) CCSD
0.005949 0.009130 0.013466 0.014099 0.014469 0.015539 0.016682 0.021481 0.031807 0.04519 0.0604 0.0721 0.001788 0.001669 0.001666 0.001671 0.001675 0.001687 0.001702 0.001779 0.001988 0.00226 0.0028 0.0035
RMRCCSD MRCCSD(4ex)
0.001747 0.001640 0.001696 0.001715 0.001728 0.001769 0.001820 0.002104 0.002998 0.00427 0.0047 0.0028 0.001730 0.001579 0.001520 0.001516 0.001515 0.001513 0.001513 0.001528 0.001618 0.00179 0.0020 0.0021
MRCCSD MRCISD
)108.624700 )109.167573 )109.276528 )109.278135 )109.278339 )109.276583 )109.271915 )109.238397 )109.160305 )109.08621 )109.0303 )108.9948
0.005545 0.005733 0.005834 0.005843 0.005849 0.005864 0.005879 0.005937 0.006070 0.00628 0.0065 0.0066
EFCI R (a.u.)
1.500 1.800 2.068 2.100 2.118 2.168 2.218 2.400 2.700 3.000 3.300 3.600
Table 3 FCI energies and deviations from FCI for various wave function approximations as a function of inter-nuclear distancea
0.000022 0.000074 0.000193 0.000215 0.000229 0.000272 0.000323 0.000583 0.001357 0.00241 0.0026 0.0005
J.W. Krogh, J. Olsen / Chemical Physics Letters 344 (2001) 578±586
585
using the cc-pVDZ basis and various correlation methods and compared with FCI. The correlation methods include the single-reference methods CCSD, CCSD(T), CCSDT, CCSDTQ, MRCISD and MRCCSD. For the multi-reference calculations, the smallest complete active space with correct dissociation was employed, 6 electrons in six orbitals 3rg ; 3ru ; 1pg ; 1pu . Including all single and double excitations out from the active space, the CC and CI methods have deviations in the equilibrium distance of and 1 cm 1 in the harmonic frequency. 0.0001 A, Restricting the excitations included in the CC calculations to at most fourfold excitations increases the deviation for the frequency to about 5 cm 1 , whereas freezing the triple and quadruple amplitudes to those derived from the MRCISD calculation does not aect the accuracy of the equilibrium distance. The deviations from FCI are signi®cantly larger for the single-reference methods and only using the CCSDTQ method is a frequency with an error of 5 cm 1 obtained. Thus even for equilibrium properties like bond distances and harmonic frequencies, it may be important to use multi-reference rather than single-reference CC methods. The potential curves obtained with the multi-reference methods are in signi®cantly better agreement with the FCI curve than are the curves obtained with single-reference methods. Although the multi-reference CI curve has deviations that are three times larger than obtained with the full multi-reference CC method, the NPE of these two schemes are comparable (0.0010 and 0.0006 Eh ). The restrictions of the excitation manifold in the MRCCSD calculations to include at most fourfold excitation, produces signi®cant deviations for inter-nuclear distances larger than 2.5 a.u. Freezing the triple and quadruple excitations to those deduced from the MRCISD calculations produces a curve that is more parallel to the FCI curve than obtained by actually re-optimizing the triple and quadruple excitations in the CC calculations including at most quadruple excitations. This is not surprising as the frozen triple and quadruple excitations originated from MRCISD calculations that include all single and double excitations from the active space.
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The present paper demonstrates that we now have coupled cluster methods that can describe complete potential curves. However, as the number of parameters present in the full MRCCSD calculations corresponds to the number of parameters in the full MRCISD calculations, the present approach is not easily extended to large molecules. The restriction to include only triple and quadruple excitations degrades signi®cantly the accuracy of the potential curves. Alternative restrictions of the active space that allow correct dissociation are included in the GAS formalism, and may give better potential curves. An alternative possibility is to use an internal contracted formalism. Work on developing an internally contracted formalism using commuting operators spanning the full ®rst-order interacting subspace is currently in progress. Acknowledgements This work has been supported by the Danish Research Council (Grant No. 9901973). References [1] R.J. Bartlett, in: D.R. Yarkony (Ed.), Modern Electronic Structure Theory, Part I, World Scienti®c, Singapore, 1995, p. 1047. [2] T. Helgaker, P. Jùrgensen, J. Olsen, Molecular ElectronicStructure Theory, Wiley, New York, 2000. [3] J. Paldus, X. Li, Adv. Chem. Phys. 110 (1999) 1.
