Variational second order density matrix study of F3-: importance of subspace constraints for size-consistency

Variational second order density matrix theory under "two-positivity" constraints tends to dissociate molecules into unphysical fractionally charged products with too low energies. We aim to construct a qualitatively correct potential energy surface for F(3)(-) by applying subspace energy constraints on mono- and diatomic subspaces of the molecular basis space. Monoatomic subspace constraints do not guarantee correct dissociation: the constraints are thus geometry dependent. Furthermore, the number of subspace constraints needed for correct dissociation does not grow linearly with the number of atoms. The subspace constraints do impose correct chemical properties in the dissociation limit and size-consistency, but the structure of the resulting second order density matrix method does not exactly correspond to a system of noninteracting units.     

Fractional spin in reduced density-matrix functional theory

We study the behavior of different functionals of the one-body reduced density matrix (1RDM) for systems with fractional z-component of the total spin. We define these systems as ensembles of integer spin states. It is shown that, similarly to density functional theory, the error in the dissociation of diatomic molecules is directly related to the deviation from constancy of the atomic total energies as functions of the fractional spin. However, several functionals of the 1RDM show a size inconsistency which leads to additional errors. We also investigate the difference between a direct evaluation of the energy of an ensemble of integer-spin systems and a direct minimization of the energy of a fractional-spin system.     

Superconducting first‐order pairing: Finite‐temperature simulations

A recently developed first‐order mechanism for superconducting pairing has been extended from T = 0 K to finite temperatures. On the basis of quantum statistical considerations, we have suggested a direct pairing interaction that does not necessarily involve second‐order elements, such as the electron–phonon coupling or specific magnetic interactions submitted by spin fluctuations. The driving force for the (energy‐driven) first‐order pairing is an attenuation of the destabilizing influence of the Pauli antisymmetry principle (PAP). Only the moves of unpaired fermions are controlled by the PAP, while the moves of superconducting Cooper pairs are not. The quantum statistics of Cooper pairs is of a mixed type, as it combines fermionic on‐site and bosonic intersite properties. The strong correlation between the strength of PAP constraints and system topology in combination with the electron number has been discussed for some larger clusters. Detailed finite‐temperature simulations on first‐order pairing have been performed for four‐center–four‐electron clusters with different topologies. A canonical ensemble statistics has been employed to derive the electronic energy, the electronic configuration entropy, and the free energy of paired and unpaired states in thermal equilibrium. The simulations show that pairing can be caused by either the electronic energy or the electronic configuration entropy. The coexistence of two different sets of quantum particles in paired states (i.e., the Cooper pairs and the unpaired electrons) can lead to an enhanced configuration entropy. In this context, we discuss the possibility of an entropy‐driven high‐temperature superconductor emerging from a low‐temperature unpaired state. The charge and spin degrees of freedom of the four‐center–four‐electron systems have been studied with the help of the charge and spin fluctuations. The spin fluctuations are helpful in judging the validity of pairing theories based on magnetic interactions. The charge fluctuations are a measure for the carrier delocalization in unpaired and paired states. The well‐known proximity between Jahn–Teller activity and superconductivity is analyzed in the zero‐temperature limit. It is demonstrated that both processes compete in their ability to reduce PAP constraints. All theoretical results have been derived within the framework of the simple Hubbard Hamiltonian. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

The electronic ground-state energy problem: a new reduced density matrix approach

We present here a formulation of the electronic ground-state energy in terms of the second order reduced density matrix, using a duality argument. It is shown that the computation of the ground-state energy reduces to the search of the projection of some two-electron reduced Hamiltonian on the dual cone of N-representability conditions. Some numerical results validate the approach, both for equilibrium geometries and for the dissociation curve of N(2).     

