EOM-CC methods

Equation-of-motion (EOM) is a versatile electronic structure approach that allows one to describe many multi-configurational wave functions within a single-reference formalism. For example, EOM for excitation energies (EOM-EE) method accurately describes electronically excited states, while ionized/electron attached EOM models (EOM-IP/EA) can tackle doublet radicals, including notorious cases of symmetry breaking. We have extended EOM approach to diradicals, triradicals, and bond-breaking. In our approach, which is called the Spin-Flip (SF method) problematic low-spin states are treated as spin-flipping excitations from the high-spin reference state.

EOM-EE Slater determinants

Equation-of-motion excitation energies (EOM-EE) determinants
Ψ(Ms = 0) = R(Ms = 0)Ψ0(Ms = 0)

EOM-IP Slater determinants

Equation-of-motion ionization potential (EOM-IP) determinants
Ψ(N) = R(-1)Ψ0(N + 1)

EOM-EA Slater determinants

Equation-of-motion electron attachment (EOM-EA) determinants
Ψ(N) = R(+1)Ψ0(N - 1)

EOM-SF Slater determinants

Equation-of-motion spin-flip (EOM-SF) determinants
Ψ(Ms = 0) = R(Ms = -1)Ψ0(Ms = 1)

Diradicals and the spin-flip method

From the electronic structure point of view, diradicals are molecules in which two electrons are distributed in two nearly degenerate molecular orbitals. For such a system, six Slater determinants can be generated as in the picture:

Spin-flip Slater determinants

Determinants (a) – (d) have zero projection of the total spin (Ms = 0). High-spin determinants (e) and (f) correspond to Ms = +1 and Ms = -1 configurations, respectively. From these determinants, three singlet and three triplet wave functions can be constructed as follows (coefficient λ is large when the energy gap between orbitals is small):

Singlets
Ψs1 = (a) - λ(b)
Ψs2 = λ(a) - (b)
Ψs3 = (c) - (d)
Triplets
Ψt1 = (c) + (d)
Ψt2 = (e)
Ψt3 = (f)

All of the above singlet wave functions are two-determinantal. The Ms = 0 component of the triplet is also two-determinantal, however, the high-spin triplets (Ms = 1/Ms = -1) are single-determinantal. Note that all the Ms = 0 determinants are formally single electron excitations with a spin-flip from the Ms = 1/Ms = -1 configurations. Therefore, the Ms = 0 states can be described as spin-flipping excited states from the high-spin |α α> triplet reference. This is the essence of the Spin-Flip (SF) method. The SF method describes ground and excited states of diradicals (or potential energy surfaces along bond-breaking coordinate) as spin-flipping, e. g., α→β, excitations from a high spin |α α> triplet reference. Similarly, electronic states of triradicals are described as spin-flipping excitations from the high-spin component of the quartet state. The SF approach allows one to describe multi-configurational wave functions in a size-consistent fashion and within a single-reference formalism thus resulting in efficient, accurate, and robust computational scheme.


Modeling of charge transfer reactions by EOM methods

Electron transfer reactions are common in biological and synthetic polymers. The rates of these processes can be related to the coupling between the diabatic electronic states that correspond to reactant and product states. Calculations on these systems are difficult due to the propensity of Hartree-Fock solutions to overlocalize charge and break symmetry.

Positively charged ethylene dimer is an often-studied prototype system for perpendicular hole conductance. The reaction coordinate slowly interpolates between neutral and cationic geometries of monomers in the dimer.

Energy change during charge transfer reactions
Electron hopping in ethylene dimer

Electron hopping in ethylene dimer

The EOM-CCSD method relies on the unstable Hartree-Fock solution for the open-shell doublet system, which is the positively charged dimer. It predicts excessive charge localization and a cusp on the potential energy surface. In contrast, EOM-IP-CCSD predicts a smooth charge flow along the reaction coordinate as well as smooth PESs. This method employs stable reference wave function of the neutral, which is a closed-shell singlet system, and describes both charge-transfer states in a balanced fashion.

Electron hopping in ethylene dimer by EOM methods

Electron hopping in ethylene dimer by EOM methods


Current research

Current research includes development of reduced scaling methods, efficient tensor algorithms, as well as novel EOM models to deal with other types of open shell systems and metastable electronic states (resonances). In addition, we are developing tools for describing non-linear response properties.