[4] T.H. Dunning Jr., K.A. Peterson, D.E. Woon, in: P.v.R. Schleyer, N.L. Allinger, T. Clark, J. Gasteiger, P.A. Kollman, H.F. Schaefer III, P.R. Scheiner (Eds.), The Encyclopedia of Computational Chemistry, Wiley, New york, 1998. [5] J.M.L. Martin, Chem. Phys. Lett. 259 (1996) 669. [6] W. Klopper, K.L. Bak, P. Jùrgensen, J. Olsen, T. Helgaker, J. Phys. B: At. Mol. Opt. Phys. 32 (1999) R103. [7] N. Oliphant, L. Adamowicz, J. Chem. Phys. 94 (1991) 1229. [8] N. Oliphant, L. Adamowicz, J. Chem. Phys. 96 (1992) 3739. [9] P. Piecuch, N. Oliphant, L. Adamowicz, J. Chem. Phys. 99 (1993) 1875±1900. [10] P. Piecuch, N. Oliphant, L. Adamowicz, J. Chem. Phys. 100 (1994) 5792±5809. [11] P. Piecuch, S.A. Kucharski, V. Spirko, J. Chem. Phys. 111 (1999) 6679±6692. [12] P. Piecuch, S.A. Kucharski, R.J. Bartlett, J. Chem. Phys. 110 (1999) 6103. [13] X. Li, J. Paldus, Chem. Phys. Lett. 286 (1998) 145. [14] S.A. Kucharski, J.D. Watts, R.J. Bartlett, Chem. Phys. Lett. 302 (1999) 295. [15] J. Olsen, J. Chem. Phys. 113 (2000) 7140. [16] S. Hirata, R.J. Bartlett, Chem. Phys. Lett. 321 (2000) 216. [17] M. Kallay, P.R. Surjan, J. Chem. Phys. 113 (2000) 1359. [18] H. Larsen, J. Olsen, P. Jùrgensen, O. Christiansen, J. Chem. Phys. 113 (2000) 6677. [19] G.E. Scuseria, C.L. Janssen, H.F. Schaefer III, J. Chem. Phys. 86 p. 2881. [20] J. Noga, R.J. Bartlett, J. Chem. Phys. 1987 (1987) (7041) 3401(E). [21] B.O. Roos, Adv. Chem. Phys. 69 (1987) 399. [22] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007. [23] X. Li, J. Paldus, J. Chem. Phys. 107 (1997) 6257. [24] W. Kutzelnigg, Theoret. Chim. Acta 80 (1991) 349. [25] K. Raghavachari, G.W. Trucks, J.A. Pople, M. HeadGordon, Chem. Phys. Lett. 157 (1989) 479.
Chemical Physics Letters 344 (2001) 578±586
www.elsevier.com/locate/cplett
A general coupled cluster study of the N2 molecule Jesper Wisborg Krogh *, Jeppe Olsen Department of Chemistry, University of Aarhus, DK-8000 Aarhus, Denmark Received 28 September 2000
Abstract The equilibrium distance, harmonic frequency and potential curve of the nitrogen molecule are investigated using the cc-pVDZ basis and various single- and multi-reference coupled cluster (CC) methods. Including single and double excitations from all determinants of the smallest active space that ensures correct dissociation, the CC method gives for the equilibrium distance, 1 cm 1 for the frequency, deviations from full con®guration interaction (FCI) of 0.0001 A and a non-parallelity error (NPE) of 0:0006 Eh for the potential curve. Restricting the single and double excitations from the active space to those that are at most quadruple excited compared to the Hartree±Fock determinant, produces results that are very close to those obtained including all excitations up to quadruple excitations. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The coupled cluster (CC) method [1±3] provides geometries and relative energies with an accuracy that often is comparable to or surpasses the accuracy oered by experiment [2,4±6]. Thus, the coupled cluster method has emerged as the standard method for very accurate calculations of equilibrium geometries, atomization energies, and reaction enthalpies for small molecules. The standard formulation of the CC theory uses a single determinant as a reference state, and this has traditionally been considered to imply that the standard CC method is accurate only for wave functions that are dominated by a single con®guration. To accommodate for systems including static correlation where several con®gurations are
important, numerous multi-reference CC methods have been advanced (see [3] for a recent review). It may, however, be argued, that the standard single reference method may be turned into a multi-reference method simply by modifying the excitation manifold [7,8]. Consider for example the excitation manifold ^ sl ^ s0l ; s^1l ; s^2l
1
and a single reference determinant jrefi, so s^0l jrefi are the determinants of a complete active space (CAS), and s^1l jrefi (^ s2l jrefi) are the determinants that can be obtained by single (double) excitations from the CAS determinants. As all the excitation operators commute we may write the corresponding CC wave function as ! X X X T^ 2 2 1 1 0 0 e jrefi exp cl s^l cl s^l cl s^l jrefi l
*
Corresponding author. Fax: +45-8619-6199. E-mail address: [email protected] (J.W. Krogh).
