Analytic gradient for the multireference Brillouin-Wigner coupled cluster method and for the state-universal multireference coupled cluster method.

We present the analytic gradient theory and its pilot implementation for the multireference Brillouin-Wigner coupled cluster (BWCC) method and for the state-universal multireference coupled cluster method. The analytic gradient has been derived for three cases: (i) BWCC method without a size-extensivity correction, (ii) BWCC method with the iterative size-extensivity correction, and (iii) for the rigorously size-extensive state-universal method. The pilot implementation is based on full-configuration interaction expansions and is presently limited to single and double excitation levels; however, the resulting equations are general. For BWCC methods, they also do not contain terms explicitly mixing amplitudes of different reference configurations and can thus be implemented in an efficient way. The analytic gradients have been verified with respect to numerically computed ones on the example of CH2 molecule, and geometry optimizations of CH2 and SiH2 have been carried out.

[1]  H. Monkhorst,et al.  Coupled-cluster method for multideterminantal reference states , 1981 .

[2]  Y. Yamaguchi,et al.  A New Dimension to Quantum Chemistry: Analytic Derivative Methods in AB Initio Molecular Electronic Structure Theory , 1994 .

[3]  I. Dubois.  The absorption spectrum of the free SiH2 radical , 1968 .

[4]  Alistair P. Rendell,et al.  Analytic gradients for coupled-cluster energies that include noniterative connected triple excitations: Application to cis- and trans-HONO , 1991 .

[5]  P. Mach,et al.  Multireference Brillouin–Wigner Coupled-Cluster Theory , 1999 .

[6]  An open-shell coupled-cluster response method for static properties , 1992 .

[7]  T. Crawford,et al.  An Introduction to Coupled Cluster Theory for Computational Chemists , 2007 .

[8]  J. Gauss,et al.  Analytic energy gradients for the equation‐of‐motion coupled‐cluster method: Implementation and application to the HCN/HNC system , 1994 .

[9]  Rodney J. Bartlett,et al.  Hilbert space multireference coupled-cluster methods. I: The single and double excitation model , 1991 .

[10]  Jürgen Gauss,et al.  Coupled‐cluster methods with noniterative triple excitations for restricted open‐shell Hartree–Fock and other general single determinant reference functions. Energies and analytical gradients , 1993 .

[11]  R. Bartlett,et al.  Analytic energy gradients for the two-determinant coupled cluster method with application to singlet excited states of butadiene and ozone , 1994 .

[12]  Karol Kowalski,et al.  Extension of renormalized coupled-cluster methods including triple excitations to excited electronic states of open-shell molecules. , 2005, The Journal of chemical physics.

[13]  Jae Shin Lee,et al.  Critical comparison of single-reference and multireference coupled-cluster methods: Geometry, harmonic frequencies, and excitation energies of N2O2 , 2003 .

[14]  Ivan Hubač,et al.  Multireference Brillouin-Wigner Coupled-Cluster Theory. Single-root approach. , 1998 .

[15]  Trygve Helgaker,et al.  Configuration-interaction energy derivatives in a fully variational formulation , 1989 .

[16]  Garnet Kin-Lic Chan,et al.  Tailored coupled cluster singles and doubles method applied to calculations on molecular structure and harmonic vibrational frequencies of ozone. , 2006, The Journal of chemical physics.

[17]  J. Paldus,et al.  Truncated version of the reduced multireference coupled‐cluster method with perturbation selection of higher than pair clusters , 2000 .

[18]  J. Pittner,et al.  Multireference Brillouin-Wigner coupled clusters method with noniterative perturbative connected triples. , 2006, The Journal of chemical physics.

[19]  L. Meissner.  Fock-space coupled-cluster method in the intermediate Hamiltonian formulation: Model with singles and doubles , 1998 .

[20]  Jiří Pittner,et al.  Continuous transition between Brillouin-Wigner and Rayleigh-Schrödinger perturbation theory, generalized Bloch equation, and Hilbert space multireference coupled cluster , 2003 .

