Benchmark full-CI calculation on C2H2: comparison with (SC)2-CI and other truncated-CI approaches

22 May 1998

Chemical Physics Letters 288 Ž1998. 348–355

Benchmark full-CI calculation on C 2 H 2 : comparison with ž SC/ 2-CI and other truncated-CI approaches Nadia Ben-Amor a , Stefano Evangelisti b, Daniel Maynau a , Elda Rossi a

c

Laboratoire de Physique Quantique, URA 505, UniÕersite´ Paul Sabatier, 118, Route de Narbonne, 31062 Toulouse CEDEX, France b Dipartimento di Chimica Fisica e Inorganica, UniÕersita` di Bologna, Viale Risorgimento 4, I-40136 Bologna, Italy c CINECA, Via Magnanelli 6 r 3, I-40033 Casalecchio di Reno (BO), Italy Received 26 January 1998; in final form 9 March 1998

Abstract Large-scale full configuration interaction ŽFCI. calculations are presented on the acetylene molecule using several basis sets. The largest calculation involves more than five billion symmetry-adapted Slater determinants. The FCI results are compared with those obtained using the ŽSC. 2-CI method, that corrects the size-consistency defect of truncated CI schemes. A comparison with other truncated-CI calculations, of both contracted and uncontracted type, is also done. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction The improvements in the power of computers and in the ab initio algorithms have allowed the study of chemical problems of always increasing size. Among the various methods developed in quantum chemistry, the full configuration interaction ŽFCI. method has known large technical improvements due to direct configuration interaction ŽCI. techniques w1,2x, and molecular species of chemical interest can now be studied using non-minimal basis sets. Some potential curves are available in the literature, such as NH w3x, N2 , NO and O 2 w4x, LiF w5x, Be 2 , w6x, BH, w7x, and a few others. When the molecule is not diatomic, as in the FCI study of CH 3 w8x, it is possible to obtain a potential curve as a function of a homothetic deformation of the coordinates. Several different implementations of the FCI method have been proposed w9,10x and this has led to the possibility of calculations of large dimension

w11,12x. In recent years, an FCI algorithm has been developed and implemented in Bologna w13x. The algorithm has been used for the study of ab initio and semi-empirical Hamiltonians. It is especially suitable for an implementation on massive parallel computers w14x, where huge calculations can be performed. The FCI results are particularly interesting to calibrate and test more approximated methods. In particular, the potential curves obtained from the binding distance to infinity are useful to test the size-consistency error, which is usually associated with CI calculations. This Letter is devoted to a comparison between the recently developed sizeconsistent self-consistent configuration interaction ŽSC. 2-CI method and FCI, in the case of the acetylene molecule. We have chosen C 2 H 2 for this comparison because two different bond stretchings can be studied by varying separately the C–C and C–H distances. Due to the factorial growth of the FCI space, the

0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 2 8 9 - 9

N. Ben-Amor et al.r Chemical Physics Letters 288 (1998) 348–355

basis sets that can be used in FCI calculations are obviously smaller than those used for more approximate methods. However, the recent progresses in FCI techniques allows the use of quite ‘‘reasonable’’ basis sets, so that comparison with experimental results becomes possible. The basis sets used in this study are derived from an atomic natural orbital ŽANO. contraction. For the smallest calculation we used a w3s2 p x contraction for C and w2 s x for H, indicated as w3s2 pr2 s x. A larger one was obtained by including a d function for each carbon atom, w3s2 p1dr2 s x. Finally, for the largest calculation we were able to perform, a third s orbital was added on each hydrogen, so to have a w3s2 p1dr3s x contraction. For the last basis set the size of the FCI space is more than 5 = 10 9 symmetry-adapted Slater determinants. We performed a single calculation near the equilibrium geometry with this basis set, while the study of C 2 H 2 for various interatomic distances was done using the intermediate basis w3s2 p1dr2 s x.