exp
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 8 5 3 - 3
X l
l
c2l s^2l
X l
!
l
c1l s^1l jCASi;
2
J.W. Krogh, J. Olsen / Chemical Physics Letters 344 (2001) 578±586
P
where jCASi exp l c0l s^0l jrefi is a general CAS state that may be considered as the reference state. The distinction between single- and multi-reference CC expansions is therefore not clearcut. Note, however, that the single determinant reference still is present through the choice of the excitation manifold. Irrespective of whether the CC expansion of Eq. (2) should be considered a multior single-reference expansion, it provides a theoretically sound foundation for CC calculations. If the unlinked CC equations for the excitation manifold of Eq. (1) are linearized, one recovers the multi-reference con®guration interaction with single and double excitations (MRCISD) equations, which are independent of the choice of the reference determinant. We will in the following denote CC calculations using the excitation manifold of Eq. (1) as multi-reference CC single and double (MRCCSD) calculations, mainly to distinguish these calculations from standard CC calculations. The use of CC wave functions of the form of Eq. (2) was pioneered by Adamowicz [7±10], and such expansions have subsequently been used among others by Bartlett and coworkers [11], Piecuch and coworkers [12], and Li and Paldus [13]. A drawback of this approach is that selected operators with high excitation levels are required, and these excitation levels have traditionally been dicult to include in CC calculations. A very general CC method, the general active space CC (GASCC) method, that allows arbitrary divisions of orbital spaces and general levels of excitations, has recently been introduced by one of us [15]. Several other methods using CI-technology to enable CC calculations with general excitation rank have recently been published [16,17]. We will here apply the GASCC method and code using wave functions of the form of Eq. (2) to study the nitrogen molecule. In addition to the potential curve, the equilibrium distance and harmonic frequency are obtained and compared to the full con®guration interaction (FCI) results of Larsen et al. [18]. Several recent studies have used the CC method to study the equilibrium constants and potential curve of N2 . Li and Paldus [13] used their reduced multi reference CC (RMRCCSD) method to study the potential curve. They did not, however, con-
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sider the equilibrium properties, nor did their study include triple and higher excitations obtained within the CC framework. Using large basis sets, Kucharski et al. [14] have studied the performance of various single-reference methods for the determination of the potential curve of N2 . In the following section, we brie¯y summarize the algorithms currently used for the GASCC method, Section 3 discusses the results, and the conclusions are given in Section 4. 2. The general active space coupled cluster method 2.1. The coupled cluster equations In the CC method [1,2], the wave function is obtained from a reference wave function jrefi by applying an exponential operator as given in Eq. (2). We will in the following assume that the reference function is a single Slater-determinant, allowing a division of the molecular spin-orbitals into occupied and virtual spin-orbitals. The operator T^ is written in the spin-orbital basis as X T^ tl s^l ; 3 l
where s^l is an spin-orbital excitation operator exciting from the occupied spin-orbitals to unoccupied spin-orbitals and tl is the associated amplitude. The energy and amplitudes are obtained from the transformed Schr odinger equation ^ ^ e T H^ eT jrefi Ejrefi
by projecting jrefi; s^l jrefi
with
4 the
Slater-determinants
^ ^ hrefje T H^ eT jrefi E;
5
^ ^ hrefj^ syl e T H^ eT jrefi 0:
6
If all spin-orbital excitation operators, s^l , are included in the CC excitation operator, Eq. (3), the CC equations, Eqs. (5) and (6), constitute an alternative form of FCI. Usually, only a restricted set of excitation operators and amplitudes are included, for example all single and double excitations from the reference state.