Energy spectrum of extended Hubbard model with spin‐dependent hopping and related spin ladder model

We studied the energy spectrum of the 1‐D extended Hubbard model with spin‐dependent hopping and related spin ladder system formed by two coupled XXZ spin 1/2 chains with the interchain interaction of Ising type. It was proved that the model excitation spectrum has no gap excepting some special values of z‐projection of the ground‐state total spin. The thorough analytic consideration of two‐magnon states was given. The existence up to five bound states at specified value of quasimomentum of the pair of inverted spins was shown. We also present the results of density matrix renormalization group calculations that showed nonadequacy of the pair approximation for n‐magnon bound states of the extended model with the strong electron–electron interactions. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Use of an “ordered” electron spin state in molecular spectroscopy: Theoretical and experimental considerations of spin-ordering in excited states

The effect of the dynamics of excited states on the observed decay of an ordered electron spin state induced in an electronically-excited ensemble is presented and used to illustrate the sensitivity of this technique to experimental conditions. Considered are feeding and decay processes involving the excited state, spin—lattice relaxation between the levels that are ordered, energy transfer to both translationally equivalent and inequivalent sites, and two different types of exchange processes. The treatment shows that the observed ordered-state decay is very sensitive to exprimental conditions. In particular, experiments performed off-resonance can display severely distorted decay curves with non-decaying components. These distortions limit the information on energy exchange in the excited state that can be obtained with electron spin ordering by adiabatic demagnetization in the rotating frame (ADRF). Also presented are treatments for the production and detection of order in excited states to complete a set of basic experimental considerations that affect data quality in ADRF measurements. Experimentally observed decays are presented to illustrate the predicted effects.     

Constrained spin-density dynamics of an iron-sulfur complex: ferredoxin cofactor

The computation of antiferromagnetic exchange coupling constants J by means of efficient density-based approaches requires in practice to take care of both spin projection to approximate the low spin ground state and proper localization of the magnetic orbitals at the transition metal centers. This is demonstrated here by a combined approach where the extended broken-symmetry (EBS) technique is employed to include the former aspect, while spin density constraints are applied to ensure the latter. This constrained EBS (CEBS) approach allows us to carry out ab initio molecular dynamics on a spin-projected low spin potential energy surface that is generated on-the-fly by propagating two coupled determinants and thereby accessing the antiferromagnetic coupling along the trajectory. When applied to the prototypical model of the oxidized [2Fe-2S] cofactor in Ferredoxins, [Fe(2)S(2)(SH)(4)](2-), at room temperature, CEBS leads to remarkably good results for geometrical structures and coupling constants J.     

Surface Induced Dissociation Yields Quaternary Substructure of Refractory Noncovalent Phosphorylase B and Glutamate Dehydrogenase Complexes

Ion mobility (IM) and tandem mass spectrometry (MS/MS) coupled with native MS are useful for studying noncovalent protein complexes. Collision induced dissociation (CID) is the most common MS/MS dissociation method. However, some protein complexes, including glycogen phosphorylase B kinase (PHB) and L-glutamate dehydrogenase (GDH) examined in this study, are resistant to dissociation by CID at the maximum collision energy available in the instrument. Surface induced dissociation (SID) was applied to dissociate the two refractory protein complexes. Different charge state precursor ions of the two complexes were examined by CID and SID. The PHB dimer was successfully dissociated to monomers and the GDH hexamer formed trimeric subcomplexes that are informative of its quaternary structure. The unfolding of the precursor and the percentages of the distinct products suggest that the dissociation pathways vary for different charge states. The precursors at lower charge states (+21 for PHB dimer and +27 for GDH hexamer) produce a higher percentage of folded fragments and dissociate more symmetrically than the precusors at higher charge states (+29 for PHB dimer and +39 for GDH hexamer). The precursors at lower charge state may be more native-like than the higher charge state because a higher percentage of folded fragments and a lower percentage of highly charged unfolded fragments are detected. The combination of SID and charge reduction is shown to be a powerful tool for quaternary structure analysis of refractory noncovalent protein complexes, as illustrated by the data for PHB dimer and GDH hexamer.