[21]  Karol Kowalski,et al.  The method of moments of coupled-cluster equations and the renormalized CCSD[T], CCSD(T), CCSD(TQ), and CCSDT(Q) approaches , 2000 .

[22]  Jerzy Leszczynski,et al.  COMPUTATIONAL CHEMISTRY: Reviews of Current Trends , 2006 .

[23]  J. Pittner,et al.  Multireference coupled-cluster calculations on the energy of activation in the automerization of cyclobutadiene: Assessment of the state-specific multireference Brillouin–Wigner theory , 2000 .

[24]  Rodney J. Bartlett,et al.  Coupled-cluster methods with internal and semi-internal triply and quadruply excited clusters: CCSDt and CCSDtq approaches , 1999 .

[25]  W. D. Allen,et al.  Geometrical structures, force constants, and vibrational spectra of SiH, SiH2, SiH3, and SiH4 , 1986 .

[26]  M. Musiał,et al.  Method of moments of coupled-cluster equations: a new formalism for designing accurate electronic structure methods for ground and excited states , 2004 .

[27]  Karol Kowalski,et al.  Extensive generalization of renormalized coupled-cluster methods. , 2005, The Journal of chemical physics.

[28]  Karol Kowalski,et al.  Efficient computer implementation of the renormalized coupled-cluster methods: The R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) approaches , 2002 .

[29]  Mihály Kállay,et al.  A general state-selective multireference coupled-cluster algorithm , 2002 .

[30]  Nicholas Petraco,et al.  General-Model-Space State–Universal Coupled-Cluster Method: Diagrammatic Approach , 2004 .

[31]  Piotr Piecuch,et al.  Renormalized coupled-cluster methods exploiting left eigenstates of the similarity-transformed Hamiltonian. , 2005, The Journal of chemical physics.

[32]  Josef Paldus,et al.  Algebraic Approach to Coupled Cluster Theory , 1994 .

[33]  S. Wilson,et al.  On the use of Brillouin-Wigner perturbation theory for many-body systems , 2000 .

[34]  J. Paldus,et al.  Multi-reference Brillouin–Wigner coupled-cluster method with a general model space , 2005 .

[35]  Josef Paldus,et al.  N-reference, M-state coupled-cluster method: Merging the state-universal and reduced multireference coupled-cluster theories , 2003 .

[36]  Josef Paldus,et al.  Coupled Cluster Theory , 1992 .

[37]  J. Pittner,et al.  State-Specific Brillouin−Wigner Multireference Coupled Cluster Study of the Singlet−Triplet Separation in the Tetramethyleneethane Diradical , 2001 .

[38]  Petr Nachtigall,et al.  Assessment of the single-root multireference Brillouin–Wigner coupled- cluster method: Test calculations on CH2, SiH2, and twisted ethylene , 1999 .

[39]  R. Bartlett,et al.  Analytic energy gradients for general coupled‐cluster methods and fourth‐order many‐body perturbation theory , 1986 .

[40]  R. Bartlett.  Analytical Evaluation of Gradients in Coupled-Cluster and Many-Body Perturbation Theory , 1986 .

[41]  Ioannis S. K. Kerkines,et al.  A multireference coupled-cluster potential energy surface of diazomethane, CH(2)N(2). , 2005, The journal of physical chemistry. A.

[42]  R. Bartlett,et al.  Analytic energy gradients with frozen molecular orbitals in coupled-cluster and many-body perturbation theory methods: Systematic study of the magnitude and trends of the effects of frozen molecular orbitals , 1997 .

[43]  J. Pittner,et al.  The singlet-triplet gap in trimethylenmethane and the ring-opening of methylenecyclopropane: a multireference Brillouin-Wigner coupled cluster study. , 2006, The journal of physical chemistry. A.

[44]  Pál.  Multireference coupled-cluster response approach for the calculation of static properties. , 1989, Physical review. A, General physics.

[45]  I. Hubač,et al.  Multireference Brillouin—Wigner Coupled-Cluster Theory: Hilbert Space Approach , 1997 .

[46]  Michael Hanrath,et al.  An exponential multireference wave-function Ansatz. , 2005, The Journal of chemical physics.