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The CC wave function is obtained by an iterative scheme and in each iteration, one evaluates the CC vector function ^ ^ fl t hrefj^ syl e T H^ eT jrefi:
7
Highly ecient algorithms have been developed for evaluating the standard CC vector functions, CC with single and double excitations (CCSD) [19], and with single, double, and triple excitations (CCSDT) [20]. In the present context, a more general approach is taken. In the initial implementation, the con®guration interaction (CI) codes of LUCIA 1 have been modi®ed and extended to allow the evaluation of the CC vector function. We will here brie¯y sketch the approach and refer to [15] for further details. The evaluation of the vector function f proceeds in four steps. First the following CI-expansion is obtained ! X 1 T^ n ^ T jrefi: jai e jrefi 8 n! n0 The above step is realized by evaluating ^ T^jrefi; T^2 jrefi; . . . until all terms in eT jrefi that contribute to the vector function have been obtained. To evaluate the eect of T^ on a CI vector, the action of general n-body operators on a state must be evaluated. In the second step, the action of the Hamiltonian H^ on jai is evaluated using the standard direct CI approach jbi H^ jai:
9
The exponential e ^
jci e T jbi
T^
is then applied yielding ! X 1n n T^ jbi 10 n! n0
and ®nally the CC vector function is obtained as the general transition density matrix fl
hrefj^ syl jci:
11
As discussed in [15], jai must be evaluated in a space that is obtained by allowing single and double excitations out from jrefi; s^l jrefi, 1 LUCIA, a general CI code written by J. Olsen, University of Aarhus with contributions from H. Larsen and M. F ulscher.
whereas jbi and jci must be evaluated in the space jrefi; s^l jrefi. The use of an intermediate space that is two excitation levels higher than the chosen CC space, makes the present approach signi®cantly more time-consuming than the conventional CC approach. For a CC calculation containing all excitations up through level k, a conventional CC approach scales like Ok V k2 , where O is the number of occupied orbitals and V is the number of virtual orbitals. In the present implementation, the operation count scales as Ok2 V k2 and this initial implementation is therefore not suitable for production calculations. A new general CC code with the same scaling as conventional CC algorithms is currently being tested. The new method is based on direct manipulations of spin-strings without invoking intermediate CI-expansions. A number of features that are present in the ®nal GASCC codes have been added to the program. The single excitations, T^1 , are excluded from ^ ^ the operators eT and e T , and are instead included by replacing the Hamiltonian H^ with the similarity transformed Hamiltonian 1 H^ T e
T^1
^1 H^ eT :
12
The requirement that the CC excitation operator has singlet spin is used to halve the number of CC coecients that need to be calculated and stored. For example, the coecients for exciting from alpha-orbitals i and j to alpha-orbitals a and b, is equal to the coecient for exciting the corresponding beta-orbitals. For a CCSD calculation, the number of doubles amplitudes that needs to be calculated in the present spin-orbital basis is thereby reduced to 34O2 V 2 which is only 50% larger than the number of coecients in a spin-adapted basis. 2.2. The generalized active space In the original CC methods, the orbital space was divided into occupied and virtual orbitals. This simple division of the orbitals has been very important for the generation of black box methods, but is not general enough to allow compact descriptions of, for example, bond-breakings. To describe the breaking of bonds and other multicon®gurational wave functions, one must use at
J.W. Krogh, J. Olsen / Chemical Physics Letters 344 (2001) 578±586
least three sets of orbitals [21]: inactive orbitals that are doubly occupied in all dominating con®gurations, active orbitals that have variable occupations in the important con®gurations, and secondary orbitals that are unoccupied in all important con®gurations. This threefold division of the orbital space has allowed the generation of multi-reference CI methods that accurately describe potential curves of small molecules. The division of orbitals into three spaces is not sucient for the development of commutative CC multi-reference methods, as these assume the division of the orbitals in hole-orbitals where electrons are excited from and particle orbitals where electrons are excited to. To obtain the hole-particle division, the occupation of the reference determinant is used to divide the active orbitals into a set of active hole orbitals, where electrons are excited from, and a set of active particle orbitals, where electrons are excited to. At least four orbital spaces are thus required to set up a commutative multireference method. In the present approach we use a more general ansatz, the generalized active space (GAS), where the orbital space may be divided into an arbitrary number of subspaces, and arbitrary restrictions may be imposed on the number of electrons that may be annihilated and created in the hole and particle spaces. 