Nonstationary perturbation theory for open-shell molecules

Nonstationary perturbation theory equations have been obtained for open-shell molecules. The equations were formulated in terms of a density matrix in the MO-LCAO method. The first variant is coupled perturbation theory in the framework of the restricted Hartree-Fock method for open shells, and the second variant is variational perturbation theory for ground and excited electronic states of molecules, in which the perturbed wave function of the system is constructed in the form of a superposition of the ground and singly excited configurations composed of the Hartree-Fock orbitals of the open shell. A calculation of the Cauchy moments of the dynamic dipole polarizability of several molecules of conjugated open-shell hydrocarbons, viz., doublet states of odd alternant hydrocarbons, as well as triplet excited states and doublet states of radical ions of even alternant hydrocarbons, has been carried out in the framework of both methods.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 21, No. 1, pp. 18–27, January–February, 1985.     

Excited states of the benzyl radical

The CI method is used in the -electron approximation with orbitals for closed and open shells to calculate the properties of excited doublet states with allowance for all singly excited configurations and some doubly excited ones, and also for the first quartet and sextet states, which are calculated in the one-configuration approximation via the open-shell theory. The energies and transition moments agree satisfactorily with the available experimental evidence. A classification and assignment is given for the excited terms. Truncation of the complete set of singly excited configurations greatly distorts the calculated spectrum. Inclusion of doubly excited configurations in the CI also produces a substantial change in the spectrum; in some cases it alters the order of adjacent terms. Conversion in CI from basis closed-shell orbitals to open-shell ones produces a considerable lowering of all terms in the spectrum. As in the case of triplet terms for molecules, weakening of electron interaction brings the lowest excited term of the radical closer to the ground-state term. The electron-density and spin-density distributions are calculated for the excited states.     

Orbital optimisation in single open-shell configurations: A sequential unconstrained minimisation technique

A simple algorithm for optimising the orbitals for single open-shell configurations which utilises a sequential unconstrained minimisation technique (SUMT) suggested by Morrison is presented in this paper. As numerical illustrations, the orbitals of the lowest π – π* singlet and triplet (M_s = 0) states of three conjugated hydrocarbons are optimised by the algorithm. The advantages of using the SUMT method are also emphasised.     

Analysis of the half-projected Hartree–Fock function: Density matrix,natural orbitals,and configuration interaction equivalence

The half-projected Hartree–Fock function for singlet states (HPHF ) is analysed in terms of natural electronic configurations. For this purpose the HPHF spinless density matrix and its natural orbitals are first deduced. It is found that the HPHF function does not contain any contribution from odd-times excited configurations. It is seen in addition, in the case of the singlet ground states, this function is approximately equivalent to two closed-shell configurations, although the nature of the excited one depends on the nuclear geometry. An example is given in the case of the LiH ground state. Finally, the application of this model for studying systems of more that two atoms is criticized.     

具有辛群对称性的一类价激发n电子波函数的性质和应用

讨论了具有辛群对称性的一类价激发n电子波函数 ψ_(a,s_1,s_2)=N(multiply from t=s_1 to s_1+s_2-1)(a_1+(a_t))(multiply from k=s_1+s_2 to s_1+s_2+m-1)G+(ζ,k,V_(s+1))|0>的性质,它们对闭壳层和开壳层体系均适用.用此类函数计算了Li_2,LiH_2~-和CH_3~+,结果表明这类函数能较好地描述电子相关作用,并且变分得到的轨道都是自然轨道。     

Efficient and accurate local single reference correlation methods for high-spin open-shell molecules using pair natural orbitals

A production level implementation of the high-spin open-shell (spin unrestricted) single reference coupled pair, quadratic configuration interaction and coupled cluster methods with up to doubly excited determinants in the framework of the local pair natural orbital (LPNO) concept is reported. This work is an extension of the closed-shell LPNO methods developed earlier [F. Neese, F. Wennmohs, and A. Hansen, J. Chem. Phys. 130, 114108 (2009); F. Neese, A. Hansen, and D. G. Liakos, J. Chem. Phys. 131, 064103 (2009)]. The internal space is spanned by localized orbitals, while the external space for each electron pair is represented by a truncated PNO expansion. The laborious integral transformation associated with the large number of PNOs becomes feasible through the extensive use of density fitting (resolution of the identity (RI)) techniques. Technical complications arising for the open-shell case and the use of quasi-restricted orbitals for the construction of the reference determinant are discussed in detail. As in the closed-shell case, only three cutoff parameters control the average number of PNOs per electron pair, the size of the significant pair list, and the number of contributing auxiliary basis functions per PNO. The chosen threshold default values ensure robustness and the results of the parent canonical methods are reproduced to high accuracy. Comprehensive numerical tests on absolute and relative energies as well as timings consistently show that the outstanding performance of the LPNO methods carries over to the open-shell case with minor modifications. Finally, hyperfine couplings calculated with the variational LPNO-CEPA∕1 method, for which a well-defined expectation value type density exists, indicate the great potential of the LPNO approach for the efficient calculation of molecular properties.     