[47]  Rodney J. Bartlett,et al.  COUPLED-CLUSTER THEORY: AN OVERVIEW OF RECENT DEVELOPMENTS , 1995 .

[48]  Rodney J. Bartlett,et al.  Analytic energy derivatives in many‐body methods. I. First derivatives , 1989 .

[49]  J. Gauss.  Analytic second derivatives for the full coupled-cluster singles, doubles, and triples model: Nuclear magnetic shielding constants for BH, HF, CO, N2, N2O, and O3 , 2002 .

[50]  G. Diercksen,et al.  Methods in Computational Molecular Physics , 1983 .

[51]  Jürgen Gauss,et al.  Analytic second derivatives for general coupled-cluster and configuration-interaction models. , 2003, The Journal of chemical physics.

[52]  N. Oliphant,et al.  Coupled‐cluster method truncated at quadruples , 1991 .

[53]  R. Bartlett,et al.  Open-shell analytical energy gradients for triple excitation many-body, coupled-cluster methods: MBPT(4), CCSD+T(CCSD), CCSD(T),and QCISD(T) , 1992 .

[54]  Gustavo E. Scuseria,et al.  Analytic evaluation of energy gradients for the singles and doubles coupled cluster method including perturbative triple excitations: Theory and applications to FOOF and Cr2 , 1991 .

[55]  J. Pittner,et al.  Multireference Brillouin-Wigner coupled cluster singles and doubles (MRBWCCSD) and multireference doubles configuration interaction (MRD-CI) calculations for the Bergman cyclization reaction , 2003 .

[56]  Michael Hanrath,et al.  Initial applications of an exponential multi-reference wavefunction ansatz , 2006 .

[57]  J. Gauss,et al.  Analytic gradients for the coupled-cluster singles, doubles, and triples (CCSDT) model , 2002 .

[58]  J. Pittner,et al.  Towards the multireference Brillouin-Wigner coupled-clusters method with iterative connected triples: MR BWCCSDT-alpha approximation. , 2005, The Journal of chemical physics.

[59]  Julia E. Rice,et al.  Analytic evaluation of energy gradients for the single and double excitation coupled cluster (CCSD) wave function: Theory and application , 1987 .

[60]  So Hirata,et al.  High-order coupled-cluster calculations through connected octuple excitations , 2000 .

[61]  Peter J. Knowles,et al.  Unlimited full configuration interaction calculations , 1989 .

[62]  Jean-Paul Malrieu,et al.  Intermediate Hamiltonians as a new class of effective Hamiltonians , 1985 .

[63]  Josef Paldus,et al.  Reduced multireference CCSD method: An effective approach to quasidegenerate states , 1997 .

[64]  Mihály Kállay,et al.  Higher excitations in coupled-cluster theory , 2001 .

[65]  Ludwik Adamowicz,et al.  A state-selective multireference coupled-cluster theory employing the single-reference formalism , 1993 .

[66]  R. Mcweeny,et al.  Quantum Systems in Chemistry and Physics. Trends in Methods and Applications , 1997 .

[67]  R. Bartlett,et al.  Analytic evaluation of energy gradients at the coupled‐cluster singles and doubles level using quasi‐restricted Hartree–Fock open‐shell reference functions , 1991 .

[68]  L. Adamowicz,et al.  New scheme for solving the amplitude equations in the state-specific coupled cluster theory with complete active space reference for ground and excited states , 2000 .

[69]  A. Campargue,et al.  Absorption spectroscopy of SiH2 near 640 nm , 1998 .

[70]  Rodney J Bartlett,et al.  Coupled-cluster method tailored by configuration interaction. , 2005, The Journal of chemical physics.

[71]  R. Bartlett,et al.  Coupled‐cluster open‐shell analytic gradients: Implementation of the direct product decomposition approach in energy gradient calculations , 1991 .

[72]  J. Pittner,et al.  Multireference Brillouin−Wigner Coupled Cluster Singles and Doubles Study of the Singlet−Triplet Separation in Alkylcarbenes† , 2004 .