3. Results In the following we discuss calculations on N2 using the cc-pVDZ basis of Dunning and coworkers [22]. In all calculations, the 1rg and 1ru orbitals are doubly occupied. Canonical Hartree± Fock orbitals are used in the single reference calculations (CCSD, CCSDT, etc.) whereas the multi-reference calculations use orbitals obtained from a CASSCF calculation. In the CASSCF calculations, the orbitals 1rg ; 1ru ; 2rg ; 2ru are doubly occupied inactive orbitals, 6 electrons are distributed in 6 active orbitals (3rg ; 3ru ; 1pg ; 1pu ), and the remaining orbitals are unoccupied. This active space is the smallest active space that allows correct dissociation of the nitrogen molecule. Note that the doubly occupied orbitals were re-optimized in the CASSCF calculations. The results are
581
compared with the recent FCI potential curve of Larsen et al. [18] obtained using the canonical Hartree±Fock orbitals. As the doubly occupied orbitals were re-optimized in the CASSCF calculations, the FCI calculations employing the CASSCF orbitals dier slightly from the reported FCI calculations. From calculations at the equilibrium distance (2.068 a.u.), it was established that the dierence between the FCI energies using the two orbital sets was of no signi®cance for the following comparisons. The multi-reference CI (MRCISD) calculations included the determinants of the CAS and all determinants obtained by single and double excitations out from these determinants. Several types of multi-reference CC calculations were carried out, all using the ground state determinant as the reference state. Calculations were carried out using the same set of excitations as the multi-reference CI calculations (MRCCSD). The operator manifold in the MRCCSD calculations includes up to eightfold excitations from the reference determinant. In addition, calculations were carried out with the excitation manifold restricted to those excitations that are at most fourfold excited compared to the reference determinant (MRCCSD(4ex)). Finally, we performed coupled cluster calculations, where the triple and quadruple excitations were obtained from MRCISD calculations and frozen in the following CC calculations, that only optimized the single and double excitations. The latter calculations are examples of the use of the reduced multireference method suggested by Li and Paldus [23], and are therefore referred to as the RMRCCSD calculations. Note that the triple and quadruple excitations in the RMRCCSD calculations are obtained from the full MRCISD calculations. 3.1. Calculations at the equilibrium distance To study the convergence of the CC method for the nitrogen molecule at the equilibrium distance, various single reference coupled cluster and CI calculations were carried out. In addition to the SD and SDT expansions, expansions containing up to quadruple excitations (SDTQ), up to quintuple excitations (SDTQ5) and up to hextuple excitations (SDTQ56) were studied. This analysis
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may be considered as a continuation of a previous study of CH2 and H2 O [15]. In Table 1, we give the energies relative to FCI as a function of excitation level for the CI and CC calculations, and in Fig. 1 these data are presented in the form of a singlelogarithmic plot. The convergence of the CC hierarchy has been theoretically analyzed by Kutzelnigg [24] and later by Helgaker et al. [2]. The energy obtained from the CC wave function including up to n-fold excitation is correct through order 3n=2 in the ¯uctuation potential, where x is the largest inteTable 1 The convergence of the energy in the CC and CI hierarchies for the ground state of N2 using the cc-pVDZ basisa
a
Excitation level N
ECC N
0 2 3 4 5 6
0.3219744 0.0134653 0.0016261 0.0001925 0.0000165 0.0000013
EFCI
ECI N
EFCI
0.3219744 0.0342770 0.0243304 0.0020562 0.0007113 0.0000391
The energy is given in Eh as the deviation from the FCI energy. The full CI energy is 109:27652761 Eh .
ger not exceeding x. The deviation of the energy from the FCI value should therefore show larger decreases when an even excitation order is added, than when an odd order is added. This behavior has been observed for the water molecule [15]. For the nitrogen molecule we notice marked dierences from the theoretical predictions, with the reductions in the energy errors not depending on whether the added excitation level is even or odd. This behavior re¯ects the complicated electronic structure of N2 making an analysis based on perturbation arguments invalid. It should furthermore be noted that the CC hierarchy converges signi®cantly slower for N2 than for H2 O. Thus, whereas the CCSDTQ energy diers from the FCI energy by 0.00001 Eh [15] for H2 O using the cc-pVDZ basis, the CCSDTQ energy for N2 diers by 0.00019 Eh from the FCI energy. 3.2. Equilibrium distance and harmonic frequency For the various single- and multi-reference calculations, the equilibrium distance and harmonic frequency were determined by calculating
Fig. 1. The deviations of CI and CC energies from the FCI energy as a function of excitation level for N2 using the cc-pVDZ basis set.