Photoelectron spectroscopy and electronic structure calculations of d1 vanadocene compounds with chelated dithiolate ligands: implications for pyranopterin Mo/W enzymes

Gas-phase photoelectron spectroscopy and density functional theory have been used to investigate the electronic structures of open-shell bent vanadocene compounds with chelating dithiolate ligands, which are minimum molecular models of the active sites of pyranopterin Mo/W enzymes. The compounds Cp2V(dithiolate) [where dithiolate is 1,2-ethenedithiolate (S2C2H2) or 1,2-benzenedithiolate (bdt), and Cp is cyclopentadienyl] provide access to a 17-electron, d1 electron configuration at the metal center. Comparison with previously studied Cp2M(dithiolate) complexes, where M is Ti and Mo (respectively d0 and d2 electron configurations), allows evaluation of d0, d1, and d2 electronic configurations of the metal center that are analogues for the metal oxidation states present throughout the catalytic cycle of these enzymes. A "dithiolate-folding effect" that involves an interaction between the vanadium d orbitals and sulfur p orbitals is shown to stabilize the d1 metal center, allowing the d1 electron configuration and geometry to act as a low-energy electron pathway intermediate between the d0 and d2 electron configurations of the enzyme.     

Constrained-pairing mean-field theory. IV. Inclusion of corresponding pair constraints and connection to unrestricted Hartree-Fock theory

Our previously developed constrained-pairing mean-field theory (CPMFT) is shown to map onto an unrestricted Hartree-Fock (UHF) type method if one imposes a corresponding pair constraint to the correlation problem that forces occupation numbers to occur in pairs adding to one. In this new version, CPMFT has all the advantages of standard independent particle models (orbitals and orbital energies, to mention a few), yet unlike UHF, it can dissociate polyatomic molecules to the correct ground-state restricted open-shell Hartree-Fock atoms or fragments.     

The spin-polarized extended Brueckner orbitals

Conventional natural and Brueckner orbitals (BOs) are rather frequently used for improving active 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 Lo?wdin'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 hydrogen clusters (H(4), H(6), and H(8)), 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.     

The origin of energy functional in Roothaan open shell SCF theory

Different methods of averaging of energy over the states of electronic configurations γ^N (n_γ = 1, 2, 3 and N = 1, 2, …, 2n_γ ? 1) leading to Roothaan' energy expression are considered. The consequent values of vector coupling coefficients (VCC ) in energy functionals for various states as well as for average values of energy are presented. It is shown also that in molecular systems of cubic and tetragonal symmetry having electronic configurations t^N (N = 2–4) and e² there exist states for which VCC are dependent on the choice of basis set of degenerate open-shell molecular orbitals. The origin of such “non-Roothaan” terms and peculiarities of its calculation by the restricted Hartree–Fock method are discussed.     

Complex molecular orbital method: Open-shell theory

A single-determinant open-shell formalism for complex molecular orbitals is developed. An iterative algorithm for solving the resulting secular equations is constructed. It is based on a sequence of similarity transformations and matrix triangularizations.     

Excited state geometries within time-dependent and restricted open-shell density functional theories

Singlet excited state geometries of a set of medium sized molecules with different characteristic lowest excitations are studied. Geometry optimizations of excited states are performed with two closely related restricted open-shell Kohn–Sham methods and within linear response to time-dependent density functional theory. The results are compared to wave-function based methods. Excitation energies (vertical and adiabatic) calculated from the open-shell methods show systematic errors depending on the type of excitation. However, for all states accessible by the restricted methods a good agreement for the geometries with time-dependent density functional theory and wave-function based methods is found. An analysis of the energy with respect to the mixing angle for the singly occupied orbitals reveals that some states (mostly [n→π^*]) are stable when symmetry constraints are relaxed and others (mostly [π→π^*]) are instable. This has major implications on the applicability of the restricted open-shell methods in molecular dynamics simulations.     

