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 meth- ods for electron correlation, the question of the con- vergence of the CC equations is important. CCD (coupled-cluster doubles) [ l-4 1, CCSD (singles and doubles) [ 5 1, and various methods for including tri- ples (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 func- tions times the number of iterations required to reach convergence. Since it is not iterative, some [ lo] ar- gue that finite-order approximations to CC theory like MBPT, are preferable to converged CC ap- proaches despite the greater accuracy of the latter [9]. In order to combine the considerable advan- tages 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 re- duced linear equation (RLE) technique to acceler- ate 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. techniques for converging CC equations [ 17 1. The convergence of the singles and doubles cou- pled-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, im- plies that their approach converges faster than oth- ers. 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 ap- proximations 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 em- ploy 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 ap- proach 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- 548 0 009-26 14/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)