Convergence of the coupled-cluster singles, doubles and triples method

Volume 145. number 6

CHEMICAL PHYSICS LETTERS

22 April 1988

CONVERGENCE OF THE COUPLED-CLUSTER SINGLES, DOUBLES AND TRIPLES METHOD *

Gary W. TRUCKS, Jozef NOGA ’ and Rodney J. BARTLETT Quantum Theory Project, Departments of Chemistry and Physics, Universityof Florida, Gainesville, FL 32611, USA Received 29 January 1988

The convergence of coupled-cluster equations for several cases, CCD, CCSD, CCSDT-n and the full CCSDT is investigated. Comparisons are made between the reduced linear equation (RLE) method for accelerating convergence and simple geometric extrapolation techniques, and between energy and wavefunction convergence criteria.

With the emergence of coupled-cluster theory [ 1,2] among the most accurate quantum-chemical methods for electron correlation, the question of the convergence of the CC equations is important. CCD (coupled-cluster doubles) [ l-4 1, CCSD (singles and doubles) [ 5 1, and various methods for including triples (CCSDT-n) [6-81 and the full CCSDT [9] method have now been developed. Asymptotically, these methods have respectively an N n 6, - n6, - n’ and N n8 dependence on the number of basis functions times the number of iterations required to reach convergence. Since it is not iterative, some [ lo] argue that finite-order approximations to CC theory like MBPT, are preferable to converged CC approaches despite the greater accuracy of the latter [9]. In order to combine the considerable advantages of infinite-order CC method without requiring too much time, it is requisite to converge the CC equations rapidly. A thorough study of this problem including some systems intentionally chosen for their poor convergence has already been presented [ 111. That paper proposed a new approach termed the reduced linear equation (RLE) technique to accelerate the convergence of the CC equations, a method we have used successfully for some years [ 12- 161. Other investigators have also developed effective * This work has been supported by the US Air Force Offrce of Scientific Research.

’ Permanent address: Institute of Inorganic Chemistry, Center for Chemical Research, Slovak Academy of Sciences, 842 36 Bratislava, Czechoslovakia.

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techniques for converging CC equations [ 17 1. The convergence of the singles and doubles coupled-cluster (CCSD) model was also the topic of a recent paper by Scuseria, Lee and Schaefer (SLS) [ 18 ] which, despite a footnote to the contrary, implies that their approach converges faster than others. Furthermore, there are suggestions that CCSD might have different convergence properties than CCD, even though a detailed comparison using the reduced linear equation (RLE) method to converge CCD and the CCSD-1 approximation to CCSD, which included the most important effects of single excitations, showed no important differences [ 141. Although not explicitly considered in that paper, for many years we have been using the RLE with great success for the full CCSD [ 12-141, CCSDT-1 [ 15,161, and now multireference CC approaches [ 19 1. To ‘eliminate any further questions about the effectiveness of our convergence procedures, in this note, we consider the convergence of various approximations of the CC equations that include triple excitations, including the full CCSDT method to demonstrate what can be expected from converging the CC equations in the most general cases. We employ the reduced linear equation (RLE) method and an alternative geometric extrapolation. In addition, we demonstrate the convergence due to RLE is at least equal to that recently discussed by SLS. The approach employed by SLS is the direct inversion of the iterative subspace (DIIS) method of Pulay [ 20,2 11, which in its second version [ 2 1 ] is equiv-

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alent to RLE for most practical purposes. In the context of eigenvalue equations, the fundamental idea of all reduced space techniques seems to be due to Lanczos [ 221, who effected a partial diagonalization of a large matrix by transforming to a much smaller set of functions built upon a Krylov sequence, followed by a matrix triangularization technique. A variational perturbation approach for large scale eigenvalue problems for one or several states was presented by Bartlett and Brandas [23]. This approach, termed the reduced partitioning procedure, demonstrated the equivalents of classes of Pad& approximants and variational calculations in a reduced space. Furthermore, this work proved that functions proportional to H”&, like perturbation theory, spanned the space in an optimum (steepest descent) manner. Later, the Davidson method [ 241 improved upon the convergence of the basic Lanczos technique for eigenvalue problems. Work by Hegarty and Robb [25] and Roos and Siegbahn [26] have considered the prior developments. The same strategy is possible for linear equations, as perhaps first used by Pople et al. in the implementation of an SCF second-derivative theory [ 271. Building upon the reduced partitioning technique for eigenvalue problems, Purvis and Bartlett introduced the RLE [ 111 method, which could be used to solve the coupled-cluster equations in a quasi-linearized form and demonstrated its convergence properties, comparing the RLE to Padt approximants among other options. Unlike the work of Pople et al. [ 271 and the original DIIS method [ 201, which were both developed in the context of SCF theory, the RLE was explicitly developed for correlated methods and for CC methods, in particular. Wormer et al. have since recognized the RLE method as a pre-conditioned conjugate gradient approach [ 281. Following the development of the RLE, additions to the original DIIS procedure [ 211 incorporated a new error vector which made use of available quantities instead of a requiring the construction of an additional Fock matrix for the error vector and addressed the criticism of the earlier DIIS procedure that the DIIS equations were only solved periodically. Consequently, the difference between the RLE and DIIS appears to be details such as the choice of error vector or how many iterations are made before solving the reduced equation; and there should be no significant improve-

