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The Spin-Polarized Extended Brueckner Orbitals

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The Spin-Polarized Extended Brueckner Orbitals The spin-polarized extended Brueckner orbitals Cite as: J. Chem. Phys. 135, 094107 (2011); https://doi.org/10.1063/1.3629780 Submitted: 15 June 2011 . Accepted: 08 August 2011 . Published Online: 06 September 2011 A. V. Luzanov, and O. V. Prezhdo ARTICLES YOU MAY BE INTERESTED IN Incremental full configuration interaction The Journal of Chemical Physics 146, 104102 (2017); https://doi.org/10.1063/1.4977727 Projected Hartree–Fock theory The Journal of Chemical Physics 136, 164109 (2012); https://doi.org/10.1063/1.4705280 Hole-particle characterization of coupled-cluster singles and doubles and related models The Journal of Chemical Physics 125, 154106 (2006); https://doi.org/10.1063/1.2360262 J. Chem. Phys. 135, 094107 (2011); https://doi.org/10.1063/1.3629780 135, 094107 © 2011 American Institute of Physics. THE JOURNAL OF CHEMICAL PHYSICS 135, 094107 (2011) The spin-polarized extended Brueckner orbitals A. V. Luzanov1,a) and O. V. Prezhdo2,b) 1STC “Institute for Single Crystals,” National Academy of Sciences, Kharkov 61001, Ukraine 2Department of Chemistry, University of Rochester, Rochester, New York 14627, USA (Received 15 June 2011; accepted 8 August 2011; published online 6 September 2011) Conventional natural and Brueckner orbitals (BOs) are rather frequently used for improving ac- tive orbital spaces in various configuration interaction (CI) approaches. However, the natural and Brueckner single-determinant models per se fail to give an adequate picture of highly correlated and quasidegenerate states such as open-shell singlet and dissociative states. We suggest the use of the spin-polarized extended BOs formally defining them in the same manner as in Löwdin’s spin- extended Hartree-Fock method. Such BO orbitals turn out to be quite flexible and particularly useful for analyzing highly correlated electronic states. It is shown that the extended BOs always exist, unlike the usual unrestricted BOs. We discuss difficulties related to violation of size-consistency for spin projected determinant models. The working algorithm is proposed for computing BOs within the full CI and related complete active space methodology. The extended BOs are analyzed in terms of the special density-like matrices associated with spin-up and spin-down BO orbitals. From these density matrices, the corresponding spin-polarization diagrams are produced for effectively unpaired (essentially correlated) electrons. We illustrate the approach by calculations on cyclic hy- drogen clusters (H4,H6, and H8), certain carbene diradicals and monoradicals, and low-lying excited states. The computations show that the BO spin-projected determinant provides a strong overlap with the multi-configurational state even for quasidegenerate states and bond breaking processes. © 2011 American Institute of Physics. [doi:10.1063/1.3629780] I. INTRODUCTION paper24 an effective one-body Hamiltonian was constructed, giving approximate BOs as its eigenvectors, and the negative One of the salient features of modern quantum chemistry of ionization potentials and electron affinities as its eigenval- is a wide use of non-canonical Hartree-Fock (HF) orbitals and ues. Analogously, within DFT an appropriate computational even non-HF orbitals for ab initio post-HF techniques. This is technique25 can be formulated by using local correlation po- especially true for the virtual space, where the canonical HF tentials derived from a Brueckner double (BD) excitation cou- orbitals are very inefficient.1 In this regard, the recent paper2 pled cluster wave-function (BDCC approximation). Some in- proposes an original and promising scheme for constructing teresting applications of BDCC and related schemes can be strongly localized virtual orbitals needed for electron cor- found in the recent works.26–31 In particular, Brueckner or- relation approaches. Among other non-standard orbital sets, bitals allow one to overcome the instability inherent in typi- the Brueckner orbitals (BOs) (Refs. 3–5) and closely related cal simplified CC approaches, for low-symmetry diradicals to constructs appear important for formulating optimized non- have narrow HOMO-LUMO gaps.26, 27 All these facts demon- variational approaches such as the coupled cluster (CC) and strate the importance of further development of the BO theory related configuration interaction (CI) models.6–10 This fact is for modern quantum chemistry. not so unexpected if we recollect that BOs can be also de- Introduction of BOs implies that they produce a single fined as the best-overlap orbitals. BOs are even more prefer- Slater determinant that maximally overlaps with either a true able for interpretive purposes than the natural orbitals, which or a best approximate wave-function. However, a conven- are also extensively used in current electronic structure theory. tional spin-purity Slater determinant (even from BOs) cannot (Refs. 11–22 are only a small part of an extremely rich