[73]  Uttam Sinha Mahapatra,et al.  A size-consistent state-specific multireference coupled cluster theory: Formal developments and molecular applications , 1999 .

[74]  J. Gauss,et al.  Analytic energy derivatives for ionized states described by the equation‐of‐motion coupled cluster method , 1994 .

[75]  H. Petek,et al.  Analysis of CH2 ã 1A1 (1,0,0) and (0,0,1) Coriolis‐coupled states, ã 1A1–X̃ 3B1 spin–orbit coupling, and the equilibrium structure of CH2 ã 1A1 state , 1989 .

[76]  S. Pal,et al.  Fock space multireference coupled cluster theory: noniterative inclusion of triples for excitation energies , 1998 .

[77]  P. Schleyer.  Encyclopedia of computational chemistry , 1998 .

[78]  Josef Paldus,et al.  The general-model-space state-universal coupled-cluster method exemplified by the LiH molecule , 2003 .

[79]  R. Bartlett,et al.  Analytical gradients for the coupled-cluster method† , 1984 .

[80]  D. Herschbach,et al.  New methods in quantum theory , 1996 .

[81]  Josef Paldus,et al.  General-model-space state-universal coupled-cluster theory: Connectivity conditions and explicit equations , 2003 .

[82]  Poul Jørgensen,et al.  Geometrical derivatives of energy surfaces and molecular properties , 1986 .

[83]  Uttam Sinha Mahapatra,et al.  Development of a size-consistent state-specific multireference perturbation theory with relaxed model-space coefficients , 1999 .

[84]  P. Szalay,et al.  Analytic energy derivatives for coupled‐cluster methods describing excited states: General formulas and comparison of computational costs , 1995 .

[85]  D R Yarkony,et al.  Modern electronic structure theory , 1995 .

[86]  Roy,et al.  Stationary multideterminantal coupled-cluster response. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[87]  J. Paldus,et al.  Reduced multireference coupled cluster method with singles and doubles: Perturbative corrections for triples. , 2006, The Journal of chemical physics.

[88]  John F. Stanton,et al.  Analytic energy gradients for open-shell coupled-cluster singles and doubles (CCSD) calculations using restricted open-shell Hartree—Fock (ROHF) reference functions , 1991 .

[89]  J. Cizek.  On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods , 1966 .

[90]  Ludwik Adamowicz,et al.  New approach to the state-specific multireference coupled-cluster formalism , 2000 .

[91]  Peter J. Knowles,et al.  A new determinant-based full configuration interaction method , 1984 .

[92]  John F. Stanton,et al.  Many‐body methods for excited state potential energy surfaces. I. General theory of energy gradients for the equation‐of‐motion coupled‐cluster method , 1993 .

[93]  Francesco A Evangelista,et al.  High-order excitations in state-universal and state-specific multireference coupled cluster theories: model systems. , 2006, The Journal of chemical physics.

[94]  Ivan Hubač,et al.  Size-extensivity correction for the state-specific multireference Brillouin–Wigner coupled-cluster theory , 2000 .

[95]  Y. Ishikawa,et al.  Intermediate Hamiltonian Fock-space coupled-cluster method: Excitation energies of barium and radium , 2000 .

[96]  P. Mach,et al.  Many-body Brillouin–Wigner second-order perturbation theory using a multireference formulation: an application to bond breaking in the diatomic hydrides BH and FH , 2006 .

[97]  Trygve Helgaker,et al.  Mo/ller–Plesset energy derivatives , 1988 .

[98]  I. Hubač.  Size-Extensive Brillouin-Wigner Perturbation Theory. Size-Extensive Brillouin-Wigner Coupled Cluster Theory , 1996 .

[99]  Francesco A Evangelista,et al.  Coupling term derivation and general implementation of state-specific multireference coupled cluster theories. , 2007, The Journal of chemical physics.

[100]  Trygve Helgaker,et al.  Coupled cluster energy derivatives. Analytic Hessian for the closed‐shell coupled cluster singles and doubles wave function: Theory and applications , 1990 .