Related Publications

190. K.D. Nanda and A.I. Krylov
Visualizing the contributions of virtual states to two-photon absorption cross-sections by natural transition orbitals of response transition density matrices
J. Phys. Chem. Lett. 8, 3256 – 3265 (2017) Abstract  PDF Supporting info

188. A. Sadybekov and A.I. Krylov
Coupled-cluster based approach for core-level states in condensed phase: Theory and application to different protonated forms of aqueous glycine
J. Chem. Phys. 147, 014107 (2017) Abstract  PDF Supporting info

187. K.D. Nanda and A.I. Krylov
Effect of the diradical character on static polarizabilities and two-photon absorption cross-sections: A closer look with spin-flip equation-of-motion coupled-cluster singles and doubles method
J. Chem. Phys. 146, 224103 (2017) Abstract  PDF Supporting info

184. M. de Wergifosse, A.L. Houk, A.I. Krylov, and C.G. Elles
Two-photon absorption spectroscopy of trans-stilbene, cis-stilbene, and phenanthrene: Theory and experiment
J. Chem. Phys. 146, 144305 (2017) Abstract  PDF 

183. M. de Wergifosse, C.G. Elles, and A.I. Krylov
Two-photon absorption spectroscopy of stilbene and phenanthrene: Excited-state analysis and comparison with ethylene and toluene
J. Chem. Phys. 146, 174102 (2017) Abstract  PDF 

182. K.Z. Ibrahim, E. Epifanovsky, S. Williams, and A.I. Krylov
Cross-scale efficient tensor contractions for coupled cluster computations through multiple programming model backends
J. Parallel Distrib. Comput. 106, 92 – 105 (2017) Abstract  PDF 

180. S. Manzer, E. Epifanovsky, A.I. Krylov, and M. Head-Gordon
A general sparse tensor framework for electronic structure theory
J. Chem. Theo. Comp. 13, 1108 – 1116 (2017) Abstract  PDF 

179. T.-C. Jagau, K.B. Bravaya, and A.I. Krylov
Extending quantum chemistry of bound states to electronic resonances
Ann. Rev. Phys. Chem. 68, 525 – 553 (2017) Abstract  Full text 

178. A.I. Krylov
The quantum chemistry of open-shell species
Reviews in Comp. Chem. 30, 151 – 224 (2017) Abstract  PDF 

177. I. Kaliman and A.I. Krylov
New algorithm for tensor contractions on multi-core CPUs, GPUs, and accelerators enables CCSD and EOM-CCSD calculations with over 1000 basis functions on a single compute node
J. Comp. Chem. 38, 842 – 853 (2017) Abstract  PDF 

175. A.O. Gunina and A.I. Krylov
Probing electronic wave functions of sodium-doped clusters: Dyson orbitals, anisotropy parameters, and ionization cross-sections
J. Phys. Chem. A 120, 9841 – 9856 (2016) Abstract  PDF Supporting info

174. K.D. Nanda and A.I. Krylov
Static polarizabilities for excited states within the spin-conserving and spin-flipping equation-of-motion coupled-cluster singles and doubles formalism: Theory, implementation and benchmarks
J. Chem. Phys. 145, 204116 (2016) Abstract  PDF Supporting info

168. T.-C. Jagau and A.I. Krylov
Characterizing metastable states beyond energies and lifetimes: Dyson orbitals and transition dipole moments
J. Chem. Phys. 144, 054113 (2016) Abstract  PDF Supporting info

167. J. Brabec, C. Yang, E. Epifanovsky, A.I. Krylov, and E. Ng
Reduced-cost sparsity-exploiting algorithm for solving coupled-cluster equations
J. Comp. Chem. 37, 1059 – 1067 (2016) Abstract  PDF 

163. S. Gozem, A.O. Gunina, T. Ichino, D.L. Osborn, J.F. Stanton, and A.I. Krylov
Photoelectron wave function in photoionization: Plane wave or Coulomb wave?
J. Phys. Chem. Lett. 6, 4532 – 4540 (2015) Abstract  PDF Supporting info

161. A.I. Krylov, J.M. Herbert, F. Furche, M. Head-Gordon, P.J. Knowles, R. Lindh, F.R. Manby, P. Pulay, C.-K. Skylaris, and H.-J. Werner
What is the price of open-source software?
J. Phys. Chem. Lett. 6, 2751 – 2754 (2015) Abstract  PDF 

160. E. Epifanovsky, K. Klein, S. Stopkowicz, J. Gauss, and A.I. Krylov
Spin-orbit couplings within the equation-of-motion coupled-cluster framework: Theory, implementation, and benchmark calculations
J. Chem. Phys. 143, 064102 (2015) Abstract  PDF 

