Electronic Structure Lab
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People:
Anna Golubeva
Dr. Sergey V. Levchenko
Piotr Adam Pieniazek
Dr. Lyudmila V. Slipchenko
Dr. Tao Wang

Electronic structure methods for open shells, bond-breaking, and excited states

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.

We have implemented and benchmarked several SF models:

  • SF-CIS (based on Hartree-Fock and configuration interaction singles),
  • SF-TDDFT (time-dependent DFT),
  • SF-CIS(D) (perturbation theory), and
  • SF-OD and SF-CCSD, two coupled-cluster based models.

The current methodological developments include:

  • spin-orbit couplings by using Breit-Pauli Hamiltonian;
  • non-adiabatic couplings;
  • spin-adaptation of the SF and other EOM models;
  • extension of the SF approach to higher multiplicity references.


Related Publications

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

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)

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)