22 April 1988

ment in converging CC equations by the DIIS approach compared to our RLE method. A detailed discussion of the reduced linear equation method for solving general coupled-cluster equations has been presented elsewhere [ 111, and will not be repeated here. Briefly, in the RLE method, approximations obtained from each iteration are saved. These quantities serve the analogous role to the Krylov sequence of perturbation functions in other variants of the “reduced” space method [ 23 1, and are used in subsequent iterations to construct a subspace onto which the quasi-linear equation is projected and in which the projected system of equations is solved. The convergence of the DIIS method is based on the construction of error vectors, chosen by SLS to be the difference of subsequent amplitudes, that approximate the zero vector in the least-squares sense. RLE differs from DIIS by requiring that the best leastsquares approximation to the true error vector within the basis vanish. This can occur only if the reduced linear equation approximates the zero vector; therefore, the error vector is never explicitly considered. A more detailed discussion is given in ref. [ 111. In our implementation of the RLE method we routinely solve the reduced equation after each iteration while SLS allow the system to “relax” for n consecutive iterations in which a pure iterative procedure is used. It should be noted that the first DIIS scheme [20] was proposed to be used in this manner; however, periodic extrapolation was the main objection against the original algorithm and was addressed in the later improved version of DIIS [ 2 1 ] as is used in the RLE procedure [ II]. The number of vectors incorporated is flexible but we normally default to six. In the following, we adjust our reduced linear equation to maintain a maximum of eight vectors to be consistent with SLS. Either choice usually avoids the occurrence of singularities. SLS reported correlation energy differences to indicate convergence. Just as in SCF theory, a more reasonable criterion would be based on the accuracy of the wavefunction. That is the method that we adopt and it is reflected in the convergence of the t amplitudes. Nevertheless, we report both for comparison purposes with SLS. Table 1 compares convergence between our CCSD, SLS, and results from the Comenius code [ 29 1. The 549

Volume 145, number 6 Table 1 Convergence

CHEMICAL PHYSICS LETTERS

22 April 1988

ofCCSD model ‘). All energies are in hartree

Molecule

Basis

Number of basis functions

ESCFb'

Eec.rrC'

CCd'

NC'

Number of iterations

HB

DZ

12

-25.113673922

-0.073080 -0.0730804 -0.0730802 ‘) -0.073d803716 -0.073080 -0.0730803 ‘) - 0.0730804

e e e e t t t

6 7 7 10 6 7 7

9 11 (SLS: 11) 17 16 12 20 13

H2Be

DZ

10

- 15.536467049

-0.087723 -0.0877228 -0.0877230 f, -0.0877229973 -0.087723 -0.0877230 f, -0.0877230

e e e e t t t

6 7 7 10 6 7 7

12 13 (SLS: 14) 18 22 (SLS: 22) 14 27 16

HsO

DZ

14

- 76.009837606

-0.146238 -0.1462381 -0.1462376” -0.1462381202 -0.146238 -0.1462381 f, -0.1462381

e e e e t t t

6 7 7 10 6 7 7

8 10 (SLS: 11) 8 15 (SLS: 15) 11 14 14

DZP

26

-76.044053872

-0.219449 -0.2194490 -0.2194490” -0.2194490546 -0.219449 -0.2194490 f, -0.2194490

e e e e t t t

6 7 7 10 6 7 7

9 10 (SLS: 10) 11 14 (SLS 16) 12 14 13

a) Basis sets and geometries are the same as SLS [ 11 except for DZP Hz0 since their reported reference leads to marginally different energies. Also, SLS denote that basis set for H,Be contains 14 functions, in fact it contains 10. b, SCF energies are converged to lO_“. c, The reduced linear equation method is used unless otherwise indicated. d, Convergence criterion, where e indicates that the correlation energy differences were chosen and t indicates that the t amplitudes were chosen. e, 10VNis the tolerance for convergence. ‘) Geometric extrapolation is used.

