Local pair natural orbitals for excited states.

We explore how in response calculations for excitation energies with wavefunction based (e.g., coupled cluster) methods the number of double excitation amplitudes can be reduced by means of truncated pair natural orbital (PNO) expansions and localized occupied orbitals. Using the CIS(D) approximation as a test model, we find that the number of double excitation amplitudes can be reduced dramatically with minor impact on the accuracy if the excited state wavefunction is expanded in state-specific PNOs generated from an approximate first-order guess wavefunction. As for ground states, the PNO truncation error can also for excitation energies be controlled by a single threshold related to generalized natural occupation numbers. The best performance is found with occupied orbitals which are localized by the Pipek-Mezey localization. For a large test set of excited states we find with this localization that already a PNO threshold of 10(-8)-10(-7), corresponding to an average of only 40-80 PNOs per pair, is sufficient to keep the PNO truncation error for vertical excitation energies below 0.01 eV. This is a significantly more rapid convergence with the number doubles amplitudes than in domain-based local response approaches. We demonstrate that the number of significant excited state PNOs scales asymptotically linearly with the system size in the worst case of completely delocalized excitations and sub-linearly whenever the chromophore does not increase with the system size. Moreover, we observe that the flexibility of state-specific PNOs to adapt to the character of an excitation allows for an almost unbiased treatment of local, delocalized and charge transfer excited states.

[1]  Martin Schütz,et al.  A multistate local coupled cluster CC2 response method based on the Laplace transform. , 2009, The Journal of chemical physics.

[2]  Tatiana Korona,et al.  Local CC2 electronic excitation energies for large molecules with density fitting. , 2006, The Journal of chemical physics.

[3]  Dimitrios G Liakos,et al.  Efficient and accurate approximations to the local coupled cluster singles doubles method using a truncated pair natural orbital basis. , 2009, The Journal of chemical physics.

[4]  F. Neese,et al.  Efficient and accurate local approximations to coupled-electron pair approaches: An attempt to revive the pair natural orbital method. , 2009, The Journal of chemical physics.

[5]  Pavel Rosmus,et al.  PNO–CI and CEPA studies of electron correlation effects. III. Spectroscopic constants and dipole moment functions for the ground states of the first‐row and second‐row diatomic hydrides , 1975 .

[6]  Hans-Joachim Werner,et al.  Local perturbative triples correction (T) with linear cost scaling , 2000 .

[7]  Trygve Helgaker,et al.  Excitation energies from the coupled cluster singles and doubles linear response function (CCSDLR). Applications to Be, CH+, CO, and H2O , 1990 .

[8]  Hermann Stoll,et al.  The correlation energy of crystalline silicon , 1992 .

[9]  H. Lischka,et al.  PNO-CI (pair natural-orbital configuration interaction) and CEPA-PNO (coupled electron pair approximation with pair natural orbitals) calculations of molecular systems. , 1975 .

[10]  Richard L. Martin.  NATURAL TRANSITION ORBITALS , 2003 .

[11]  Wilfried Meyer,et al.  PNO–CI Studies of electron correlation effects. I. Configuration expansion by means of nonorthogonal orbitals, and application to the ground state and ionized states of methane , 1973 .

[12]  Walter Thiel,et al.  Benchmarks for electronically excited states: CASPT2, CC2, CCSD, and CC3. , 2008, The Journal of chemical physics.

[13]  Marco Häser,et al.  Møller-Plesset (MP2) perturbation theory for large molecules , 1993 .

[14]  F. Weigend,et al.  Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy. , 2005, Physical chemistry chemical physics : PCCP.

[15]  F. Weigend,et al.  Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations , 2002 .

[16]  H. Monkhorst,et al.  Calculation of properties with the coupled-cluster method , 2009 .

[17]  H. Lischka,et al.  PNO–CI (pair natural orbital configuration interaction) and CEPA–PNO (coupled electron pair approximation with pair natural orbitals) calculations of molecular systems. I. Outline of the method for closed‐shell states , 1975 .

[18]  S. F. Boys.  Construction of Some Molecular Orbitals to Be Approximately Invariant for Changes from One Molecule to Another , 1960 .

[19]  P. Hay.  On the calculation of natural orbitals by perturbation theory , 1973 .

[20]  H. Lischka,et al.  PNO–CI (pair natural orbital configuration interaction) and CEPA–PNO (coupled electron pair approximation with pair natural orbitals) calculations of molecular systems. II. The molecules BeH2, BH, BH3, CH4, CH−3, NH3 (planar and pyramidal), H2O, OH+3, HF and the Ne atom , 1975 .

[21]  Christian Ochsenfeld,et al.  Rigorous integral screening for electron correlation methods. , 2005, The Journal of chemical physics.

[22]  Jeppe Olsen,et al.  Excitation energies of BH, CH2 and Ne in full configuration interaction and the hierarchy CCS, CC2, CCSD and CC3 of coupled cluster models , 1995 .

[23]  Michael Dolg,et al.  Fully automated implementation of the incremental scheme: application to CCSD energies for hydrocarbons and transition metal compounds. , 2007, The Journal of chemical physics.