159. T.-C. Jagau, D.B. Dao, N.S. Holtgreve, A.I. Krylov, and R. Mabbs
Same but different: Dipole-stabilized shape resonances in CuF- and AgF-
J. Phys. Chem. Lett. 6, 2786 – 2793 (2015) Abstract  PDF Supporting info

157. A.V. Luzanov, D. Casanova, X. Feng, and A.I. Krylov
Quantifying charge resonance and multiexciton character in coupled chromophores by charge and spin cumulant analysis
J. Chem. Phys. 142, 224104 (2015) Abstract  PDF 

156. K. Nanda and A.I. Krylov
Two-photon absorption cross sections within equation-of-motion coupled-cluster formalism using resolution-of-the-identity and Cholesky decomposition representations: Theory, implementation, and benchmarks
J. Chem. Phys. 142, 064118 (2015) Abstract  PDF Supporting info

154. D. Zuev, E. Vecharynski, C. Yang, N. Orms, and A.I. Krylov
New algorithms for iterative matrix-free eigensolvers in quantum chemistry
J. Comp. Chem. 36, 273 – 284 (2015) Abstract  PDF 

152. D. Zuev, T.-C. Jagau, K.B. Bravaya, E. Epifanovsky, Y. Shao, E. Sundstrom, M. Head-Gordon, and A.I. Krylov
Erratum: "Complex absorbing potentials within EOM-CC family of methods: Theory, implementation, and benchmarks" [J. Chem. Phys. 141, 024102 (2014)]
J. Chem. Phys. 143, 149901 (2015) PDF 

151. T.-C. Jagau, D. Zuev, K.B. Bravaya, E. Epifanovsky, and A.I. Krylov
Correction to "A Fresh Look at Resonances and Complex Absorbing Potentials: Density Matrix-Based Approach"
J. Phys. Chem. Lett. 6, 3866 (2015) PDF 

150. T.-C. Jagau and A.I. Krylov
Complex absorbing potential equation-of-motion coupled-cluster method yields smooth and internally consistent potential energy surfaces and lifetimes for molecular resonances
J. Phys. Chem. Lett. 5, 3078 – 3085 (2014) Abstract  PDF Supporting info

149. S. Matsika, X. Feng, A.V. Luzanov, and A.I. Krylov
What we can learn from the norms of one-particle density matrices, and what we can't: Some results for interstate properties in model singlet fission systems
J. Phys. Chem. A 118, 11943 – 11955 (2014) Abstract  PDF Supporting info

148. K.Z. Ibrahim, S.W. Williams, E. Epifanovsky, and A.I. Krylov
Analysis and tuning of libtensor framework on multicore architectures
Proceedings of 21st Annual IEEE International Conference on High Performance Computing (HiPC 2014), 1 – 10 (2014) Abstract  PDF 

147. X. Feng, A.B. Kolomeisky, and A.I. Krylov
Dissecting the effect of morphology on the rates of singlet fission: Insights from theory
J. Phys. Chem. C 118, 19608 – 19617 (2014) Abstract  PDF Supporting info

146. S. Gozem, F. Melaccio, A. Valentini, M. Filatov, M. Huix-Rotllant, N. Ferre, L.M. Frutos, C. Angeli, A.I. Krylov, A. Granovsky, R. Lindh, and M. Olivucci
Shape of multireference, equation-of-motion coupled-cluster, and density functional theory potential energy surfaces at a conical intersection
J. Chem. Theor. Comp. 10, 3074 – 3084 (2014) Abstract  PDF Supporting info

145. D. Zuev, T.-C. Jagau, K.B. Bravaya, E. Epifanovsky, Y. Shao, E. Sundstrom, M. Head-Gordon, and A.I. Krylov
Complex absorbing potentials within EOM-CC family of methods: Theory, implementation, and benchmarks
J. Chem. Phys. 141, 024102 (2014) Abstract  PDF 

143. A.B. Kolomeisky, X. Feng, and A.I. Krylov
A simple kinetic model for singlet fission: A role of electronic and entropic contributions to macroscopic rates
J. Phys. Chem. C 118, 5188 – 5195 (2014) Abstract  PDF 

142. T.-C. Jagau, D. Zuev, K.B. Bravaya, E. Epifanovsky, and A.I. Krylov
A fresh look at resonances and complex absorbing potentials: Density matrix based approach
J. Phys. Chem. Lett. 5, 310 – 315 (2014) Abstract  PDF Supporting info