latter incorporates a geometrical extrapolation and we have chosen the default parameters. In each example, comparing our data to that of SLS, the convergence of the coupled-cluster equations are approximately the same, our implementation converging an iteration or two faster in some cases, probably due to our choice of solving the reduced linear equation approximation to the amplitudes after each iteration. Inclusion of the Comenius CCSD indicates that the RLE method is preferable in most cases. The 550

RLE converged much more rapidly for the slowly convergent cases (HB and H,Be), for example. Table 2 contains convergence information for the CCD model. Comparison of table 2 with table 1 demonstrates that inclusion of T, has little effect on the convergence rate for a coupled-cluster wavefunction that is built upon an SCF reference. The convergence of well-behaved examples (Hz0 DZ and Hz0 DZP) are only one to two iterations different. Even using more general (non-SCF ) reference func-

Volume 145, number 6

CHEMICAL PHYSICS LETTERS

22 April 1988

Table 2 Convergence of CCD model. All energies are in hartree Molecule

Basis

Number of basis functions

E SCF

E corr

cc a’

Nb’

Number of iterations

HB

DZ

12

-25.113673922

-0.072487 -0.0724871 -0.0724871345 -0.072487 -0.0724871

e e e t t

6 7 10 6 7

9 10 13 12 13

HaBe

DZ

10

- 15.536467049

-0.085532 -0.0855320 -0.0855320094 -0.085532 -0.0855320

e e e t t

6 7 10 6 7

12 15 21 15 17

Hz0

DZ

14

- 76.009837606

-0.145435 -0.1454351 -0.1454351765 -0.1454352 -0.14543518

e e e t t

6 7 10 6 7

8 9 13 11 12

Hz0

DZP

26

- 76.044053872

-0.218489 -0.2184895 -0.2184896122 -0.218490 -0.2184896

e e e t t

6 7 10 6 7

8 9 13 11 12

‘) Convergence criterion, where e indicates that the correlation energy differences were chosen and t indicates that the t amplitudes were chosen. w 1O-“’is the tolerance for convergence.

tions such as a crude reference composed of simple bond-orbital functions we have typically found that RLE converges CCSD very well [ 301. Since the implementation of CCSD, we have developed a series of approximations of CCSDT termed CCSDT-n [ 8,311, and the full-CCSDT model for closed-shell systems [ 91. Table 3 presents the correlation energies and convergence information for various approximations up to the full CCSDT. It should be noted that except for CCSDT- la the RLE method is not used for the approximations of CCSDT, instead we employ a geometrical extrapolation for convergence of t amplitudes as in Comenius. The reason for not using the RLE in these examples is the storage requirements needed to save the CCSDT vectors to construct the RLE solution, since the triple amplitudes potentially require * n :WlvZi* values. In table 4 the BeH, data for CCSDT-la indicate the poorer convergence of the geometric extrapolation compared to the RLE; however, there is generally not any unusual difficulty with

converging the triple-excitation contributions using a geometrical extrapolation procedure and, usually, there is little change in the number of iterations required for convergence. In addition to a normal iterative scheme, in tables 3 and 4 we also consider starting each model from converged amplitudes of the previous model. The analogous procedure in CCD e.g. would be to converge the linearized (LCCD) equations first, and then start the non-linear CCD convergence. Since this procedure tends to collect terms of predominantly one sign before introducing the second group of terms of opposite sign (linear terms are predominantly negative, non-linear predominantly positive), it is not generally recommended, It is usually better to use a “locked-step” type of iteration that combines. terms of both signs. The superiority for the energy of only a single CCSDT approximation is desired, it is advantageous to use a normal convergence route; however, to obtain a spectrum of different CC models with triples we find a savings ranging from 29 to 74 551

CHEMICAL PHYSICS LETTERS

Volume 145, number 6

iterations for these examples for the calculation of each model from CCSD to CCSDT. The number of iterations required for convergence starting from the converged t amplitudes of the previous model is given in the rightmost column of tables 3 and 4. Finally, since the convergence of all triple excita-

22 April 1988

tion models is approximately the same, as an example, table 4 offers a comparison between CCSDTla convergence using RLE, the geometric approximation, and the geometric approximation starting from a converged result for CCSD. The first option is best.