A ligand-field study of the ground spin-state magnetic anisotropy in a family of hexanuclear Mn(III) single-molecule magnets.

A ligand field analysis of two structurally related hexanuclear Mn(iii) coordination complexes reveals that the observed difference in their ground spin-state anisotropy originates from the difference in projection coefficients of the single-ion anisotropy to spin states of different total spin quantum-number, S, rather than the geometrical distortions of the metal ions. Furthermore we show that the single-ion second order anisotropy induces fourth and higher order anisotropy terms to the ground spin states of the studied systems, as a consequence of spin-state mixing effects due to the comparable magnitude of the single-ion second order anisotropy and the isotropic exchange parameters.     

An ab initio multireference perturbation theory study on the manganese dimer

The potential energy curves of the ground state and of some excited states of the manganese dimer have been calculated over a wide range of internuclear distances using the second order n-electron valence state perturbation theory applied to a complete active space self-consistent field reference wave function. The ground state of Mn(2), for which also the third order NEVPT has been used, is calculated to be a singlet belonging to the Sigma(g) (+) symmetry, characterized by a large equilibrium internuclear distance R(e) of 3.7-3.8 A, by a low dissociation energy D(e) of 0.07-0.08 eV, and by a small harmonic frequency omega(e) of 43 cm(-1). The experimental evidence that Mn(2) is a van der Waals molecule is thus confirmed. Among the excited states, (11)Pi(u), which is usually indicated as the ground state by density functional theory studies, appears as a low-lying state with R(e)=2.50 A, D(e)=1.35 eV, and omega(e)=246 cm(-1).     

Projected Hartree-Fock theory

Projected Hartree-Fock (PHF) theory has a long history in quantum chemistry. PHF is here understood as the variational determination of an N-electron broken symmetry Slater determinant that minimizes the energy of a projected state with the correct quantum numbers. The method was actively pursued for several decades but seems to have been abandoned. We here derive and implement a "variation after projection" PHF theory using techniques different from those previously employed in quantum chemistry. Our PHF methodology has modest mean-field computational cost, yields relatively simple expressions, can be applied to both collinear and non-collinear spin cases, and can be used in conjunction with deliberate symmetry breaking and restoration of other molecular symmetries like complex conjugation and point group. We present several benchmark applications to dissociation curves and singlet-triplet energy splittings, showing that the resulting PHF wavefunctions are of high quality multireference character. We also provide numerical evidence that in the thermodynamic limit, the energy in PHF is not lower than that of broken-symmetry HF, a simple consequence of the lack of size consistency and extensivity of PHF.     

Electron delocalization and magnetic state of doubly-reduced polyoxometalates.

Different mechanisms of spin pairing in doubly reduced polyoxometalates are studied on the basis of quantum-chemical DFT calculations. Using the nitrosyl derivative of decamolybdate [Mo(10)O(25)(OMe)(6)(NO)](-) (I) as an example, we elucidate an important role of the delocalization of "blue electrons". The charge distributions and spin states are studied for the series of isomers of I differing by positions of methyl groups (modeled by hydrogens). Three different states are calculated for each isomer: spin triplet, spin-restricted singlet, and a broken symmetry state. If the quasihomogeneous distribution of the "blue electrons" density is weakly perturbed by protonation, the delocalization mechanism is responsible for the spin pairing. It is evidenced by the singlet ground state given by a spin-restricted solution. If the perturbation of charge distribution is strong enough and the "blue electrons" density is localized at several metal centers, the exchange mechanism becomes active. A lowest energy broken symmetry state indicates the antiferromagnetic nature of the singlet ground state. The modulation of magnetic interactions in reduced polyoxoanions by external perturbations provides new possibilities for design of molecular magnetic materials.     