litera- provide a strong overlap with a highly correlated electronic ture on this subject.) state, especially when quasidegenerate states appear. It would We stress that although BOs are not frequently employed be more advantageous if we could definitively interpret the for routine quantum-chemical calculations because they are exact state vector in terms of a generalized independent parti- rather demanding computationally, they remain an essential cle model, even in the case of degenerate and quasidegenerate part of the current orbital methodology. For example, BOs states. As it turns out, it is indeed possible to do so directly for turn out to be closely related to the Kohn-Sham orbitals of several simple systems. For instance, the two-electron prob- density functional theory (DFT) and the corresponding gen- lem can be correctly treated by the Coulson-Fischer orbitals.32 eralized Koopmans’ orbitals.23 Furthermore, in the recent In fact, the two-electron Coulson-Fischer wave function can be envisaged as a specific case of Löwdin’s extended 33 a)Electronic mail: [email protected]. Hartree-Fock (EHF) function. The latter is just the general- b)Electronic mail: [email protected]. ized independent particle model that preserves physical clar- 0021-9606/2011/135(9)/094107/14/$30.00135, 094107-1 © 2011 American Institute of Physics 094107-2 A. V. Luzanov and O. V. Prezhdo J. Chem. Phys. 135, 094107 (2011) ity and, at the same time, includes correlation effects. In other The standard variational procedure gives the first varia- words, the new different orbitals for different spins (DODS) tion as should be constructed to ensure the maximal overlap between = | | the high-quality CI function under study and the appropri- δJ 2 δ , (2.4) ate spin-projected determinant. The orbitals so obtained will where, for simplicity, these and subsequent quantities are as- be termed the spin-polarized extended (or simply, extended) sumed to be real. The prefactor | in Eq. (2.4) ensures BOs. Thus, the main goal of our paper is to present a new the correct sign of δJ by accounting for the phase factors of interpretive scheme based on the extended. We would like the current orbitals. Therefore, the prefactor is retained de- to stress that up to now the spin-polarized unrestricted MOs spite its appearance as a common multiplier. and related BOs remain a physically attractive basis for treat- In order to obtain explicit equations for the determination 34–41 ing electron correlation effects. Nevertheless, preserving of BOs, the maximum of Eq. (2.4) is found by taking the first spin-purity of electronic states is a principal point, and in our variation of the Slater determinant, |δ, under the fixed |. study we eliminate spin contamination in a proper way. The variation |δ is of the conventional form The paper is organized as follows. In Sec. II the starting definitions and new computational equations concerning BOs N | = + | are given. Section III contains the key equations for construct- δ aiaa ai , = ≥ + ing the extended BOs from full CI (FCI) and related models i 1 a N 1 + of the complete active space self-consistent field (CASSCF) with ai and aa being the annihilation and creation operators, type. Section IV proposes a new interpretive tool for elec- respectively, and ai arbitrary parameters. The equivalent ex- tron correlation by spin-polarization diagrams derived from pression (e.g., see Eq. (4.27) in Ref. 42 for a general rule) is BOs. Section V demonstrates how the given BOs and spin- polarization diagrams work for rather typical hydrogen clus- | = | ters and carbon-containing systems, especially those having δ (k) , (2.5) open-shell ground states (singlet diradicals) or those under- 1≤k≤N going bond breaking. Some results for excited states are also where the sum runs over all N electrons, and the one-electron presented here. The last section concludes the paper, and Ap- operator is pendices furnish the details on derivation of the main ex- pressions and some algorithmic formulas. Furthermore, in = ai|ϕaϕi |. Appendix E we offer a plausible reasoning for the existence i;a of the extended BOs for any level of electron-correlation (in This is in fact a particle-hole component of the first varia- contrast to the unrestricted BOs). tion δρ, that is = (I − ρ)δρρ. Here and elsewhere we use the customaty notation Z(k) to denote the given operator Z II. DEFINITION OF BOs AND STARTING EXPRESSIONS acting on electron k. Then Here we briefly sketch the main concepts and derive + new computational expressions for BOs in the usual (non- δ | =| (k)|, (2.6) extended) formulation. Let | be the given N-electron state 1≤k≤N | + + vector with a total spin s. For practical purposes, is un- where = ρδρ(I − ρ). Treating in Eq. (2.6) as the derstood to be approximated from a “correlated” level of the- usual one-electron matrix, we see that ory, i.e., CI or CASSCF. Moreover, a Slater determinant, |, + can be constructed from a set of (unknown) orthonormal spin- δ | =Tr D,, (2.7) orbitals, where D, is the ordinary one-electron transition density {|ϕi }1≤i≤N .
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