Table 3 Convergence of various CCSDT models. All energies are in hartree Molecule (basis)

Approximation

E,,

NB’

Number of iterations b,

Successive iterations ‘)

HB (DZ)

CCSD

-0.073080 -0.0730803 -0.073662 -0.0736626 -0.073658 -0.073658s -0.073631 -0.0736308 -0.073633 -0.0736330 -0.073950 -0.0739503 -0.073910 -0.0739099

6 7 6 7 6 7 6 7 6 7 6 7 6 7

17 20 18 23 18 23 18 23 18 23 20 24 17 18

17 20 10 12 4 7 6 9 7 8 11 15 7 9

-0.087723 -0.0877230 -0.088219 -0.0882195 -0.088227 -0.0882270 -0.088243 -0.0882430 -0.088225 -0.0882254 -0.088508 -0.0885077 -0.088448 -0.0884479

6 7 6 7 6 7 6 7 6 7 6 7 6 7

2s 27 26 30 26 30 29 34 29 34 26 34 26 29

25 27 20 26 13 17 14 20 14 21 17 21 14 17

-0.146238 -0.1462381 -0.147577 -0.1475768 -0.147580 -0.1475803 -0.147459 -0.1474589 -0.147450 -0.1474501 -0.147613 -0.1476128 -0.147592 -0.1475921

6 7 6 7 6 7 6 7 6 I 6 7 6 7

11 14 11 12 11 12 11 14 11 14 11 14 11 12

11 14 8 11 4 6 7 6 4 6 7 12 6 8

CCSDT-la CCSDT-lb CCSDT-2 CCSDT-3 CCSDT-4 CCSDT HzBe (DZ)

CCSD CCSDT-la CCSDT-lb CCSDT-2 CCSDT-3 CCSDT-4 CCSDT

Hz0 (DZ)

CCSD CCSDT-la CCSDT-lb CCSDT-2 CCSDT3 CCSDT-4 CCSDT

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Volume 145, number 6 Table 3 (continued)

Molecule (basis)

Approximation

E,,

N’=’

Number of iterations b,

Successive iterations ‘)

HZ0 (DZP)

CCSD

-0.219449 -0.2194490 -0.222817 -0.2228711 -0.222881 -0.2228806 -0.222616 -0.2226166 -0.222614 -0.2226140 -0.222994 -0.2229942 -0.222916 -0.2229159

6 7 6 7 6 7 6 7 6 7 6 7 6 7

11 14 11 14 I1 14 11 14 11 14 11 13 I1 14

11 14 8 11 4 6 6 8 4 7 7 11 8 10

CCSDT- 1a CCSDT-lb CCSDT-2 CCSDT-3 CCSDT-4 CCSDT

‘) t amplitudes converged to 1O-N. b, Calculation started with no previous I amplitudes. ‘) Triples approximation started with previous model’s converged amplitudes.

In conclusion, the improved DIIS method for converging CCSD [ 2 11 is effectively equivalent to our RLE method [4], differing by details of implementation. The RLE has been used in all of our published works since 198 1 involving CCD, CCSD, and CCSDT- 1 and has proven itself as an effective scheme to accelerate convergence of the coupled-cluster equations. Though SLS have limited themselves to a closed-shell CCSD method, RLE converges our

different orbital for different spin, UHF, and ROHF based open-shell CC methods [ 321 with no apparent deleterious effect on the convergence of the coupledcluster equations [ 12- 161. To illustrate with a more realistic example, C, exists as a linear molecule in a 3Cs state. It also has a closely lying rhombic form in a closed-shell ‘AIBstate [ 151. A DZP basis consists of 64 CCSD functions. For the open-shell UHF it requires 13 iterations for convergence to 1O- 7 in the

Table 4 Convergence of CCSDT- 1a. All energies are in hartree Molecule (basis)

E cOrr

NQ

Number of iterations geometric extrapolations b,

geometric extrapolation ‘)

RLE

HB (DZ)

-0.073662 -0.0736626

6 7

27 32

18 23

12 13

HzBe (DZ)

-0.088219 -0.0882195

6 7

45 53

26 30

15 16

Hz0 (DZ)

-0.147577 -0.1475768

6 7

19 25

11 12

12 13

Hz0 (DZP)

-0.222877 -0.2228771

6 7

19 25

11 14

12 13

‘) f amplitudes converged to IO-‘? b, Calculation started from converged CCSD amplitudes. ‘) Calculation started without previous t amplitudes.

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energy ( lo-’ in t). For CCSDT-1, it requires 14 iterations to the same tolerance. For the closed shell, 13 RLE iterations converge CCSD to 1O-’ in energy and 1Om5in t, while CCSDT-1 requires 14. Furthermore, our recently developed multireference CC approaches use the same RLE approach with great success [ 191.

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