Phase Space View of Ensembles of Excited States

The density functional theory and its extension to ensembles of excited states can be formalized as thermodynamics. However, these theories are not unique because one of their key quantities, the kinetic energy density,can be defined in several ways. Usually, the everywhere positive gradient form is applied; however, other forms are also acceptable, provided they integrate to the true kinetic energy. Recently, a kinetic energy density of the ground-state theory maximizing the information entropy has been proposed. Here, ensemble kinetic energy density, leading to extremum information entropy, is derived via constrained search. The corresponding ensemble temperature is found to be constant.     

Molecular dynamics investigations of ozone on an ab initio potential energy surface with the utilization of pattern-recognition neural network for accurate determination of product formation

The singlet-triplet transformation and molecular dissociation of ozone (O(3)) gas is investigated by performing quasi-classical molecular dynamics (MD) simulations on an ab initio potential energy surface (PES) with visible and near-infrared excitations. MP4(SDQ) level of theory with the 6-311g(2d,2p) basis set is executed for three different electronic spin states (singlet, triplet, and quintet). In order to simplify the potential energy function, an approximation is adopted by ignoring the spin-orbit coupling and allowing the molecule to switch favorably and instantaneously to the spin state that is more energetically stable (lowest in energy among the three spin states). This assumption has previously been utilized to study the SiO(2) system as reported by Agrawal et al. (J. Chem. Phys. 2006, 124 (13), 134306). The use of such assumption in this study probably makes the upper limits of computed rate coefficients the true rate coefficients. The global PES for ozone is constructed by fitting 5906 ab initio data points using a 60-neuron two-layer feed-forward neural network. The mean-absolute error and root-mean-squared error of this fit are 0.0446 eV (1.03 kcal/mol) and 0.0756 eV (1.74 kcal/mol), respectively, which reveal very good fitting accuracy. The parameter coefficients of the global PES are reported in this paper. In order to identify the spin state with high confidence, we propose the use of a pattern-recognition neural network, which is trained to predict the spin state of a given configuration (with a prediction accuracy being 95.6% on a set of testing data points). To enhance the prediction effectiveness, a buffer series of five points are validated to confirm the spin state during the MD process to gain better confidence. Quasi-classical MD simulations from 1.2 to 2.4 eV of total internal energy (including zero-point energy) result in rate coefficients of singlet-triplet transformation in the range of 0.027 ps(-1) to 1.21 ps(-1). Also, we find very low dissociation probability up to 2.4 eV of internal energy during the investigating period (5 ps), which suggests that dissociation does not occur directly from the singlet ground-state, but it involves the excited triplet-state as an intermediate step and requires more reaction time to occur.     

Spin‐free quantum chemistry: What one can gain from fock's cyclic symmetry

The spin‐free wave function due to Fock (Zh Eksp Teor Fiz, 1940, 10, 961) is re‐examined with a stress on the reduced density matrix (RDM) theory. The key notion of the Fock approach is the cyclic symmetry of wave functions. It is a specific algebraic identity involving transpositions of numbers taken from two different columns of the corresponding Young tableau. We show first how to construct symmetry adapted states by accounting for high‐order cyclic symmetry conditions. For Young's projectors, it gives a new expression including nothing but antisymmetrizers. Next, transforming the Fock spin‐free state by a duality operator (the star operator in exterior algebra), we arrive at the representation closely related to spin‐flip models. In such spin‐flip models, a coupling operator is the basic object for which we show that the cyclic symmetry is transformed into a tracelessness of the coupling operator. The main results are related to the spin‐free theory of spin properties. In particular, the theorem previously stated (Luzanov and Whyman, Int J Quantum Chem, 1981, 20, 1179) is refined by an explicit general representation of spin density operators through spin‐free (charge) RDMs. Some applications implicating high‐order RDMs (collectivity numbers, the unpaired electron problem, cumulant spin RDMs, spin correlators, etc.) are also considered. For spin‐free RDM components, a new projection procedure without constructing any symmetry adapted state is proposed. An unsolved problem of constructing orthogonal representation matrices within the Fock theory is raised. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Interplay among aromaticity, magnetism, and nonlinear optical response in all-metal aromatic systems