J.W. Krogh, J. Olsen / Chemical Physics Letters 344 (2001) 578±586
the energy at ®ve points around the experimental distance and performing quartic interpolation. The resulting equilibrium geometries and harmonic frequencies, and the corresponding FCI results from [18] are given in Table 2. The table also contains the results from [18] obtained using the CCSD(T) method, which augments CCSD with a perturbative estimate of triples contributions [25]. The equilibrium distance is accurately obtained using all the multi-reference methods, deviating at from the FCI distance. Neither the most 0.0002 A use of con®guration interaction instead of CC nor the use of restricted multi-reference CC expansions eects the accuracy of the equilibrium distance. For expansions using smaller active spaces, one may expect larger dierences between the results obtained using CI and CC expansions. For the single-reference calculations, the geometries are similar to those obtained by Kucharski et al. [14], where calculations using CCSD, CCSDT, CCSDTQ in the cc-pVDZ basis were reported. The distance obtained with the CCSD method deviates signi®cantly from the FCI values, and also the full CCSDT methods predict a distance that is inferior to that obtained by the multi-reference methods. It is only with the inclusion of all triple and quadruple excitations in CCSDTQ that the accuracy of the distance becomes comparable to that obtained by the multi-reference methods. The CCSD(T) method predicts a distance very close to the CCSDT distance with a deviation from This deviation is in line the FCI result of 0.0014 A. with the high accuracy generally reported for the Table 2 The equilibrium distance Req and harmonic frequency xe for N2 in the cc-pVDZ basis obtained using FCI and various correlation methods Model
Req (A)
xe (cm 1 )
FCI MRCISD MRCCSD MRCCSD(4ex) RMRCCSD CCSD CCSD(T) CCSDT CCSDTQ
1.1201 1.1200 1.1202 1.1199 1.1201 1.1128 1.1187 1.1185 1.1198
2323 2324 2324 2328 2325 2408 2349 2346 2328
583
CCSD(T) method [2], but is about an order of magnitude larger than the presented deviations of the multi-reference methods. For the harmonic frequency, the MRCISD, MRCCSD, and RMRCCSD methods all produce values within 2 cm 1 of the FCI value, whereas the MRCCSD(4ex) expansion gives a deviation of 5 cm 1 . For the single-reference calculations, the CCSDT method gives a deviation of more than 20 cm 1 which is twenty times larger then the best multi-reference methods. Even the full CCSDTQ calculation gives a frequency that is about 5 cm 1 too high. It is interesting to note that the calculations including all triple and quadruple excitations (CCSDTQ) and the multi-reference calculation including only selected triple and quadruple excitations (MRCCSD(4ex)) give nearly identical results for the equilibrium distance and frequency. We will return to this in the discussion of the potential curve. 3.3. The potential curve In Fig. 2 we have plotted the deviations from the FCI energy as a function of the inter-nuclear distance for the single- and multi-reference methods. The curves are based on the calculations reported in Table 3. In Table 3 we also report the various non-parallelity errors (NPE) de®ned as the dierence between the largest and smallest deviations from FCI. All the multi-reference methods have deviation curves with nearly zero curvature at the equilibrium distance which explains the accurate frequencies obtained with these methods. The energies obtained with the multi-reference CI method have deviations from the FCI values that are 3±4 times larger than the deviations obtained with the multi-reference coupled cluster method. However, the NPE obtained with the MRCISD method (0.0010 Eh ) is lower than obtained with the RMRCCSD(0.0018 Eh ) and MRCCSD(4ex) (0.0031 Eh ) methods, and is only surpassed by the NPE of the MRCCSD method (0.0006 Eh ). The full MRCCSD method gives the smallest energy deviations with absolute errors of less than 0.002 Eh and the smallest NPE of 0.0006 Eh . The MRCCSD(4ex) deviations increase signi®cantly
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Fig. 2. The potential curves for N2 using various correlation methods and the cc-PVDZ basis. The energies are in hartrees and the inter-nuclear distances are given in atomic units.