140. X. Feng, A.V. Luzanov, and A.I. Krylov
Fission of entangled spins: An electronic structure perspective
J. Phys. Chem. Lett. 4, 3845 – 3852 (2013) Abstract  PDF Supporting info

139. E. Epifanovsky, D. Zuev, X. Feng, K. Khistyaev, Y. Shao, and A.I. Krylov
General implementation of resolution-of-identity and Cholesky representations of electron-repulsion integrals within coupled-cluster and equation-of-motion methods: Theory and benchmarks
J. Chem. Phys. 139, 134105 (2013) Abstract  PDF Supporting info

138. S. Gozem, F. Melaccio, R. Lindh, A.I. Krylov, A. Granovsky, C. Angeli, and M. Olivucci
Mapping the excited state potential energy surface of a retinal chromophore model with multireference and EOM-CC methods
J. Chem. Theor. Comp. 9, 4495 – 4506 (2013) Abstract  PDF Supporting info

135. E. Epifanovsky, M. Wormit, T. Kus, A. Landau, D. Zuev, K. Khistyaev, P. Manohar, I. Kaliman, A. Dreuw, and A.I. Krylov
New Implementation of high-level correlated methods using a general block-tensor library for high-performance electronic structure calculations
J. Comp. Chem. 34, 2293 – 2309 (2013) Abstract  PDF Supporting info

131. K.B. Bravaya, D. Zuev, E. Epifanovsky, and A.I. Krylov
Complex-scaled equation-of-motion coupled-cluster method with single and double substitutions for autoionizing excited states: Theory, implementation, and examples
J. Chem. Phys. 138, 124106 (2013) Abstract  PDF Supporting info

130. D. Ghosh, D. Kosenkov, V. Vanovschi, J. Flick, I. Kaliman, Y. Shao, A.T.B. Gilbert, A.I. Krylov, and L.V. Slipchenko
Effective Fragment Potential method in Q-Chem: A guide for users and developers
J. Comp. Chem. 34, 1060 – 1070 (2013) Abstract  PDF Supporting info

129. S. Gozem, A.I. Krylov, and M. Olivucci
Conical intersection and potential energy surface features of a model retinal chromophore: Comparison of EOM-CC and multireference methods
J. Chem. Theor. Comp 9, 284 – 292 (2013) Abstract  PDF Supporting info

127. A.I. Krylov and P.M.W. Gill
Q-Chem: An engine for innovation
WIREs Comput. Mol. Sci. 3, 317 – 326 (2013) Abstract  PDF 

123. T. Kus and A. I. Krylov
De-perturbative corrections for charge-stabilized double ionization potential equation-of-motion coupled-cluster method
J. Chem. Phys. 136, 244109 (2012) Abstract  PDF 

122. Y.A. Bernard, Y. Shao, and A.I. Krylov
General formulation of spin-flip time-dependent density functional theory using non-collinear kernels: Theory, implementation, and benchmarks
J. Chem. Phys. 136, 204103 (2012) Abstract  PDF Supporting info

121. D. Ghosh, A. Roy, R. Seidel, B. Winter, S.E. Bradforth, and A.I. Krylov
First-principle protocol for calculating ionization energies and redox potentials of solvated molecules and ions: Theory and application to aqueous phenol and phenolate
J. Phys. Chem. B 116, 7269 – 7280 (2012) Abstract  PDF Supporting info

107. T. Kus and A. I. Krylov
Using the charge stabilization technique in the double ionization potential equation-of-motion calculations with dianion references
J. Chem. Phys. 135, 084109 (2011) Abstract  PDF Supporting info

86. A. Landau, K. Khistyaev, S. Dolgikh, and A.I. Krylov
Frozen natural orbitals for ionized states within equation-of-motion coupled-cluster formalism
J. Chem. Phys. 132, 014109 (2010) Abstract  PDF 

85. C.M. Oana and A.I. Krylov
Cross sections and photoelectron angular distributions in photodetachment from negative ions using equation-of-motion coupled-cluster Dyson orbitals
J. Chem. Phys. 131, 124114 (2009) Abstract  PDF 

84. P.U. Manohar, J.F. Stanton, and A.I. Krylov
Perturbative triples correction for the equation-of-motion coupled-cluster wave functions with single and double substitutions for ionized states: Theory, implementation, and examples
J. Chem. Phys. 131, 114112 (2009) Abstract  PDF 