All-metal aromatic molecules are the latest inclusion in the family of aromatic systems. Two different classes of all-metal aromatic clusters are primarily identified: one is aromatic only in the low spin state, and the other shows aromaticity even in high-spin situations. This observation prompts us to investigate the effect of spin multiplicity on aromaticity, taking Al(4)(2-), Te(2)As(2)(2-), and their copper complexes as reference systems. Among these clusters, it has been found that the molecules that are aromatic only in their singlet state manifest antiaromaticity in their triplet state. The aromaticity in the singlet state is characterized by the diatropic ring current circulated through the bonds, which are cleaved to generate excess spin density on the atoms in the antiaromatic triplet state. Hence, in such systems, an antagonistic relationship between aromaticity and high-spin situations emerges. On the other hand, in the case of triplet aromatic molecules, the magnetic orbitals and the orbitals maintaining aromaticity are different; hence, aromaticity is not depleted in the high-spin state. The nonlinear optical (NLO) behavior of the same set of clusters in different spin states has also been addressed. We correlate the second hyperpolarizability and spin density in order to judge the effect of spin multiplicity on third-order NLO response. This correlation reveals a high degree of NLO behavior in systems with excess spin density. The variance of aromaticity and NLO response with spin multiplicity is found to stem from a single aspect, the energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), and eventually the interplay among aromaticity, magnetism, and NLO response in such materials is established. Hence, the HOMO-LUMO energy gap becomes the cornerstone for tuning the interplay. This correlation among the said properties is not system-specific and thus can be envisaged even beyond the periphery of all-metal aromatic clusters. Such interplay is of crucial importance in tailoring novel paradigm of multifunctional materials.     

The interference in the decay of two ½ spins in a molecular medium,studied by the Nakajima–Zwanzig technique

This paper studies the decay, due to the spin-lattice coupling, of two ½ spins with slightly different Zeeman energies when the lattice is thermally excited. The analysis is based on obtaining, by means of the Nakajima–Zwanzig projection operator technique, an equation for the evolution of the reduced density operator of the spin system which manifests the influence of one spin on the relaxation process of the other. The zero-order solutions obtained for the evolution of the expectation values of the spin dynamics operators are essentially equivalent to the Bloch equations; higher order solutions are also obtained.     

Necessary conditions for the N‐representability of pair distribution functions

A necessary condition for the N‐representability of the electron pair density proposed by one of the authors (E. R. D.) is generalized. This shows a link between this necessary condition and other, more widely known, N‐representability conditions for the second‐order density matrix. The extension to spin‐resolved electron pair densities is considered, as is the extension to higher‐order distribution functions. Although quantum mechanical systems are our primary focus, the results are also applicable to classical systems, where they reduce to an inequality originally derived by Garrod and Percus. As a simple application, bounds to the average angle between an electron pair are derived. It is shown that computational methods based on variational minimization of the energy with respect to the electron pair density can give extremely poor results unless robust N‐representability constraints are considered. For reference, constraints for the N‐representability of the pair density are summarized. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Density matrix renormalization group calculations on relative energies of transition metal complexes and clusters

The accurate first-principles calculation of relative energies of transition metal complexes and clusters is still one of the great challenges for quantum chemistry. Dense lying electronic states and near degeneracies make accurate predictions difficult, and multireference methods with large active spaces are required. Often density functional theory calculations are employed for feasibility reasons, but their actual accuracy for a given system is usually difficult to assess (also because accurate ab initio reference data are lacking). In this work we study the performance of the density matrix renormalization group algorithm for the prediction of relative energies of transition metal complexes and clusters of different spin and molecular structure. In particular, the focus is on the relative energetical order of electronic states of different spin for mononuclear complexes and on the relative energy of different isomers of dinuclear oxo-bridged copper clusters.