when the inter-nuclear distance is increased with errors of more than 0.004 Eh for distances larger than 2.5 a.u. The RMRCCSD curve is closer than the MRCCSD(4ex) curve to the MRCCSD curves, but starts to show signi®cant deviations for distances larger than 3.0 a.u. It is thus only the MRCISD and MRCCSD methods that are able to provide energies of high accuracy over the complete potential curve. It may seem surprising that the RMRCCSD curve is more accurate than the MRCCSD(4ex) curve, as these calculations employ the same set of CC excitations. However, as discussed above, the RMRCCSD calculation uses triple and quadruple excitations obtained from a complete MRCISD calculation, and should therefore be considered as an approximation to the full MRCCSD calculation and not as an approximation to the MRCCSD(4ex) calculation. Not surprisingly, the single reference methods show deviations from FCI that are very geometry dependent, and they have therefore signi®cant NPEs. Thus for the shorter geometries, the CCSD(T), CCSDT, and CCSDTQ methods have
smaller deviations from the FCI energy than the multi-reference methods, but the deviations increase as the bond is stretched and have a maximum around 2.75 a.u. Furthermore, the curvature of the CCSD(T) and CCSDT deviation curves are signi®cant already at the equilibrium distance, which explains why these methods give harmonic frequencies that dier by more than 20 cm 1 from the FCI value. It is interesting to note that the curves for CCSDTQ and MRCCSD(4ex) are nearly parallel. The triple and quadruple excitations included in the MRCCSD(4ex) calculations are thus those that are most dependent on the geometry. On the other hand, the ®ve-fold and higher excitations excluded in the CCSDTQ and the MRCCSD(4ex) calculations are necessary to obtain an NPE of less than 0.001 Eh . 4. Conclusion The equilibrium distance, harmonic frequency, and potential curve for N2 have been investigated
0.0026 0.0099 0.0070 0.0662 0.0018 0.0031 Energies and deviations are in hartrees.
± NPE
0.0010
0.0006 a
CCSDTQ CCSDT
0.000400 0.000844 0.001627 0.001759 0.001839 0.002080 0.002351 0.003631 0.006834 0.01031 0.0081 0.000638 0.001076 0.001565 0.001807 0.001879 0.002180 0.002475 0.003087 0.005182 0.00770 0.0076
CCSD(T) CCSD
0.005949 0.009130 0.013466 0.014099 0.014469 0.015539 0.016682 0.021481 0.031807 0.04519 0.0604 0.0721 0.001788 0.001669 0.001666 0.001671 0.001675 0.001687 0.001702 0.001779 0.001988 0.00226 0.0028 0.0035
RMRCCSD MRCCSD(4ex)
0.001747 0.001640 0.001696 0.001715 0.001728 0.001769 0.001820 0.002104 0.002998 0.00427 0.0047 0.0028 0.001730 0.001579 0.001520 0.001516 0.001515 0.001513 0.001513 0.001528 0.001618 0.00179 0.0020 0.0021
MRCCSD MRCISD
)108.624700 )109.167573 )109.276528 )109.278135 )109.278339 )109.276583 )109.271915 )109.238397 )109.160305 )109.08621 )109.0303 )108.9948
0.005545 0.005733 0.005834 0.005843 0.005849 0.005864 0.005879 0.005937 0.006070 0.00628 0.0065 0.0066
EFCI R (a.u.)
1.500 1.800 2.068 2.100 2.118 2.168 2.218 2.400 2.700 3.000 3.300 3.600
Table 3 FCI energies and deviations from FCI for various wave function approximations as a function of inter-nuclear distancea
0.000022 0.000074 0.000193 0.000215 0.000229 0.000272 0.000323 0.000583 0.001357 0.00241 0.0026 0.0005
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using the cc-pVDZ basis and various correlation methods and compared with FCI. The correlation methods include the single-reference methods CCSD, CCSD(T), CCSDT, CCSDTQ, MRCISD and MRCCSD. For the multi-reference calculations, the smallest complete active space with correct dissociation was employed, 6 electrons in six orbitals 3rg ; 3ru ; 1pg ; 1pu . Including all single and double excitations out from the active space, the CC and CI methods have deviations in the equilibrium distance of and 1 cm 1 in the harmonic frequency. 0.0001 A, Restricting the excitations included in the CC calculations to at most fourfold excitations increases the deviation for the frequency to about 5 cm 1 , whereas freezing the triple and quadruple amplitudes to those derived from the MRCISD calculation does not aect the accuracy of the equilibrium distance. The deviations from FCI are signi®cantly larger for the single-reference methods and only using the CCSDTQ method is a frequency with an error of 5 cm 1 obtained. Thus even for equilibrium properties like bond distances and harmonic frequencies, it may be important to use multi-reference rather than single-reference CC methods. The potential curves obtained with the multi-reference methods are in signi®cantly better agreement with the FCI curve than are the curves obtained with single-reference methods. Although the multi-reference CI curve has deviations that are three times larger than obtained with the full multi-reference CC method, the NPE of these two schemes are comparable (0.0010 and 0.0006 Eh ). The restrictions of the excitation manifold in the MRCCSD calculations to include at most fourfold excitation, produces signi®cant deviations for inter-nuclear distances larger than 2.5 a.u. Freezing the triple and quadruple excitations to those deduced from the MRCISD calculations produces a curve that is more parallel to the FCI curve than obtained by actually re-optimizing the triple and quadruple excitations in the CC calculations including at most quadruple excitations. This is not surprising as the frozen triple and quadruple excitations originated from MRCISD calculations that include all single and double excitations from the active space.