76. A.A. Golubeva, P.A. Pieniazek, and A.I. Krylov
A new electronic structure method for doublet states: configuration interaction in the space of ionized 1h and 2h1p determinants
J. Chem. Phys. 130, 124113 (2009) Abstract  PDF Supporting info

73. D. Casanova, L.V. Slipchenko, A.I. Krylov, and M. Head-Gordon
Double spin-flip approach within equation-of-motion coupled cluster and conguration interaction formalisms: Theory, implementation and examples
J. Chem. Phys.  130, 044103 (2009) Abstract  PDF 

70. P.U. Manohar and A.I. Krylov
A non-iterative perturbative triples correction for the spin-flipping and spin-conserving equation-of-motion coupled-cluster methods with single and double substitutions
J. Chem. Phys. 129, 194105 (2008) Abstract  PDF 

67. P.A. Pieniazek, S.E. Bradforth, and A.I. Krylov
Charge localization and Jahn-Teller distortions in the benzene dimer cation
J. Chem. Phys. 129, 074104 (2008) Abstract  PDF 

62. A. I. Krylov
Equation-of-motion coupled-cluster methods for open-shell and electronically excited species: The hitchhiker's guide to Fock space
Ann. Rev. Phys. Chem. 59, 433 – 462 (2008) Abstract  Full text 

59. C.M. Oana and A.I. Krylov
Dyson orbitals for ionization from the ground and electronically excited states within equation-of-motion coupled-cluster formalism: Theory, implementation, and examples
J. Chem. Phys. 127, 234106 (2007) Abstract  PDF (873 kB) 

58. A.A. Golubeva, A.V. Nemukhin, S.J. Klippenstein, L.B. Harding, and A.I. Krylov
Performance of the spin-flip and multi-reference methods for bond-breaking in hydrocarbons: A benchmark study
J. Phys. Chem. A 111, 13264 – 13271 (2007) Abstract  PDF (162 kB) 

57. P. A. Pieniazek, S. A. Arnstein, S. E. Bradforth, A. I. Krylov, and C. D. Sherrill
Benchmark full configuration interaction and EOM-IP-CCSD results for prototypical charge transfer systems: Noncovalent ionized dimers
J. Chem. Phys. 127, 164110 (2007) Abstract  PDF (1542 kB) 

55. E. Epifanovsky and A. I. Krylov
Direct location of the minimum point on intersection seams of potential energy surfaces with equation-of-motion coupled-cluster methods
Mol. Phys. 105, 2515 – 2525 (2007) Abstract  PDF (240 kB) 

48. Y. Shao, L. F. Molnar, Y. Jung, J. Kussmann, C. Ochsenfeld, S. Brown, A. T. B. Gilbert, L. V. Slipchenko, S. V. Levchenko, D. P. O'Neil, R. A. Distasio Jr., R. C. Lochan, T. Wang, G. J. O. Beran, N. A. Besley, J. M. Herbert, C. Y. Lin, T. Van Voorhis, S. H. Chien, A. Sodt, R. P. Steele, V. A. Rassolov, P. Maslen, P. P. Korambath, R. D. Adamson, B. Austin, J. Baker, E. F. C. Bird, H. Daschel, R. J. Doerksen, A. Drew, B. D. Dunietz, A. D. Dutoi, T. R. Furlani, S. R. Gwaltney, A. Heyden, S. Hirata, C.-P. Hsu, G. S. Kedziora, R. Z. Khalliulin, P. Klunziger, A. M. Lee, W. Z. Liang, I. Lotan, N. Nair, B. Peters, E. I. Proynov, P. A. Pieniazek, Y. M. Rhee, J. Ritchie, E. Rosta, C. D. Sherrill, A. C. Simmonett, J. E. Subotnik, H. L. Woodcock III, W. Zhang, A. T. Bell, A. K. Chakraborty, D. M. Chipman, F. J. Keil, A. Warshel, W. J. Herhe, H. F. Schaefer III, J. Kong, A. I. Krylov, P. M. W. Gill, and M. Head-Gordon
Advances in methods and algorithms in a modern quantum chemistry program package
Phys. Chem. Chem. Phys. 8, 3172 – 3191 (2006) Abstract  PDF (863 kB) 

43. L.V. Slipchenko and A.I. Krylov
Efficient strategies for accurate calculations of electronic excitation and ionization energies: Theory and application to the dehydro-meta-xylylene anion
J. Phys. Chem. A 110, 291 – 298 (2006) Abstract  PDF (150 kB) 