2. Summary of the employed methods 2.1. FCI We used the parallel version of our FCI algorithm, which is adapted for CRAY T3D and T3E computers. In its present implementation, the method requires only two arrays ŽX and Y. of the dimension of the FCI space: one for the coefficients of the wavefunction
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to perform large calculations, because the storage on disk of several vectors would require an unreasonable amount of space. Moreover, the InputrOutput time would become extremely long. A complete account of the parallel implementation of the algorithm, with a detailed description of the technical solutions adopted to perform the acetylene calculation, can be found in Ref. w17x. We briefly report here some technical results concerning our largest FCI. The calculation was run on the 128-processor CRAY T3E of CINECA, with a global memory of 2048 Mwords Ž16 Mwords per processor.. Only two symmetry blocks at a time, one of X and one of Y, are resident in memory, split over the different processors. Since for a singlet state both X and Y are symmetric arrays, the total disk occupation is of about 40 Gbytes, with a global IrO per iteration of more than 250 Gbytes. The wall-clock time is about 33000 seconds per iteration, mainly because of the huge IrO load. In fact, the cpu time needed to produce Y from X is about 12000 seconds, while 21000 seconds are spent in IrO operations. In order to reduce the IrO time, we computed an accurate guess by selecting those components of X and Y that are larger than given thresholds. In this way, the real FCI process actually reduces to a small number of iterations, while the preliminary iterations needed to compute the guess require limited IrO resources. For instance, by selecting those components of X whose absolute value is larger than 10y8 , the IrO time reduces to about 1000 seconds only. 2.2. (SC) 2-CI To clearly understand what is the aim of the ŽSC. 2-CI method w18x, let us consider two aspects of the approximation due to the truncation of the FCI matrix. The first aspect is that the energy obtained by the diagonalization of the truncated matrix is not equal to the FCI result and the second one is that even basic properties like size-consistency are no more verified. Obviously, the first drawback cannot be corrected and is the price to pay for easier Žor simply, feasible. calculations. Quite fortunately, the chemists are in general not interested with total energies and this defect is not really crucial. The second consequence of the truncation is much more

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problematic, because for large systems the size-consistency error may be large. Fortunately again, this defect is not impossible to correct. The goal of the ŽSC. 2-CI formalism only concerns this second aspect. For the sake of simplicity, we shall consider the case of the ŽSC. 2-CI dressing of a single reference configuration of the singly and doubly excited determinants ŽSDCI.. The exact FCI wavefunction
A Di value does exist, even if we do not know it, which gives the FCI projection onto the SD space. Di can be added to the first H0 i term. In this case the dressing is called ‘‘column dressing’’. As an alternative, the quantity Dirc j can also be added to any other Hi j term of line i. In the ‘‘diagonal’’ dressing, it is the diagonal matrix element to be modified and the correction is written

Di i ' Dirc i s Ž c i .

y1

Ý Hi ca a

Ž 7.

a


Ž 1.

a

j

where Ø <0: is a reference determinant Že.g. the Hartree– Fock determinant. Ø the index jruns over the singly- and doubly-excited determinants ŽSD space. Ø the index a runs on the outer space i.e. over triply-, quadriply-, etc. excited determinants. For the line iof a singly or doubly excited determinant, the molecular Schrodinger equation H
Ž 3.

i

and Hi0 q Ý Hi j c j s Ec i

Ž 4.

j

The c i coefficients that solve Eqs. Ž4. are obtained by the diagonalization of the SDCI matrix. We shall now demonstrate that a small modification Ža ‘‘dressing’’. of the Hamiltonian would permit to obtain the coefficients c i of the FCI. Indeed, if we write

Di ' Ý Hiaca

Ž 5.

a

Eq. Ž2. may be re-written as Hi0 q Ý Hi j c j q Di s Ec i j

Ý Hi j c j q Ž Hi i q Di i . c i s Eci

Ž 8.

j/i

In general, the FCI solution is unknown and therefore it is not possible to find the Di i that give exactly the FCI result, but it is possible to improve the SDCI results. The exact Di i cannot be obtained by Eq. Ž7., since it is impossible to know the exact ca s without resolving the FCI problem. However, approximate values can be found: by using the coupled cluster formula ca s Ý c i c j

where the indices j and a run respectively over the SD and outer spaces, as for Eq. Ž1.. In a SDCI, Eqs. Ž1. and Ž2. may be re-written, without the sum over the outer space:
H0 i q

Ž 2.

a

j

which gives

Ž 6.