Determination of diabatic states through enforcement of configurational uniformity

. An electronic structure-based construction of diabatic states from adiabatic states is formulated that is applicable when individual diabatic states contain several dominant configurations. It is accomplished by maximizing the electronic uniformity of the diabatic states with respect to their dominant configurations throughout the entire nuclear coordinate region. The configurations are generated from unambiguously defined diabatization-adapted molecular orbitals. The orthogonal transformation from adiabatic to diabatic states is deduced by an intrinsic analysis of the adiabatic CI coefficients, without calculating matrix elements of additional, derivative or non-derivative operators. The practicality of the method is demonstrated by applying it to the conical intersection region of the 1¹ A ₁ and 2¹ A ₁ states of ozone.     

Electron correlation in the 3 (1)Sigma(g)+ and 2 (1)Sigma(u)+ excited state lithium molecule

Electron correlation effects in the two excited states of Li(2), 3 (1)Sigma(g) (+) and 2 (1)Sigma(u) (+), one with a shelf shape and another with double minima in their potential energy curves, have been studied with the aid of the calculated electron pair density distribution as a function of the internuclear distance and the analysis of the natural orbitals. Both states show increased electron pair densities at intermediate interelectronic distances around the second minimum of their potential energy curves. Since the bond breaks homolitically this observation runs contrary to regular expectations. Analysis of the electron pair density distributions and the natural orbitals provides mechanisms to account for this abnormal behavior.     

Full configuration interaction for the benzyl radical

The electronic structure of the benzyl radical in its ground state has been computed using a model Hamiltonian due to Pariser–Parr with full configuration interaction as well as with different truncated configurational sets built on SCF open-shell orbitals. The correlation energy corresponding to this model was found to be equal to –0.929722 eV. With the singly excited configurations only 18% of this energy is taken into account. By extending the basis to include the doubly excited configurations one can account for 94% of the correlation energy. An analysis of the accuracy of the proton hyperfine splitting calculation caused by inaccurate computation of the wave function is given. If only singly and even doubly excited configurations are taken into account one cannot hope to obtain splittings with an accuracy of more than 0.5 g. Inclusion of triply excited configurations lowers this error by one order. In addition, the use of the simple McConnell relation may lead to an error in splitting calculations of no less than 1.5 g.     

四同芳香性转烯及其羰基硼衍生物

运用G03W程序, 在高精度理论水平(B3P86/6-311＋G**)下, 对母体转烯(Hypostrophene)及其BCO衍生物的单态、三态、开壳层单态的Cope重排体系进行了理论研究: 对体系进行了相应的结构优化和频率计算, 并进一步计算了体系的重排势垒、反应能量、核独立化学位移值等理论参数. 文中首次提出具有四同芳香性的实例: 转烯的Cope重排过渡态. 计算同时表明BCO取代CH的行为使得进行Cope重排的反应物和过渡态的离域性、芳香性以及稳定性都得到很大的促进, 这可以从前线轨道的成键以及延伸方面得到合理的解释. 所得结果进一步验证了BCO基团的稳定性效应.     

(Hyper)polarizability density analysis for open-shell molecular systems based on natural orbitals and occupation numbers

We have developed a method for analyzing the (hyper)polarizabilities of open-shell molecular systems. This method employs the (hyper)polarizability densities based on the natural orbitals and occupation numbers, which enables us to analyze the contributions of odd electrons having various open-shell (diradical) characters. Within broken-symmetry, i.e., spin-unrestricted, single-determinant molecular orbital and density functional theory approaches, we can also remove the spin contamination effects on these quantities through spin projection. To do that, an approximate spin projected method has been elaborated and applied to the analysis of the (hyper)polarizability of multi-radical systems. As examples, typical open-shell singlet systems, 1,3-dipoles and rectangular graphene nanoflakes, are examined.     

Covalent bond orders revisited: the open-shell case

This article states the concept of covalent bond order for open-shell systems from the invariance properties of the first- and second-order reduced density matrices for all the components of a multiplet state. A general bond order definition is formulated in the framework of the electronic population analyses in the Hilbert space of atomic orbitals.