42. L. V. Slipchenko and A. I. Krylov
Spin-conserving and spin-flipping equation-of-motion coupled-cluster method with triple excitations
J. Chem. Phys. 123, 84107 (2005) Abstract  PDF (173 kB) 

39. A. I. Krylov
The spin-flip equation-of-motion coupled-cluster electronic structure method for a description of excited states, bond-breaking, diradicals, and triradicals
Acc. Chem. Res. 39, 83 – 91 (2006) Abstract  PDF (246 kB) 

38. S. V. Levchenko, T. Wang, and A. I. Krylov
Analytic gradients for the spin-conserving and spin-flipping equation-of-motion coupled-cluster models with single and double substitutions
J. Chem. Phys. 122, 224106 (2005) Abstract  PDF (146 kB) 

34. S. V. Levchenko and A. I. Krylov
Equation-of-motion spin-flip coupled-cluster model with single and double substitutions: Theory and application to cyclobutadiene
J. Chem. Phys. 120, 175 – 185 (2004) Abstract  PDF (195 kB) 

31. J. S. Sears, C. D. Sherrill, and A. I. Krylov
A spin-complete version of the spin-flip approach to bond breaking: What is the impact of obtaining spin eigenfunctions?
J. Chem. Phys. 118, 9084 – 9094 (2003) Abstract  PDF (150 kB) 

28. Y. Shao, M. Head-Gordon, and A. I. Krylov
The spin-flip approach within time-dependent density functional theory: Theory and applications to diradicals
J. Chem. Phys. 118, 4807 – 4818 (2003) Abstract  PDF (185 kB) 

27. A. I. Krylov, L. V. Slipchenko, and S. V. Levchenko
Breaking the curse of the non-dynamical correlation problem: The spin-flip method
ACS Symposium Series 958, 89 – 102 (2007) PDF (657 kB) 

26. L. V. Slipchenko and A. I. Krylov
Singlet-triplet gaps in diradicals by the spin-flip approach: A benchmark study
J. Chem. Phys. 117, 4694 – 4708 (2002) Abstract  PDF (237 kB) 

24. A. I. Krylov and C. D. Sherrill
Perturbative corrections to the equation-of-motion spin-flip SCF model: Application to bond-breaking and equilibrium properties of diradicals
J. Chem. Phys. 116, 3194 – 3203 (2002) Abstract  PDF (122 kB) 

23. A. I. Krylov
Spin-flip configuration interaction: An electronic structure model that is both variational and size-consistent
Chem. Phys. Lett. 350, 522 – 530 (2001) Abstract  PDF (148 kB) 

20. A. I. Krylov
Size-consistent wave functions for bond-breaking: The equation-of-motion spin-flip model
Chem. Phys. Lett. 338, 375 – 384 (2001) Abstract  PDF (114 kB) 

19. A. I. Krylov
Spin-contanination in coupled-cluster wavefunctions
J. Chem. Phys. 113, 6052 – 6062 (2000) Abstract  PDF (144 kB) 

18. A. I. Krylov, C. D. Sherrill, and M. Head-Gordon
Excited states theory for optimized orbitals and valence optimized orbitals coupled-cluster doubles models
J. Chem. Phys. 113, 6509 – 6527 (2000) Abstract  PDF (212 kB) 

17. J. Kong, C. A. White, A. I. Krylov, C. D. Sherrill, R. D. Adamson, T. R. Furlani, M. S. Lee, A. M. Lee, S. R. Gwaltney, T. R. Adams, C. Ochsenfeld, A. T. B. Gilbert, G. S. Kedziora, V. A. Rassolov, D. R. Maurice, N. Nair, Y. Shao, N. A. Besley, P. Maslen, J. P. Dombroski, H. Daschel, W. Zhang, P. P. Korambath, J. Baker, E. F. C. Bird, T. Van Voorhis, M. Oumi, S. Hirata, C.-P. Hsu, N. Ishikawa, J. Florian, A. Warshel, B. G. Johnson, P. M. W. Gill, M. Head-Gordon, and J. A. Pople
Q-Chem 2.0: A high performance ab initio electronic structure program package
J. Comp. Chem. 21, 1532 – 1548 (2000) Abstract  PDF (283 kB) 

16. S. R. Gwaltney, C. D. Sherrill, M. Head-Gordon, and A. I. Krylov
Second order perturbation corrections to singles and doubles coupled-cluster methods: General theory and applications to the valence optimized doubles model
J. Chem. Phys. 113, 3548 – 3560 (2000) Abstract  PDF (173 kB)