Ž 9.

i, j

Žwhere < a : is obtained by the product of the excitaq < : < : tion operators Dq i and Dj giving i and j from <0:., Eq. Ž7. gives Di i and the resolution of Ž8. gives the CCSD wavefunction w19x. If one considers two fragments without interaction and using localized orbitals, only one c i c j product of Eq. Ž9. does not vanish. For this non-zero product, q Dq i and Dj are localized on each fragment. In this case ca s c i c j

Ž 10 .

If one uses relation Ž10. in Eq. Ž7., the only thing we can say about the Di i dressing is that the wavefunction of the two fragments without interaction is a product of the wavefunctions of the fragments and therefore the energy of the supersystem is the sum of the energies of the separated fragments. No more, but it is sufficient to ensure that the asymptotes of the potential curves are correctly positioned and therefore the whole curve has a correct behavior. The ŽSC. 2-CI formalism has no other goal.

N. Ben-Amor et al.r Chemical Physics Letters 288 (1998) 348–355

In the case of a multi-reference CI ŽMRCI. with the single reference dressing, the formalism is exactly the same. The SD space is replaced by a model space obtained by all the single and double excitations on the MR or CAS space. Some difficulties appear in defining the outer space, but the Di i diagonal dressing is obtained in the same way. The resolution of Eq. Ž8. gives a more accurate wavefunction, since the model space has been enlarged.

3. Computational details The ANO basis sets optimized by Widmark et al. w20x were used in the present work. The contraction coefficients were the standard ones, reported in Ref. w20x. Calculations at different geometries were done with the w3s2 p1dr2 s x contraction Žon carbon and hydrogen, respectively. of the ANO basis set. This corresponds to a FCI space of 2538565366 symmetry-adapted Slater determinants in D 2 h symmetry. At the equilibrium geometry only, a calculation was also done by adding a third ANO contraction on each hydrogen atom, which gives a w3s2 p1dr3s x basis set. In this case, the FCI space contains 5069065864 Slater determinants. For completeness, also a smaller basis set, the w3s2 pr2 s x ANO contraction, was used at the equilibrium geometry. For the largest calculation, the geometry of the molecule was chosen with C–C and C–H bond lengths of 2.33 and 2.05 bohr, respectively. These ‘‘reference’’ values are close to the equilibrium values obtained at the contracted-MRCI level using the same basis set. Therefore, they should be close to the FCI equilibrium values with this basis set. The symmetry was assumed to be D`h , i.e. only symmetric stretches of the bonds were allowed. The actual group used in full- and truncated-CI calculations was the D 2 h subgroup of the full group of the molecule. With a smaller ANO basis set, the w3s2 p1dr2 s x contraction, we computed four points at the FCI level, corresponding to 1.0 and 1.5 times the reference C–C and C–H bond lengths. Therefore, C–C bonds of 2.330 and 3.495 bohr and C–H bonds of 2.050 and 3.075 bohr were considered. Geometries with the atoms at infinite distance were also investigated. At the FCI level, this was obtained by taking

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advantage of the size extensivity of the method, i.e. by adding the energies of the separate fragments. Truncated CI methods are notoriously not size extensive and a calculation on the complete system at large distances between the fragments is required. For practical reasons, it is not advisable to use interatomic distances larger than a few tens of bohr, because this could give rise to convergency problems. In the present work, for infinite C–C and C–H distances we chose values of 100 and 40 bohr, respectively. Since the system separates into neutral fragments, such distances are certainly sufficient to have energies stable beyond 10y6 hartree. We studied different geometries of the form w R CC rR CH x: by indicating with w R CC rR CH x the reference ‘‘equilibrium’’ geometry Žwith R CC s 2.33 and R CH s 2.05 bohr, respectively., we studied the nine geometries

w R CC rR CH x ' j CC R CCrj CH R CH

Ž 11 .