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The present paper demonstrates that we now have coupled cluster methods that can describe complete potential curves. However, as the number of parameters present in the full MRCCSD calculations corresponds to the number of parameters in the full MRCISD calculations, the present approach is not easily extended to large molecules. The restriction to include only triple and quadruple excitations degrades signi®cantly the accuracy of the potential curves. Alternative restrictions of the active space that allow correct dissociation are included in the GAS formalism, and may give better potential curves. An alternative possibility is to use an internal contracted formalism. Work on developing an internally contracted formalism using commuting operators spanning the full ®rst-order interacting subspace is currently in progress. Acknowledgements This work has been supported by the Danish Research Council (Grant No. 9901973). References [1] R.J. Bartlett, in: D.R. Yarkony (Ed.), Modern Electronic Structure Theory, Part I, World Scienti®c, Singapore, 1995, p. 1047. [2] T. Helgaker, P. Jùrgensen, J. Olsen, Molecular ElectronicStructure Theory, Wiley, New York, 2000. [3] J. Paldus, X. Li, Adv. Chem. Phys. 110 (1999) 1.
[4] T.H. Dunning Jr., K.A. Peterson, D.E. Woon, in: P.v.R. Schleyer, N.L. Allinger, T. Clark, J. Gasteiger, P.A. Kollman, H.F. Schaefer III, P.R. Scheiner (Eds.), The Encyclopedia of Computational Chemistry, Wiley, New york, 1998. [5] J.M.L. Martin, Chem. Phys. Lett. 259 (1996) 669. [6] W. Klopper, K.L. Bak, P. Jùrgensen, J. Olsen, T. Helgaker, J. Phys. B: At. Mol. Opt. Phys. 32 (1999) R103. [7] N. Oliphant, L. Adamowicz, J. Chem. Phys. 94 (1991) 1229. [8] N. Oliphant, L. Adamowicz, J. Chem. Phys. 96 (1992) 3739. [9] P. Piecuch, N. Oliphant, L. Adamowicz, J. Chem. Phys. 99 (1993) 1875±1900. [10] P. Piecuch, N. Oliphant, L. Adamowicz, J. Chem. Phys. 100 (1994) 5792±5809. [11] P. Piecuch, S.A. Kucharski, V. Spirko, J. Chem. Phys. 111 (1999) 6679±6692. [12] P. Piecuch, S.A. Kucharski, R.J. Bartlett, J. Chem. Phys. 110 (1999) 6103. [13] X. Li, J. Paldus, Chem. Phys. Lett. 286 (1998) 145. [14] S.A. Kucharski, J.D. Watts, R.J. Bartlett, Chem. Phys. Lett. 302 (1999) 295. [15] J. Olsen, J. Chem. Phys. 113 (2000) 7140. [16] S. Hirata, R.J. Bartlett, Chem. Phys. Lett. 321 (2000) 216. [17] M. Kallay, P.R. Surjan, J. Chem. Phys. 113 (2000) 1359. [18] H. Larsen, J. Olsen, P. Jùrgensen, O. Christiansen, J. Chem. Phys. 113 (2000) 6677. [19] G.E. Scuseria, C.L. Janssen, H.F. Schaefer III, J. Chem. Phys. 86 p. 2881. [20] J. Noga, R.J. Bartlett, J. Chem. Phys. 1987 (1987) (7041) 3401(E). [21] B.O. Roos, Adv. Chem. Phys. 69 (1987) 399. [22] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007. [23] X. Li, J. Paldus, J. Chem. Phys. 107 (1997) 6257. [24] W. Kutzelnigg, Theoret. Chim. Acta 80 (1991) 349. [25] K. Raghavachari, G.W. Trucks, J.A. Pople, M. HeadGordon, Chem. Phys. Lett. 157 (1989) 479.