[24]  S. F. Boys,et al.  Canonical Configurational Interaction Procedure , 1960 .

[25]  Christof Hättig,et al.  CC2 excitation energy calculations on large molecules using the resolution of the identity approximation , 2000 .

[26]  Martin W. Feyereisen,et al.  Use of approximate integrals in ab initio theory. An application in MP2 energy calculations , 1993 .

[27]  H. Monkhorst,et al.  Some aspects of the time-dependent coupled-cluster approach to dynamic response functions , 1983 .

[28]  J. Olsen,et al.  An analysis and implementation of a general coupled cluster approach to excitation energies with application to the B2 molecule , 2001 .

[29]  T. H. Dunning.  Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen , 1989 .

[30]  Hans-Joachim Werner,et al.  Low-order scaling local electron correlation methods. IV. Linear scaling local coupled-cluster (LCCSD) , 2001 .

[31]  Branislav Jansík,et al.  Linear scaling coupled cluster method with correlation energy based error control. , 2010, The Journal of chemical physics.

[32]  S. Grimme,et al.  Assessment of TD-DFT methods and of various spin scaled CIS(D) and CC2 versions for the treatment of low-lying valence excitations of large organic dyes , 2010 .

[33]  J. Almlöf,et al.  Integral approximations for LCAO-SCF calculations , 1993 .

[34]  A. Schäfer,et al.  Fully optimized contracted Gaussian basis sets of triple zeta valence quality for atoms Li to Kr , 1994 .

[35]  P. C. Hariharan,et al.  Refined abinitio calculation of the potential energy surface of the He–H2 interaction with special emphasis to the region of the van der Waals minimum , 1980 .

[36]  Christian Ochsenfeld,et al.  Tighter multipole-based integral estimates and parallel implementation of linear-scaling AO-MP2 theory. , 2008, Physical chemistry chemical physics : PCCP.

[37]  Marco Häser,et al.  Improvements on the direct SCF method , 1989 .

[38]  H. Werner,et al.  Local treatment of electron excitations in the EOM-CCSD method , 2003 .

[39]  Henrik Koch,et al.  Coupled cluster response functions , 1990 .

[40]  M. Krauss,et al.  Pseudonatural Orbitals as a Basis for the Superposition of Configurations. I. He2 , 1966 .

[41]  R. Mata,et al.  An incremental correlation approach to excited state energies based on natural transition/localized orbitals. , 2011, The Journal of chemical physics.

[42]  Wilfried Meyer,et al.  Ionization energies of water from PNO‐CI calculations , 2009 .

[43]  Jürgen Gauss,et al.  Calculation of excited-state properties using general coupled-cluster and configuration-interaction models. , 2004, The Journal of chemical physics.

[44]  Trygve Helgaker,et al.  The integral‐direct coupled cluster singles and doubles model , 1996 .

[45]  Peter Pulay,et al.  Localizability of dynamic electron correlation , 1983 .

[46]  Hans-Joachim Werner,et al.  Local treatment of electron correlation in coupled cluster theory , 1996 .

[47]  Ove Christiansen,et al.  Response functions in the CC3 iterative triple excitation model , 1995 .

[48]  Philippe Y. Ayala,et al.  Linear scaling second-order Moller–Plesset theory in the atomic orbital basis for large molecular systems , 1999 .

[49]  Paul G. Mezey,et al.  A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions , 1989 .

[50]  R. Ahlrichs,et al.  Efficient molecular numerical integration schemes , 1995 .

[51]  Wilfried Meyer,et al.  PNO-CI and CEPA studies of electron correlation effects , 1974 .

[52]  A. V. Luzanov,et al.  Application of transition density matrix for analysis of excited states , 1976 .

[53]  Manabu Oumi,et al.  A doubles correction to electronic excited states from configuration interaction in the space of single substitutions , 1994 .

[54]  Poul Jørgensen,et al.  The second-order approximate coupled cluster singles and doubles model CC2 , 1995 .

[55]  S. Dancoff.  Non-Adiabatic Meson Theory of Nuclear Forces , 1950 .

[56]  Debashis Mukherjee,et al.  A response-function approach to the direct calculation of the transition-energy in a multiple-cluster expansion formalism , 1979 .

[57]  Holger Patzelt,et al.  RI-MP2: optimized auxiliary basis sets and demonstration of efficiency , 1998 .

[58]  T. Dunning,et al.  Electron affinities of the first‐row atoms revisited. Systematic basis sets and wave functions , 1992 .

[59]  T. Daniel Crawford,et al.  Locally correlated equation-of-motion coupled cluster theory for the excited states of large molecules , 2002 .

[60]  Hideo Sekino,et al.  A linear response, coupled‐cluster theory for excitation energy , 1984 .

[61]  R. Ahlrichs,et al.  Direct determination of pair natural orbitals , 1975 .

[62]  W. Thiel,et al.  Basis set effects on coupled cluster benchmarks of electronically excited states: CC3, CCSDR(3) and CC2 , 2010 .