with

j CC ,

j CH s 1.0, 1.5, `

We note that this choice of the geometries is totalsymmetric, i.e. does not destroy the D`h symmetry of C 2 H 2 . The zeroth-order description of the wavefunction is done through a complete-active space self-consistent field ŽCAS-SCF. calculation. Since we are interested in the breaking of all the five bonds of the molecule, all the molecular orbitals coming from the hydrogen 1 s and carbon 2 s, 2 p atomic orbitals must belong to the active space. This corresponds to an active space of ten electrons in ten orbitals, indicated as CASŽ1010.. The two lowest molecular orbitals Žone of sg and one su symmetry, respectively, corresponding to the atomic 1 s orbitals of carbon. do not belong to the active space, i.e. they are optimized but kept doubly occupied at the CAS-SCF level. At CI level, they are frozen and therefore excluded from the calculation. This means that no inactive electrons exist in the truncated-CI calculations. This Letter is mainly devoted to the comparison of the size-extensive ŽSC. 2-CI results with the exact FCI values. The MRCI results, obtained from all single- and double-excitations from the CAS space, are also reported. For the sake of completeness, we

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performed the same calculation with the internallycontracted MRCI approach ŽC-MRCI., developed by Werner and Knowles w23x and implemented in the ab initio program MOLPRO w24x. For the C-MRCI case, the Davidson and Pople corrections, based on general considerations about the effect of quadruple excitations on the total energy, are also indicated.

Table 2 CAS-SCF and Full-CI total energies Žhartree. and energy differences with respect to FCI Žmhartree.: different geometries, using the w3s2 p1d r2 s x ANO contraction

j CH

1.0 1.0 ECA S - SCF E FC I

Table 1 Size of the different CI spaces: w3s2 pr2 s x, w3s2 p1d r2 s x and w3s2 p1d r3s x basis sets Method

w3s2 pr2 s x

w3s2 p1d r2 s x

w3s2 p1d r3s x

CAS-SCF CAS-SCF Žconf. C-MRCI Žconf. MRCI MRCI Žconf. FCI

8152 2640 38894 1033472 252702 30047616

8152 2640 75860 3916472 932492 2538565366

8152 2640 83754 4716000 1122298 5069065864

1.5 ECA S - SCF E FC I

ECA S - SCF E FC I

`

122.520 6.561 y0.467 0.730 4.047 1.702

133.713 7.553 y1.914 y0.369 4.766 2.611

y76.816479 y76.601277 y76.530943 y76.928086 y76.726574 y76.663743

CAS-SCF 111.610 C-MRCI 5.577 C-MRCI q Davidson y0.045 C-MRCI q Pople 0.934 MRCI 3.579 ŽSC. 2 -CI 1.275 `

1.5

y76.977833 y76.743413 y76.610593 y77.093711 y76.865936 y76.744311

CAS-SCF 115.883 C-MRCI 5.325 C-MRCI q Davidson y0.126 C-MRCI q Pople 0.834 MRCI 3.695 ŽSC. 2 -CI 1.346

4. Results and discussion The present study obviously does not intended to obtain an accurate description of acetylene. However, it is important to compare our results with the experimental values, in order to check the quality of the basis set. In particular, the experimental energy associated to the breaking of the C–C triple bond is 230 kcalrmol w21x, while we obtain 219 kcalrmol. Although in our value the relaxation of the C–H bond length is not taken into account, we believe that a difference of only 11 kcalrmol is acceptable. A second check is about bond lengths. A rough geometry optimization done at the C-MRCI level gives bond a C–C bond of 2.33 bohr and a C–H bond of 2.05 bohr, while the experimental bond lengths for acetylene are 2.273 and 2.003 bohr, respectively w22x. Both lengths are too long by about 0.05 bohr. Table 1 reports the size of the CI matrices in the different cases. Due to the exponential growth of the FCI space, its size increases fast with the dimension of the basis set. The largest FCI space contains more than five billion determinants. The corresponding MRCI space, although much smaller, is still quite large, being of almost five million Slater determinants. The contraction has a dramatic effect, reduc-

j CC

125.290 7.645 y0.411 0.929 4.320 1.858

132.797 7.774 y2.092 y0.501 4.416 2.368

y76.626979 y76.485598 y76.406144 y76.723266 y76.615155 y76.527769

CAS-SCF 96.291 C-MRCI 3.731 C-MRCI q Davidson y0.534 C-MRCI q Pople 0.224 MRCI 3.059 ŽSC. 2 -CI 1.236

129.562 6.752 y1.916 y0.484 3.937 2.304

121.628 6.287 y2.476 y1.051 4.135 2.731

ing the dimension to less, about 80000 configurations. In Table 2 the energy differences with respect to FCI, for different geometries and with the 3s2 p1dr2 s basis set, are shown. Since the FCI results can be used as benchmarks for future calculations, the CAS-SCF and FCI total energies are also reported in the table. With respect to MRCI, the ŽSC. 2-CI results are closer to FCI by about a factor of two. Even if the ŽSC. 2-CI formalism does not guarantee an upper bound to the FCI energy, ŽSC. 2-CI results always lie above the FCI. This is not surprising, since many linked contributions, due to the more excited configurations of the CI matrix, are lacking.

N. Ben-Amor et al.r Chemical Physics Letters 288 (1998) 348–355

The Davidson and Pople corrections give total energies closer to the FCI result, but the values are not so regular when the geometry changes. As a consequence, the potential curves would be less parallel to the FCI result than ŽSC. 2-CI. In particular, if we consider the maximum variation with respect to FCI, we obtain for ŽSC. 2-CI a value of 1.495 mhartree for the nine geometries we investigated. On the same geometries, both Davidson and Pople corrections give a larger variation: 2.350 ŽDavidson. and 1.884 ŽPople.. Moreover, Davidson correction tends to overcorrect the SDCI results. As shown in Table 3, this happens especially for the largest basis sets. In Table 3 we present energies obtained at the equilibrium geometry with the three ANO contractions w3s2 pr2 s x, w3s2 p1dr2 s x and w3s2 p1dr3s x. For each basis set we report the CAS-SCF and FCI total energies and the energy differences with respect to FCI. For what concerns the comparison between different methods as a function of the basis set, we note that adding a third s orbital to the hydrogen basis does not change the results. At the MRCI level, the error with respect to the FCI energy is much larger for the two large basis sets. This error is largely corrected when the ŽSC. 2-CI correction is included. For this geometry, the Davidson and Pople correction give results closer to the exact value, even if the Davidson energies lie below the FCI. An interesting product of large-size FCI calculations is the opportunity to study the dispersion of the coefficients in the FCI vector. We report in Table 4

Table 3 CAS-SCF and Full-CI total energies Žhartree. and energy differences with respect to FCI Žmhartree.: equilibrium geometry with the three ANO basis sets w3s2 pr2 s x, w3s2 p1d r2 s x and w3s2 p1d r3s x Method

w3s2 pr2 s x w3s2 p1d r2 s x w3s2 p1d r3s x

ECA S - SCF EFC I

y76.959872 y76.977833 y76.983786 y76.998410 y77.093711 y77.103980

CAS-SCF C-MRCI C-MRCI q Davidson C-MRCI q Pople MRCI ŽSC. 2-CI

38.538 1.103 0.235 0.399 0.664 0.336

115.878 5.327 y0.131 0.829 3.695 1.346

120.194 5.657 y0.155 0.864 3.990 1.434

353

Table 4 Population of the FCI vector with the w3s2 p1d r3s x basis set, equilibrium geometry Class Threshold 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

00

Population

0.1=10 3 0.1=10y0 1 184 0.1=10y0 2 3094 0.1=10y0 3 67071 0.1=10y0 4 894068 0.1=10y0 5 8831527 0.1=10y0 6 63165211 0.1=10y0 7 290400449 0.1=10y0 8 865918188 0.1=10y0 9 1493284047 0.1=10y1 0 1415352464 0.1=10y1 1 694274827 0.1=10y1 2 188322088 0.1=10y1 3 40172592 0.0 8380051

total

5 51 1.17347732 3.25859248 8.20274348 14.48635690 22.81574050 22.67406962 17.79195074 8.80592742 2.88300373 0.55440304 0.05890022 0.00323100 0.00009369 0.00000201 0.00000004

5 52 0.88691345 0.07526284 0.03222839 0.00461254 0.00088611 0.00008869 0.00000757 0.00000040 0.00000001 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000

5069065864 102.70849219 1.00000000

the dispersion of the FCI vector in the case of the w3s2 p1dr3s x basis set. The table shows that the number of small coefficients is extremely large, reaching its maximum in the 10y8 –10y9 class. For each class of coefficients, we computed the contribution to the norm of the vector, defined as 5 X 5 2 ' Ý < CI < 2 I

As one could expect, the largest contribution to the 5 X 5 2 is concentrated in the highest class and goes below 10y8 for thresholds smaller than 10y8 . We computed also the sum 5 X 5 1 of the absolute values of the coefficients, 5 X 5 1 ' Ý < CI < I

With respect to 5 X 5 2 , 5 X 5 1 is much more dispersed, reaching its maximum in the 10y4 –10y5 class and going slowly to zero. The dispersion of the FCI vector as a function of the bond stretchings is reported in Table 5 for the smaller w3s2 p1dr3s x basis set. Only results corresponding to the four geometries for which we explicitly performed a FCI calculation are shown in the table. The dispersion of the FCI vector is strongly related to the length of the

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Table 5 Population of the FCI vector with the w3s2 p1d r2 s x basis set, four different geometries, corresponding to different values of the w j CC r j CH x pairs ClassThreshold w1.0r1.0x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 total

0.1=10 00 0.1=10y0 1 0.1=10y0 2 0.1=10y0 3 0.1=10y0 4 0.1=10y0 5 0.1=10y0 6 0.1=10y0 7 0.1=10y0 8 0.1=10y0 9 0.1=10y1 0 0.1=10y1 1 0.1=10y1 2 0.1=10y1 3 0.0

w1.0r1.5x

3 173 2948 64662 807799 7742995 51317088 212503407 560298064 800284289 612857144 232901869 48847206 9333407 1604312

w1.5r1.0x

4 179 4060 83362 1083434 10029037 63484282 244208067 606745702 789246779 563889393 210151560 42347665 6449816 842026

w1.5r1.5x

8 192 6215 100538 1478167 13102159 82929652 314815693 705375267 819097989 452239927 110895541 16152751 6992904 15378363

9 280 7842 154150 2037842 18144349 104759985 371832448 754592583 790723192 382141995 80260023 9407553 1502387 23000728

2538565366253856536625385653662538565366

C–C bond. This is not surprising because of the nature of the bond that implies a strong correlation for a correct description.

5. Conclusion We have shown that it is now possible to study small polyatomic molecules at the FCI level. Although the size of the matrices to be diagonalized is large, we did not face particular convergence problems. The comparison with experiment shows that such calculations are not optimal to study real chemical problems. However, it is now possible to use non minimal basis sets, with physically relevant results. The ŽSC. 2-CI calculations show an improvement of the results with respect to MRCI. The aim of the ŽSC. 2-CI formalism is to correct size-consistency errors, so that studies concerning small systems which can be treated at the FCI level are not the most demonstrative. Direct comparisons with experimental values for larger molecules show that the ŽSC. 2-CI correction is highly efficient w25x

Acknowledgements This work was supported by the EEC within the ‘‘Human Capital Mobility Program’’, contract No. ERBCHRXCT96r0086. We wish to thank CINECA for a generous supply of computer time on the CRAY T3D and T3E.

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